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NEW FRONTIERS IN ROBOTICS Series Editors: Miomir Vukobratovic (“Mihajlo Pupin” Institute, Serbia) Ming Xie (Nanyang Technological University, Singapore)
Published Vol. 1
Haptics for Teleoperated Surgical Robotic Systems by M. Tavakoli, R. V. Patel, M. Moallem & A. Aziminejad
Vol. 2
Miomir Vukobratovic
“Mihajlo Pupin” Institute, Belgrade, Serbia
Dragoljub Surdilovic
Fraunhofer Institute for Production Systems and Design Technology IPK, Berlin, Germany
Yury Ekalo
Research and Engineering Center of St Petersburg Electrotechnical University, Russia
Dusko Katic
“Mihajlo Pupin” Institute, Belgrade, Serbia
World Scientific NEW JERSEY
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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
DYNAMICS AND ROBUST CONTROL OF ROBOT-ENVIRONMENT INTERACTION New Frontiers in Robotics — Vol. 2 Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN-13 978-981-283-475-1 ISBN-10 981-283-475-3
Printed in Singapore.
Preface
When writing a book that is concerned with the robot’s physical contact and interaction with an environment, first and foremost we ought to consider the following contradictory situation. During the last two decades, the control of robot contact motion (also referred to as compliant motion) has emerged as one of the most attractive and fruitful research areas in robotics. The initial investigations in the field were motivated by the practical needs for automating complex tasks mainly performed by humans, such as assembly, deburring, etc. The control of physical robot interaction is still a challenging research issue, recently addressing the emerging fields of human-robot interaction systems, human augmentations and enhancements, haptic rendering, rehabilitation robotics, etc. However, in spite of considerable research efforts and results achieved, the applications of compliance control in the industry and service fields are still insignificant in comparison with widespread free-space robot applications, such as pick-and-place or seam-tracking tasks. The majority of industrial robot assembly applications utilize passive compliance devices (Remote Center of Compliance - RCC) compensating for misalignments of parts with specific geometry. Other applications employ additional passive or active axes with simple compliance control algorithms. More sophisticated robotic systems that would involve the programming and control of the interaction with a complex, dynamic and variable environment are still missing in practice. There are many different reasons for such a situation. Based on almost twenty years of research and experience with implementing the compliantmotion control algorithms in industrial and other robotic systems we would try to identify the most critical causes which in our opinion mainly inhibited a more widespread application of interactive robotic systems. First, the development of a controller for contact tasks has proven to be quite difficult, largely due to the stability problems that arise in the dynamic interaction during robot’s physical contact with an environment. The interaction control problems are still insufficiently specified, and their structural relationships to classical servo control design problems and methodologies are not completely clear. One
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Dynamics and Robust Control of Robot-Environment Interaction
further limitation is the absence of a widely accepted framework for the synthesis of the interaction control parameters that would ensure the stability of both the contact transition and interaction processes, and guarantee the desired contact performance. The existing design procedures based on robot passivity appear to be exceedingly conservative in the applications in which the interaction between an industrial robot and a stiff environment should be controlled. The established interaction control algorithms are mostly concerned with particular problems, usually at the lowest servo-control layer, and their integration into a complete control system, which appears to be a very tedious task, is still missing. Further, many of the proposed control algorithms are based on the computedtorque method and are closely related to direct-drive robotic systems. The directdrive technology is, however, still seldom used in robotic practice, due to several serious problems related to the large mass, overheating, and high costs of directdrive actuators. On the other hand, direct-drive robots appear to be quite suitable for advanced experimental investigations of robot control in research laboratories. The popular computed-torque control technique requires real-time computation of complete dynamic models of the robot and environment, which makes its realization rather complex. This approach works reasonably well in direct-drive robots when their dynamic parameters have been correctly identified. In industrial robots, however, the performance improvements which can be achieved with these algorithms are not in proportion with the implementation efforts. Due to quite different performance, the results obtained for direct-drive robots, although experimentally verified, cannot be applied onto industrial robots. The investigations of compliant motion control are usually concerned with the nonlinear effects in robot and environment dynamics, rather than with the problems encountered in conventional robotic systems, such as Coulomb friction, control time delay, practical limitations of computer and sensory systems, etc. Still, there is another problem concerning the knowledge of the environment model. The majority of proposed algorithms exhibit good performance only under the assumption that an accurate model of the environment and its parameters (e.g. contact location, stiffness) are available. In real stiff environments, however, this condition is quite restrictive and non-realistic, since it is difficult to identify the parameters using available sensory and computer control systems. All these problems make the existing results of compliant motion control difficult to implement from a practical point of view. This requires further
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investigation efforts on reliable and simple, but nevertheless robust, control solutions. Despite the progress that has been made during the last decade, there are still research issues insufficiently investigated, such as: • the stability of contact transition is not clearly addressed in the literature. The reliable necessary and sufficient conditions that would ensure the maintenance of a stable contact during transition are still missing. • the problem of handling an inadvertent loss of contact has not been solved yet. Furthermore, of special practical importance are the following topics, deserving further computational/experimental study: Design of robust compliance motion control to improve disturbance rejection capabilities; • Definition of measures and criteria to evaluate the compliant motion capabilities of industrial robots in relation to performing a task, taking into account the distortion of friction and other similar disturbances in the arm; • Comparison of the available algorithms and definition of benchmark tests; • Development of reliable control schemes based on a unified approach to force, position and impedance control, which can be applied in conventional industrial robotic systems. Therefore, this book is aimed at considering the interaction control problems from a broad and comprehensive point of view. The problem of robotenvironment interaction control is tackled taking into account different issues, such as: mathematical modeling of contact and interaction kinematics and dynamics, stability of coupled systems and contact transition process, various interaction control algorithms and techniques, robustness against disturbances and model uncertainties, programming and planning of simple contact tasks, sampled systems control effects, as well as practical control synthesis and design for industrial implementations. Last but not least, practical knowledge and experience gained in the developing and implementing various interactive robotic systems is reflected in this book. The contents of the book are organized as follows. Chapter 1 provides a comprehensive review of various compliant motion control methods proposed in the literature. It covers some early ideas and their later improvements, as well as new control concepts and recent trends in this field. Before reviewing many of the results, a categorization of compliant motion tasks and proposed control concepts is established based on various
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classifying criteria. In this survey, particular attention is paid not only to traditional indices of control performance, but also to the reliability and applicability of algorithms and control schemes in industrial robotic systems. These systems are widely employed in practice, providing a reasonable background for compliance motion control implementation. However, compliance control is a very attractive control approach in the new emerging technologies, such as service robotics (e.g. surgical and rehabilitation robots), virtual reality and haptics, telemanipulation, human augmentation and assistance. These fields apply new and quite different robotic structures in comparison to conventional industrial robotic systems (e.g. direct-drive robots, parallel and wire manipulators, etc.). This chapter provides a historical perspective, summarizes contributions of the most relevant or representative investigations and methods, and identifies the interaction control problems that are still open, requiring additional research efforts. It provides useful information, especially for younger roboticians having no previous work experience. Chapter 2 is devoted to the unified approach to dynamic control of the robot interacting with a dynamic environment. The unified position-force control differs essentially from the conventional hybrid position/force control schemes. A dynamic approach to controlling simultaneously both the position and force in an environment with completely dynamic reactions has been established. The approach of dynamic interaction control defines two control subtasks responsible for the stabilization of robot position and interaction force. The both control subtasks utilize dynamic models of the robot and environment in order to ensure tracking of both the nominal motion and force. Special attention is given not only to the synthesis of control laws ensuring stability of robot’s desired motions and desired interaction forces of the robot and environment, but also to the definition of possible motions of the robot and its possible interaction forces in contact tasks. The concept of the family of transient responses with respect to the robot’s motion and its force of interaction with environment is formulated. It allows one to set and then solve the problem of the synthesis of control laws that not simply stabilize the motion and force of interaction of the robot with its environment, but also solves the problem of stabilization with the preset quality of transient processes. Significant attention is given to the analysis of the influence of the constraints imposed on the state, control, and interaction force on transient responses, taking into account the inadequacy of dynamics models of the robot and environment and/or external perturbations. The adaptive control scheme proposed in this chapter enables one to solve contact tasks for robots with both stationary and nonstationary dynamics. The elaborated stability test
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may be used either to check the stability of the specified control laws, or to establish procedures for the synthesis of parameters of different control laws. Hence, the control synthesis becomes much more accurate and effective, i.e. higher robustness of the control to the uncertainties in the robot and environment models can be ensured, which is one of the most relevant aspects in the potential industrial applications of robots in numerous technological tasks where the robot is interacting with the environment (e.g. in cutting, deburring, etc.). The developed control algorithms also appear very promising for interaction with a virtual environment in high dynamic haptic systems, as well as for controlling novel dual-arm robots and bimanual contact tasks. Chapter 3 is concerned with the design of compliance control algorithms that are reliable and robust for implementing in industrial robots and advanced interaction systems (e.g. haptic interfaces, surgical and rehabilitation robots, human enhancers, collaborative robots, etc.). The problems and research issues that are associated with the design of robust impedance control algorithms for stable interaction with a passive environment are in the main focus of this chapter. The basic control development problems, such as stability, performance and robustness of impedance control algorithms are addressed. These problems are considered at the lowest servo control layer. For the sake of simplicity, the impedance control design problem is split into two subproblems concerning the realization of the target impedance and selection of target impedance parameters which ensure specific desired task performance, as well as common control design requirements, such as stability, fast reaction and robustness. The stability of the interaction between the robot and environment in contact, which is essential for the impedance control synthesis, is defined by means of the coupled stability. For the examination of stability we have applied a common approach utilizing the properties of the system at equilibrium, and various modern control techniques (e.g. positivity and H ∞ control concepts, etc.). We have further considered the stability of the contact transition process. Although recognized to be most fundamental in contact tasks control synthesis, this problem has not been addressed appropriately until now. Especially, the contact stability in industrial robotic systems has not been explored adequately. Several practical contact stability definitions are proposed in order to clearly make distinction between the contact stability and coupled stability, often mistaken in the literature. Based on these definitions, new stability conditions are proposed. The concept of dynamic systems passivity and robust stability analysis are applied to obtain the reliable conditions for ensuring contact stability. The established stability criteria provide the basis for examining the effects of impedance control
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parameters on the transition process stability. The analysis/synthesis oriented stability examinations allow the tuning of target impedance parameters in order to meet both interaction performance and stability. Within the robust stability analysis framework, the generalized contact stability condition is also derived, ensuring both contact and coupled system stability. The stability analysis is pursued in discrete-time and sampled-data interaction systems. In order to evaluate the derived contact stability criteria a set of several hundred contact transition experiments was designed and realized using an industrial robot. The aim was to capture a practical parametric contact stability limit in a typical real robot-environment interaction system. On the basis of tests performed it is demonstrated that the most accurate contact stability results (the parametric limits closest to the experimental bounds) provides the passivity-based criterion for the sampled-data system. Robust contact stability always ensures a safe transition and appears to be very practical for the control synthesis in an uncertain robot-environment interaction system. Chapter 4 addresses the synthesis of the adopted second-order target impedance model for a generic contact task. The contact task consists of the realization and maintenance of a stable contact with the environment. The interaction force should be kept within the prescribed limits, dependent on the position tolerances and environmental stiffness. This assignment is intrinsically involved in almost all robot interaction tasks. The chapter considers the algorithms for the practical impedance control design in industrial robotic systems. The developed algorithms integrate the theoretical and practical stability results dealt with in the previous chapter. The considered impedance control synthesis addresses basic control design problems at the servo-control layer. The impedance control design has been established for a reliable decoupled compliance geometric model that allows a relatively simple parameterization of the target impedance behavior. For the fundamental and common interaction tasks, the compliance parameters can be chosen independently of the interaction system configuration. More complex robotenvironment interactions were also considered based on the spatial compliance model. The control synthesis consists of the straightforward steps of computing the target impedance parameters and impedance compensator gains. All input parameters to the design algorithm have been explicitly specified. The feasibility of the developed algorithms was demonstrated using experiments with two industrial robot systems. Finally, a reliable geometric and control framework for the implementation of compliance control in industrial and other advanced robotic systems has been developed and presented. Several practical and robust
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control algorithms at higher planning and programming control layers were designed and tested. The essential algorithms support setting of the compliance parameters, such as the C-frame location and impedance gains, as well as continuous switching of compliance control and variation of parameters. These features are proven to be essential for a stable and robust execution of the compliance control tasks. Powerful sets of control functions, also presented in this chapter, integrate the basic compliance control algorithms in the forward robot control. These functions perform all of the computations and management of the parameters between the convenient robot position control system and impedance control kernel. Finally, some new commands, providing a flexible user-interface, are designed and implemented in a high-level robot programming environment. The new programming language commands are illustrated by means of several examples. An essential design requirement was to combine the user’s experience with robot motion programming and simple understandable physical behavior of the impedance control which mimics a variable spatial mass-damper-spring system. The experimental testing within the space control system SPARCO has clearly proven the reliability and robustness of the presented high-level compliance control algorithms. Certainly, a basic precondition for the implementation of compliant motion control is the design of a robust servo impedance controller, which ensures stable transition and coupling with the environment. However, the control integration and programming issues, which are often underestimated in the literature, are essential for a customary and efficient application of impedance control in practical contact tasks. A proper selection of the C-frame location and target impedance gains is crucial for a successful execution of the impedance control tasks. This selection should be compatible with the very nature of the motion constraints, i.e. contact task geometry and physical task characteristics (e.g. force-motion relationships). The experience gained in performing the compliance tasks presented here is essential for compliance control design and implementation in a wide range of tasks in industrial and service robotics. The robust control framework and the new contact stability theory established for the control synthesis of the interaction between an impedancecontrolled robot and a passive environment are expanded in Chapter 5 to the control and synthesis of haptic interfaces interacting with a virtual environment. This rapidly emerging technology imposes high requirements on the interaction stability and robustness of the control system in spite of considerable control computation efforts and time lags. Recently, the new interactive systems concerning the interaction between a human and a robotic device, as well as with robot’s physical or virtual dynamic environments, have aroused a strong research
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interest. To the novel interactive systems belong kinesthetic displays and haptic interfaces, teleoperation systems, human enhancers and augmentation devices, rehabilitation robots, robot assistants and collaborative robots, etc. These systems are designed to produce/receive kinesthetic stimuli for/from human movements, as well as to provide the user with a realistic feeling of the contact and dynamic interaction with the close, remote or virtual environments. The advanced interaction systems have recently found very attractive applications in surgical and rehabilitation robotics, power assist-devices, training simulation systems, etc. The most critical issue in these systems is how to ensure stable and safe interaction with a high fidelity of reproduction of a virtual environment. This is a challenging task when taking into account serious problems such as unknown and variable human dynamics, commonly non-linear environmental characteristics, as well as various disturbances in computer-controlled systems. This chapter considers the stability of the interaction of a human-robotenvironment (real or virtual) system based on the robust control design approach. The proposed new interaction stability paradigm ensures contact stability during all phases of the interaction. Moreover, the new design framework realizes low-impedance performance allowing considerable reduction of high apparent industrial robot inertia and stiffness. The defined stability indices take into account the relevant effects in the robot control systems, such as time lags and sampling data effects, as well as the uncertainties in the environment and realized target admittance models. The synthesis of robust control laws is confirmed to be very efficient for the stabilization of the interaction between a robot and a stiff and force-delayed environment taking into account the desired interaction performance. The testing of this approach in various robotic systems demonstrates the feasibility and reliability of the interaction control approach even for relatively high control rates and lags. The advantage of robust stability is particularly demonstrated in the interaction control of novel intelligent power-assist handling systems with significant perturbations in the force and position measurements. This shows the practical applicability of the novel stability criteria for haptic systems. Chapter 6 covers some advanced control techniques. As robotic systems make their way into standard practice, they have opened the door to a wide spectrum of complex applications. Such applications usually demand highly intelligent robots. Future robots are likely to have greater sensory capabilities, more intelligence, higher levels of manual dexterity, and the mobility, compared to humans. In order to ensure high-quality control and performance in robotics, new intelligent control techniques must be developed that will be capable of
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coping with task complexity, multi-objective decision making, large volumes of perception data and substantial amounts of heuristic information. Soft Computing paradigms consisting of complementary elements of Fuzzy Logic, Neural Computing and Evolutionary Computation are viewed as the most promising methods towards intelligent robotic systems. The specific emphasis in research is given on the development of efficient learning rules for robotic connectionist training and synthesis of neural learning algorithms for low-level control in the domain of robotic compliance tasks. The synthesis of new advanced learning algorithms for robotic contact tasks by nonrecurrent and recurrent connectionist structures is presented in this chapter as the main research contribution. The main concern of this chapter, which provides a survey of connectionist algorithms for robotic contact tasks, is the development of learning control algorithms as an upgrade of conventional non-learning control laws for robotic compliance tasks (algorithms for stabilization of robot motion, stabilization of robot interaction force and impedance algorithms). In view of the important influence of the robot environment, a new comprehensive learning approach, based on simultaneous classification of robot environment and learning of robot uncertainties, is also presented. The book is addressed to a wide audience of scientists, practitioners and scholars dealing with interactive robotic systems. It is our hope that the material presented in this book will be useful to a wide range of readers, ranging from undergraduate and graduate students, new and advanced academic researchers, to the technical specialists (mechanical, electrical, computer or systems engineers). It can also be adopted as a textbook for a graduate course on advanced robotic systems.
The authors Belgrade, December 2007
Contents
Preface .......................................................................................................................................
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1. Control of Robots in Contact Tasks: A Survey ................................................................... 1.1 Introduction ............................................................................................................... 1.2 Contact Tasks ............................................................................................................ 1.3 Classification of Constrained Motion Control Concepts ........................................... 1.4 Model of the Robot Performing Contact Tasks ......................................................... 1.5 Passive Compliance Methods .................................................................................... 1.5.1 Non-adaptable compliance methods ............................................................. 1.5.2 Adaptable compliance methods .................................................................... 1.6 Active Compliant Motion Control Methods.............................................................. 1.6.1 Impedance control......................................................................................... 1.6.1.1 Force-based impedance control ...................................................... 1.6.1.2 Position-based impedance control .................................................. 1.6.1.3 Other impedance-control approaches ............................................. 1.6.2 Hybrid position/force control........................................................................ 1.6.2.1 Explicit force control...................................................................... 1.6.2.2 Position-based (implicit) force control ........................................... 1.6.2.3 Other force control approaches....................................................... 1.6.3 Force-impedance control............................................................................... 1.6.4 Unified position-force control....................................................................... 1.7 Contact Stability and Transition ................................................................................ 1.8 Compliance Planning................................................................................................. 1.9 Haptic Systems Control ............................................................................................. 1.10 New Robot Application ............................................................................................. 1.11 Conclusion................................................................................................................. Bibliography........................................................................................................................
1 1 1 2 10 14 14 15 18 18 22 24 27 28 30 37 39 40 43 46 56 60 64 65 67
2. A Unified Approach to Dynamic Control of Robots........................................................... 2.1 Introduction ............................................................................................................... 2.2 Dynamic Environments ............................................................................................. 2.2.1 Model of a dynamic environment ................................................................. 2.2.1.1 Kinematic-dynamic constraints ...................................................... 2.2.1.2 Pure dynamic environment............................................................. 2.2.1.3 Linear impedance model ................................................................ 2.3 Synthesis of Control Laws for the Robot Interacting with Dynamic Environment ... 2.3.1 Stabilization of motion with the preset quality of transients......................... 2.3.2 Stabilization of interaction force with the preset quality of transients .......... 2.3.3 Concluding discussion ..................................................................................
77 77 80 81 82 82 83 84 86 91 100
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Dynamics and Robust Control of Robot-Environment Interaction 2.4
Analysis of Transient Processes................................................................................. 2.4.1 Task setting ................................................................................................... 2.4.2 Motion transient processes............................................................................ 2.4.3 Force transient processes............................................................................... 2.4.4 Numerical example ....................................................................................... 2.4.5 Effect of sensor errors on the transient processes ......................................... 2.5 Adaptive Stabilization of Motion and Forces ............................................................ 2.5.1 Introduction................................................................................................... 2.5.2 Task setting ................................................................................................... 2.5.3 General scheme of robot adaptive control in contact tasks ........................... 2.5.4 Adaptive stabilization of programmed motions and forces........................... 2.6 Position-Force Control – A Generalization ............................................................... 2.6.1 Models of robot and environment dynamics. Task setting ........................... 2.6.2 Control laws stabilizing the interaction force................................................ 2.6.3 Example ........................................................................................................ 2.6.4 Conclusion .................................................................................................... 2.7 Position-Force Control in Cartesian Space ................................................................ 2.7.1 Introduction................................................................................................... 2.7.2 Task setting ................................................................................................... 2.7.3 Relation to previous results........................................................................... 2.7.4 Control laws for specified force dynamics.................................................... 2.7.5 Example ........................................................................................................ 2.7.6 Conclusion .................................................................................................... 2.8 New Realization of Hybrid Control........................................................................... 2.8.1 Introduction................................................................................................... 2.8.2 Revised hybrid control procedure ................................................................. 2.8.3 Case study ..................................................................................................... 2.9 Impedance Control – A Special Case of the Unified Approach ................................ 2.9.1 Introduction................................................................................................... 2.9.2 Improved impedance control......................................................................... 2.9.3 Case study ..................................................................................................... 2.9.4 Concluding remarks ...................................................................................... 2.10 Stability of Robots Interacting with Dynamic Environments .................................... 2.10.1 Introduction................................................................................................... 2.10.2 Practical stability of robots interacting with dynamic environment.............. 2.10.3 Mathematical model...................................................................................... 2.10.4 Formulation of the control task ..................................................................... 2.10.5 Control law ................................................................................................... 2.10.6 Practical stability analysis ............................................................................. 2.10.7 Example ........................................................................................................ 2.10.8 Conclusion .................................................................................................... 2.10.9 Practical stability - A remark ........................................................................ Appendix A Proof of Theorem 1 ....................................................................................... Appendix B Proof of Theorem 2 ....................................................................................... Appendix C Proof of Theorem 3 ....................................................................................... Appendix D Proof of Theorem 4 ....................................................................................... Appendix E Proof of Theorem 5 ....................................................................................... Appendix F Proof of Theorem 6 ....................................................................................... Appendix G Proof of Lemma ............................................................................................
101 102 104 110 119 121 128 128 129 134 139 146 146 149 155 157 158 158 159 160 161 166 169 170 170 173 177 182 182 182 186 190 191 191 194 195 196 198 199 203 207 208 211 215 217 219 221 223 225
Contents
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Appendix H Proof of Theorem 7 ....................................................................................... Appendix I Some Basic Relations.................................................................................... Appendix J Proof of Theorem 8 ....................................................................................... Appendix K Prior Assertions and Proof of Theorem 10.................................................... Bibliography........................................................................................................................
229 232 237 242 250
3. Impedance Control .............................................................................................................. 3.1 Introduction ............................................................................................................... 3.2 Control Objectives..................................................................................................... 3.3 Impedance Control Scheme ....................................................................................... 3.4 Impedance Control Synthesis .................................................................................... 3.4.1 Effects of impedance control ........................................................................ 3.4.2 Common impedance control law................................................................... 3.4.3 Interactive system behavior - coupled stability ............................................. 3.4.3.1 Linearized interaction models......................................................... 3.4.3.2 Coupled system stability................................................................. 3.4.3.3 Coupling of passive systems........................................................... 3.4.3.4 Robust coupled system stability ..................................................... 3.4.3.5 Coupled system performance ......................................................... 3.4.3.6 Performance of the controller Gf = Gt-1 .......................................... 3.5 Improved Impedance Control .................................................................................... 3.5.1 Improved control law .................................................................................... 3.5.2 Coupled system performance ........................................................................ 3.5.3 Target impedance model realization ............................................................. 3.6 Typical Impedance Contact Behavior........................................................................ 3.7 Contact Transition Stability....................................................................................... 3.7.1 Definition of the contact stability.................................................................. 3.8 Contact Stability Conditions...................................................................................... 3.8.1 Time domain analysis ................................................................................... 3.8.1.1 Constant velocity phase contact ..................................................... 3.8.1.2 Constant acceleration/deceleration phase contact........................... 3.8.2 Passivity-based contact transition stability analysis...................................... 3.8.3 Robust transition stability - generalized contact stability.............................. 3.8.4 Equivalence of robust- and passivity-based contact stability........................ 3.9 Influence of Non-Linear Effects on Contact-Stability ............................................... 3.9.1 Coulomb’s friction ........................................................................................ 3.9.2 Control lags and sampling effects ................................................................. 3.9.2.1 Ideal target system with force delay ............................................... 3.9.2.2 Robust and passivity-based contact stability of discrete-time system............................................................................................. 3.9.2.3 Contact stability of sampled-data system ....................................... 3.10 Evaluation of Contact Transition Stability Conditions.............................................. 3.10.1 Contact transition performance indices......................................................... 3.10.2 Contact transition assessment........................................................................ 3.10.3 Upper limits on target impedance frequency ................................................ 3.11 Conclusion................................................................................................................. Bibliography........................................................................................................................
255 255 256 261 266 267 268 269 270 272 277 280 284 287 288 290 294 296 301 319 320 332 333 335 339 340 344 347 351 351 352 354 365 369 375 379 381 394 398 401
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4. Practical Synthesis of Impedance Control ........................................................................... 4.1 Introduction ............................................................................................................... 4.2 Influence of the Target Parameters on the Impedance Control Performance............. 4.2.1 Influence of target frequency and mass on contact transition ....................... 4.2.2 Reduction of force overshoots – Hogan’s target impedance model (generalized stiffness control) ....................................................................... 4.3 Selection of Target Impedance Parameters – Impedance Control Design at Lower Control Layer ................................................................................................. 4.3.1 Specification of impedance control geometry............................................... 4.3.2 Specification of input design parameters - user interface.............................. 4.3.3 Design algorithm........................................................................................... 4.4 Synthesis of Impedance Control at Higher Layers .................................................... 4.4.1 World model for the impedance control ....................................................... 4.4.2 Impedance control integration into forward industrial robot position control 4.4.3 Impedance control functions and algorithms ................................................ 4.4.3.1 End-effector position modification................................................. 4.4.3.2 Selection and initialization of impedance gain parameters............. 4.4.3.3 Computation of path correction...................................................... 4.4.3.4 Delta x to delta T transformation (deltax_2_deltaT) ...................... 4.4.3.5 Compensation for payload and inertial effects ............................... 4.4.3.6 Sensor frame to compliance frame transformation (Sf2cf)............. 4.4.3.7 Force setpoint check ....................................................................... 4.4.3.8 Force sensor data preprocessing ..................................................... 4.4.3.9 Contact check ................................................................................. 4.4.4 Higher control layers algorithms................................................................... 4.4.4.1 Impedance control operating modes............................................... 4.4.4.2 Change of impedance gains (relax) ................................................ 4.4.5 Actions and tasks control algorithms ............................................................ 4.4.5.1 Grasping/detach action ................................................................... 4.4.5.2 Insertion/extraction......................................................................... 4.5 Conclusion................................................................................................................. Bibliography........................................................................................................................
405 405 406 406
5. Robust Control of Human-Robot Interaction in Haptic Systems ........................................ 5.1 Introduction ............................................................................................................... 5.2 Haptic System Structures........................................................................................... 5.3 Haptic Rendering ....................................................................................................... 5.4 Robust Control of Haptic Systems Interaction .......................................................... 5.4.1 Admittance display control ........................................................................... 5.4.2 Impedance display control ............................................................................ 5.5 Conclusion................................................................................................................. Bibliography........................................................................................................................
491 491 492 496 498 499 510 516 517
6. Intelligent Control Techniques for Robotic Contact Tasks ................................................. 6.1 Introduction ............................................................................................................... 6.2 The Role of Learning in Intelligent Control Algorithms for Compliant Tasks ......... 6.3 A Survey of Intelligent Control Techniques for Robotic Contact Tasks ...................
519 519 523 532
409 414 416 426 431 451 452 457 461 461 463 463 464 465 466 467 467 468 468 468 469 471 473 476 485 487
Contents The Synthesis of New Connectionist Learning Control Algorithms for Robotic Contact Tasks............................................................................................... 6.4.1 The background of the new connectionist control synthesis......................... 6.4.2 Model of robot interacting with dynamic environment – task setting........... 6.4.3 Factors affecting task performance and stability in robotic compliance control........................................................................................ 6.4.4 The comprehensive connectionist control algorithm based on learning and classification for compliance robotic tasks............................... 6.4.5 The genetic-connectionist algorithm for compliant robotic tasks ................. 6.4.6 GA tuning of PI force feedback gains........................................................... 6.4.7 Case studies................................................................................................... 6.5 Connectionist Reactive Control for Robotic Assembly Tasks by Soft Sensored Grippers ..................................................................................................................... 6.5.1 Analysis of the assembly process with soft fingers....................................... 6.5.2 Assembly process.......................................................................................... 6.5.3 Learning compliance methodology by neural networks ............................... 6.5.4 Experimental results...................................................................................... 6.6 Intelligent Control of Contact Tasks in Humanoid Robotics..................................... 6.6.1 Introduction................................................................................................... 6.6.2 Definition of control problem and advanced control methods for humanoid robots ........................................................................................... 6.6.3 The model of the system ............................................................................... 6.6.3.1 Model of the robot’s mechanism .................................................... 6.6.3.2 Definition of control criteria........................................................... 6.6.3.3 Gait phases and indicator of dynamic balance................................ 6.6.4 Hybrid integrated dynamic control algorithm with reinforcement structure. 6.6.4.1 Dynamic controller of trajectory tracking ...................................... 6.6.4.2 Compensator of dynamic reactions based on reinforcement learning structure ............................................................................ 6.6.4.3 Impact-force controller ................................................................... 6.6.4.4 Conflict between controllers........................................................... 6.6.5 Simulation studies......................................................................................... 6.7 Conclusion................................................................................................................. Bibliography........................................................................................................................
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552 552 554 557 559 567 569 570 575 577 578 579 580 584 584 587 593 593 594 596 598 600 601 605 606 608 616 618
Instead of Conclusion ................................................................................................................
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Index ..........................................................................................................................................
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Chapter 1
Control of Robots in Contact Tasks: A Survey
1.1 Introduction This chapter provides the state of the art in compliant motion control. It covers some early ideas and their later improvements, as well as some new control concepts and recent trends in this field. A comprehensive review of various compliant motion control methods proposed in the literature would certainly be very voluminous, since the research in this area has rapidly been growing in the recent years. Therefore, for practical reasons, a limited number of the most relevant or representative investigations and methods are discussed. Before reviewing many of the results, a categorization of compliant motion tasks and proposed control concepts will be made based on various criteria. In this survey particular attention is paid not only to traditional indices of control performance, but to the reliability and applicability of algorithms and control schemes in industrial robotic systems. These systems are widely employed in practice, and they provide a reasonable background for compliance motion control implementation. However, compliance control is a very interesting control approach in the new emerging technologies such as service robotics (e.g. surgical and rehabilitation robots), virtual reality and haptics, telemanipulation, human augmentation and assistance. These fields apply new and quite different robotic structures in comparison to conventional industrial robotic systems (e.g. direct-drive robots, parallel and wire manipulators, etc.). The development of interaction control algorithms for new robotic applications will also be addressed in this chapter. 1.2 Contact Tasks Regarding the nature of the interaction between a robot and its environment robotic applications can be categorized in two classes. The first one is related to non-contact tasks, i.e. unconstrained motion in a free space, without any relevant environmental influence exerted on the robot. In the non-contact tasks the robot’s own dynamics has a crucial influence upon its performance. A limited number of 1
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Dynamics and Robust Control of Robot-Environment Interaction
most frequently performed simple robotic tasks in practice such as pick-andplace, spray painting, gluing or welding, belong to this group. In contrast to these tasks, many complex advanced robotic applications such as assembly and machining require the manipulator to be mechanically coupled to the other objects. In principle, two basic contact task subclasses can be distinguished. The first one covers essential force tasks whose very nature requires the end-effector to establish the physical contact with the environment and exert a process-specific force. In general, these tasks require a simultaneous control of both the end-effector position and interaction force. Typical examples of such tasks are machining processes such as grinding, deburring, polishing, bending, etc. In these tasks, the force is an inherent part of the process and plays a decisive role for its fulfillment (e.g. metal cutting or plastic deformation). In order to prevent overloading or damage to the tool during operation, this force must be controlled in accordance with some definite task requirements. The prime emphasis of the tasks within the second subclass lies on the endeffector motion, which has to be realized close to the constrained surfaces (compliant motion). A typical representative of such tasks is the part-mating process. The problem of controlling the robot during these tasks is, in principle, the problem of accurate positioning. However, due to imperfections inherent in the process, sensing and control system, these tasks are inevitably accompanied by the occurrence of contact with constrained surfaces, which results in the appearance of reaction forces. The measurement of interaction force provides useful information for error detection and an appropriate modification of the prescribed robot motion. Compliance, i.e. accommodation [1], can be considered as a measure of the ability of a manipulator to react to interaction forces. This term refers to a variety of different control methods in which the end-effector motion is modified by contact forces. The future will certainly hold more tasks for which the interaction with the environment is fundamental. Recent medical robot applications in surgery (e.g. spine surgery, neurosurgical and microsurgical operations, knee and hip joints replacement) may also be considered as being essentially contact tasks. Comprehensive research programs in automated construction, agriculture and food industry focus on the robotization of several representative contact tasks such as underground excavation, meat deboning, etc. 1.3 Classification of Constrained Motion Control Concepts The previous classification of elementary robotic tasks provides a framework for further systematization of the concepts concerning the robotized compliant
Control of Robots in Contact Tasks: A Survey
3
motion control. The problems encountered in controlling compliant motion have been extensively investigated and several control strategies and schemes have been proposed and elaborated. These methods can be systematized according to the different criteria. The primary systematization can be made in relation to the sort of compliance. According to this criterion, two basic groups of control concepts for compliant motion can be distinguished (Fig. 1.1): i)
ii)
Passive compliance, in which the end-effector position is accommodated by the contact forces themselves, due to the compliance inherent in the manipulator structure, servos or special compliant devices; Active compliance, in which the compliance is provided by constructing a force feedback in order to achieve a programmable robot reaction, either by controlling interaction forcea or by generating task-specific compliance at the robot end-point.
Regarding the possibility of adjusting the system compliance to specific process requirements, passive compliance methods can be classified as adaptable and non-adaptable. Based on the dominant sources of compliance, the following subgroups of methods within these groups can be distinguished (Fig. 1.2): i) • •
ii) • •
Fixed passive compliance: Methods based on the compliance inherent in the robot’s mechanical structure, such as elasticity of the arm, joints and end-effectors [2]; Methods that use specially constructed passive deformable structures attached close to the robot end-effectors and specially designed for particular problems. The best known is the “Remote Center Compliance”-RCC element [3]. Adaptable passive compliance: Methods based on the devices with tunable compliance [4], Compliance achieved by the adjustment of joint servo-gains [5].
The basic classification of active compliance control methods can be done based on the previous underlying contact task classification into essential and potential. Using the terminology of bond-graph formalisms, the robot behavior
a By force we mean force and torque and accordingly position should be interpreted as position
and orientation.
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Dynamics and Robust Control of Robot-Environment Interaction
in the essential contact tasks can be generalized as a source of the effort (force) that should raise a flow (motion) reaction of the environment objects. The robot behavior associated with the second task’s subclass corresponds to the impedance, since this is characterized by the robot’s motion reaction to external forces exerted by the environment.
Task-Oriented Compliance of Robots
passive
active
Solution: controller or structural inherent compliance (system response in accordance to arising contact force)
Solution: force feedback controlled compliance (system response in accordance to task specific criteria)
Fig. 1.1 Basic classification of robot compliance
In view of the above, the active control force methods can be classified into the following two groups (Fig. 1.3): i)
ii)
Force, i.e. position/force control in general, or admittance control, where both desired interaction force and robot position are controlled. In force control, a desired force trajectory is commanded and force should be measured to realize the feedback control; Impedance control [6], which uses the different relationships between the acting forces and manipulator position to adjust the mechanical impedance of the end-effector to the external forces. The impedance control problem can be defined as a requirement for designing a controller so that the interaction forces govern the difference between nominal and actual positions of the end-effector according to the target impedance law. Impedance control is essentially based on position control and requires position commands and position measurements in order to close the feedback loop. In addition, force measurements are needed to realize the target impedance behavior.
Position/force control methods can be divided into:
Control of Robots in Contact Tasks: A Survey
i)
ii)
5
Hybrid position/force control, where position and force are controlled in a non-conflicting way in two orthogonal subspaces defined in a task-specific frame (compliance or constraint frame). In the force controlled end-effector degrees of freedom (DOFs) the contact force is essential for performing the task, while in the position DOFs, the motion is most important. Force is imposed and controlled along directions that are constrained by the environment, while position is controlled in those directions in which the manipulator is free to move (unconstrained). Note that the term hybrid control is usually referred to the method of Raibert and Craig [7]. However, in accordance with the view of Mason [1], the definition of this term is used here in a more general sense and it refers to any controller that is based on the division into force- and position-controlled directions. Unified position-force control, which differs essentially from the above conventional hybrid control schemes. Vukobratovic and Ekalo [8] have established a dynamic approach, to control simultaneously both the position and force in an environment with completely dynamic reactions. The approach of dynamic interaction control [8] defines two control subtasks responsible for the stabilization of robot position and interaction force. Both control subtasks utilize dynamic models of the robot and environment in order to ensure tracking of both the nominal motion and force. Parallel position/force control [9] is based on the appropriate tuning of position and force controllers. The force loop is designed to dominate the position control loop along constrained task directions where a force interaction is expected. From this viewpoint, the parallel control can be considered as a combination of impedance and force control.
Taking into account the way in which the force information is included in the forward control path, the following position/force control schemes can be distinguished: i)
Explicit or force-based [7, 10, 11] algorithms, where force control signals (i.e. the difference between the desired and the actual force) are used to generate the torque inputs for the actuators at the robot’s joints.
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Dynamics and Robust Control of Robot-Environment Interaction
ii)
Implicit or position-based algorithms [12, 13], where the force control error is converted to an appropriate robot motion adjustment in force-controlled directions and then added to the positional control loop.
Similar to the above classification, impedance control methods can also be divided in accordance with the way in which the robotic mechanism is treated, either as an actuator (i.e. source) of position or a force actuator. However, as already noted, the aim in impedance control is rather to provide specific relationships between the effort and motion than to follow a prescribed force trajectory as in case of force control. Considering the arrangement of position and force sensor and control signals within control loops (inner or outer), the following two common approaches to the issue of providing task-specific impedance via feedback control can be distinguished [14]: i)
ii)
Position-mode or outer-loop control, where a target impedance control block relating the force exerted on the end-effector and its relative position is added in an additional control loop around the position-controlled manipulator. Here, an inner loop is closed on the basis of the position sensor with an outer loop closed around it, based on the force sensor [15, 16]; Force-mode or inner-loop control, where position is measured and force command is computed to satisfy target impedance objectives [14].
Regarding the force-motion relationship, i.e. the impedance order, impedance control schemes can be further categorized into: stiffness control [17], damping control [18], and general impedance control [19, 20], using the zero-, first- and second-order impedance model respectively. There are also additional criteria for a further detailed classification of active compliant motion control concepts. For example, we can categorize methods with respect to the source of force information (with or without direct interaction force sensing), allocation of force sensor (wrist, torque sensor at joints, force sensing pedestal, force sensor placed at the contact surface, sensors at robot links, fingers, etc.). In order to avoid the problems associated with noncollocation between measurement of contact forces and actuation at robot joints, which can cause instability [21], it was also proposed to use redundant force information, combining the joint force sensing with one of the above force sensing principles.
Control of Robots in Contact Tasks: A Survey
7
Regarding the space in which the active force control is performed, one can distinguish: i)
ii)
Operational space control techniques, where the robot control is taking place in the same frame in which robot actions are specified [22, 23]. This approach requires the construction of a model describing the system dynamic behavior as perceived at the point of an end-effector, where the task is specified (operational point, i.e. coordinate frame). Traditionally, compliant motion is specified using the task or compliance frame approach [1]. This geometrical approach introduces a Cartesian compliant frame with orthogonal force and position (velocity) controlled directions. In order to overcome the limitations of this approach, new methods were proposed [24, 25]. These approaches, referred to as explicit task specification of compliant motion, are based on the model of the constraint topology for every contact configuration and utilize projective geometry metrics in order to define a hybrid contact task. Joint space control, where control objectives and actions are mapped into joint space [26]. Associated with this control approach are transformations of action attributes, compliance and contact forces from the task to the joint space.
Further, considering control issues, such as variations of control parameters (gains) during execution, one can distinguish: i)
ii) iii)
Non-adaptive active compliance control algorithms which use fixed gains, assuming small variations in the robot and environment parameters, Adaptive control, which can adapt to the process variations [27, 28], and Robust control approaches that can handle model imprecision and parametric uncertainties within specified bounds [29, 30].
Depending on the extent the system dynamics is involved in the applied control laws, it is further possible to distinguish: i)
Non-dynamic, i.e. kinematic model-based algorithms, such as hybrid control [7], stiffness control [17], etc., which approximate the contact problem considering its static aspects only;
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Dynamics and Robust Control of Robot-Environment Interaction
ii)
Dynamic model-based control schemes, such as resolved acceleration control [31], dynamic hybrid control [11], constrained robot control [32], dynamic position/force control in contact with dynamic environment [8, 33], based on complete dynamic models of the robot and environment that take into account all dynamic interactions between position- and force- controlled directions.
Although relatively low velocities are involved in contact motion, high dynamic interaction (i.e. exchange of energy) between the robot and its environment affects the control system significantly and can jeopardize the stability of the control system [34]. Consequently, the roles of the robot dynamics [35] and environment dynamics [8, 33] in the control of compliant motion are of essential importance. Kinematic algorithms are mostly based on Jacobian matrix computation, while the complexity of dynamic methods is much higher [36]. The seminal hybrid control method proposed by Raibert and Craig [7] provides essentially a quasi-static approach to compliance control based on an idealized simple geometric model of a constrained motion task (Mason’s constraint-frame formalism). In the hybrid control, the dynamics of both the robot and environment (i.e. dynamic interaction) is neglected. The dynamic hybrid control [11] and constrained motion control [32] consider constraints on the robot motion described in the form of algebraic equations defining a hyper surface. These methods take both the robot dynamic models and the model of environment into account in order to synthesize the dynamic control laws that ensure an admissible robot motion under the constraint and realize desired interaction forces. A further generalization of the constrained motion problem leads to introducing active dynamic contact forces (dynamic environment), also described by differential equations. In a dynamic environment, the interaction forces are not compensated for by constraint reactions, they rather produce active work on the environment. Obviously, the contact with a dynamic environment requires considering the overall system dynamics, involving the robot and interaction models, in order to obtain an admissible robot motion and interaction forces. In the papers dealing with the dynamic control of robots interacting with dynamic environment [8, 33] Vukobratovic and Ekalo considered “purely dynamic” interaction without passive reactions. De Luca and Manes [37] have proposed a convenient model structure that handles a more general case in which purely kinematic constraints on the robot effector exist together with dynamic interactions.
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Control of Robots in Contact Tasks: A Survey
Passive Compliance
fixed
inherent compliance of robot structure
adaptable
additional compliance devices (RCC)
additional adjustable compliance devices
servo gain adjustment
Fig. 1.2 Passive compliance classification
Active Compliance
Force Control Solution: direct control of interaction force
Impedance Control Solution: control of dynamic robot reaction to contact forces according to target force/motion relationship
hybrid position/force control
unified postion/force control
stiffness
position and force control in two orthogonal subspaces
control of both position and force along each task space direction
position proportional
F=k ⋅∆ x
damping general (accomodation) impedance ⋅ ⋅ ⋅∆ x F=k ⋅∆ x F=M ⋅ ∆⋅⋅ x+D ⋅∆ x+k velocity proportional
position, velocity and acceleration proportional
Fig. 1.3 Active compliance control methods
Although being very inclusive, the above classification cannot encompass all of the concepts that have been proposed up to now. Namely, some of the elaborated approaches combine two or more different methods categorized in distinct groups, and attempt to use benefits of each to compensate specific disadvantages of single solution strategies. To such methods belong compliant motion control approaches that combine force and impedance control [12, 38]. Some methods integrate control and mechanical system design [39]. This approach is based on micro-macro manipulator structures, providing an inherently stable and well-suited subsystem for high-bandwidth active force
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Dynamics and Robust Control of Robot-Environment Interaction
control. Note that the above terminology represents, in some measure, a trade-off among different nomenclature used in the literature. For instance, Mason [1] denominates the control concepts specifying a linear relation between the effector force and position as explicit feedback, while Whitney [6] refers the explicit control to techniques having a desired force input rather than position or velocity input. The introduced classification and terminology reflect, in our opinion, in a most suitable way the essential aspects of appropriate control strategies. The above classification is summarized in (Figs. 1.1, 1.2, 1.3). 1.4 Model of the Robot Performing Contact Tasks Here we consider in brief the simplified models of robot constrained motion to be used for the analysis of contact motion control concepts. In order to form the mathematical model which describes the dynamics of a closed-configuration manipulator, let us consider an open robotic structure whose terminal link (endeffector) is subjected to a generalized external force (Fig. 1.4). The dynamics model of a rigid manipulation robot interacting with the environment is described by the vector differential equation in the form:
H (q )qɺɺ + h(q, qɺ ) = τ a + J T (q ) F
(1.1)
where q = q(t ) is an n-dimensional vector of robot generalized coordinates; H ( q ) is an n × n positive definite matrix of inertia moments of the manipulator mechanism; h( q, qɺ ) is an n-dimensional nonlinear vector function involving centrifugal, Coriolis and gravitational moments, i.e. h(q, qɺ ) = h (q, qɺ ) + g ( q ) , where g ( q ) is the vector of gravitational moments and h ( q, qɺ ) is the Coriolis and centrifugal vector component; τ a = τ a (t ) is an n-dimensional vector of generalized joint-axes driving torques; J T ( q ) is an n × m Jacobian matrix relating joint space velocity to task space velocity; F = F (t ) is an mdimensional vector of external forces and moments acting on the end-effector. The dynamic model of the actuators (we confine ourselves to robot manipulators driven by DC motors) that drive the robot joints has to be added to the above equations. It is convenient to adopt this model in a linear form. Taking into account that electric time constants of actual DC motors driving almost all commercial robotic systems are very low, we shall adopt a second-order model of actuators
Control of Robots in Contact Tasks: A Survey
n 2j I mj qɺɺmj + n2j bmj qɺmj + τ aj = n jτ mj
11
(1.2)
where qmj is the output angle of motor shaft after reducer; n j is the gear ratio; I mj is the inertia of the motor actuator; bmj is the motor viscous friction coefficient; τ mj is the control input to the actuator j (i.e. motor torque), while j denotes the j -th local subsystem (m stands for “motor”). The torque produced by the motor is proportional to the armature current, that is
τ mj = kmj imj
(1.3)
where kmj is the torque constant. If we assume the stiffness at the joints (gears) to be infinite, the relations between the mechanism coordinate q j coincide with those of the actuator coordinate qmj . The dynamic models of the actuators and mechanical part of the robot are related by joint torques (loads). If we introduce τ aj from (1.2) into (1.1) we get the overall model of the robotic mechanism in the joint coordinate space
H (q) qɺɺ + Bm qɺ + h (q, qɺ ) + g (q) = τ q + J T (q) F
(1.4)
where
H (q ) = H (q ) + I m = H (q ) + diag (n 2j I mj ) (1.5)
Bm = diag (n 2j bmj )
and τ q is a nx1 vector of input torques at joint shaft (after reducer), which for n = 6 has the form T
τ q = [ n1τ m1.... n6τ m 6 ] . The above dynamic model can be transformed into an equivalent form which is more convenient for the analysis and synthesis of a robot controller for contact tasks. When the manipulator interacts with the environment it is very convenient to describe its dynamics in the space where the manipulation task is described, rather than in joint coordinate space (also termed configuration space). The endeffector position and orientation with respect to a reference coordinate system can be described by a 6-dimensional vector x. Using the robot Jacobian matrix we can transform the robot dynamic model (1.4) from the joint into the endeffector coordinate system
Λ ( x ) ɺɺ x + B ( x ) xɺ + µ ( x, xɺ ) + p ( x ) = τ + F
(1.6)
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Dynamics and Robust Control of Robot-Environment Interaction
Fig. 1.4 Open kinematic chain exposed to the action of an external force
where the relationships between the corresponding matrices and vectors from equations (1.4) and (1.6) are given by the following equations
Λ ( x)
=
J − T ( q ) H ( q ) J −1 ( q )
B( x)
=
J −T (q ) Bm J −1 (q )
µ ( x, xɺ ) =
J −T (q ) h (q, qɺ ) − Λ ( x) Jɺ (q ) qɺ −T
p( x)
=
J
τ
=
J −T ( q ) τ q
(1.7)
(q) g (q )
Description, analysis and control of manipulator systems with respect to the dynamic characteristics of their end-effectors are referred to as the operational space formulation [22]. Analogous to the joint space quantities, Λ ( x ) is the operational space inertia matrix, µ ( x, xɺ ) is the vector of Coriolis and centrifugal forces, p ( x ) is the vector of gravity terms, and τ is the applied input control force in the operational space. The above Cartesian dynamic model covers a large class of different robotic structures such as industrial robots, parallel manipulator, wire robots, etc.
Control of Robots in Contact Tasks: A Survey
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Since the mechanical interaction process is generally very complex and difficult to describe mathematically in an exact way, we are compelled to introduce certain simplifications and thus partly idealize the problem. In practice the interaction force F is commonly modeled as a function of the robot dynamics, i.e. end-effector’s motion (position, velocity and acceleration) and control input
F = F ( x, xɺ , ɺɺ x ,τ , d , d e )
(1.8)
where d and d e denote the sets of robot and environment model parameters, respectively. The following general work environment models have been mostly applied in the literature for describing robot constrained motion: rigid hypersurface [11, 32], dynamic environment [8, 37], and compliant environment [40]. In the case when the environment does not possess the displacements (DOFs) that are independent from the robot motion, the mathematical model of the environment dynamics in the frame of robot coordinates can be described by nonlinear differential equations [8]
M ( q )qɺɺ + L( q, qɺ ) = S T ( q ) F
(1.9)
where M ( q) is a non-singular n × n matrix; L( q, qɺ ) is a nonlinear ndimensional vector function; S T ( q) is an n × n matrix with rank equal to n. Then, the system (1.4)-(1.9) describes the dynamics of robot interacting with dynamic environment. We assume that for the contact cases all the mentioned matrices and vectors are continuous functions of the arguments. In the operational space the model of a purely dynamic environment has the form [40]
M( x) ɺɺ x + L( x, xɺ ) = − F
M( x) = − S −T (q) M (q) J −1 (q) L( x, xɺ ) = − S −T (q) L(q, qɺ ) + M( x) Jɺ (q)qɺ In effect, a general environment model involves geometrical (kinematic) constraints plus dynamic constraints [37]. An example of such dynamic environment is when the robot is turning a crank or sliding a drawer whose dynamics is relevant for the robot motion and cannot be neglected. For the control design purposes it is customary to utilize the linearized model of manipulator and environment. The applicability of linearized model in constrained motion control design, especially in industrial robotic systems, was
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Dynamics and Robust Control of Robot-Environment Interaction
demonstrated in [41, 42]. Neglecting the nonlinear Coriolis and centrifugal effects due to relatively low operating velocities (“rate linearization”) during contact and assuming the gravitational effect to be ideally compensated for, we obtain the linearized model around a nominal trajectory in the Cartesian space x0 in the form
Λ( x0 ) xɺɺ + B ( x0 ) xɺ = τ ( x0 ) + F
(1.10)
In passive linear environments, it is convenient to adopt the relationship between forces and motion around the contact point in the form (linear elastic environment)
− F = M e ɺɺ p + Be pɺ + K e p
(1.11)
where p denotes the end-effector penetration through the surface defined by p = x − xe , xe is the contact point location, M e , Be and K e are the inertial, damping and stiffness matrices, respectively. 1.5 Passive Compliance Methods In accordance with the above classification, we shall firstly review the compliant control methods based on passive accommodation (with no actuator involved). Passive compliance in general is a concept often used in practice to overcome the problems arising from positional and angular misalignments between the manipulator and its working environment. 1.5.1 Non-adaptable compliance methods The passive compliance method, which is based on the inherent robots structural elasticity, is more interesting as a theoretical solution than as a feasible approach. This method supposes that the compliance of the mechanical structure has a determining role in the compliance of the entire system. However, this assumption is opposite to the real performance of commercial robotic systems, which are designed to achieve high positioning accuracy, whereas elastic properties of the arms are not significant. This method does not offer any possibility to adapt system compliance to the various task requirements. The idea to utilize flexible manipulator arms as an instrumented compliant system [2] is coupled with additional problems due to complex modeling and controlling of elastic robots.
Control of Robots in Contact Tasks: A Survey
15
The method based on mechanical compliance devices, in principle, also utilizes structural compliance, but most influential source of multi-axes compliance in this case is a specially constructed device, whose behavior is known and sufficiently repeatable. By this means relatively good performances, especially in the robotic assembly, have been achieved. Different types of such devices have been developed, the best known being the RCC, “Remote Center Compliance” [3], developed in the Charles Stark Draper Laboratory. RCC is designed to achieve the workpiece rotation around a defined center of compliance. The compliance center is referred to as a point such that the force applied at this point causes only translation, while the torque applied around an axis through this point will cause pure rotation of the workpiece (Fig. 1.5). A crucial feature of the RCC is that it consists of translational and rotational parts, which allows lateral and angular errors to be accommodated independently. An improvement of RCC represents IRCC, “Instrumented Remote Center Compliance” [43], which provides the fast error absorption characteristic of an RCC and measurement characteristic of a multi-DOF sensor. The information about contact forces and deformations can be used for task monitoring, calibration, contour following, or for positioning feedback.
Fig. 1.5 Remote Center Compliance (RCC)
1.5.2 Adaptable compliance methods Further development of the RCC has led to adaptable compliant devices [4] which enable the location of the center of compliance to be automatically controlled to a prescribed extent in accordance with involved parts of different
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Dynamics and Robust Control of Robot-Environment Interaction
length and weight. These devices are usually instrumented so to provide information about end-point deflections for robot control.
Fig. 1.6 Passive gain adjustment scheme
The controller gain adjustment method is based on the compliance of the robotic controller and attempts to provide a universally programmable passive compliance at the end-point, by a relatively simple adjustment of the servo gains. The basic principle is to tune the positional servo gains in order to make the robot behave as a linear six-dimensional spring in Cartesian space with programmable stiffness. Therefore, taking into account the relationship between the forces exerted upon the robot and its reaction (stiffness-like behavior), the gain adjustment method is often considered as being equivalent to the impedance (i.e. stiffness) control. The basic gain adjustment control scheme is sketched in Fig. 1.6, where x0 and q0 are the nominal Cartesian and joint position vectors respectively; f −1 ( x ) denotes the inverse kinematic transformation; q is the actual joint position; gˆ ( q ) is the computed gravitational torque, while the control torque τ q is defined as
τ q = k p ( q0 - q ) + gˆ ( q )
(1.12)
where k p represents the joint stiffness matrix which should be tuned in order to achieve the arm to behave with the desired stiffness K S . The relationship between the joint and Cartesian stiffness matrices is given by
k p = J T KS J K S = J − T k p J −1
(1.13)
Control of Robots in Contact Tasks: A Survey
17
where J (q ) is a Jacobian matrix relating velocities (i.e. forces) between a Cartesian frame fixed at the compliance center and joint coordinate space. At the center of compliance the Cartesian stiffness matrix is diagonal, but, according to (1.13), the corresponding joint stiffness k p is a fully symmetric matrix. This means that the joint stiffness matrix is highly coupled and a position error at one joint will affect the commanded torque at all other joints. This equation represents the central formulation of active gain adjustment methods. However, although this stiffness-like behavior could be theoretically adjusted on-line during task running, we have classified this method as being passive compliance, because the compliant motion is performed in a purely passive way by the action of external forces, rather than by the force feedback as in the active stiffness control. In reality, this concept is coupled with several problems. The desired springlike behavior cannot be accurately realized by the major of contemporary robotic systems. The nonlinearities such as friction and backlash in mechanical transmission, or process frictional phenomena like jamming, can largely destroy the force-position causality based on the stiffness law. Furthermore, by setting the control gains in some directions to be very low, we make the entire system more sensitive to perturbations. All these facts make the performance of this control approach uncertain, imposing the necessity to introduce additional sensor information in order to monitor the task execution. The principle of adaptable control gains is more suitable for the direct drive, multifingered or wrist hands, but in this case the method appears to be very close to the above ones that use special adaptable compliant-devices. Concerning the use of passive gain adjustment concept in industrial practice, it should be taken into account that conventional robotic systems are nonbackdriveable, because of high gear ratios and Coulomb friction/stiction effects at joints (the order of equivalent friction force in the Cartesian space is about 102 N). Hence, although a compliant control is applied, a force exerted at the end-effector will not cause a corresponding detectable displacement at joints. Therefore, the method may be applied only in the manipulation tasks permitting large interaction forces. Due to relatively high costs and low robustness of force sensors, however, there has been evident recently an increasing interest of industrial robot manufacturers to apply this method in specific tasks such as
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Dynamics and Robust Control of Robot-Environment Interaction
handling of castings (e.g. Soft-servo or Soft-float industrial robot control functions). 1.6 Active Compliant Motion Control Methods The active compliance control methods utilize in a best way the reprogrammability of manipulation robots, representing their major characteristic (i.e. the ability to switch from one production task to another). 1.6.1 Impedance control First implementations of force feedback control of a manipulator belong to the impedance control [6]. The impedance control problem can be defined as designing a controller so that interaction forces govern the error between desired and actual positions of the end-effector. The control input describing a desired target impedance relation may, in principle, have an arbitrary functional form, but it is commonly adopted in the form of a linear second-order differential equation describing the simple six-dimensional decoupled mass-spring-damper mechanical system. The reason for this lies in the fact that the dynamics of a second-order system is well understood and familiar. Lee and Lee [44] have developed a control algorithm, referred to as generalized impedance control, by introducing a higher order impedance relation between position and force errors, which includes force derivatives. In other words, impedance control is a general approach to contact task control in which the robot behaves as a mass-spring-dashpot system whose parameters can be specified arbitrarily. This can be achieved by feedback control using position and force sensing. The following control objective should be attained
F = M t ( ɺɺ x − ɺɺ x0 ) + Bt ( xɺ − xɺ0 ) + K t ( x − x0 ) = M t ɺɺ e + Bt eɺ + K t e
(1.14)
or in the s-domain
F ( s ) = Z t ( s ) ( x − x0 ) = Z t ( s ) e = ( M t s 2 + Bt s + K t ) ( x − x0 )
(1.15)
2 where Z t ( s ) = M t s + Bt s + K t is the target robot impedance in Cartesian space; x0 describes the desired position trajectory, x is the actual position
Control of Robots in Contact Tasks: A Survey
19
vector; e is the position control error; F is the external force exerted upon the robot, and M t , Bt , and K t are positive definite matrices which define the target impedance, where K t is the stiffness matrix, Bt is the damping matrix and M t is the inertia matrix. The diagonal elements of these target model matrices describe the desired robot mechanical behavior during the contact. One of the most common approaches for representing robot’s and object’s positions in robot programs is based on coordinate frames. Therefore it is convenient to describe the robot impedance reaction to the external forces also with respect to a frame, referred to as compliance or C-frame. Along each of Cframe directions, the target model describes the mechanical system with the programmable impedance (mechanical parameters), presented in Fig. 1.7, where, for simplicity sake, only spring elements are depicted. The model describes a virtual spatial system consisting of mutually independent spatial mass-damperspring subsystems in six Cartesian directions.
Fig. 1.7 Target stiffness model in the C-frame
The target impedance matrices can be selected to correspond to various objectives of the given manipulation task [14]. Obviously, high stiffness is selected in the directions where the environment is compliant and positioning accuracy is important. Low stiffness is selected in the directions where small interaction forces have to be maintained. Large Bt values are specified when
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Dynamics and Robust Control of Robot-Environment Interaction
energy is to dissipate, and M t is used to provide smooth transient behavior in the system response during the contact. In order to assess how well a designed impedance controller meets the above control objective, it is customary to specify various performance criteria. A reasonable measure capable to express the performance of the impedance control represents the difference between the target model and the real system behavior described by actual robot motion and interaction forces [45]. Depending on which of these physical quantities is used to characterize the system behavior (force or position), the impedance control error can be expressed by means of force measure (force mode error)
e f = M t ( ɺɺ x − ɺɺ x0 ) + Bt ( xɺ − xɺ0 ) + K t ( x − x0 ) − F
(1.16)
or by position measure (position model error)
e p = x − x0 − δ x f
(1.17)
where the target model-based position deviation δ x f is obtained as the solution of the target model differential equation
F = M tδ ɺɺ x f + Btδ xɺ f + K tδ x f
(1.18)
for the initial conditions: F ( t0 ) = 0; δ x f ( t0 ) = δ x0 .
Fig. 1.8 Damping control
The control goal defined above can be achieved using various control strategies. Impedance control represents a strategy for constrained motion rather
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than a concrete control scheme. Various control concepts and schemes have been established and proposed for controlling the relation between robot motion and interaction force. One of the first approaches to impedance control was proposed by Whitney [18] (Fig. 1.8). In this approach, referred to as damping or accommodation control, the force feedback is closed around the velocity control loop. The interaction force is converted into velocity modification command by a constant damping coefficient K f . On a simplified example of discrete time force control, Whitney defined the condition for system stability during the contact
0 < T K f Ke < 1
(1.19)
where T is the sampling period, K f is the force control gain (damping coefficient), and K e is the stiffness of the environment. This condition implies that if K e is high, the product TK f must be small. To avoid large contact forces, a very high sampling rate, i.e. small T , is required. Alternatively, for the contact with a very stiff object Whitney proposed to introduce a passive compliance in order to make the equivalent environmental stiffness (includes the stiffness of robot structure, environment, sensor, etc.) K e smaller. Salisbury [17] proposed to modify the end-effector position in accordance with the interaction force (Fig. 1.9). This concept is based on a generalized stiffness formulation F = Kδ x , where δ x is a generalized displacement from the nominal commanded end-effector position, and K is a six-dimensional stiffness matrix. Based on the difference between desired and actual end position, a nominal force is computed and converted into the joint torque using the transpose of the Jacobian matrix. This force is then used to determine the torque error at each joint, which is further used to correct the applied torque, so that a desired force (i.e. stiffness) is maintained at the robot hand. The requirements on the stiffness matrix elements and their design for specific tasks are considered in [6]. The above impedance control schemes are simple and relatively easy to implement. However, in these approaches the achieved closed-loop impedance behavior in the Cartesian space depends on robot configuration. Obviously, to replace a highly nonlinear robot dynamic model with a linear time-invariant target system (e.g. mass-damper-spring system), requires in general a control law that will compensate for the relevant system nonlinearities (model-based dynamic control). The most common and general impedance control concept was
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Dynamics and Robust Control of Robot-Environment Interaction
proposed by Hogan [19]. The author has defined a unified theoretical framework for understanding the mechanical interactions between physical systems.
Fig. 1.9 Stiffness control
1.6.1.1 Force-based impedance control Most of the impedance control algorithms utilize the computed torque method to cancel the nonlinearity in robot dynamics in order to achieve linear target impedance behavior. This popular approach requires computation of a complete dynamic model of the robot’s constrained motion, which makes its realization rather complex. An important drawback of this approach is also the sensitivity to model uncertainties and parameter variations. Performance improvements that can be achieved with the algorithms in industrial robotics are not in proportion to the implementation efforts. Hogan [46] has proposed several techniques with and without force feedback for modulating the end-point impedance of a general nonlinear manipulator. Supposing that the Cartesian dynamic model perfectly matches the real system, Hogan proposed the following nonlinear control law
τ = Λˆ M t −1 [ K t ( x0 − x ) − Bt xɺ + F ] + µˆ ( x, xɺ ) + pˆ ( x ) − F
(1.20)
to be applied in order to attain a reasonable target impedance behavior in the ideal case in the form
F = M t ɺɺ x + Bt xɺ + K t ( x − x0 )
(1.21)
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The control scheme corresponding to the above control law is sketched in Fig. 1.10. In this figure a distinction is made between the active force exerted by the robot ( F ) and the reactive external force ( F ), which can be computed assuming a simple spring-like environmental model
F = K e ( x − xe ) = − F
(1.22)
where K e is the stiffness of the environment. This control law represents essentially a nonlinear control algorithm which combines the inverse control technique [47] (terms also used are computed torque method, nonlinear decoupling) and force-based (inner loop) impedance control. In the force-based impedance control algorithms (Fig. 1.10), in general, an expected reference force is computed to satisfy the desired impedance specification based on the position error and target impedance FC ( s ) = Gt ( s )( x0 − x)(Gt ( s ) = Z t ( s )). The expected contact force FC is compared with the actual force sensed by the force sensor and a force error is computed. This error is further multiplied
ˆ M −1 . Finally, the product is summed with dynamic with the inertia matrices Λ t compensation terms (Coriolis and gravitation vectors) and feed-forward force F to obtain Cartesian control force, which is further transferred to the robot joint T via the transposed Jacobian J , to get the actuator torque control inputs. It is relatively easy to prove that the control law
τ = Λˆ {ɺɺ x0 + M t −1 [ K t ( x0 − x ) + Bt ( xɺ0 − xɺ ) + F ]} + µˆ ( x, xɺ ) + pˆ ( x ) − F
(1.23)
realizes the impedance control behavior specified in (1.15). The reason why impedance control methods based on force control input cannot be suitably applied in commercial robotic system lies in the fact that commercial robots are designed as “positioning devices”. In the above methods the driving torque vector ensuring the desired target impedance behavior has been computed and then multiplied by the transpose of the Jacobian matrix, to be realized around the actuated robot joints. However, the realization of computed torque in commercial robotic systems is not accurate because the local servos are position-controlled and there is no force feedback with respect to the torques around the robot’s joints. Consequently, the realization of desired torques is poor, since high friction and other nonlinearities in the transmission mechanisms contribute significantly to the inaccuracy of current-torque causality. The implementation of force-based impedance control under above-mentioned
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Dynamics and Robust Control of Robot-Environment Interaction
difficulties can be only successful in a new generation of direct drive robotic systems [48] with accurate joint torque control. It may also be remarked that force-based impedance control requires a completely new control system to be implemented.
Fig. 1.10 Force-based dynamic impedance control
1.6.1.2 Position-based impedance control In commercial robotic systems it is feasible to implement only the position-mode impedance control by closing a force-sensing loop around position controller. Position- based impedance control is most reliable and suitable for implementation in industrial robot control systems since it does not require any modification of conventional positional controller. Practically, two basic impedance control schemes with an internal position control can be distinguished [49]. The first scheme is sketched in Fig. 1.11. In this control system, an inner position control loop is closed based on the position sensing, with an outer loop closed around it, based on the force sensing. The force loop is naturally closed when the end-effector encounters the environment. The outer loop includes a force feedback compensator GF , basically representing admittance since its role is to shape the relation between the contact force and corresponding nominal position modifications ∆x f . This block is imposed on the system to regulate the force response to the commanded and −1 actual motion according to the target admittance Z t . Other control blocks in Fig. 1.11 represent a common industrial robot position control system involving
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the following transfer function matrices: Gr -position control regulator, Gs robot plant and Ge -environment. The position correction ∆x F is subtracted from the nominal position x0 and the command input vector for the positional controller, referred to as reference position xr , is computed. A good tracking of the reference position has to be realized by the internal position controller. −1 Practically, assuming GF = Z t , the position error input to the position controller ∆xr becomes
∆xr = xr − x = x0 − ∆xF − x = x0 − x − Z t −1F = e p
(1.24)
This means that the control system depicted in Fig. 1.11 utilizes the positionrelated impedance model error e p (1.17) to realize the target impedance behavior. Practically, the impedance model error e p is fed forward to the position controller Gr in order to be nullified within the internal position control loop. Since the purpose of the control system in Fig. 1.11 is to control position, it will be referred to as position impedance model error control.
Fig. 1.11 Position model error impedance control
Another position-based impedance control structure is depicted in Fig. 1.12. This scheme provides a generalization of the original scheme proposed by Maples and Becker [15] and is referred to as outer/inner loop stiffness control. The control scheme consists of two parallel feedback loops superimposed to the internal position control and closed using measurements from both the wrist force sensor and position sensors. By analyzing the control scheme one can see that the position error e = ∆x0 = x0 − x is multiplied by the task-specific target impedance G t ( s ) = Z t ( s ) to provide a nominal (reference) force F0 which corresponds to the target impedance behavior at the output. The tracking of this force is realized by the next feedback loop closed on the sensed force F . In the
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Dynamics and Robust Control of Robot-Environment Interaction
ideal case we have F0 = F , describing the target behavior. Thus, Fig. 1.12 shows basically a force control system with target impedance added to regulate the motion response to the interaction force. Following the control flow we see that the force-error in this control scheme corresponds to the previously defined force impedance model error (1.17)
∆F = F0 − F = Gt ( x0 − x ) − F = e f
(1.25)
Therefore, the control system shown in Fig. 1.12 we will refer to as force model error impedance control. Similarly to the previous system (Fig. 1.11), the model error (1.25) is further relayed to the internal control part, to reduce this error to zero. However, in contrast to the position model error control shown in Fig. 1.11, where the position model error is eliminated by the internal position control, in the control system depicted in Fig. 1.12 the regulation of the model error is realized by means of the compensator GF . In order to retain the internal position control loop, the implicit force control structure is implemented by passing the force error ∆F through the admittance filter GF , providing the nominal path modification ∆x F . The position correction is further added to the Cartesian nominal position x0 , and via reference position xr fed forward to the position servo. Obviously, to achieve ∆F = e f → 0 as t → ∞ which ensures a steady-state position deviation ( x0 − x )∞ = e∞ corresponding to the target impedance (stiffness) model, the regulator GF has to include an integral control term. Both control approaches (schemes) utilize basically similar concepts to realize the target impedance model by reducing the impedance model errors e p and e f to zero. Each of them has specific advantages and disadvantages [49]. The e p -based scheme (Fig. 1.11) is essentially simpler and easier for implementation. This scheme, under some circumstances, allows the realization of different target impedances. However, their realization is done by setting the compensator GF to the target admittance, while the feedback control is to be undertaken by the position controller. This is similar to an “open-loop” target impedance control. Contrarily to this, in the force model-error control scheme (Fig. 1.12), the target impedance is specified in the outer-loop using the Gt block, while the role of internal loop compensator GF is to ensure tracking of
Control of Robots in Contact Tasks: A Survey
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the selected model using the force feedback. The main problem with the e f based scheme lies in the transition from and to the contact (constrained motion). The external impedance loop in this scheme is closed even in the free space, when the actual contact force is zero, and thus affects the position control performance. The compensator GF has to be tuned to achieve the required control goal in the contact with a stiff environment, e.g. a large amount of damping, in order to ensure a stable transition. However, this is usually contrary to the position control performance needed in the free space. In the e p -based scheme, however, the force feedback loop is closed naturally by the physical contact. The position-based impedance approach, in general, suffers from the inability to provide very soft impedance because of the limits in the accuracy of the position control system and sensor resolution. This approach is mainly suitable for the applications where high position accuracy is required in some Cartesian directions, which is realized by a stiff and robust joint control. The force- (i.e. torque)-based approach is better suited to providing small impedance (stiffness and damping) while reducing the contact force. From a computational viewpoint, this approach is reasonable for the applications where manipulator gravity is small, and slow motion is required. In other cases, manipulator modeling details (i.e. complete dynamic models) are needed. Contrary to the position-based impedance control, the force-based control is mainly intended to be applied in robotic systems with relatively good causality between joint torques and endeffector forces such as in direct-drive manipulators. A detailed consideration of the synthesis of position-based impedance control for industrial robots is presented in Chapter 3. 1.6.1.3 Other impedance-control approaches Considerable research efforts have been made to develop adaptive impedance control algorithms. Daneshmend et al. [27] have proposed a model-reference adaptive control scheme with Whitney’s damping control loop. Several authors have pursued Craig’s adaptive inverse dynamic control algorithms [50] and expanded it to suit the application for contact motion. Lu and Goldenberg [45] proposed a sliding-mode-based control law for impedance control. The proposed
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Dynamics and Robust Control of Robot-Environment Interaction
controller consists of two parts: nominal dynamic model to compensate for nonlinearities in the robot dynamics, and a compensator ensuring the impedance error (i.e. the difference between the nominal target model and the actual impedance) converges asymptotically to zero on the sliding surface. In order to cope with the chattering effects in the variable structure sliding-mode control, a continuous switching algorithm in a small region around sliding surface is proposed. Al-Jarah and Zheng [51] proposed an interesting adaptive impedance control algorithm intended to minimize the interaction force between the manipulator and the environment.
Fig. 1.12 Force-model-error-based impedance control
Dawson et al. [30] have developed a robust position/force control algorithm based on the impedance approach. The control scheme consists of two blocks: a “desired trajectory generator”, which computes the modified command position (based on the target impedance model, the nominal position and force measurements), and a controller involving a PD regulator and a robust controller. The purpose of the robust controller is to ensure that the control tracking error (i.e. difference between target and actual robot impedance) converges asymptotically to zero in spite of model uncertainties within specified bounds. 1.6.2 Hybrid position/force control This approach is based on a theory of compliant force and position control formalized by Mason [1] and it concerns with a large class of tasks involving partially constrained motion of the robot. Depending on the specific mechanical and geometrical characteristics of the given contact problem, this approach makes a distinction between two sets of constraints upon the robot’s motion and
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contact forces. A set of constraints that occur as a natural consequence of the task configuration, i.e. of the nature of the desired contact between an endeffector held by robot and constrained surface, is called natural constraints. Physical objects impose natural constraints. As already mentioned, a suitable frame in which the task to be performed is easily described, i.e. in which constraints are specified, is referred to as the constraint-frame (terms also used are task- or compliance-frame) [52]. For example, in the “surface sliding” contact task it is customary to adopt the Cartesian constraint-frame as sketched in Fig. 1.13. Assuming an ideally rigid and frictionless contact between the endeffector and the constraint surface, it is obvious that the natural constraints restrict the end-effector motion in the -z-direction, as well as rotations about the x and y axes. The frictionless contact prevents the forces in these directions and the torque around z-axis to be applied. In order to specify the task to be realized by the robot with respect to a compliant frame it is necessary to introduce the so-called artificial constraints, and they have to be imposed by the control system. These constraints essentially partition the possible DOFs of motion in those that must be position-controlled and those which should be force-controlled, in order to perform the given task. It is obvious to define an artificial constraint with respect to force when there is a natural constraint on the end-effector motion in this direction (i.e. DOF), and vice versa (Fig. 1.13) (the index “-z” denotes constraints in the negative z direction).
Fig. 1.13 Specification of “surface sliding” hybrid position/force control task
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Dynamics and Robust Control of Robot-Environment Interaction
To implement hybrid position/force control, a diagonal Boolean matrix S, called the compliance selection matrix [7], has been introduced in the feedback loops in order to filter out sensed end-effector forces and displacements which are inconsistent with the contact task model. In accordance with the specified artificial constraints the i-th diagonal element of this matrix has the value 1 if the i-th DOF with respect to the task-frame is to be force-controlled, and the value 0, if it is position-controlled. According to Mason [1], to specify a hybrid contact task, the following information sets have to be defined i) ii) iii)
Position and orientation of the task frame. Denotation of position- and force-controlled directions with respect to the task frame (selection matrix). Desired position and force setpoints expressed in the task frame.
Once the contact task is specified, the next step is to select the appropriate control algorithms. The relevant methods are discussed below. 1.6.2.1 Explicit force control The most important method within this group is certainly the algorithm proposed by Raibert and Craig [7]. Figure 1.14 presents the control scheme that illustrates the main idea of this method. The control consists of two parallel feedback loops, the upper one for the position and the lower one for the force. Each of these loops utilizes separate sensor systems. The positional loop utilizes the information obtained from the positional sensors at the robot joints, and the force loop is based on force sensing data. Separate control laws are adopted for each loop. The central idea of this hybrid control method is to apply two outwardly independent control loops assigned to each DOF in the task frame. Both control loops cooperate synchronously to control each of the manipulator joints.
Fig. 1.14 Explicit hybrid position/force control
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At first glance, this concept appears to be ideally suited to solve the hybrid position/force control problems. However, a deeper insight into the method reveals some essential difficulties related to this concept. The first problem is related to the opposite requirements of the hybrid control concept concerning position and force control subtasks. Precisely, it is required that the position control is highly stiff in order to keep the positioning errors in the selected directions as small as possible, while the force control requires a relatively low stiffness of the robot (corresponding to the desired force) in the force-controlled direction with respect to the task-frame, in order to ensure that the end-effector behaves compliantly with the environment. As already explained above, the explicit hybrid control attempts to solve this problem by control decoupling in two independent parts that are position- and forcecontrolled (Fig. 1.14). In the force-controlled directions the position errors are put to zero by multiplication with the selection matrix orthogonal complement (position selection matrix) defined as S = I − S b. This would mean that the position control part does not interfere with the force control loop. However, that is not the actual case. The joint space nature of robot control realization results in the coupling between the position- and force-control loops which have been previously mathematically decoupled in the task-frame. Assuming a proportional plus differential (PD) position control law, and that the force control consists of a proportional plus integral controller (PI) with the respective gains K Fp and K Fi , as well as a force feedforward part, the control law according to the scheme in Fig. 1.14 can be written in the Cartesian space as t
τ = K p S ∆x + K v S ∆xɺ + K Fp S∆F + K Fi S ∫ ∆Fdt + F0
(1.26)
0
Based on the relationships between the Cartesian and joint space gains, Zhang and Paul [26] have proposed an equivalent hybrid control law in the joint space t T ɺ τ q = J τ = k p J S J∆q + kv J S J∆q + J K Fp S∆F + K Fi S ∫ ∆Fdt + F0 0 T
−1
−1
(1.27)
b Note that for sake of simplicity it is assumed that the task-frame coincides with the Cartesian
frame. Generally, the selection matrix S is not diagonal in Cartesian space [35].
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Dynamics and Robust Control of Robot-Environment Interaction
Since each robot joint contributes to the control of both position and force, couplings in the manipulators mechanical structure (implied in the Jacobian matrix) cause that a control input to the actuator, corresponding to the force loop (e.g. force-controlled directions), produces additional forces in positioncontrolled directions in the task frame, and vice versa. It is obvious from (1.26) that by setting the position errors in the force-controlled directions to zero (i.e. by filtering the position error through S ), the position feedback gains in all directions are actually changed in comparison with the position control in free space because the entire system becomes more sensitive to perturbations. As a consequence, the performance of a robot is not unique with this scheme for all robotic configurations and for all position/force commanded directions. Moreover, one can find certain robot configurations for which, depending on selected force and position directions, the robot becomes unstable with the control law (1.26). This can be easily demonstrated on a simplified linearized robot model (1.10)
Λ ( x ) ɺɺ x =τ + F
(1.28)
Let us analyze the case when the manipulator is in free space and a noncontacting environment (e.g. in the transition phase when the force-controlled robot is approaching a contact surface after being switched from the positioncontrol mode). Let us assume that some directions (e.g. those orthogonal to the contact surface) have been selected for force control and the remaining ones for position control. Taking into account that the force is zero, substituting (1.26) in (1.28) yields
Λ ( x ) ɺɺ x + K v S xɺ + K p S x = K v S xɺ0 + K p S x0
(1.29)
with a robot closed-loop system matrix
0 A = −1 Λ K pS
I Λ K v S −1
(1.30)
To analyze the stability of this system we should determine the eigenvalues of A. As was shown in [53] there are a number of configurations in which the closed-loop matrix becomes unstable. Even if we introduce the feedback loops with respect to the integral of position errors in directions which are position-
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controlled, it is always possible that an unstable configuration may appear. It must be noted that these unstable configurations build working subspaces quite far away from singular positions where the system matrix A is intrinsically unstable due to the degeneration of the Jacobian matrix. Moreover, it was found that in a robot position only alterations of the selection matrix can cause switching of robot’s behavior from stable to unstable, and vice versa. The instability was experimentally tested and proven using the industrial robot control systems [53]. Although the above stability analysis was based on a linearized model and, therefore, has some limitations, it provides a simple explanation in revealing the nature of stability problems in hybrid position/force control. Since the instability is influenced only by the robot’s position and selection matrix, this phenomenon is referred to as kinematic instability [54]. This phenomenon does not depend on whether the robot is in contact with the constraint surface or not. However, if it is in contact, the analysis of this problem is complicated by the force/position relationship and experimental tests become highly dangerous. It may be concluded that the kinematic instability problem encountered in the considered explicit hybrid position/force control represents a serious deficiency of this method and reduces significantly its applicability. In order to overcome the difficulties related to the kinematic instability, Zhang [55] has proposed to introduce an additional selection of input forces. In other words, the input torques from the position and force control parts (Fig. 1.14) are decoupled in the task-frame before they are applied to the joints. In the above case, when the robot is in free space, the joint torque from the position control part (1.27) is initially transferred into the Cartesian compliant frame, then multiplied with the selection matrix and again transferred back using the static force transformation (i.e. Jacobian matrix), which provides the following control law for the position loop
τ q p = J T S J −T k p J −1S J ∆q + J T S J −T kv J −1 S J ∆qɺ
(1.31)
It is relatively easy to prove that the linearized model (1.29) becomes kinematically stable with this control law. However, similarly to the original control scheme, the eigenvalues of the system change not only with variation of the robot’s configuration, but already with the given task, i.e. selection matrix.
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Dynamics and Robust Control of Robot-Environment Interaction
This causes the robot performance to be strongly dependent on the configuration and selection of controlled directions. Fisher and Mujtaba [56] have shown that the kinematic instability is not inherent to the explicit hybrid position/force control scheme, but it is rather a result of an inappropriate mathematical formulation of position/force decomposition via the selection matrix S. It was demonstrated that in the original hybrid control formulation (1.26)-(1.27), the position control loop is responsible −1 for inducing the instability, precisely the term J S J in (1.27). The crucial error in the position control loop is, in the authors’ opinion, made by the decomposition of the robot coordinate (DOF) into the position- and forcecontrolled. Instead to compute the selected position-controlled DOF and the corresponding selected joint errors, respectively, based on
x p = Sx
(1.32)
∆q p = J −1∆x p = J −1S ∆x = J −1SJ ∆q
(1.33)
and the authors proposed to use the “correct” relationship between the selected Cartesian errors and the joint errors
∆x p = ( SJ ) ∆q
(1.34)
Taking into account the selection matrix structure, it is obvious that ( SJ ) is a singular matrix (with zero rows corresponding to the force DOFs). Hence, the selected joint errors equivalent to the selected Cartesian position error are obtained as the minimal 2-norm solution +
+
+
+
∆q p = ( SJ ) ∆x p = ( SJ ) S ∆x = ( SJ ) ∆x = ( SJ ) J ∆q
(1.35)
or, in general, when the robot is in a singular position, or it has a redundant number of joints, with an additional term from the null-space of the Jacobian J +
∆q p = ( SJ ) ∆x + I − J + J zq
(1.36)
where zq is an arbitrary vector in the joint space, the sign “+” denotes the MoorPenrose pseudo-inverse matrix. Thus, for the case (1.35) the control law of the position hybrid control part becomes
Control of Robots in Contact Tasks: A Survey +
35
+
τ q p = k p ( S J ) J ∆q + kv ( S J ) J ∆qɺ
(1.37)
To determine how the above kinematic transformations can induce instability of the hybrid control, the authors have defined a sufficient condition for kinematic stability. From the viewpoint of control, this criterion prevents the second-order system gain matrices (1.26) become negative definite, which is a necessary and sufficient condition for the system instability [55]. Testing the kinematic stability conditions for both original and “correct” selection and position error transformation solutions, the authors have proven that in the first case the instability can occur. The new hybrid control scheme, however, always satisfies the kinematic stability condition (it is always possible to find a vector zq to ensure the kinematic stability). The second problem is related to some dynamic stability issues in force control [57]. These effects concern: high-gain effect of force sensor feedback (caused by high environment stiffness), unmodeled high frequency dynamic effects (due to the arm and sensor elasticity), contact with a stiff environment, non-collocated sensing and control, etc. To overcome the dynamics problems of hybrid position/force control several researchers have pursued the idea of including the robot dynamics model into the control law. The resolved acceleration control originally formulated for the position control [58] belongs to the group of dynamic position control algorithms. Shin and Lee [31] have extended this approach to the hybrid position/force control. The joint space implementation of the proposed control scheme is sketched in Fig. 1.15.
Fig. 1.15 Resolved acceleration-motion force control
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Dynamics and Robust Control of Robot-Environment Interaction
In this scheme, the driving torque compensates for the gravitational, centrifugal and Coriolis effects, and the feedback gains are adjusted according to the changes in the inertial matrix. An acceleration feed-forward term is also included to compensate for the changes along the nominal motion in position directions. Finally, the control inputs are computed by
τ = Λˆ ɺɺ x ∗ + µˆ ( x, xɺ ) + pˆ ( x ) + Sf ∗
(1.38)
where ɺɺ x ∗ is the commanded equivalent acceleration
ɺɺ x ∗ = ɺɺ x0 + K v ( xɺ0 − xɺ ) + Κ p ( x0 − x )
(1.39)
and f ∗ is the command vector from the force control parts, whose form depends on the applied control law. To minimize the force error it is convenient to introduce the PI force regulator of the form t
f ∗ = K fp ( F0 − F ) + Κ fi ∫ ( F0 − F ) dt
(1.40)
0
Khatib [22] has introduced an “active damping” term into the force control part to avoid bouncing and minimize force overshoots during transition (impact effects)
τ f = Sf ∗ − Λˆ SK vf xɺ
(1.41)
where K vf is a diagonal Cartesian damping matrix. Bona and Indri [59] have proposed further modifications of the control scheme. To compensate for the coupling between force and position control loops, as well as for the disturbance on the position controller due to reaction force, the authors modified the position control law to
τ p = Λˆ S ɺɺ x ∗ − Λˆ −1 ( Sf ∗ − F ) + µˆ ( x, xɺ ) + pˆ ( x )
(1.42)
If the dynamic modeling used for computation of the control law is exact, the above control law provides the complete decoupling between position and force control in the task frame, i.e. the following closed-loop behavior
ˆ −1Sf ∗ − S Λ ˆ −1 F − SK xɺ ɺɺ x = Sxɺɺ∗ + S Λ vf
(1.43)
A comprehensive experimental evaluation and comparison of explicit force control strategies has been presented in [60].
Control of Robots in Contact Tasks: A Survey
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1.6.2.2 Position-based (implicit) force control The practical reason why the methods based on explicit force control can not be suitably applied in commercial robotic system is the same as in the force-based impedance control and lies in the fact that commercial robots are designed as “positioning devices”. In addition, since there is no position feedback loop in the force-controlled direction, the robot will move due to various disturbances acting upon it, such as the controller and sensor drifts, etc. The implementation of explicit force control under above difficulties, as already mentioned, could be only successfully performed by a new generation of direct-drive robots. In commercial robotic system it is most promising to implement implicit or position-based force control by closing a force-sensing loop around the position controller as in the scheme shown in Fig. 1.16. This scheme involves an additional explicit force block, proposed in [61], whose role will be explained in the text to follow. In the implicit force control the input to the force controller is the difference between desired and actual contact force in the task frame. The output is an equivalent position in force-controlled directions that is used as the reference input to the position controller. According to the hybrid position/force control concept the equivalent position in force direction x0F is superimposed onto the orthogonal vector x0P in the compliance frame, which defines the nominal position in the orthogonal position-controlled directions. The robot behavior in the force direction is practically affected only by the acting force. The position controller remains unchanged, except for the additional transformations between the Cartesian and task frames, which have to be introduced since in general case these two frames do not coincide. Since the position controller provides a basis for realization of force control, this concept is referred to as implicit or positionbased force control [15], or external force control [13]. The role of the force control block in this scheme is twofold: firstly, to compensate for the effects of the environment (contact process), and secondly, to realize tracking of the desired force. Another important feature of the forcecontrolled manipulator is the ability to respond to positional variations of the contact surfaces. Commonly, a PI force controller is applied. A more complex force controller, including the compensation of the internal position control effects, has been proposed in [61], where an explicit force control block is included in the control scheme (Fig. 1.16). This scheme combines the implicit and explicit controls with the aim of using the benefits of the both (robustness and reliability of implicit force control, and fast reaction of the explicit one) and compensating for specific disadvantages of single-force control approaches.
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Dynamics and Robust Control of Robot-Environment Interaction
Fig. 1.16 Implicit hybrid position/force control (an explicit control block is included according to [61])
The main features of implicit force control scheme are its reliability and robustness. Implemented in commercial robotic systems, this scheme is neither configuration dependent, nor sensitive to parameter variation. However, this scheme also exhibits some drawbacks. The accuracy of contact forces is mainly limited by the precision of robot positioning (sensor resolution). It can be especially disturbed when a contact with a very stiff environment is involved. Fortunately, an inherent compliance of the robot structure or force sensor is always present, and it reduces the equivalent system stiffness. The performance of implicit force control is significantly limited by the bandwidth of the position controller. A slightly higher bandwidth can be achieved using a compensator of a higher order. However, due to coupling between the position- and forcecontrolled DOFs, it remains questionable whether force control may become significantly faster or not.
Control of Robots in Contact Tasks: A Survey
39
1.6.2.3 Other force control approaches The next group of algorithms refers to more complex constraints on the robot motion which are described by a set of rigid hypersurfaces in the space of the end-effector Cartesian coordinates [11], or in the joint coordinate space [32]. The system model is described by a typical set of linearly implicit second-order differential-algebraic equations (mechanical differential-algebraic equations). This model is used to compute the control law whose functions are to linearize and decouple the system dynamics, as well as to decompose the control problem into position-and force-controlled directions. To improve the concept reliability, the dynamic hybrid control has been extended to the case of unknown environments, consisting of hypersurfaces [62]. The improved control schemes involve on-line identification algorithms based on force and position measurements, and adaptive control mechanisms. However, as already mentioned, the adaptive constrained motion control is theoretically attractive, but still impractical in reality. The hybrid position-force task specification has also been a subject of considerable investigations. Lipkin and Duffy [24] showed that the Mason’s position-force decomposition approach based on “geometrical orthogonality” is in fact erroneous. The resulting planning for hybrid control is invariant neither with respect to the origin translation nor the change of unit length. The authors have proposed a more general and mathematically consistent invariant hybrid task formulation based on screws algebra. In this approach, the complementarities between motion (modeled by a twist) and force (represented by a wrench) is expressed via the reciprocity relationship that is independent of the coordinate frame, scaling or units. In certain simple tasks and reference frames, both conventional and reciprocity-based decomposition, show the same results. However, the reciprocity-based approach provides a more general decomposition, applicable to the cases when the freedom and constraint subspaces do not span a six-dimensional space or have nonzero intersections, but also to the case when a manipulator has less than six DOFs [63]. If the specified twist and wrench are consistent with the environment (i.e. the freedom and constraint equations are satisfied), the specified task is feasible for hybrid control. In the opposite case, it is necessary to filter the specified twist and wrench to obtain a kinestatically realizable control action (so-called kinestatic filtering). A procedure to apply the reciprocity-based task decomposition to manipulator dynamics in order to obtain equations of motion relevant for hybrid control has been presented in [64]. Several model-based tools for tasks
40
Dynamics and Robust Control of Robot-Environment Interaction
specification using this approach are presented in [22]. It has been shown that the reciprocity concept is not only well suited for nominal specification of arbitrary motion constraints, but also serves definition of possible uncertainties and on-line identification and observation of real motion constraints. An overall hybrid position-force control scheme based on the general decomposition formalism including identification of geometrical uncertainties has been proposed in [25]. 1.6.3 Force-impedance control There have been several attempts to combine impedance and force control with the aim of using benefits and overcoming specific disadvantages of single control approaches. Although it is possible, under some circumstances, to demonstrate a correspondence between force and impedance control laws [65], there are essential differences between these main constrained motion control concepts. The main advantage of the impedance control over the force control is the easier task specification and programming. A contact task is practically specified in terms of motion sequences, hence the impedance control does not require any modifications of conventional free-space planning control concepts and algorithms (the programmer can take advantage of his own experience and existing off-line programming tools). Moreover, the impedance control can be activated in free space during the approach motion. Thus, it can be applied for the transition to and from the constrained motion, without specific control switching algorithms. As mentioned above, the impedance control realizes the closed loop position control in free space, while in case of the contact with rigid environments it offers force open-loop capabilities. In contrast to this, the force (admittance) control approach allows closed-loop force control capabilities in contact, but exhibits open-loop position control characteristics in free space. Therefore, the activation of force control in free space is only possible under specific circumstances. In general, however, a discontinuous control strategy is required for the transition from non-contact to contact motion phase, or vice versa. The control structure changes in the most critical phase when the manipulator is in contact with the environment, and this represents a major drawback of force control. To cope with unexpected collisions, additional sensors (e.g. distance)
Control of Robots in Contact Tasks: A Survey
41
have to be integrated in the control system. The fundamental superiority of the force control implies, however, from the fact that the interaction force is a result of the control action, rather than of the actual deviation of environment position and chosen target impedance. In Goldenberg’s algorithm [38], force control is closed around an internal impedance control loop. Desired force and force error are used to compute an equivalent desired relative motion of the end-effector. In addition, the impedance control is included with the aim to achieve a suitable relationship between force and relative motion during contact. This is realized in the internal velocity loop by adjusting compensator gains to obtain the target impedance. A similar reliable position-based force-impedance control scheme suitable for implementation in industrial robots has been proposed in [66]. In this scheme an external implicit force controller loop is closed around an internal position-based impedance controller (Fig. 1.17). The main goal of the internal loop is to achieve target impedance while the external loop takes care of desired force realization. In this scheme, the selection between position- (i.e. impedance) and force-controlled directions is not needed. Indeed, both impedance and force control affect all directions. The disproportion between motion and force planning is not critical in the control scheme (Fig. 1.17), since the internal control loop behaves as a low-stiffness target impedance system allowing relatively large differences between the input position command and real robot position at the output. In contrast to this, the internal loop in the implicit force control (Fig. 1.16) is a very stiff position control, so that the selection is inevitable.
Fig. 1.17 Position-based force-impedance control scheme
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Dynamics and Robust Control of Robot-Environment Interaction
Anderson and Spong [12] have proposed an approach to controlling contact forces, referred to as hybrid-impedance control algorithm. The kernel part of the algorithm is Raibert-Craig’s hybrid position/force control scheme, with the selection matrix applied to decompose position- and force-controlled subspaces. Both control parts use the feedback of contact force to realize desired system impedance (position-based and force-based impedance control) along each DOF. A controller that combines an internal position control, a position-based impedance compensator, and a desired force feedforward has been described in [67]. The authors proposed integral control actions to be applied for both impedance (damping control) and force filters to ensure the compliance and desired steady-state force. A conceptually different approach to position/force control, referred to as parallel control (Fig. 1.18), has been proposed in [9]. Contrary to the hybrid control, the key feature of the parallel approach is to have both force and position control along the same task space direction without any selection mechanism. Generally, both position and force cannot be effectively controlled in an uncertain environment. Therefore, the logical conflict between the position and force actions is managed by imposing the dominance of the force control action over the position one along the constrained task direction where force interaction is expected. The force control is designed to prevail over the position control in constrained motion directions. This means that the force tracking is dominant in the directions where interaction with the environment is expected, while the position control loop allows the compliance, i.e. deviation from the nominal position in order to attain the desired forces. For this reason the parallel control method can be considered as a force-impedance control approach.
Fig. 1.18 Parallel position-force control
For a parallel controller case, consisting of a PD action in the position loop, and a PI control in the force loop, together with the gravity compensation and
Control of Robots in Contact Tasks: A Survey
43
desired force feedforward, a set of sufficient local asymptotic stability conditions has been derived in [68]. Stability analysis and simulation results on an industrial robot also are included. These conditions imply a relatively high damping (i.e. velocity gains) to ensure the system’s stability. 1.6.4 Unified position-force control Vukobratovic and Ekalo [8] have established a unified approach to control simultaneously position and force in an environment with completely dynamic reactions. This fully dynamic approach to the control of robot’s interacting with dynamic environment will be presented in a very condensed way. In the case when the environment does not possess displacements (DOFs) that are independent of robot motion, the model (1.9) provides a mathematical description of the environment dynamics in terms of the robot coordinates (motion). Then the system (1.1)-(1.9) describes the dynamics of the robot interacting with dynamic environment. It is assumed that in the case of contact all mentioned matrices and vectors are continuous functions of the arguments. It is also assumed that the robot is constantly in a unilateral contact with the environment. Further, it is taken that n=m, where n is the number of robot DOFs and m is the number of contact force components. The general case, when n>m, has been considered in [69]. In the case of contact with the environment, the robot control task can be described as the robot motion along a programmed trajectory q p (t ) representing a twice-continuously differentiable function, when a desired force of interaction Fp (t ) acts between the robot and the environment. Based on the nonlinear model, the programmed motion q p (t ) and desired force Fp (t ) must satisfy the relations:
Fp (t ) ≡ f ( q p (t ), qɺ p (t ), qɺɺp (t )) f ( q, qɺ , qɺɺ) = ( S T ( q)) −1[ M ( q)qɺɺ + L( q, qɺ )]
(1.44)
The control goal of a robot interacting with dynamic environment can be formulated in the following way. Let us define the control τ (t ) for t ≥ t0 that is to satisfy the target conditions:
q(t ) → q p (t ), F (t ) → Fp (t ), as
t→∞
(1.45)
Two alternative questions can be phrased concerning the control design problem. Can we choose such a control law which, by satisfying the preset robot
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Dynamics and Robust Control of Robot-Environment Interaction
motion quality, would enable the attainment of the control goals that satisfy the relation (1.45)? Is it possible to choose the control law in such a way as to ensure the preset quality of the robot interaction force, and also the attainment of the control goals? The answer to the first question is quite simple [8, 33]. The inverse dynamics methods ensure a desired motion quality and at the same time guarantee a stable interaction force. The answer to the second question depends on the environmental dynamics. The task of stabilizing the programmed interaction force ( PFI ) Fp (t ) can be tackled by considering a family of transient processes with respect to force, in the form µɺ = Q ( µ ), µ = F (t ) − Fp (t ) and choosing a continuous vector function Q (Q (0) = 0) of dimension n, such that the asymptotic stability as a whole is ensured for the trivial solution of µ (t ) ≡ 0 . Let us consider the “pure force control” according to the assumption that m = n , i.e. when the number of the contact force components is equal to the number of the powered DOFs of the robot. For convenience, when describing the quality of transient processes with reference to perturbation force dynamics, µɺ = Q ( µ ) , we shall use an equivalent relation of the form: t
µ (t ) = µ0 + ∫ Q ( µ (ω ))d ω
(1.46)
t0
Without loss of generality, we can adopt µ0 ≡ 0 , because the stabilization of µ in the sense of preset quality (1.46) directs stabilization according to the preset quality µɺ = Q ( µ ) , independently of the value of µ0 . Let us consider only one of the possible control laws with the feedback loops with respect to q, qɺ and F of the form [8, 33]
t
τ = H (q ) M −1 ( q) − L(q, qɺ ) + S T ( q) Fp + ∫ Q( µ (ω )) dω + h(q, qɺ ) − J T ( q) F (1.47)
t0 By applying this control law to the robot dynamics model (1.1) we obtain the following law for the robot operating in contact with the environment:
t M ( q ) qɺɺ + L ( q, qɺ ) = S T ( q ) Fp + ∫ Q ( µ (ω ))d ω t0
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Control of Robots in Contact Tasks: A Survey
Taking into account the environment dynamics model (1.9), we obtain the following closed-form control system: t S ( q ) µ (t ) − ∫ Q ( µ (ω ))d ω = 0 t0 T
(1.48) t
∫
and, because rank(S) = n , (1.48) is equivalent to: µ (t ) = Q ( µ (ω ))d ω , from t0
where, µɺ (t ) = Q ( µ (ω )) follows directly. In this way, the control law (1.47) ensures the desired quality of stabilization of ( PFI ) Fp (t ) . The stability of the real motion (position) when asymptotic stability of the contact force is fulfilled has been considered in [8, 33, 69]. Sufficient conditions for constrained motion stability based on generalized Lyapunov’s theorem on stability in the first approximation of the system with perturbation have been derived. The theorem defines conditionally the properties of “internal stability” of the environment because the fulfillment of stability conditions depends in general not only on the environment dynamics but also on the nature of the programmed motion. It should be emphasized that without knowing a sufficiently accurate environment model it is not possible to determine the nominal contact force. Besides, the insufficiently accurate environment dynamics model can significantly influence the contact task performance. Inaccuracies of the robot and environment dynamic models, as well as dynamic control robustness will be considered in Chapter 2. The problems arising from the uncertainty of parameters may also be resolved by applying the knowledge-based techniques. Taking into account external perturbations, which do not expire with time, as well as model and parameter uncertainties, it may be difficult to achieve asymptotic (exponential) stability of the system. Therefore, it may be of practical interest to demand a more relaxed stability condition, i.e. to consider the socalled practical stability of the robot around the desired position and force trajectories by specifying the finite regions around them within which the robot actual position and force have to be during the task execution. More details on the synthesis and stability of dynamic control of the robot’s interacting with dynamic environment, called unified approach, will be presented in Chapter 2.
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Dynamics and Robust Control of Robot-Environment Interaction
1.7 Contact Stability and Transition The type of contact tasks may vary substantially in dependence of specific requirements, but in all cases the robot realizing a contact task has to perform three kinds of motion: · · ·
gross motion, related to robot’s movement in free space (freemotion mode), compliant or fine motion, related to robot’s movement constrained by an environment, and transient motion, representing all transition phases between the freespace and compliant motions.
From a practical point of view, the contact transition can be considered as stable if the contact is not lost after the manipulator hits the environment. A stable contact transition can be characterized by non-zero force (after the contact is detected), positive penetration of the manipulator end-point into the environment, nonappearance of bouncing, etc. A most critical issue in transition control is the initial impact against a stiff environment. Obviously, a stable controller should ensure the passage through the transition phase while maintaining the contact until all impact energy has been absorbed. Many researchers have shown that in most of the proposed control algorithms instability occurs when the contact between the end-effector and environment is stiff. However, the investigations have primarily been concerned with the question of coupled stability (i.e. will the robot remain stable when it is interconnected with the environment?) of robots and the environment under various control algorithms, while assuming the manipulator being initially and constantly remaining in contact with the environment. Surprisingly, relatively little research has addressed the problem of contact transition stability (i.e. will the robot during transition from free- to contact-motion establish a continuous contact with the environment without multiple impacts?), which is most fundamental for performing contact tasks. The contact transition stability problem is important for both unilateral (i.e. force) and bilateral (geometric) constraints. Namely, a bilateral constraint is usually achieved by closing the gripper, which is due to position misalignment usually realized through a unilateral contact between the gripper jaws and grasping object. In the impedance control, contact stability issues have mainly been considered on the basis of simplified models of the interaction between a target impedance system and the environment. Colgate and Hogan [70] have defined
Control of Robots in Contact Tasks: A Survey
47
necessary and sufficient conditions to ensure the stability of a linear robotic system coupled to a linear environment. The authors have applied the network theory to describe the manipulator and environment interactive behavior at the equilibrium point. In a passive stationary environment, two time-invariant networks coupled along interaction ports (Fig. 1.19) can represent the interactive model around the equilibrium p0 ( ∞ ) = p0∗ , where p0 denotes the penetration into the environment. The coupling imposes the velocities of robot and environment at contact point to be equal, while the forces acting upon the robot and environment have opposite signs (action and reaction). If the environmental transfer matrix Ge ( s ) s is positive real, representing any passive Hamiltonian environment, then a necessary and sufficient condition to ensure stability of linearized robotic control system is that the realized admittance sGt −1 ( s ) be positive real [70]. In other words, it should represent the driving point impedance of a passive network. Considering a SISO system, the coupled stability has been proven using the Nyquist criterion and the property of positive real transfer function having a limited phase by ± 90 deg. [70]. Then, it is relatively easy to prove that the mapping of the Nyquist contour of a positive real environmental impedance
Ge ( s ) s through an, also positive real, admittance sGˆ t −1 ( s ) , altering the phase by ± 90 deg. and changing the magnitude by a factor 0 to ∞ , provides a stable system, i.e. a stable Nyquist plot of the open-loop coupled system transfer function.
Fig. 1.19 Robot-environment interaction model
System passivity concept provides a relatively simple test for the assessment of coupled system stability. In this test, only passivity of the environment should be proven, without an accurate knowledge of parameters. Assuming again that an ideal target impedance response (1.15) is being realized, the passivity of target
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Dynamics and Robust Control of Robot-Environment Interaction
admittance sGt ( s ) implies positive definite matrices M t , Bt and K t , and consequently, the closed-loop system should be stable in contact with any passive environment to which it is directly coupled. The explicit design of a positive-real robot control system, however, may in practice become cumbersome [71]. Moreover, various practical control implementation effects, including computational time delay and sampling effects, as well as unmodeled dynamics (e.g. high-order actuator and arm dynamic effects), may result in a non-passive real impedance control response. The above stability results can be practically extended to nearly-passive control systems. However, in this case a passive environment can be found which destabilizes the coupled system. In order to simplify the coupled stability analysis, Colgate and Hogan [70] have introduced the term worst or the most destabilizing environment, denoting the most critical environmental for the coupled system stability. Such −1 environmental impedance Ge ( s ) s shapes the Nyquist contour of sGˆ t ( s ) by minimizing the distance from the critical point -1 to the nearest point on the −1 Nyquist plot of the loop transfer function Ge ( s ) Gt ( s ) . Taking into account that the driving point impedance of simple passive environmental models, such as mass or spring ( M e s and sK e ), perform the maximum rotation in the Nyquist plane, the authors have found that the worst passive environment for the coupled stability consists of a set of pure masses and springs. The beauty of the passivity theory is that it guarantees the phase lead or lag of any passive block to be no more than 90 deg. Thus, when two passive blocks are connected together as shown in Fig. 1.19, the maximum phase lead or lag of the overall open-loop system will be limited to no more than 180 deg. Therefore, it guarantees stability regardless of the overall gain. This result can be obtained from either the Nyquist or Bode analysis. The problem with any practical application such as a robot performing impedance control is that additional phase lags caused by cut-off (analogue and digital) filters and sampling delays make the overall phase lag to be more than 180 deg. at high frequencies. However, the realization of a passive system in a real digital robot controller imposes a fundamental limit on the reduction of apparent robot inertia to a maximum of 50% [72]. In industrial robots with apparent Cartesian end-point masses of several hundred kilograms, that is an exceedingly conservative condition. Newman [73] proposes a natural admittance control (NAC) approach that provides considerable compensation for friction, however, without −1
Control of Robots in Contact Tasks: A Survey
49
significant improvement of achievable target admittance reduction which does not violate the passivity constraints. To solve this critical limitation in the implementation, the author proposed in [74] the insertion of a mechanical filter between the manipulator and the environment in order to achieve low-impedance performance. However, this is a specific solution, rather than a general approach. The reduction of apparent inertia appears also to be essential in human-robot systems. In order to achieve valuable performance of the human-robot interaction, Buerger and Hogan have suggested [75] relaxing the restrictive passivity condition and designing the interaction system either by taking into account the limited knowledge of the particular environment or by lowering the target inertia beyond the passivity threshold. If both the environment and realized admittance are stable, the coupled stability of the interactive system in Fig. 1.19 can also be assessed by means of small gain theorem. This theorem states that a feedback loop composed of stable operators will certainly remain stable if the product of all operator gains is smaller than unity
Ge ( jω ) Gˆ t −1 ( jω )
∞
<1
(1.49)
The small gain theorem provides a quite general law, valid for continuous- or discrete-time, SISO and MIMO, linear and nonlinear systems, and it is also a convergence criterion that is used in many iterative processes. However, the small gain theorem only gives the sufficient stability conditions, which in many cases are too conservative to be of much use in practical contact tasks. For example, assuming that ideal second-order target impedance has been achieved, the condition (1.49) implies the admissible target stiffness to be K t ≥ K e in order to ensure stable interaction. In real stiff environments this is without any practical relevance. This result is similar to the stability analysis performed by Kazerooni et al. [16]. The established interaction stability criterion practically implies that the gain of feedback compensator (i.e. the target admittance) should be limited by the magnitude of the sum of environmental admittance and robot position control sensitivity. For a SISO system this imposes in the steady state
K t ≥ min ( K p , K e )
(1.50)
In direct-drive robotic systems with a significantly lower position control stiffness (due to the elimination of transmission) than in industrial robots, this
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Dynamics and Robust Control of Robot-Environment Interaction
condition might provide reliable target models for practical tasks. The sufficiency of stability condition (1.50) has been experimentally demonstrated on a lightweight direct-drive robot (University of Minnesota robot). However, from the viewpoint of industrial robots performance, it may be stated that this condition is very conservative and practically useless. Practically, in industrial robots, with quite stiff servos gains (e.g. position control gains usually have the order 10 6 N/m) K p > K e the above condition also requires the target stiffness to be higher than the environmental one. Moreover, no target model, i.e. the compliance feedback compensator G f (Fig. 1.11), can be found to allow interaction with an infinitely rigid environment ( K e → ∞ ). Therefore, one of the main conclusions in [16] is related to the practical need for an intrinsic compliance, either in the robot or in the environment for achieving interactive stability. The problems with control design taking into account environmental models and parameters are the uncertainties and nonlinearities in the real systems. A promising approach to cope with these problems is provided by robust-control theory. The robust control framework considers simplified linearized models while taking into account specific nonlinear effects by means of weighting functions in the frequency domain. In [42, 76], this approach is applied to derive new stability criteria for the design of low-impedance interaction in industrial robot systems. The robust interaction stability paradigm ensures contact stability during all phases of interaction. Moreover, the new design framework realizes low-impedance performance, allowing considerable reduction of high apparent industrial robot inertia and stiffness. The novel stability criteria are established based on robust control theory and taking into account the estimates of environmental stiffness, tolerating thereby large uncertainties and variations in the industrial environments. These criteria are proved by extensive tests involving industrial and space robots, and have been extended to control synthesis of human robot interaction systems (haptic admittance displays and rehabilitation robots) [77]. The contact transition stability conditions require interaction force, i.e. actual penetration to be positive p(t ) = x − xe > 0, t ≥ t0 , or the position deviation to be smaller than the nominal penetration e (t ) ≤ p0 (t ) i.e.
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Control of Robots in Contact Tasks: A Survey
e(t ) p (t ) − p (t ) x0 (t ) − x(t ) = 0 = ≤1 p0 (t ) p0 (t ) x0 (t ) − xe
(1.51)
In other words, this relation implies the actual end-effector position during a stable contact transition to be always located between the position of environment and the nominal position. Since this contact stability condition is based on a simple geometric consideration it is referred to as geometric criterion. The advantage of the geometric criterion is that it compares two time signals. The norm comparison offers the possibility to apply relatively simple and efficient system theory formalisms for the contact stability analysis. In [76, 77], this criterion has been utilized to derive robust contact stability condition ensuring both coupled and contact transition stability based on
sup
e (t ) 2 p0 ( t )
< W ( s ) ( I + Ge −1 ( s ) Gt ( s ) )
2
−1
≤1
(1.52)
∞
where stable weighting transfer function matrix W ( s ) describes the uncertainties of environmental model and target impedance realization. In a SISO system, this condition implies
ξt ≥
1 2
(
)
1 + 2κ − 1
(1.53)
where ξ t = Bt 2 M t K t and κ = K e K t are target damping and stiffness ratios, respectively. However, in spite of an effective and simple formulation, this criterion only ensures sufficient contact stability conditions, but not the necessary ones. Consequently, the obtained contact stability indices may be conservative. It should be mentioned that the damping ratio bound (1.53) is still smaller than in the most commonly applied in practice dominant real-pole solution [78], imposing ξ t ≥ 1 + κ . A very important advantage of the input/output criterion (1.52) is that it can be applied for the continuous as well as discrete systems, including the time lags. The control time lags have been identified in [76, 77] as the critical destabilizing contact transition effect. In general, the retarded system requires a significantly higher amount of damping in order to stabilize the transition process with delayed force signals. Robust contact stability analysis and control synthesis are dealt with in detail in Ch. 3.
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Dynamics and Robust Control of Robot-Environment Interaction
Typical transition results obtained experimentally in position-based impedance and force control during contact with a stationary environment are presented in (Fig. 1.20) [42]. In order to compare force and impedance control, a nominal force corresponding to the steady-state impedance interaction force F∞ = p0∞ K e K t ( K e + K t ) , where p0∞ denotes the stationary value of the nominal penetration, has been selected, and an integral force controller with the same bandwidth as the impedance control has been synthesized. As expected, it can be remarked that the force transition in hybrid control is characterized by lower overshoots. The reason for this is that the force control represents an explicit goal in the hybrid control that is realized by an appropriate control structure and design. In the impedance control, however, the aim is to passively modify a preplanned motion in accordance to the interaction forces. Therefore, the force transition in the impedance control is to a great extent influenced by the selected target impedance parameters and nominal motion [42]. In contrast to impedance control algorithms, which provide the same control structure for the three motion phases, in force control schemes the transition to and from contact motion is usually based on discontinuous control. There are two basic concepts for the change of the control strategy from the position control to the force control, operating (a) in free space, where the transition is realized in the force control mode, and (b) after the contact has been established. Most of force control algorithms execute the transition control in the force mode. The reason for this is that the impact force can be very large, especially due to the relatively high approach velocities and delay in the stiff position controller. One of methods to reduce impact is to use soft force sensors [79], i.e. passive compliance, but this reduces the position accuracy during position control. The underlying idea of the majority of methods concerned with the impact problem is to increase damping in the collision direction [22]. Assuming a simplified stiffness model of the environment, the damping effect can be achieved by utilizing either the force derivatives or the approach velocity feedbacks. However, both methods have practical limits. The force signals are usually noisy and the derivation is inaccurate. Qian and De Schutter [80] have proposed lowpass sensor filtering and nonlinear damping to cope with the transition problem. However, in the commonly slow approach motion before contact the velocity sensing is not reliable. Moreover, in a stiff environment, relatively fast oscillations in force and robot velocity can cause the instability due to time (phase) lags between the sensing and control actions. These difficulties have been addressed in several works, proposing design of a stable force controller without velocity
Control of Robots in Contact Tasks: A Survey
53
measurement [81], and even without sensing the end-effector contact force [82], for a system with well-known dynamics. However, these innovative schemes are relatively complex to implement and require further experimental tests.
Fig. 1.20 Comparison of contact transition performance: impedance vs. implicit hybrid control (experiments with Manutec R3 robot; target impedance parameters: Mt = 10 kg, ξt = 8, Kt = 1500 N/m; environment: Ke = 60000 N/m; bandwidth of the integral force control 2 Hz)
In the other transition control concept, the approach phase is realized in the position control mode, and when contact is established, the control is switched to force control mode. Numerous discontinuous transition control algorithms have
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been elaborated and experimentally tested using simple test-systems. Considering discontinuous controller as an entire generalized system, Mills and Lokhorst [83] have proposed a discontinuous control scheme which guarantees the following performance: global asymptotic stability of the closed-loop system, asymptotic trajectory tracking of position and force inputs; and reestablishment of contact after an inadvertent loss. Wu et al. [84] have proposed to add a positive acceleration feedback to the force control in the impact direction. In addition, a switching control strategy is introduced to eliminate unexpected bouncing. A similar control strategy for the transition problem in both force and impedance control has been developed by Volpe and Khosla [85]. The authors recommended a positive force feedback to be used during transition, and integral force control after a stable contact is established. A force-regulated switch triggers the transition from position control to impact control, while several options are proposed for the further switch to the integral force control. Based on the equivalency between the force and impedance control, the authors have established the transition stability condition for the impedance control that imposes the mass ratio (robot inertia over target inertia) to be less than one. Based on the experiments with this control approach, realized on a direct-drive robot at very high impact velocity (0.7 m/s) with a relatively stiff environment (104 N/m), the authors have demonstrated the reliability of the established criteria. However, these results on direct-drive robots are not applicable to industrial robots. The performance of industrial robot systems, such as very high Cartesian inertia (> 500 kg), extremely stiff position controller (several million N/m equivalent stiffness in Cartesian space) and considerable time lags (from 2 to 20 ms), causes switching algorithms to be quite difficult to implement. Gorinevski et al. [86] have examined the transition problem in both impedance and general force control during the contact with stationary and dynamic environments. The authors tested two control approaches: linear control and sliding-mode control. The influence of several effects, such as the time delay and elasticity of robot end-effector, transmissions and mechanical structure, on the contact stability has been examined. The contact stability criteria for singleand two-DOF systems are derived in the explicit closed form in terms of control gains and limits on the robot and environment velocities. Several authors have considered transition control as a short-impulse dynamic problem. This model, however, is valid for very fast systems (e.g. micro-macro manipulators), but they are still rare in practice. In industrial robotic systems, the transition problem can be accurately analyzed in a finite time period. The
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features of almost all industrial control systems still do not provide mechanisms to control short-impulse impact effects. McClamorch and Wang [32] emphasized the important role of constraints in contact dynamics. They presented global conditions for tracking based on a modified computed torque and local conditions for feedback stabilization using a linear controller. The closed- loop properties in the case of force disturbances, dynamics in the force feedback loops, or uncertainties in constrained functions have also been investigated. Eppinger and Seering [87] studied the influence of unmodeled dynamics on contact task stability, introducing additional (elastic) DOFs of both the robot and environment. A treatment of the contact stability that considers the environment as a nonlinear dynamic system is given in [34]. It is shown that if the impedance control is applied, enabling the robot to be asymptotically stable in free space, the robot interacting with the environment is a passive system and is stable in isolation. However, the conclusion is valid only if the robot in contact is at rest, and for this reason the result cannot be considered complete. The stability issue, i.e. the establishment of the conditions under which a particular control law guarantees the stability of the robot in contact with the environment, is of essential importance. In [8, 33, 88], the control laws stabilizing simultaneously the motion of the robot and the forces of its interaction with a dynamic environment are synthesized, ensuring the exponential stability of the closed-loop systems (based on the analysis of a complete dynamic model of the robot and dynamic environment). The papers formulated conditions ensuring asymptotically stable position of the system in the first approximation (local stability). It has been emphasized that the character of the mentioned position stability depends particularly on the nature of programmed (desired) motion. In spite of sufficient conditions of the linearized system asymptotic stability being conservative, the fact is that the dynamic character of interaction of the environment with the robot can lead to positional instability. This problem deserves full attention of the researchers, as well as of the designer of robotic controllers dedicated to the diverse contact tasks. Therefore, it should be emphasized that linear analysis gives a very important criterion that must be fulfilled by any force-based law. However, the model uncertainties representing a crucial problem in control of the robot’s interacting with a dynamic environment have not been appropriately addressed yet. Therefore, it can be difficult to achieve the asymptotic (exponential) stability of the system (unless robust control laws including factors for compensating these perturbations and uncertainties are used). Inaccuracies of the robot and environment dynamic models, as well as dynamic control robustness, have been considered in [89, 90, 91]. The problem
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arising from the uncertainties of parameters may also be resolved by applying the knowledge-based techniques [92]. Taking into account external perturbations and model uncertainties, it may be difficult to achieve asymptotic (exponential) stability. Therefore it is of practical interest to require less restrictive stability conditions, i.e. to consider the so-called practical stability of the system. An approach to the analysis of the practical stability of manipulation robots interacting with a dynamic environment based on a centralized model of the system is presented in [93]. The conditions for testing practical stability of a robot interacting with its dynamic environment were given in [93, 94] and the test were performed using two very representative control laws. The first one is the pure position dynamic control (based on the so-called inverse dynamic technique, or computed-torque method), where the desired position trajectories are calculated based on the desired position and force trajectories using the dynamic model of the environment. The other control law considered belongs to the hybrid position/force control schemes, where the complete dynamic model of the interactive system is taken into account: in the directions in which the desired position trajectories are specified the control law attempts to stabilize position, while in the directions in which force trajectories are specified, the control law focuses on the force. The elaborated stability test may be used either to check the stability of the specified control laws, or to establish procedures for the synthesis of parameters of different control laws. The derived stability may be too conservative due to a number of linearizations (approximations) made. More refined approximations, e.g. by taking into account possible dependence of the model elements on the parameters, may lead to a less conservative test. 1.8 Compliance Planning In spite of the existence of numerous sophisticated compliance and interaction control strategies and schemes, these advanced control capabilities have not been implemented yet in commercial robotic control systems. There are several reasons for this. The majority of compliance control concepts are concerned with particular problems, mainly at lower control levels. Combining various algorithms and control concepts and their integration in conventional robot position control systems is complex and tedious. Most of the studies on the impedance control are primarily related servo control. Except the seminal works on the compliance control [1, 18] the contact task planning and programming
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issues have usually been neglected in the research studies. However, these subjects are essential for practical application of compliance control. The planning of complex impedance control tasks in terms of spatial geometrical and compliance parameters is still an insufficiently treated problem. In complex geometrical tasks, the relationship between forces and displacements becomes rather complex. Practically, in the common linear second-order impedance target system (1.14)-(1.15) the mass and damping elements specify system dynamics (i.e. transition processes), while the stiffness provides apparently simple relationship between displacements and forces (steady-state behavior). However, due to the configuration dependence of the spatial stiffness it is difficult to specify and select compliance parameters as well to easily understand the spatial compliance relationships. The reasoning about admissible or realizable interaction forces during design is not a simple task, and the practical goal is to explore means of facilitating it. As proposed by Fasse and Broenink [95], the specification would be simplified if the compliance parameters could be chosen independently of the interaction system configuration (e.g. reference frame, contact point, etc.). In the compliance control design it is quite desirable to specify target stiffness matrix independently of the robot/environment configuration. The authors proposed the application of principal (eigen) characteristic, i.e. (singular) values, as quite suitable for specification of stiffness (compliance) submatrices. Since the stiffness submatrices are positive semi-definite, the eigen values are identical to the singular ones. The characteristic values are independent of the rigid-body rotation occurring in the transformations. Hence, the appropriate parameterization of stiffness and compliance matrices involves two sets of parameter: the set of non-spatial parameters consisting of principal values of compliance submatrices and the set of spatially-dependent parameters describing rigid body transformation (rotation matrix and displacement vector). This parameterization provides the basis for computation of the centers of stiffness and compliance [96]. Dynamic impedance control algorithms based on resolved acceleration control approach are proposed in [97, 98, 99] to control six-dimensional impedance interaction in industrial robot control systems. The orientation
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representation based on quaternion was identified as a most reliable and robust approach for specification of impedance for rotational DOFs [97]. Considerable research efforts have been devoted to the synthesis of admittance for a reliable and robust assembly [100, 101]. The main goal is to replace spatial intuitive reasoning with more systematic synthesis approaches. A reliable admittance design, which defines an appropriate mapping of interaction forces into robot velocities for a particular assembly task, requires the contact forces always reduce the existing position misalignments. This approach is referred to as force-assembly and has been proven [100] to be quite efficient for insertion of workpieces at infinitesimal errors. A practical compliance control planning and programming approach was realized in the ESA Space robot control system (SPARCO) [102, 103]. The SPARCO control system provides an integrated compliance control framework based on the industrial robots standards. In order to avoid the nominal motion perturbation it is recommended to keep the nominal position constant during force realization using this control approach. Such strategy was realized in the SPARCO control system by means of the “apply-force” control function. Hence, in order to retain the simple and obvious impedance control structure, it is more reliable to activate the force control in the contact realized by the impedance control. This strategy is integrated in the SPARCO control system, and was proven in numerous experiments to be quite reliable and robust [102]. The SPARCO control system involves impedance- and integral-force control algorithms at the servo control level, as well as compliance control monitoring, planning and programming control functions at higher control levels. This control system utilizes extended PDL2 language commands to specify and program a compliance task, such as to define compliance frame (relative to the tool frame), to select pre-designed impedance control gains (e.g. highimpedance, medium-damping, etc.) or to explicitly specify target impedance parameters, to activate and deactivate impedance control, to control interaction force, as well as to monitor the contact and force, etc. A specific control algorithm referred to as relax supports a continuous change of impedance control gains (i.e. target impedance models). The SPARCO system utilizes simple and robust position controllers of industrial robotic systems that exhibit quite
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desirable control performance, such as diagonal dominance, spatial roundness (Fig. 1.4) and normality, in order to realize simple and efficient position-based impedance and force control. Safe and robust executions of SPARCO interaction tasks in numerous tests demonstrated the feasibility and reliability of the applied interaction control concepts in an uncertain and stiff environment ( K e ≈ 100000 N/m, position misalignments to 10 mm) (Fig. 1.21). The SPARCO is implemented in a space control platform that is used in the European space programs. A more comprehensive presentation of the impedance control synthesis at a higher control level based on spatial intuitive reasoning, as well as integration of impedance control in forward industrial robot controllers, are presented in Chapter 3.
Fig. 1.21 SPARCO testbed
The SPARCO system considers a six-dimensional decoupled compliance model with respect to a compliance C-frame of reference (Fig. 1.7). The origin of this frame represents the compliance (impedance) center. Interaction forces and moments move this frame relative to the initial unloaded position according to the selected compliance model. The angle-axis representation has been applied to compose elemental rotation displacements. A specific planning
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problem is to specify the location of the C-frame so that equilibrium can be achieved [102]. In the space automation tasks with relatively simple geometry, such as grasping and mating of cylindrical parts, drawer open/close tasks, etc., this approach was demonstrated to be reliable and robust. 1.9 Haptic Systems Control Considerable research interest is related to new interactive systems designed for the interaction between human and a robotic device, or with remote or virtual dynamic environments. To the new interactive systems belong kinesthetic displays and haptic interfaces, teleoperation systems, human enhancers and augmentation devices, etc. These systems are designed to produce/receive kinesthetic stimuli for/from human movements, as well as to render a realistic feeling of contact and dynamic interaction with nearby, remote or virtual environments. The advanced interaction systems have found very attractive applications in surgical and rehabilitation robotics, power assist-devices, training simulation systems, etc. The most critical issue in these systems is to ensure stable and safe interaction with a high rendering performance. This is a challenging task, when taking into account serious problems, such as unknown and variable human dynamics, common nonlinear environmental characteristics, as well as various disturbances in computer-controlled systems. Essentially, the basic interaction chain in a haptic display consists of three principal elements (Fig. 1.22): human operator (H), haptic device (D) and virtual environment (VE). Common model of a haptic interface is presented in (Fig. 1.23). The middle element in this model is a haptic device, which is, based on an analogy with electrical network circuits, represented as a so-called two-port network. A haptic device interconnects the human with the virtual environments (both linked as one-port network) via force and velocity I/O signal pairs, describing the exchange of energy between the blocks. This representation has been demonstrated to be very useful in the analysis of teleoperation and haptic systems [12, 104]. Since the haptic device is computer-controlled, critical SD effects on the interaction system stability (control delay and sample-and-hold effects) must be also introduced in the interaction model. The main role of SD control system is to measure and render I/O signals via the haptic interface, and thus provide the operator with an enforced sense of haptic (or kinesthetic) presence in a virtual environment. Depending on the signals measured in the haptic interface, two system classes may be distinguished: impedance and admittance displays. In the
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impedance display the velocity (position) of the haptic mechanism is measured and a command force is rendered. The admittance display ensures tracking of a command position using interaction force/moment measurements in the handle. Impedance displays are commonly lightweight back-drivable active mechanisms (e.g. Phantom), while position-controlled high-inertia manipulators have often been used as admittance displays. An impedance-controlled industrial robot could be considered as an admittance display rendering the target admittance. Although study and modeling of human motor control and spatial limbs dynamics are fundamental challenges in biomechanics and neuroscience, the understanding of human interaction with a dynamic environment is still insufficient. The key quantity describing human arm dynamic interaction is the end-point impedance [105]. Numerous studies have demonstrated surprising human capabilities to adapt the arm impedance to variable interaction conditions and perturbations, even so to perform mechanically unstable tasks [106]. The Cartesian end-point arm impedance is nonlinear and non-symmetric spatial impedance, combining passive and active components [105]. However, in the control analysis human impedance is commonly, for the sake of simplicity, considered as linear variable impedance, often with one or two DOFs. Likewise, the haptic display dynamics can be considered as a linear admittance, while the environment is generally represented by nonlinear impedance. However, the performance obtained with the basic interaction system shown in Fig. 1.22 is commonly poor and, therefore, such an interaction structure is not feasible. The essential interaction problems, such as unpredictable and unknown human behavior and nonlinear and highly variable environment, cannot be successfully resolved with this interaction system architecture. Such a haptic system could interact well with a particular environment and for a specific human behavior, but in other cases performance might be bad, moreover the interaction might be unstable (especially in the contact with a stiff virtual wall). Generally, it is not possible to guarantee the stability of interaction with the simple haptic interface control system presented in Fig. 1.22. Obviously, a more sophisticated controller is required to take into account the human and environmental dynamics. The synthesis of such a controller is, however, an extremely complex task. In order to simplify design and improve the stability of the haptic interaction system, Colgate et al. [107] have proposed to couple an additional block, referred to as virtual compliance or virtual coupling (Fig. 1.23), between the haptic device and the virtual environment. The virtual coupling is commonly selected as impedance, i.e. admittance. The virtual coupling provides a simple but nevertheless stable and robust haptic controller. For a particular haptic
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device a corresponding virtual coupling might be designed regardless of the simulated virtual worlds and real human behavior. The main design goal is to ensure passive behavior of the coupled subsystem consisting of the virtual coupling and haptic display, thereby also taking into account critical SD effects (sampling and control delay). By these means, when taking into account that the human performs almost passive and stable interaction with a passive system, the stability of the entire haptic system may be ensured under all operating conditions if the virtual environment is passive. A haptic interface operates as a connection between human and virtual environments. The transparency and the Z-width are the main measures of performance of a haptic display. Transparency represents the degree to which velocities and forces (at the human and environment sides) match each other. The Z-width [108] of a haptic interface can be defined as the achievable range of impedance that can stably be presented to the operator. An ideal haptic interface can simulate free motion without inertia or friction, as well as render infinitely rigid and massive virtual objects. The primary concern in haptic systems is to achieve stable interaction under all operating conditions and for all simulated virtual environments, without undesired oscillations that would degrade virtual surface rendering. However, that is a challenging goal, because several destabilizing effects tend to jeopardize interaction stability. In order to ensure stability, almost all modern haptic systems utilize the advantages of the virtual coupling concept and implement the control architectures similar to the on sketched in Fig. 1.23.
Fig. 1.22 Elemental network model of a haptic system
Fig. 1.23 Haptic interface with virtual coupling (admittance)
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The stability of a haptic interaction system is commonly considered on the basis of the passivity theory. Explicit conditions for the passivity of a haptic system including a linear haptic device, a virtual coupling and a virtual environment, have been derived in [109] taking into account sampling and computation delay effects. The authors argued the essential relevance of physical damping parameters for the enhancement of system passivity and interaction stability. For a simple SISO coupling system consisting of the haptic-device, i.e. admittance Z D = 1 ( ms + b ) , and the virtual-coupling impedance ZV = Bs + K , the stability (passivity) condition imposes
b>
KT +B 2
(1.54)
where T is the control sampling time. In this elemental case of haptic interface, the virtual coupling represents a virtual wall, which consists of the parallel connection of the virtual stiffness K and the virtual damping B, to be rendered to the human. Hence, the condition (1.54) means that physical damping must be involved in the system in order to ensure a stable interaction with the virtual wall. Higher sampling rates (i.e. smaller T) facilitate the implementation of stiffer walls. Brown and Colgate [110] derived similar expressions for the minimum mass of the virtual wall that can be simulated passively. However, the stability conditions that were obtained appear to be quite conservative. Moreover, these criteria imply physics-based system design that is not always reliable. For example, higher additional damping (1.54) allows higher virtual impedance to be realized, but thereby the impedance of the haptic device must also be increased. Adams [111] has proposed an approach for a virtual coupling design based on the network stability that appears to be less conservative than the passivity-based synthesis. The stability of two-port network consisting of the haptic device and the virtual coupling guarantees the stability of a haptic interface when coupled with any passive virtual environment and human operator. Miller et al. [112] have extended the passivity-based approach to haptic systems involving nonlinear and time-delayed virtual environments. Hannaford and Ryu [113] have applied time-domain passivity analysis in order to improve the system performance in contact with a very stiff and delayed environment. However, numerous experiments clearly showed that the contact stabilization with stiff, delayed and nonlinear environments still represents the crucial problem in haptic interfaces. A specific problem in haptic interfaces is the lack of objective stability testing. Human exhibits good capability to stabilize (damp)
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the interaction with a slightly oscillating environment. Therefore, the loss of contact and bouncing in the haptic interfaces appear to be less critical [113], comparing with contact stability problems in industrial robots (see Chapter 3 and Chapter 5). However, these oscillations can jeopardize the interaction fidelity. In the majority of experiments the increasing of sampling rates and reducing of force magnitudes have been recognized as promising measures to reduce bouncing. Chapter 5 presents an extension of the new robust control design framework established for the control synthesis of interaction between an impedancecontrolled robot and a passive environment [76, 77] to other interactive systems with physical or virtual interfaces (admittance and impedance displays). 1.10 New Robot Applications Several authors have considered the development of interaction control algorithms based on soft-computing techniques [114, 115]. Fuzzy and neuralnetwork-based control algorithms provide promising approaches to cope with nonlinear environment effects. However, further investigations are needed to improve robustness, design and reliability of soft-computing methods based compliance control. The development of specific interaction control algorithms for new robotic structures, such as parallel robots [116], dual and cooperative arms [117], wire robots [118], etc., has also attracted considerable research attention. The design of light-weight robotic arms with integrated torque sensors at robot joints has been demonstrated to be a very promising concept for future safe robotic systems suitable for dynamic interaction control [119]. Passivity-based control design techniques have demonstrated good potential for interaction stabilization of rigid [120] and flexible-joints robots [121, 122] with a stiff environment. Specific interaction control algorithms suitable for the application in surgical and rehabilitation robotics have also attracted research interest [123]. The impedance control has also appeared as a very promising approach to dynamic interaction control in humanoid gait [124, 125]. The compliance control represents the basic control approach in new robot applications in rapidly emerging fields requiring physical interaction between the robot and operator such as human extenders and power-assists devices, rehabilitation robots, etc (see only a few of selected papers [126, 127, 128]). The control algorithms applied in these fields are essentially based on the above control approaches. The integral force control is basically applied in hand-driven
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intelligent assistant devices [128] to realize the movement of large payloads by a human operator handling a force sensor interface. The synthesis of passive and robust control laws, discussed above, is crucial for ensuring a stable interaction between human, robot and a passive environment in intelligent assistance devices. It is worth mentioning an experimental comparison of robot interaction control schemes [129]. Position based force and impedance control has been implemented and tested in an industrial experimental robotic system. The authors have pointed out superior performance of the dynamic-model-based control schemes in comparison to so-called “static-model” algorithms designed to achieve desired steady-state performance. Good performance of parallel position-force control has also been reported. However, further investigations are still required concerning practical needs and benefits of the complete robot dynamic model computations in the interaction control algorithms intended for industrial robot control systems interacting with a passive environment and in relatively slow contact tasks, which are most frequently performed in practice. Definitions of benchmark tests and objective criteria for comparison of various compliance control schemes are also required. Significant research efforts have also been made in the direction of practical design of interaction control [130]. Computer-aided procedures for design of hybrid position/force and impedance control were developed in [131] and [132], respectively. Publication of the first monographs focusing on interaction control [133, 134] has also contributed to the further investigations in this field. Finally, the integration of compliance and visual control [135, 136] should also be mentioned as a very attractive research topic that is quite promising for further improvement of interaction control capabilities and performance. 1.11 Conclusion During the past two decades, compliant motion control has emerged as one of the most attractive and fruitful research areas in robotics. The control of constrained motion of robots is still a challenging research area, whose successful solution will considerably influence further applications of robots in the industry and emerging service fields. Widespread applications of impedance control in industrial robotic systems are still a challenging problem. One of limitations is the absence of a widely-accepted framework for the synthesis of impedance control parameters that would ensure the stability of both contact transition and interaction processes and guarantee desired contact performance.
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In conclusion, it is perhaps of interest to indicate some of possible future research subjects. A clear formulation and specification of hybrid control and impedance control are further required in order to integrate these control approaches in commercial industrial robotic systems. Especially, the synthesis of compliance control for specific contact tasks at higher motion planning and programming control levels is essential for the realization of first industrial robot systems with compliance control capabilities. Further simulation and experimental tests of the proposed compliant motion control algorithms, such as parallel position-force control, adaptive and variable structure algorithms, and, particularly, dynamic control of a robot interacting with a dynamic environment, are also of interest. Particular attention should be paid to solving contact task control problems caused by uncertainties and nonlinear effects in the environment and robotic system, such as friction, multipoint contact, elasticity, time lags in computer controlled systems, etc. In the impedance control, further advances are to be expected in the adaptation of target impedance to complex task requirements. Design and optimization of robot admittance for specific tasks represents a challenging practical task. The compliant motion capability analysis of industrial robots and requirements on the next robot generation from the viewpoint of contact-task applications are of interest to designers. Robust control provides a practical approach for control design of compliance motion. Research on human-robot interaction, understanding of human motoric functions and diseases, modeling and estimation of human impedance, as well as the development of stable and safe interaction algorithms, represent further very attractive topics in new robotic applications supporting physical interactions with a human. Nonlinear control concepts might be useful to bridge the gap between geometric and dynamic constraints. Sophistication of the robustness concept is another worthwhile theoretical challenge. From a more practical point of view, future research should produce systems with improved intermediate and high level performance and user friendliness and safety. Comparison of the available algorithms, definition of benchmark tests, investigation of compliant control in uncertain and dynamic environment, examination of nonlinear effects in the robot and environment dynamics, specification and design of force and impedance control tasks, solving control problems at higher control levels, etc., are certainly some areas deserving further computational/experimental studies.
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Chapter 2
A Unified Approach to Dynamic Control of Robots
2.1 Introduction The difficulties encountered in solving the problem of simultaneous stabilization of programmed motion and desired contact force of the robot interacting with its environment, have probably been the reason for introducing the simplifying idea of splitting the task into the motion control and interaction force control. This idea enabled Raibert and Craig [1] and Mason [2] to formulate the approach to manipulator control called the hybrid position/force control. The basic idea of this approach is to divide the control task into two independent subtasks in a certain coordinate space {Z } . One subtask is the robot’s motion control along a predetermined part of the coordinates (directions), and the other is the control of the interaction force of the robot and environment along the rest of the coordinates (directions). From the mathematical point of view, the essence of the hybrid control concept is in the following. It is assumed that there exists a coordinate space {Z } and its partition into a direct sum of the subspaces {X } and {Y } : {Z } = { X } ⊕ {Y } , so that the dynamic model of manipulation robot written in the coordinate space {Z } is split into two independent equations:
U1 ( x, xɺ, ɺɺ x) = τ x U 2 ( Fy ) = τ y where τ x , τ y are the corresponding control inputs. However, there are no real examples of a robot interacting with its dynamic environment in which such partition would be possible. Even for a simplest manipulation mechanism interacting with the environment, the following partition is obtained: 77
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Dynamics and Robust Control of Robot-Environment Interaction
U1 ( x, xɺ, ɺɺ x, Fx ) = τ x U 2 ( y, yɺ , ɺɺ y, Fy ) =τ y In Section 1.6.2 of Chapter 1, the reader can find a detailed discussion of the position/force hybrid control. The orthogonality between the constraint force and the direction of unconstrained motion is assumed and has been used in the majority of the works. The weakness of this approach related to the notion of “orthogonality” lies not only in the fact that it is not correct to use the term “orthogonality” itself, but also in the fact that, in finding directions along which motion and force are “orthogonal”, the followers of hybrid control commit a mistake. Namely, for the task of stabilization in these directions they use feedback loops with respect to motion or force only. The criticism refers to the basic idea of the position/force stabilization based on traditional hybrid control concept, and not to the realization of this possibility itself by means of a certain procedure. However, there have been no attempts to realize the hybrid control concept by some other means, save the “orthogonal complements”. The idea of splitting the task of the robot interacting with the environment into the task of position control in certain directions and force control in the other, represents by itself a more profound idea than the idea of hybrid control based on the “orthogonal complements”. This Chapter is devoted to the unified approach to dynamic control of the robots interacting with a dynamic environment, which differs essentially from the above conventional hybrid control schemes. A dynamic approach to control simultaneously both the position and force in an environment with completely dynamic reactions has been developed. The approach of dynamic interaction control defines two control subtasks responsible for the stabilization of robot position and interaction force. Both control subtasks utilize dynamic model of the robot and the environment in order to ensure the tracking of both the nominal motion and force. Special attention is paid not only to the synthesis of the control laws that ensure stability of robot’s desired motions and desired interaction forces of the robot and environment, but also to the definition of possible motions of the robot and its possible interaction forces in contact tasks. These motions and interaction forces are called programmed. The concept of the family of transient processes with respect to the robot’s motion and its force of interaction with environment
A Unified Approach to Dynamic Control of Robots
79
is formulated. It allows one to set and solve the problem of synthesis of control laws that not simply stabilize the motion and force of the robot’s interacting with its environment, but also enables solving of the stabilization problem with the preset quality of transient processes. Thus, two types of control laws can be synthesized. One type of control laws solve the problem of exponential stabilization of programmed motions of the robot with the predetermined quality of transient processes with respect to position, and simultaneously stabilize exponentially the programmed interaction forces of the robot and environment. The other type of control laws solve the problem of exponential stabilization of programmed interaction forces of the robot and environment with the predetermined quality of transient processes with respect to force, and simultaneously stabilize exponentially the programmed motions of the robot. It is necessary to note that the laws of former type solve the problem of robot’s control both in free space and at its contact with the environment. In the general case, for the control laws of second type it was possible to obtain only a sufficient condition of exponential stability of motion when the exponential stability of the contact force is achieved. The special case of the environment dynamics model for which this type of control laws gives both necessary and sufficient conditions of exponential stability of motion is also considered. For the control laws of both types significant attention is paid to the analysis of the influence on transient processes of the constraints imposed on the state, control and interaction force, taking into account the inadequacy of dynamics models of the robot and environment and/or external perturbations. In view of the fact that in the course of operation, the robot parameters, especially the viscous friction coefficients at the manipulator’s drives, may vary with time, the unknown drifts of parameters are of interest. The adaptive control scheme proposed in this chapter enables one to solve contact tasks for robots with both stationary and nonstationary dynamics. One of the parameters of the scheme, such as the algorithm adaptation processing speed, determines the “speed” of the parameters’ drift to which the adaptive control system has time to adapt without violating the a priori constraints of the control, motion, and interaction forces. Classes of stabilized motions and forces and their stabilization accuracy are considered in reference to the level of initial and external perturbations of dynamics of the robot and its environment, as well as of the sensors errors, processing speed of the adaptation algorithm, and of other parameters of the adaptive control scheme.
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Dynamics and Robust Control of Robot-Environment Interaction
As was mentioned above, the control laws that stabilize simultaneously both the robot motions and interaction forces are presented. These control laws possess exponential stability of closed-loop systems and ensure the preset quality of transient processes of motions and interaction forces. However, for the control laws stabilizing the desired interaction force with preset quality of transient process only sufficient conditions for the exponential stability of motion are given. In the cases when the environment dynamics can be approximated sufficiently well by a linear time-invariant model in Cartesian space, necessary and sufficient conditions for the exponential stability of both motion and force will be derived and the corresponding control laws will be defined. In addition to the schemes that use adaptive control laws, the problem arising from the uncertainties of parameters may also be resolved by applying the knowledge-based techniques. Taking into account external perturbations that do not expire with time and model uncertainties, it may be difficult to achieve asymptotic (exponential) stability. Therefore, it is of practical interest to demand less restrictive stability conditions, i.e. to consider the so-called practical stability of the system. The practical stability of the robot around the desired position and force trajectories is defined by specifying the finite regions around the desired position and force trajectories within which the robot position and interaction force have to be during the task execution. Practical stability tests will be demonstrated in this chapter using two very representative control laws. 2.2 Dynamic Environments Control of commercial robotic systems in free space does not usually require so accurate knowledge of the system’s dynamics since simple servo controllers are often capable of ensuring satisfactory positioning of industrial robots. As already mentioned, however, to control the robots involving constrained motion, sophisticated and accurate models of the entire system, encompassing the robotic mechanism and its environment, need to be studied carefully. During the execution of a contact task, the kinematic structure of the robot changes from an open to closed kinematic chain. The contact with the environment imposes some kinematic and dynamic constraints on the motion of the robot end-effector. One of the most difficult aspects of dynamic modeling is related to the interactions with the bodies in contact.
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81
2.2.1 Model of a dynamic environment The reaction force is influenced not only by the robot’s motion, but also by the very nature of the environment. Since the mechanical interaction process is generally very complex and difficult to describe mathematically in an exact way, we are compelled to introduce certain simplifications, and thus partly idealize the problem. An ideal frictionless environment, imposing pure kinematic constraints on the end-effector motion is characterized by a static force balance at the contact. On the other hand, the interaction forces in contact with a dynamic environment are not completely compensated for by the constraint reactions. These forces produce active work on the environment, causing also its motion. The relationship between the interaction force and environment dynamics is usually described by nonlinear second-order ordinary differential equations:
M ( s )ɺɺ s + L( s, sɺ) = ϕ sT ( s ) F
(2.1)
and by algebraic equations connecting the end-effector and environment coordinates
x = ϕ ( s ) ; s = ϕ −1 ( x) xɺ =
(2.2)
∂ϕ sɺ = ϕ s ( s ) sɺ ∂ sT
where x ∈ R 6
– vector of the end-effector coordinates (position and orientation),
s∈ Rd
ϕ ( s) ∈ R
– vector of the environment coordinates (displacements), 6
– nonlinear kinematic function,
M (s)
– d × d nonsingular mass matrix,
L( s, sɺ) ∈ R d
– nonlinear dynamic function,
F =− F
– 6-dimensional vector of generalized forces acting on the environment,
ϕs (s)
– 6 × d Jacobian matrix,
d
– dimension of the coordinates’ vector s .
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Dynamics and Robust Control of Robot-Environment Interaction
The general multibody environment topology involves rigid and/or flexible bodies with passive elements, such as inertia, springs and dampers, but without actuators. Furthermore, the friction and contact forces can be included. The case when the environment DOFs are powered by additional actuators leads to a specific redundant control problem, referred to as cooperative manipulation, which has been studied in a number of works (see, for example, [3, 4]). An example may be two cooperating robots handling an object. Depending on the dimension d of the coordinates’ vector s , as well as on the rank of the Jacobian matrix ϕ s , several passive dynamic environment cases can be distinguished. 2.2.1.1 Kinematic-dynamic constraints In effect, a general model of the environment involves geometrical (kinematic) constraints plus dynamic constraints [5]. An example of such dynamic environment would be a robot turning a crank or sliding a drawer (Fig. 2.1), the dynamics of which is relevant to the robot’s motion and cannot be neglected.
Fig. 2.1 Dynamic environment interactions
2.2.1.2 Pure dynamic environment Pure dynamic environment would correspond to an environment consisting of a mechanical system rigidly coupled to the end-effector. Thus, no kinematic
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constraints on the robot’s end-effector motion are imposed, which results in a relatively simple mapping between the robot and environment coordinates. Such environment was considered in [6, 7]. This model may represent an elastodynamic environment, which is in a linearized form (linear impedance model) considered below. Another representation of this general environment model combines a passive mechanical system and real deformable constraints (joints) (Fig. 2.1) along the different DOFs. 2.2.1.3 Linear impedance model The linear impedance model provides a significant simplification of the environment dynamics modeling. In principle, this concept is based on the assumption of small movement in the neighborhood of the equilibrium contact point xe ∈ R 6 describing a contact frame (position and orientation). Thus, the dynamic interaction can be described by the convenient and well-studied linear multi-DOF system model (impedance causality).
M e ( ɺɺ x − ɺɺ xe ) + Be ( xɺ − xɺe ) + K e ( x − xe ) = − F
(2.3)
where M e is a positive-definite environment mass or inertia matrix; Be and K e are semi-definite environment damping and stiffness matrices, respectively. The term
p = x − xe
(2.4)
refers to the end-effector penetration into the environment. Accordingly
p0 = x0 − xe
(2.5)
will be termed as the nominal position, and could be taken to mean a position planning failure due to tolerances, or a desired ingoing into the environment. It should be mentioned that the computation of the penetration vector components associated with rotational DOFs is dependent on the form used to describe the end-effector orientation. If the orientation vector has been applied, the subtraction of orientation subvectors in equations (2.4), (2.5) is symbolic and denotes special algebraic formalisms for subtracting orientation vectors [8].
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Dynamics and Robust Control of Robot-Environment Interaction
2.3 Synthesis of Control Laws for the Robot Interacting with Dynamic Environment The model of the dynamics of a robot interacting with its environment is described by vector differential equation of the form:
H (q )qɺɺ + h(q, qɺ ) = τ + J T (q ) F
(2.6)
where, q = q (t ) is the n -dimensional vector of the robot generalized coordinates; H ( q ) is the n × n positive definite matrix of inertia moments of the manipulation mechanism; h( q, qɺ ) is the n -dimensional nonlinear function of centrifugal, Coriolis’ and gravitational moments; τ = τ (t ) is the n dimensional vector of the input control; J T (q ) is the n × m Jacobian matrix connecting the velocities of the robot end-effector and velocities of the robot generalized coordinates; F = F (t ) is the m -dimensional vector of generalized forces or generalized forces and moments acting on the end-effector from the environment. The dimension of the vector F can be adopted in the Cartesian coordinates to be 3 or 6. Presently, for simplicity sake and without loss of generality it will be assumed that n = m (in general n ≥ m ). Because the inverse of the matrix H ( q ) exists, let us write the equation of the robot dynamics model (2.6) in the form solved with respect to the higher derivative:
qɺɺ = − H −1 (q ) h(q, qɺ ) + H −1 (q )τ + H −1 (q ) J T (q ) F
(2.7)
and denote the right-hand side of (2.7) by Φ ( q, qɺ , τ , F ) . Then, this equation can be written in the form:
qɺɺ = Φ ( q, qɺ ,τ , F ) , q(t0 ) = q0 , qɺ (t0 ) = qɺ0
(2.8)
which also includes the initial conditions q (t0 ) = q0 , qɺ (t0 ) = qɺ0 . It is assumed that the function Φ satisfies all the conditions needed for the existence and uniqueness of the solution of the system of differential equations (2.8) on [t0 , +∞) . The robot dynamics model (2.6) can also be solved with respect to the control variable τ :
τ = U ( q, qɺ , qɺɺ, F )
(2.9)
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A Unified Approach to Dynamic Control of Robots
ɺɺ, F ) = H (q)qɺɺ + h(q, qɺ ) − J (q) F . where U ( q, qɺ , q Let us consider now the mathematical model of environment. It represents one of the most complex and delicate problems in the contact tasks control. As we have seen, to describe the environment we should generally use a different set of generalized coordinates, say q e , when the environment has autonomous DOFs. In the case when the environment does not possess displacements that are independent of the robot’s motion (the environment is firmly attached to the robot’s end-effector), the environment dynamics can be described by the nonlinear differential equationa: T
M 1 ( s )ɺɺ s + L1 ( s, sɺ) = − F
(2.10)
s = ϕ (q) or in the frame of generalized coordinates of the robot (joint variables):
M (q )qɺɺ + L(q, qɺ ) = S T (q ) F
(2.11)
where M (q ) is a nonsingular n × n matrix; L( q, qɺ ) is a nonlinear n dimensional vector function; S T ( q ) is the n × n matrix of rank n , i.e. rank ( S ) = n . We also assume that all the mentioned matrices and vectors are continuous functions of their arguments. By using the inverse matrix M, equation (2.11) can be written in the form:
qɺɺ =ψ ( q, qɺ , F )
(2.12)
where ψ ( q, qɺ , F ) = − M (q) L(q, qɺ ) + M (q) S (q) F . We assume that the vector function ψ satisfies all the conditions for the existence and uniqueness of the solution in an initial value problem for the system of differential equation (2.12) on [t0 + ∞) . Let us solve (2.11) with respect to the force F. Due to the fact that rank ( S ) = n , we have: −1
F = ( S T (q) )
−1
−1
T
[ M (q)qɺɺ + L(q, qɺ )]
a For difference’s sake, when the environment possesses autonomous DOFs, we introduce the
coordinates of passive environment instead of the coordinates
qe .
s
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Dynamics and Robust Control of Robot-Environment Interaction
In the general case ( n ≥ m ), it is supposed that rank ( S ) = m and, instead of the −1 inverse to the matrix S T ( q ) its left inverse matrix S ( q ) S T ( q ) S ( q ) is used. Evidently, S ( q ) S T ( q ) is the Gramm matrix. ɺɺ) , it can be By denoting the right-hand side of this equation with f ( q, qɺ , q written in the form:
(
)
F = f ( q, qɺ , qɺɺ)
(2.13)
where f is a continuous function of its arguments. Any specific force F (t ) by which robot acts on the environment causes a unique motion of the robot q (t ) in this environment for the prescribed initial conditions q (t0 ) = q0 , qɺ (t0 ) = qɺ0 . The solvability property of the model equation (2.11) with respect to the force action presumes that if the robot moves in the environment in conformity with the function q (t ) , then the force:
F (t ) ≡ f ( q(t ), qɺ (t ), qɺɺ(t ) ) is the only force that causes this motion. In the case of contact with the environment the robot control task can be described as the robot’s motion along a programmed (desired) trajectory q p (t ) when a programmed (desired) force of interaction Fp (t ) is acting between the robot and the environment. The desired programmed motion ( PM ) q p (t ) and desired programmed force of interaction ( PFI ) Fp (t ) cannot be arbitrary. The two functions must satisfy the following relation:
Fp (t ) ≡ f ( q p (t ), qɺ p (t ), qɺɺp (t ) ) ,
∀ t ≥ t0
(2.14)
The control goal of the robot interacting with dynamic environment can be formulated in the following way: Synthesize the control τ (t ) for t ≥ t0 so that to satisfy the target conditions:
q (t ) → q p (t ) as t → ∞, F (t ) → F p (t ) as t → ∞
(2.15)
2.3.1 Stabilization of motion with the preset quality of transients Let us describe the deviation of the real motion q (t ) from the programmed one by the transient process:
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η (t ) = q (t ) − q p (t ) It is natural to determine in advance the requirements which the transient process should obey. These requirements allow certain quality of the transient processes to be set in advance. As certain initial perturbations are always present, i.e. η0 =η (t0 ) = q (t0 ) − q p (t0 ) ≠ 0 , these requirements should not refer to a particular function η (t ) , but to the whole family of functions {η (t )} , each of which possesses its concrete value at the initial moment. Therefore, the family of desired transient processes can be given by the vector differential equation [9, 10]:
ηɺɺ = P(η ,ηɺ )
(2.16)
where P is an n -dimensional vector function continuous over the whole set of arguments, such that equation (2.16) has a unique trivial solution η (t ) ≡ 0. Then, the fulfillment of the requirements that have been imposed on the transient process is ensured by the choice of the function P , and the robot control should be designed in such a way that in the absence of any perturbations except for the initial ones, the robot dynamics equation coincides with equation (2.16). We shall a priori adopt that the choice of the function P in (2.16) ensures asymptotic stability in the whole of trivial solution of the system (2.16). The function P can be adopted, for example, in the form: P(η ,ηɺ ) = Γ1ηɺ + Γ2η . Then, (2.16) acquires the form:
ηɺɺ = Γ1ηɺ + Γ2η
(2.17)
where Γ1 and Γ2 are the constant n × n matrices. Let us turn from equation (2.17) to a system of 2n -order differential equations of the form:
xɺ = Γ x
(2.18)
where
0 η x= , Γ= n ηɺ Γ 2
In . Γ1
0 n and I n are the respective zero and unit matrices of dimension n × n . Then, the matrices Γ1 and Γ2 can be chosen in such a way that the eigenvalues
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Dynamics and Robust Control of Robot-Environment Interaction
of the matrix Γ possess negative real parts. In this way asymptotic stability as a whole of the solution of equation (2.18), and thus of equation (2.17), is achieved. The task of stabilizing the PFI Fp (t ) can be tackled in an analogous way, by considering the family of transient processes with respect to force in the form [10]:
µɺ (t ) = Q( µ ) , µ (t ) = F (t ) − F p (t )
(2.19)
and by choosing a continuous vector function Q (Q (0) = 0) of dimension n , such that the asymptotic stability of the trivial solution of µ (t ) ≡ 0 as a whole, is ensured. At this point, a crucial question is whether one can generally achieve that both the perturbed robot’s motion and perturbed robot’s force of interaction with the environment satisfy simultaneously equations (2.16) and (2.19). The answer is negative. Let us explain this. Let τ = τ (t ), t ≥ t0 be a control law, such that the robot’s dynamics equation (2.6) with the control law is equivalent to the reference equation (2.16). Then, it follows from (2.16):
qɺɺ = qɺɺp + P (η ,ηɺ )
(2.20)
On the other hand, by comparing (2.13) and (2.14), we obtain the following relation between the real and programmed force of interaction, irrespective of the control law:
F − Fp = f ( q, qɺ , qɺɺ) − f ( q p , qɺ p , qɺɺp )
(2.21)
or, by taking into account (2.20):
µ (t ) = f (η + q p , ηɺ + qɺ p , qɺɺp + P(η ,ηɺ ) ) − f ( q p , qɺ p , qɺɺp )
(2.22)
This shows that the character of change of µ (t ) is fully defined by the righthand side of (2.22) and depends on the function f , determining the model of the external environment dynamics, on the function P that determines the transient process character of the robot’s motion and on the control τ , via q and qɺ . In this way, µ (t ) is actually a function which cannot be arbitrary and which, being the solution of equation (2.19), is dependent of the function Q that has been chosen in advance, and which is relatively arbitrary. This means that the quality
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of transient processes is controlled only in an implicit way, via the function P , and is strongly dependent on the model of the environment. Let us formulate two alternative questions: Is it possible to choose such control law that would attain the control goals, defined by the relations (2.15), and at the same time ensure the preset quality of robot’s motion, specified by (2.16)? Is it possible to choose a control law that would ensure the preset quality of mutual force interaction between the robot and environment, defined by (2.19), and also attaining of the control goals? If the answer to the first question is positive (which will be discussed in the text to follow), the answer to the second question depends on the environment dynamics model, and this is going to be considered in the next section. Let us synthesize the control law τ (t ) in such a way to ensure the desired quality of the robot’s motion (2.16). Then the relation (2.20) has to be satisfied in the closed loop. This can be achieved by adopting the control law that has feedback loops with respect to q, qɺ and F [6]:
τ = H (q ) qɺɺp + P(η ,ηɺ ) + h ( q, qɺ ) − J T (q ) F
(2.23)
By substituting (2.23) into the robot’s dynamics model (2.6), we obtain the equation of a closed-form system:
H (q ) qɺɺ − qɺɺp − P (η ,ηɺ ) = 0
(2.24)
Because H ( q ) is a positive definite matrix, det H ≠ 0 , and consequently, (2.24) is equivalent to (2.16). Due to the property of the robot’s function P , which ensures asymptotic stability of the system (2.16) as a whole with respect to (η ,ηɺ ) , we will have:
η (t ) → 0 and ηɺ (t ) →0 t →∞ t →∞
(2.25)
and, since P is a continuous function and P (0, 0) = 0 , it follows that
ηɺɺ (t ) →0 t →∞
(2.26)
By taking into account the previously obtained relations (2.21) or (2.22), and also the continuity of the function f , we obtain:
µ (t ) →0 t →∞
(2.27)
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Dynamics and Robust Control of Robot-Environment Interaction
Thus, the control goal is achieved, i.e. the real robot’s motion is close to the programmed motion and simultaneously the real force F (t ) is approaching the programmed force of interaction F p (t ) between the robot and environment. Let us consider one more control law of the robot motion stabilization without utilizing force feedback when the interaction force between the robot and environment is exists. For this purpose, let us modify the control law (2.23) using the relations (2.13) and (2.20). We obtain:
τ = H (q ) qɺɺp + P(η ,ηɺ ) + h(q, qɺ ) − J T (q ) f ( q, qɺ , qɺɺp + P(η ,ηɺ ) ) (2.28) Let us show that the control law (2.28), with the feedback loops only with respect to q and qɺ , solves the task of stabilizing PM (2.25), (2.26) and, consequently, in accordance with (2.27), the control task itself under the condition of the robot’s contact with the environment and under some supplementary assumption. By substituting the control (2.28) into the robot’s dynamics model (2.6) we obtain:
(
H (q ) [ηɺɺ − P (η ,ηɺ ) ] = J T (q ) F − f ( q, qɺ , qɺɺp + P (η ,ηɺ ) )
)
(2.29)
Taking into account the equality (2.13) we have:
(
H (q ) [ηɺɺ − P (η ,ηɺ ) ] = J T (q ) f ( q, qɺ , qɺɺ) − f ( q, qɺ , qɺɺp + P (η ,ηɺ ) )
)
(2.30)
or, by taking into account the form of the function f : −1
H (q ) [ηɺɺ − P (η ,ηɺ ) ] = J T (q ) ( S T (q ) ) M (q ) [ηɺɺ − P (η ,ηɺ )]
(2.31)
Therefore
( H (q) − J (q) ( S (q) ) T
T
−1
)
M (q ) [ηɺɺ − P (η ,ηɺ ) ] = 0
(2.32)
Let us suppose that for all q , i.e. for an arbitrary robot configuration, the supplementary condition −1 det H (q ) − J T (q ) ( S T (q ) ) M (q ) ≠ 0
(2.33)
is fulfilled. Then, the equation of the closed-loop system (2.32) is equivalent to the reference equation (2.16). Although the control law (2.28) does not use
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A Unified Approach to Dynamic Control of Robots
force feedback, it is explicitly dependent on the environment dynamics:
(
τ = U q, qɺ , qɺɺp + P(η ,ηɺ ), f ( q, qɺ , qɺɺp + P(η ,ηɺ ) )
)
(2.34)
through the function f , and cannot be used to control the manipulator in free space. 2.3.2 Stabilization of interaction force with the preset quality of transients As shown in the preceding section, the answer to the first question was quite simple: the inverse dynamics method ensures achievement of a desired motion quality and at the same time guarantees stability of the interaction force. Let us consider now the issue of stabilizing the force of interaction of the robot with the environment. Let us discuss first several control laws ensuring the solution of this task. For convenience, when describing the quality of transient processes with respect to force (2.19), we shall use an equivalent relation of the form [6, 10, 11]: t
µ (t ) = µ0 + ∫ Q ( µ (ω ) ) d ω
(2.35)
t0
Without loss of generality we can adopt that µ0 = 0 , since the stabilization of µ in the sense of preset quality (2.35) implies the stabilization according to the preset quality (2.19). Let us consider the control law with the feedback with respect to q, qɺ and F , of the form: t T τ = H ( q ) M (q ) − L (q, qɺ ) + S ( q ) Fp + ∫ Q ( µ (ω ) ) d ω + t0 + h(q, qɺ ) − J T (q ) F −1
(2.36)
By substituting this control law into the robot’s dynamics model (2.6) we obtain the following law of robot’s functioning in the contact with the environment: t M (q )qɺɺ + L(q, qɺ ) = S (q ) Fp + ∫ Q ( µ (ω ) ) d ω t0 T
(2.37)
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Dynamics and Robust Control of Robot-Environment Interaction
Taking into account the environment dynamics model (2.11) we obtain the following closed-loop control system: t S T (q ) µ (t ) − ∫ Q ( µ (ω ) ) d ω t0
=0
and, because rank ( S ) = n , this equation is equivalent to the equation: t
µ (t ) = ∫ Q ( µ (ω ) ) dω
(2.38)
t0
wherefrom, equation (2.19) follows right away. In this way the control law (2.36) ensures a desired quality of stabilization of PFI Fp (t ) . Let us note that the control law (2.36) can be rewritten in a more compact form as: t τ = U q, qɺ ,ψ q, qɺ , Fp + ∫ Q ( µ (ω ) ) dω , F t0
(2.39)
Let us consider another control law of the form t τ = U q, qɺ ,ψ (q, qɺ , F ), Fp + ∫ Q ( µ (ω ) ) dω (2.40) t0 involving feedback loops with respect to q, qɺ and F , and let us demonstrate that it also ensures a desired quality of stabilizing PFI Fp (t ) . In a developed
form, this control law can be written as
τ = H (q) M −1 (q) − L(q, qɺ ) + S T (q) F + h(q, qɺ ) − t − J ( q) Fp + ∫ Q ( µ (ω ) ) d ω t0 T
(2.41)
Substituting τ from (2.41) into (2.6) we obtain the closed-form equation:
H (q )qɺɺ = H (q ) M −1 (q ) − L(q, qɺ ) + S T (q ) F + t + J T (q ) µ − ∫ Q ( µ (ω ) ) d ω t0
(2.42)
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A Unified Approach to Dynamic Control of Robots
Taking into account that for the contact of the robot with environment the equation (2.11) holds, i.e.
− L(q, qɺ ) + S T (q ) F ≡ M (q )qɺɺ the relation (2.42) is equivalent to the relation: t J (q ) µ − ∫ Q ( µ (ω ) ) d ω t0 T
=0
Assuming that rank J ( q ) = n for all robot configurations {q} , we obtain a closed-form equation of the system in the form (2.35). The requirement rank J ( q ) = n is quite natural if it is borne in mind that from the point of view of physics this means that the magnitude of the force of interaction of the robot (2.6) with the environment (2.11) should be uniquely determined by the function of control and by the robot’s motion. Moreover, it can be stated that in the case of robot’s contact with environment no configurations {q} for which rank J ( q ) < n can arise, as this would mean that the interaction force satisfying the equation of environment’s dynamics (2.13) would not be uniquely determined by the function of motion q (t ). Finally, let us consider one more control law stabilizing PFI Fp (t ) with a preset quality of transient processes, using no explicit force feedback, but the implicit one, via the integral of the difference F (t ) − F p (t ). Let the control law be given in the form:
t
t
τ = U q, qɺ ,ψ q, qɺ , Fp + ∫ Q ( µ (ω ) ) dω , Fp + ∫ Q ( µ (ω ) ) dω
t0
t0
(2.43)
or, in a more developed form, as t t T T τ = HM − L + S Fp + ∫ Q( µ )d ω + h − J Fp + ∫ Q( µ )dω . t0 t0
−1
The closed form of the control system is then t t Hqɺɺ = HM −1 − L + S T Fp + ∫ Q( µ )dω + J T µ − ∫ Q( µ )dω t0 t0
(2.44)
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Dynamics and Robust Control of Robot-Environment Interaction
ɺɺ from equations (2.44) and (2.12) we obtain By eliminating q t t − L + S T F = − L + S T Fp + ∫ Q( µ )d ω + MH −1 J T µ − ∫ Q( µ )d ω t0 t0
or
t S T µ − ∫ Q( µ )dω t0
t −1 T = MH J µ − Q( µ ) d ω . ∫ t0
Wherefrom, t T −1 T S − MH J µ − ( ) ∫ Q(µ )dω t0
=0
(2.45)
Let us set the supplementary requirement as
rank S T (q ) − M (q ) H −1 (q ) J T (q ) = n
(2.46)
for all possible robot configurations {q} . Then, the relation (2.45) will be identical to (2.19) and, consequently, under the supplementary assumption (2.46), the control law (2.43) also ensures stabilization of the PFI Fp (t ) with a preset quality of transient processes. Let us notice that because of the continuity of the robot’s motion, the violation of the conditions (2.46) over a finite or countable set of points does not entail violation of the closed-loop control functioning of the form (2.38). The violation of this condition may generally arise only if the inequality
rank S T (q ) − M (q ) H −1 (q ) J T (q ) < n has over some time interval twice continuously differentiable solution q (t ), t ∈ (ta , tb ), ta < tb . This issue requires additional investigations of dynamics models of the robot and environment. However, attention has to be paid to the possibility of appearance of the regimes analogous to sliding regimes in the tasks of dynamic objects control [12]. So, we have synthesized three control laws, (2.39), (2.40), and (2.43), each of which ensures a desired quality of stabilizing the PFI Fp (t ) . Now let us consider the question of whether there is a sufficient condition (and what it should be) for the convergence of the real motion q (t ) to the
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programmed one q p (t ) , when asymptotic stability of the contact force takes place. To discuss this issue, let us return to the environment dynamics model (2.11) and, by taking into consideration the relation between PM q p (t ) and PFI
FP (t ) in the form (2.14), we obtain: M ( q) qɺɺ − M (q p ) qɺɺp + L(q, qɺ ) − L( q p , qɺ p ) − S T ( q) − S T (q p ) Fp = S T ( q) ( F − Fp ) or in the deviation form
M (η + q p )ηɺɺ + M (η + q p ) − M (q p ) qɺɺp − S T (η + q p ) − S T ( q p ) Fp +
+ L(η + q p ,ηɺ + qɺ p ) − L(q p , qɺ p ) = S T (η + q p ) ( F − Fp )
(2.47)
Let us introduce the notation:
K (η ,ηɺ, t ) = M −1 (η + q p ) {L(η + q p ,ηɺ + qɺ p ) − L(q p , qɺ p ) + + M (η + q p ) − M (q p ) qɺɺp − S T (η + q p ) − S T (q p ) Fp } Then, the relation (2.47) can be written in a more compact form
ηɺɺ + K (η ,ηɺ, t ) = M −1 (η + q p ) S T (η + q p ) ( F − Fp )
(2.48)
Let us note that the continuous vector function K satisfies the property K (0, 0, t ) ≡ 0, ∀t ≥ t0 . Consider the system of differential equations:
ηɺɺ + K (η ,ηɺ , t ) = 0
(2.49)
Evidently, η (t ) ≡ 0 is the trivial solution of equation (2.49). In fact, this equation represents the environment dynamics equation (2.12) written in the form of deviations for F (t ) = F p (t ). It is clear that the environment dynamics should satisfy the property of asymptotic stability (desirable in the whole) of the trivial solution of the system (2.49) which, along with the stability of η (t ) , should ensure fulfillment of the limiting condition:
η (t ) → 0 as t → ∞
(2.50)
This means that in the most favorable case of robot control, when the real interaction force of the robot with environment, F (t ) , coincides all the time
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Dynamics and Robust Control of Robot-Environment Interaction
with PFI FP (t ) , the real motion q (t ) should be close to q p (t ) , and converge to it when t → ∞ . In the opposite case, F (t ) ≠ F p (t ) , the stabilization of the force of interaction of the robot with environment does not solve the contact task as a whole in a satisfactory way. Let us consider sufficient conditions (in general form) for the stabilization of PM q p (t ) under which asymptotic stability of the solutions of (2.49) entails the asymptotic stability of the perturbed motion η (t ) in (2.48) and the fulfillment of the limiting condition (2.50). For this purpose, assuming the vector function K is continuously differentiable, let us consider the first approximation of this function in the neighborhood of the point (η ,ηɺ ) = (0, 0) :
K (η ,ηɺ , t ) =
∂K (t )ηɺ ∂ηɺ
+
∂K (t )η ∂η
(η ,ηɺ ) = (0,0)
where
2
2
α 0 (η ,ηɺ , t ) = o ( η + ηɺ ) as
+ α 0 (η ,ηɺ , t )
(η ,ηɺ ) = (0,0)
η → 0, ηɺ → 0 . Because of
K (0, 0, t ) ≡ 0, ∀t ≥ t0 it is evident that α 0 (0, 0, t ) ≡ 0, ∀t ≥ t0 . Assuming that the vector function α 0 (η ,ηɺ, t ) is smooth, let us introduce the notation:
x x1 =η , x2 =ηɺ , x = 1 , x2 In On ∂K ∂K A(t ) = − − (t ) (t ) ∂ηɺ ∂η (η ,ηɺ ) = ( 0 , 0 ) (η ,ηɺ ) = ( 0 , 0 ) 0 0 α ( x, t ) = β x t = , ( , ) , −1 T α 0 ( x1 , x2 , t ) M ( x1 + q p (t ) ) S ( x1 + q p (t ) ) where A is a continuous 2n × 2n matrix function, α is a smooth 2n vector function, and β is a 2n × n matrix function. The system of differential equations (2.48) can be rewritten in the form:
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A Unified Approach to Dynamic Control of Robots
xɺ = A(t ) x + α ( x, t ) + β ( x, t ) µ (t )
(2.51)
where α ( x, t ) = o ( x ) as x → 0 , α (0, t ) ≡ 0, ∀t ≥ t0 and sup A(t ) < ∞ t
because of q p (t ) ∈ Vq , qɺ p (t ) ∈ Vqɺ , Fp (t ) ∈VF . Here, Vq , Vqɺ , VF are some known bounded sets. This system is the one with the perturbation β ( x, t ) µ (t ) for the system of differential equations (2.49) which, with the adopted notation, has the form:
xɺ = A(t ) x + α ( x, t )
(2.52)
Here, x(t ) ≡ 0 is the trivial solution of (2.52). Sufficient conditions for asymptotic stability of the system of differential equation (2.51), together with the fulfillment of the condition x(t ) → 0 as t → ∞ , are given by the following theorem. Theorem 1. Let the environment dynamics satisfy the following conditions: 1. The first approximation system:
xɺ = A(t ) x
(2.53)
is regular [13], i.e. there exists the limit t
2n 1 Sp A ( ω ) d ω = σ and σ = αk , ∑ 0 0 t →∞ t ∫ k =1 t0
lim
where α k ( k = 1, 2,… , 2n) are characteristic indicesb of the solutions of the system (2.53), and Sp A is the trace of matrix A ; 2. All characteristic indices α k ( k = 1, 2,… , 2n) are negative.
t ∈ [t0 , + ∞) then the number 1 (or the symbols +∞ or −∞ ) defined by the formula χ [ f ] = lim ln f (t ) is called x →∞ t Lyapunov’s characteristic index (shortly the characteristic index) of the function f .
b If
f
is the complex-valued function of the real variable
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Dynamics and Robust Control of Robot-Environment Interaction
Let equation (2.19) that define the quality of transient processes with respect to interaction force be such that the following estimate holds for an arbitrary solution µ (t ) :
µ (t ) ≤ C2e − λ (t −t ) µ (t0 ) 0
(2.54)
with the positive constant C2 and index λ > 0 . Let the number γ satisfy the inequalities:
max α k < −γ < 0 k
−λ < −γ
(2.55)
sufficiently small initial perturbations x(t0 ) and µ (t0 ) = F (t0 ) − Fp (t0 ) , the transient processes defined by the system of differential equations (2.51) with respect to motion and its derivative behave according to the inequality Then,
with
b µ (t0 ) − γ (t −t0 ) x(t ) ≤ a x(t0 ) + , ∀t ≥ t0 e λ −γ with the positive constants a and b which consequently fulfill the limiting condition:
x(t ) → 0 as t → ∞
(2.56)
which represents the exponential stability of the system (2.51) under the condition of small initial perturbations. The proof of Theorem 1 is given in Appendix A. In a sense, this theorem represents the Lyapunov’s generalized theorem on stability in the first approximation for a system with perturbations [13, 14]. In this way, with the conditions of the theorem being fulfilled, the control laws (2.39), (2.40), (2.43) ensure a desired quality of stabilization of PFI FP (t ) and also of PM q p (t ) , due to the fulfillment of the following conditions, comes out from (2.56):
η (t ) → 0, ηɺ (t ) → 0, as t → ∞ . Let us notice that conditions 1 and 2 of Theorem 1 define the property of the environment which might be called “internal stability”. Besides, the stabilization
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99
of PM in the case of using force stabilization control laws takes place only if the rate of stabilization of force interaction λ is higher than the guaranteed rate γ (inequality (2.55)) of stabilization of motion. Here, sufficient conditions for the exponential stability of the first approximation of the system (2.51) under condition of small perturbations are given by conditions 1 and 2. If these conditions are fulfilled, the system will be stable. It has to be emphasized that the fulfillment of the conditions of asymptotic stability depends not only on the environment dynamics but also on the programmed displacement (motion). For instance, let us adopt a nonlinear model of environment dynamics with the nonlinearity in the form of a cubic spring:
F (t ) = me ɺɺ x(t ) + be xɺ (t ) + k1 x(t ) + k2 x3 (t )
(2.57)
where me > 0 is a parameter representing the equivalent environment mass;
be > 0 is the corresponding environment damping; k1 ≥ 0 and k2 > 0 are the environment stiffness coefficients. According to equation (2.51), the deviation form of (2.57) becomes: 2
k + 3k2 x p (t ) b ηɺɺ(t ) = − e ηɺ (t ) − 1 η (t ) − me me
(2.58)
k 1 µ (t ) − 2 (η 3 (t ) + 3η 2 (t ) x p (t ) ) + me me It is obvious that
k2 3 (η (t ) + 3η 2 (t ) x p (t ) ) = o( η me
1+ε
) as η → 0 .
Here 0 < ε < 1 . For simplicity let us suppose that desired displacement x p (t ) = x p = const . According to equation (2.57), the desired force of interaction must be equal to Fp (t ) = Fp = k1 x p + k2 x3p . The characteristic polynomial of the linear part of the system (2.58) is given by 2
k1 + 3k2 x p b det( sI − A) = s + e s + me me 2
(2.59)
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Dynamics and Robust Control of Robot-Environment Interaction
where
0 A = k1 + 3k2 x 2p − me
1 be . − me
The eigenvalues of the characteristic equation corresponding to polynomial (2.59) are
s1,2 =
−be ± be2 − 4me (k1 + 3k2 x 2p ) 2me
.
Let the parameters be , me , k1 of the cubic spring meet the condition b < 4me k1 . Then, independently of desired displacement x p , characteristic indices of the linear part of the system (2.58) : α1 = Re s1 , α 2 = Re s2 will be negative. Therefore the second condition of Theorem 1 is fulfilled. Let us consider the first condition of Theorem 1. It is also fulfilled because: 2 e
t
t
∫
∫
0
0
b b 1 1 be lim Sp( A) dω = lim − dω = − e and α1 + α 2 = − e t→∞ t t→∞ t m me me e t t
(2.60)
Based on Theorem 1 we can conclude that any control law asymptotically stabilizing the desired force of interaction Fp for the cubic spring considered guarantees asymptotic stabilization of the corresponding desired displacement x p as well. 2.3.3 Concluding discussion We have focused our attention on the problem of position stabilization when asymptotic stability of the contact force was ensured. This task is a basic issue in the control of a robot interacting with its dynamic environment. The theorem ensuring asymptotically stable position of the system in the first approximation (local stability) formulates sufficient conditions under which such stability is achieved. It should be emphasized that the character of this position (displacement) stability depends particularly on the nature of programmed motion. Nevertheless, it should be pointed out that the presented linear analysis gives a very important criterion that must be verified for any force-based control
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101
law. However, if the environment model (2.11) is not known to a sufficiently accurate degree, it is not possible to determine the nominal contact force F p (t ) . Besides, an insufficiently accurate environment dynamics model can significantly affect the contact task execution. The inaccuracies of the robot and environment dynamics models, as well as dynamic control robustness, are considered in [6, 15-17]. The problems arising from the parametric uncertainties may also be resolved by applying the knowledge-based techniques (fuzzy logic and neural networks) [18]. Taking into account external perturbations, which do not expire with time, as well as the model and parameter uncertainties, it may be difficult to achieve asymptotic (exponential) stability of the system unless robust and adaptive control laws that include a factor for compensating these perturbations and uncertainties are used. Therefore, it may be of practical interest to require more relaxed stability conditions, i.e. to consider the so-called practical stability of the robot around the desired position and force trajectories by specifying the finite regions around them within which the robot’s actual position and force have to be during the task execution, and by assuming that the inaccuracies of the model parameters (of both the robot and environment) are bounded. The conditions for practical stability of the robot interacting with dynamic environment enable study of the model uncertainty issue in control of robots in this class of tasks without any approximation, i.e. to correctly examine the influence of these uncertainties upon different control laws. The test conditions for practical stability of the robot interacting with dynamic environment have been derived in [19, 20]. A wider discussion of stability of the robot interacting with dynamic environment is given in Section 2.10. 2.4 Analysis of Transient Processes In this section we give the estimates of transient processes with respect to both the robot’s motion and its force of interaction with the environment. Here we shall formulate sufficient conditions for the fulfillment of the constraints imposed on the state, control, and interaction force, taking into account the inadequacy of the dynamics models of the robot and environment and/or external perturbations.
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Dynamics and Robust Control of Robot-Environment Interaction
2.4.1 Task setting Dynamics model of the robot interacting with the environment is described by the vector differential equation (2.6). This equation can be written in the form (2.8). Let the real robot dynamics be written in the form of the vector differential equation:
qɺɺ = Φ ( q, qɺ ,τ , F ) + r (t )
(2.61)
where the n -dimensional continuous vector function r (t ) represents the inaccuracy of robot’s dynamics description and/or the uncontrollable external perturbations. We shall assume that the vector function Φ satisfies conditions sufficient for the existence and uniqueness of the solution for equations (2.8) and (2.61) with the initial conditions q (t0 ) = q0 , qɺ (t0 ) = qɺ0 . Let the dynamics model of the environment with which robot is interacting be described by the nonlinear differential equation (2.11). Let the real environment dynamics be described by the vector differential equation:
M (q)qɺɺ + L(q, qɺ ) = S T (q) [ F − ρ (t ) ]
(2.62)
where the n -dimensional vector function ρ (t ) represents the inaccuracy of describing environment dynamics by the mathematical model (2.11) and/or the uncontrollable external perturbations of the environment. The real environment can be represented by the equation:
F = f ( q, qɺ , qɺɺ) + ρ (t )
(2.63)
The goal of the control of the robot in contact with its environment is to realize the programmed motion ( PM ) q p (t ) in the presence of the programmed force acting on the environment. Namely,
Fp (t ) = f ( q p (t ), qɺ p (t ), qɺɺp (t ) ) Since
some
initial
perturbations
are always present, i.e. q (t0 ) ≠ q p (t0 ), qɺ (t0 ) ≠ qɺ p (t0 ), F (t0 ) ≠ Fp (t0 ), the control goal is transformed into the task of stabilizing PM q p (t ) :
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103
q (t ) → q p (t ), qɺ (t ) → qɺ p (t ), as t → ∞ and the task of stabilizing PFI Fp (t ) :
F (t ) → F p (t ), as t → ∞ . In Section 2.3, these two tasks are solved for the case of the absence of the model’s inaccuracy and external perturbations r (t ) and ρ (t ) . Because of the uncontrollable behavior of the introduced functions r (t ) and ρ (t ) , the control goal will be formulated in the form of the following two goal conditions: 2
2
q (t ) − q p (t ) + qɺ (t ) − qɺ p (t ) < ε 2
(2.64)
F (t ) − Fp (t ) < δ
(2.65)
where ε , δ > 0 are the given numbers. These goal conditions should be fulfilled for any t ≥ t p , where t p is a time instant, t p − t0 is the time interval of the transient process. Let us note that the goal condition (2.64) is equivalent to the inequality:
x(t ) − x p (t ) < ε ,
(
T
)
(
T
)
where x(t ) = qT (t ), qɺ T (t ) , x p (t ) = qTp (t ), qɺ Tp (t ) . Here, and in the text to follow, the norm ⋅ denotes the Euclidean norm. The robot’s motion q (t ), qɺ (t ), the control action τ (t ) , and the force of interaction of the robot with environment F (t ) , are usually constrained by the conditions of the technological task to be performed. Let us preset these constraints in the form of the following relations:
q(t ) ∈ Vq ⊂ R n , ∀t ≥ t0 qɺ (t ) ∈ Vqɺ ⊂ R n , ∀t ≥ t0 n
τ (t ) ∈ Vτ ⊂ R , ∀t ≥ t0 n
F (t ) ∈ VF ⊂ R , ∀t ≥ t0
(2.66) (2.67) (2.68) (2.69)
where Vq , Vqɺ , Vτ , V F are the prescribed, open, bounded, and simply connected sets in the corresponding spaces, and Vτ is the closure of the set Vτ in R n . We will assume that the levels of inaccuracy of the robot and environment dynamics models and/or external perturbations are bounded, i.e.
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Dynamics and Robust Control of Robot-Environment Interaction
r (t ) ≤ Cr , ∀t ≥ t0 ,
(2.70)
ρ (t ) ≤ Cρ , ∀t ≥ t0
(2.71)
Under these conditions, the control task can be formulated in the following way. For the complete equations of the robot model (2.61) and the external environment (2.62), including the disturbances r (t ) and ρ (t ) satisfying the constraints (2.70) and (2.71), the admissible control action τ (t ), t ≥ t0 , i.e. the action satisfying the constraint (2.68), should be synthesized in such a manner that the real motion q (t ), qɺ (t ) satisfies the constraints (2.66) and (2.67), the generated interaction force satisfies the constraint (2.69), and the goal conditions (2.64), (2.65) are fulfilled starting from the time instant t p . 2.4.2 Motion transient processes In Section 2.3.1, two control laws were synthesized that stabilize robot’s motion in accordance with the given quality of transient processes (2.16), (2.17). The first control law (2.23) has a more general character, ensuring the solution of the motion stabilization task q p (t ) not only when the robot is in contact with the environment but also in a free space. The second control law, (2.28), is applied only when the robot is in contact with the environment. The control law (2.28), in contrast to (2.23), does not use force feedback. There is a constraint on the applicability of this control law, i.e. the condition (2.33) has to be satisfied for all possible configurations {q} of the robot in contact with the environment. Both control laws considered ensure also a simultaneous stabilization of F p (t ) , but the quality of this stabilization cannot be set a priori in an arbitrary way. In this section the classes of the programmed motions q p (t ) and
{
programmed interaction forces
{F (t )} p
}
for which the control laws (2.23) and
(2.28) solve the control tasks (2.64) and (2.65) in the presence of the constraints (2.66) – (2.69), will be described. Stability of these control laws under external perturbations of the robot’s and environment dynamics, or subject to their
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A Unified Approach to Dynamic Control of Robots
inaccuracies, will be demonstrated, and the estimate of transient process time in the case of motion and force stabilization will be presented. Let us consider the task of robot control subject to the constraints (2.66) – (2.69). The PM q p (t ) and PFI Fp (t ) have to satisfy the constraints (2.66), (2.67) and (2.69), i.e. the conditions:
q p (t ) ∈ Vq , ∀t ≥ t0
(2.72)
qɺ p (t )∈ Vqɺ , ∀t ≥ t0
(2.73)
Fp (t )∈ VF , ∀t ≥ t0
(2.74)
must be fulfilled. For the existence of the control task solution it is necessary to demand the control:
τ p (t ) = U ( q p (t ), qɺ p (t ), qɺɺp (t ), Fp (t ) )
(2.75)
be admissible. Let us suppose that there exists such a bounded open set Vqɺɺ in R n , that:
U : Vq × Vqɺ × Vqɺɺ × V F → Vτ
(2.76)
Then, if PM q p (t ) satisfies the constraints (2.72), (2.73) and the condition:
qɺɺp (t ) ∈ Vqɺɺ ⊂ R n , ∀t ≥ t0 and PFI
(2.77)
Fp (t ) satisfies (2.74), then, in accordance with (2.76), the control
τ p (t ) = U ( q p , qɺ p , qɺɺp , Fp ) will be admissible for any t ≥ t0 . Consequently, the existence of the set Vqɺɺ that ensures fulfillment of the relation (2.76) is a natural sufficient condition for the existence of the solution of the task of robot control in contact with environment in the presence of the constraints (2.66) – (2.69). In view of the fact that some initial perturbations are always present, the exact realization of PM q p (t ) and PFI Fp (t ) is not possible. Hence, we shall require PM q p (t )
together with its first two derivatives qɺ p (t ) and
qɺɺp (t ), to be distant from the bounds of the corresponding sets Vq , Vqɺ , Vqɺɺ for some “reserves” δ1 , δ 2 , δ 3 > 0 ; i.e.
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Dynamics and Robust Control of Robot-Environment Interaction
q p (t )∈ Vqδ1 ⊂ Vq , ∀t ≥ t0
(2.78)
qɺ p (t ) ∈ Vqɺδ 2 ⊂ Vqɺ , ∀t ≥ t0
(2.79)
qɺɺp (t ) ∈ Vqɺɺδ3 ⊂ Vqɺɺ ,
(2.80)
∀t ≥ t0
and, PFI Fp (t ) be distant from the bound of the set V F for a “reserve” σ > 0 , i.e.:
Fp (t ) ∈ VFσ ⊂ VF , ∀t ≥ t0
(2.81)
Here, Vqδ1 ,Vqɺδ 2 , Vqɺɺδ 3 ,VFσ respectively, are the δ1 , δ 2 , δ 3 and σ narrowings of the regions Vq ,Vqɺ , Vqɺɺ , VF . The ε -narrowing of the open set A in R n is defined as a such non-empty subset Aε that:
A = ∪ Bε ( x) x∈ Aε
where Bε (x) is an open solid sphere in R n with the center at the point x and the radius ε . For instance, if A = Bc ( x0 ) is an open solid sphere in R n , with the center at the point x0 and the radius c > ε , then: Aε = Bc −ε ( x0 ). Let the deviations of the real robot motion q (t ) and its derivatives qɺ (t ) and
qɺɺ(t ) from q p (t ), qɺ p (t ), qɺɺp (t ) , in the sense of Euclidean norm, be not greater than δ1 , δ 2 , δ 3 respectively, and let the deviation of the real interaction force
F (t ) of the robot in contact with the environment from Fp (t ) be not greater than σ . Then, the real robot motion and the real interaction force will satisfy the constraints (2.66), (2.67), (2.77) and (2.69). Let us consider the control laws (2.23) and (2.28) that stabilize PM q p (t ). We will suppose that the function P , determining the family of transient processes (2.16), is of the form (2.17):
P(η ,ηɺ ) = Γ1ηɺ + Γ2η As the real robot’s dynamics is described by (2.61) and the real environment’s dynamics by (2.62), the closed-loop system functioning with the control laws (2.23) and (2.28) applied, in contrast to (2.17), will have the form:
ηɺɺ = Γ1ηɺ + Γ2η + i (t )
(2.82)
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A Unified Approach to Dynamic Control of Robots
where i (t ) = r (t ) in the case when the control law (2.23) is applied, and
i (t ) = D1 ( q (t ) ) r (t ) + D2 ( q (t ) ) ρ (t ) in the case of applying the control law (2.28). Here −1
D1 (q ) = H (q ) − J T (q ) ( S T (q )) −1 M (q ) H (q ) −1
D2 (q ) = H (q ) − J T (q ) ( S T (q )) −1 M (q ) J T (q ) Let all eigenvalues λ1 , λ2 ,..., λ2 n of the matrix
0 Γ = n Γ 2
In Γ1
be real, different and negative. Let us estimate the solution of the system (2.82). For that purpose let us rewrite it in the form of a 2n -dimensional system of differential equations:
xɺ = Γx + iɶ ( t ) η
(2.83)
0 . Then, the solution (2.83) can be written in the i(t )
where x = , iɶ (t ) = ɺ
η
following form: t
x(t ) = eΓ (t −t0 ) x(t0 ) + ∫ eΓ (t −ω ) iɶ (ω )d ω
(2.84)
t0
Let T be a non-singular transformation that transforms the matrix Γ into its diagonal form:
T −1Γ T = diag(λ1 , λ2 ,..., λ2 n ) Then
(
)
eΓ (t −ω ) = T diag eλ1 ( t −ω ) , eλ2 (t −ω ) ,..., eλ2 n (t −ω ) T −1 and consequently: 1/ 2
e
Γ ( t −ω )
≤ T T
−1
2 n 2 λi ( t −ω ) ∑e i =1
≤ T T −1
2n max eλi ( t −ω ) = CΓ e − λ (t −ω ) i
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Dynamics and Robust Control of Robot-Environment Interaction
where
C Γ = 2n T T −1 , λ = − max λi . i
By using the obtained inequality, let us estimate by norm the right-hand side of equation (2.84), taking into account that iɶ = i , and
i(t ) ≤ Ci ,
∀t ≥ t0 ,
where Ci = Cr , if the control law (2.23) is used, and:
Ci = d1Cr + d 2Cρ if the control law (2.28) is applied. Here d1 = sup D1 (q) , d 2 = sup D2 (q ) where the suprema are taken over all q ∈ Vq . Let us note that Cr and C ρ are the estimates of perturbation levels given by (2.70) and (2.71). We shall, then, have: t
x(t ) ≤ e
Γ ( t − t0 )
x(t0 ) + ∫ eΓ (t −ω ) i(ω ) d ω ≤ CΓ e− λ (t −t0 ) x(t0 ) + t0
(2.85)
t
+ CΓ Ci ∫ e
− λ ( t −ω )
d ω ≤ CΓ e
− λ ( t −t0 )
x(t0 ) +
CΓ Ci
λ
t0
This gives: 2
2
2
2
η (t ) ≤ CΓ η (t0 ) + ηɺ (t0 ) e− λ (t −t ) + CΓCiλ −1 0
ηɺ (t ) ≤ CΓ η (t0 ) + ηɺ (t0 ) e− λ (t −t ) + CΓCiλ −1 0
(2.86) (2.87)
Let us estimate now the transient processes in the case of stabilizing the
PFI Fp (t ). As the real environment dynamics is described by (2.62) and the functioning of the closed-loop control system proceeds according to (2.82), the difference between the real and programmed interaction force
µ (t ) = F (t ) − F p (t ) satisfies the relation
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A Unified Approach to Dynamic Control of Robots
µ (t ) = f ( q, qɺ , qɺɺp + Γ1ηɺ + Γ 2η + i(t ) ) + ρ (t ) − f ( q p , qɺ p , qɺɺp )
(2.88)
We will assume that the function f , describing the environment dynamics, satisfies the Lipschitz conditions with the constants Li (i = 1, 2,3) for any i -th variable on the set of Vq × Vqɺ × Vqɺɺ . Then, from (2.88) we will have
µ (t ) ≤ L1 η + L2 ηɺ + L3 Γ1ηɺ + Γ 2η +i(t ) + ρ (t ) ≤ ≤ ( L1 + L3 Γ 2
) η +(L
2
+ L3 Γ1
) ηɺ
+ L3Ci + Cρ
By using the estimates (2.86) and (2.87), from the last inequality we obtain: 2
2
µ (t ) ≤ L CΓ η (t0 ) + ηɺ (t0 ) e− λ (t −t ) + L CΓCi λ −1 + L3Ci + Cρ where
0
(2.89)
L = L1 + L2 + L3 ( Γ1 + Γ2 ).
The obtained estimates of transient processes (2.86), (2.87), (2.89) show that sufficiently small inaccuracy levels of the robot and environment models, and/or of the external perturbations, C r and C ρ , after a certain time of transient processes, ensure fulfillment of the goal conditions (2.64) and (2.65). If, in addition, the initial perturbations are sufficiently small, the a priori constraints (2.66) – (2.69) can be fulfilled, too. More precisely, the following theorem holds.
Theorem 2. Let PM q p (t ), together with its derivatives qɺ p (t ) and qɺɺ p (t ) , satisfy the inclusions (2.78) – (2.80), and let PFI FP (t ) satisfy (2.81). If the initial perturbations η (t0 ), ηɺ (t0 ) and the inaccuracy levels of the models and/or external perturbations C r and C ρ satisfy the inequalities: 2
2
CΓ η (t0 ) + ηɺ (t0 ) + CΓ Ci λ −1 < min {δ1 , δ 2 }
(Γ
1
+ Γ2
)
(C 2
2
Γ
2
)
η (t0 ) + ηɺ (t0 ) + CΓCi λ −1 < δ 3 , 2
(2.90) (2.91)
LCΓ η (t0 ) + ηɺ (t0 ) + LCΓ Ci λ −1 + L3Ci + Cρ < σ ,
(2.92)
CΓ Ci λ −1 < ε ,
(2.93)
−1
Ci ( LCΓ λ + L3 ) + Cρ < δ ,
(2.94)
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Dynamics and Robust Control of Robot-Environment Interaction
then, for any t ≥ t0 and any control law (2.23) or (2.28): 1. the real robot’s motion q (t ), qɺ (t ) will satisfy the constraints (2.66) and (2.67); 2. the control law will be admissible, i.e. it will satisfy the constraint (2.68); 3. the real interaction force F (t ) will satisfy the constraint (2.69). Besides, the goal condition (2.64) will be realized not later than: 2 2 C η (t0 ) + ηɺ (t0 ) t + 1 ln Γ , if CΓ tp1 = 0 λ ε − CΓCi λ −1 t0 , if not and the goal condition (2.65) not later than:
2
η (t0 ) + ηɺ (t0 )
2
+ CΓCi λ −1 ≥ ε
2 2 LCΓ η (t0 ) + ηɺ (t0 ) 1 t + ln , 0 λ δ − Ci ( LCΓ λ −1 + L3 ) − Cρ 2 2 tp2 = if LCΓ η (t0 ) + ηɺ (t0 ) + Ci ( LCΓ λ −1 + L3 ) + Cρ ≥ δ t0 , if not
(2.95)
(2.96)
Consequently, the both conditions (2.64) and (2.65) will be fulfilled not later
{
than: tp = max tp1 , tp2
}.
The proof of Theorem 2 is given in Appendix B. Let us note that in the conditions of Theorem 2, the initial force perturbation µ (t0 ) = F (t0 ) − Fp (t0 ) is not present. This is not unexpected because, in view of the relation (2.88), which is fulfilled when using the control laws (2.23) and (2.28), the initial perturbation µ (t0 ) is uniquely determined by the initial perturbations η (t0 ) and ηɺ (t0 ) .
2.4.3 Force transient processes Let us consider the control laws stabilizing the robot-environment interaction force with the preset quality of transient processes defined by the relation [10, 11, 16]:
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A Unified Approach to Dynamic Control of Robots t
µ (t ) = ∫ Q ( µ (ω ) ) d ω
(2.97)
t0
where µ (t ) = F (t ) − F p (t ) , Q is a vector function characterizing quality of the transient processes. Due to equation (2.97), a differential equation of the form (2.19) will hold:
µɺ (t ) = Q ( µ (t ) ) Usually, the choice of the function Q , which defines the character of the transient processes, relies upon the relation (2.19), and (2.97) is considered as the reference equation in the system of control laws. Of the three control laws, (2.36), (2.41) and (2.43), considered in [10, 16], the first two: t τ = U q, qɺ ,ψ q, qɺ , Fp + ∫ Q ( µ (ω ) ) dω , F t0 t τ = U q, qɺ ,ψ ( q, qɺ , F ) , Fp + ∫ Q ( µ (ω ) ) d ω t0 utilize the feedback with respect to q, qɺ and force F , and they are not limited
by the supplementary conditions of their applicability. At the same time, the third control law, (2.43):
t
t
t0
τ = U q, qɺ ,ψ q, qɺ , Fp + ∫ Q ( µ (ω ) ) dω , Fp + ∫ Q ( µ (ω ) ) dω
t0
has a constraint on its applicability in the form of the supplementary condition (2.46):
rank S T (q ) − M (q ) H −1 (q ) J T (q ) = n which should be fulfilled for all possible configurations {q} that appear in its contact with the environment. The control laws (2.36), (2.41), (2.43) ensure a desired quality of realization of the robot-environment interaction force F p (t ) . At the same time, stabilization of the robot’s motion q p (t ) is also achieved, but the character of that motion cannot be prescribed in advance in an arbitrary way.
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Dynamics and Robust Control of Robot-Environment Interaction
{
}
In this section we will describe the class of programmed motions q p (t ) , the class of programmed interaction forces F p (t ) and the class of environment’s dynamics equations for which the control laws (2.36), (2.41), (2.43) solve the control task (2.64), (2.65) under the constraints (2.66) – (2.69). We will demonstrate the stability of these control laws in the presence of external perturbations of the robot and environment dynamics, as well as in respect to the models inaccuracies in any finite time interval, giving also an estimation of the time of transient processes in the stabilization of the motion and force. Let us consider the task of robot control under the constraints (2.66) – (2.69). Let PM q p (t ) and PFI Fp (t ) satisfy the conditions (2.78) – (2.81), and let the set Vqɺɺ ⊂ R n be defined by the relation (2.76). It is easy to prove (and this will be done in Theorem 4) that, if the control laws (2.36), (2.41), (2.43) ensure the necessary closeness of q (t ) to q p (t ) and F (t ) to F p (t ) for any t ≥ t0 , they are admissible and will ensure fulfillment of the constraints (2.66), (2.67) and (2.69). Hence, the basic issue in using these control laws is the influence of the robot and environment models inaccuracies and/or external perturbations r (t ) and ρ (t ) on the transient processes. For simplicity, we will assume that the function Q , determining the family of transient processes in (2.19), has the form:
{
}
Q ( µ ) = Rµ where R is a constant n × n matrix. Since the real robot’s dynamics is described by (2.61) and the real environment dynamics by (2.62), the functioning of the control laws (2.36), (2.41), (2.43), in contrast to (2.97), will correspond to: t
µ (t ) = R ∫ µ (ω )dω + D1 ( q(t ) ) r (t ) + D2 ( q(t ) ) ρ (t )
(2.98)
t0
where the n × n matrix D1 and the n × n matrix D2 are of the form: −1
D1 (q ) = ( S T (q ) ) M (q ) D2 (q ) = I n in the case of the control law (2.36),
(2.99)
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A Unified Approach to Dynamic Control of Robots −1
D1 (q ) = − ( J T (q ) ) H (q)
(2.100)
D2 (q) = D1 (q) M −1 (q ) S T (q) in the case of using the control law (2.41), and −1
D1 (q ) = ( D(q) ) M (q)
(2.101)
−1
D2 (q) = ( D(q) ) S T (q) in the case of using the control law (2.43), where:
D ( q ) = S T ( q ) − M ( q ) H −1 ( q ) J T ( q ) . −1
Let us note that, due to the condition (2.46), the inverse matrix ( D ( q ) ) exists. Assume the vector functions r and ρ are differentiable for t ≥ t0 . If we introduce the vector function:
g (t ) = D1 ( q(t ) ) r (t ) + D2 ( q (t ) ) ρ (t ) then, equation (2.98) can be written in the form: t
µ (t ) = R ∫ µ (ω ) dω + g (t ) t0
By differentiating this expression with respect to t , we obtain the equation:
µɺ = R µ (t ) + gɺ (t ) Its solution can be written in the form t
µ (t ) = e R (t −t ) µ (t0 ) + ∫ e R (t −ω ) gɺ (ω )d ω 0
t0
Integrating the expression e R ( t −ω ) gɺ (ω ) by parts, we obtain t
µ (t ) = e
R ( t − t0 )
( µ (t0 ) − g (t0 ) ) + g (t ) + R ∫ e R (t −ω ) g (ω )dω
(2.102)
t0
Let on the set Vq the vector functions D1 (q ) and D2 (q ) be bounded:
Di (q) ≤ di , i = 1,2
(2.103)
The inequalities (2.103) give the estimate of the vector function norm g (t ) :
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Dynamics and Robust Control of Robot-Environment Interaction
g (t ) ≤ d1Cr + d 2Cρ
(2.104)
Let all eigenvalues λ1 , λ 2 ,..., λ n of the matrix R be real, different and negative, and let λ = − max λi . Based on the inequality (2.104), analogously to i the estimation of x(t ) for equation (2.84), we obtain the estimate of the transient processes with respect to force:
µ (t ) ≤ CR e− λ (t −t ) ( µ (t0 ) + g (t0 ) ) + g (t ) + 0
(2.105)
+ R CR λ −1 ( d1Cr + d 2Cρ ) ≤ CR e − λ (t −t0 ) µ (t0 ) + Cµ where
CR = 2n R R −1 , Cµ = (1 + CR + R CR λ −1 ) ( d1Cr + d 2Cρ ) .
Let us estimate now the transient process η (t ) = q (t ) − q p (t ) when using the control laws (2.36), (2.41), (2.43), which stabilize PFI Fp (t ) . Since the real environment dynamics is described by (2.62), and PM q p (t ) and PFI Fp (t ) satisfy the constraint equation (2.14), or, which is the same, the equation:
M (q p )qɺɺp + L(q p , qɺ p ) = S T (q p ) F p then, by subtracting it from (2.62), we obtain:
ηɺɺ + K (η ,ηɺ , t ) = M −1 (η + q p ) S T (η + q p ) F − Fp − ρ (t ) where:
K (η ,ηɺ, t ) = M −1 (η + q p ) {L(η + q p ,ηɺ + qɺ p ) − L(q p , qɺ p ) + + M (η + q p ) − M ( q p ) qɺɺp − S T (η + q p ) − S T ( q p ) Fp
}
We are not going to repeat here the procedure of obtaining the system in its first approximation for the vector function K in order to examine asymptotic stability of the perturbed motion η (t ) in (2.48) for the case of an ideal model of the environment with which the robot is in dynamic interaction. In contrast to that case, in this case we deal with the real environment, so that the perturbation equation in its first approximation differs from the previous case (2.51) only by one perturbation term β ( x, t ) ρ (t ) , and it has the form:
xɺ = A(t ) x + α ( x, t ) + β ( x, t ) µ (t ) − β ( x, t ) ρ (t )
(2.106)
To estimate the transient process x (t ), let us consider the behavior of the solution of equation (2.106) in an arbitrary finite time interval [t0 , T ] on the set:
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A Unified Approach to Dynamic Control of Robots
Ω = {( x, t ) ∈ R 2 n × R / x < h, t0 ≤ t ≤ T } where
h = min{δ 1 , δ 2 }
(2.107)
and the values δ 1 and δ 2 determine the measure of the closeness of the real motion q (t ), qɺ (t ) to the programmed one q p (t ), qɺ p (t ) in (2.78) and (2.79). Because α ( x, t ) = o ( x ) at x → 0 , α (0, t ) ≡ 0, ∀t ≥ t0 and due to smoothness of the vector function α ( x, t ) , a positive constant C1 and constant p > 1 can be found, such that: p
α ( x, t ) ≤ C1 x , ∀( x, t ) ∈ Ω
(2.108)
Besides, due to the continuity of the function β ( x, t ) , a positive constant M can be found, such that:
β ( x, t ) ≤ M , ∀( x, t ) ∈ Ω
(2.109)
Let us assume that the system of first approximation of equation (2.106):
xɺ = A(t ) x
(2.110)
is regular, and its characteristic indices α 1 ,..., α 2 n are negative. Let us carry out transformation of the system (2.106):
x = y e −γ (t −t0 )
(2.111)
where the value γ satisfies the inequality:
max α k < −γ < 0 k
Then, we will have:
yɺ = B (t ) y + αɶ ( y, t ) + βɶ ( y, t ) µ (t ) − βɶ ( y, t ) ρ (t ) where:
B (t ) = A(t ) + γ I 2 n ,
αɶ ( y, t ) = eγ (t −t ) α ( y e −γ (t −t ) , t ) , 0
0
βɶ ( y, t ) = eγ ( t −t ) β ( y e −γ (t −t ) , t ) . 0
Evidently, x (t0 ) = y (t0 ).
0
(2.112)
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Dynamics and Robust Control of Robot-Environment Interaction
In Appendix A (proof of Theorem 1) we show that the first approximation system of (2.112):
yɺ = B (t ) y
(2.113)
is also regular, and its characteristic indices are negative. Let us replace (2.112) by the equivalent integral equation: t
y (t ) = H (t ) y (t0 ) + ∫ K (t , ω ) αɶ ( y (ω ), ω ) + βɶ ( y (ω ), ω ) µ (ω ) − t0
(2.114)
− βɶ ( y (ω ), ω ) ρ (ω )] d ω where H (t ) is the normed (i.e. H (t0 ) = I 2 n ) fundamental matrix of the system (2.113), and K (t , ω ) = H (t ) H −1 (ω ) is the Cauchy matrix. Since all characteristic indices of (2.113) are negative, the following estimate of its fundamental matrix H (t ) at any t ≥ t0 is valid:
H (t ) ≤ C 2 (C 2 ≥ 1)
(2.115)
By taking into account the regularity of the system (2.113) it can be stated that a constant C 3 can be found, such that:
K (t , ω ) ≤ C 3
(2.116)
t0 ≤ ω < t ≤ T . Then, the following theorem on estimation of the transient processes norm in (2.106) is justified.
at
Theorem 3. Let the environment dynamics be such that (a) the first approximation system (2.110) is regular; (b) all characteristic indices α k ( k = 1, 2,..., 2n) in (2.110) are negative. Let the index λ of stabilizing PFI Fp (t ) in the estimate (2.105) satisfy the inequality:
−λ < −γ Then, if the initial perturbation levels
x(t0 ) , µ (t0 )
and the levels of
inaccuracy of the models and/or the external perturbations C r and C ρ satisfy the inequality:
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A Unified Approach to Dynamic Control of Robots
C2 x(t0 ) +
≤2
−
1 p −1
MCRC3 MC3 Cµ + Cρ ) eγ (T −t0 ) − 1 ≤ µ (t0 ) + ( λ −γ γ
(
)
1 γ p −1 min , h C1C3
(2.117)
the following estimation of transient processes in the time interval [t0 , T ] is valid:
b µ (t0 ) − γ (t −t0 ) + C eγ (T −t0 ) − 1 x(t ) ≤ a x(t0 ) + e λ −γ
(
1
where:
1
)
(2.118)
1
a = 2 p −1 C2 , b = 2 p −1 MCR C3 , C = 2 p −1 MC3γ −1 ( Cµ + Cρ ) .
From the obtained estimation of transient processes (2.118) on a finite time interval [t0 , T ] , it follows that for any t ∈ [t0 , T ] holds:
max { η (t ) , ηɺ (t )
}≤
b µ (t0 ) − γ (t −t0 ) 2 2 ≤ a η (t0 ) + ηɺ (t0 ) + + C eγ (T −t0 ) − 1 e λ −γ
(
)
(2.119)
The proof of Theorem 3 is given in Appendix C. Let us show that in the case when the conditions of Theorem 3 are fulfilled one can choose sufficiently small levels of inaccuracy of the robot and environment dynamics models and/or external perturbations C r , C ρ , as well as of the initial perturbations with respect to position and force so as the fulfillment of the goals (2.64) and (2.65) after some time instant smaller than T − t0 will be guaranteed, as well as the fulfillment of the a priori constraints (2.66) – (2.69) for all t ∈ [t0 , T ] .
Theorem 4. Let (a) the function ψ , describing the environment dynamics, satisfy the Lipschitz conditions with respect to any of the i -th variables with the constants Li (i = 1, 2,3) ;
ɺɺ p (t ) , satisfy the (b) PM q p (t ), together with its derivatives qɺ p (t ) and q inclusions (2.78), (2.79), (2.80), and PFI Fp (t ) the inclusion (2.81);
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Dynamics and Robust Control of Robot-Environment Interaction
(c) the conditions of Theorem 3 be fulfilled. If the levels of initial perturbation η (t0 ) , ηɺ (t0 ) , µ (t0 ) and the levels of inaccuracy of the models and/or external perturbations C r and C ρ , in addition to the inequality (2.117), satisfy also the inequalities:
C R µ (t 0 ) + C µ < d1Cr + d 2Cρ <
σ 2
σ 2
,
(2.120)
,
(2.121)
Cµ < δ ,
(2.122)
(
)
C eγ (T −t0 ) − 1 < ε , 1 a max ln γ
(2.123) 2
2
η (t0 ) + ηɺ (t0 ) + b µ (t0 ) (λ − γ ) −1 , ε − C (eγ (T −t ) −1) 0
(2.124)
1 CR µ (t0 ) ln < T − t0 λ δ − Cµ and the values σ and δ 3 of the narrowing of the sets VF and Vqɺɺ are connected by the inequality: a
2
2
η (t0 ) + ηɺ (t0 ) +
b µ (t0 )
λ −γ
+ C eγ (T − t0 ) − 1 ( L1 + L2 ) + L3σ < δ 3
(
)
(2.125)
then, for all t ∈ [t0 , T ] , and for each of the control laws (2.39), (2.40), (2.43): 1) the real robot motion q (t ), qɺ (t ) will satisfy the constraints (2.66) and (2.67); 2) the control τ (t ) will be admissible, i.e. it will satisfy the constraint (2.68); 3) the real interaction force F (t ) will satisfy the constraint (2.69). Besides, the goal condition (2.64) will be fulfilled not later than the time instant:
A Unified Approach to Dynamic Control of Robots 2 2 a η (t0 ) + ηɺ (t0 ) + b µ (t0 ) (λ − γ ) −1 1 t + ln , 0 γ ε − C eγ (T −t0 ) − 1 tp1 = if a η (t ) 2 + ηɺ (t ) 2 + b µ (t0 ) + C eγ (T −t0 ) − 1 ≥ ε , 0 0 λ −γ t0 , if not
(
119
)
(
)
(2.126)
and the goal condition (2.65) will be fulfilled not later than:
1 CR µ (t0 ) , t0 + ln λ δ − Cµ tp2 = t , if not 0
if CR µ (t ) + Cµ ≥ δ
(2.127)
Consequently, both goal conditions (2.64) and (2.65) will be fulfilled not later than:
{
}
tp = max tp1 , tp2 < T
(2.128)
The proof of Theorem 4 is given in Appendix D. The estimates obtained in Theorems 2-4, connecting the levels of initial perturbation and the levels of external perturbation and/or the inadequacy of the models with the magnitudes of “reserves” δ1 , δ 2 , δ 3 , σ , indicating the distances ɺɺp (t ) and PFI Fp (t ) from the bounds of the sets of PM q p (t ), qɺ p (t ), q Vq , Vqɺ , Vqɺɺ V F enable us to make a practical conclusion. The narrower the classes of the stabilized PM { q p (t ) } and PFI Fp (t ) are (i.e. the greater are the values δ 1 , δ 2 , δ 3 , σ ), the less bounded are the conditions of the “smallness” of the initial and external perturbation levels.
{
}
2.4.4 Numerical example To illustrate the obtained theoretical results, let us consider the following hypothetical contact task: A 2-DOF sliding joint manipulator has to move the workpiece over a support that behaves as a system with distributed parameters. Using the relation explained in Fig. 2.2, the robot dynamics model is:
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Dynamics and Robust Control of Robot-Environment Interaction
Fig. 2.2. Robot in contact with dynamic environment
(m1 + m2 ) ɺxɺ = τ x − Fx
(2.129)
m2 ɺyɺ = τ y − Fy
(2.130)
The contact force component Fy is a sum of the inertial, friction and elastic terms:
Fy = m y ɺyɺ + h y yɺ + K y y
(2.131)
while the component of contact force along the x -axis, Fx , is the sum of the inertial and friction terms
Fx = mx ɺɺ x + hx xɺ +ν Fy sgn xɺ
(2.132)
It is adopted in (2.131) and (2.132) that m x = m , where m is the workpiece mass and m y = m + me . Here me is the equivalent mass representing the contribution of the environment to the inertia terms. Further, hy , hx , K y ,ν denote viscous friction coefficients, environment stiffness, and static friction coefficient, respectively. The control goal is the realization of the programmed (nominal) motion along the x -axis
x p (t ) = x 0 (t ) = V0 t , V0 = const > 0 and of programmed (nominal) interaction force along the y -axis
(2.133)
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A Unified Approach to Dynamic Control of Robots
Fy p (t ) = Fy0 = F 0 = const The complement force and motion components are obtained from (2.131), (2.132)
Fx p (t ) = Fx0 (t ) = hx V0 +ν F 0 ,
y p (t ) = y 0 (t ) =
1 0 F Ky
(2.134)
In order to stabilize nominal motion and force, the control law (2.23) is applied with closed-loop dynamics equation (2.24), and therefore (2.16) is expressed in the form:
ηɺɺx + 2ς xcω xcηɺ x + ω xc2 η x = 0
(2.135)
ηɺɺy + 2ς ycω ycηɺ y + ω yc2 η y = 0
(2.136)
It is obvious that unless the models of robot and environment have some uncertainties, (2.135), (2.136) represent the closed-loop system behavior. In order to study the influence of the inaccuracy of the robot and environment dynamics the following functions
rx 0 sin (Ω x t ) r (t ) = ry 0 sin (Ω y t )
ρ x 0 cos (Ω x t ) ρ y 0 cos (Ω y t )
ρ (t ) =
are introduced into the robot and environment equations (see (2.61), (2.62)). The bounds on the motion and force deviations from their nominal values in the presence of the inaccuracies r (t ) and ρ (t ) are given by (2.85), (2.89). For some parameter values the tracking of the nominal motion and force is shown in Fig. 2.3, while Fig. 2.4 shows the bounds along with the motion and force deviations from their nominal values.
2.4.5 Effect of sensor errors on the transient processes In this section we deal with the stability of the considered control laws in respect of the errors of the sensors of position, velocity and force, as well as with the effect of these errors on the accuracy of transient processes in the stabilization of PM q p (t ) and PFI Fp (t ) under the conditions of the constraints (2.66) – (2.69).
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Dynamics and Robust Control of Robot-Environment Interaction
Fig. 2.3 Tracking the nominal force and motion
Fig. 2.4 Norm and norm bounds for the motion and force
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A Unified Approach to Dynamic Control of Robots
Let the functions of sensor errors for any t ≥ t0 be described by the equalities
∆ q (t ) = qˆ (t ) − q (t ) , ∆ qɺ (t ) = qˆɺ (t ) − qɺ (t ) ,
(2.137)
∆ F (t ) = Fˆ (t ) − F (t ) , where qˆ, qɺˆ , Fˆ are the sensors’ readings of position, velocity and force. Let the accuracy of these sensors be determined by the values δ q , δ qɺ , δ F , so that ∆ q (t ) ≤ δ q ,
∀t ≥ t0 ,
∆ qɺ (t ) ≤ δ qɺ ,
∀t ≥ t0 ,
∆ F (t ) ≤ δ F ,
∀t ≥ t0 .
(2.138)
Let us consider the control laws (2.23) and (2.28), stabilizing PM q p (t ) with the given quality of transient processes, which, after taking into account the sensors’ errors assume the form:
( ) ( ) τ = U ( qˆ , qɺˆ , qɺɺ + Γ ( qɺˆ − qɺ ) + Γ ( qˆ − q ) , f ( qˆ , qɺˆ , qɺɺ + Γ ( qɺˆ − qɺ ) + Γ ( qˆ − q ) ) ) τ = U qˆ , qˆɺ , qɺɺp + Γ1 qɺˆ − qɺ p + Γ 2 ( qˆ p − q p ) , Fˆ 1
p
1
p
p
p
2
(2.139)
p
2
(2.140)
p
Obviously, the control τ (2.139) satisfies the equation
(
)
(
qɺɺp + Γ1 qˆɺ − qɺ p + Γ 2 ( qˆ − q p ) = Φ qˆ , qɺˆ ,τ , Fˆ
)
Subtracting this equation from the equation of robot’s dynamics (2.61) we obtain the following closed-loop system
ηɺɺ = Γ1ηɺ + Γ 2η +i(t )
(2.141)
where
(
i(t ) = r (t ) + Γ1∆ qɺ (t ) + Γ 2 ∆ q (t ) + Φ ( q, qɺ ,τ , F ) − Φ qˆ , qˆɺ ,τ , Fˆ
)
(2.142)
The same form as (2.141) has the equation of closed-loop control system when using the control law (2.140). Thereby, the function i(t ) is of the form
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Dynamics and Robust Control of Robot-Environment Interaction
i(t ) = D1 ( q (t ) ) r (t ) + D2 ( q (t ) ) ρ (t ) + Γ1∆ qɺ (t ) +
(
+ Γ 2 ∆ q (t ) + ϕ ( q, qɺ ,τ ) − ϕ qˆ , qˆɺ ,τ
(2.143)
)
where the vector function ϕ is determined by the expression −1
ϕ ( q, qɺ ,τ ) = [ H (q) − N (q ) M (q) ]
( −h(q, qɺ ) + τ + N (q) L(q, qɺ ) )
and the vector functions D1 ( q ), D2 ( q ) are determined in the section 2.4.2,
(
and N ( q ) = J T ( q ) S T ( q )
)
−1
.
Indeed, by eliminating the force F from the equations of the robots dynamics (2.61) and environment dynamics (2.62), we obtain:
qɺɺ = D1 ( q(t ) ) r (t ) + D2 ( q(t ) ) ρ (t ) + ϕ ( q, qɺ ,τ )
(2.144)
On the other hand, the control law (2.140) satisfies obviously the equation
(
)
(
qɺɺp + Γ1 qˆɺ − qɺ p + Γ 2 ( qˆ − q p ) = ϕ qˆ , qˆɺ ,τ
)
(2.145)
By subtracting (2.145) from (2.144) we obtain the desired representation (2.141), (2.143). As the dependence of the estimates of transient processes (2.85), (2.86), (2.87), (2.89) on the applied control laws (2.23), (2.28) is fully determined by the estimate Ci of the perturbation function i(t ) in equation (2.82), then, on the basis of (2.142), (2.143) it can be concluded that the analogous relations hold also in respect of the control laws (2.139), (2.140). Hence, the following theorem, similar to Theorem 2, is justified
Theorem 5. Let PM q p (t ) , together with its derivatives qɺ p (t ), qɺɺp (t ) , satisfy the inclusions: δ +δ q
q p (t ) ∈ Vq 1
δ 2 +δ qɺ
qɺ p (t )∈ Vqɺ
,
∀t ≥ t0 ,
(2.146)
,
∀t ≥ t0 ,
(2.147)
δ + Γ1 δ qɺ + Γ 2 δ q
qɺɺp (t ) ∈ Vqɺɺ 3
,
∀t ≥ t0 ,
(2.148)
and let PFI Fp (t ) satisfy the inclusion σ + max {δ F ,( L1 + L3 Γ 2 )δ q + ( L2 + L3 Γ1 ) δ qɺ }
Fp (t ) ∈ VF
,
∀t ≥ t0
(2.149)
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A Unified Approach to Dynamic Control of Robots
Let on the sets Vq , Vqɺ , Vτ , VF of the values of arguments the vector functions Φ and ϕ satisfy the Lipschitz conditions with respect to each variable with the constants LΦq , LΦqɺ , LΦτ , LΦF for the function Φ and Lϕq , Lϕqɺ , Lϕτ for the function ϕ . Then, for the control laws (2.139), (2.140) Theorem 2 holds, with the constant Ci being:
Ci = Cr + Γ1 δ qɺ + Γ 2 δ q + LΦq δ q + LΦqɺ δ qɺ + LΦF δ F for the control law (2.139) and
Ci = d1Cr + d 2Cρ + Γ1 δ qɺ + Γ 2 δ q + Lϕq δ q + Lϕqɺ δ qɺ for the control law (2.140). The proof of Theorem 5 is given in Appendix E. Let us consider the control laws (2.39), (2.40), (2.43) ( Q ( µ ) = R µ ), stabilizing PFI Fp (t ) with the preset quality of transient processes, which, after taking into account sensors’ errors assume the form: t ˆ ˆ τ = U qˆ , qɺ , ψ qˆ , qɺ , Fp + R ∫ Fˆ − Fp (ω )dω , Fˆ t0
(
(
)
t
)
(
)
τ = U qˆ , qˆɺ ,ψ qˆ , qɺˆ , Fˆ , Fp + R ∫ Fˆ − Fp (ω ) d ω
t0
t ˆ ˆ τ = U qˆ , qɺ ,ψ qˆ , qɺ , Fp + R ∫ Fˆ − Fp (ω ) d ω , t0 t Fp + R ∫ Fˆ − Fp (ω )d ω t0
(
(
(2.150)
(2.151)
)
(2.152)
)
ɺɺ from the equation of robot’s After eliminating the second derivative q dynamics (2.61) and environment’s dynamics (2.62), we obtain
F = D1 (q) r (t ) + D2 (q) ρ (t ) + G ( q, qɺ ,τ , F ) where
D1 (q ), D2 (q ) are given by (2.99) and
(2.153)
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Dynamics and Robust Control of Robot-Environment Interaction
G ( q, qɺ ,τ , F ) = (2.154)
−1
= ( S T (q ) ) M (q ) H −1 (q ) (τ − h(q, qɺ ) + J T (q ) F ) + L(q, qɺ ) for the control law (2.150); D1 ( q ), D2 ( q ) are determined by (2.100) and
G ( q, qɺ ,τ , F ) = −1
= ( J T (q ) ) H (q ) M −1 (q ) ( S T (q ) F − L(q, qɺ ) ) + h(q, qɺ ) − τ
(2.155)
for the control law (2.151); D1 ( q ), D2 (q ) are determined by (2.101) and −1
G ( q, qɺ ,τ , F ) = ( D(q) ) M (q) H −1 (q) (τ − h(q, qɺ ) ) + L(q, qɺ )
(2.156)
for the control law (2.152). On the other hand, the control laws (2.150) – (2.152) satisfy the equation t
(
)
(
Fp + ∫ Fˆ − Fp (ω ) dω = G qˆ , qˆɺ ,τ , Fˆ
)
(2.157)
t0
with the function G determined by the respective formulas (2.154), (2.155), (2.156). By subtracting equation (2.157) from (2.153) we obtain that for each of the control laws (2.150), (2.151), (2.152) the closed-loop control system obeys the equation t
t
t0
t0
µ (t ) = R ∫ µ (ω )dω + R ∫ ∆ F (ω )dω + D1 ( q(t ) ) r (t ) + D2 ( q(t ) ) ρ (t ) +
(
+ G ( q(t ), qɺ (t ),τ (t ), F (t ) ) − G qˆ (t ), qˆɺ (t ),τ (t ), Fˆ (t )
(2.158)
)
which differs from equation (2.98) because of the presence of the additional terms on the right-hand side of (2.158). Let us consider the behavior of the system (2.158) over a finite time interval [t0 , T ] on the set
Ω = Vq × Vqɺ × Vτ × VF . Equation (2.158) can be written in the form: t
t
µ (t ) = R ∫ µ (ω )dω + R ∫ ∆ F (ω )dω + g* (t ) t0
t0
(2.159)
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A Unified Approach to Dynamic Control of Robots
g* (t ) = D1 ( q(t ) ) r (t ) + D2 ( q(t ) ) ρ (t ) + where
(
)
+ G ( q(t ), qɺ (t ),τ (t ), F (t ) ) − G qˆ (t ), qɺˆ (t ),τ (t ), Fˆ (t ) .
By differentiating (2.159) with respect to t , we obtain the equation:
µɺ (t ) = Rµ (t ) + R∆ F (t ) + gɺ* (t ) Its solution can be written in the form t
µ (t ) = e R (t −t ) µ (t0 ) + ∫ e R (t −ω ) ( R∆ F (ω ) + gɺ* (ω ) ) dω 0
t0
Integrating the last expression by parts, we obtain
µ (t ) = e R (t −t ) ( µ (t0 ) − g* (t0 ) ) + g* (t ) + 0
t
+ R∫ e
t
R ( t −ω )
∆ F (ω )d ω + R ∫ e R ( t −ω ) g* (ω )d ω
t0
(2.160)
t0
Let on the set Ω the vector-function G satisfy the Lipschitz conditions with respect to each variable with the constants LGq , LGqɺ , LGτ , LGF . Then, we will have
g* (t ) ≤ C g* , where
∀t ∈ [t0 , T ]
C g* = d1Cr + d 2Cρ + LGq δ q + LGqɺ δ qɺ + LGF δ F .
From the equality (2.160) we obtain the following estimation of the transient processes with respect to force:
µ (t ) ≤ CR e− λ (t −t ) ( µ (t0 ) + g* (t0 ) ) + g* (t ) + 0
+ R CR λ −1δ F + R CR λ −1Cg* ≤ CR e− λ (t −t0 ) µ (t0 ) + Cµ where
(2.161)
Cµ = C g* (1 + CR + R CR λ −1 ) + R CR λ −1δ F .
Because the estimates (2.161), (2.105) differ only by the value of the constant Cµ which depends on the control law applied, and Cµ → 0 when Cr , Cρ , δ q , δ qɺ , δ F → 0 the following theorem, analogous to Theorem 4, is justified.
Theorem 6. Let the vector-function G satisfy the Lipschitz conditions on the set Ω with respect to the each variable q, qɺ , F with the constants LGq , LGqɺ , LGF correspondingly.
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Dynamics and Robust Control of Robot-Environment Interaction
ɺɺp (t ) , satisfy the Let PM q p (t ) , together with its derivative qɺ p (t ), q inclusions δ +δ q
q p (t ) ∈ Vq 1
δ +δ qɺ
qɺ p (t ) ∈ Vqɺ 2
δ3 +( qɺɺp (t ) ∈ Vqɺɺ
,
∀t ≥ t0
,
∀t ≥ t0
L1 + L3 LGq
)δ +( q
L2 + L3 LGqɺ
(2.162) (2.163)
)δ
qɺ + L3δ F
max
{
LGF
} , ∀t ≥ t 0
,1
(2.164)
and let PFI Fp (t ) satisfy the inclusion
{
σ + max δ F , LGq δ q + LGqɺ δ qɺ + LGF δ F
Fp (t ) ∈ VF
} , ∀t ≥ t 0
(2.165)
Then, for the control laws (2.150), (2.151), (2.152) Theorem 4 is justified. The proof of Theorem 6 is given in Appendix F. As a supplement to the above discussion let us point out that the application of the control laws (2.139), (2.140), (2.150), (2.151), (2.152), which take into account the errors of the sensors of position, velocity, and force under the conditions of a priori constraints (2.66) – (2.69) and in the presence of external perturbations narrows the classes of stabilized PM q p (t ) and PFI Fp (t ) and worsens the accuracy of stabilization in comparison with the ideal control laws (2.23), (2.28), (2.39), (2.40), (2.43). This is related to the fact that sensors’ inaccuracies introduce additional components into the estimates used in the conditions of Theorems 2 and 4. The preservation of these estimates may be ensured by means of an increase of the quantities δ 1 , δ 2 , δ 3 , σ that narrow the classes of stabilized PM and PFI in addition to their narrowing caused by the conditions of Theorems 5 and 6, and also by lowering the accuracy of stabilization of ε and δ in the goal conditions (2.64), (2.65).
2.5 Adaptive Stabilization of Motion and Forces 2.5.1 Introduction In adaptive cases, the goal of stabilization of motion and contact forces of the manipulators interacting with dynamic environment especially for the robots with nonstationary dynamics is attained by designing a special adaptive control scheme and using finite-convergence adaptation algorithms [21].
A Unified Approach to Dynamic Control of Robots
129
In view of the fact that in the course of technological operation the robot parameters, especially the viscous friction coefficients at the manipulator’s drives, may vary with time, the unknown drifts of parameters are of interest. There is a sufficiently broad array of adaptive control laws [6, 22-24] for solving contact tasks by manipulators with stationary dynamics. However, there are practically no results for the robots with nonstationary dynamics. The adaptive control scheme proposed here enables solving contact tasks for robots with both stationary and nonstationary dynamics. One of the parameters of the scheme, such as the algorithm adaptation processing speed, determines the “speed” of the parameters’ drift to which the adaptive control system has time to adapt without violating the a priori constraints that are usually imposed on the control, motion, and interaction forces. Let us notice that an additive component of the external perturbation of the robot dynamics in the proposed scheme can be considered as a time-dependent robot parameter. Therefore this control scheme can be used for adaptation to the essentially inadequate description of the robot’s dynamics by its mathematical model. The extent of parameters’ drift and the degree of inadequacy of the robot model are determined by the class of functions of uniformly bounded variation. Here we consider the classes of stabilized motions and forces and their stabilization accuracies subject to the level of initial and external perturbations of dynamics of the robot and its environment, as well as of the sensors errors, processing speed of the adaptation algorithm, and of other parameters of the adaptive control scheme.
2.5.2 Task setting Let us consider a manipulation robot whose mechanism model is described by the vector differential equation of the form
H (q, ξ1 )qɺɺ + h(q, qɺ , ξ1 ) = τ + J T (q, ξ1 ) F
(2.166)
where, q = q (t ) is the n -vector of the generalized coordinates; H ( q, ξ1 ) is the n × n positive-definite matrix of inertia moments of the manipulation mechanism; h( q, qɺ , ξ1 ) is the n -vector of the centrifugal, Coriolis’ and gravitational moments; τ is the n -vector of the torques at the robot joints; J T (q, ξ1 ) is the n × m matrix connecting the robot end-effector velocities with the velocities of the generalized coordinates; F is the m -vector of the generalized forces, or of the generalized forces and moments acting on the robot
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Dynamics and Robust Control of Robot-Environment Interaction
end-effector, and ξ1 is the k1 -vector of the manipulation mechanism parameters. We will assume that n ≥ m (general case). The robot actuators considered are the electromechanical drives with reducers, whose dynamics is described with a high degree of accuracy by the following system of differential equations [4]:
LR
diR + RiR + Eqɺ = u dt
(2.167)
J R qɺɺ + K v qɺ + τ = DiR
(2.168)
where LR , R, K v , E and D are the n × n diagonal matrices representing the respective parameters of the actuators such as the electric motor armature inductivity, armature resistances, coefficients of viscous friction at the joints, and electromechanical and mechanoelectrical constants, including the reduction ratio of the reducers; J R is the diagonal n × n matrix of inertia moments of the actuators, reduced to the output shafts; u is the n -vector of the electric motor controlling voltage, and iR is the n -vector of the electromotor rotor current. To obtain a generalized model of the robot dynamics, let us eliminate the vector of moments τ from (2.166) and (2.168). Then, the vector of currents iR will be described via the generalized coordinates q :
iR = D −1 ( J R + H (q, ξ1 ) ) qɺɺ + K v qɺ + h ( q, qɺ , ξ1 ) − J T ( q, ξ1 ) F . Substituting this expression into (2.167) we obtain an equation that connects the vector of the controlling voltages u to the vector of generalized coordinates q in the form:
u = U ( q, qɺ , qɺɺ, ɺɺɺ q , F , Fɺ , ξ )
(2.169)
where:
U ( q, qɺ , qɺɺ, ɺɺɺ q , F , Fɺ , ξ ) = A ( q, ξ ) ɺɺɺ q + B ( q, qɺ , ξ ) qɺɺ + C ( q, qɺ , ξ ) + + a ( q, ξ ) Fɺɺ + b ( q, qɺ , ξ ) F ,
(
A ( q, ξ ) = LR D
−1
) ( J R + H ( q, ξ1 ) ) ,
(
)
n ∂H ∂h −1 B ( q, qɺ , ξ ) = LR D −1 ∑ qɺi + K v + + RD ( J R + H ( q, ξ1 ) ) , ∂ qɺ i =1 ∂ qi
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A Unified Approach to Dynamic Control of Robots
C ( q, qɺ , ξ ) = RD −1 ( K v qɺ + h ( q, qɺ , ξ1 ) ) + Eqɺ + LR D −1
∂h qɺ , ∂q
a ( q, ξ ) = − LR D −1 J T ( q, ξ1 ) , n ∂ JT b ( q, qɺ , ξ ) = − D −1 LR ∑ qɺi + R J T ( q, ξ1 ) . i =1 ∂ qi
(
Here, the k -vector of robot parameters ξ = ξ1T , ξ 2T
T
)
consists of the k1 -
vector ξ1 of the manipulation mechanism parameters and the k 2 -vector ξ 2 of the actuator parameters ( k = k1 + k 2 ) . Let us note that the presented form of the matrices A, B, a, b and of the vector C , correspond to a constant value of the vector ξ 2 of the parameters of the actuators. If this vector is a time function, these matrices will have a more unwieldy form, but the form of robot dynamics equation (2.169) will be preserved. In that case, the vector of parameters ξ should be supplemented with the components corresponding to the derivatives of the vector ξ 2 (t ) . On the basis of the positive-definiteness of the matrix A( q, ξ ), let us write (2.169) in the form solved with respect to the highest derivative:
ɺɺɺ q = Φ ( q, qɺ , qɺɺ, F , Fɺ , u , ξ )
(2.170)
where
Φ ( q, qɺ , qɺɺ, F , Fɺ , u , ξ ) = = A−1 ( q, ξ ) u − B ( q, qɺ , ξ ) qɺɺ − C ( q, qɺ , ξ ) − a ( q, ξ ) Fɺ − b ( q, qɺ , ξ ) F . Let the real robot dynamics satisfy the differential equation
ɺɺɺ q = Φ ( q, qɺ , qɺɺ, F , Fɺ , u , ξ ) + r (t )
(2.171)
where the n -vector r (t ) represents the uncontrollable external perturbation, which can also be treated as the inadequacy of description of the robot dynamics by the mathematical model (2.170). We will assume that the vector functions Φ and r satisfy conditions sufficient for the existence and uniqueness of the solutions of (2.170) and ɺɺ(t0 ) = qɺɺ0 . (2.171) for the initial conditions q (t0 ) = q0 , qɺ (t0 ) = qɺ0 , q
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Dynamics and Robust Control of Robot-Environment Interaction
Let us describe the dynamics model of the environment interacting with the robot by a nonlinear vector differential equation:
M (q )qɺɺ + L(q, qɺ ) = S T (q ) F
(2.172)
with the nonsingular and continuous n × n matrix M (q ), continuous n -vector function L( q, qɺ ) , and continuous n × m matrix S T (q ) of the environment, reduced to the robot coordinates frame
{q} .
The environment dynamics
parameters have been described in [25]. In the Cauchy form, the equation of the environment dynamics model (2.172) can be written as
qɺɺ = ψ ( q, qɺ , F ) where ψ ( q, qɺ , F ) = M −1 ( q ) S T ( q ) F − L( q, qɺ ) . We will assume that (2.172) has the following property:
rank S (q ) = m,
∀q ∈ R n
which means that the equation is uniquely solvable with respect to force:
F = f ( q, qɺ , qɺɺ) where
−1
f ( q, qɺ , qɺɺ) = ( S (q ) S T (q ) ) S (q ) [ M (q )qɺɺ + L(q, qɺ )] .
Let the real environment dynamics satisfy the vector differential equation
F = f ( q, qɺ , qɺɺ) + ρ (t )
(2.173)
where the vector function ρ (t ) of the dimension m represents the uncontrollable perturbations of the environment. The goal of the control of the robot in contact with its environment is the realization of the programmed (desired) motion q p (t ) and the programmed (desired) force of robot’s interaction with environment Fp (t ) . The programmed robot motion ( PM ) q p (t ) and the programmed force of interaction ( PFI ) Fp (t ) must satisfy the constraint equation
Fp (t ) ≡ f ( q p (t ), qɺ p (t ), qɺɺp (t ) ) ,
∀t ≥ t 0
(2.174)
Because of the existence of initial perturbations and uncontrollable external perturbations r (t ) and ρ (t ) , exact realization of PM q p (t ) and PFI Fp (t )
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A Unified Approach to Dynamic Control of Robots
is not possible. Hence, we will formulate the control goal in the form of the following two conditions on the analogy of (2.64), (2.65)
x (t ) − x p (t ) < ε F (t ) − Fp (t ) < δ
(2.175)
where ε , δ > 0 are the given numbers, T
T
x (t ) = ( qT (t ), qɺ T (t ), qɺɺT (t ) ) , x p (t ) = ( q p T (t ), qɺ pT (t ), qɺɺpT (t ) ) . Here, and below, the norm of a vector signifies the Euclidean norm. The goal conditions must be fulfilled starting from a time instant t p ≥ t0 .
ɺɺ(t ), the control action u (t ) , and the force of The robot motion q (t ), qɺ (t ), q interaction of the robot and environment F (t ) , are usually constrained by the conditions of the technological task to be performed. Let us set up these constraints in the form of the relations:
q(t ) ∈ Vq ⊂ R n , qɺ (t ) ∈ Vqɺ ⊂ R n , qɺɺ(t ) ∈ Vqɺɺ ⊂ R n , ∀t ≥ t0 u (t ) ∈ Vu ⊂ R n ,
∀t ≥ t0
F (t ) ∈ VF ⊂ R m , ∀t ≥ t0
(2.176) (2.177) (2.178)
where Vq , Vqɺ , Vqɺɺ , Vu , and VF are the preset open, bounded and simply connected sets of the corresponding spaces, and Vu is the closure of the set Vu in
Rn . We will assume that the robot parameters are unknown and can change as time functions ξ = ξ (t ) in an unpredictable manner. We will also assume that the closed convex set Vξ ∈ R k , constraining the parameter variations is known:
ξ (t ) ∈ Vξ ,
∀t ≥ t0
(2.179)
Also, we assume that the levels of external perturbation of the robot and environment are bounded:
r (t ) ≤ Cr ,
ρ (t ) ≤ Cρ , ∀t ≥ t0
(2.180)
and that the function ρ (t ) is differentiable and the norm of its derivative is also bounded:
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Dynamics and Robust Control of Robot-Environment Interaction
ρɺ (t ) ≤ Cρɺ ,
∀t ≥ t0
(2.181)
Under these conditions, the task of controlling robot in contact with its environment can be formulated in the following way. For the complete equations of dynamics of the robot (2.171) and of the environment (2.173), it is necessary to synthesize the admissible (i.e. satisfying the constraint (2.177)) control action u (t ), t ≥ t0 , which, in solving the contact task, will ensure that the real robot
ɺɺ(t ) satisfies the constraints (2.176), that the real force of motion q (t ), qɺ (t ), q interaction of the robot with environment F (t ) satisfies the constraint (2.178), and that the goal conditions (2.175) are fulfilled starting from a time instant t p ≥ t0 . 2.5.3 General scheme of robot adaptive control in contact tasks We will assume that the function f , describing the environment dynamics, possesses continuous partial derivatives with respect to each variable. After differentiating, the environment dynamics equation (2.173) becomes
Fɺ = f ( q, qɺ , qɺɺ, ɺɺɺ q ) + ρɺ (t )
(2.182)
where
f ( q, qɺ , qɺɺ, ɺɺɺ q ) = N (q) M (q )ɺɺɺ q + m(q, qɺ )qɺɺ + l (q, qɺ ) .
Here
N (q ) = S (q ) S T (q ) S (q ),
−1
n
m(q, qɺ ) = ∑ i =1 n
l (q, qɺ ) = ∑ i =1
∂ [ N (q ) M (q) ] ∂ qi
qɺi + N (q )
∂ L(q, qɺ ) , ∂ qɺ
∂N (q) ∂ L(q, qɺ ) qɺi L(q, qɺ ) + N (q) qɺ . ∂ qi ∂q
Substituting F and Fɺ from (2.173) and (2.182) into the robot dynamics equation (2.171), and solving it with respect to the highest derivative, we obtain the following equation of interaction of the robot and environment:
ɺɺɺ q = ϕ ( q, qɺ , qɺɺ, u , ξ ) +
+ A −1 ( q, ξ ) A ( q, ξ ) r (t ) − a ( q, ξ ) ρɺ (t ) − b ( q, qɺ , ξ ) ρ (t )
(2.183)
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A Unified Approach to Dynamic Control of Robots
where
A ( q , ξ ) = A ( q, ξ ) + a ( q, ξ ) N ( q ) M ( q ) ,
ϕ ( q, qɺ , qɺɺ, u, ξ ) = A −1 ( q, ξ ) u − ( B ( q, qɺ , ξ ) + a ( q, ξ ) m(q, qɺ ) ) qɺɺ − −C ( q, qɺ , ξ ) − a ( q, ξ ) l (q, qɺ ) − b ( q, qɺ , ξ ) f ( q, qɺ , qɺɺ) . We will assume that ∀(q, ξ ) ∈ Vq × Vξ , the matrix A( q, ξ ) is invertible, and the matrix and vector functions continuous. Then the quantities:
ϕ (⋅), A −1 (⋅), A(⋅), a (⋅), and b (⋅) are
Cϕ = sup ϕ ( q, qɺ , qɺɺ, u , ξ ) , C A = sup A −1 ( q, ξ ) , C A = sup A ( q, ξ ) , Ca = sup a ( q, ξ ) , Cb = sup b ( q, qɺ , ξ ) ,
ɺɺ ∈Vqɺɺ , u ∈Vu , and ξ ∈Vξ , where the suprema are taken over all q ∈ Vq , qɺ ∈Vqɺ , q are bounded. Hence, for an arbitrary admissible control u (t ), t ≥ t0 , ensuring fulfillment of the inclusion
( q(t ), qɺ (t ), qɺɺ(t ) )∈Vq × Vqɺ × Vqɺɺ ,
∀t ≥ t0 ,
(2.184)
because of (2.183) and inequalities (2.180), (2.181), the inequality
ɺɺɺ q (t ) ≤ Cɺɺɺq ,
∀t ≥ t0
(
(2.185)
)
is satisfied, where Cɺɺɺq = Cϕ + C A C ACr + Ca Cρɺ + Cb Cρ . Furthermore, we will assume that on the set Vq × Vqɺ × Vqɺɺ × R n × R m × R m × R k the vector function U is linear with respect to the parameters:
U ( q, qɺ , qɺɺ, ɺɺɺ q , F , Fɺ , ξ ) = G ( q, qɺ , qɺɺ, ɺɺɺ q , F , Fɺ ) ξ
(2.186)
Here, G (⋅) is a known continuous matrix function of the dimension n × k . The property (2.186) of the robot dynamics model (2.169) is well known, and is frequently used in the synthesis of adaptive control laws [9, 22-24]. We will also assume that there exist the open, simply connected and bounded sets Vɺɺɺq ⊂ R n and VFɺ ⊂ R m such that
U : Vq ×Vqɺ ×Vqɺɺ ×Vɺɺɺq ×VF ×VFɺ ×Vξ → Vu
(2.187)
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Dynamics and Robust Control of Robot-Environment Interaction
This property is a sufficient condition for the existence of the admissible control in the realization of PM q p (t ) and PFI Fp (t ) by the robot under constraint conditions (2.176) – (2.178) and in the absence of initial, external, and parametric perturbations. Let us define a scheme of the function of robot’s control system in the following way. Let ξ 0 be an arbitrary vector in Vξ . Let the set of time constants
t0 , t1 ,… , tk and the set of vectors ξ0 , ξ1 ,… , ξ k in Vξ be determined. For t ≥ tk let us consider the control law of the form
(
) )
(
uk (t ) = U qˆ , qɺˆ , qɺɺˆ , q∗ , Fˆ , f qˆ , qˆɺ , qɺɺˆ , q∗ , ξ k
(
)
(
)
(
(2.188)
)
where q ∗ = ɺɺɺ q p + Γ1 qɺɺˆ − qɺɺp + Γ 2 qɺˆ − qɺ p + Γ 3 qˆ − q p , Γ1 , Γ 2 , Γ 3 are the
n × n matrices, such that the eigenvalues of the matrix 0n Γ = 0n Γ3
In 0n Γ2
0n I n Γ1
ɺɺˆ , Fˆ are negative, real and different quantities; qˆ, qˆɺ , q
are the respective
indications of the position, velocity, acceleration and force sensors. Here, 0 n and I n are the respective zero and unit matrices of dimension n × n . It is easy to see, that the matrix Γ will have eigenvalues λ1 , λ 2 ,..., λ3 n if its submatrices Γ1 , Γ 2 , Γ 3 have the following representation
Γ1 = diag ( λ1 λ2 λ3 , … , λ3n − 2 λ3n −1 λ3n ) , Γ 2 = diag ( −λ1λ2 − λ1λ3 − λ2 λ3 , … , − λ3n − 2 λ3n −1 − λ3n − 2 λ3n − λ3n −1λ3n ) , Γ3 = diag ( λ1 + λ2 + λ3 , … , λ3n − 2 + λ3n −1 + λ3n ) . Let us for t ≥ t k determine the auxiliary function
(
(
) )
uɶk (t ) = U qˆ , qˆɺ , qɺɺˆ , ɺɺɺ qˆ , Fˆ , f qˆ , qˆɺ , qɺɺˆ , ɺɺɺ qˆ , ξ k
where ɺqɺˆɺ is the measured or calculated estimate of ɺqɺɺ . Consider the inequality of the form
(2.189)
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A Unified Approach to Dynamic Control of Robots
uk (t ) − uɶk (t ) < h ,
t ≥ tk
(2.190)
where h > 0 is a control scheme parameter. Let tk ′ ≥ tk be the first time instant in which the inequality (2.190) is violated, that is, at that time instant
uk (tk ′ ) − uɶk (tk ′ ) ≥ h
(2.191)
Let
(
ξ k +1 = Α ξ k , tk ′
)
(2.192)
be a correction algorithm defining the estimate ξ k +1 of the current value of the unknown parameters vector ξ (t ) at the time instant tk ′ , such that ξ k +1 ∈ Vξ . The time instant t k +1 , following t k in the sequence t0 , t1 , t2 ,… , tk , is determined by the equality
tk +1 = tk ′ + θ where θ is the time needed to calculate a new estimate of the parameters vector ξ k +1 according to the algorithm (2.192). In this way, over the time interval [tk , tk +1 ) , the control low is defined by (2.188), while the duration of this time interval is determined by the duration of fulfillment of the inequality (2.190), and by the time interval θ . The number θ plays the role of the control scheme parameter characterizing the processing speed of the adaptation algorithm under consideration. Note that a similar scheme has been utilized for adaptive robot control in free space [9, 22-24]. To solve the inequalities (2.190) one can use any algorithm out of the class of finite-convergence algorithms. As an example of the correction algorithm of estimates (2.192), we will consider an algorithm of the form [6]
G T (tk ′ ) uk (tk ′ ) − uɶk (tk ′ ) ξ k +1 = PrVξ ξ k + 2 G (tk ′ )
(
)
(2.193)
where
(
(
G (tk ′ ) = G qˆ (tk ′ ), qɺˆ (tk ′ ), qɺɺˆ (tk ′ ), ɺɺɺ qˆ (tk ′ ), Fˆ (tk ′ ), f qˆ (tk ′ ), qɺˆ (tk ′ ), qɺɺˆ (tk ′ ), ɺɺɺ qˆ (tk ′ ) and PrV is the orthogonal projector onto the set Vξ . ξ
))
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Dynamics and Robust Control of Robot-Environment Interaction
Let us denote by
∆ q (t ) = qˆ (t ) − q(t ), ∆ qɺ (t ) = qˆɺ (t ) − qɺ (t ), ∆ qɺɺ (t ) = qɺɺˆ − qɺɺ(t ),
qˆ (t ) − ɺɺɺ q (t ), ∆ F (t ) = Fˆ (t ) − F (t ) ∆ɺɺɺq (t ) = ɺɺɺ the deviation of the corresponding estimates of motion and force from their real values. Let the accuracy of these estimates be bounded by the quantities δ q , δ qɺ , δ qɺɺ , δɺɺɺq and δ F , such that:
∆ q (t ) ≤ δ q , ∆ qɺ (t ) ≤ δ qɺ , ∆ qɺɺ (t ) < δ qɺɺ ,
(2.194)
∆ɺɺɺq (t ) ≤ δɺɺɺq , ∆ F (t ) ≤ δ F , ∀t ≥ t0 Let us denote by
CG = sup G ( q, qɺ , qɺɺ, ɺɺɺ q , F , f ( q, qɺ , qɺɺ, ɺɺɺ q ))
(2.195)
where the supremum is taken over all
q , F ) ∈ Vq × Vqɺ × Vqɺɺ × Y ×VF , ( q, qɺ, qɺɺ, ɺɺɺ
Y = { y ∈ R n / y ≤ Cɺɺɺq + δ ɺɺɺq } .
Because the matrix and vector functions G and f are continuous, the number
C G is finite. We
assume n
that m
the
vector
m
k
function
U
satisfies
on
the
set
Vq ×Vqɺ ×Vqɺɺ × R × R × R × R the Lipschitz conditions for each variable, with the Lipschitz constants Luq , Luqɺ , Luqɺɺ , Luɺɺɺq , LuF , LuFɺ and Luξ , respectively. Due to the linearity of U with respect to ɺɺɺ q , F , Fɺ and ξ this function unconditionally satisfies the Lipschitz conditions with respect to these variables if ( q, qɺ , qɺɺ) ∈Vq × Vqɺ ×Vqɺɺ . In view of the assumption on continuous differentiability of the function f we can state that this function satisfies for the set Vq × Vqɺ × Vqɺɺ the Lipschitz conditions for each variable, with some Lipschitz constants Lqf , Lqfɺ , Lqɺfɺ . We will assume that the vector function f also satisfies the Lipschitz conditions for each variable with the constants equaling the respective Lqf , Lqfɺ , Lqɺɺf and Lɺɺɺqf on the set Vq ×Vqɺ ×Vqɺɺ × R n .
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139
2.5.4 Adaptive stabilization of programmed motions and forces Let us give the following accepted definition. The variation (or the total variation) of the real-valued function g (t ) , defined on the interval [a, b ] is the quantity N
Var ( g ) = sup ∑ g (tk ) − g (tk −1 ) [a , b]
k =1
where the supremum is taken over all partitions a = t0 < t1 < … < tn = b of the interval [a, b ] . If there exists a constant M such that Var ( g ) ≤ M , the function [a , b]
g is called the function of bounded variation over the interval [a, b ] . The function g :[t0 , ∞ ) → R will be called the function of uniformly bounded variation of order p if on an arbitrary interval [ a, b ] ⊂ [t0 , ∞ ) of the length p the function g is a function of bounded variation with one and the same constant M ( p ). Consider the scheme of adaptive control presented in the preceding section. Let us denote by r( a, b) the number of corrections (the number of time instants
tk ′ ) of the algorithm (2.193) on the interval [a, b] . We will prove first an auxiliary statement asserting that under certain conditions the number of corrections of the algorithm (2.193) is bounded on any finite time interval [a, b ] by the majorant independent of the adaptation algorithm processing speed θ , if the components ξ j (t ), j = 1,2,..., k of the vector function ξ (t ) are the functions of bounded variation on [a, b ] .
Lemma: Let [a, b ] be an arbitrary interval contained in [t0 , ∞) , and for every time instant t ∈ [a, b] during the functioning of the closed-loop control system defined by the control law (2.188), the following conditions are fulfilled: 1. 2.
(qˆ (t ), qɺˆ (t ), qɺɺˆ (t ), ɺɺɺ q p (t ) + Γ1 (qɺɺˆ (t ) − qɺɺp (t )) + + Γ 2 (qɺˆ (t ) − qɺ p (t )) + Γ3 (qˆ (t ) − q p (t ))) ∈Vq × Vqɺ ×Vqɺɺ ×Vɺɺɺq
( q(t ), qɺ (t ), qɺɺ(t ) )∈Vq × Vqɺ × Vqɺɺ
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Dynamics and Robust Control of Robot-Environment Interaction
3. 4.
Fˆ (t ) ∈ V F f (qˆ (t ), qˆɺ (t ), qɺɺˆ (t ), ɺɺɺ q p (t ) + Γ1 (qɺɺˆ (t ) − qɺɺp (t )) + + Γ 2 (qɺˆ (t ) − qɺ p (t )) + Γ 3 (qˆ (t ) − q p (t ))) ∈ VFɺ
Let the adaptation scheme parameter h be chosen in such a way that the real number h 0 = h − 2 h1 > 0 , where
h1 = Luq δ q + Luqɺ δ qɺ + Luqɺɺ δ qɺɺ + Luɺɺɺq (δɺɺɺq + Cr ) + LuF δ F +
(
+ LuFɺ Lqf δ q + Lqfɺ δ qɺ + Lqɺɺf δ qɺɺ + Lɺɺɺqf δɺɺɺq + Cρɺ
)
If the components ξ j (t ), j =1,..., k of the vector function ξ (t ) are the functions of bounded variations on [a, b ] , the number of corrections r ( a, b) of the algorithm (2.193) is bounded from above by the quantity
k Κ ∑ Var (ξ j ) = j =1 [ a , b]
( diam V ) ξ
2
k
+ 4 Cξ
(ξ ) ∑ Var [ ] j
a,b
j =1 −2 G
(2.196)
hh 0 C
which is not dependent on the parameter θ of the control scheme. Here Cξ = max ξ , diamVξ is the diameter of the set Vξ . ξ ∈Vξ
The proof of Lemma is given in Appendix G. Let us consider the transient processes occurring during the realization of the adaptive control low (2.188). Evidently, the control u k (t ) satisfies the equation
(
)
(
)
(
ɺɺɺ q p + Γ1 qɺɺˆ − qɺɺp + Γ 2 qɺˆ − qɺ p + Γ 3 ( qˆ − q p ) = ϕ qˆ , qˆɺ , qɺɺˆ , Fˆ , uk , ξ k
)
(2.197)
where
( (
(
)
)
ϕ qˆ , qɺˆ , qɺɺˆ , Fˆ , uk , ξ k = A −1 ( qˆ , ξ k ) uk − B qˆ , qˆɺ , ξ k + a ( qˆ , ξ k ) m (qˆ , qˆɺ ) qɺɺˆ −
(
)
(
)
)
−C qˆ , qˆɺ , ξ k − a ( qˆ , ξ k ) l (qˆ , qˆɺ ) − b qˆ , qˆɺ , ξ k Fˆ and the auxiliary function uɶk (t ) corresponding to this control (relation (2.189)) satisfies the equation
(
ɺɺɺ qˆ = ϕ qˆ , qɺˆ , qɺɺˆ , Fˆ , uɶk , ξ k
)
(2.198)
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A Unified Approach to Dynamic Control of Robots
By extracting (2.197) from (2.198) we obtain the relation
(
)
(
)
ɺɺɺ qˆ − ɺɺɺ q p − Γ1 qɺɺˆ − qɺɺp − Γ 2 qˆɺ − qɺ p − Γ 3 ( qˆ − q p ) =
(
)
(
= ϕ qˆ , qɺˆ , qɺɺˆ , Fˆ , uɶk , ξ k − ϕ qˆ , qɺˆ , qɺɺˆ , Fˆ , uk , ξ k
)
which is equivalent to the equation
ηɺɺɺ = Γ1ηɺɺ + Γ 2ηɺ + Γ3η + i (t )
(2.199)
where η (t ) = q (t ) − q p (t ) ,
i(t ) = −∆ɺɺɺq (t ) + Γ1∆ qɺɺ (t ) + Γ 2 ∆ qɺ (t ) + Γ3 ∆ q (t ) +
(
) (
)
+ ϕ qˆ , qˆɺ , qɺɺˆ , Fˆ , uɶk , ξ k − ϕ qˆ , qˆɺ , qɺɺˆ , Fˆ , uk , ξ k . Let the vector function ϕ satisfy the Lipschitz condition for the variable u on the set Vq ×Vqɺ ×Vqɺɺ ×V F × R n × R k with the Lipschitz constant Lu . Then,
i(t ) ≤ δɺɺɺq + Γ1 δ qɺɺ + Γ 2 δ qɺ + Γ3 δ q + Lu uk − uɶk
(2.200)
If the lemma conditions are fulfilled ∀t ≥ t0 , the inequality ɺɺɺ qˆ (t ) ≤ Cɺɺɺq + δɺɺɺq is true, and inasmuch as q ∗ (t ) ∈ Vɺɺɺq , then q ∗ (t ) ≤ Cɺɺɺq . Hence, in addition to the estimate (2.200) for the norm of the function i(t ) , the following estimate holds:
i(t ) ≤ 2 Cɺɺɺq + 2 δɺɺɺq + Γ1 δ qɺɺ + Γ 2 δ qɺ + Γ3 δ q , ∀t ≥ t0
(2.201)
Let us split the interval of integration [t0 , t ] into two sets:
At = { t ′∈[t0 , t ] / uk (t ′) − uɶk (t ′) < h } Bt = { t ′∈[t0 , t ] / uk (t ′) − uɶk (t ′) ≥ h } Then, according to the inequalities (2.200), (2.201), we will have
δɺɺɺq + Γ1 δ qɺɺ + Γ 2 δ qɺ + Γ 3 δ q + Lu h , i(t ′) ≤ 2 Cɺɺɺq + 2 δɺɺɺq + Γ1 δ qɺɺ + Γ 2 δ qɺ + Γ 3 δ q ,
(
Let us introduce the notation x = η T ,ηɺ T ,ηɺɺT
if t ′ ∈ At if t ′ ∈ Bt T
)
(2.202) T
, iɶ (t ) = ( 0, 0, iT (t ) ) .
Then, (2.199) can be rewritten in the form
xɺ (t ) = Γx(t ) + iɶ (t )
(2.203)
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Dynamics and Robust Control of Robot-Environment Interaction
Its solution has the form t
x(t ) = eΓ (t −t0 ) x(t0 ) + ∫ eΓ (t −ω ) iɶ (ω ) dω
(2.204)
t0
If λ = − max λi , where λi are the different, real and negative eigenvalues of the i
matrix Γ , then
eΓ (t −t0 ) x (t0 ) ≤ CΓ x (t0 ) e− λ ( t −t0 ) , CΓ ≥ 1 Hence, the following estimate of the solution of equation (2.203), x(t ) = x (t ) − x p (t ) , holds:
x(t ) ≤ CΓ e− λ (t −t0 ) x(t0 ) + CΓ ∫ e − λ (t −ω ) i (ω ) d ω + CΓ ∫ e− λ (t −ω ) i (ω ) d ω < At
< CΓ x(t0 ) e
− λ ( t −t0 )
Bt
+ CΓ (δɺɺɺq + Γ1 δ qɺɺ + Γ 2 δ qɺ + Γ3 δ q ) λ −1 +
+ Lu CΓ ∫ e − λ (t −ω ) uk − uɶk d ω +
(2.205)
At
+ CΓ ( 2 Cɺɺɺq + 2 δɺɺɺq + Γ1 δ qɺɺ + Γ 2 δ qɺ + Γ3 δ q ) µ es ( Bt ) < < CΓ x (t0 ) e − λ (t −t0 ) + δ + Lu CΓ λ −1h + C µ es ( Bt ) where
δ = CΓ (δɺɺɺq + Γ1 δ qɺɺ + Γ 2 δ qɺ + Γ3 δ q ) λ −1 , C = CΓ ( 2 Cɺɺɺq + 2 δɺɺɺq + Γ1 δ qɺɺ + Γ 2 δ qɺ + Γ 3 δ q ) ,
µ es ( Bt ) is the measure of the set Bt . Let the components ξ j (t ) of the vector function ξ (t ) be the functions of uniformly bounded variations of order p > 0 , with the constants M j ( p ) . Then, when the lemma conditions are fulfilled the number of corrections of the adaptation algorithm on the interval [t0 , t0 + p ] is bounded:
k r(t0 , t0 + p) < Κ ∑ M j ( p) j =1
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A Unified Approach to Dynamic Control of Robots
and, as the processing speed of the adaptation algorithm equals θ , it is evident that
k
j =1
µ es ( Bt ) < Κ ∑ M j ( p) θ . Consequently, in accordance with the inequality (2.205) on the interval [t0 , t0 + p ] the following estimate holds:
x(t ) < CΓ x(t0 ) e − λ (t −t0 ) + δ + Lu CΓ λ −1h + C Κ
k
∑M j =1
j
( p) θ
(2.206)
In accordance with the lemma, the function Κ (⋅) is not dependent of θ . Then the parameters h and θ of the control scheme can be chosen in such a way that the sum Lu CΓ λ −1h + C Κ (⋅) θ will be as small as desired. By subtracting (2.174) from (2.173), we obtain the following expression for the deviation µ (t ) = F (t ) − F p (t ) :
µ (t ) = f ( q, qɺ , qɺɺ) + ρ (t ) − f ( q p , qɺ p , qɺɺp )
(2.207)
Taking into account the Lipschitz conditions for the vector function f , and also the inequality (2.206), we obtain the following estimate for transient process with respect to force for all t ∈ [ t0 , t0 + p ] :
µ (t ) ≤ Lqf η + Lqfɺ ηɺ + Lqɺɺf ηɺɺ + Cρ ≤ L x(t ) + Cρ < k < LCΓ x(t0 ) e − λ (t −t0 ) + Lδ + LLu CΓ λ −1h + LC Κ ∑ M j ( p ) j =1 f f f where L = Lq + Lqɺ + Lqɺɺ .
(2.208) θ + C ρ
An α -narrowing of the open set A ⊂ R n is defined as the non-empty subset Aα such that A = ∪ Bα ( x) , where the uniting is taken over all x ∈ Aα . Here,
Bα ( x) is an open solid sphere in R n of the radius α > 0 , with the center at the point x . Theorem 7. Let the components ξ j (t ), j = 1,..., k of the robot parameters ξ (t ) be functions of the uniformly bounded variations of order p > 0 with the constants M j ( p ) .
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Dynamics and Robust Control of Robot-Environment Interaction
Let the accuracies of stabilization ε and δ satisfy the inequalities
a <ε < 1
(2.209)
La + Cρ < δ
(2.210)
where
k a = δ + Lu CΓ λ −1 h + C Κ ∑ M j ( p) θ . j =1 Let us determine the class PM
{q (t )} by means of the inclusions:
(q
×Vqɺ qɺ
(t ), qɺ p (t ), qɺɺp (t ) ) ∈ Vq
δ q +δ 0
p
p
δ +δ 0
δ +δ 0
×Vqɺɺ qɺɺ
, ∀t ≥ t0
(2.211)
ɺɺɺ q p (t ) ∈ VɺɺɺqRδ0 + r , ∀t ≥ t0
(2.212)
where δ 0 > CΓε + a and
R = Γ1 + Γ 2 + Γ3 , r = Γ1 δ qɺɺ + Γ 2 δ qɺ + Γ3 δ q . Let us determine the class PFI
{F (t )} by means of the inclusions: p
L δ 0 + Cρ + δ F F
Fp (t ) ∈ V , Fɺ (t ) ∈ V ɺC1δ 0 + C2 , p
where
∀t ≥ t0
F
(2.213)
∀t ≥ t0
(2.214)
C1 = Lqf + Lqfɺ + Lqɺɺf + Lɺɺɺqf R , C2 = Lqf δ q + Lqfɺ δ qɺ + Lqɺɺf δ qɺɺ + Lɺɺɺqf r .
Let the levels of the external perturbation C r , C ρ , C ρɺ , the sensors’ errors
δ q , δ qɺ , δ qɺɺ , δ ɺqɺɺ , δ F , and the parameters h and θ of the adaptation scheme be such that these classes of the h 0 = h − 2h1 > 0 .
PM
and
PFI
are nonempty and
If the level of initial perturbation x (t0 ) satisfies the inequality
x (t0 ) < ε and the order p satisfies the equality
C LCΓ 2 2 p = max p1 = ln Γ , p2 = ln λ ε −a λ δ − La − Cρ
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then, at every t ≥ t0 , for the transient processes of motion and force determined by action of the adaptive control law (2.188), the following estimates hold: − λ ( t −t0 ) + a, if t ∈ [t0 , t0 + p ] C x(t0 ) e x(t ) = x (t ) − x p (t ) < Γ ε , if t ≥ t0 + p
(2.215)
LCΓ x(t0 ) e− λ (t −t0 ) + La + Cρ , if t ∈ [t0 , t0 + p ] µ (t ) = F (t ) − Fp (t ) < (2.216) δ , if t ≥ t0 + p Here, for every t ≥ t0 , the following statements hold: ɺɺ(t ) satisfies the constraints (2.176); (a) the real robot motion q (t ), qɺ (t ), q (b) the control u (t ) = u k (t ) is admissible, that is, the constraint (2.177) is fulfilled; (c) the real force of interaction of the robot with the environment, F (t ) , satisfies the constraint (2.178). Besides, the goal conditions (2.175) will be fulfilled for robot’s motion not later than the time instant
1 CΓ x(t0 ) , if CΓ x(t0 ) + a ≥ ε t0 + ln tp1 = λ ε −a t , if not 0
(2.217)
and for the force of interaction of the robot with environment not later than
1 LCΓ x(t0 ) , if LCΓ x(t0 ) + La + Cρ ≥ δ t0 + ln λ δ − La − Cρ tp2 = t , if not 0
(2.218)
The proof of Theorem 7 is given in Appendix H. The adaptive control scheme proposed here enables one to solve the contact task for robot with both stationary and nonstationary dynamics. We consider the classes of stabilized motions and forces and their stabilization accuracies depending on the level of initial and external perturbations of dynamics of the robot and its environment, as well as of the sensors’ errors, processing speed of the adaptation algorithm, and other parameters of the adaptive control scheme. Because the results presented are based on the realistic assumptions about the type and character of perturbations acting on the environment-robot system to
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Dynamics and Robust Control of Robot-Environment Interaction
which the adaptive control scheme is to be applied, the obtained solution gives a justification of the possibility of its practical implementation in the controllers with adaptive features. It should also be emphasized that the proposed adaptive control laws provide stability of both interaction force and motion in a theoretically correct way, so that simulation experiments are not needed. The issues concerning practical implementation of the results presented in this section and the possible implications of these results for modification and generalization of the standard procedure of impedance control deserve further systematic research.
2.6 Position-Force Control – A Generalization 2.6.1 Models of robot and environment dynamics. Task setting Consider a robot interacting with its dynamic environment, assuming that the robot coordinates q and velocities qɺ are sufficient to represent the state of the environment, and that the number of interacting forces is smaller than the number of DOFs. The robot interacting with environment (see Appendix I and also [26]) can be modeled as:
H (q )qɺɺ + h(q, qɺ ) = τ + J T (q ) F
(2.219)
Mɶ (q )qɺɺ + Lɶ (q, qɺ ) = − J T (q ) F
(2.220)
where q is an n -dimensional vector of generalized robot coordinates; H ( q ) is an n × n matrix of inertia of the manipulator mechanism; Mɶ (q ) is n × n matrix representing environment inertia, h( q, qɺ ), Lɶ ( q, qɺ ) are n -dimensional nonlinear functions of centrifugal, Coriolis’ and gravitational terms; τ is an n dimensional vector of inputs; F is an m -dimensional vector of generalized forces acting on the end-effector from the environment side; J T ( q ) is an n × m full-rank matrix connecting the vector F with the generalized forces associated to the generalized robot coordinates q . Note that, by assumption, the interaction force F is uniquely determined by the robot motion q (t ) . However, the converse is not true. Namely, the matrix
Mɶ (q ) is positive semidefinite and rank Mɶ (q ) = m , so equation (2.220) contains only m independent differential equations. Hence, for a given interaction force F (t ) equation (2.220) does not have a unique solution with
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147
ɺɺ , and the robot motion q (t ) cannot be determined by integrating respect to q equation (2.220). However, the interaction force F (t ) determines in full the motion of the environment (see Remark 1 in Appendix I). For these reasons it is more convenient to use an equivalent representation of the environment dynamics in the form: M (q )qɺɺ + L(q, qɺ ) = S T (q ) F (2.221) ɺ where M (q ) is a continuous m × n full rank matrix, L( q, q ) is a continuous m -vector function, and S (q ) is a nonsingular continuous m × m matrix. Note: For simplicity and without loss of generality in Sections 2.1 – 2.4 the case when the number n of generalized coordinates is equal to the number m of generalized forces is considered. A general case ( n ≥ m ) dealt with in Section 2.5 differs from the case n = m only by the representation of environment equation in the form solved with respect to force: −1
F = ( S (q ) S T (q ) ) S (q ) [ M (q )qɺɺ + L(q, qɺ )] as against
F = ( S T (q) )
−1
[ M (q)qɺɺ + L(q, qɺ )] .
In this section we make generalization of the environment vector equation for the case when the number of its one-dimensional linearly independent equations is smaller than n and equal to the number m which simultaneously coincide with the number of generalized forces. The matrix M (q ) can be partitioned into the submatrices M 1 , M 2 such that: M ( q ) = M 1 ( q ) M 2 ( q ) . Here M 1 ( q ) is m × ( n − m) matrix and
M 2 ( q ) is m × m matrix with rank M 2 (q ) = m, ∀q ∈ R n . Then for the q (1) , where q (1) ∈ R n − m , q (2) ∈ R m , the expression M (q )qɺɺ can (2) q
vector q =
be written as follows
M (q )qɺɺ = M 1 (q )qɺɺ(1) + M 2 (q )qɺɺ(2) It is assumed that the conditions for the existence and uniqueness of the solutions of (2.219), (2.221), with the initial conditions q (t0 ) = q0 , and
qɺ (t0 ) = qɺ0 are satisfied. As we have seen in the previous sections, the goal of the control of a robot in contact with its environment is to realize the programmed motion ( PM ) q p (t )
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Dynamics and Robust Control of Robot-Environment Interaction
in the presence of the programmed force interaction ( PFI ) Fp (t ) with the environment, where q p (t ) and Fp (t ) must satisfy (2.219), (2.221). It is possible to choose q p (t ) and then determine Fp (t ) from (2.221), or alternatively, to (2) specify Fp (t ) and programmed motion q (1) p (t ) , whereby the subvector q p (t )
can be obtained by integrating (2.221). The control task is to define the control τ , which for all of the initial conditions q (t0 ) ≠ q p (t0 ), qɺ (t0 ) ≠ qɺ p (t0 ), F (t0 ) ≠ Fp (t0 ) , satisfies the following goal conditions:
q (t ) → q p (t ), qɺ (t ) → qɺ p (t ), t → ∞
(2.222)
F (t ) → Fp (t ), t → ∞
(2.223)
with the quality of transient processes specified in advance. However, the transient processes for both motion and force cannot be arbitrarily specified. Namely, because of equation (2.221), the motion transient process fully determines the force transient process. On the other hand, the force transient process determines the motion transient process only partially. For a given force transient process, part of motion coordinates q (1) (t ) has to be specified in order to determine the rest of motion coordinates. For these reasons, two subtasks can be identified. The first one is the attaining of the control goal (2.222), (2.223) with specified motion transient processes, and the second one is the attaining of the control goal (2.222), (2.223) with the specified processes for the force F (t ) and motion coordinates q (1) (t ) . The task of Fp (t ) stabilization can be defined as follows: Ensure the fulfillment of the condition (2.223) with the preset quality of transient processes specified by the integral equation: t
µ (t ) = ∫ Q ( µ (ω ) ) d ω
(2.224)
t0
where µ (t ) = F (t ) − Fp (t ) ; Q is a continuously differentiable vector function characterizing the quality of transient processes. Equation (2.224) can be presented in an equivalent form as: µɺ (t ) = Q ( µ (t ) ) (2.225)
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149
However, the achievement of Fp (t ) stabilization can ensure the stabilization of m robot DOFs only, represented by the vector q (2) . So, for the force stabilization task to be well defined, it is also required to set the condition:
ɺ (1) ɺ (1) q (1) (t ) → q (1) p (t ), q (t ) → q p (t ), t → ∞
(2.226)
which should be realized in such a way to ensure the desired quality of transient processes defined by the equation:
ηɺɺ1 = P1 (η1 ,ηɺ1 )
(2.227)
where η1 (t ) = q (1) (t ) − q (1) p (t ) ; P1 is ( n − m ) -dimensional vector function continuously differentiable with respect to all arguments, such that equation (2.227) has the trivial solution η1 = 0 . The control should be synthesized in such a way that the perturbed robot motion and interaction force in closed-loops satisfy (2.225), (2.227). It is a priori adopted that the choice of the functions P1 and Q ensures exponential stability (thus implying asymptotic stability) in the whole of the trivial solution of the systems (2.225), (2.227). The control laws stabilizing the motion with the required quality of transient processes were presented in [10, 11]. The answer to the question how to ensure the contact force stability is quite simple: computed torque method [27, 28] ensures that the desired motion quality is achieved and, at the same time, guarantees that interaction force is asymptotically stable. In the text to follow we will focus our attention only on the control laws that stabilize the interaction force and on the way how to satisfy the stable position requirements. For the case when the number of contact force components is greater than the number of robot DOFs, this delicate task needs new analyses.
2.6.2 Control laws stabilizing the interaction force The control laws proposed in [10, 11] ensure force stabilization not only in case when m = n but also in case when m < n . These control laws can be easily modified to suit the case considered in Section 2.6.1. To define control laws, we introduce the notation: t
∆F = ∫ Q ( µ (ω ) ) dω t0
(2.228)
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Dynamics and Robust Control of Robot-Environment Interaction
ɺ Ψ (q, qɺ , F ) = M 2−1 (q ) S T (q ) F − L(q, qɺ ) − M 1 (q ) ( qɺɺ(1) p + P1 (η1 ,η1 ) ) qɺɺ(1) + P1 (η1 ,ηɺ1 ) q( F ) = p Ψ ( q, qɺ , F )
(2.229) (2.230)
and formulate the following assertion.
Assertion: The robot and environment dynamics equations (2.219), (2.221) for each of the following control laws:
ɺ qɺɺ(1) p + P1 (η1 ,η1 ) τ I = H (q ) + h(q, qɺ ) − J T (q) F Ψ ( q, qɺ , Fp + ∆F ) ɺ qɺɺ(1) p + P1 (η1 ,η1 ) T + h(q, qɺ ) − J (q ) ( Fp + ∆F ) ɺ Ψ q , q , F ( )
τ II = H (q )
τ III
ɺ qɺɺ(1) p + P1 (η1 ,η1 ) = H (q) + h(q, qɺ ) − J T (q) ( Fp + ∆F ) ɺ Ψ ( q, q, Fp + ∆F )
(2.231)
(2.232)
(2.233)
in a closed loop are equivalent to the reference equations (2.225), (2.227). Due to notation (2.230) the control laws (2.231) – (2.233) can be rewritten in the following way
τ I = H (q)q( Fp + ∆F ) + h(q, qɺ ) − J T (q) F
(2.234)
τ II = H (q) q( F ) + h(q, qɺ ) − J T (q) ( Fp + ∆F )
(2.235)
τ III = H (q)q( Fp + ∆F ) + h(q, qɺ ) − J T (q ) ( Fp + ∆F )
(2.236)
The three control laws defined by the above assertion stabilize the desired interaction force with a specified quality of transient processes. These control laws utilize the feedback with respect to q, qɺ and force F . Let us show that these control laws ensure force stabilization. To illustrate this, let us present the proof of Assertion for the control law (2.231) separately. Substituting the control law (2.231) into the robot dynamics equation (2.219) we obtain:
H (q ) ( qɺɺ − q ( Fp + ∆F ) ) = 0
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151
which, in view of nonsingularity of the matrix H ( q ) , directly implies:
ɺ qɺɺ(1) − qɺɺ(1) p − P1 (η1 , η1 ) = 0
(2.237)
qɺɺ(2) = Ψ ( q, qɺ , Fp + ∆F )
(2.238)
Since rank M 2 ( q ) = m the environment dynamics (2.221) can be written in the form:
qɺɺ(2) = M 2−1 (q ) S T (q ) F − L(q, qɺ ) − M 1qɺɺ(1)
(2.239)
Let us eliminate qɺɺ(2) from (2.238) and (2.239). Then, in accordance with the notation (2.229), we will have
ɺ S T (q ) ( F − Fp − ∆F ) − M 1 ( qɺɺ(1) − qɺɺ(1) p − P1 (η1 , η1 ) ) = 0 Taking into account the identity (2.237) we obtain:
S T (q ) ( F − Fp − ∆F ) = 0 Since S T ( q ) is by assumption nonsingular, it follows that the closed-loop robot dynamics is described by the equations:
ηɺɺ1 = P1 (η1 ,ηɺ1 ) µɺ (t ) = Q ( µ (t ) ) and the control goal is fulfilled. Below we give the integral proof of Assertion. Let us add to each of the control laws (2.234) – (2.236) one by one three evident identities. Then we obtain three pairs of the following expressions:
H (q )q ( Fp + ∆F ) + h(q, qɺ ) = τ I + J T (q ) F T M (q )q ( Fp + ∆F ) + L(q, qɺ ) = S (q ) ( Fp + ∆F )
(2.240)
H (q) q( F ) + h(q, qɺ ) = τ II + J T (q) ( Fp + ∆F ) M (q) q( F ) + L(q, qɺ ) = S T (q) F
(2.241)
H (q )q ( Fp + ∆F ) + h(q, qɺ ) = τ III + J T (q ) ( Fp + ∆F ) M (q )q ( Fp + ∆F ) + L(q, qɺ ) = S T (q ) ( Fp + ∆F )
(2.242)
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Dynamics and Robust Control of Robot-Environment Interaction
Subtracting the equations of the expressions (2.240), (2.241), (2.242) from (2.219) and (2.221), respectively, we obtain the relations:
0 qɺɺ − q ( Fp + ∆F ) H (q) M (q) − S T (q) =0 µ − ∆F
(2.243)
H (q ) − J T (q) qɺɺ − q( F ) =0 0 µ − ∆F M (q)
(2.244)
H (q) − J T (q) qɺɺ − q( Fp + ∆F ) =0 T µ − ∆F M (q) − S (q)
(2.245)
Since all the system matrices in the above equations are nonsingular (see Remarks 4-6 in Appendix I), it follows that for all control laws τ I ,τ II ,τ III one of the following conditions is fulfilled:
qɺɺ − q ( Fp + ∆F ) = 0 qɺɺ − q ( F ) = 0 or µ − ∆F = 0 µ − ∆F = 0
(2.246)
Using (2.228) – (2.230) we obtain:
ɺ qɺɺ(1) − qɺɺ(1) p − P1 (η1 , η1 ) = 0
(2.247)
t
µ = ∫ Q ( µ (ω ) ) dω
(2.248)
t0
which finally proves that the robot environment with each of the control laws (2.231) – (2.233) behaves according to the reference equations:
ηɺɺ1 = P1 (η1 ,ηɺ1 )
(2.249)
µɺ (t ) = Q ( µ (t ) )
(2.250)
Consider now the question: is it sufficient to stabilize Fp (t ) and q (1) p (t ) in
(
order to have the real motion converging to the programmed one q (t ) → q p (t ), when t → ∞ ) ? Note: The above assertion ensures that the conditions (2.223), (2.226) are satisfied. To have stability of the closed-loop system ensured (i.e. the condition (2.222) fulfilled) it is necessary (and sufficient too) to ensure that
A Unified Approach to Dynamic Control of Robots
ɺ (2) (t ) → qɺ (2) q (2) (t ) → q (2) p (t ), q p (t ), t → ∞
153
(2.251)
Let any of the control laws (2.231) – (2.233) be applied onto the system (2.219), (2.221). Using the environment dynamics model (2.221) we obtain:
M (q)qɺɺ − M (q p )qɺɺp + L(q, qɺ ) − L(q p , qɺ p ) = S T (q) F − S T (q p ) Fp Since, according to Assertion, the system (2.219), (2.221) in a closed loop with the control laws (2.231) – (2.233) is equivalent to the system in deviation form (2.225), (2.227), the preceding relation reduces to the equation:
ηɺɺ2 + K (η ,ηɺ , t ) = M 2−1 (η + q p ) S T (η + q p ) ( F − Fp )
(2.252)
where
η2 (t ) = q (2) (t ) − q (2) p (t ), η (t ) = q (t ) − q p (t ) and:
K (η ,ηɺ , t ) = M 2−1 (η + q p ) {L (η + q p ,ηɺ + qɺ p ) − L (q p , qɺ p ) + T T + M (η + q p ) − M (q p ) qɺɺp − S (η + q p ) − S (q p ) Fp +
+ M 1 (q p +η ) P1 (η1 ,ηɺ1 )} . If F (t ) = Fp (t ) , this equation reduces to:
ηɺɺ2 + K (η ,ηɺ , t ) = 0
(2.253)
Evidently, the trivial solution η 2 (t ) = 0 is stable only if the environment has such properties that ensure fulfillment of the condition q (2) (t ) → q (2) p (t ) , as
t → ∞ Hence, if F (t ) ≠ Fp (t ), ∀t ≥ t0 , the stabilization of Fp (t ) does not necessarily ensure stability of the motion in contact with the environment. It is shown that the motion of the robot and environment in a closed loop is described by equations (2.249), (2.252):
ηɺɺ1 = P1 (η1 ,ηɺ1 ) ηɺɺ2 + K (η ,ηɺ , t ) = M 2−1 (η + q p ) S T (η + q p ) ( F − Fp ) Let us formulate sufficient conditions for stabilization of q (2) p (t ) , i.e. the conditions for which exponential stability of the trivial solution of the system (2.253) induces exponential stability of the trivial solution of the system (2.252).
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Introduce the notation:
η η x ηɺ1 x1 = 1 , x 2 = 2 , x = 1 , P ( x1 ) = P(η1 ,ηɺ1 ) ηɺ1 ηɺ 2 x2 0 ηɺ2 , β ( x2 , x1 , t ) = −1 K( x2 , x1 , t ) = T − K (η ,ηɺ , t ) M 2 (η + q p (t ) ) S (η + q p (t ) ) Then, the system (2.249), (2.252) can be rewritten as:
xɺ1 = P ( x1 )
(2.254)
xɺ2 = K( x2 , x1 , t ) + β ( x2 , x1 , t ) µ
(2.255)
On linearizing the equation (2.255) in the neighborhood of x = 0 , we get:
xɺ 2 = A(t ) x 2 + B(t ) x1 + α ( x 2 , x1 , t ) + β ( x 2 , x1 , t ) µ
(2.256)
where A(t ) and B (t ) are 2m × 2m and 2m × 2( n − m) matrices, respectively,
α ( x2 , x1 , t ) = o ( x ) when x → 0 and sup A(t ) < ∞ since q p (t ), qɺ p (t ), Fp (t ) t
belong to the bounded regions. We assume here that all vector functions possess the same smoothness as in Section 2.3.2. Let the vector functions P1 and Q in (2.225), (2.227) ensure that for their arbitrary solutions µ (t ), η1 (t ) the following estimates hold:
µ (t ) ≤ D1e − λ (t −t ) µ (t0 )
(2.257)
x1 (t ) ≤ D2 e − λ (t −t0 ) x1 (t0 )
(2.258)
0
with the constants D1 , D2 , λ > 0 . Sufficient conditions for the exponential stability of the solution of the system of differential equations (2.254), (2.255) are given by the following theorem analogous to the Theorem 1:
Theorem 8. Let the following conditions be satisfied [10, 11]: 1. The linear system
xɺ 2 = A(t ) x 2 is regular, i.e. there exists the limit
(2.259)
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155
t
2m 1 lim ∫ Sp A(ω ) dω = σ 0 and σ 0 = ∑ α k , t →∞ t i =1 t0
where α k ( k = 1,2,..., 2m) are characteristic indices of the solution of the system (2.259), Sp A is the trace of matrix A . 2. All characteristic indices α k ( k = 1, 2,… , 2m) are negative. 3. The index λ in (2.257), (2.258) and the number γ > 0 satisfy the inequalities:
max α k < −γ < 0 , k
−λ < −γ
then, for sufficiently small initial perturbations x (t0 ) and µ (t0 ) , the transient process of the system (2.255) satisfies the inequality:
b x1 (t0 ) + c µ (t0 ) − λ (t −t0 ) , x2 (t ) ≤ a x2 (t0 ) + e λ −γ
∀t ≥ t0
(2.260)
for some positive constants a, b, c , and consequently, the system (2.254), (2.255) is exponentially stable. The proof of Theorem 8 is given in Appendix J. The control laws (2.231) – (2.233) ensure a desired quality of stabilization of the robot interaction force Fp (t ) with the environment. At the same time, the stabilization of the robot motion q p (t ) can also be achieved for sufficiently small perturbations.
2.6.3 Example To illustrate the obtained theoretical results, let us consider the following hypothetical contact task: the 2-DOF sliding manipulator has to move a tool over the support which behaves as a system with distributed parameters (Fig. 2.5). The control goal is to realize the nominal motion along the x -axis x p (t ) = x 0 (t ) = V0 t , V0 = const and nominal force along the y -axis
Fy p (t ) = Fy0 = F 0 = const. The component of contact force along the x -axis is the sum of the friction terms:
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Dynamics and Robust Control of Robot-Environment Interaction
Fx = hx xɺ +ν Fy sgn xɺ
(2.261)
while the contact force component Fy is the sum of inertial, frictional and elastic terms:
Fy = m y ɺyɺ + h y yɺ + k y y
(2.262)
Fig. 2.5 Robot in contact with dynamic environment
It is adopted in (2.261), (2.262) that m y = me , where me is the equivalent mass representing the contribution of the environment inertia. Further, h y , hx denotes viscous friction, k y is the environment stiffness and ν is the static friction coefficient. Using the notation given in Fig. 2.5 the robot dynamics model is:
(m1 + m 2 ) ɺxɺ = τ x − Fx
(2.263)
m 2 ɺyɺ = τ y − Fy
(2.264)
After introducing the notation
τ x (m + m2 ) 0 x q = , τ = , F = Fy , H (q ) = 1 , 0 m2 y τ y h xɺ −ν sgn xɺ h(q, qɺ ) = x , J (q ) = , 0 −1
M (q ) = 0 m y , L(q, qɺ ) = hy yɺ + k y y
S T (q ) = [1]
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equations (2.261) – (2.264) reduce to (2.219), (2.221). Determine now the remaining nominal motion component from (2.262):
y p (t ) = y 0 (t ) =
1 0 F ky
(2.265)
and choose equations specifying the desired motion and force transient processes in the form:
where
ηɺɺx + 2 ς xcω xcηɺ x + ω xc2 η x = 0
(2.266)
µɺ + T yc µ = 0
(2.267)
η x = x − x p = x − x 0 , µ = F − Fy = F − F 0 p
and
ς xc , ω xc , T yc
are
constants. Let us apply the control law (2.233), which in this case is:
0 t 0 2 ɺ ɺɺ ɺ τ x = (m1 + m2 ) ( x − 2ς xcω xcη x − ωxcη x ) + hx x +ν F + ∫ µ d ω sgn xɺ (2.268) t0 m m2 0 t τ y = + 1 F + ∫ µ d ω − 2 (hy yɺ + k y y ) m t0 y my
(2.269)
It is obvious that with this control law equations (2.266), (2.267) represent the closed-loop system behavior. In order to examine the nominal motion and force stability, the environment equation (2.262) has to be considered in the deviation form:
m yηɺɺy + h yηɺ y + k yη y = µ
(2.270)
It is easy to check that if the conditions of Theorem are satisfied, i.e. equation (2.270) has characteristic values α 1 , α 2 < − γ < 0 , for some γ , and
ς xc , ω xc , T yc are chosen such that the characteristic values of (2.266), (2.267) are smaller than α 1 , α 2 , then the closed-loop system is stable. 2.6.4 Conclusion The control methodology presented in this section is applicable to a wide class of robotic tasks. It exhibits the same features as in the case when the dimension of the control input vector and contact force vector are equal. Namely, the synthesis of control laws is performed in order to ensure local asymptotic
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Dynamics and Robust Control of Robot-Environment Interaction
stability of the desired robot behavior. Stabilization of both desired robot motion and interaction force is realized simultaneously, for the cases in which environment dynamics is described by second-order differential equations. All necessary modifications of control laws stabilizing the interaction force of the robot and environment have been performed and the corresponding stability theorem has been proved. The presented theorem formulates sufficient conditions for asymptotic stability of the system in the first approximation (local stability). It should be emphasized that the character of this position (displacement) stability depends substantially on the nature of programmed motion and programmed force of interaction. Nevertheless, the presented linear analysis gives a very important criterion that must be verified for any force-based control law. It should be pointed out that the inadequate accuracy of the environment dynamics model can significantly influence the contact task performance. Inaccuracies of the robot and environment dynamics models, as well as dynamic control robustness are considered in Sections 2.4 and 2.5.
2.7 Position-Force Control in Cartesian Space 2.7.1 Introduction All the results obtained previously on the basis of dynamic models of the robot and environment in the system of generalized coordinates can be, without loss of generality, directly transposed into the Cartesian coordinate frame of the robot’s end-effector. In this operation, the representations of the robot and environment dynamics models as the systems of differential equations (2.6) and (2.11) retain their form. In the previous sections, the control laws stabilizing simultaneously the robot motion and interaction force have been synthesized. These control laws possess exponential stability of closed-loop systems and ensure the preset quality of transient processes of motion and interaction force. However, for the control laws stabilizing the desired interaction force with a preset quality of transient processes, only sufficient conditions for the exponential stability of motion were given. This section presents a more detailed description of the environment dynamics. In the cases when the environment dynamics can be approximated sufficiently well by a linear time-invariant model in the Cartesian space, necessary and sufficient conditions for the exponential stability of both motion and force are derived, and the corresponding control laws are defined.
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2.7.2 Task setting Consider the model of the robot interacting with its environment (2.6):
H (q )qɺɺ + h(q, qɺ ) =τ + J T (q ) F The model of environment dynamics can be described by equation (2.11):
M (q )qɺɺ + L(q, qɺ ) = S T (q ) F In this section, as in all previous sections except for Sections 2.5, 2.6, to simplify the analysis, we adopted that m = n , which means that the number of components of interactive force is equal to the number of powered DOFs. Using the operational space approach [29], the end-effector equations of motion can be written in the task frame. Let z be an n -dimensional vector of external coordinates specifying position and orientation in the chosen task frame, and let Τ be a vector of the control force coordinates. If the relation between z and q is defined by the bijective function χ :
z = χ (q) then zɺ =
(2.271)
∂χ (q )qɺ = J (q )qɺ , ɺɺ z = J (q )qɺɺ + Jɺ (q )qɺ and equations (2.6), (2.11) ∂q
can be transformed so to acquire the forms:
where:
Λ ( z ) ɺɺ z + v ( z , zɺ) = Τ + F
(2.272)
M( z ) ɺɺ z + L( z , zɺ ) = − F
(2.273)
Λ ( z ) = J −T ( χ −1 ( z ) ) H ( χ −1 ( z ) ) J −1 ( χ −1 ( z ) ) ,
(
)
v ( z , zɺ ) = J −T ( χ −1 ( z ) ) h χ −1 ( z ), J −1 ( χ −1 ( z ) ) zɺ − − Λ ( z ) Jɺ ( χ −1 ( z ) ) J −1 ( χ −1 ( z ) ) zɺ,
Τ = J − T ( χ −1 ( z ) ) τ ,
M( z ) = − S −T ( χ −1 ( z ) ) M ( χ −1 ( z ) ) J −1 ( χ −1 ( z ) ) ,
(
)
L( z , zɺ) = − S −T ( χ −1 ( z ) ) L χ −1 ( z ), J −1 ( χ −1 ( z ) ) zɺ + + M( z ) Jɺ ( χ −1 ( z ) ) J −1 ( χ −1 ( z ) ) zɺ.
( )
Here J −T (⋅) = J −1
T
(⋅) .
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Dynamics and Robust Control of Robot-Environment Interaction
It is assumed that all the functions in (2.6), (2.11), i.e. (2.272), (2.273) are continuously differentiable with respect to all variables, ensuring thus the existence and uniqueness of the solutions for (2.272), (2.273), with the initial conditions z (t0 ) = z0 , zɺ (t0 ) = zɺ0 . Let z p (t ) be the desired motion and let Fp (t ) be the desired interaction force, satisfying (2.273). The control objective is to realize the asymptotically stable z p (t ), Fp (t ) , i.e. to achieve:
( z (t ), zɺ(t ), F (t ) ) → ( z p (t ), zɺ p (t ), Fp (t ) ) ,
t →∞
It is additionally required that the closed-loop system possesses contact force transient processes (force dynamics) specified by the equation: µɺɺ = Q( µ , µɺ ) (2.274) where µ (t ) = F (t ) − Fp (t ) . Alternatively, it may be required that the closedloop system possesses motion transient processes (motion dynamics) specified by the equation: ηɺɺ = P(η ,ηɺ ) (2.275) where η (t ) = z (t ) − z p (t ) . The vector functions P and Q are assumed to be continuously differentiable with respect to all arguments, so that equations (2.274) and (2.275) have the asymptotically stable trivial solutions η = 0 and µ = 0 , respectively. Note that because of (2.273) Q ( µ , µɺ ) and P (η ,ηɺ ) can not be arbitrarily specified at the same time. Thus the force stabilization task and motion stabilization task should be distinguished. 2.7.3 Relation to previous results The computed torque-based control law has been proposed to stabilize the desired motion z p (t ) with the quality of transient processes specified by (2.275), and also to stabilize the contact force Fp (t ) :
(
Τ = U z , zɺ, ɺɺ z p + P (η ,ηɺ ), f ( z , zɺ, ɺɺ z p + P (η ,ηɺ ) )
)
(2.276)
where U ( z , zɺ, ɺɺ z , F ) = Λ ( z ) ɺɺ z + v ( z , zɺ ) − F , f ( z , zɺ, ɺɺ z ) = −M( z ) ɺɺ z − L( z , zɺ ) .
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161
By assumption, the vector function P (η ,ηɺ ) has the property of assuring an asymptotic stability of the entire trivial solution η = 0 of the system (2.275), i.e. η (t ) → 0, ηɺ (t ) → 0, t → ∞ . Because of the continuity of the function f we have: F (t ) → Fp (t ), t → ∞ . We give here the control laws: t t T = U z , zɺ, Ψ z , zɺ, Fp + ∫ Q ( µ (ω ) ) dω , Fp + ∫ Q ( µ (ω ) ) dω t0 t0
(2.277)
where Ψ ( z , zɺ, F ) = −M−1 ( z ) [ F + L( z , zɺ) ] , to stabilize a desired interaction force with the quality of transient processes specified by the equation:
µɺ = Q ( µ )
(2.278)
The control laws are applicable only if the matrix M( p ) is invertible. However, these control laws allow stable motion of the robot in contact with environment if the environment possesses the properties that ensure fulfillment of the limiting condition z (t ) → z p (t ), t → ∞ , i.e. the environment stabilizes motion of the robot in contact. 2.7.4 Control laws for specified force dynamics Let us observe first the relation between the closed-loop motion and force dynamics. Assume that the robot is in closed loop and that the corresponding motion and force dynamics are described by (2.275), (2.274), i.e. by the vector functions P (η ,ηɺ ) and Q ( µ , µɺ ) , respectively. If the motion dynamics is specified (vector function P (⋅, ⋅) given), then, because of (2.273), the vector function Q ( µ , µɺ ) is uniquely determined. Furthermore, if η = 0, ηɺ = 0 , then µ = 0 . However, in the converse case, having Q( µ , µɺ ) specified, the vector function P (η ,ηɺ ) is not generally uniquely determined. Moreover, µ = 0, µɺ = 0 does not imply η = 0 , therefore the force stabilization does not necessarily imply the motion stabilization. To simplify derivation, let us focus on a particular class of control laws (2.276) (obtained after subtracting equation (2.273) from (2.272) and compensating for all nonlinearities)
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Dynamics and Robust Control of Robot-Environment Interaction
(
)
τ = J T ( χ −1 ( z ) ) ( Λ ( z ) + M( z ) ) ( ɺɺz p + Γ1ηɺ + Γ 2η ) + v ( z , zɺ) + L( z , zɺ) (2.279) where z = χ ( q ) , which enable linear closed-loop dynamics
ηɺɺ = Γ1ηɺ + Γ2η
(2.280)
where Γ1 and Γ2 are n × n constant matrices. If Γ1 , Γ 2 are chosen so that
det ( − I λ 2 + Γ1λ + Γ 2 ) is Hurwitz, equation (2.280) represents a class of
admissible motions, and the convergence to zero of the errors is ensured. However, the force dynamics is determined by Γ1 , Γ2 . Our aim is to attain the given force transient processes (2.274), still ensuring stability of the motion defined by (2.280) with some matrices Γ1 , Γ 2 . However, the problem is to determine if any of the corresponding matrices exist, and how to find them. To obtain the correspondence between the motion and force dynamics, assume that in the neighborhood of the nominal trajectory the environment dynamics equation (2.273) can be approximated sufficiently well by the equation
µ = Mηɺɺ + Lhηɺ + Lkη
(2.281)
where M , Lh , Lk are n × n constant matrices of inertia, viscous friction and stiffness, respectively. This approximation is often used in Cartesian space. Note also that (2.281) does not impose additional restrictive conditions such as real positiveness of the corresponding transfer function [30].
η
µ
Defining x = , Γ = Γ 2 Γ1 , L = Lk Lh , y = , equations ηɺ µɺ (2.280), (2.281) can be written as:
0 xɺ = n Γ 2
In x ≡ Γη x Γ1
µ = ( MΓ + L) x ≡ Cx
(2.282) (2.283)
From (2.283) we have
C η y = ɺ ≡ α x C Γ η η In a more expanded form, the matrix α can be written as follows:
(2.284)
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MΓ + L
MΓ + L
2 k 1 h α = ( M Γ + L ) Γ ( M Γ + L ) Γ + M Γ 2 + Lk 1 h 2 1 h 1
If the matrix α is nonsingular, then (2.282), (2.284) yield
yɺ = Qµ y , where Qµ = α Γηα −1
(2.285)
Introducing the notation:
α 0 = Lk + M Γ 2 , α i +1 = βi Γ 2 β0 = Lh + M Γ1 , βi +1 = α i + βi Γ1 , i = 0,1
(2.286)
µɺɺ = Q1 µɺ + Q2 µ
(2.287)
we obtain:
where
[Q2 |
α β Q1 ] = [α 2 | β 2 ]α −1 , α = 0 0 . α1 β1
Therefore, the linear closed-loop motion dynamics (2.280) induces the linear force dynamics given by (2.287). Furthermore, the motion and force transient processes are the same. It follows from the fact that because of (2.285) the matrices Γη , Qµ have equal eigenvalues. Note: When the matrix α is singular, the force dynamics in closed loop is still linear, but it can not be represented in the form of (2.287). In fact, the following cases are possible: (i) rank α < n . Force variables µ are not independent (see (2.283), (2.284)). However, this contradicts the assumption that there are n independent forces. (ii) rank α = n . It is easily seen from (2.283), (2.284) that in this case the variable µɺ is dependent on µ and the induced force dynamics equation reduces to the degenerative form:
Q1 µɺ + Q2 µ = 0
(2.288)
Namely, let Yi , i = 1, 2,..., 2n be the rows of α . Then n of them (say first
n ) are linearly independent, and the matrix Q2 = qij containing no zero n× n row, such that Yi + n =
n
∑q Y j =1
ij
j
, exists. Therefore
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Dynamics and Robust Control of Robot-Environment Interaction n
n
j =1
j =1
µɺ i = Yi + n x = ∑ qijY j x = ∑ qij µ j . When any n rows of α are linearly independent, (2.288) is obtained. (iii) n < rank α < 2n . Induced force dynamics is represented by the linear system consisting of k equations of the type (2.287) and n − k equations of the type (2.288), where k = rank α − n . Singular matrix α usually appears when the force vector µ , still being linearly independent, effectively depends on some (but not on all) DOFs. For example, if η i is the i -th generalized coordinate, then µ j = k jη i , µ i = hiηɺ i are linearly independent, but they are only influenced by the i -th DOF. Although in this case there are n independent forces to be controlled, it can be seen that the environment configuration space is a subspace of the robot configuration space, which contradicts the assumption that the environment possesses m = n DOFs. The relation between stability characteristics of the motion and force dynamics equations (2.280), (2.287) is given by the following theorem based on the well-known linear system theorem [31]. Theorem 9. A nonsingular coordinate transformation (preserving eigenvalues) T
T
between the motion η T , ηɺ T and force µ T , µɺ T variables will exist if and only if the system (2.282), (2.283) is completely observable and the observability index is 2. A corollary of the theorem is that if a desired force dynamics is specified in the form of equation (2.287) and all characteristic roots are stable, then the corresponding motion dynamics equation will have a stable solution with the same characteristic roots, provided that the matrix α is nonsingular. Finally, let us answer the question on how to find the corresponding matrices Γ1 , Γ2 , having the matrices Q1 , Q2 specified, ensuring stable force dynamics. Let us introduce the matrices:
c0 = Lk
| Lh − I n , c1 = [ I n | M ]
then the matrix C can be written as C = c0 + c1Γη . It follows from (2.287) and (2.284)
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165
C η µ µɺɺ = [Q2 | Q1 ] = [Q2 | Q1 ] µɺ C Γη ηɺ ɺɺ yields the relation: Differentiating (2.283) twice and substituting µ
C C Γη2 = [ Q2 | Q1 ] C Γη
(2.289)
which can be reduced to:
(c0 + c1Γη ) Γη2 = Q1 (c0 + c1Γη )Γη + Q2 (c0 + c1Γη )
(2.290)
c1Γη3 + (c0 − Q1c1 )Γη2 − (Q1c0 + Q2 c1 )Γη − Q2 c0 = 0
(2.291)
i.e.
Given the environment parameters Lk , Lh , M (i.e. c0 , c1 ) and desired (stable) force dynamics specified by Q1 , Q2 , the solution (if it exists) of the obtained equation yields the matrices Γ1 , Γ2 . If there exists a matrix Γη , satisfying equation (2.291), and if the matrix α is nonsingular, then the desired force dynamics specified by (2.287) can be realized by the control law (2.279). At the same time, the closed-loop robot motion will be stable, with the motion dynamics satisfying (2.280) and force dynamics satisfying (2.287). The proposed control law is applicable under sufficient condition that the matrix α is nonsingular. However, this condition is also necessary for the existence of the solution of the problem posed in Section 4.2. Namely, according to Note, the matrix α is singular if either m < n (not considered) or closedloop force dynamics equation has a degenerative form. Since the matrix α depends on Γ1 , Γ2 and thus on Q1 , Q2 , the latter case arises from an inappropriate specification of desired force dynamics. A question should be posed: Is it possible to have the matrices Q1 , Q2 always chosen so as to ensure nonsingularity of the matrix α ? The answer gives the following theorem. Theorem 10. Given the n × n matrices M , Lh , Lk , with rank [ Lk | Lh | M ] = n , the matrices Q1 , Q2 can always be chosen so that the corresponding matrices Γ1 , Γ2 exist and the matrix α is nonsingular.
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Dynamics and Robust Control of Robot-Environment Interaction
Proof of the theorem is given in Appendix K. Note that due to the correspondence between the closed-loop motion and force dynamics, the control law stabilizes the overall system with either motion or force of transient processes being preset. 2.7.5 Example To illustrate the obtained theoretical results let us consider the following hypothetical contact task: The planar two-link revolute manipulator, whose workspace is the horizontal plane, has to move the workpiece over a flat support which may rotate about the horizontal axis (Fig. 2.6).
p
x p
y
q
m
2
y R
m2 ϕ
m1
l2 l1
q
1
Fg
O h R ϕ mg
y
0
Fig. 2.6 Robot in contact with dynamic environment
The task is to realize the nominal motion along the x -axis:
x p (t ) = x 0 (t ) = V0 (t ), V0 = const > 0
(2.292)
and nominal contact force between the end-effector and the workpiece along the y -axis:
A Unified Approach to Dynamic Control of Robots
Fy p (t ) = Fy0 = F 0 = const > 0
167
(2.293)
Adopting that q = [ q1 q2 ] , neglecting the joints friction, and using the T
notation given in Fig. 2.6, the matrices in the robot dynamics model (2.6) can be written as:
(m1 + m2 )l12 + m2l2 (l2 + 2l1 cos q2 ) H (q) = m2l2 (l2 + 2l1 cos q2 )
m2l2 (l2 + 2l1 cos q2 ) m2l22
(2.294)
− qɺ (qɺ + 2qɺ1 ) h(q, qɺ ) = m2 l1l 2 sin q 2 2 22 qɺ 2
(2.295)
− (l sin q1 + l 2 sin (q1 + q 2 )) − l 2 sin (q1 + q 2 ) J (q) = 1 l1 cos q1 + l 2 cos (q1 + q 2 ) l 2 cos (q1 + q 2 )
(2.296)
Introduce the Cartesian position vector of the manipulator end-effector
z =[x
y ] and contact force vector F = Fx Fy . The relation between z T
T
and q is:
l1 cos ( q1 ) + l2 cos ( q1 + q2 ) z = χ (q) = l1 sin ( q1 ) + l2 sin ( q1 + q2 ) − y0 The contact force component Fy (assuming y small, y < 0 ) may be approximated by a sum of the inertial Fi = ( m + me ) ɺɺ y , the frictional Fh = hy yɺ , and the gravitational Fg = − k y y terms:
Fy = (m + me ) ɺyɺ + h y yɺ − k y y
(2.297)
while the component of contact force along the x -axis, Fx , is a sum of the inertial and frictional terms:
Fx = mxɺɺ + hx xɺ + ν x ( Fy − me ɺɺ y )sgn ( xɺ )
(2.298)
It has been adopted that m in (2.297), (2.298) is the workpiece mass, me is the support equivalent mass, whereas hy , hx and ν x denote the viscous friction coefficients and static friction coefficient respectively; k y = hme g / R 2 , where
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Dynamics and Robust Control of Robot-Environment Interaction
R and h are the distances shown in Fig. 2.6, and g is the gravity acceleration. Let us determine the remaining force and motion components from (2.297), (2.298):
Fx p (t ) = Fx0 (t ) = hxV0 + ν x F 0
y p (t ) = y 0 (t ) = −
(2.299)
1 0 F ky
(2.300)
and set the control goal be exponentially stable nominal system behavior, with the force transient processes specified by the equation:
µɺɺ = Q1 µɺ + Q2 µ
(2.301)
with µ being exponentially stable. The linearization of (2.297), (2.298) for
V0 + ηɺx > 0 gives (2.281) with
hx ν x hy 0 −ν x k y m ν x m M = L = L = , , 0 − k h k hy y 0 m + me 0 Choose the matrices Q1 , Q2 such that the matrix Qµ has the eigenvalues
λ1 , λ 2 , ..., λ 2 n and the conditions of Lemmas 5 and 6, given in Appendix K, are satisfied. Then the matrix α is nonsingular. Solving (2.291) yields the matrices Γ1 , Γ2 . As was shown in Section 2.7.4, the control law
z = χ (q)
(
τ = J T (q ) ( Λ ( z ) + M( z ) ) ( ɺɺz p + Γ1ηɺ + Γ 2η ) + v ( z , zɺ ) + L( z , zɺ )
(2.302)
)
(2.303)
ensures that in the closed-loop, force and motion deviations µ behave according to (2.301). In fact, the environment dynamics equations in deviation form (2.297), (2.298) become:
µ y = (m + me )ηɺɺy + h yηɺ y − k yη y
(2.304)
µ x = mηɺɺx + hxηɺx +ν x ( µ y − meηɺɺy )
(2.305)
Since the control law (2.303) ensures (ηɺɺ,ηɺ ,η ) → 0 as t → ∞ , it is clear that (2.304), (2.305) implies ( µɺ , µ ) → 0 .
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169
Now we consider the case when the control law (2.277) is applied. Let us specify the desired force transient processes by the equations:
µɺ x = − Q11µ x , Q11 > 0
(2.306)
µɺ y = − Q22 µ y , Q22 > 0
(2.307)
As shown in [10], the control law (2.277) will certainly ensure that in a closed loop the force deviations µ x , µ y satisfy the above equations, i.e. the desired force dynamics will be achieved. However, the motion will be unstable. Namely, the behavior of the closed-loop system will be described by equations (2.303) – (2.307). Since equations (2.304), (2.307) are independent of the variables η x , µ x , the characteristic polynomial associated to the overall system will have the factor:
Py (λ ) = ( (m + me )λ 2 + hy λ − k y ) ( λ + Q22 )
and unstable roots because k y > 0 . Therefore, in spite of achieving ( µɺ , µ ) → 0 as t → ∞ , the motion of the robot in contact is unstable. Note that if the environment dynamics equations had stable solutions (the case when − k y > 0 ) the conditions of Theorem 1 in Section 2.3 would be satisfied, and the motion would also be asymptotically stable. 2.7.6 Conclusion The task of stabilizing both the robot motion and interaction force simultaneously in Cartesian space, within the scope of the unified approach to control laws synthesis for robot manipulator in contact with dynamic environment, has been considered. The task is solved under less restrictive conditions imposed on environment dynamics than in Section 2.3, where some particular environment properties are required to ensure the overall system stability. The one-to-one correspondence between the closed-loop motion and force dynamics equations is obtained and a unique control law that ensures system’s stability and preset either motion or force transient processes, is proposed. The perturbed environment dynamics equation was assumed to be linear in Cartesian space, so that the obtained results are applicable to nonlinear environment dynamics in the range of validity of the linear approximation.
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Dynamics and Robust Control of Robot-Environment Interaction
2.8 New Realization of Hybrid Control 2.8.1 Introduction In Chapter 1 we presented sufficiently clear hybrid control. In the introduction to Chapter 2 we also gave a brief account of the basic idea of such control. As we have already pointed out, the basic idea of this control is that the control task can be divided into two independent subtasks in a certain coordinate space. One is the robot motion control along a predetermined part of the coordinates (directions) and the other is the control of the interaction force of the robot and environment along the rest of the coordinates (directions):
U1 ( x, xɺ, ɺɺ x ) =τ x
(2.308)
U 2 ( Fy ) = τ y
(2.309)
where τ x , τ y are the corresponding control inputs. However, there are no instances of the robot’s interaction with the environment in which the possibility of such partition would exist. Even for the simplest manipulation mechanism interacting with environment, the following partition is obtained:
U1 ( x, xɺ, ɺɺ x, Fx ) = τ x
(2.310)
U 2 ( y, yɺ , ɺɺ y, Fy ) = τ y
(2.311)
where equations (2.310) and (2.311) represent the robot’s dynamics model in two subspaces; τ x and τ y are the driving forces or torques acting in the corresponding subspaces. The holonomic constraints in the traditional hybrid control are not the only limitations that are important. It should be mentioned that for the Cartesian coordinate robots the controllers with first and higher order integrators can compensate to a certain extent for the perturbations due to friction forces. For the non-Cartesian robots the compensation for friction forces is practically impossible. All these problems become evident if we consider the simplest example of a two-link planar manipulator with the joints of sliding type. In the coordinate frame illustrated in Fig. 2.7, the equations of robot dynamics can be written in the form: (m1 + m2 ) ɺɺ x = τ 1 − kFsgn ( xɺ ) (2.312)
A Unified Approach to Dynamic Control of Robots
m2 ɺyɺ = m2 g + τ 2 − F
171
(2.313)
Fig. 2.7 Simplest planar manipulator
where mi is the mass of the i -th ( i = 1, 2 ) link of the manipulator, F is the reaction force by which the environment is acting on the manipulator’s endeffector, k is the friction coefficient and g is the gravitational constant. Let us suppose that the manipulator has to perform the following simple technological operation. A pressure force F p (t ) should be generated onto the environment along the coordinate y , and the preset motion x p (t ) = vt realized with a constant velocity v along the coordinate x . Evidently, equations (2.312), (2.313) represent the partitioned forms of the robot dynamics equations (2.310) and (2.311), and there is no any other form of partitioning. It is easy to see that for k ≠ 0 , the task of stabilizing the motion x p (t ) using the hybrid control concept does not have a solution. Namely, in this case there is no the control action τ 1 that is independent of the force that realizes PM x p (t ) , even in case of the absence of initial perturbations. In fact, the control action should then satisfy the equation:
(m1 + m 2 ) ɺxɺp = τ 1 − kF (t )
x p (t ) ≡ 0 , this equation becomes: and, because of ɺɺ
τ 1 = kF (t ) Evidently, no equivalent replacement of force increments with position increments is possible because these increments occur in the mutually independent orthogonal directions.
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Dynamics and Robust Control of Robot-Environment Interaction
As for the stabilization of interaction force along the coordinate y , even in the simplest case, the task is unsolvable. Let e.g. the environment dynamics in the coordinate frame considered be described by the relation:
F = k1 y
(2.314)
where k1 is the environment stiffness coefficient, and PFI is constant, i.e.
F p (t ) = F p . Then, the coefficients α 1 and α 2 of the control law with the force
feedback only: t
τ 2 = − m2 g + α1 ( F − Fp ) + α 2 ∫ ( F − Fp ) (ω ) dω + Fp
(2.315)
0
cannot be chosen in such a way that F (t ) → F p as t → ∞ . In fact, by substituting (2.315) into (2.313) we obtain the following equation of the closedloop control system: t
m2 ɺɺ y = (α1 − 1) ( F − Fp ) + α 2 ∫ ( F − Fp ) (ω )d ω
(2.316)
0
ɺɺ = k ɺyɺ . Since Fɺɺ (t ) = 0 , by differentiating From (2.314) it follows that F 1 p (2.316) we obtain
m2 ɺµɺɺ = (α 1 − 1) µɺ + α 2 µ k1
(2.317)
where µ (t ) = F (t ) − F p . The characteristic equation is of the form:
m2 3 λ − (α 1 − 1)λ − α 2 = 0 k1 It is obvious that since the coefficient at λ 2 is equal to zero, the equation roots λ1 , λ 2 , λ3 satisfy the identity λ1 + λ2 + λ3 ≡ 0 . Therefore, the equation (2.317) has no stable solutions (not to mention asymptotically stable solutions). Thus we can not determine any of the preset roots λ1 , λ 2 , λ3 ensuring µ (t ) → 0 when t → ∞ independently of initial perturbations of the force. Let us notice that in the case of a sufficiently small friction coefficient k (e.g. the environment consists of homogeneous ice), the task of motion
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173
stabilization can be solved with satisfactory accuracy if, instead of the dynamics equation (2.312), we solve the same equation but without perturbations:
(m1 + m2 ) ɺxɺ = τ 1 Let us note that the presence of friction during the contact with the environment does not represent only problem of hybrid control. Let the environment equation have a more general form than equation (2.314):
F = k1 ɺɺ y + k2 ( y, yɺ ) Then, the task of synthesis of the control law stabilizing force of the robot interacting with the environment in the y -axis direction using force feedback only becomes a task of the same level of complexity as the task of contact control itself. It is evident that for the motion stabilization along the x -axis and force stabilization along the y -axis it is necessary to use both the position and force feedback loops. In this case the control laws:
τ 1 = (m1 + m2 ) ɺɺ x p + γ 1 ( xɺ − xɺ p ) + γ 2 ( x − x p ) + kFsgn ( xɺ ) t m m2 τ 2 = + 1 Fp + r ∫ ( F − Fp ) (ω )d ω − 2 k2 ( y, yɺ ) − m2 g k1 k1 t0
can easily solve the stabilization task of PM x p (t ) along the x -axis and
PFI Fp (t ) along the y -axis, respectively, if γ 1 , γ 2 and r are negative constants. 2.8.2 Revised hybrid control procedure The critical points emphasized above concern the very idea of position/force stabilization based on the conventional hybrid control concept rather than the special algorithm implementing it. One such algorithm is based on the concept of “orthogonal complements”. Its fallacy has been rightly pointed out by Duffy [32]. No attempts have been made to realize hybrid control on the basis of a concept other than the orthogonal complements. The idea of splitting the control task for a robot interacting with environment into the task of position control in certain directions and the task of force control in the others seems to be more profound than the idea of the hybrid control based on orthogonal complements.
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Dynamics and Robust Control of Robot-Environment Interaction
Here we assume again that the environment dynamics model is described in the robot’s coordinate frame by a system of nonlinear differential equations of the form (2.11), i.e.
M (q )qɺɺ + L(q, qɺ ) = S T (q ) F where M (q ) is a nonlinear n × n matrix; L(q, qɺ ) is a nonlinear n -vector function; S T (q ) is an n × n matrix with rank ( S ) = n . We can represent the equation describing the environment dynamics as follows: qɺɺ = Ψ ( q, qɺ , F ) (2.318)
Ψ ( q, qɺ , F ) = − M −1 (q) L(q, qɺ ) + M −1 (q) S T (q ) F . Here we assume that m = n , i.e. the number of interaction force components
where
is equal to the number of powered DOFs. Let us suppose that there exists a function of coordinate transformation w :{ z} → {q} which transforms the robot dynamic model (2.6) into two equations (2.310) and (2.311). The same function of coordinate transformation ensures partition of the environment dynamics model (2.318) into two independent equations:
ɺɺ x = Ψ1 ( x, xɺ , Fx ) , ɺɺ y = Ψ 2 ( y, yɺ , Fy )
(2.319)
which admit a unique representation in the form:
Fx = f1 ( x, xɺ , ɺɺ x) ,
Fy = f 2 ( y, yɺ , ɺɺ y)
(2.320)
Let us consider two systems of equations taking into account (2.310) and (2.311), and let the first system
U1 ( x, xɺ, ɺɺ x, Fx ) = τ x ,
ɺɺ x = Ψ1 ( x, xɺ , Fx )
(2.321)
describe the robot’s interaction with the environment in the coordinate space {X }. Let the second system
U 2 ( y, yɺ , ɺɺ y, Fy ) = τ y ,
ɺɺ y = Ψ 2 ( y, yɺ , Fy )
(2.322)
describe the robot’s interaction with the environment in the coordinate space {Y } . It is obvious that both systems (2.321) and (2.322) are completely independent of each other. The former system stabilizes the robot motion
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A Unified Approach to Dynamic Control of Robots
(
(
x p (t ) ɺɺ x p (t ) ≡ Ψ1 x p (t ), xɺ p (t ), Fx p (t ) force
))
independently of the magnitude of
Fx (t ), t ≥ t0 , whereas the latter stabilizes the interaction force
(
(
Fy p (t ) ɺɺ y p (t ) ≡ Ψ 2 y p (t ), yɺ p (t ), Fy p (t )
))
independently of the position
magnitude y (t ), t ≥ t0 . The contact task can then be solved by stabilizing the position (motion) in the coordinate space { X } and the interaction force in the coordinate space {Y } . In contrast to the conventional hybrid control concept, the control laws ensuring stabilization of the motion and interaction force utilize both the position and force feedback loops. Let us write equations (2.310) and (2.311) solved with respect to second derivatives:
ɺɺ x = Φ1 ( x, xɺ ,τ x , Fx ) ɺɺ y = Φ 2 ( y, yɺ ,τ y , Fy ) Then, using the control laws (2.23), (2.28) or (2.34) from Sections 2.3.1 we solve the task of stabilizing x p (t ) , whereas the laws (2.39), (2.40), (2.43) from Section 2.3.2 solve the task of stabilizing the interaction force Fy p (t ) . In addition, the stabilization of x p (t ) is carried out with the preset quality of transients determined by (2.16) and stabilization of Fy p (t ) with the preset quality of transients determined by the relation (2.19). The mentioned control laws that stabilize PM x p (t ) are:
(
τ x = U1 x, xɺ, ɺɺx p + P ( x − x p , xɺ − xɺ p ) , Fx
(
)
(
τ x = U1 x, xɺ, ɺɺ x p + P ( x − x p , xɺ − xɺ p ) , f1 x, xɺ , ɺɺ x p + P ( x − x p , xɺ − xɺ p )
(2.323)
))
(2.324)
The control laws (2.39), (2.40), (2.43) that stabilize PFI Fy p (t ) are of the form: t τ y = U 2 y, yɺ , Ψ 2 y, yɺ , Fy + ∫ Q Fy − Fy dω , Fy t
(
p
0
p
)
(2.325)
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Dynamics and Robust Control of Robot-Environment Interaction
t τ y = U 2 y, yɺ , Ψ 2 ( y, yɺ , Fy ) , Fy p + ∫ Q Fy − Fy p dω t0 t τ y = U 2 y, yɺ , Ψ 2 y, yɺ , Fy p + ∫ Q Fy − Fy p d ω , Fy p + t0 t + ∫ Q Fy − Fy p d ω t0
(
)
(
(
(2.326)
)
(2.327)
)
It is easy to see that analogous control laws can also be applied in a more general case, when 1. representations (2.310), (2.311) are of the form
U1 ( x, xɺ, ɺɺ x, y, yɺ , Fx , Fy ) = τ x
(2.328)
U 2 ( y, yɺ , ɺɺ y, x, xɺ, Fx , Fy ) = τ y
(2.329)
under the condition of their unique solvability via the functions Φ1 and
Φ 2 with respect to second derivatives ɺxɺ and ɺyɺ . 2. the equation of the environment dynamics model (2.11) or (2.12) can be presented in the form of the two equations
ɺɺ x = Ψ1 ( x, xɺ, y, yɺ , Fx , Fy )
(2.330)
ɺɺ y = Ψ 2 ( y, yɺ , x, xɺ, Fx , Fy )
(2.331)
under the condition of their unique solvability with respect to Fx and Fy :
Fx = f1 ( x, xɺ , ɺɺ x, y, yɺ , Fy )
(2.332)
Fy = f 2 ( y, yɺ , ɺɺ y, x, xɺ, Fx )
(2.333)
In this case PM x p (t ), y p (t ) and PFI Fx p (t ), Fy p (t ) should satisfy the system of equations
( ) (t ) = f ( y (t ), yɺ (t ), ɺɺ y (t ), x (t ), xɺ (t ), F (t ) )
Fx p (t ) = f1 x p (t ), xɺ p (t ), ɺɺ x p (t ), y p (t ), yɺ p (t ), Fy p (t )
(2.334)
Fy p
(2.335)
2
p
p
p
p
p
xp
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A Unified Approach to Dynamic Control of Robots
Then, to stabilize PM x p (t ) in the space { X } we can apply, for instance, the control law (an analogue of the control law (2.323))
(
τ x = U1 x, xɺ, ɺɺ x p + P ( x − x p , xɺ − xɺ p ) , y, yɺ , Fx , Fy
)
(2.336)
and, to stabilize PFI Fy p (t ) in the space {Y } we can apply, for instance, the control law (an analogue of the control law (2.325)) t τ y = U 2 y, yɺ , Ψ 2 y, yɺ , x, xɺ, Fx , Fy + ∫ Q Fy − Fy dω , x, xɺ, Fx , Fy (2.337) t
(
p
p
)
0
One of these control laws is used to solve the task of stabilizing PM and PFI in the considered example of the two-link manipulator given in Fig. 2.7. 2.8.3 Case study In order to get a better insight into the efficiency of the proposed hybrid dynamic position-force control laws in this section we present simulation results of the manipulation robot’s model with two DOFs (Fig. 2.8).
Fig. 2.8 Model of a 2-DOF planar manipulator
We consider a typical robot task in which the robot end-effector realizes the desired motion x p (t ) along the x -axis and the required contact force Fy p in the y -direction. The environment model defined in the direction perpendicular to the sliding surface (the y -direction in Fig. 2.8) is assumed to be in the form of linear impedance, while the model of force in the horizontal plane ( x -
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Dynamics and Robust Control of Robot-Environment Interaction
direction in Fig. 2.8) is defined by the friction force model of the end-effector. The models of active forces acting upon the tip of the robot’s end-effector are represented by the following relations:
− Fy = M ɺɺ y + B yɺ + K y
(2.338)
Fx = −ν Fy sgn ( xɺ (t ))
(2.339)
where M , B and K are the environment equivalent mass, damping and stiffness parameters, respectively. For a relatively rigid environment we assume the following hypothetical values of its parameters: M = 0.5 [ kg ] ,
K = 10 5 [N / m ] , whereas Coulomb’s friction coefficient of the robot end-effector, ν = 0.3 . The returned motion of the robot end-effector is imposed from X 0 to X 1 and back, with the periodically varying
B = 268 [ N /(m / s) ]
and
velocity v(t ) = v max sin (ωt ) . The maximal length of the end-effector path along the y -axis: lmax = 0.5 [ m] . Maximal feedrate of the manipulator’s end-effector:
v max = 0.2 [ m / s ] , while the frequency of the imposed end-effector feedrate xɺ p (t ) = v(t ) is assumed to be ω = 0.8 [rad / s ] . Maximal output torque of the
actuator shaft at both mechanism’s joints is τ max = ± 50 [Nm] . The robot contact task is to synthesize such control law which would ensure the desired motion precision x p (t ) and the interaction force Fy p (t ) = 11[ N ] . The robot links are l1 = l 2 = 1[m] , and their masses m1 = 1.3 [kg ] and
m2 = 0.9 [kg ] . The initial joint position deviations are: ∆q1 (0) = 0.0306 [rad ] and ∆q 2 = − 0.0611[rad ] . This causes the end-effector position displacement in the x -direction ∆x(0) = − 0.045 [m] . As a consequence, the contact force error in the initial time instant is ∆Fy (0) = −6.98 [N ] . In this planar example, the entire manipulator workspace {Z } (Fig. 2.8) can
be decoupled in two “orthogonal subspaces”,
{X}
and {Y } . Thus z = [ x, y ]T ,
F = Fx , Fy . It should be noticed that the robot’s motions in the planes { X } and {Y } are mutually interconnected (see the models (2.338) and (2.339)). This T
fact is of crucial importance for the understanding of the differences between the
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A Unified Approach to Dynamic Control of Robots
conventional hybrid position/force control and the proposed dynamic positionforce control. The former is based on the concept of the so-called “orthogonal complements”, while the proposed control laws are based on the simultaneous control of robot position and interaction contact force. Below we present simulation results that emphasize the necessity of applying the “corrected” position-force control laws. Using the environment model (2.338) and the model of manipulation robot in Cartesian space ((2.272) in Section 2.7)), the conventional hybrid control law in the robot task space is expressed by the general relation (1.41) – (1.43) in Chapter 1. For comparison sake, along with the conventional hybrid control, we apply dynamic position-force control laws (2.336), (2.337), which control simultaneously the end-effector position and contact interaction force during the robot’s operation. More concretely, the conventional hybrid control law in robot task space is described by the expression [29]:
τ = J T (q ) {( I 2 − S ) [ Λ( z ) ɺɺzc + v ( z , zɺ) ] − SFc }
(2.340)
where
ɺzɺc = ɺzɺp − K v ( zɺ − zɺ p ) − K p ( z − z p ) , t
Fc = Fp − K fp ( F − Fp ) − K fi ∫ ( F − Fp )d ω + Λ ( z ) K fv zɺ 0
The subscript “p” denotes programmed values of the particular variables. The unit matrix I 2 and the selectivity matrix S are defined in the form:
1 0 0 0 I2 = ; S = 0 1 0 1 The matrices K (⋅) in the control law (2.340) represent the corresponding diagonal matrices of the regulator of control gains k(⋅) . Instead of the implementation of the conventional hybrid control (2.340) we propose implementation of the above dynamic position-force control laws (2.336), (2.337). In this case the contact force Fy (t ) is controlled explicitly. Then, the laws (2.336), (2.337) can be written in the compact vector form:
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Dynamics and Robust Control of Robot-Environment Interaction
τ x τ = = J T (q ) {Λ ( z ) ɺɺzc + v ( z, zɺ ) − Fc } τ y
(2.341)
where
x p − kv ( xɺ − xɺ p ) − k p ( x − x p ) ɺɺ , ɺɺ zc = − 1 ( B yɺ + K y + Fy ) M
Fx t ω Fc = Fy p + ∫ −κ fp ( Fy − Fy p ) − κ fi ∫ ( Fy − Fy p ) dω1 t0 t0
. dω
Simulation results obtained by implementing the control laws (2.340) and (2.341) under the imposed motion conditions are presented in Fig. 2.9. The following values of parameters were used: k p = 631 [ s −2 ] ,
k v = 40 [ s −1 ] , κ fp = 38 [ s −1 ] , κ fi = 355 [ s −2 ] , k fp = 2.76 , k fi = 35.53 and k fv = 0.1 [ N s / kg m] . A comparative analysis shows that in this case the conventional hybrid position/force control (2.340) can ensure neither the desired stable motion of the robot’s end-effector nor the required contact force. On the contrary, the proposed dynamic control law (2.341) with respect to position and contact force ensures a stable programmed motion x p (t ) of the end-effector in the horizontal direction and a desired contact force Fy p (t ) in the vertical direction. Besides, the control law (2.341) ensures both the stability of the desired quality of dynamic behavior in all directions of motion. The dynamic position control law (2.336), combined with explicit control (2.337)c in the y -direction, ensures simultaneous position-force control of the end-effector. Which control law will be implemented with the force control law (2.336) in a particular robot contact task it will depend on the concrete requirements concerning the quality of robot task performance. We have to
c In addition to the explicit force control law (2.337), there are two more variants of this control
such as implicit and explicit/implicit force control, analogous to those represented by (2.326) and (2.327) for a somewhat simpler case of coupling in respect of force and position.
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underline again that the stability of robot motion (positions), as well as of contact forces, is guaranteed by the application of any of the proposed control laws pairs: (2.336) with (2.337), (2.326) or (2.327), respectively [33].
Fig. 2.9 Accuracy indices of the nominal trajectory tracking and desired contact force during motion of the end-effector along the x -axis and y -axis using control laws: a) (2.340) and b) (2.341)
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2.9 Impedance Control – A Special Case of the Unified Approach 2.9.1 Introduction Impedance control represents a strategy for constrained motion rather than a concrete control scheme. The objective of this control concept is to achieve specific mechanical impedance at the manipulator end-effector. This objective imposes a desired relationship between the position error and force acting on the end-effector. Various control concepts and schemes have been established and proposed for controlling the relation between the robot’s motion and interaction force. It should be emphasized that despite of the existence of a large number of impedance control algorithms only a few of them can be applied in the existing industrial robot control systems. In Section 1.6.1 of Chapter 1 we gave a brief survey of the state-of-the-art in the domain of impedance control, whereas impedance control as active compliant motion control will be presented in a much wider form in Chapter 3. 2.9.2 Improved impedance control Let us start with the general impedance control concept presented in [34]. The basic idea of this approach consists in generating a control law for the robot, in which the closed-loop control system functions in accordance with the differential equation:
F = M ′( ɺɺ x − ɺɺ x0 ) + B′( xɺ − xɺ0 ) + K ′( x − x0 )
(2.342)
where the constant m × n matrices M ′, B′, K ′ represent the matrices of inertia, damping and stiffness of the overall interactive system. These matrices can be chosen by the control system designer in dependence of the goals to be achieved by the robot in the technical task to be performed. Equation (2.342) can be written in the robot’s joint coordinates in the form:
F = M (q ) (qɺɺ − qɺɺp ) + B (q ) (qɺ − qɺ p ) + K (q ) (q − q p )
(2.343)
where the joint-coordinates dependent m × n matrices M ( q ), B ( q ), K ( q ) represent the matrices of inertia, damping and stiffness of the whole interactive system, too. Equation (2.343) is derived from (2.342) by transpose Jacobian matrix multiplication, and using the known kinematic relations between the Cartesian and joint coordinates.
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For the sake of simplicity and without loss of generality, the impedance control will be considered in the coordinate frame q by putting m = n . The goal of impedance control is then to ensure the preset impedance of the closed-loop system defined by (2.342). Hereby, it is assumed that the environment dynamics model is also given in the impedance form:
F = Mɶ ′ ɺɺ x + Bɶ ′ xɺ + Kɶ ′ x
(2.344)
or in its alternative form in the joint coordinate space:
F = Mɶ (q )qɺɺ + Bɶ (q )qɺ + Kɶ (q )q (2.345) ~ ~ ~ where M ( q ), B ( q ), K ( q ) and Mɶ ′, Bɶ ′, Kɶ ′ are the corresponding sets of matrices of inertia, damping and stiffness of the environment. The same notation for F , used in (2.342) and (2.343), has evidently a different physical meaning, i.e. force and moment, respectively. In accordance with the definitions of PM (programmed motion) and PFI (programmed force of interaction), these components of the desired dynamic behavior have to satisfy the equation of environment dynamics, i.e.:
Fp (t ) = Mɶ (q p ) qɺɺp (t ) + Bɶ (q p ) qɺ p (t ) + Kɶ (q p ) q p (t ), ∀t ≥ t0
(2.346)
Let us assume that the task of impedance control has been solved, i.e. that such control τ has been synthesized that ensures the closed-loop control system functioning in accordance with (2.342). Assuming the differences in the environment dynamics matrices for the actual and nominal motion to be negligible, it follows from (2.344) and (2.345) that the differences between the actual and programmed robot force interaction with the environment are determined by the equation:
F − Fp = Mɶ (qɺɺ − qɺɺp ) + Bɶ (qɺ − qɺ p ) + Kɶ (q − q p )
(2.347)
By subtracting this equation from (2.343) we find that the functioning of the closed-loop system is governed by the equation:
Fp = ( M − Mɶ )ηɺɺ + ( B − Bɶ )ηɺ + ( K − Kɶ )η
(2.348)
where η (t ) = q (t ) − q p (t ). Evidently, the control system designer can choose the matrices M , B, K in such a way that the trivial solution of the unperturbed motion:
( M − Mɶ )ηɺɺ + ( B − Bɶ )ηɺ + ( K − Kɶ )η = 0
(2.349)
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is, say, exponentially stable, whereby all its solutions comply with the inequality:
max { η (t ) , ηɺ (t ) } ≤ C e − λ (t −t0 )
η (t0 ) + ηɺ (t0 ) 2
2
(2.350)
where C is a constant and the index λ > 0. In that case, an arbitrary solution of the differential equation (2.348) with the perturbation F p (t ) will be stable, whereby the estimation of stabilizing accuracy implies the following:
max { η (t ) , ηɺ (t ) } ≤ Ce− λ (t −t0 )
η (t0 ) + ηɺ (t0 ) + CCF / λ 2
2
where C F is a constant which bounds the norm of PFI Fp (t ) for every t ≥ t0 , i.e. F p (t ) ≤ C F . It is clear that for a large λ , or, which is the same, for a small
C F , the impedance control solves satisfactorily the task of stabilizing motion of the robot in contact with its environment (2.345). A question arises as to whether this condition implies also that the interaction force F (t ) is close to F p (t ) , i.e. whether the impedance control solves fully the control task of the robot in contact with the dynamic environment. Evidently, if F p (t ) is changing slowly with time, and its norm F p (t ) is small, the answer is positive. In fact, because η and ηɺ are small, it follows from (2.347) that for a small F p (t ) , ηɺɺ will be small too. And, because the connection between the deviation of the force from its programmed value F (t ) − F p (t ) and deviation of the motion from the programmed one η (t ) = q (t ) − q p (t ) is determined by (2.347), the small values η , ηɺ , ηɺɺ cause µ (t ) = F (t ) − F p (t ) to be small as well. It is easy to see that all these supplementary requirements related to the small value of C F and to the slowly changing F p (t ) are explained by the presence of the perturbation F p (t ) on the left-hand side of equation (2.348), describing the behavior of the closed-loop control system. It is therefore fully justified to introduce a correction into the impedance equation (2.343), by replacing it with the following equation:
F − F p = M (qɺɺ − qɺɺp ) + B (qɺ − qɺ p ) + K (q − q p )
(2.351)
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Thus the closed-loop system will evidently operate according to (2.349) without the perturbation F p (t ) . In that case the estimation (2.350) will hold, i.e.
η (t ), ηɺ (t ) → 0 when t → ∞ and, because of equation (2.349), the relation ηɺɺ(t ) → 0 when t → ∞ will also hold. Hence, bearing in mind (2.347), it immediately follows that F (t ) → F p (t ) at t → ∞ . In this way, the correction carried out by the impedance yields an ideal solution of the contact task. Let us now focus our attention on the fact that in the case when impedance equation is of the form (2.351), any impedance control solving the task of attaining the preset impedance of the closed-loop system is actually a special case of the control law (2.23), with a suitably chosen vector function P in the case when environment dynamics is described by the differential equation (2.345). This relates to the fact that the realization of the preset impedance goal in the form (2.351) leads to the closed-system equation in the form of (2.349), which is a special case of the equation ηɺɺ = P (η ,ηɺ ) (see (2.16)), specifying the preset quality of stabilization, where:
P (η ,ηɺ ) = ( M − Mɶ )−1 ( Bɶ − B )ηɺ + ( Kɶ − K )η
(2.352)
The impedance equation (2.351), solvable with respect to second derivative, is of the form:
ηɺɺ = M −1 (− Bηɺ − Kη + F − Fp ) Therefore,
( M − Mɶ )−1 ( Bɶ − B)ηɺ + ( Kɶ − K )η = P(η ,ηɺ ) = ηɺɺ = M −1 (− Bηɺ − Kη + F − Fp ). In that case, the robot control law (2.23), based on the impedance equation (2.351), can be written as
τ = H (q) qɺɺp + M −1 (− Bηɺ − Kη + F − Fp ) + h(q, qɺ ) − J T (q) F
(2.353)
This relation provides an ‘ideal’ solution of the task of motion and force stabilization. The control law synthesized on the basis of impedance control (2.343), i.e. on the basis of the equation without correction, has a form analogous to (2.353), if the term − F p is omitted:
τ = H (q) qɺɺp + M −1 (− Bηɺ − Kη + F ) + h(q, qɺ ) − J T (q) F Indeed, this control law has been used for the synthesis of adaptive impedance control of robots and manipulators [35].
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2.9.3 Case study Simulation studies were conducted on the MANUTEC-r3 industrial robot (Fig. 2.10) to verify the described impedance control scheme. The parameters of the mechanical structure and actuators were taken from [36]. Tables 2.1–2.3 show the values of kinematic and dynamic parameters of the robot. The actuators’ parameters can be found in [36].
Fig. 2.10 MANUTEC-r3 robot: (a) View, (b) Kinematic scheme
Table 2.1 Vector from the joint center to the center of mass of the augmented link i (link i and electric motor rotor i + 1), resolved in the body fixed frame of link i
Rxi Ryi Rzi
i
1
2
3
4
5
6
[m] [m] [m]
0.000 -0.172 0.000
-0.295 0.000 0.172
0.000 -0.064 -0.034
0.000 -0.410 0.000
0.000 0.000 0.023
0.000 0.000 -0.020
Table 2.2 Mass of the link (mi) (i = 1, n) i mi [kg]
1
2
3
4
5
6
20.000
56.500
26.400
28.700
5.200
1500
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Table 2.3 Inertia tensor of the augmented link i (link i and electric motor rotor i + 1), with respect to the center of mass of the augmented link, resolved in the body fixed frame of link i i Ixxi [kg m2] Iyyi [kg m2] Izzi [kg m2]
1
2
3
4
5
6
0.0000 1.1600 0.0000
2.5800 2.7300 0.6400
0.2790 0.4130 0.2400
1.6700 1.6700 0.0810
1.2500 1.5300 0.8100
0.0000 0.0000 0.0002
Fig. 2.11 Robot machining process under consideration
The example of robotic deburring has been selected for simulation experiments using the conventional and improved impedance control schemes. The rotational-milling work tool of the robot performs surface processing in the π-plane (Fig. 2.11). The robot end-effector (tool) moves along a prescribed, nominal trajectory 1-2 (Fig. 2.11). The total trajectory is 200 [mm] long. The robot’s task is to carry out machining of the work-surface along the prescribed trajectory with a prescribed contact force FN0 = 5 [ N ] and the prescribed
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velocity v f = 25 mms −1 . The adopted general model of impedance was the environment model (2.344). The corresponding matrices in the environment model were adopted in the form: Inertia matrix: Mɶ ′ = diag {mɶ i′}[ kg ], mɶ 1′ = 28.173, mɶ ′2 = 101.424, mɶ 3′ = 28.173, mɶ ′4 = 10.142, mɶ 5′ = 10.142, mɶ 6′ = 10.142.
(2.354)
Damping matrix:
Bɶ ′ = diag {bɶi′}[ N /(ms −1 )] bɶ1′ = 1061.571, bɶ2′ = 4458.599, bɶ3′ = 1061.571, bɶ4′ = 445.859, bɶ′ = 445.859, bɶ′ = 445.859. 5
(2.355)
6
Stiffness matrix: Kɶ ′ = diag {kɶi′}[ Nm −1 ] kɶ1′ = 10 4 , kɶ2′ = 10 5 , kɶ3′ = 10 4 , kɶ4′ = 10 4 , kɶ5′ = 10 4 , kɶ6′ = 10 4 ,
(2.356)
The matrices of the overall interactive closed-loop control system (2.342) were cast in the following forms:
M ′ = diag {mi′}[kg ], m1′ = 100, m′2 = 100, m3′ = 100, m4′ = 10,
m5′ = 10,
m6′ = 10,
B ′ = diag {bi′}[ N / ms −1 ], b1′ = 6000, b2′ = 6000, b3′ = 6000, b4′ = 600, K ′ = diag {k i′}[ Nm −1 ], k1′ = 10 4 ,
b5′ = 600,
b6′ = 600,
k 2′ = 10 5 , k 3′ = 10 4 ,
k 4′ = 10 4 , k 5′ = 10 4 , k 6′ = 10 4.
(2.357)
(2.358)
(2.359)
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Fig. 2.12 Simulation results: (a) for conventional impedance control; (b) for improved impedance control
The following initial conditions were used in the simulation: initial deviation of the end-effector tip ∆y = + 30 [µ m] and ∆z = + 50 [µ m ] ; the robot gripper started with a zero initial velocity; the position deviations resulted in an initial deviation of the contact force of ∆FN = + 4.375 [N ] , which means that the robot actuator at the instant t = 0 was overloaded because the nominal force value was FN (t ) (t = 0) = 0 [N ] . The setting time of the desired contact force was
t = 0.5 [sec] . The simulation results are presented in Fig. 2.12. The relevant
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indices of control quality are the shapes of transient processes of force components FN , FT and F f and the deviation of the positions ∆q, ∆x, ∆y and
∆z of the internal and external robot coordinates. The results presented in Fig. 2.12 show that both control laws ((2.343) and (2.351)) ensured a satisfactory quality of dynamic behavior. The control law realizing conventional impedance (2.343) gave worse dynamic feature than the control law realizing improved impedance (2.351); in the latter case the overshoot of the contact force in the setting time interval was reduced. The control law corresponding to the improved impedance equation (2.351) exhibited a decrease in the mean force error squares of about 8 %. It should be noticed that the parameters of the systems in contact were selected in a way that yielded no phenomena that would strongly favor the improved impedance control.
2.9.4 Concluding remarks In concluding the discussion of impedance approach to control tasks let us note that even without the mentioned correction of impedance equation, in contrast to hybrid control, the control laws synthesized on the basis of target impedance (2.343), solve satisfactorily the task of control of robot in contact with its environment. Indeed, because of the equivalence of the impedance equations (2.343) and (2.348) for the environment (equation (2.345)), the matrices M , B and K can be selected on the basis of equation (2.348), to ensure the convergence η (t ) and ηɺ (t ) to zero for t → ∞ . In principle, this can always be achieved because the left-hand side of equation (2.348) represents the known time function F p (t ) . But in this case, for the different PFI Fp (t ) it is necessary to consider each time different matrices M , B and K . However, even so, it is not possible to ensure that the second derivative ηɺɺ(t ) tends to zero for t → ∞ , and this can in some cases worsen significantly the quality of stabilization of the contact force. For this reason, as well as from the point of view of obtaining “ideal” control laws, the use of the target impedance with correction (2.351) seems to be unavoidable. From a theoretical point of view it was shown that the impedance control is only a special case of active compliant control scheme. This leads to the disappearance of the principal difference between the active compliance tasks in which impedance control is much different from the position-force control. A
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direct aim is to improve the transient dynamic process immediately after the contact. In this way, the impedance control performance is improved in the very instant when its quality is of the decisive practical importance, and thus the impedance control can also become competitive in other contact tasks. Finally, it should be added that it is not possible to determine the nominal contact force Fp without having a sufficiently accurate environment model. Moreover, an insufficiently accurate environment dynamics model can significantly affect the contact task execution. Inaccuracies of the robot and environment dynamics models have been considered in [6, 15]. The problems arising from parameter uncertainties may also be resolved by applying knowledge-based techniques [37]. Let us also note that the adaptive control laws given in Section 2.5 can cope with the task of motion and force stabilization even in case of an essential inadequacy of the models of dynamics of the robot and environment, and thus with an inaccurate calculation of the force Fp .
2.10 Stability of Robots Interacting with Dynamic Environments 2.10.1 Introduction The synthesis of control of robots in the so-called constrained motion control tasks, i.e. the tasks in which manipulation robots are coming into contact with the environment, has attracted much attention in the last two decades or so. One of the most delicate problems in position-force control of robots interacting with dynamic environment is the stability of both desired motions and interaction forces. A multitude of various control approaches such as hybrid control, stiffness control, impedance control, etc. points to the control stability tasks as a problem which is not yet satisfactorily solved, both from the theoretical and practical standpoint. In almost all approaches when considering specific contact tasks, simplifications are introduced in the modeling of robot and environment dynamics in order to obtain simpler control algorithms. A very popular approach is to describe the environment by a set of algebraic equations, assuming that the motion of the robot in contact is kinematically constrained [1, 2]. Colgate and Hogan consider the environment to be a linear time-invariant dynamic system [30]. In both cases experimental verification led to the discovery of instability caused by the environment dynamics [38]. Ann and Holerbach have identified a new form of instability in the force control of manipulators [39]. That form of instability occurs only in multi-joint
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manipulators and is caused by an inverse kinematic transformation in the feedback path. It has been shown that kinematic instability occurs not only at the points of kinematic singularities, where the Jacobian inverses are not defined, but in a wide range of the manipulator work space, where the Jacobian inverses are well defined. Mc Clamroch and Wang emphasized the important role of constraints in the constrained dynamics [40]. They presented global conditions for the tracking based on a modified computed torque controller, and local conditions for feedback stabilization using a linear controller. The closed-loop properties in the case of force disturbances, dynamics in the force feedback loops, or uncertainty in the constraint functions, were also investigated. Eppinger and Seering have studied the influence of unmodeled dynamics on contact task stability, introducing additional (elastic) DOFs of both robot and environment [41]. Nevertheless, they concluded that the environment dynamics cannot cause instability. A treatment of the contact task stability, considering the environment as a nonlinear dynamic system is given in [42]. It is shown that if impedance control is applied, enabling the robot to be asymptotically stable in free space, then the robot interacting with environment is a passive system and is stable in the isolation. However, the conclusion is valid only if the robot in contact is at rest and for this reason the result can not be considered complete. The stability of the robot and environment taken as a whole, using unstructured models for the dynamic behavior of the robot and the environment, has been investigated in [43]. The objective was to define sufficient conditions for stability of the closed-loop system. The basic idea has been to start from the very general (“unstructured”) model of the robot in contact with environment, where both the robot dynamics and environment dynamics are presented by general functions, for which only input/output characteristics are taken into account (structure is totally ignored). This approach is then applied to the “structured” model of the robot and environment, where infinitely rigid environment model is assumed. Due to the fact that the structure of the model is totally ignored, the obtained results are extremely conservative. In the case of structured model the same “conservativism” is preserved, since structural features of the model are nowhere specifically used to derive the stability conditions. Both robot and environment dynamics are “approximated” by the “gain” of mapping of the input trajectory into the force vector. This means that the stability conditions set upon the “allowed” control law “size” take into account robot dynamics, environment dynamics and the position control law –
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all together – through one “gain”. No attempt has been made to show how the results of the stability analysis can actually be used for control synthesis when the desired force trajectory has to be achieved, and when the model of environment dynamics is to be assumed. In order to extend the hybrid strategy to the cases when the environment also exhibits a dynamic behavior De Luca and Manes [25, 44] introduced a suitable modeling of the robot-environment interaction. The paper [44] describes how to use this modeling technique to design hybrid control laws in the presence of environment dynamics. The fundamentals of the dynamic control of robotic manipulators interacting with dynamic environment [6, 10, 11, 15, 16, 45], are presented in Sections 2.3.2 and 2.3.3. Attention was focused just on the stabilization of position when asymptotic stability of the contact force was ensured. This task is the basic problem of controlling a robot interacting with dynamic environment. The theorem ensuring the asymptotically stable position of the system in the first approximation (local stability) formulates the sufficient conditions under which it is achieved. However, without knowing sufficiently accurate environment model it is not possible to determine the nominal contact force Fp (t ) . Besides, an insufficiently accurate environment dynamics model can significantly affect the contact task performance. Inaccuracies of the robot and environment dynamics models, as well as dynamic control robustness are considered in [6, 15, 16]. The problems arising from parametric uncertainties may also be resolved by applying the knowledge-based techniques (fuzzy logic and neural networks) [46]. Taking into account external perturbations, which do not expire with time, and model and parameter uncertainties, it may be difficult to achieve asymptotic (exponential) stability of the system unless use is made of robust and adaptive control laws that include a factor to compensate for these perturbations and uncertainties. Therefore, it may be of practical interest to demand more relaxed stability condition, i.e. to consider the so-called practical stability of the robot around the desired position and force trajectories by specifying the finite regions around them within which the robot actual position and force have to be during the task execution, and by assuming that the inaccuracies of model parameters (of the robot and environment) are bounded. The conditions for the practical stability of the robot interacting with dynamic environment enable one to study the model uncertainty issue in control of robots in this class of tasks without any approximations, i.e. to correctly examine the influence of these uncertainties upon different control laws. The test conditions for practical stability of the robot interacting with dynamic environment will be presented in the text to follow.
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2.10.2 Practical stability of robots interacting with dynamic environment Numerous applications of robots in the industry, agriculture, mining, health care, etc. require effective solutions for the control of robots in the so-called constrained motion control tasks, i.e. the tasks in which manipulation robots are coming into contact with the environment. The synthesis of control of robots for such tasks has attracted significant attention in the last 25 years (see e.g. [1, 2, 5, 29, 47, 48]). A special problem represents the stability issue, i.e. the establishment of the conditions under which a particular control law guarantees stability of the robot in contact with its environment. A number of papers have considered the stability aspects assuming different approximate models of the robot and the environment [6, 10, 38, 43]. In [6, 10], the control laws stabilizing simultaneously the robot motion and its interaction force with dynamic environment have been synthesized to ensure exponential stability of the closedloop systems (based on the analysis of the complete dynamics models of the robot and dynamic environment). However, the model uncertainties, representing the crucial problem in control of robots interacting with the dynamic environment, have not been appropriately addressed yet. The uncertainties of the environment dynamics model in certain technological tasks may have especially strong influence because of the difficulties in the identification/prediction of the environment parameters and behavior of the environment. Therefore, it may be difficult to achieve asymptotic (exponential) stability of the system (unless robust and adaptive control laws including factors for compensating these perturbations and uncertainties are used). It is of practical interest to require a more relaxed stability condition, i.e. to consider the so-called practical stability of the system. An approach to the analysis of practical stability of manipulation robots interacting with the dynamic environment based on centralized model of the system has been presented in [19, 49]. In [50], an approach to stability analysis of the robots in such control tasks based on the decomposition of the system into two subsystems (one associated to the ‘position’- controlled and the other to the ‘force’-controlled part) has been presented. In this section, a new approach is presented following the basic idea of the decomposition-aggregation method for the stability analysis of large-scale systems in which the system is decomposed into ‘subsystems’, each associated to one DOF in Cartesian (or, task) space. The dynamic interactions among these ‘subsystems’ are taken into account within the ‘aggregation’ phase of the method. The objective is to establish for the first time less conservative conditions for
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robot’s practical stability, suitable for the analysis and synthesis of decentralized control laws. The conditions are derived using the method of Michael [51], which has been already effectively applied for the practical stability analysis of the robot moving in a free space [52]. The inaccuracy in modeling, constraints upon motion, control, and interaction forces are considered in a general form. The control law considered belongs to the so-called hybrid position-force control schemes, where the overall dynamics model of the system is taken into account.
2.10.3 Mathematical model We will consider a robot in contact with dynamic environment, assuming that the contact is constantly maintained. The issue of keeping of contact and impact between the robot and the environment (i.e. the discontinuity of the model, since the robot can only push and not pull the environment) is out of the scope of this section. The overall dynamics model of the robot with n ≤ 6 DOFs and the dynamics model of the environment are considered in Cartesian space. The model of dynamics of the mechanical part of the robot can be written in the form: ɺ d ) = ( J -1 )T ( z, d )τ + F Λ ( z, d ) ɺɺ z + ρ ( z, z, (2.360) where z = χ ( q ) is the n -vector of the robot Cartesian coordinates (see (2.271)); q is the n -vector of the robot internal coordinates; Λ ( z, d ) is the
ɺ d ) is the n -dimensional nonlinear vector function n × n inertia matrix; ρ ( z, z, of Coriolis’, centrifugal and gravity moments; d is the l -vector of parameters which belongs to the constrained set D ; J ( z, d ) is the n × n Jacobian matrix; τ is the n -vector of driving torques (inputs); F is the n -vector of Cartesian forces, generalized interaction forces (forces and moments) acting upon the endeffector of the robot. For the sake of simplicity we shall: (1) consider the second order models of actuators, which are assumed to be included in the robot dynamics model, and (2) write the functions without arguments. We shall assume that the environment does not introduce any extra DOFs in the overall system. The model of dynamic environment can be written as:
ɺ d)= − S F M ( z, d ) ɺɺ z + L ( z, z,
(2.361)
where M is the n × n matrix; L is the n -dimensional nonlinear vector function, and S is an n × n matrix function of Cartesian coordinates, representing transformation between forces acting upon the robot end-effector
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and forces (moments) of the dynamic environment, with a rank equal n. (This is the property of the environment model (2.361), whose solution is unique with respect to the force F .) We shall assume that S = I . It is assumed that all mentioned matrices and vectors are continuous functions of their arguments. The model of the robot in the state space can be defined as:
xɺ = f ( x, d ) + B ( x, d )τ + G ( x, d ) F
(2.362)
where x = ( z T , zɺT )T is a 2n -dimensional state vector; f ( x, d ) is a 2n -vector; B ( x, d ) and G ( x, d ) are 2n × n matrices.
2.10.4 Formulation of the control task Let us assume that in m1 directions ( m1 < n ) the desired force trajectories
Fp(1) (t ) are specified, where Fp(1) (t ) is m1 -vector, while in n1 ( n1 < n ) (1) directions, the desired trajectory z (1) p (t ) is specified, where z p (t ) is n1 -vector
and n1 + m1 = n . This means that in some directions are specified only desired forces and in some directions only desired position trajectories. Let us introduce the following notation: z p (t ) =
(
)
( z (1)p ) (t ), ( z (2)p ) (t ) , where z (2)p (t ) is the T
T
T
m1 -vector of the nominal trajectories of the Cartesian coordinates in the directions in which the force trajectories are specified. Note that the trajectories z (2) p (t ) are not specified in advance, but have to be determined on the basis of the environment model. Similarly, the vector of desired force trajectories can be denoted as Fp (t ) =
(( F )
(1) T p
)
(t ), ( Fp(2) ) (t ) , where Fp(2) (t ) is the n1 -vector T
T
of nominal force trajectories in the directions in which forces are acting upon the robot, but the nominal trajectories of the Cartesian space coordinates are specified. The force trajectories Fp(2) (t ) are not specified in advance, but have to be calculated based on the dynamics model of the environment. The nominal trajectories of the forces and Cartesian coordinates must satisfy the environment model, i.e.:
M ( z p (t ), d ) ɺɺ z p (t ) + L ( z p (t ) , zɺ p (t ), d ) = − Fp (t )
(2.363)
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Let us introduce the following notations:
M M = 11 M 21
M 12 T and L = L1T ,L2T M 22
(2.364)
where the dimensions of the matrix M 11 are m1 × n1 , of M 12 m1 × m1 , of M 21
n1 × n1 , and of M 22 n1 × m1 , while L1 and L2 are the vectors of dimensions m1 and n1 respectively. The desired trajectory in the state space is denoted by
x p (t ) = ( z pT (t ), zɺ pT (t ) ) . Now, the control task is specified as a task of T
practical stability of the robot around the nominal trajectory x p (t ) in the following form: Let t1 be the predefined time period and Τ = [0, t1 ] . The robot
ɶ control has to ensure that ∀x0 ∈ R 2 n and ∀d ∈ D x(0) = x0 ∈ X 0
imply
ɶ t ), ∀ t ∈Τ , where X( ɶ t ) are the finite regions in the state space around x(t )∈ X( ɶ is the prescribed nominal trajectory x (t ) defined for every point of time t , X p
0
the finite region in the state space around the prescribed nominal trajectory ɶ , x (t ) ∈ X( ɶ t) , x p (0) in the initial instant. It is assumed that x p (0) ∈ X 0 p
ɶ ⊂ X(0) ɶ ∀ t ∈Τ , X . This formulation of the control task can be interpreted in 0 the following way. Given the desired trajectory x p (t ) , an initial error around the nominal trajectory is allowed such that the initial state x(0) = x0 must belong to
ɶ around x (0) ; the control has to ensure the robot’s the predefined region X 0 p state follows the desired trajectory x p (t ) with an allowed error which is constrained by the requirement that the state of the robot x(t ) in each point of
ɶ t ) around the nominal trajectory time must belong to the predefined region X(
x p (t ) in the predefined time period Τ . This must be provided for all admissible parameters d . Because of (2.363), and, as was shown in [10], the fulfillment of the specified control task also guarantees tracking of the desired force trajectories Fp (t ) , i.e. it guarantees that ∀ F0 ∈ R n and ∀ d ∈ D
F (0) = F0 ∈ Fɶ0
ɶ (t ) , ∀ t ∈ Τ , where Fɶ (t ) are the finite imply F (t ) ∈ F
regions in the n -dimensional space around the nominal trajectory
Fp (t )
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ɶ is a region in the n -dimensional space defined for every point of time t , and F 0 around the nominal trajectory Fp (0) in the initial time. Note that the regions
ɶ and X( ɶ t ) . In order Fɶ0 and Fɶ (t ) must correspond to the respective regions X 0 to simplify the stability analysis let us consider specific forms of the finite ɶ and X( ɶ t) : regions X 0
{
}
{
}
ɶ t ) = x ∈ R 2 n / x − x (t ) < X e −α t , X( p t
ɶ = x ∈ R 2 n / x − x (0) < X , X 0 p 0
∀ t ∈ Τ , where X t > X 0 > 0 , α > 0 . Here X t , X 0 , α denote real-valued positive numbers, ⋅ denotes Euclidean norm of the corresponding vector. Let
(
us denote by ∆x(t ) = x(t ) − x p (t ) = ∆z T (t ), ∆zɺT (t )
)
T
a 2n -dementional
vector of the state deviation of the real trajectory x(t ) from the desired nominal
ɶ and x(t )∈ X( ɶ t ), ∀ t ∈Τ trajectory x p (t ) . Then the inclusions x(0) = x0 ∈ X 0 will be fulfilled if ∆x(0) < X 0 and ∆x(t ) < X t e −α t , ∀ t ∈Τ .
2.10.5 Control law The dynamic position-force control law is considered (see Fig. 2.13) in the form:
τ = U * ( z , zɺ, ɺɺzc , F ) = ( J * ) (Λ*ɺɺzc + ρ * − F ) T
(2.365)
where J * , Λ* , ρ * denote the matrices and vector corresponding to J , Λ, ρ from the model (2.360) but with the assumed parameter values d = d 0 ∈ D . This means that we assume that the parameters values are not accurately known. The variable zc in (2.365) is governed by the equations: (1) ɺɺ ɺ (1) ) z (1) p + P1 ( ∆z , ∆z ɺɺ zc = * , (1) ɺ (1) ), Fp(1) + ∆F (1) ) z (1) p + P1 ( ∆z , ∆z W ( z , zɺ, ɺɺ (1) * (1) * (1) * ɺɺ W * ( z , zɺ, ɺɺ z (1) p , Fp ) = ( M 12 ) ( Fp − M 11 z p − L1 ) ,
−1
∆F (1) = K 1F ∫ ( F (1) (t ) − Fp(1) (t ))dt , P1 (∆z (1) , ∆zɺ (1) ) = K11∆z (1) + K 21 ∆zɺ (1) ,
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where
M 11* , M 12* , L*1 denote the matrices and vector corresponding to
M 11 , M 12 , L1 from the model (2.364) but with the assumed parameter values d 0 ; F (1) is an m1 -subvector of the force vector F for which the nominal trajectories
are
specified
in
advance;
F=
(( F ) , ( F ) (1) T
),
T (2) T
1F ∆z (1) = z (1) (t ) − z (1) is an m1 × m1 matrix of force p (t ) is an n1 -vector; K
feedback gains; K11 and K 21 denote the n1 × n1 matrices of the position and velocity feedback gains, respectively. It is assumed that the quadratic matrix M 12* is nonsingular. For the sake of simplicity we shall assume that both matrices are diagonal, K11 = diag ( K1ii ) and K 21 = diag ( K 2ii ) . Note that, contrary to the so-called classical hybrid control schemes [1, 2], the control law (2.365) takes into account complete dynamic models of the robot and environment, as well as the interaction among directions in which position is controlled and directions in which force is controlled. This means that both position and force feedback loops are used in all directions. The control law may be considered in a more general form, i.e. P1 and ∆F (1) can be defined in a more general form. Note also that the control law (2.365) implies that the contact forces are directly measured by appropriate force sensors. The force sensors are normally based on measuring the ‘deformation’ and, therefore, their ability to measure highly dynamic changes of forces is limited. Therefore, certain limits on the allowable frequency of change of forces should be taken into account.
2.10.6 Practical stability analysis Here we will establish a procedure for the analysis of practical stability of the robot’s interacting with dynamic environment, based on the decompositionaggregation approach. The closed-loop model of the robotic system (model of deviation around the desired nominal trajectory x p (t ) ) in the state space is obtained by combining the robot model in the state space (2.362) and the corresponding control law (2.365):
∆xɺ = ∆f (∆x, x*p , d ) + ∆G (∆x, d ) F
(2.366)
where ∆f (∆x, x*p , d ) is a 2n -vector and ∆G ( ∆x, d ) is a 2n × n matrix. In [6, 10], it has been shown that the application of the control law (2.365) ensures
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desired tracking of the prescribed nominal force trajectory (i.e. desired transient behavior of the force F (t ) ) when an ideal model of the system used in the control law is assumed. Since the model of the system (robot + environment) in (2.365) is not ideal (because of the parameters’ uncertainties) the desired force transient process cannot be perfectly achieved. However, it can be shown that by an appropriate selection of force feedback gains, and under assumption of limited deviations of the assumed model parameters from the actual values, the control law (2.365) can guarantee the force transient process satisfies the conditions of practical stability, i.e.
ɶ t ), ∀ t ∈ Τ F (t ) ∈ F(
(2.367)
Fig. 2.13 Dynamic position-force control law
{
}
where Fɶ (t ) = F ∈ R n / F − Fp (t ) < F t e− β t , ∀ t ∈ Τ . Here F t and β are positive numbers. If we denote by ∆F (t ) = F (t ) − Fp (t ) an n -dimensional vector-function of the deviation of the force vector F (t ) around the desired nominal trajectory Fp (t ) , then the inequality ∆F (t ) < F t e − β t will be fulfilled for each t ∈ Τ . The values F t and β correspond to the values X t and α , since
ɶ = ∪ Fɶ (t ) corresponds to the region X ɶ = ∪ X( ɶ t) . the region F t ∈Τ
t ∈Τ
Starting from the assumption that the force transient process satisfies (2.367), it has to be examined whether the control law (2.365) can ensure the practical stability of the overall system. This means that the conditions under which the
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proposed control law fulfills the specified control task, i.e. the conditions of practical stability of the robotic system have to be derived. On the other hand, if the system satisfies the conditions of practical stability with respect to the ɶ and X( ɶ t ) , then the force satisfies also the condition (2.367). For regions X 0 this reason, in what follows we will check the practical stability of the system ɶ , X( ɶ t ) , Τ ), which implies that the force satisfies (2.367). with respect to ( X 0 It can be shown [51, 52] that the system is practically stable with respect to ɶ ɶ t ) , Τ ) as defined above if there exists a real-valued continuously ( X 0 , X( differentiable function v(t , x) and a real-valued function of time Ψ (t ) which is integrable over the time interval Τ such that
vɺ(t , x) ≤ Ψ (t ), t
∫ Ψ(t ′)dt ′ < v
∀x ∈ X(t ), ∀t ∈T
ɶ t) ∂X( m
ɶ
− vM∂X0 ,
∀t ∈T
(2.368) (2.369)
0
ɶ denotes the boundary of the corresponding region and ∂X ɶ t ) \ X e−α t . In (2.368) vɺ denotes the time derivative of the function X(t ) = X( t
where
v(t , x) along the solution of the closed-loop system. The terms vm and vM in (2.369) denote the respective minimum and maximum values of v at the corresponding boundaries of the regions. Note that (2.368) and (2.369) are sufficient but not necessary conditions for the practical stability of the system. For justification of this method for testing the practical stability, see [51, 52]. Following the decomposition-aggregation approach to system stability analysis one can decompose the system (2.366) into a set of n subsystems: the position-controlled subsystems (i = 1,...,n1) and the force-controlled subsystems (i = n1 +1,…,n):
∆xɺ i = Aii ∆x i + ∆f i (∆x, x*p , ∆F , Fp , d ) + ∆Gi (∆x, d ) F , i = 1,…, n
(2.370)
where ∆xi (t ) = x i (t ) − xip (t ) = (∆z i (t )T , ∆ɺz i (t )T )T , i = 1,… , n and ∆f i and
∆Gi are the vectors and matrices of the appropriate dimensions, while the 2 × 2 matrices Aii are given by:
0 Aii = ii ii Q11K1
, i = 1,… , n1 ; Q K 1
ii 11
ii 2
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Dynamics and Robust Control of Robot-Environment Interaction
0 1 Aii = ii Pi , i = n1 + 1,… , n ii Vi Q22 L1 Q22 L1 and Q22ii are the diagonal (constant) elements of the n1 × n1
where Q11ii
and n2 × n2 matrices representing estimates of Q11 = Λ−11 J −T J * Λ*1 , and
Q22 = Λ−21 J −T J *Λ*2 ( Λ−11 , Λ−21 , Λ*1 , Λ*2 are the corresponding submatrices of the P matrices Λ −1 and Λ* ); L1Pi and LVi 1 represent elements of the vectors L1 and
LV1 estimating stability factors in the dynamics model of the environment and integral force feedback, i.e.
(M )
* −1 12
L*1 − ( M 120 ) L10 + ( M 12* ) K 1F ∫ ( F (1) (t ) − Fp(1) (t ))dt = −1
−1
= L1P ∆z (2) + LV1 ∆zɺ (2) + L1R Let us consider the practical stability of the decoupled subsystems
∆xɺ i = Aii ∆xi . We may assume that the regions of practical stability can be presented in the form:
ɶ =X ɶ (1) × ..... × X ɶ (n) X 0 0 0 where
{
and
ɶ t) = X ɶ (1) (t ) × ..... × X ɶ ( n ) (t ) X(
}
ɶ (i ) = x (i ) ∈ R / x (i ) − x (i ) (0) < X (i ) , X 0 p 0
{
}
ɶ (i ) (t ) = x (i ) ∈ R / x (i ) − x ( i ) (t ) < X (i ) e −αi t , ∀ t ∈ Τ , and X p t
X t(i ) > X 0( i ) > 0, α i > 0, ∀ t ∈Τ, ∀ i = 1,… , n . Let
us
select
the
vi (t , x) = ( ∆x i T H i ∆x matrices
of
the
)
i 1/ 2
functions
vi (t , x) , i = 1,… , n
in
the
form:
, i = 1,… , n , where H i are the positive definite
appropriate
dimensions
and
∆xi (t ) = x (i ) (t ) − x (pi ) (t ) ,
t ∈Τ, i = 1,… , n . The derivative of the functions vi along the solutions of the decoupled subsystems can be written as:
vɺi (t , x) = ( grad vi ) Aii ∆xi ≤ − γ i vi ≤ − γ i′ ∆xi T
(2.371)
where γ i = min λ ( Aii ) , λ ( Aii ) denotes the eigenvalues of the corresponding matrix and γ i′ = γ i λmin ( H i ) . The conditions (2.371) are valid provided
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that the matrices H i are selected so as to satisfy H i Aii + Aii T H i ≤ − 2γ i H i . The coupling members in the subsystems (2.370) can be estimated in the following form (summation indices being omitted): (1) 1 (1) 2 (1) ( grad vi )T ∆fi (∆x, x (1) p , ∆F , Fp , d ) < ∑ ξ ij ( x p ) (t ) + ∑ ξ ij ( Fp ) (t ) + j
j
4 5 + ∑ ξij3 ( xɺ (1) p ) (t ) + ∑ ξ ij ∆x j + ∑ ξ ij ∆F j
(2.372) (2.372)
j
( grad vi )T ∆Gi (∆x, d ) F < < ∑ ξij6 ∆x j + ∑ ξij7
( (F ) p
j
)
(2.373)
+ ∆Fj , i = 1,… , n
where ξ ijk , k = 1, 2,… , nk ( nk denotes an appropriate number corresponding to
k ) are the real numbers (note that these numbers may also be negative). The ɶ t ) , ∀ t ∈Τ and inequalities (2.372) and (2.373) must be valid for ∀ x ∈ X( ∀ d ∈ D . The practical stability conditions of the overall system can be established by considering the derivative of the function vi along the solutions of the coupled subsystems (2.370) (aggregation principle). Based on (2.371) – (2.373), the candidates for the functions Ψ i (t ) for each subsystem i can be obtained in the form:
( )
Ψ i (t ) = −γ i Xt( i ) e −αi t + ∑ (ξij4 + ξij6 )Xt e−α t + ∑ ξij1 x (1) p
(
+ ∑ ξij2 Fp(1)
)
j
( )
(t ) + ∑ ξij5 F t
j
j
( )
(t ) + ∑ ξij3 xɺ (1) p
( )
e − β t + ∑ ξij7 ( Fp ) (t ) + F t j
j
e− β t ,
j
(t ) +
i = 1, 2
(2.374) By substituting (2.374) into (2.369) we obtain the practical stability test for the robot interacting with dynamic environment when the control law (2.365) is applied.
2.10.7 Example Let us consider the following simple example: n = 2 DOFs, acting upon the dynamic environment (see Fig. 2.14). The model of the environment in the x direction is assumed to be of the form:
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Dynamics and Robust Control of Robot-Environment Interaction
me ɺxɺ + k D xɺ + ke ( x − xe ) = Fx
(2.375)
while in the y -direction the model is given by:
µe Fx = Fy where me is the equivalent mass of the environment in the x -direction; k D is the equivalent damping (not shown in the figure); ke is the equivalent stiffness,
xe is the equivalent position, and µe represents the equivalent friction. The system parameters include those of the robot (masses of the links m1 and m2 , moments of inertia of the links J1 and J 2 , lengths of the links l1 and l2 , and the distances between the joints and the mass centers of the links l01 and l02 ) and the environment parameters ( me , k D , ke , xe and µe ).
Fig. 2.14 Robot interacting with the environment
For the sake of simplicity let us assume that the parameters of the robot are precisely known, and only parameters of the environment are not sufficiently well known, but the lower and upper bounds of their allowable values are being known. This means that the set D of the allowable values of the parameters is defined as:
{
D = d : m1 = m10 , m2 = m20 , J1 = J10 , J 2 = J 20 , l1 = l10 , l2 = l20 , l01 = l010 , l02 = l020 ,
mel < me < meu , k Dl < k D < k Du , kel < ke < keu , xel < xe < xeu , µel < µe < µeu } (2.376) where superscript "0" denotes the ‘nominal’ values of the parameters and the indices l and u stand for the lower and upper bounds of the allowable
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parameters values. Let us consider the following simple task: let in the x direction be defined the desired force Fp(1) (t ) = Fx0 (t ) = F 0 c = const , while in 0 0c the y -direction the desired position trajectory is imposed z (1) p (t ) = y (t ) = v t ,
where v 0 c = const . Let us consider how the dynamic position-force control law (2.365) can fulfill this control task. In the y -direction, the position control law is applied. Let us assume that the position, K111 , and the velocity, K 211 , feedback gains are being selected in such a way that the solutions of the characteristic equation of the closed-loop system K111 + K 211s + s 2 = 0 are −σ 1 and −σ 2 . In this case
K 211 = σ 1 + σ 2 and K111 = σ 1 σ 2 . In the x -direction, the force control law is applied. According to (2.365) this control law has the form:
τ = U * ( z, zɺ, ɺɺzc , F ) = ( J * ) (Λ* ɺɺzc + ρ * − F ) T
(2.377)
K111 ( y − v 0 c t ) + K 211 ( yɺ − v 0 c ) ɺɺ zc = −1 ( me* ) [ F 0 c + K 1F ∫ ( Fx − F 0 c )dt − k D* xɺ − ke* ( x − xe* )] where me* , k D* , ke* and xe* represent the prescribed ‘nominal’ values of the environment model (which must be within the allowed region of values of these parameters). From the assumption of the precise knowledge of robotic parameters it follows that J * = J , Λ* = Λ, ρ * = ρ . Therefore, one may assume that the control law perfectly compensates for the robot dynamics and for the forces in both directions. However, the control of the force Fx is realized using unknown parameters of the environment model. To apply the method proposed for practical stability analysis, the system can be ‘decomposed’ in two subsystems. The first subsystem is associated to the y direction in which the position of the robot tip is controlled, while the second subsystem is associated to the force-controlled x -direction. In this particular case the subsystem matrices Aii get the simple forms:
0 A11 = 11 K1
1 K 211
(2.378)
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Dynamics and Robust Control of Robot-Environment Interaction
0 * k A22 = − e + K 1F k Dl m* me* e ii
K 1F
1 mel k D* − me* me*
(2.379)
ii
since the elements Q11 and Q 22 are equal to unity. To define the matrices H i , which enable one to estimate the stability degree of the matrices Aii , it is convenient to introduce the transformation of the state coordinates ∆xi = T1i ∆xˆ i in such a way that the stability test of the decoupled subsystems in the transformed state space leads to (2.371). Assuming that the matrix H i in the function vi (t , x) is equal to I 2 , it is obtained H i Aˆ iiT + Aˆ iiT H i ≤ −2γ i H i , where
γ 1 = min { σ 1 , σ 2 } and γ 2 = min λ ( A22 ) . The stability test in this case leads to the following relations: (1)
ˆ −γ 1 X 0
1 − e −α1t
α1
1− e ˆ (2) −γ 2 X 0
−α 2 t
α2
(2)
(1)
(1)
ˆ e−α1t − X ˆ , ∀ t ∈Τ <X 0 t k* max 1 − e 0 * ke max xe − x ke + + < 2 me*t me* F 0 ct
(2.380)
(2.381)
(2)
ˆ e −α 2t − X ˆ , ∀ t ∈Τ <X 0 t where ‘max’ is to be defined over the allowable parameter values. The test (2.381) shows explicitly the way how the environment parameter uncertainties affect the practical stability of the system. If we assume that all parameters of the environment model are precisely known, then the test leads to the relation: (i )
ˆ −γ i X 0
1 − e −α i t
αi
(i )
(i )
ˆ e −α i t − X ˆ , i = 1, 2 <X t 0
(2.382)
The relation (2.382) represents a test of whether or not the characteristics of the environment model and the force feedback law can provide the practical stabilization of the global system, i.e. it represents a generalization of the stability conditions presented in [6, 10] to the practical stability case for this particular example. Obviously, the test can be reduced to checking of the (i )
(i )
ˆ =X ˆ ). This means that the position control part must condition γ i > α i (if X t 0
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207
provide a stabilization rate which is higher than the required exponential shrinkage of the practical stability regions, as well as that the environment together with the force feedback law must have stability properties such that the −α t system in the x -direction is stabilized faster than e 2 . Stability is tested with respect to the properties of both the environment model ( ke / me , k D / me ) and integral force feedback loop with the feedback gain K 1F .
2.10.8 Conclusion In this section are presented the new, less conservative, test conditions for the practical stability of the robot interacting with dynamic environment. The test has been presented for one specific control law, but the procedure can be easily extended to other control laws. It is specifically convenient (i.e. less conservative) for the analysis of the so-called decentralized control laws. The test enables one to study the influence of the different model uncertainties upon the system behavior, as one of the most relevant aspects for practical implementation of robots in the numerous potential applications in which the robot has to interact with the dynamic environment. This enables an effective comparison of the different control laws and study of the effects of uncertainties. Unlike the test given in [19, 49], this test examines explicitly the stability of the position-controlled DOFs and force-controlled DOFs. The stability test of the force-controlled subsystems shows explicitly the conditions under which is possible to ensure the practical stability if the force is practically stabilized. However, the test takes into account dynamic coupling of the position-controlled and force-controlled DOFs, both through the dynamics of the robot and the dynamics of the environment. The considered control law (2.365) represents a generalization of the so-called computed torque method to the case when the robot is interacting with the dynamic environment. Obviously, if the model parameters are precisely known, and if the environment model possesses stability properties, this control law can practically stabilize the system. However, if the approximate model of the environment is used in the control loop (e.g. where dynamic coupling among different directions is ignored, M * = diag (mi* ) ), the control law is simplified, but it should be tested (using the established test) whether such control can ensure the practical stability. If the dynamics of the robot and environment are fully ignored ( L* = I , ρ * = 0, M * = I , L* = 0 ), the control law (2.365) reduces practically to the so-called classical hybrid control law [1, 2], and the established stability test
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Dynamics and Robust Control of Robot-Environment Interaction
can be applied to investigate which conditions provide the possibility of stabilizing the robot with such hybrid control law. Therefore, the obtained test can be used to analyze which of the dynamic elements may be ignored but still to get ensured the fulfillment of the desired control task. This means that this test may serve as a procedure for the identification of a ‘minimal’ dynamic control law, which may practically stabilize the robot. Such procedure is of a great importance in the synthesis of control laws for the robots interacting with a dynamic environment, since, on the one hand, ignoring the dynamics (both of the robot and the environment) may lead to an inappropriate performance (and even instability), while, on the other hand, involving all dynamic factors in the control law can lead to a high complexity and an insufficient robustness of the control system because of the high uncertainties in modeling of the environment. More refined approximations (e.g. by taking into account possible dependence on the parameters d, etc.) may lead to a less conservative test. The tests presented assume that the structure of the environment model is known. In a general case, this structure may not be known, and this will be the subject of further studies.
2.10.9 Practical stability - A remark It would be natural to give here an autonomous text that would be self-sufficient for understanding the practical stability and provide a proof of its testing with robotic systems. However, in order to have a self-contained text in view of the understanding practical stability and its testing, it would be necessary to make an appendix that would be significantly longer than the basic text of Section 2.10. For this reason, we direct the readers interested in getting information on practical stability and its testing to the research monograph: Non-Adaptive and Adaptive Control of Manipulation Robots, by Vukobratovic M., Stokic D. and Kircanski N., Vol. 5 of the monograph series “Scientific Fundamentals of Robotics”, Springer-Verlag, 1985., with a special emphasis on Appendix. 2B: Practical Stability of Manipulation Robots. In view of the above we will give here a basic remark about practical stability. Practical stability is primarily concerned with the system’s behavior in a limited time interval (Fig. 2.15). ɶ = ∪ X( ɶ t ) around x (t ) is defined as a region to which the The region X p t ∈Τ
system state must belong during the tracking of x p (t ) . Perturbations that are
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209
acting upon the robotic system can be reduced, in principle, to perturbations of the initial conditions type. However, the initial error of the robot state can be assumed to be bounded, i.e. we can assume that the initial state x(0) = x0 must
ɶ in the state space X ɶ ⊂ R N . Thus, our belong to some finite region (set) ∈ X 0 0 control task at the executive level is to synthesize the control which will ensure ɶ it must be satisfied x(t )∈ X( ɶ t ), ∀ t ∈Τ , where that for all x0 ∈ X 0
ɶ , x (t )∈ X( ɶ t ), ∀ t ∈Τ and X ɶ ⊂ X(0) ɶ x p (0)∈ X must be satisfied (Fig. 2.15). 0 p 0
Fig. 2.15 Regions of practical stability of the robotic system
This control task can be viewed as a task of practical stabilization of the robotic system over a finite time interval around the given nominal trajectory x p (t ) . Similar control has already been considered in the research monograph: Control of Manipulation Robots: Theory and Application, by Vukobratovic M, and Stokic D., Vol. 2, of the monograph series “Scientific Fundamentals of Robotics”, Springer-Verlag, 1982. However, we have previously assumed that all parameters of the robot are well identified and known in advance. Now, we shall consider the case when all parameters of the robots d , θ can not be defined in advance or some of them are variable. We shall divide the robot parameters in two groups: d parameters of the mechanical part of the robot are assumed to be variable and unknown, and we shall assume that their variation is relatively fast; θ parameters of actuators are assumed to vary very slowly. Thus, we shall assume that the variation of the
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Dynamics and Robust Control of Robot-Environment Interaction
actuator parameters θ may be neglected, and their identification can be performed off-line. So we can synthesize the control assuming that the models of actuators are constant and precisely defined. During the robot’s operation, the control system must occasionally check the parameters of actuators θ , to identify their values, and, if they significantly vary, either synthesize new feedback gains or indicate variation of the parameters. As we have already noted, we assume that d ∈ D , i.e. the parameter values must belong to some bounded set D of the allowable values of parameters. Thus, our control task at the executive level can be stated as follows (practical stabilization of the robotic system): Definition of control task: A control u (t ) has to be synthesized which will ensure that ∀x0 ∈ R N
and ∀ d ∈ D
ɶ x(0) = x0 ∈ X imply 0
ɶ t ), ∀ t ∈Τ , where X( ɶ t ) are the finite regions in the state x ( t , x0 , d , u (t ) )∈ X(
space around the prescribed nominal trajectory x p (t ) defined for every point of
ɶ ⊂ X(0) ɶ is a bounded region of the initial perturbations. time t and X 0
A Unified Approach to Dynamic Control of Robots
211
Appendix A
Proof of Theorem 1
Let us consider the behavior of the system (2.51) in the region
Ω = {( x, t ) ∈ R 2 n × R / x < h, t0 ≤ t < ∞} ,
where h > 0 is a fixed number. Because of α ( x, t ) = o ( x ) , for x → 0 , α (0, t ) ≡ 0, ∀t ≥ t0 , and due to the smoothness of the vector function α ( x, t ) , a positive constant C1 can be found, such that:
α ( x, t ) ≤ C1 x , p > 1, ∀( x, t ) ∈ Ω p
(A.1)
Due to the continuity of the function β ( x, t ), which has a special form, such constant M can be determined that
β ( x, t ) ≤ M , ∀( x, t ) ∈ Ω
(A.2)
If in the system (2.51) we carry out the transformation
x = y e−γ (t −t0 )
(A.3)
yɺ = B(t ) y + αɶ ( y, t ) + βɶ ( y, t ) µ (t )
(A.4)
we will get where
B(t ) = A(t ) + γ I 2 n
αɶ ( y, t ) = eγ (t −t ) α ( ye −γ (t −t ) , t ) 0
0
(A.5)
βɶ ( y, t ) = eγ (t −t ) β ( ye −γ (t −t ) , t ) 0
0
whereby x(t0 ) = y (t0 ) . Then the system
yɺ = B(t ) y
(A.6)
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Dynamics and Robust Control of Robot-Environment Interaction
is regular. Indeed, if β k (k = 1, 2,… , 2n) are characteristic indices of the linear system (A.6), it is obvious that
β k = α k + γ < 0 (k = 1, 2,… , 2n) Taking into account the regularity of the system (2.52) and formula (A.5) we get t
t
2n 2n 1 1 Sp ( ) lim Sp ( ) 2 2 B ω d ω = A ω + n γ d ω = α + n γ = βk [ ] ∑ ∑ k t →∞ t ∫ t →∞ t ∫ k = 1 k = 1 t0 t0
lim
Therefore, the system (A.6) is regular. Let H (t ) ( H (t0 ) = I 2 n ) be the normed fundamental matrix of the system (A.6). The system of equations (A.4) can be replaced by the integral equation t
y (t ) = H (t ) y (t0 ) + ∫ K (t , ω ) αɶ ( y (ω ), ω ) + βɶ ( y (ω ), ω ) µ (ω ) dω
(A.7)
t0
where the matrix K (t , ω ) = H (t ) H −1 (ω ) is a Cauchy matrix [13, 53]. As all the characteristic indices β k (k = 1,2,...,2n) of the linear system (A.6) are negative, there is a positive constant C3 such that
H (t ) < C3 for t ≥ t0
(C3 ≥ 1)
Besides, we can introduce the following estimate [13] of the Cauchy matrix K (t , ω ) for the regular system with negative characteristic indices
K (t , ω ) < C4 eε 0 (ω −t0 ) for t0 ≤ ω < t < ∞ and for an arbitrarily small constant ε 0 > 0 . On the basis of (A.1), (A.2), (2.54) we have the following estimate
αɶ ( y, t ) + βɶ ( y, t ) µ (t ) ≤ eγ (t −t ) α ( ye −γ (t −t ) , t ) + β ( ye−γ (t −t ) , t ) µ (t ) ≤ 0
0
0
≤ eγ (t −t0 )C1e −γ p (t −t0 ) y + eγ ( t −t0 ) MC2e − λ ( t −t0 ) µ (t0 ) = p
= C1e − ( p −1)γ (t −t0 ) y + MC2 e − ( λ − γ ) (t −t0 ) µ (t0 ) p
Taking into account the relation (2.55) let us select the positive number ε 0 so small to satisfy the inequalities
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A Unified Approach to Dynamic Control of Robots
δ 0 = ( p − 1)γ − ε 0 > 0 and λ − γ − ε 0 > 0 . Hence, estimating with respect to the norm the right and left side of the integral equation (A.7) we have t
y (t0 ) + ∫ K (t , ω ) αɶ ( y (ω ), ω ) + βɶ ( y (ω ), ω ) µ (ω ) d ω ≤
y (t ) ≤ H (t )
t0 t
≤ C3 y (t0 ) + ∫ C4C1e[ 0
ε − ( p −1) γ ] (ω −t0 )
p y (ω ) + MC4C2 e − ( λ −γ −ε 0 )(ω −t0 ) µ (t0 ) dω =
t0
t
= C3 y (t0 ) + ∫ C5e[ 0
ε − ( p −1) γ ] (ω − t0 )
y (ω ) d ω + p
t0
(
)
C6 1 − e − ( λ −γ −ε 0 ) (t −t0 ) µ (t0 ) ≤ λ −γ − ε0 C6 µ (t0 ) t p ε − ( p −1) γ ] (ω −t0 ) ≤ C3 y (t0 ) + + ∫ C5e[ 0 y (ω ) dω , λ − γ − ε 0 t0 +
where
(A.8)
C 5 = C 4 C1 , C 6 = MC 4 C 2 .
Then the estimate (A.8) implies t C6 µ (t0 ) p y (t ) ≤ C3 y (t0 ) + + C5 ∫ e −δ 0 (ω −t0 ) y (ω ) d ω λ −γ −ε0 t0
(A.9)
From the inequality (A.9), according to the Bihari lemma [13, 54], we have
C3 y (t0 ) + y (t ) ≤
C6 µ (t0 ) λ −γ −ε0
C µ (t0 ) 1 − ( p − 1) C3 y (t0 ) + 6 λ − γ − ε 0
p −1
C5 δ0
1 p −1
(A.10)
if only
C µ (t0 ) ( p − 1) C3 y (t0 ) + 6 λ −γ −ε0
p −1
C5
δ0
<1
(A.11)
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Dynamics and Robust Control of Robot-Environment Interaction
It is obvious that the inequality (A.11) is satisfied if y (t0 ) and µ (t0 ) are sufficiently small. Let us choose these quantities so as to satisfy the inequality
C µ (t0 ) ( p − 1) C3 y (t0 ) + 6 λ −γ −ε0
p −1
C5
δ0
≤
1 2
(A.12)
Fulfillment of the inequality (A.12) automatically entails the fulfillment of the inequality (A.11). Then, from the inequality (A.10) we have
y (t ) ≤ 2
1 p −1
C3 y (t0 ) +
2
1 p −1
C6 µ (t0 )
λ −γ −ε0
for the sufficiently small y (t0 ) and µ (t0 ) . The latter inequality ensures the correctness of using the estimates (A.1) and (A.2). Indeed, as x(t ) ≤ y (t ) , ∀ t ≥ t0 then ∀ t ≥ t0 ( x(t ), t ) ∈ Ω . Taking into account that y (t0 ) = x(t0 ) , from (A.3) we get
b µ (t0 ) −γ (t −t0 ) x(t ) ≤ a x(t0 ) + e λ −γ −ε0 1 1
(A.13)
where a = 2 p −1 C3 , b = 2 p −1 C6 . Since the inequality (A.13) is not strict and is satisfied for any small ε 0 > 0 , the limit inequality is also justified:
b µ (t0 ) x(t ) ≤ a x(t0 ) + λ −γ
−γ (t −t0 ) , ∀t ≥ t0 e
Let us notice that the theorem was proved using the methodology of proving Lyapunov’s stability theorem in the first approximation, given in [13]. Note: The assumption of the smoothness of the function α ( x, t ) can be replaced by the more weak condition (A.2) that must be fulfilled in the vicinity of the point x = 0 (not in Ω ) with some constant Cɶ1 .
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A Unified Approach to Dynamic Control of Robots
Appendix B
Proof of Theorem 2
It follows from expressions (2.86), (2.87) that for each t ≥ t0 , η ( t )
and
ηɺ ( t ) are bounded by the magnitude of CΓ η (t0 )2 + ηɺ (t0 ) + CΓCi λ −1 . 2
Hence, because of the condition (2.90) of Theorem 2, for every t ≥ t0 the following conditions will be fulfilled:
q (t ) ∈ Vq , qɺ (t ) ∈ Vqɺ
(B.1)
Thus the constraints (2.66), (2.67) are satisfied. Hence, it follows that for every t ≥ t0 the constraint
qɺɺp (t ) + Γ1ηɺ (t ) + Γ 2η (t ) ∈ Vqɺɺ
(B.2)
will be satisfied. Indeed, on the basis of the condition (2.91) of Theorem 2 we conclude that
Γ1ηɺ (t ) + Γ 2η (t ) ≤ Γ1 ηɺ (t ) + Γ 2 η (t ) ≤ ( Γ1 + Γ 2
(C
Γ
)
)
η (t0 ) + ηɺ (t0 ) + CΓ Ciλ −1 < δ 3 2
2
Besides, from (2.89) we have that for any t ≥ t0 , the norm µ (t ) is bounded by the value
L CΓ η (t0 ) + ηɺ (t0 ) + LCΓ Ci λ −1 + L3Ci + Cρ 2
2
(B.3)
Because of the condition (2.92) of Theorem 2, the value of (B.3) is smaller than σ , and since Fp (t ) ∈ VFσ , ∀t ≥ t0 , then
F (t ) ∈ VF , ∀t ≥ t0 ,
(B.4)
that is, the interaction force F (t ) satisfies for every t ≥ t0 the constraint (2.69). Let us consider the control law (2.23)
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Dynamics and Robust Control of Robot-Environment Interaction
τ (t ) = U ( q, qɺ , qɺɺp + Γ1ηɺ + Γ 2η , F ) As the components of the vector function U for every t ≥ t0 satisfy the constraints (B.1), (B.2), (B.4), on the basis of the relation (2.76) we conclude that
τ (t ) ∈ Vτ
∀t ≥ t0 ,
that is, the control law (2.23) represents an admissible control. Let us consider the control law (2.28)
(
τ (t ) = U q, qɺ , qɺɺp + Γ1ηɺ + Γ 2η , f ( q, qɺ , qɺɺp + Γ1ηɺ + Γ 2η )
)
As the function f satisfies the Lipschitz conditions with the constants Li with respect to the i -th variable (i = 1, 2, 3), then, in view of the estimates (2.86), (2.87) and condition (2.92) of Theorem 2, we have
Fp − f ( q, qɺ , qɺɺp + Γ1ηɺ + Γ 2η ) = f ( q p , qɺ p , qɺɺp ) − f ( q, qɺ , qɺɺp + Γ1ηɺ + Γ 2η ) ≤ ≤ L1 η + L2 ηɺ + L3 Γ1ηɺ + Γ 2η ≤ ( L1 + L3 Γ 2
+ ( L2 + L3 Γ1
) ηɺ
≤ L CΓ η (t0 ) + ηɺ (t0 ) 2
2
)η
+
<σ .
Therefore, for every t ≥ t0 there will be the inclusion
f ( q, (t ), qɺ (t ), qɺɺp (t ) + Γ1ηɺ + Γ 2η ) ∈ VF
Analogously to the previous case, it comes out that the control law (2.28) is also admissible. Now we will prove the estimates (2.95), (2.96). As the right-hand side of the inequality (2.85) is a monotonously decreasing function of t , solving the inequality with respect to t
CΓ e − λ (t −t0 ) x(t0 ) + CΓ Ci λ −1 < ε along with the inequality (2.93), yields the estimate (2.95). Analogously, from the inequality (2.89) under the condition (2.94) we obtain the estimate (2.96).
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A Unified Approach to Dynamic Control of Robots
Appendix C
Proof of Theorem 3
On the basis of (2.108), (2.109), (2.105) we have the following estimate:
αɶ ( y, t ) + βɶ ( y, t ) µ (t ) − βɶ ( y, t ) ρ (t ) ≤ C1e− ( p −1) γ (t −t ) y (t ) + 0
p
+ MCR e− ( λ −γ )( t −t0 ) µ (t0 ) + Meγ (t −t0 ) ( Cµ + Cρ )
(C.1)
By estimating with respect to the norm the left and right sides of the integral equation (2.114), and taking into account the estimations (2.115), (2.116), we arrive at t
y (t ) ≤ H (t ) ⋅ y (t0 ) + ∫ K (t , ω ) αɶ (⋅) + βɶ (⋅) µ (ω ) − βɶ (⋅) ρ (ω ) dω ≤ t0 t
≤ C2 y (t0 ) + C3 ∫ C1e− ( p −1) γ (ω −t0 ) y (ω ) + MCR e − ( λ −γ )(ω −t0 ) µ (t0 ) + p
(C.2)
t0
+ Me
γ (ω −t0 )
t
( Cµ + Cρ ) dω ≤ A + C1C3 ∫ e−( p−1)γ (ω −t0 ) y(ω ) p dω t0
where the constant A is
A = C2 y (t0 ) +
MCR C3 µ (t0 ) MC3 + Cµ + Cρ ) eγ (T −t0 ) − 1 ( λ −γ γ
(
)
(C.3)
Therefore, in accordance with the Bihari lemma [13, 54], we obtain
y (t ) ≤
A C1C3 p −1 1 − ( p − 1) A ( p − 1) γ
1 p −1
(C.4)
Because of the condition (2.117) of Theorem 3 and since x(t0 ) = y (t0 ) , we have
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Dynamics and Robust Control of Robot-Environment Interaction
A≤2
−
1 p −1
1
γ p −1 C1C3
(C.5)
Then, it comes out that C1C3γ −1 A p −1 ≤
1 − A p −1C1C3γ −1 ≥
1 2
1 and 2 (C.6)
Therefore, the estimate (C.4) is correct and, taking into account the inequality (C.6), we get
y (t ) ≤ 2
1 p −1
A
(C.7)
On the other hand, because of the condition (2.117) we also have 1
y (t ) ≤ 2 p −1 A < h,
∀t ∈ [t0 , T ]
(C.8)
Therefore, the solution y (t ) of equation (2.114) is defined over the whole interval [t0 , T ] . On returning to the variable x from (2.111) and (C.7) we obtain for every t ∈ [ t0 , T ] : 1
x(t ) ≤ 2 p −1 Ae −γ (t −t0 ) ≤
b µ (t0 ) −γ (t −t0 ) MC3 ≤ a x(t0 ) + + 2 p −1 C µ + C ρ ) eγ ( T − t 0 ) − 1 ( e λ −γ γ 1 1 1
(
)
(C.9)
where a = 2 p −1 C2 , b = 2 p −1 MCRC3 . Thus, the formula (2.118) of the theorem is proved. As is obvious that
x(t ) ≤ y (t ) < h
(C.10)
Then, due to the formula (2.107), for every t ∈ [t0 , T ] , the real motion has the property that q (t ) ∈ Vq and qɺ (t ) ∈ Vqɺ . Hence, all the estimates based on this property are correct as well as the derivation of (2.118).
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A Unified Approach to Dynamic Control of Robots
Appendix D
Proof of Theorem 4
Before starting direct proving of the theorem we are going to demonstrate the correctness of its conditions related to the existence of solutions of the inequalities (2.117), (2.120) – (2.125). Indeed, in accordance with the determination of the quantities Cµ = Cµ (Cr , C ρ ), C = C (Cr , C ρ ) , they converge to zero when Cr , Cρ → 0 . Because of that, it is obvious that the conditions of existence of the small quantities Cr , Cρ under which the inequalities (2.117), (2.120) – (2.125) hold, are fulfilled. The assertion 1) of Theorem 4, i.e. fulfillment of the inclusions
q(t ) ∈ Vq , qɺ (t ) ∈ Vqɺ , ∀t ∈ [t0 , T ]
(D.1)
comes directly out from Theorem 3 (see the inequality (C.10)). From the inequality (2.105) and the condition (2.120) of Theorem 4, we have
F (t ) − Fp (t ) < Hence,
σ
2
< σ , ∀t ∈ [t0 , T ]
(D.2)
∀t ∈ [t0 , T ]
F (t ) ∈ VF ,
(D.3)
which proves assertion 3) of Theorem 4. From the inequality (D.2), condition (2.121) of Theorem 4, and equation (2.98) we obtain the following estimate: t
t
t0
t0
Fp (t ) + R ∫ µ (ω )d ω − Fp (t ) ≤ Fp (t ) − F (t ) + F (t ) − Fp (t ) − R ∫ µ (ω )d ω < <
σ 2
+ d1Cr + d 2Cρ <
δ 2
+
δ 2
=σ
(D.4)
Therefore, t
Fp (t ) + R ∫ µ (ω ) dω ∈ VF , t0
∀t ∈ [t0 , T ]
(D.5)
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Dynamics and Robust Control of Robot-Environment Interaction
Besides, from the inequality (2.125) of Theorem 4, taking into account the fulfillment of Lipschitz conditions for the function ψ and the inequalities (2.119), (D.2), (D.4), it comes out that on the interval [t0 , T ] the following two estimates hold t t ψ q, qɺ , Fp + R ∫ µ (ω )d ω − qɺɺp = ψ q, qɺ , Fp + R ∫ µ (ω ) dω −ψ ( q p , qɺ p , Fp ) ≤ t0 t0 t
≤ L1 η + L2 ηɺ + L3 R ∫ µ (ω )d ω ≤ ( L1 + L2 ) max { η , ηɺ } + L3σ < δ 3 . t0
ψ ( q, qɺ , F ) − qɺɺp ≤ ( L1 + L2 ) max { η , ηɺ } + L3 F − Fp < < ( L1 + L2 ) max { η , ηɺ } + L3σ < δ 3 Therefore, for every t ∈ [t0 , T ] there exist the inclusions t ψ q, qɺ , Fp + R ∫ µ (ω )dω ∈ Vqɺɺ t0
(D.6)
ψ ( q, qɺ , F ) ∈ Vqɺɺ
(D.7)
Then, taking into account the relations (D.1), (D.3), (D.5), (D.6), (D.7), from the condition (2.76) we have that all the control laws (2.39), (2.40), (2.43) on the interval [t0 , T ] are admissible. Thus, assertion 2) of Theorem 4 is also proved. Using the estimates of transient processes (2.105), (2.118), solving the inequalities with respect to t
b µ (t0 ) a x(t0 ) + λ −γ
−γ (t −t0 ) + C eγ (T −t0 ) − 1 < ε e
(
)
CR µ (t0 ) e − λ (t −t0 ) + Cµ < δ and taking into account the conditions (2.122), (2.123) of Theorem 4, we obtain the corresponding estimates of the time of transient processes in the form of (2.126), (2.127). Obviously, the requirement (2.124) of Theorem 4 ensures the fulfillment of the inequality (2.128).
A Unified Approach to Dynamic Control of Robots
221
Appendix E
Proof of Theorem 5
To prove this theorem it suffices to check whether the control laws (2.139), (2.140) are admissible. On the basis of the proof of Theorem 2 and relations (2.146) – (2.149) we conclude that for any t ≥ t0 , the following inclusions are justified δ
δ
q (t ) ∈ Vq q , qɺ (t ) ∈ Vqɺ qɺ , qɺɺp + Γ1 (qɺ − qɺ p ) + Γ2 (q − q p ) ∈ Vqɺɺ
Γ1 δ qɺ + Γ2 δ q
(E.1)
and additionally,
{
max δ F , ( L1 + L3 Γ 2 )δ q + ( L2 + L3 Γ1 )δ qɺ
F (t ) ∈ VF
}
(E.2)
From the inequalities (2.138) and relation (E.1) we have
qˆ (t ) ∈ Vq ,
qˆɺ (t ) ∈ Vqɺ ,
∀t ≥ t0
(E.3)
and because of
(
)
qɺɺp + Γ1 qɺˆ − qɺ p + Γ 2 ( qˆ − qˆ p ) − qɺɺp − Γ1 ( qɺ − qɺ p ) − Γ 2 ( q − q p ) ≤ ≤ Γ1 δ qɺ + Γ 2 δ q , we obtain
(
)
qɺɺp (t ) + Γ1 qɺˆ (t ) − qɺ p (t ) + Γ 2 ( qˆ (t ) − q p (t ) ) ∈ Vqɺɺ , ∀t ≥ t0
(E.4)
From (E.2) and the last inequality in (2.138) it follows that
Fˆ (t )∈ VF ,
∀t ≥ t0
(E.5)
Because of that, on the basis of (2.76) and conditions (E.3), (E.4), (E.5) we obtain that the control law (2.139) is admissible. Further, analogously to the proof of Theorem 2, we have the estimates:
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Dynamics and Robust Control of Robot-Environment Interaction
(
(
)
)
Fp − f qˆ , qˆɺ , qɺɺp + Γ1 qɺˆ − qɺ p + Γ 2 ( qˆ − q p ) ≤ ( L1 + L3 Γ 2 + ( L2 + L3 Γ1
) qɺˆ − qɺ
p
) qˆ − q
p
+
≤ σ + ( L1 + L3 Γ 2 ) δ q + ( L2 + L3 Γ1 ) δ qɺ
Hence, on the basis of the conditions (2.149) we conclude that
(
(
)
)
f qˆ , qˆɺ , qɺɺp + Γ1 qɺˆ − qɺ p + Γ 2 ( qˆ − q p ) ∈VF , ∀t ≥ t0 . The last inclusion, together with conditions (E.3), (E.4) and relation (2.76) prove the admissibility of the control law (2.140).
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A Unified Approach to Dynamic Control of Robots
Appendix F
Proof of Theorem 6
To prove Theorem 6 it suffices to check admissibility of the control laws (2.150), (2.151), (2.152). Relying on the proof of Theorem 4 and taking into account the conditions (2.162), (2.163), we obtain the following inclusions
q (t ) ∈ Vq q , qɺ (t ) ∈ Vqɺ qɺ , ∀t ∈ [t0 , T ] , δ
δ
wherefrom, according to (2.138), we have
qˆ (t ) ∈ Vq , qˆɺ (t ) ∈ Vqɺ , ∀t ∈ [t0 , T ]
(F.1)
Besides, from the condition (2.165), we have
F (t ) ∈ VFδ F , ∀t ∈ [t0 , T ] and therefore,
Fˆ (t ) ∈VF ,
∀t ∈ [t0 , T ] .
(F.2)
Taking into account Lipschitz conditions for the function G , it comes out from equation (2.158) that t
F − Fp − R ∫ ( Fˆ − Fp ) (ω )d ω ≤ d1Cr + d 2Cρ + G (q, qɺ ,τ , F ) − G (qˆ , qɺˆ ,τ , Fˆ ) ≤ t0
≤ d1Cr + d 2Cρ + LGq δ q + LGqɺ δ qɺ + LGF δ F <
σ 2
+ LGq δ q + LGqɺ δ qɺ + LGF δ F ,
where the last inequality follows from the condition (2.121) of Theorem 4. Because of that, taking into account the inequality (D.2) from the proof of Theorem 4, we obtain the following estimate t
Fp + R ∫ t0
(
)
t
(
)
Fˆ − Fp (ω )d ω − Fp ≤ F − Fp + F − Fp − R ∫ Fˆ − Fp (ω ) d ω < t0
224
<
Dynamics and Robust Control of Robot-Environment Interaction
σ 2
+
σ 2
+ LGq δ q + LGqɺ δ qɺ + LGF δ F = σ + LGq δ q + LGqɺ δ qɺ + LGF δ F
(F.3)
Therefore, according to the condition (2.165) of Theorem 6 we have t
∀t ∈ [t0 , T ]
Fp + R ∫ ( Fˆ − Fp ) (ω ) dω ∈ VF ,
(F.4)
t0
Taking into account Lipschitz conditions for the function ψ and also the inequalities (2.119), (2.125), and (F.3), we obtain the following two estimates
t
t0
(
)
ψ qˆ , qˆɺ , Fp + R ∫ Fˆ − Fp (ω )dω − qɺɺp =
t = ψ qˆ , qˆɺ , Fp + R ∫ Fˆ − Fp (ω )d ω −ψ ( q p , qɺ p , Fp ) ≤ t0
(
)
t
(
)
≤ L1 qˆ − q p + L2 qˆɺ − qɺ p + L3 R ∫ Fˆ − Fp (ω )d ω ≤ t0
≤ L1 q − q p + L1δ q + L2 qɺ − qɺ p + L2δ qɺ + L3 (σ + LGq δ q + LGqɺ δ qɺ + LGF δ F ) < < δ 3 + ( L1 + L3 LGq ) δ q + ( L2 + L3 LGqɺ ) δ qɺ + L3 LGF δ F ,
(
)
(
)
ψ qˆ , qˆɺ , Fˆ − qɺɺp = ψ qˆ , qˆɺ , Fˆ −ψ ( q p , qɺ p , Fp ) ≤ L1 q − q p + L2 qɺ − qɺ p + + L1δ q + L2δ qɺ + L3 F − Fp + L3δ F < δ 3 + L1δ q + L2δ qɺ + L3δ F Hence, from the condition (2.164) of Theorem 6 we have that for any t ∈ [t0 , T ] hold the inclusions t ˆ ˆ ɺ ψ q, q, Fp + R ∫ Fˆ − Fp (ω )dω ∈ Vqɺɺ t0
(
(
)
ψ qˆ , qɺˆ , Fˆ ∈ Vqɺɺ
)
(F.5) (F.6)
The relations (F.1) – (F.6), together with the condition (2.76), ensure the admissibility of the control laws (2.150), (2.151), (2.152).
225
A Unified Approach to Dynamic Control of Robots
Appendix G
Proof of Lemma
On the basis of (2.187), taking into account Lemma conditions 1, 3, 4 and since ξ k ∈ Vξ , ∀k (formula (2.193)), we obtain that the control law (2.188) is admissible on [a, b] . Since Lemma condition 2 corresponds to the condition (2.184) on the interval [a, b] , the inequality (2.185) is fulfilled for all t ∈ [a, b] . Hence, ɺɺɺ q (t ) ∈ Y , ∀t ∈ [a, b] . From the inequality ɺɺɺ qˆ (t ) − ɺɺɺ q (t ) ≤ δɺɺɺq , we have
ɺɺɺ qˆ (t ) ∈ Y , ∀t ∈ [a, b] . Therefore, the norm G (tk′ ) is bounded by the constant CG (formula (2.195)) for every tk′ ∈ [a, b] :
G (tk′ ) ≤ CG
(G.1)
Let us denote by
ξ = ξ (tk′ ), G = G (tk′ ), uk = uk (tk′ ), uɶk = uɶk (tk′ ), pk = uk − uɶk ,
(
(
)
)
v(t ) = U qˆ (t ), qˆɺ (t ), qɺɺˆ (t ), ɺɺɺ qˆ (t ), Fˆ (t ), f qˆ (t ), qˆɺ (t ), qɺɺˆ (t ), ɺɺɺ qˆ (t ) , ξ (t ) − −U ( q (t ), qɺ (t ), qɺɺ(t ), ɺɺɺ q (t ) − r (t ), F (t ), f ( q (t ), qɺ (t ), qɺɺ(t ), ɺɺɺ q (t ) ) + ρɺ (t ), ξ (t ) ) . Because of Lemma conditions 1, 2, the Lipschits conditions for the functions U and f are fulfilled. Then, due to the inequalities (2.194), we have
v(t ) ≤ h1 , ∀t ∈ [a, b] . Consequently, the following inequality takes place: vk = v(tk ′ ) ≤ h1 , ∀tk ′ ∈ [a, b]
(G.2)
Let us denote by
Gɶ = G q (tk ′ ), qɺ (tk ′ ), qɺɺ(tk ′ ), ɺɺɺ q (tk ′ ) − r (tk ′ ), F (tk ′ ),
(
f (q (tk ′ ), qɺ (tk ′ ), qɺɺ(tk ′ ), ɺɺɺ q (tk ′ )) + ρɺ (tk ′ )
)
226
Dynamics and Robust Control of Robot-Environment Interaction
Then, because of the linearity of the vector function U with respect to the vector of the parameters ξ (property (2.186)), we can write: vk = (G − Gɶ )ξ . Since Vξ is a convex set, for the vector
(
)
ξ k +1′ = ξ k + G T (tk ′ ) uk (tk ′ ) − uɶk (tk ′ ) GT (tk ′ )
−2
we will have: 2
ξ − ξ k +1 ≤ ξ − ξ k +1′ . 2
Therefore, 2
ξ − ξ k − ξ − ξ k +1 ≥ ξ − ξ k − ξ − ξ k +1′ = ξ − ξ k − ξ − ξ k − 2
=
2
2 ξ − ξ k , GT pk G
2
−
2
G T pk G
4
2
≥
2
2 G ( ξ − ξ k ) , pk − pk
Here, and from now on, the sign
G
,
G T pk G
2
2
=
2
2
.
(G.3)
denotes the scalar product of the two
vectors. For the equations of robot’s dynamics (2.171) and environment dynamics (2.182), for an arbitrary control u , the following identity is fulfilled:
u (t ) ≡ U (q (t ), qɺ (t ), qɺɺ(t ), ɺɺɺ q (t ) − r (t ), F (t ), f ( q (t ), qɺ (t ), qɺɺ(t ), ɺɺɺ q (t ) ) + ρɺ (t ), ξ (t )) Hence, for the control law uk (t ) we have: uk ≡ Gɶ ξ . According to the notation adopted:
G (ξ − ξ k ) = Gξ − uɶk = Gɶ ξ − uɶk + (G − Gɶ )ξ = uk − uɶk + vk = pk + vk . Since the violation of inequalities (2.190) takes place for every time instant
tk ′ then, in accordance to (2.191), the inequality holds:
pk ≥ h . From
h − 2h1 = h 0 > 0
it follows that
inequality (G.2) and Lemma condition
pk − 2 vk ≥ h 0 .
227
A Unified Approach to Dynamic Control of Robots
Let us go on to estimate (G.3). Taking into account the inequality (G.1), we obtain
(
ξ − ξ k − ξ − ξ k +1 ≥ 2 pk + vk , pk − pk 2
≥
(p
2
k
2
)
− 2 pk vk CG
−2
= pk
(
2
)C
−2
G
≥
pk − 2 vk ) CG ≥ hh 0CG −2
Let us denote Vk (t j′ ) = ξ (t j′ ) − ξ k
(G.4)
−2
2
. From inequality (G.4) we have
Vk (tk ′ ) − Vk +1 (tk ′ ) ≥ hh 0CG −2
(G.5)
for the time instants tk ′ ∈ [ a, b] . Let us consider the sum of N + 1 inequalities (G.5) for k = 0,1,… , N . Then, N
(
V0 (t0′ ) − VN +1 (t N ′ ) ≥ ( N + 1)hh 0CG −2 + ∑ Vk (tk −1′ ) − Vk (tk ′ ) k =1
)
Therefore, N
V0 (t0′ ) − VN +1 (t N ′ ) ≥ ( N + 1)hh 0CG −2 − ∑ Vk (tk −1′ ) − Vk (tk ′ ) (G.6) k =1
Besides,
Vk (tk −1′ ) − Vk (tk ′ ) = ξ (tk −1′ ) − ξ k 2
2
− ξ (tk ′ ) − ξ k
2
=
2
= ξ (tk −1′ ) − ξ (tk ′ ) + 2 ξ k , ξ (tk ′ ) − ξ (tk −1′ ) ≤ 2
≤ ξ (tk −1′ ) − ξ (tk ′ )
2
k
+ 2Cξ ∑ ξ j (tk ′ ) − ξ j (tk −1′ ) . j =1
Hence, because of the inequality (G.6), we obtain
( )
V0 (t0′ ) ≥ ( N + 1)hh 0CG −2 − Var ξ [ a ,b ]
2
− 2Cξ ∑ Var (ξ j ) k
j =1
[ a ,b ]
(G.7)
Since, for an arbitrary partition a = t0 < t1 < … < t N = b of the interval [ a, b]
228
Dynamics and Robust Control of Robot-Environment Interaction
N
N
k =1
k =1
∑ ξ j 2 (tk ) − ξ j 2 (tk −1 ) ≤ ∑ ξ j (tk ) − ξ j (tk −1 ) ξ j (tk ) + ξ j (tk −1 ) ≤ N
≤ 2Cξ ∑ ξ j (tk ) − ξ j (tk −1 ) , k =1
then
( )
Var ξ [ a ,b ]
2
≤ 2Cξ ∑ Var (ξ j ) . k
j =1
[ a ,b ]
Therefore, from the inequality (G.7) we have
V0 (t0′ ) + 4Cξ ∑ Var (ξ j ) ≥ ( N + 1)hh 0CG −2 k
j =1
[ a ,b ]
which means that the inequality
( diamVξ ) + 4Cξ ∑ Var (ξ j ) ≥ ( N + 1)hh 0CG −2 2
k
j =1
[ a ,b ]
(G.8)
is fulfilled. Because the variations of the functions ξ j are bounded on the interval [ a, b] , the number of corrections of the algorithm (2.193) r( a, b) = N can not be infinite. Indeed, the left-hand side of the inequality (G.8) is bounded, and is not dependent on N . The needed estimated value (2.196) follows directly from the inequality (G.8). Evidently, it is not dependent on the parameter θ .
229
A Unified Approach to Dynamic Control of Robots
Appendix H
Proof of Theorem 7
It is evident ( CΓ ≥ 1 ) that the estimate (2.215) is fulfilled for t = t0 . Let us prove that the estimate (2.215) is fulfilled for ∀t ∈ [t0 , t0 + p ] . Let us suppose the contrary. Then, because of the continuity of x(t ) , it is possible to find the first time instant t ′ > t0 , such that
x(t ′) = CΓ x(t0 ) e− λ (t ′−t0 ) + a
(H.1)
Consequently, taking into account the theorem conditions CΓε + a < δ 0 ,
x (t0 ) < ε , for ∀t ∈ [t0 , t ′] holds the inequality: x(t ) ≤ CΓ x(t0 ) e − λ ( t −t0 ) + a < CΓε + a < δ 0 Hence, according to (2.211), for ∀t ∈ [t0 , t ′] holds the inclusions:
( qˆ (t ), qˆɺ (t ), qɺɺˆ (t ) ) ∈V ×V ×V
( q (t ), qɺ (t ), qɺɺ(t ) ) ∈Vq × Vqɺ × Vqɺɺ , Because of the inequality
q* (t ) − ɺɺɺ q p (t ) ≤ Γ1
( ηɺɺ + δ ) + Γ ( ηɺ qɺɺ
2
+ δ qɺ ) + Γ3
q
qɺ
(η
+ δq ) ≤
≤ R x(t ) + r < Rδ 0 + r
qɺɺ
(H.2)
(H.3)
and according to (2.212), for ∀t ∈ [t0 , t ′] , the following inclusion is fulfilled:
(
)
(
)
q* (t ) ≡ ɺɺɺ q p (t ) + Γ1 qɺɺˆ − qɺɺp + Γ 2 qɺˆ − qɺ p + Γ3 ( qˆ − q p ) ∈ Vɺɺɺq
(H.4)
From the inequality (H.4) follows that for ∀t ∈ [t0 , t ′] the inequality analogues to (2.208) is fulfilled:
µ (t ) < L ( CΓ x(t0 ) e − λ (t −t ) + a ) + Cρ < Lδ 0 + Cρ . 0
According to (2.213) we will have:
F (t ) ∈ VF , Fˆ (t ) ∈ VF
(H.5)
230
Dynamics and Robust Control of Robot-Environment Interaction
On the basis of (H.3), (H.4) we obtain:
(
)
(
) (
)
(
)
f qˆ , qˆɺ , qɺɺˆ , q* − Fɺp = f qˆ , qɺˆ , qɺɺˆ , q* − f q p , qɺ p , qɺɺp , ɺɺɺ q p ≤ Lqf η + δ q +
(
)
(
)
ɺɺ + δ qɺɺ + Lɺɺɺfq q* − ɺɺɺ + Lqfɺ ηɺ + δ qɺ + Lqɺɺf η q p ≤ C1 x(t ) + C2 < C1δ 0 + C2 Hence, in accordance with (2.214), for ∀t ∈ [t0 , t ′] :
(
)
f qˆ (t ), qˆɺ (t ), qɺɺˆ (t ), q* (t ) ∈VFɺ
(H.6)
On the basis of the mapping (2.187), relations (H.2), (H.4) – (H.6) and due to the fact that ∀k ξ k ∈ Vξ , we conclude that for ∀t ∈ [t0 , t ′] the following inclusion is fulfilled:
uk (t ) ∈ Vu
(H.7)
The relations (H.2), (H.4) – (H.6) guarantee the fulfillment of the Lemma conditions on the interval [t0 , t ′] and, consequently, of the estimate (2.206). In particular, it holds for t = t ′ :
x(t ′) < CΓ x(t0 ) e − λ (t ′−t0 ) + a which contradicts the equality (A.1). In this way, the estimate (2.215) has been proven for ∀t ∈ [t0 , t0 + p ] . It necessitates the fulfillment on this closed interval of the inequality (2.216) and of the relations (H.2), (H.4) – (H.7). As x(t0 ) < ε < 1 , then from the estimate (2.215) we have:
x(t0 + p / 2) < CΓ x(t0 ) e
−λ
p 2
+ a ≤ CΓ x(t0 ) e
−λ
p1 2
+ a = CΓ x(t0 ) e
ln
ε −a CΓ
+
+ a = x(t0 ) (ε − a ) + a < ε
x(t ) < ε , ∀t ∈ [t0 + p / 2, t0 + p]
(H.8)
In particular, the inequality (H.8) is fulfilled for t = t0 + p . Hence, the time instant t0 + p can be regarded as the initial one in the analysis of the robot’s behavior in contact with the environment. Since functions ξ j (t ) are the functions of uniformly bounded variations of order p , the relations (H.2), (H.4) – (H.7) and the Lemma’s conditions are fulfilled on the interval [t0 + p, t0 + 2 p ] again. Then ( ε < 1 ):
231
A Unified Approach to Dynamic Control of Robots
x(t0 + 3 p / 2) < CΓ x(t0 + p ) e
−λ
p 2
+ a < x ( t0 + p ) (ε − a ) + a <
< ε (ε − a ) + a < ε , x(t ) < ε , ∀t ∈ [t0 + 3 p / 2, t0 + 2 p]
(H.9)
In particular, the inequality (H.9) is fulfilled for t = t0 + 2 p . By induction, we obtain that the relations (H.2), (H.4) – (H.7) are fulfilled for ∀t ≥ t0 . Hence, the conclusions (a), (b), (c) of Theorem have been proven. Let us prove the second inequality in the formula (2.215). On every closed interval of the length p the following estimate holds: ɶ x(t ) < CΓ x(tɶ ) e− λ (t −t ) + a, t ∈ [tɶ, tɶ + p], tɶ ≥ t0 .
Then, taking into account (2.209), from inequality (H.8) we obtain for
∀tɶ ∈ [t0 + p / 2, t0 + p ] :
x(tɶ + p / 2) < CΓ x(tɶ ) e This means that
−λ
p 2
+ a < x ( tɶ ) (ε − a) + a < ε (ε − a) + a < ε .
x(t ) < ε , ∀t ∈ [t0 + p, t0 + 3 p / 2] . Together with the
inequality (H.9) we have x(t ) < ε , ∀t ∈ [t0 + p, t0 + 2 p ] . By repeating this procedure we can conclude that x(t ) < ε , ∀t ≥ t0 + p . Hence, the estimate (2.215) is completely proven. As the estimate (2.216) holds for ∀t ∈ [t0 , t0 + p ] then taking into account the formula (2.209), we obtain:
µ (t0 + p / 2) ≤ LCΓ x(t0 ) e = LCΓ x(t0 ) e
ln
−λ
p 2
+ La + Cρ ≤ LCΓ x(t0 ) e
−λ
p2 2
+ La + Cρ =
δ − La − Cρ LCΓ
+ La + Cρ = x(t0 ) (δ − La − Cρ ) + La + Cρ <
< ε (δ − La − Cρ ) + La + Cρ < (δ − La − Cρ ) + La + Cρ = δ ,
µ (t ) < δ , ∀t ∈ [t0 + p / 2, t0 + p ]. By analogy with estimation of the norm
x(t )
we will have:
µ (t ) < δ , ∀t ≥ t0 + p . Hence, the estimate (2.216) is also completely proven. The estimates (2.217), (2.218) follow directly from the validity of the inequalities (2.215), (2.216).
232
Dynamics and Robust Control of Robot-Environment Interaction
Appendix I
Some Basic Relations
Consider a robot with n DOFs, whose configuration space is an n -dimensional differentiable manifold DR . Let the robot interact with the dynamic environment possessing its own DOFs, i.e. the robot’s generalized coordinates and velocities are sufficient to represent the state of the environment. Then the environment configuration is generally an m -dimensional submanifold DE of the manifold
DR , with m ≤ n . In the text to follow, all relations used are assumed to be valid everywhere on DR and DE , respectively. Let q be the generalized robot coordinates (local coordinates on DR ) and let
s be the local coordinates on DE . The robot’s dynamics can be described by the differential equation:
H (q )qɺɺ + h(q, qɺɺ) =τ + F
(I.1)
where q is an n -dimensional vector of generalized robot coordinates; H (q ) is an n × n positive definite matrix of the manipulator; h( q, qɺ ) is an n dimensional nonlinear function of centrifugal, Coriolis’ and gravitational terms; τ is an n -dimensional vector of control input; F is a vector of the generalized forces acting on the end-effector from the environment side. The environment dynamics satisfies the equation:
M( s )ɺɺ s + L( s, sɺ) = − F
(I.2)
where M( s ) is a continuous m × m positive definite matrix, L( s, sɺ) is a continuous m -vector function, and F is an m -vector of generalized interaction force in local coordinates on DE . Since DE ⊆ DR , there exists a function
φ : DR → DE such that the following relations hold:
A Unified Approach to Dynamic Control of Robots
s = φ (q) sɺ =
∂φ (q )qɺ = J (q )qɺ ∂q
rank
∂φ (q ) = rank J (q ) = m ∂q
ɺsɺ = J (q )qɺɺ + Jɺ (q )qɺ
233
(I.3) (I.4) (I.5) (I.6)
Further, equating the expressions for virtual work in the corresponding spaces, we obtain the well-known relation:
F = J T (q ) F
(I.7)
Substituting (I.3) – (I.6) into (I.2), the environment dynamics equation (I.2) can be written in terms of generalized coordinates q as:
M (φ (q) ) J (q)qɺɺ + L (φ (q), J (q)qɺ ) + M (φ (q) ) Jɺ (q)qɺ = − F (I.8) Left multiplying by J T (q ) and taking into account (I.7), the obtained equation reduces to:
Mɶ (q )qɺɺ + Lɶ (q, qɺ ) = −F
(I.9)
where:
Mɶ (q) = J T (q)M (φ (q) ) J (q)
(I.10)
Lɶ (q, qɺ ) = J T (q ) ( L (φ (q ), J (q )qɺ ) + M (φ (q ) ) Jɺ (q ) qɺ )
(I.11)
Equations (I.1), (I.9) describe the robot and environment dynamics in generalized coordinates. It is often useful to retain the force F in external coordinates and use the alternative form of the equations (I.1), (I.9):
H (q )qɺɺ + h(q, qɺɺ) = τ + J T (q ) F Mɶ (q )qɺɺ + Lɶ (q, qɺ ) = − J T (q ) F
(I.12)
Remark 1. The matrix Mɶ (q ) , representing the environment inertia in the robot’s generalized coordinates, is positive-semidefinite. In fact, the following conclusion can be drawn from (I.10). Since rank J (q ) = m , there exists a non-zero vector q such that J (q )q = 0 , so
Mɶ (q ) is not positive definite. Due to the positive definiteness of the m × m
234
Dynamics and Robust Control of Robot-Environment Interaction
matrix M (φ (q ) ) , it follows that the expression qT Mɶ ( q ) q can not be negative,
~
so the matrix M (q ) is positive semidefinite. Moreover, from (I.10) we have:
rank Mɶ (q ) = m
(I.13)
Having in mind Remark 1, the environment dynamics equation can be rearranged in the following way. Because of (I.5), there is a submatrix S (q ) of − J (q ) such that:
rank S (q ) = m
(I.14)
and the matrix − J ( q ) can be partitioned into the submatrices:
− J (q ) = S (q ) S (q )
(I.15)
by rearranging the coordinates of the vector q . Substituting (I.4), (I.6) and (I.7) into (I.1), (I.2) and left multiplying by − S T ( q ) , the robot and environment dynamics equations reduce to the form:
H (q )qɺɺ + h(q, qɺɺ) = τ + J T (q ) F
(I.16)
M (q )qɺɺ + L(q, qɺ ) = S T (q ) F
(I.17)
where
M (q ) = M 1 (q ) M 2 (q ) = S T (q )M (φ (q ) ) S (q ) S T (q ) M (φ (q ) ) S (q ) (I.18)
L(q, qɺ ) = − S T (q ) ( L (φ (q ), J (q )qɺ ) + M (φ (q ) ) Jɺ (q ) qɺ )
(I.19)
M 2 (q) = S T (q)M (φ (q) ) S (q) is positive definite.
Remark 2. The matrix:
−1 Hɶ (q ) = H (q ) − J T (q ) ( S T (q ) ) M (q )
Remark 3. The matrix:
is positive
definite. This statement can be easily verified by eliminating F from (I.16), (I.17), which yields:
( H (q) − J (q) ( S (q) ) T
T
−1
)
M (q ) qɺɺ + h(q, qɺ ) − J T (q ) ( S T (q ) ) L(q, qɺ ) = τ −1
(I.20) The matrix of inertia Hɶ ( q ) of this system is positive definite by definition.
235
A Unified Approach to Dynamic Control of Robots
Remark 4. The matrix:
0 n× m H (q) Λ I (q) = is nonsingular. T M (q ) − S ( q )
Since H (q ) and S T (q ) are nonsingular, from the identity:
det Λ I (q ) = − det H (q ) det S (q ) , we have that det Λ I (q) ≠ 0 . Remark 5. The matrix:
H (q) − J T (q) Λ II (q) = 0m M (q)
is nonsingular.
Let us prove this statement. If q1 , q 2 are subvectors of the vector q :
q q = 1 such that (see (I.4), (I.15)): q 2 sɺ = − S (q ) qɺ1 − S (q ) qɺ2
(I.21)
the matrix Λ II (q ) can be partitioned into submatrices in the following way:
H11 (q ) H12 (q ) S T (q ) Λ II (q ) = H 21 (q ) H 22 (q ) S T (q ) M 1 ( q ) M 2 (q ) 0 m
(I.22)
Introduce the nonsingular matrix
In−m 0( n − m )×m −1 T (q) = − S (q) S (q) Im 0 m ×( n − m ) 0m Then:
0( n − m )×m 0m I m
rank Λ II (q ) = rank (T T (q )Λ II (q ) T (q ) )
(I.23)
(I.24)
Further, keeping in mind (I.18), we obtain
h11 (q ) T (q )Λ II (q )T (q ) = h21 (q ) 0 m ×( n − m ) T
where the matrix:
0( n − m )×m h22 (q ) S T (q ) M 2 (q) 0m h12 (q )
(I.25)
236
Dynamics and Robust Control of Robot-Environment Interaction
T −1 h11 (q ) h12 (q ) In − m − ( S (q ) S (q ) ) = Im h21 (q ) h22 (q ) 0m×( n − m )
I 0( n − m )×m H (q ) −1 n − m − S (q) S (q) Im
(I.26) is positive definite, having all principal minors positive definite. From (I.25) we have:
det (T T (q )Λ II (q ) T (q ) ) = − det h11 (q ) det M 2 (q ) det S (q )
(I.27)
Since the matrices M 2 ( q ), h11 ( q ) are nonsingular it is finally obtained:
det Λ II (q ) = det (T T (q )Λ II (q )T (q ) ) ≠ 0 Remark 6. The matrix:
H (q) − J T (q) Λ III (q) = T M (q) − S (q )
(I.28)
is nonsingular.
The proof is straightforward. From the identity:
In 0n×m H ( q ) − J T (q ) −1 T T M (q ) − S (q ) ( S (q) ) M (q ) I m H (q ) − J T (q ) ( S T (q ) )−1 M (q ) − J T (q) = T 0m×n − S (q )
(I.29)
we have immediately:
det Λ III (q ) = − det ( H (q ) − J T (q ) ( S T (q )) −1 M (q ) ) det S T (q )
(I.30)
Because of Remark 3 and relation (I.14) it is obtained:
det Λ III (q) ≠ 0
(I.31)
A Unified Approach to Dynamic Control of Robots
237
Appendix J
Proof of Theorem 8
Consider the solution of equation (2.256) on the set:
Ω = {( x, t ) ∈ R 2 n × R / x < h, t0 ≤ t < ∞ }
(J.1)
where h > 0 is a fixed number. Because of the smooth vector function α ( x 2 , x1 , t ) = o( x ) at x → 0 , the positive constants C1 , C 2 and the constant
p > 1 can be found, such that:
α ( x2 , x1 , t ) ≤ C1 x1 + C2 x2 p
p
∀( x, t ) ∈Ω
(J.2)
Because of the continuity of the functions B (t ), β ( x2 , x1 , t ) in (2.256) it is possible to find the positive constants M 1 , M 2 such that:
B(t ) ≤ M 1 ,
β ( x2 , x1 , t ) ≤ M 2 , ∀( x, t )∈Ω
(J.3)
Because of theorem conditions the system of the first approximation of equation (2.256):
xɺ 2 = A(t ) x 2
(J.4)
is regular and characteristic indices α1 ,… , α 2 m of (J.4) are negative. Let us carry out the transformation of (2.256):
x2 = ye −γ ( t −t0 )
(J.5)
where the number γ satisfies the inequality:
max α k < −γ < 0 k
Then, we will have:
yɺ = Aɶ (t ) y + Bɶ (t ) x1 + αɶ ( y, x1 , t ) + βɶ ( y, x1 , t ) µ where:
Aɶ (t ) = A(t ) + γ I 2 m ,
(J.6)
238
Dynamics and Robust Control of Robot-Environment Interaction
Bɶ (t ) = eγ (t −t0 ) B (t ) ,
(J.7)
αɶ ( y, x1 , t ) = eγ (t −t )α ( ye −γ (t −t ) , x1 , t ) , 0
0
βɶ ( y, x1 , t ) = eγ (t −t ) β ( ye −γ ( t −t ) , x1 , t ) . 0
0
Evidently, x2 (t0 ) = y (t0 ) . The linear system:
yɺ = Aɶ (t ) y
(J.8)
is also regular, and all its characteristic indices are negative. In fact, if β k (k = 1, 2,… , 2m) are the characteristic indices of the linear system (J.8), then we have:
βk = α k + γ < 0
(k = 1, 2,… , 2m)
Taking into account the regularity of the system (2.259) and relation (J.7) we have: t
t
2m 2m 1 1 lim ∫ SpAɶ (ω )dω = lim ∫ (SpA(ω ) + 2mγ )dω = ∑ α k + 2mγ = ∑ β k t →∞ t t →∞ t k =1 k =1 t0 t0
Hence, the system (J.8) is regular. Let H (t ) be the normed ( H (t0 ) = I 2 m ) fundamental matrix of the system (J.8). Then, equation (J.6) is equivalent to the integral equation: t
y (t ) = H (t ) y (t0 ) + ∫ K (t , ω )[ Bɶ (ω ) x1 (ω ) + αɶ ( y (ω ), x1 (ω ), ω ) + t0
(J.9)
+ βɶ ( y (ω ), x1 (ω ), ω ) µ (ω )] d ω where K (t , ω ) = H (t ) H −1 (ω ) is the Cauchy matrix. Since all characteristic indices of (J.8) are negative, the following estimate of its fundamental matrix H (t ) for all t ≥ t0 is valid:
H (t ) ≤ C3 , (C3 ≥ 1)
∀t > t0
(J.10)
The Cauchy matrix K (t , ω ) for a regular system with negative characteristic indices can be estimated in the following way [13]:
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A Unified Approach to Dynamic Control of Robots
K (t , ω ) ≤ C4 eε 0 (ω −t0 ) , t0 ≤ ω < t < ∞
(J.11)
where C 4 > 0 is a suitably chosen constant, and ε 0 > 0 is an arbitrarily small constant. On the basis of (J.2), (J.3) and (2.257), (2.258) we obtain the following estimates:
Bɶ (t ) x1 = eγ (t −t0 ) B (t ) x1 ≤ eγ (t −t0 ) M 1 D2 e − λ (t −t0 ) x1 (t0 ) ,
αɶ ( y, x1 , t ) = eγ ( t −t ) α ( x2 , x1 , t ) ≤ eγ (t −t ) (C1 x1 + C2 ye −γ (t −t 0
p
0
0)
p
)≤
≤ eγ (t −t0 ) (C1 D2p e − λ p ( t −t0 ) x1 (t0 ) + C2 e −γ p (t −t0 ) y ), p
p
βɶ ( y, x1 , t ) µ (t ) = eγ (t −t ) β ( ye−γ (t −t ) , x1 , t ) µ (t ) ≤ 0
0
≤ eγ (t −t0 ) M 2 D1e− λ (t −t0 ) µ (t0 ) , Bɶ (t ) x1 + αɶ ( y, x1 , t ) + βɶ ( y, x1 , t ) µ (t ) ≤ Bɶ (t ) x1 + αɶ ( y, x1 , t ) +
(
p + βɶ ( y, x1 , t ) µ (t ) ≤ eγ (t −t0 ) M 1 D2 e− λ (t −t0 ) x1 (t0 ) + C1 D2p e − λ p (t −t0 ) x1 (t0 ) +
+ C2 e − γ p ( t − t 0 ) y Since e
p
+ M 2 D1e − λ (t −to ) µ (t0 )
− λ ( p −1)( t −t0 )
).
≤ 1 , choosing the constant M 3 such that:
M 1 D2 + C1 D2p x1 (t0 )
p −1
≤ M3
it is finally obtained:
Bɶ (t ) x1 + αɶ ( y, x1 , t ) + βɶ ( y, x1 , t ) µ (t ) ≤ ≤ e − ( λ −γ ) (t −t0 ) ( M 3 x1 (t0 ) + M 2 D1 µ (t0 ) ) + C2 e− ( p −1) γ (t −t0 ) y . p
Let us choose a positive number ε 0 , small enough to satisfy the inequalities:
δ 0 = ( p − 1) γ − ε 0 > 0 and λ − γ − ε 0 > 0 Therefore, by estimating the norms of both sides of the integral equation (J.9) we will have:
240
Dynamics and Robust Control of Robot-Environment Interaction t
y (t ) ≤ H (t ) y (t0 ) + ∫ K (t , ω ) Bɶ (ω ) x1 (ω ) + αɶ ( y (ω ), x1 (ω ), ω ) + t0 t
p ε − ( p −1) γ ] (ω − t0 ) + βɶ ( y (ω ), x1 (ω ), ω ) µ (ω ) dω ≤ C3 y (t0 ) + ∫ C4C2 e[ 0 y (ω ) +
+C4 ( M 3 x1 (t0 ) + M 2 D1 µ (t0 ) ) e
t0
− ( λ −γ −ε 0 )( ω −t0 )
d ω = C3 y (t0 ) + (J.12)
t
+ ∫ C5 e[ 0
ε − ( p −1) γ ] (ω −t0 )
y (ω ) dω + p
t0
C6 x1 (t0 ) + C7 µ (t0 ) 1 − e − ( λ −γ −ε 0 ) (t −t0 ) ≤ λ − γ − ε0
(
)
C6 x1 (t0 ) + C7 µ (t0 ) t p ε − ( p −1) γ ] (ω −t0 ) ≤ C3 y (t0 ) + + ∫ C5 e[ 0 y (ω ) dω λ − γ − ε0 t0 where C 5 = C 4 C 2 , C 6 = C 4 M 3 , C 7 = C 4 M 2 D1 . Then, the estimate (J.12) can be written as follows:
y (t ) ≤ C3 y (t0 ) +
t C6 x1 (t0 ) + C7 y (t0 ) p + C5 ∫ e −δ 0 (ω −t0 ) y (ω ) dω (J.13) λ − γ − ε0 t0
From the following inequality, in accordance with the Bihari lemma [13, 54], it follows that:
C3 y (t0 ) + y (t ) ≤
C6 x1 (t0 ) + C7 µ (t0 ) λ − γ − ε0
C x (t ) + C7 µ (t0 ) 1 − ( p − 1) C3 y (t0 ) + 6 1 0 λ − γ − ε0
p −1
C5 δ0
1 p −1
(J.14)
only if
C x (t ) + C7 µ (t0 ) ( p − 1) C3 y (t0 ) + 6 1 0 λ − γ − ε0
p −1
C5
δ0
<1
(J.15)
It is clear that the inequality (J.15) will be fulfilled if y (t0 ) , x1 (t0 ) and
µ (t0 ) are sufficiently small. Let us choose these values such that the following inequality is satisfied:
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A Unified Approach to Dynamic Control of Robots
C x (t ) + C7 µ (t0 ) ( p − 1) C3 y (t0 ) + 6 1 0 λ − γ − ε0
p −1
C5
δ0
≤
1 2
(J.16)
The fulfillment of the inequality (J.16) automatically ensures fulfillment of the inequality (J.15). Then, from the inequality (J.14) we have:
y (t ) ≤ 2
1 p −1
C3 y (t0 ) +
2
1 p −1
(C
6
x1 (t0 ) + C7 µ (t0 )
λ − γ − ε0
)
(J.17)
with the sufficiently small y (t0 ) , x1 (t0 ) and µ (t0 ) . The latter inequality ensures the correctness of using the estimates (J.2) and (J.3). By taking into account that y (t0 ) = x2 (t0 ) we obtain from (J.5):
b x1 (t0 ) + c µ (t0 ) −γ (t −t0 ) x2 (t ) ≤ a x2 (t0 ) + e λ − γ − ε0 1
1
(J.18)
1
where a = 2 p −1 C 3 , b = 2 p −1 C 6 , c = 2 p −1 C 7 . Since the inequality (J.18) is not strong, and is fulfilled with an arbitrarily small ε 0 > 0 , then the following inequality is also justified:
b x1 (t0 ) + c µ (t0 ) x2 (t ) ≤ a x2 (t0 ) + λ −γ
−γ (t −t0 ) . e
(J.19)
242
Dynamics and Robust Control of Robot-Environment Interaction
Appendix K
Prior Assertions and Proof of Theorem 10
Consider the system:
ηɺɺ = Γ1ηɺ + Γ 2η , η ∈ R n
(K.1)
µ = Mηɺɺ + Lhηɺ + Lkη , µ ∈ R n
(K.2)
satisfying the condition
rank [ Lk | Lh | M ] = n
(K.3)
Denote the class of diagonal matrices of order n by:
ϒ n = { diag (λ1 , λ2 ,… , λn ) / λi ≠ 0, i ≠ j ⇒ λi ≠ λ j , i, j =1, 2,… , n }
(K.4)
Let us introduce for convenience the following definition:
Definition.
Let A, B, C be arbitrary n × n matrices. Consider the n × 3n
( 3n × n ) block matrix [ A | B | C ] whose columns (rows) are numbered by 1, 2,… ,3n . Let us say that the n -th order minor of this block matrix possesses the property ℜ if it consists of the columns (rows) (i1 , i2 , …, in ) such that, if k1 ≠ k2 , then all of the differences ik1 − ik2 are not multiple to the number ± n . It is evident that for all matrices Λ ∈ ϒ n the following lemma is valid. A B C
I Lemma 1. All of the n -th order minors of the matrix Λ possessing the Λ 2 property ℜ are nonzero and all of the others n -th order minors are equal to zero.
A Unified Approach to Dynamic Control of Robots
243
Lemma 2. Let A, B, C be arbitrary n × n matrices. If all of the n -th order
minors of the matrix [ A | B | C ] possessing the property ℜ are equal to zero, then for all matrices Λ ∈ ϒ n
I det [ A | B | C ] Λ = 0 Λ 2
(K.5)
Proof: In compliance with the Cauchy-Binet formula [55] the determinant of the
I matrix [ A | B | C ] Λ is equal to the sum of every possible products of the n Λ 2 th order minors of the matrix [ A | B | C ] on the corresponding n -th order I minors of the matrix Λ . Due to Lemma 1 the only those n -th order minors Λ 2 I of the matrix Λ are nonzero that possesses the property ℜ but at the same Λ 2 time the corresponding n -th order minors of the matrix [ A | B | C ] are equal to zero. Then the equality (K.5) is fulfilled.
Lemma 3. Let A, B, C be arbitrary n × n matrices. If the matrix [ A | B | C ] has at least one nonzero n -th order minor possessing the property ℜ , then for almost all matrices Λ ∈ ϒ n the following expression is fulfilled:
I det [ A | B | C ] Λ ≠ 0 2 Λ Proof: In compliance with the Cauchy-Binet formula we have
(K.6)
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Dynamics and Robust Control of Robot-Environment Interaction
I det [ A | B | C ] Λ = R (λ1 , λ2 ,… , λn ) + det A 2 Λ
(K.7)
where R (λ1 , λ2 ,… , λn ) is the multinomial of the variables λ1 , λ2 ,… , λn . If det A = 0 , then, due to the condition of the lemma R (λ1 , λ2 ,… , λn ) ≡/ 0 , i.e. the order of the multinomial is greater than or equal 1. Let λ2∗ ,… , λn∗ be the arbitrary fixed values of the variables
λ2 ,… , λn . Then, the equality
R (λ1 , λ ,… , λ ) = 0 is possible only if λ1 is the root of the polynomial ∗ 2
∗ n
R ∗ (λ1 ) = R (λ1 , λ2∗ ,… , λn∗ ) . Discarding the finite numbers ( ≤ 2 ) of these roots we obtain the assertion of the lemma. If det A ≠ 0 , then there are two possibilities: a) det A is the only one nonzero n -th order minor of the matrix
[ A | B | C ] . In
this case R (λ1 , λ2 ,… , λn ) ≡ 0, ∀Λ ∈ ϒ n . Then, the left side of the formula (K.6) is equal to det A ≠ 0 . b) in addition to det A ≠ 0 there exists another nonzero n -th order minor of the matrix [ A | B | C ] . In this case R (λ1 , λ2 ,… , λn ) ≡/ 0 . Taking into account the equality R ∗ (λ1 ) = R (λ1 , λ2∗ ,… , λn∗ ) + det A we can repeat the proof that was made when det A = 0 .
Lemma 4. There exists an infinite number of nonsingular n × n matrices X ( Λ ) such that
X (Λ) det [ Lk | Lh | M ] X (Λ )Λ ≠ 0 X (Λ )Λ 2 where Λ ∈ ϒ n .
Proof: Let Λ = diag (λ1 , λ2 ,… , λn ) and
(K.8)
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A Unified Approach to Dynamic Control of Robots
λ1 λ4 1 X (Λ ) = λ17 ⋯ λ13n − 2
λ2 λ24 λ27 ⋯
λ23n− 2
λn λn4 λn7
⋯ ⋯ ⋯ ⋯ λn3n − 2 ⋯ ⋯
(K.9)
Then the products X ( Λ ) Λ, X ( Λ ) Λ 2 can be written as follows
λ12 λ22 5 λ1 λ25 X (Λ)Λ = ⋯ ⋯ 3 n −1 λ23n −1 λ1 λ13 6 λ 2 X (Λ)Λ = 1 ⋯ 3n λ1
⋯ ⋯
⋯ ⋯ λn3n −1 ⋯
λ23 λ26
⋯
⋯
⋯
λ
3n 2
λn2 λn5
, (K.10)
λn3 ⋯ λn6
⋯ ⋯ λn3n
It is evident that the different rows
(λ
k 1
λ2k … λnk )
where
X (Λ ) k ∈ K = {1, 2,… ,3n} of the matrix B(Λ ) = X (Λ )Λ have the different X (Λ )Λ 2 exponents k . It is evident that for Λ ∈ ϒ n the determinant of the matrix X ( Λ ) is not equal to zero (this determinant is analogues to the Vandermonde determinant). ( k , k ,…, k ) Let J 1 2 n (λ1 , λ2 ,… , λn ) be the n -th order minor of the matrix B ( Λ ) of the form
λ1k1 k2 λ ( k1 , k2 ,…, kn ) J (λ1 , λ2 ,… , λn ) = det 1 ⋯ kn λ1
λ2k λ2k
1
2
⋯
λ2k
n
⋯ ⋯ ⋯ ⋯
λnk λnk 1
2
⋯ λnkn
246
Dynamics and Robust Control of Robot-Environment Interaction
where k1 , k2 ,… , kn ∈ K . Then this minor is
J ( k1 , k2 ,…, kn ) (λ1 , λ2 ,… , λn ) =
∑ ( ±λ
i1 1
( i1 ,i2 ,…,in )
λ2i ⋅… ⋅ λni 2
n
)
(K.11)
Here the summation is made for all permutations (i1 , i2 , …, in ) on the
(k1 , k2 , …, kn ) elements. The sign of the monomial λ1i1 λ2i2 ⋅… ⋅ λnin is defined by the parity of the permutation (i1 , i2 , …, in ) . The minor J
( k1 , k2 ,…, kn )
(λ1 , λ2 ,… , λn ) is a multinomial of the variables
λ1 , λ2 ,… , λn . It
is
easy
to
see
that
if
J ( k1′ ,k2′ ,…, kn′ ) (λ1 , λ2 ,… , λn )
and
(λ1 , λ2 ,… , λn ) are two different minors of the matrix B (Λ ) , i.e. (k1′, k2′ , …, kn′ ) ≠ (k1 , k2 , …, kn ) , then they do not have coincident monomials at J
( k1 , k2 ,…, kn )
the corresponding expansions (K.11). Let
A = [ Lk | Lh | M ]
and
P(λ1 , λ2 ,… , λn ) = det ( AB (Λ ) ) ,
where
P (λ1 , λ2 ,… , λn ) is the multinomial of the variables λ1 , λ2 ,… , λn . Then, taking into account (K.3) we can conclude that at least one of the n -th order minors of the matrix A is nonzero and, due to the Cauchy-Binet formula, the multinomial P (λ1 , λ2 ,… , λn ) is not identically equal to constant, i.e. the multinomial P (λ1 , λ2 ,… , λn ) really depends on the variables λ1 , λ2 ,… , λn . Let λɶ2 ,… , λɶn be arbitrary fixed numbers. Then the polynomial of the variable λ1 : Pɶ (λ1 ) = P (λ1 , λɶ2 ,… , λɶn ) is the polynomial whose order is not greater than 3n . The polynomial Pɶ (λ1 ) can be equal to zero only at the finite numbers of it roots. Let λɶ1 be an arbitrary number except for the root of Pɶ (λ1 ) .
P(λɶ1 , λɶ2 ,… , λɶn ) ≠ 0 and for all matrices ɶ = diag(λɶ , λɶ ,… , λɶ ) the expression (K.8) is fulfilled. Λ
Therefore,
1
2
ɶ) X (Λ
where
n
Lemma 5. Given Λ ∈ ϒ n , there exists an infinite number of nonsingular n × n matrices X that are not dependent of Λ , such that
A Unified Approach to Dynamic Control of Robots
247
X rank [ Lk | Lh | M ] X Λ = n (K.12) X Λ 2 ɶ ∈ ϒ be an arbitrary fixed matrix that satisfies (K.8). Denote Proof: Let Λ n ɶ ) . Let us show that the matrix [ L X | L X | MX ] has the n -th order X = X (Λ k
h
nonzero minor possessing the property ℜ . Suppose the contrary. Then, in compliance with Lemma 2,
I det [ Lk X | Lh X | MX ] Λ = 0, 2 Λ
∀Λ ∈ ϒ n
(K.13)
ɶ . But In particular, (K.13) is fulfilled for the matrix Λ = Λ
I ɶ )| L X (Λ ɶ ) | MX (Λ ɶ ) Λ ɶ det Lk X (Λ h = ɶ 2 Λ ɶ ) X (Λ ɶ )Λ ɶ ≠ 0 = det [ Lk | Lh | M ] X (Λ ɶ )Λ ɶ 2 X (Λ We obtain a contradiction. Thus, it is proved that the matrix [ Lk X | Lh X | MX ] has the n -th order nonzero minor possessing the property
ℜ . The assertion of Lemma 3 completes the proof. Let the system (K.1) be of a diagonal canonical form
wɺ = Λw , Λ ∈ ϒ 2 n
(K.14)
µ = [C1 | C2 ] w
(K.15)
and where C1 ,C 2 are nonsingular n × n matrices. Let
Λ 0 Λ = 1 , Λ1 , Λ 2 ∈ ϒ n , 0 Λ 2
248
Dynamics and Robust Control of Robot-Environment Interaction
then the following lemma is valid.
Lemma 6. Matrix Λ ∈ ϒ 2 n can always be chosen such that the matrix
C1 C2 C1Λ1 C2 Λ 2
αɶ =
(K.16)
is nonsingular.
Proof: Denoting Y = C1−1C 2 , the matrix αɶ can be written as
C1 0 0 C1
αɶ =
I Λ 1
0 I n
Y I 0 Y Λ − Λ Y 2 1
and
det αɶ = det 2 C1 det (Y Λ 2 − Λ1Y ) Let τ i , i = 1, 2, ..., n be eigenvalues of the matrix Y −1Λ1−1Y Λ 2 . If τ i = 1 for
(
)
some i , then the equality det Y −1Λ1−1Y Λ 2 − I = 0 implies the fulfillment of the equality det ( Y Λ 2 − Λ1Y ) = 0 . However, choosing the constant c such that
cτ i ≠ 1, i = 1, 2,… , n and scaling Λ 2 with 1 / c we ensure that the matrix Y Λ 2 − Λ 1Y is nonsingular, and therefore det αɶ ≠ 0 . Proof of Theorem 10. Choose an n × n matrix X , det X ≠ 0 and the matrices
Λ1 Λ1 , Λ 2 ∈ ϒ n such that the matrix Λ = 0n
0n ∈ϒ 2 n and (according to Λ 2
Lemma 5) the matrices
C1 = Lk X + Lh X Λ 1 + MXΛ21 C 2 = Lk X + Lh X Λ 2 + MXΛ22 are nonsingular. Let the form of the matrix αɶ be given by (K.16). Define
[Q2 |
Q1 ] = C1Λ12 | C2 Λ 22 αɶ −1
(K.17)
A Unified Approach to Dynamic Control of Robots
249
Let us show the existence of the matrices Γ1 , Γ2 such that the corresponding motion dynamics induces force dynamics specified by Q1 ,Q2 . Write (K.17) in the form:
C
[Q2 | Q1 ] C Λ1
1
1
C2 = [C1Λ1 | C2 Λ 2 ] Λ C2 Λ 2
(K.18)
Combining it with the equality
[ 0n
C2 C | In ] 1 = [C1 | C2 ] Λ C1Λ1 C2 Λ 2
(K.19)
we obtain
0n Q 2
I n C1 C2 C1 C2 = Λ Q1 C1Λ1 C2 Λ 2 C1Λ1 C2 Λ 2
(K.20)
which can be written as
0 Qµ = n Q2
In = αɶΛαɶ −1 Q1
(K.21)
Let
Γη = T ΛT −1 X X Λ1
where the matrix T =
I T n 0n
(K.22)
X is nonsingular. Indeed, X Λ 2
−In X = I n X Λ1
0n X (Λ 2 − Λ1 )
Therefore, det T = det 2 X det ( Λ 2 − Λ1 ) ≠ 0 . On inverting the matrix T −1 and calculating the matrix product in (K.22) it can be shown that the matrix Γη has the form:
0 Γη = n Γ 2 where
In Γ1
(K.23)
250
Dynamics and Robust Control of Robot-Environment Interaction
Γ2 = − X Λ 1 Λ 2 X −1 Γ1 = X (Λ 1 + Λ 2 ) X −1 Let us note that because of (K.21), (K.22), the matrices Γη , Qµ are similar to the matrix Λ . Consequently, they have equal eigenvalues. It can be easily verified that the matrices Γη , Qµ , having the same canonical
form, satisfy (2.285), which is equivalent to (2.291), since the matrix α , given by
α = αɶT −1 is nonsingular because of the nonsingularity of αɶ and T .
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Dynamics and Robust Control of Robot-Environment Interaction
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Chapter 3
Impedance Control
3.1 Introduction Impedance control provides a fundamental approach to controlling a stiff industrial robot to interact with the environment. Impedance control mainly addresses the contact tasks for which the control of interaction force is not essential for the successful task execution. These contact tasks, such as an insertion task, require a specific motion of the work piece to be realized closely to external constraints in the presence of possible contact with the environment. This kind of motion is referred to as constrained or compliant robot motion. In essence, compliant motion tasks concern motion control problems. The objective of impedance control is thereby to reduce very high contact impedance (stiffness) of the position controlled robot by controlling dynamic robot reaction to the external contact forces (robot compliance) in order to compensate for uncertainties and tolerances in the relative robot/environment position, while maintaining acceptable force magnitudes. The interaction force between the robot and a fixed environment depends on the robot motion and the achieved target impedance. Under certain circumstances the impedance control may also be applied to realize a desired force, too. To ensure a successful accomplishment of a constrained motion task, the stiff robot position control behavior must be replaced with a compliant target impedance model. An impedance control task is specified in terms of desired motion trajectory and relationships between position error and interaction force exerted at the end-effector. The process of designing a control system generally involves many steps, such as i) ii) iii) iv) vi) vii)
specification of control problem and study of systems; system modeling, model simplification and analysis; desired performance specification; controller selection and design to meet the specifications; design tests using simulations, optimization and if needed redesign; implementation, experimental verification and application. 255
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Dynamics and Robust Control of Robot-Environment Interaction
The steps i) and ii) were considered in the previous chapters, while the current chapter will be concerned with the impedance control specification and elemental design to meet coupled robot/environment stability, as well as with the initial testing of the position based impedance control. 3.2 Control Objectives The control objective of the impedance control differs from the conventional control goals in the sense that the main control issue is not to ensure tracking of a reference input signal (e.g. nominal position or force), but rather to realize a reference target model specifying the interaction between robot and environment, i.e. the desired relationship between acting forces and robot motion reaction. A conventional control system is usually analyzed for its ability to track standard input signals (e.g. step, ramp) within the time. The main impedance control performance specification, however, addresses the capability to achieve the target model. The control input describing a desired target impedance relation may, in principle, have an arbitrary functional form, but it is commonly adopted in the linear second-order differential equation form, describing the simple and wellunderstood six-dimensional decoupled mass-spring-damper mechanical system
F = M t ( ɺxɺ − ɺxɺ0 ) + Bt ( xɺ − xɺ0 ) + K t ( x − x0 )
(3.1)
or in s-domain
F ( s ) = ( M t s 2 + Bt s + K t )( x − x0 ) = − Z t ( s )( x-x0 ) = Z t ( s )( x0 − x)
(3.2)
where Z t ( s ) = −( M t s 2 + Bt s + K t ) is the target robot impedance in Cartesian space, x0 describes the desired position trajectory, x is the actual position vector, F is the external force exerted upon the robot, and M t , Bt , and K t are positive definite matrices which define the target impedance, where K t is the stiffness matrix, Bt is the damping matrix and M t is the inertia matrix. One of the most common approaches to the representation of robot and objects position in robot programs is to use coordinate frames. Therefore, it is convenient to describe the robot impedance reaction to the external forces also with respect to a frame, referred to as a compliance or C-frame. It is convenient to assume a decoupled target impedance model in the C-frame. Thus, in this 6× 6 frame M t ∈ R 6 x 6 , Bt ∈ R , K t ∈ R 6 x 6 become positive diagonal matrices.
Impedance Control
257
The diagonal elements of these target model matrices describe the desired robot mechanical behavior during contact. Along each C-frame direction, the target model describes a mechanical system (presented in Fig. 3.1) with the programmable impedance (variable mechanical parameters) interacting with an environment (represented by a spring with stiffness K e ). Comparing this model with the mechanical model of the linearized robot control system sketched in (Fig. 3.2), it can be stated that the main objective of impedance control is to replace the stiff position robot control system (consisting of Cartesian inertia Λ at the robot end-point, position and damping gains K p and KV , and system damping BV ) with the target system (3.1). In the six-dimensional space, the target model is sketched (Fig. 3.3), where for simplicity only spring elements are depicted. The model describes a virtual spatial system consisting of mutually independent spatial mass-damper-spring subsystems in six Cartesian directions. A corresponding decoupled physical system is quite difficult to realize (for example, by combining Cartesian linear axes and Cardan frames). By appropriate selection of target impedance parameters along specific axes, various practical cases can be realized using active impedance control.
Fig. 3.1 The desired effect of impedance control
It should be mentioned that in terms of a mechanical/electrical analogy (force/voltage and velocity/current), impedance is defined as the relationship between force and velocity. In the frequency domain this is represented in the linear case by
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Dynamics and Robust Control of Robot-Environment Interaction
Fig. 3.2 Mechanical model of a position controlled robot in contact with the environment
Fig. 3.3 Target stiffness model in C-frame
Z (s ) =
F (s ) F (s ) = v(s ) sx(s )
which for the second-order system gives
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Impedance Control
Z t (s ) = − M t s − Bt −
Kt s
However, in contact tasks control, the mechanical impedance is commonly expressed in terms of the position deviation from a nominal position rather than in terms of robot velocity
Z t (s) =
F ( s) = − M t s 2 + Bt s + K t x0 − x
(
)
(3.3)
In order to assess how well a designed impedance controller meets the above control objective, it is customary to specify various performance criteria. A reasonable measure pertinent to express the performance of the impedance control is the difference between the target model and the real system behavior described by actual robot motion and interaction forces [1]. Depending on which of these physical values is used to characterize the system behavior (force or position), the impedance control error can be expressed by means of force measure (force model-error)
e f = M t ( ɺxɺ − ɺxɺ0 ) + Bt ( xɺ − xɺ0 ) + K t ( x − x0 ) − F
(3.4)
or by position measure (position model-error)
e p = x − x0 − δx f
(3.5)
where the target position deviation δx f is obtained as a solution of the target model differential equation
F = M t δɺxɺf + Bt δxɺ f + K t δx f
(3.6)
for the initial conditions: F (t0 ) = 0; δx f (t0 ) = δx0 . The computing of the model errors requires both the force and the robot position to be measured. According to the introduced performance criteria, the impedance control problem could also be considered as the model-matching problem [2], concerned with the control design minimizing the deviation norm between the model and the real plant
e f opt := min M t ( ɺxɺ − ɺxɺ0 ) + Bt ( xɺ − xɺ0 ) + Κ t ( x − x0 ) − F Gf
where the minimum is taken over all stable impedance controllers G f . Since the design is often carried out in the frequency space, it is convenient to express the model errors in the frequency domain and to constrain the
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Dynamics and Robust Control of Robot-Environment Interaction
evaluation to some interval of interest within the impedance, i.e. position control bandwidth
(
)
e f = M t s 2 + Bt s + Kt ( x0 ( s ) − x ( s )) − F ( s )
(
2
e p = x0 ( s ) − x ( s ) − M t s + Bt s + Kt
)
−1
(3.7)
F (s )
The relationship between e f (s ) and e p (s ) is clearly
e f (s ) = Z t (s )e p (s ) Assuming the relation between the actual force and position deviation in the form
F (s ) = Zˆt (s )( x(s ) − x0 (s )) where Zˆ t (s ) = Gˆ t (s ) is the realized impedance model transfer function, we can express the model errors in terms of transfer function norms using the following theorem [3] Theorem 3.1 (Input-output energy relationship): If u
2
< ∞ , y (s ) = G (s )u (s )
and G (s ) is stable and proper and has no poles on the imaginary axis, then
sup u
y u
2
= G
∞
= sup σ max (G ( jω ))
(3.8)
ω
2
Considering that the 2-norm of a signal corresponds to the energy 1
∞ 2 2 u 2 = ∫ u (t ) dt −∞ it is possible to define relative impedance model errors based on (3.8) as
⌢ ep =
ep
2
x0 − x
ef ⌢ ef = F
2 2
= I − Zt 2
= Z t (s )Zˆt
−1
(s )Zˆt (s ) ∞ (3.9)
−1
(s ) − I
∞
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261
Similar to the conventional systems performance analysis, it is possible further to divide the performance examination into steady-state (as t → ∞ ) and transient time domains. As the main purpose of using feedback control is to reduce the effects of uncertainty, the impedance controller should be capable of following the reference model in face of the robot and environment inaccuracies and external disturbances (robust control design). Thus, we can say that the impedance control objective concerns standard control design goals: stability, performance and robustness. It should be mentioned that the above defined impedance control objective represents by far the most common approach to impedance control design. However, in the literature we can find slightly different impedance control ideas. Namely, the view of some authors is that the main issue in impedance control is to achieve compliance, but not exactly a desired model. This simplification is motivated by the fact that is always not obvious and easy to deduce the desired target model from a given contact task. Furthermore, contact interaction depends not only on the target model, but also on the environment which is often uncertain or unknown. However, as will be shown later, the realization of a target impedance model represents in our opinion a reliable control goal, which assures the prediction (qualitatively as well as quantitatively) of the interaction behavior in different contact tasks.
3.3 Impedance Control Scheme The above defined control goal can be achieved using various control strategies. Impedance control represents a strategy for constrained motion control rather than a concrete control scheme. However, as mentioned in Chapter 1, depending on the features of the robotic system the implementation is usually reduced to the two basic operating procedures: • •
position-mode with an outer force loop (position based impedance control), and force-mode with an inner force loop (force based impedance control).
Force-based impedance control is mainly intended to be applied in robotic systems with relatively good causality between joint and end-effector forces such as in direct-drive manipulators. In commercial robots, this causality is significantly destroyed by the effects of non-linear friction in transmission
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Dynamics and Robust Control of Robot-Environment Interaction
systems with high-gear ratios. Therefore, in commercial robotic systems it is feasible to implement only the position-mode impedance control by closing a force-sensing loop around the position controller. Position-based impedance control is most reliable and suitable for implementation in industrial robot control systems since it does not require any modification of conventional positional controller. Practically, two basic impedance control schemes with an internal position control could be distinguished [4]. The first scheme is based on the general impedance control with an internal position controller. For the adopted position control structure this scheme is presented in Fig. 3.4. In this control system an inner loop is closed based on the position sensor with an outer loop closed around this based on the force sensing. The force loop is naturally closed when the end-effector encounters the environment. The outer loop includes a force feedback compensator G f , basically representing admittance since its role is to shape the relation between contact force and corresponding nominal position modifications ∆x f . This block is imposed on the system to regulate the force response to the commanded and actual motion according to the target admittance −1
Z t . Other control blocks in (Fig. 3.4) represent a common industrial robot position control system involving the following transfer function matrices: Gr position control regulator, Gs -robot plant and Ge -environment. The output signal from the impedance control block G f , position correction ∆x f , is subtracted from the nominal position x0 and the command input vector for the positional controller, referred to as the reference position xr is computed. A good tracking of the reference position has to be realized by the internal position controller. Practically, assuming G f = Z t−1 , the position error input to the position controller ∆xr (Fig. 3.4) becomes −1
∆xr = xr − x = x0 − ∆x f − x = x0 − x − Z t F
(3.10)
which is, in view of Equation (3.7), equal to the position impedance model error
∆xr = e p That means that the control system in (Fig. 3.4) utilizes the position related impedance model error e p to realize the target impedance behavior. Practically,
Impedance Control
263
the impedance model error e p is fed forward to the position controller Gr in order to be nullified within the internal position control loop. Since the purpose of the control system in Fig. 3.4 is to control position, it will be referred to as position model-error control. The target admittance G f is added to the control structure to regulate the force response to the motion.
Fig. 3.4 Position model-error impedance control
The second position based impedance control structure is depicted in Fig. 3.5. This scheme provides a generalization of the original scheme proposed by Maples and Becker [5] referred to as outer/inner loop stiffness control. The control scheme consists of two parallel feedback loops superimposed to the internal position control and closed using measurements from both the wrist force sensor and position sensors (joint encoders). Analyzing the control scheme, it can be seen that the position error ∆x0 (i.e. the difference between nominal and actual position) is multiplied by the task specific target impedance, Gt (s ) = Z t (s ) , providing a nominal (reference) force F0 , which corresponds to the target impedance behavior, on the output. The tracking of this force is realized by the next feedback loop closed on the sensed force F . In the ideal case we have F0 = F , providing the target behavior. Thus Fig. 3.5 basically represents a force control system with target impedance added to regulate the motion response to the interaction force. Following the control flow we see that the force-error in this system corresponds to the previously defined force impedance model-error (3.4)
∆F = F0 − F = Gt ( x0 − x ) − F = e f
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Dynamics and Robust Control of Robot-Environment Interaction
Therefore we will refer to the control system in Fig. 3.5 as force model-error control. Similarly to the previous system (Fig. 3.4), the model error (3.10) is further relayed to the internal control part in order to regulate this error to zero as time increases. However, unlike the position model-error control in Fig. 3.4, where the position model-error is eliminated by the internal position control, the regulation of the model-error in the control system in Fig. 3.5 is realized by means of the compensator G f . In order to retain the internal position control loop, the implicit force control structure is implemented by passing the force error ∆F through the admittance filter G f providing the nominal path modification ∆x f . The position correction is further added to the Cartesian nominal position x0 , and via the reference position xr fed forward to the position servo system. Obviously to achieve ∆F = e f → 0 as t → ∞ which ensures a steady state position deviation ( x0 − x )∞ corresponding to the target impedance (stiffness) model, the regulator G f has to involve an integral control term.
Fig. 3.5 Force model-error based impedance control
As already mentioned, this scheme is originally developed as a position based realization of Salisbury’s stiffness control algorithm [6]. In the seminal scheme [5], Gt block was a diagonal stiffness matrix that allows the user to specify compliance along Cartesian directions, while the compensator G f was realized as a pure integrator ensuring the desired stiffness steady-state.
Impedance Control
265
Both control approaches (schemes) basically utilize the same concept to realize the target impedance model by lowering the impedance model errors e p and e f to zero. Each approach has its own advantages and disadvantages [4]. The e p -based scheme (Fig. 3.4) is essentially simpler and easier for implementation. This scheme allows under some circumstances, different target impedances to be realized. However, this realization is done by settling the compensator G f to the target admittance, while the position controller undertakes the feedback control. This is similar to an “open loop” target impedance control. Contrary to this, in the force model-error control scheme (Fig. 3.5), the target impedance is specified in the outer loop using the Gt block, while the role of internal loop compensator G f is to ensure the tracking of the selected model using the force feedback. The internal position control loop in this scheme is kept to achieve a robust position-based force control , i.e. implicit force control, as well as to control the robot motion in the free-space. It seems that this scheme offers more possibilities to adjust the system contact behavior by choosing Gt and tuning G f . However, the opportunities to arbitrarily select the target model and dynamically maintain the force/motion relationship is quite limited by complex cascade structure of this scheme. The main problem with the e f -based scheme lies in the transition to and from contact (constrained motion). The external impedance loop in this scheme is closed even in the free-space, when the actual contact force is zero, and thus affects the position control performance. Although the magnitude of the position deviation can be insignificant, considering that the stiffness of the position control is essentially higher than the target one, as well as that the inner position loop is faster than the external impedance loops, this effect is not desirable in practice. The compensator G f has to be tuned to achieve the required control goal in contact with a stiff environment, e.g. a large amount of damping in order to ensure a stable transition. However, that is commonly contrary to the position control performance needed in the free space. In the e p -based scheme (Fig. 3.4), the force feedback loop is closed naturally by the physical contact and interaction force sensing. In the free space, only the forward position control is active. In order to avoid this shortcoming of the e f -based impedance control manifested by deviations of position control performance in the free space
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Dynamics and Robust Control of Robot-Environment Interaction
through the impedance control blocks Gt and G f (Fig. 3.5), the outer part of the control scheme providing the position modification ∆x f can be deactivated in the free space and only activated upon contact with the environment (control switching). The contact state can be observed using the force sensor information and a force threshold, which should prevail over the noise effects in the force sensing (e.g. offsets, high-frequency oscillations, gripper inertial forces during robot motion, etc.). However, as a rule, the switching algorithms are not easy to implement. This causes the force model-error control scheme to be even more difficult for the realization and integration in today’s industrial controllers. Moreover, in conjunction with control delays, the change of control structure can cause chattering, quite undesirable in the contact task, which can cause contact and system instability. Thus, the design of a smooth stable impedance controller becomes very complex with this scheme. The main advantage of the position model-error scheme over the force model scheme lies in its reliability and much simpler design and implementation. The achieved system behavior is obvious and easy to understand. Furthermore, taking into account reliable performance of the industrial robot position control, sufficiently accurate and robust desired impedance behavior can be achieved with this scheme. Therefore, we will select the e p -based control scheme for the further considerations, as well as implementation and experiments with convenient industrial robotic systems.
3.4 Impedance Control Synthesis Following the above defined impedance control objectives, the control synthesis task can be split into two steps (subtasks). The first one concerns the selection and realization of the target impedance model, while the second subtask focuses on the tuning of the model parameters in order to achieve desired contact behavior and to meet specific contact task requirements. Indeed these steps depend on another, since the realization of the target impedance poses some constraints on effective target model structure, which otherwise also determines the robot-environment interaction. However, in order to simplify the control synthesis, both steps will be considered separately. It should be also mentioned that the control design will concentrate on an adopted reasonable second-order
267
Impedance Control
target model (3.1), whereas the realization of others model structure will only be briefly discussed.
3.4.1 Effects of impedance control Considering the control scheme in Fig. 3.4, we can write the following relations
∆F = F0 − F = Gt ( x0 − x ) − F = e f Gs
−1
(s )x = Gr (s )[xr − x] − F
xr = x0 − ∆x f
(3.11)
∆x f = G f (s )F Substituting the last two relations into the first one yields the closed-loop position behavior of a robot under the impedance control G f ( s ) in contact with the environment.
x = G p (s )x0 − G p (s )G f (s )F − S p (s )Gs (s )F
(3.12)
where
[
G p (s ) = [I + Gs (s )Gr (s )] Gs (s )Gr (s ) = Gs −1
[
S p (s ) = [I + Gs (s )Gr (s )] = Gs −1
−1
−1
(s ) + Gr (s )]
(s ) + Gr (s )]
−1
−1
Gs
−1
Gr (s )
(s )
Before synthesizing the impedance compensator, let us analyze the effect of the controller G f ( s ) on the robot interaction with the environment. From the closed loop contact behavior (3.12) it is apparent that the effect of the controller G f ( s ) is to shape the sensitivity transfer function, i.e. the relationship between external force disturbance and the position tracking error
e = x0 − x = S p (s )x0 + S p (s )Gs (s )F + G p (s )G f (s )F
(3.13)
which differs from the regular position control error in the free-space
~ e = S p (s )x0 + S p (s )Gs (s )F
(3.14)
by the new term G p (s )G f (s )F introduced by the impedance control loop in order to neutralize very stiff disturbance attenuation effects of industrial robotic systems S p (s )Gs (s ) ≈ 0 . By these means, in the selected impedance control scheme (Fig. 3.4) a compliant robot reaction to the interaction forces during
268
Dynamics and Robust Control of Robot-Environment Interaction
contact is achieved, without affecting the tracking performance of the internal position controller (3.11)-(3.13), i.e. the closed loop position control transfer function G p ( s ) . From this viewpoint we can refer to G f ( s ) as disturbance controller. A physical system representing the impedance control effects based on the scheme in Fig. 3.4 is presented in Fig. 3.6.
3.4.2 Common impedance control law Following the main idea of the scheme in (Fig. 3.4) to reduce the position model measure (3.7) to zero by means of an internal position controller, almost in all position based impedance control laws proposed in the literature, the compensator G f ( s ) has been adopted in the form of the target admittance
G f (s ) = Gt
−1
(s ) = Z t −1 (s )
(3.15)
Substituting (3.15) in (3.12) provides the impedance model error that is equivalent to the expected position control error (3.10)
e p = xr − x = S p (s )[xr + Gs (s )F ] = S p (s )x0 − S p (s )Gt
−1
(s )F + S p (s )Gs (s )F
(3.16) If the internal position control system behaves as an ideal stiff servo ( G p ≈ I , S p ≈ 0 ) within a relatively large bandwidth of interest 0 ≤ ω < ωb , the compensator (3.15) ensures the desired impedance control goal e p ≈ 0 [7]. However, in real systems the bandwidth of the positional controller is limited, and the realization, i.e. tracking, of the target impedance model is, according to (3.16) also influenced by the position control transfer functions. Since the quality of the impedance control is measured by (3.16), it is of interest to keep e p as small as possible. The first and third components of e p (3.16) correspond to the conventional position control error in the free-space (3.14). Therefore, this part of the impedance model error represents an inherent feature of the selected control scheme structure, and can only be affected by changing the control scheme. However, since the conventional industrial robot control systems are designed to ensure high position accuracy, the error (3.14) is generally very small. It becomes prominent in dynamic trajectories with high velocities and accelerations. However, this is not typical for contact tasks. Moreover, a quite good disturbance attenuation of the position controller
Impedance Control
269
( S p ( s ) Gs ( s ) is very small), in common contact task makes the component
S p ( s ) Gs ( s ) F to be negligible in comparison to S p ( s ) x0 . This component together with the second error term in (3.16), produced by force feedback and the chosen impedance compensator (3.15), is relevant for the impedance control performance (realization of the target model).
Fig. 3.6 Mechanical model of the impedance control
3.4.3 Interactive system behavior - coupled stability In practical applications, however, the controller (3.15) has not demonstrated the expected performance. This discrepancy is caused in fact by erroneous judgment of the model error significance for attaining the impedance control objectives. Practically, the compensator (3.15) was selected bearing in mind a good positioning accuracy of the internal industrial position controller, which anticipates small position or force errors, i.e. the performance metrics (3.4) proposed by Lu and Goldenberg [1]. However, a deeper analysis of the error effects on the entire system behavior was missing. The most critical issue in a contact task is the interaction with environment, characterized by the exchange of the mechanical work. By decomposing the control synthesis into the realization of the target impedance (performance achieving) and the system stabilization, we have simplified the design, but the dynamic interaction between the robot and the environment is fundamental to ensure a well-behaved control. From this view point, the deviations between the realized and desired impedance, even small in magnitude, upon contact may cause quite undesirable
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Dynamics and Robust Control of Robot-Environment Interaction
effects, such as sensitivity to high frequent disturbances, contact instability, bouncing, or whatever coupled system instability.
Therefore, the analysis of the interactive behavior is essential for understanding and evaluating the attained impedance control performance. 3.4.3.1 Linearized interaction models There are several principally equivalent linear models that are pertinent for the analysis of the coupled robot-environment system behavior. The first penetration model describes the relationship between real p and nominal penetration p0, and can be derived from the realized impedance model
− F (s ) = Gˆ t (s ) e(s )
(3.17)
by substituting
− F (s ) = Ge (s ) p (s )
(3.18)
e(s ) = x0 (s ) − x(s ) = p0 (s ) − p (s ) which leads to penetration model expression
[
]
[
]
−1 −1 −1 p(s ) = Gˆ t (s ) + Ge (s ) Gˆ t (s ) p0 (s ) = I + Gˆ t (s )Ge (s ) p0 (s )
(3.19)
An equivalent model can be derived by rewriting the robot interaction model under impedance control (3.12) in terms of the penetration vectors p0 and p
p = G p (s ) p0 − G p (s )G f (s )F − S p (s )Gs (s )F − S p (s )xe
(3.20)
and neglecting last two terms corresponding to the force and contact point disturbance. This leads to
[
p(s ) = I + G p (s )Gt
−1
(s )Ge (s )]
−1
G p (s ) p0 (s )
(3.21)
Practically, the target impedance realized with the control law (3.19) can be derived from the error-form closed loop model (3.12)-(3.14) −1 e = e~ + G p (s )Gt (s )F
(3.22)
after neglecting the intrinsic free-space position error component ~ e , providing the realized admittance −1 −1 Gˆ t (s ) = G p (s )Gt (s )
Impedance Control
271
and assuming that the inverse of G p (s ) theoretically exists, we obtain the realized impedance −1 Gˆ t (s ) = Gt (s )G p (s )
(3.23)
Equations (3.19) and (3.21) thus differ only by a position control “filtering effect”, and become identical in an ideal position servo. According to the adopted model of environment the reaction force depends on the parameters of the environment and end-effector penetration
F = Ge ( s ) p = ( M e s 2 + Be s + K e ) p
(3.24)
In order to derive remaining linearized interaction models, let us first introduce the coupled or contact impedance Z c expressing the relationship between the interaction force and nominal penetration p0
F = Z c ( s ) p0 Rewriting (3.12) in terms of the penetration vectors p0 and p
p = G p (s ) p0 − G p (s )G f (s )F − S p (s )Gs (s )F − S p (s )xe
(3.25)
and substituting p0 from the linearized environment model (3.25)
p = Ge
−1
(s )F
one obtains, keeping in view the complementary relationship between closedloop and sensitivity transfer functions G p (s ) + S p (s ) = I
[Ge
−1
(s ) + G p (s )G f (s ) + S p (s )Gs (s )]F = G p (s ) p0 − S p (s )xe
(3.26)
In a stationary environment, xe = const , the second term on the right-hand side can be neglected, considering a very small and bounded magnitude S p ( jω) ≈ 0 as well as the differential nature of the sensitivity function at lower frequencies of interest ( 0 ≤ ω < ω0 ) up to which the interaction force has a significant content, This term also vanishes at the steady state, i.e. S p (s )xe → 0 as s → 0 . Note that ω0 must be less than the bandwidth ωb of the position feedback loop in order to be capable of controlling the interaction using the impedance effect and the internal position control (Fig. 3.4). Thus, we obtain the coupled impedance of the robot under impedance control in contact with the environment as
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Dynamics and Robust Control of Robot-Environment Interaction
Z c ( s ) = [Ge−1 ( s ) + G p ( s )G f ( s ) + S p ( s )Gs ( s )]−1 G p ( s )
(3.27)
The coupled impedance model expresses the relationship between the force and nominal penetration by means of the contact impedance (3.27)
[ F (s ) ≈ G (s )[I + Gˆ
] (s )G (s )]
−1 −1 F (s ) = Ge (s ) I + Gˆ t (s )Ge (s ) G p (s ) p0 (s )
e
−1
t
−1
e
[
p0 (s ) = I + Ge (s )Gˆ t
−1
(s )]
(3.28)
−1
Ge (s ) p0 (s )
Based on (3.17)-(3.18) this model can be transformed to express the relationship between position deviation and nominal penetration (deviation model)
e(s ) = Gˆ t
−1
(s )Ge (s )[I + Gˆ t −1 (s )Ge (s )]
e(s ) ≈ Gˆ t
−1
(s )Ge (s )[I + Gˆ t (s )Ge (s )]
−1
−1
−1
G p ( s ) p0 ( s )
[
p0 (s ) = I + Ge
−1
(s )Gˆ t (s )]
−1
(3.29)
p0 ( s )
3.4.3.2 Coupled system stability The most important issue in the robot-environment interaction is the stability. We are interested in the stability of the coupled interactive system described by the above linear models. For this purpose the following equilibrium points will be defined
p ∗ = p (t;t → ∞ ) = x∗ − xe p0∗ = p0 (t;t → ∞ ) = x0∗ − xe
(3.30)
e∗ = e(t;t → ∞ ) = p0∗ − p ∗ = x0∗ − x ∗ F ∗ = F (t;t → ∞ )
For the adopted linear impedance and environment models defined by (3.1) and (3.24) respectively, the equilibriums can be obtained from (3.28)-(3.29) using the final value theorem for Laplace transforms
[ (0)] G (0)G (0) p = [I + K K ] p = [I + K K ] p e = [I + K K ] K K p F ∗ = I + Ge (0)Gˆ t
−1 −1
∗
e
−1
−1
t
e
∗ 0
−1 −1
∗
e
t
−1
e
t
∗ 0
p
∗ 0
−1 −1
e
t
K e p0∗ (3.31)
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Impedance Control
Assuming diagonal environment and target stiffness matrices along each C-frame direction i (i=1,...,6) yields
pi∗ =
1 p0∗i 1 + κi
ei∗ =
κi p0∗i 1 + κi
Fi ∗ =
(3.32)
K ei ∗ p0 i 1 + κi
where the stiffness ratio
κi =
K ei K ti
(3.33)
is commonly κ i >> 1 . Now we can simply formulate the notion of coupled stability. Definition 3.1 (Coupled system stability) The interaction between a robotic manipulator under impedance control and a passive environment is said to be stable if the equilibriums p* and F* are stable (in the sense of Liapunov). Examining the interaction system models (3.19)-(3.29), it is obvious that for a stable position control system G p ( s ), consisting of a stable and proper regulator and a stable robotic plant, the coupled stability will be ensured if the transfer matrices
[I + Gˆ
t
−1
(s )Ge (s )]
−1
[
and/or I + Ge (s )Gˆ t
−1
(s )]
−1
(3.34)
a
are stable, assuming thereby both environment and target impedance to be also stable. Feedback system configurations suitable for the analysis of the coupled stability problem (3.34) based on the above interactive models are presented in (Fig. 3.7). To attain more realistic models, the force disturbance df is introduced to represent unmodeled effects, sensory noise etc. Practically, the feedback system in (Fig. 3.7a), corresponding to the coupled impedance model (3.28), can a A rational transfer-function matrix is stable if it is proper and has no poles in the closed right
half-plane (Maciejowski, 1989). The properness is required in order to ensure bounded outputs with bounded inputs.
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Dynamics and Robust Control of Robot-Environment Interaction
be obtained directly from the adopted impedance control scheme (Fig. 3.4) by assuming an ideal position servo system (Fig. 3.8). The remaining configurations corresponding respectively to the penetration and deviation models are not immediately correlated to the physical control scheme on (Fig. 3.4). Therefore, we will first concern the configuration on (Fig. 3.7a). Definition 3.2 (Internal stability) The feedback system shown in (Fig. 3.7) is internally stable if and only if the transfer function matrix
(
)
−1 I + Gˆ t −1 ( s ) Ge ( s ) p ˆ = −1 F I + Ge ( s ) Gˆ t −1 ( s ) Ge ( s )
(
)
( I + Gˆ (s ) G (s )) Gˆ ( I + G (s ) Gˆ (s )) −1
−1
t
e
−1
e
−1
t
t
−1
( s ) p0
d f (3.35)
is exponentially stable. The general purpose of this definition is to exclude unstable pole-zero cancellation between blocks in (Fig. 3.7) (hidden unstable modes), which cannot be detected by Nyquist-like tests. In general case, it is necessary to check all four of these transfer matrices to ensure internal stability. However, if the environment and realized target admittance blocks are both stable, or at least Gt−1 ( s ) is stable, it is enough to check the exponential stability of (3.34). Generally this is fulfilled if and only if i)
det[ I + Ge ( s )Gˆ t−1 ( s )] has no zeros in the closed right half-plane (CRHP);
ii)
[ I + Ge ( s )Gˆ t−1 ( s )]−1 Ge ( s ) is analytic (i.e. has non zeros) at every CRHP pole of Ge ( s ) including infinity.
In an ideal environment (3.24) it is sufficient only to check the first condition. In the ideal case when both Gt ( s ) and Ge ( s ) are diagonal matrices, it is quite easy to check the stability of (3.34). In multivariable systems, however, det [ I + Ge(s) Gt-1(s) ] becomes much more complicated and difficult to use for the synthesis of a coupled stabilizing impedance controller (i.e. target impedance). The problem is also the uncertainty of the environment, and the
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Impedance Control
question remains whether the adopted model will provide acceptable performance upon real contact (robust control problem).
Fig. 3.7 Feedback configurations for investigation of coupled stability
If both the environment and the realized admittance are stable, the coupled stability of the interactive system in (Fig. 3.7) can also be assessed by means of the small gain theorem [8]. The small gain theorem states that a feedback loop composed of stable operators will certainly remain stable if the product of all operator gains is smaller than unity −1 Ge ( jω)Gˆ t ( jω) < 1
(3.36)
∞
It is relatively easy to prove that the stable loop transfer matrix
(
)
−1 −1 −1 Ge ( jω) Gˆ t ( jω) in conjunction with (3.36) implies I + Ge (s ) Gˆ t (s ) also
to be stable [2]. The small gain theorem provides a quite general law, valid for
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Dynamics and Robust Control of Robot-Environment Interaction
continuous- or discrete-time, SISO and MIMO, linear and non-linear systems, and it is also the convergence criterion used in many iterative processes. Furthermore, the norm inequality criterion (3.36) can easily be extended to maintain the uncertainties in the target system and environment models.
Fig. 3.8 Impedance control in an ideal position servo system
However, the small gain theorem only gives the sufficient stability conditions, which in many cases are too conservative to be of much use in practical contact tasks. For example, assuming that ideal second-order target impedance has been achieved Gˆ t = Gt , the condition (3.36) implies the admissible target stiffness to be K t ≥ K e in order to ensure stable interaction. In real stiff environments this is without any practical relevance. This result is similar to the stability analysis performed by Kazerooni et al. [7]. The established interaction stability criterion practically implies that the gain of feedback compensator (i.e., the target admittance for the control law 3.15) should be limited by the magnitude of the sum of environmental admittance and robot position control sensitivity. For a SISO system this imposes the condition
K t ≥ min (K p ,K e ) in the steady state. In a direct drive robotic system having significantly lower position control stiffness (due to elimination of the transmission) in comparison to industrial robots, this condition might provide reliable target models for practical tasks. However, in industrial robots, with quite stiff servos gains ( K p > K e and S p ≈ 0 ), the above condition also requires the target stiffness to be higher than the environmental one. Moreover, no target model, i.e. the compliance feedback compensator G f can be found to enable interaction with
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277
an infinitely rigid environment ( K e → ∞ ). Therefore, one of the main conclusions in [7] has pointed out the practical needs for intrinsic compliance either in the robot or in the environment for the interactive stability. The conservativeness of this result was observed in experiments performed by McCormick and Schwartz [9], and will also be confirmed later in this work based on both stability analysis and experiments. 3.4.3.3 Coupling of passive systems A power tool for the analysis of the coupled stability problem is provided by passivity theory [10], i.e. the theory of positive systems, which plays a key role in network theory (passive networks).
Definition 3.3 (Real Positivity of a Matrix) A matrix H (s ) of real rational functions is positive real if i) All elements of H (s ) are analytic in the open RHP ( Re[s ] > 0 ) (i.e. they have no poles in RHP); ii) The eventual poles on the axis Re[s] = 0 are distinct, and the associated residue matrix of H (s ) is a positive semidefinite Hermitian; iii) The matrix H ( jω ) + H T (− jω ) is a positive semidefinite Hermitian for all real values of ω which are not a pole of any element of H (s ) . If positive real H (s ) is the closed loop transfer matrix of a linear timeinvariant multivariable system
xɺ = Ax + Bu y = Cx + Du
(3.37)
i.e.
H (s ) = D + C(sI − A)−1 B
(3.38)
then the system (3.37) is a positive system. For x (0) = 0 and for any input vector function u (t ) and corresponding solutions of the system (3.37), the following inequality represents an equivalent time-space definition of positive systems [10]
278
Dynamics and Robust Control of Robot-Environment Interaction t1
∫y
T
(t )u (t )dt ≥ 0
(3.39)
0
Assume p0 to be constantb and consider a static environment. Then the interactive model configuration (Fig. 3.7a) may be modified to present two timeinvariant networks coupled along interaction ports (Fig. 3.9). The coupling requires the velocities of robot and environment at contact point to be equal, while the forces acting upon robot and environment have opposite signs (action and reaction). If the environmental transfer matrix Ge ( s ) / s is positive real, representing any passive Hamiltonian environment, then the following theorem holds.
Theorem 3.2 (Passivity and coupled stability): A necessary and sufficient condition to ensure stability of a linearized robotic control system under impedance control, when coupled with any passive Hamiltonian environment (linear or non-linear), is that the realized admittance sGˆ t−1 ( s ) be positive real.
Proof : It is based on Popov’s lemma [11] stating that any block obtained from the feedback combination of two positive real blocks also is positive real, i.e. hyperstable. In other words, the feedback interconnection of passive systems is again passive. For the considered interaction system in (Fig. 3.9) consisting of a t1
passive environment, satisfying
∫F
T
(t ) pɺ (t )dt ≥ 0 , the sufficient stability
0
condition implies t1
∫F
T
(t )eɺ(t )dt ≥ 0
0
Considering the input/output relationship between the interaction force F(t) and the motion deviation eɺ(t ) , expressed by the admittance sGˆ t−1 ( s ) , it follows
t ∫0 F (t ) ∫0 K (τ − t ) F (τ )dτ dt ≥ 0 t1
T
b That corresponds to the analysis of coupled system stability around the equilibrium point.
(3.40)
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279
where K (τ − t ) is integral kernel. A necessary and sufficient condition for the input/output inner product (3.40) to be positive or null is that K (τ − t ) be a positive definite kernel [10], i.e. that the corresponding Laplace transform ∞
sGˆ t −1 ( s ) = K(t)e− st dt
∫ 0
need be a positive real matrix of rational functions of the complex variable s = jω. Initially, the passivity was utilized in works of Colgate and Hogan [12-13] to examine the robot contact instability phenomena associated with force feedback. However, the passivity referred to the entire robot control system instead of the realized admittance. Considering a SISO system, the coupled stability has been proven using the Nyquist criterion and the property of positive real transfer function having a limited phase by ± 90 degrees [10]. Then, it is relatively easy to prove that the mapping of the Nyquist contour of a positive real environmental impedance Ge ( s ) / s through an also positive real admittance sGˆ t−1 ( s ) , altering the phase by ± 90o and changing the magnitude by a factor 0 to ∞ , provides a stable system, i.e. a stable Nyquist plot of the open loop coupled system transfer function [12].
Fig. 3.9 Passivity model of manipulator/environment interaction
The system passivity concept provides a relatively simple test for the assessment of coupled system stability. In this test only passivity of the environment should be proven, without an accurate knowledge of parameters. Assuming again that ideal target impedance response (3.1)-(3.2) is realized, the
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Dynamics and Robust Control of Robot-Environment Interaction
passivity of the target admittance sGt−1 ( s ) implies positive definite matrices M t , Bt and K t , and consequently, the closed loop system should be stable in contact with any passive environment to which it is directly coupled. The explicit design of a positive real robot control system, however, practically may become cumbersome [12]. Moreover, various practical control implementation effects, including computational time delay and sampling effects, as well as unmodeled dynamics (e.g. high order actuator and arm dynamic effects), may result in a non-passive real impedance control response [14]. The above stability results practically can be extended to nearly-passive control systems. However, in this case a passive environment can be found which destabilizes the coupled system. In order to simplify the coupled stability analysis, Colgate and Hogan [12] have introduced the term worst or most destabilizing environment denoting the most critical environmental for the coupled system stability. Such environmental impedance Ge ( s ) / s shapes the Nyquist contour of sGˆ t−1 ( s ) by minimizing the distancec from the critical point -1 to the nearest point on the Nyquist plot of the loop transfer function Ge ( s )Gt−1 ( s ) . Taking into account that driving point impedances of simple passive environmental models, such as mass or spring ( M e / s and sKe), perform the maximum rotation in the Nyquist plane, the authors have found that the worst passive environment for the coupled stability consists of a set of pure masses or springs. 3.4.3.4 Robust coupled system stability An efficient and useful framework to cope with the uncertainties in control system analysis and synthesis is provided by the robust control concept [2]. The uncertainties in the coupled system in (Fig. 3.7) occur both in the environment (e.g. uncertain model, parameters) and in the realized admittance (e.g. due to sampling effects, unmodeled dynamics, etc.).
c The distance from the critical point -1 to the nearest point on the Nyquist plot of the loop transfer
function is inversely proportional to the infinity norm of sensitivity transfer function
1/ s ∞ .
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281
Fig. 3.10 Interaction model with target system uncertainty
Generally, for the majority of practical contact problems, the assumption of a passive interactive environment yields a reliable model. In a passive environment, the simplest approach to handling environmental model and parameter uncertainties provides the above mentioned worst environment concept, identifying certain types of environment as being most destabilizing for a specific target system. The effects of uncertainties in the realized admittance on the coupled system stability can be examined by means of the basic feedback configuration in (Fig. 3.7c). This scheme is represented by the deviation-coupled model in which the resulting admittance can be considered as a “plant”. Basically, there are two approaches to dealing with the problem of plant uncertainty in a closed loop system: i)
ii)
unstructured [15], which usually assumes additive or multiplicative frequency-dependent perturbations of bounded magnitudes, but without considering physical origins and parametric models of the perturbations; structured [16-17] which accounts for the structure of the perturbations by assuming the parametric variations in the plant physical model to be within some range.
The first method we are going to apply, provides a simple input-output oriented approaches, which is similar to classical feedback-system design and analysis methods and can easily can be extended to practical sampled control problems. However, the price that may be paid is conservativeness, since in
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Dynamics and Robust Control of Robot-Environment Interaction
general the unstructured concepts are more conservative than the structured methods, which consider the physical plant model. Consider the interaction model on (Fig. 3.9) and assume that the achieved admittance Gˆ t−1 ( s ) can be represented as the target one Gt−1 ( s ) perturbed by a multiplicative perturbation (Fig. 3.10) −1 −1 Gˆ t (s ) = (I + ∆t (s ))Gt (s )
(3.41)
Conveniently, the perturbation can be presented in the form [2]
∆ t (s ) = ∆(s )Wt (s )
(3.42)
where Wt ( s ) is a fixed stable transfer function matrix, and ∆(s ) is a variable stable transfer function matrix satisfying
∆ (s ) ∞ ≤ 1
(3.43)
Commonly Wt ( s ) is chosen as a diagonal stable, proper and minimum-phase transfer function matrix. For a SISO system (3.41) provides
Gt ( jω ) −1 ≤ Wt ( jω ) Gˆ t ( jω )
(3.44)
This inequality describes a disk in the complex plane with center at 1 and radius Wt ( jω ) (disk uncertainty). The purpose of ∆ is to account for the phase uncertainties and to scale the magnitude of perturbation. For some uncertainties models it is possible to allow ∆(s ) to be unstable, but then it is important to assume that the nominal and perturbed plant (target admittances) have the same number of unstable poles. Perturbation, which does not cancel unstable poles, is said to be allowable. Now, the following robust stability test can be introduced Theorem 3.3 (Robust coupled stability): A sufficient condition to guarantee that instability cannot occur for any possible allowable multiplicative perturbations of the target admittance satisfying (3.43), in contact with a passive stable environment is
(
Wt (s ) I + Ge
−1
(s )Gt (s ))
−1
≤1 ∞
(3.45)
∆
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Impedance Control
Proof : Since for an allowable perturbation the generalized Nyquist theorem implies that no locus of Gt−1 ( s )Ge ( s ) passes through -1 as the elements of
Gt−1 ( s ) vary, it follows
(
det I + Gˆ t
−1
( jω )Ge ( jω )) ≠ 0
which is the same as
(
)
σ I + Gˆ t −1 (s )Ge (s ) > 0 or
σ ( I + Gt −1 ( jω) Ge ( jω) + ∆t ( jω) Gt −1 ( jω) Ge ( jω))
{
}
−1 = σ (∆t Gt −1Ge ) + ∆t −1 + I ∆t Gt −1 Ge > 0
which will hold if
{
}
−1 σ (Gt −1Ge ) + I ∆ t −1 + I > 0
or
σ
(∆ ) −1
t
−1 σ (Gt −1Ge ) + I > 1
where σ and σ denote smallest and largest principal gains (singular values) of transfer matrices, respectively. Hence the largest singular value is bounded by
σ
( ∆t ) σ
( Ι + G −1G )−1 < 1 e t
and concerning (3.42)-(3.43) −1 −1 σ ( ∆Wt )σ ( Ι + Ge −1Gt ) < σ ( ∆ ) σ (Wt )σ ( Ι + Ge −1Gt ) <
+
< σ W σ( Ι + G −1G) −1 < 1 ( t ) ( e t) from which taking into account the definition and basic properties of the ⋅ operator norm, one obtains the desired result (3.45). Condition (3.45) implies the bounds on the coupled system norm
(I + G
e
−1
( jω )Gt ( jω ))
−1 ∞
≤ Wt
−1
( jω )
∞
(3.46)
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Dynamics and Robust Control of Robot-Environment Interaction
or on the perturbation norm
(
Wt ( jω ) ∞ ≤ I + Ge
−1
( jω )Gt ( jω )) ∞
(3.47)
in order to ensure robust coupled stability. It should be pointed out that the transfer function matrix
(I + G
e
−1
( jω )Gt ( jω ))
−1
(3.45)-(3.47) represents the
deviation model (3.29), i.e. the relationship between nominal penetration p0 and position deviation e. There is a practical control engineering graphical interpretation of the norm bounds (3.45)-(3.46) [2]. For a SISO system
Wt ( jω )
1 W ( jω )Gt ( jω) Ge ( jω) = t 1 + Gt ( jω ) Ge ( jω ) ∞ 1 + Gt ( jω) Ge ( jω)
≤1 ∞
imposes
Wt ( jω )Gt ( jω) Ge ( jω) < 1 + Gt ( jω ) Ge ( jω ) which says that at every frequency ω, the critical point (-1,0) lies outside a disk with center at the open-loop transfer function L( jω) = Ge ( jω ) Gt ( jω ) of the coupled system, with radius Wt ( jω )L( jω ) . Practically, Theorem 3.3 represents an application of the basic robust stability theorem [15] to the coupled system. As will be demonstrated later, this theorem provides essential and quite usable results for both coupled system control analysis and synthesis. Moreover, we will extend this theorem to an examination of the stability of contact transition process to and from constrained motion, based on a simple physical interpretation of the condition (3.46). 3.4.3.5 Coupled system performance The basic coupled system configurations in (Fig. 3.7-3.10) may also be used to analyze the performance of the interactive system, taking into account the uncertainties of the achieved impedance, i.e. admittance. Consider again the deviation coupled model sketched in (Fig 3.7c) and (Fig, 3.10). Using this model we can specify the performance of the coupled system in a similar way as in common feedback control systems by assessing the tracking of a reference signal. However, the specific goal is thereby not to achieve the model output
285
Impedance Control
(position deviation e) to perfectly track the input (nominal penetration p0), but rather to bound their normed difference
e − p0 p = ≤ε p0 p0
(3.48)
This inequality practically restrains the penetration i.e. the interaction force during contact, which is an essential practical control design constraint. Taking into account the input-output Theorem 3.1 and that the control error is shaped by the sensitivity functions, the performance specification can be given as
(
S ( jω ) ∞ = I + Gt
−1
( jω)Ge ( jω))
−1
≤ε
(3.49)
∞
or in general
(
W ( jω)S ( jω) ∞ = W ( jω ) I + Gt
−1
( jω)Ge ( jω))
−1
≤1
(3.50)
∞
where W (s ) is a frequency dependent weighting function. The above inequality specifies so-called nominal coupled system performance. Based on (3.50) we can define the performance of the perturbed system, focusing again on multiplicative perturbations (Fig. 3.11). Analogously to the robust stability, we introduce the robust performance test specified by the following condition.
Theorem 3.4 (Robust coupled performance): A necessary and sufficient condition for the robust performance of the interactive system is
(
W1 (s ) I + Gt
−1
(s )Ge (s ))
−1
(
+ W2 (s ) I + Ge
−1
(s )Gt (s ))
−1
≤1
(3.51)
∞
The proof is given in [2] for a general SISO feedback system. The significance of the Theorems 3.3 and 3.4 is that they define robust stability and performance in terms of acceptable forms (shapes) of a nominal target system taking into account possible destabilizing environments. Based on modern H ∞ synthesis techniques [3, 18], these theorems allow us to optimize the weighted performance objectives (3.45) and (3.50-51) over the set of admissible target models. This general control shaping technique is referred to as singular value shaping. However, as demonstrated by Stein and Doyle [15], the singular value loop shaping techniques are effective for control design of spatially round systems, but they can be excessively conservative in the so-called skewed systems,
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Dynamics and Robust Control of Robot-Environment Interaction
characterized by large condition numbers of system matrices. Therefore, an alternate design paradigm has been proposed to handle skewed problems [18, 19], which utilizes the equivalent performance criteria but tighter matrix measure robustness norms. This approach is based on the new matrix functions
Fig. 3.11
M −∆
block diagram
Km called diagonally perturbed multivariable stability margin (MSM) introduced by Safonov and Athans [20], the reciprocal of which is known as µ, i.e. the structured singular value (SSV) [21]. For a general nominal feedback system with a block-structured transfer matrices M (s) and the block-diagonal perturbations ∆ (so called robust M - ∆ diagram, Fig. 3.11), these functions are defined as
K m (M ) =
1 = inf σ (∆) det ( I − M∆) = 0 µ (M ) ∆
{
}
(3.52)
or in other words, K m (M ) is the smallest σ (∆ ) which can destabilize the system ( I − M∆ ) −1 . If no ∆ exists such that det ( I − M∆ ) = 0 , then
K m (M ) = ∞ i.e. µ (M ) = 0 [18]. The measure can also be expressed as M
µ
= sup µ (M ( jω )) ω
Since the function µ (M ) accounts for the structure of the perturbations and systems in order to assess the instability and has similar properties as singular values, the terms structured singular value is suggestive but rather misleading since ⋅
µ
is not a norm. Doyle [21] has proved a number of properties of µ(⋅) .
The most important of these for stability analysis are as follows
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Impedance Control
µ ( AB ) ≤ σ ( A)µ (B ) ρ (M ) ≤ µ (M )
(3.53)
Using the structured singular values, the robust coupled stability theorem can be expressed as Theorem 3.5 (Structured robustness of coupled stability): A sufficient condition to guarantee that instability cannot occur for any possible allowable multiplicative perturbations of the target admittance satisfying (3.43), in contact with a passive stable environment is
(
Wt (s ) I + Ge
−1
(s )Gt (s ))
−1
≤1
(3.54)
µ
(
)
Proof : Denote M (s ) = Wt (s ) I + Ge −1 (s )Gt (s ) write based on (3.53)
−1
. Since σ (∆ ) ≤ 1 , we can
sup[ρ (M∆ )] ≤ sup[µ (M∆ )] ≤ sup[µ (M )σ (∆ )] ≤ M ω
Thus, if M
ω
µ
ω
µ
≤ 1 then ρ (M∆ ) ≤ 1 and I − M∆ cannot be singular (on the
imaginary axis). Hence the coupled system remains stable. In order to provide practical tests for stability robustness using structured singular values, several numerical procedures for computing µ(⋅) have been proposed [19]. The advantage of µ control analysis and synthesis procedures is that they combine both structured and unstructured uncertainties. However, they are coupled with very demanding computations. 3.4.3.6 Performance of the controller Gf = Gt-1 Now let’s analyze the performance of the convenient impedance control law (3.19) by assessing the achievable interactive system behavior based on the above introduced evaluation criteria and measures. As already mentioned, in the considered linearized robot/environment control model, the control laws G f ( s ) = Gt−1 ( s ) provides the following admittance/impedance behavior
288
Dynamics and Robust Control of Robot-Environment Interaction −1 −1 Gˆ t (s ) = G p (s ) Gt (s )
(3.55)
−1 Gˆ t (s ) = Gt (s ) G p (s )
For the sake of simplicity consider a SISO system. The first observation is that the realized admittance has a higher order than the desired second-order model (3.2). Generally, such a system is much more difficult to stabilize when is coupled to a stiff environment. Concerning the passivity of the realized admittance, if we select passive target admittance sGt−1 ( s ) , the passivity of (3.55) is not ensured, since a passive position control behavior G p ( s ) is difficult to achieve in practice. Commonly implemented manipulator position control laws (e.g. integral feedback) violate the passivity condition. That means that there are some passive environments (pure stiffness and masses) that can destabilize the coupled system. Considering the robust stability, we can write −1 −1 Gˆ t (s ) = (I − S p (s ))Gt (s )
and taking the position control sensitivity S p ( s ) as the model perturbation weight function, the robust coupled stability Theorem 3.3 implies
(
S p (s ) I + Ge Since
S p (s)
∞
−1
(s )Gt (s ))
−1
≤1
(3.56)
∞
≤ 1 , the coupled stability with the controller G f = Gt−1
is
guaranteed if
(I + G
e
−1
(s )Gt (s ))
−1
≤1
(3.57)
∞
is fulfilled. This is a quite conservative condition, which is difficult to realize. In effect, it allows a multiplicative uncertainty of the target admittance to be large almost 100% before instability occurs.
3.5 Improved Impedance Control The above analysis and example have clearly demonstrated that in spite of simple and straightforward control law providing relatively small model error, the most frequently applied impedance compensator (3.19) exhibits poor coupled
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289
performance. Although, following the robust control concept, we can select the target admittance model, i.e. the compensator (3.19), which will guarantee the coupled stability in contact with a stiff environment based on (3.56)-(3.57), this could be a very conservative design providing a sluggishd system behavior and large contact forces. Therefore, in the impedance control approach pursued in this Chapter a more accurate realization of the target model becomes quite important for improving the coupled system performance. As already mentioned, by decomposing the impedance control synthesis problem into the selection of target model and its realization, we have consequentially simplified the design. A propriety target model can be selected to meet robust coupled performance in contact with an environment. Then the impedance compensator should be designed to realize this target model as accurately as possible since, as demonstrated above, relatively small perturbances caused by the position controller still can jeopardize the desired contact behavior. An outstanding reason to follow this approach is that in an industrial robot control system we can improve the accuracy of the achieved impedance control in a rather simple way. As discussed in the Chapter 1, in order to realize a given target impedance system it is necessary to cancel robot dynamic and position control effects, for example using the inverse dynamics control law. However, the implementation of the non-linear dynamic control law in industrial robots is still very complex. Even more, the benefits of this control are not clear, since at usually low velocities in contact tasks, the friction effects in high-gear joint transmission systems play a dominant role in the dynamic behavior, rather than non-linear inertial effects. The joint friction is quite difficult to be compensated for due to the highly complex non-linear and variable nature [22]. As already mentioned, the industrial position controller is sufficiently reliable and robustly designed to cope with friction and other non-linear control effects. This was the main reason to retain the internal position controller in the considered compliance control systems. Utilizing performance of the industrial robot position control system in the Cartesian space (i.e. compliance frame), expressed by diagonal dominance and spatial roundness of position control transfer matrix, we can relatively simple implement a dynamic impedance control approach based on linear compensation technique.
d As will be demonstrated later, the robust stability conditions (3.56-57) imply relatively large
target damping.
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Dynamics and Robust Control of Robot-Environment Interaction
3.5.1 Improved control law Substituting
x − x0 = Gt
−1
(s )F
(3.58)
in (3.11) we derive the expression for the position modification ∆x f which ensures the realization of the target model in the form
∆x f = G p
−1
(s )[(Gt −1 (s ) − S p (s )Gs (s ))F − S p (s )x0 ]
(3.59)
The corresponding impedance control scheme (Fig. 3.12) includes the compensator
G f (s ) = G p
−1
(s )(Gt −1 (s ) − S p (s )Gs (s )) = G p −1 (s )Gt −1 (s ) − Gr −1 (s )
(3.60)
and an additional nominal position feed-forward term
Gp
−1
(s )S p (s )x0 = Gr −1 (s )Gs −1 (s )x0
(3.61)
The position control output corresponding to (3.59)-(3.61) is
(
)
τ = Gr ( s ) x0 − ∆x f − x = Gr ( s ) ( x0 − x )
(
)
+ Gs −1 ( s ) + Gr ( s ) Gt −1 ( s ) F + Gs −1 ( s ) x0 + F
(3.62)
It is relatively easy to show that in the linearized robot control system, this control law provides an equivalent effect to that of the computed-torque based impedance control (3.23). Essentially, the main issue in (3.60) is to compensate for dynamic effects in the forward position control in order to achieve the given target model, the effect of which is similar to the non-linear control goal. The difference is that the control law defined in (3.59)-(3.61) is based on linearized compensation techniques, which are less complex than the computation of nonlinear robot dynamics. However, the impedance compensator (3.60) includes the inverse position control system G p−1 ( s ) , which generally is not well suited for use in a compensator, since the inverse systems principally produce large control signals, amplify high-frequency noise and may introduce unstable pole-zero cancellations. In spite of this, we can require that G p−1 ( s ) shows inverse characteristics over some finite frequency range only. In order to obtain a realizable (proper) compensator we can employ a low-pass filter (by inserting some more poles), or even utilize low-pass performance of the target admittance
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Impedance Control
Gt−1 ( s ) . Moreover, taking the advantages of the diagonally dominant industrial robot position control system we can easily determine G p−1 ( s ) by inverting the dominant diagonal part of the position control transfer matrix −1 G p −1 ( s ) = diag ( Gɶ pii ( s ) )
for (i=1,...6).
Fig. 3.12 Target model realizing scheme
Furthermore, G p (s ) is spatially round in industrial robots, assuring that the behavior of the position control system is independent of either the Cartesian direction or compliance frame selection in a large work area far from singularities. These characteristics, quite important in position control in order to ensure uniform performance in Cartesian space, allow the impedance control to be implemented in a simple way, in the same spirit as the position controller, using a constant compensator (3.60) for a given target admittance. For the considered position controller (3.4)-(3.9) and adopted target impedance (3.1), in the case of a SISO system the compensator (3.60) has the form
G f (s ) =
( Λ − M t )s 3 + (B − Bt )s 2 + (K p − K t )s + K I
(K
p
(
s + K I ) M t s 2 + Bt s + K t
)
(3.63)
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Dynamics and Robust Control of Robot-Environment Interaction
For a stable target admittance and position regulator the above transfer function also is stable. However, the occurrence of zeros in (3.63) can produce interesting effects upon position-correction/contact-force relationship. Most frequently in practice the aim is to reduce the large industrial robot impedance and to realize a compliant behavior by selecting a target system with M t , Bt , K t << Λ, B, K p . In this case, the zeros of G f ( s ) are close to the
(
) (
)
poles of the closed loop position control transfer function (in the closed left half plane) and the compensator exhibits minimum-phase behavior. However, if target parameters take higher values, it can produce nonminimum phase right-half plane zeros. As well known [23-24], an odd number of real RHP zeros causes the transfer function to exhibit an undershooting step response, i.e. an initial reverse reaction of position modification to external forces. This initial reaction in the opposite direction to the reaction force (in surface direction) can be very challenging to stabilize the contact transition and will be discussed later within consideration of contact transition stability. As a matter of fact, if M t , Bt , K t > Λ, B, K p the
(
) (
)
feedback system in (Fig. 3.12) becomes positive. That means, if it is required to achieve an impedance higher than that of the robot control system, the positive force feedback should be applied. It is a quite interesting result, which can be generalized to the overall impedance control. The problem with the compensator law (3.60), however, is that its second term depends on the actual robot configuration and compliance frame directions. Practically, we have demonstrated that the performance of industrial robot position transfer matrix G p ( s ) is quite robust with respect to the change of the robot configuration and compliance frame selection, but that is not valid for separated regulator Gr (s ) and robot plant matrices Gs ( s ) . However, considering that the purpose of this term is to compensate for the disturbance force, the effect of which on impedance control model error is due to stiff position control system insignificant, we can simply neglect this term and get
G f (s ) = G p
−1
(s )Gt −1 (s )
(3.64)
or in the SISO system case
G f (s ) =
Λs 3 + Bs 2 + K p s + K I
(K
(
2 ps + K I ) M t s + Bt s + K t
)
(3.65)
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293
The role of the last feedforward term G p−1 ( s ) S p ( s ) x0 in the control law (3.59) is to compensate for the convenient position control error during acceleration and deceleration motion phases, which also influences the impedance model error (3.20). Considering the expressions for the closed loop position control and sensitivity transfer matrices (3.10)-(3.12), one obtains the feedforward prefilter in (Fig. 3.12) in the form
Gp
−1
(s )S p (s ) = Gr −1 (s )Gs −1 (s )
(3.66)
which gives the SISO system
Λs 3 + Bs 2 Λs + B ɺxɺ0 x0 = K ps + KI K ps + KI
(3.67)
This transfer function can be realized as a phase-lag filter of nominal acceleration obtained from the interpolator. However, the problem is again the dependence of the transfer function (3.66) on the robot position. This requires changing the impedance control parameters dependent not only on the current task, which prescribes corresponding target impedance behavior and C-frame location, but also in the function of robot configuration. Practically, the control law (3.59) ensures good model tracking close to a working position for which the gains (3.64) have been computed. However, if this position is changing during a task, it is necessary to compute on-line Cartesian robot dynamic and position control parameters in order to compensate for variable dynamic effects. This causes the impedance control law implementation to become quite complex, similar to the case the dynamic robot control (e.g. feedforward compensation or computed torque method). Computing the entire non-linear robot dynamic model in the real time, we lose the advantages of the simple position control structure and performance. However, assuming that the nominal motion exhibits slow acceleration/ deceleration in the vicinity of constraints and during contact, which is a reliable premise due to in general unknown constraints, we can also neglect the feedforward term (3.66)-(3.67), and thus substantially simplify the control law
~ −1 −1 G f (s ) = G p (s )Gt (s )
(3.68)
~
−1 where G p−1 = diag (G pii ) is the inverse of the estimated position control
transfer matrix, i.e. of its dominant diagonal part. The controller (3.68) practically consists of a diagonal and for a given task constant compensator. In order to obtain a realizable compensator G f ( s ) , the convolution
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Dynamics and Robust Control of Robot-Environment Interaction
~ −1 −1 G pii (s )Gtii (s ) should provide a proper and stable transfer function. In other ~ words G pii (s ) has to be a stable minimum-phase function fulfilling the following condition
(
)
( (
))
(
) + size ( num (Gɶ ( s )))
size den (Gtii ( s )) + size den Gɶ pii ( s ) ≤ size num (Gtii ( s ))
(3.69)
pii
where num(⋅) and den (⋅) denote the numerator and denumerator of transfer functions respectively. 3.5.2 Coupled system performance The benefits of the new control law are obvious. By keeping the simple diagonal control structure and constant gains values (for a selected target model), the control law provides more accurate target model realization and better coupledperformance. In accordance with equations (3.23)-(3.27), the target impedance/ admittance achieved by the control law (3.68) has the form
~ −1 −1 −1 Gˆ t (s ) = G p (s )G p (s ) Gt (s ) (3.70) ~ −1 Gˆ t (s ) = Gt (s )G p (s ) G p (s ) ~ which in the ideal case G p (s ) ≡ G p (s ) , provides Gˆ t (s ) = Gt (s ) . As already demonstrated in practice, we can relatively easily and accurately estimate the robot position control function by step response or dynamic parameter estimation experiments [25]. In any method, due to identification errors and variation of the position dependent inertia matrix, the diagonal constant gain
~
estimation matrix G p ( s ) differs from the actual closed loop control transfer matrix. However, taking into account the performance of the commercial industrial robot controllers, especially the diagonal-dominance of constant rotor inertia terms, these variations could be considered to be small, i.e.
~ −1 −1 G p (s ) = G p ( I + ∆ )
with ∆ << 1 , providing the realized admittance −1 −1 Gˆ t (s ) = (I + ∆)Gt (s )
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Impedance Control
Considering ∆ as model perturbation, the robust coupled stability criterion implies
(I + G
e
−1
(s )Gt (s ))
−1
≤ ∞
1 ∆
(3.71)
which is a much more reliablee condition than (3.57), provided by the common impedance control law (3.19). Moreover, in the improved control law (3.68), by
~
selecting G p ( s ) and Gt (s ) we can get the realized admittance (3.68) to exhibit passive behavior, ensuring the coupled stability in contact with any passive environment. The achieved coupled and penetration models with the control law (3.68) are −1
p ( s ) = I + G p ( s ) Gɶ p −1 ( s ) Gt −1 ( s ) Ge ( s ) G p ( s ) p0 ( s ) −1
≈ I + Gt −1 ( s ) Ge ( s ) G p ( s ) p0 ( s ) F ( s ) = Ge
−1
−1 ( s ) + G p ( s ) Gɶ p −1 ( s ) Gt −1 ( s ) G p ( s ) p0 ( s )
(3.72)
−1
≈ Ge −1 ( s ) + Gt −1 ( s ) G p ( s ) p0 ( s ) Substituting the improved control law in (3.15) and assuming thereby ideal position control compensation yields
p = G p (s ) p0 − G p (s )G f (s )F − S p (s )Gs (s )F − S p (s )xe = G p (s ) p0 − Gt
−1
(s )F − S p (s )Gs (s )F − S p (s )xe
from which neglecting small internal position control perturbation due to interaction force and contact point location gives
p = G p (s ) p 0 − G t
−1
(s )F −1 e = S p (s ) p0 + Gt (s )F
(3.73)
It is obvious that the effect of the new controller is to shape the sensitivity transfer functions, i.e. the relationship between external interaction force disturbance and the position tracking error according to the desired target impedance model, without influencing the nominal position control performance in the free-space. It should be pointed out that only the sensitivity transfer function to the interaction force sensed by the force sensor and used in the e In the ideal case, the coupled stability at contact with a stiff environment is ensured for any
second order target impedance with positive Mt, Bt and Kt.
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Dynamics and Robust Control of Robot-Environment Interaction
external control loop is modified by the impedance control. The impedance controller does not influence the robust and good perturbation rejection properties of the position controller to other disturbance effects, such as friction.
3.5.3 Target impedance model realization The following experiments demonstrate the accuracy of the target impedance model realization in an industrial robot control system using the control law (3.68). The conditions for a stable transition to and from contact, however, will be considered in the next section.
Experiment 3.1:
Target model realization tests using Manutec r3 and SMART-3 6.12R industrial robots
To demonstrate the realization of the target model by the improved control law, several experiments with Manutec r3 [23] and SMART-3 6.12R [26] industrial robots have been performed. In the experiments, the robots have been positioned in various configurations within the working space, and the impedance control laws (3.68) with selected target models and diagonal position control transfer functions have been activated. The robots stand still during experiments, i.e. initial poses (x0 = const) have been achieved and maintened by the position controller. The external interaction forces were realized by catching the endeffector (force sensor) by hand and manually pushing and pulling it in different Cartesian directions (Fig. 3.13). By this means robot compliant reaction (motion) around the initial pose has been produced according to the realized impedance. In order to evaluate the impedance control quality, i.e. the realization of selected target impedances, the force model error measure ef (3.4) has been used and both the interaction force F (t) and robot motion x (t) have been measured. The model error ef asserts the matching between real force and the ideal force computed according to the target model (3.1). To obtain this ideal force, x (t) has been differentiated twice in order to compute the end-effector velocity and acceleration xɺ (t ) and ɺxɺ(t ) respectively. Substituting these vectors, as well as initial position and target model parameters in (3.1) yields the nominal target force, which applied upon ideal target mechanical system (Fig. 3.1) would produce the same motion x (t) as obtained in experiments. Finally, the computed nominal force has been compared with the measured one.
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The following target impedances have been applied in experiments for translational and rotational DOFs respectively:
Gt ( x , y , z ) = 20 s 2 + 2078.5s + 1500 Gt (ϕ x ,ϕ y ,ϕ z ) = 0.45s 2 + 46.8s + 33.75
(3.74)
Fig. 3.13 Target model realization experiment
These target impedances were combined with zero-impedance (infinite i.e. 10 very large target stiffness K t = 10 N / m ) in order to realize various tests, starting from the impedance only in one Cartesian direction, till six-DOF impedance case. The estimated position control transfer function was adopted as
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Dynamics and Robust Control of Robot-Environment Interaction
~ Gp =
3463 s + 75s + 3463 ~ 38.4 Gp = s + 38.4 2
(3.75)
for the Manutec-r3 and SMART-3 respectively. A typical result of a target model realization experiment with Manutec r3 robot in the configuration in which the robot end-point position is x0 = [-0.18 0.91 0.1 90 0 0 ]T, is presented in (Fig. 3.14). It should be noted that in this experiment the robot compliance was only selected in the translational directions x, y and z, while rotational DOFs were stiff. The target impedance parameters selected in experiment were: M t = 20 (kg), Bt = 2078.5 (Ns/m) and K t = 1500 (N/m). To compare the target model realization with the improved and common impedance control laws, similar experiments were performed using the impedance compensator in the form (3.19). The results with the conventional impedance control law are presented in (Fig. 3.15). Illustrative experimental results with the SMART-3 6.12R robot in the configuration x0 = [0.91 0 1.69 30.3 0.4 -120]T and all six compliant Cartesian DOFs are given in (Fig. 3.16).
Fig. 3.14 Target model (solid) and measured (dashed) forces (improved law)
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Fig. 3.15 Target model (solid) and measured (dashed) forces (common control)
Obviously a very good matching of model and experimental contact forces has been achieved in all experiments. This result confirms the feasibility and reliability of the proposed impedance control approach. Slightly larger force deviations can be noticed in the quite dynamic situation that the end-effector is pushed very fast or suddenly left alone (for example at the end of experiment where F=0, see Fig. 3.14). The reason for this is that in these cases the measured force at the force-sensor, with a very high bandwidth, rises or disappears immediately, while the impedance controlled system with significantly lower bandwidth reacts significantly slower. The bandwidth of target systems used in experiments (3.74) is ωt ≈ 9 rad/s, i.e. ft ≈ 1.5 Hz. Therefore, such situations are not well suited for target model realization testing. A better correspondence is achieved in experiment in (Fig. 3.16) with continuous force/motion behavior. The bandwidth of the impedance controller is limited by the selected control scheme architecture (Fig. 3.3-3.4). This issue will be discussed in detail in the following sections.
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Dynamics and Robust Control of Robot-Environment Interaction
Fig. 3.16 Comparison of target model (solid) and measured (dashed) forces (SMART-3 robot)
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Impedance Control
Concerning the target model realization with common (3.19) and improved control law (3.66), the difference between diagrams in (Fig. 3.14) and (Fig. 3.15) seems not to be relevant. An explanation for this is that the conventional impedance control law (3.19) in precise industrial robot position control systems also assures a relatively good target model realization, providing a relatively small position model error measure. The position model error e p achieved by the improved control law is even smaller, since the second term in (3.20), i.e. the component epII is close to zero due to better target model realization. This diminishing of the target impedance position model error is, according to the linearized environmental model, manifested by a significant reduction of force model error in a relatively stiff environment. However, it is not well expressed in the compliant hand-robot interaction used in experiments. Nevertheless, we can demonstrate better model realization using the improved control in experiments by computing absolute and relative force model errors according to (3.7) and (3.9), respectively. For the examples in Fig. 3.14 and Fig. 3.15 the result is presented in Table 3.1.
3.6 Typical Impedance Contact Behavior Before pursuing the contact stability problem in the next section, in the following examples we will examine typical robot motion phase transition behavior (from the free space to contact) using the improved impedance control law (3.68). Table 3.1 Force model errors Target model error
Conventional impedance control (Fig. 3.15)
Improved impedance control (Fig. 3.14)
e fx
198.3 N
153.4 N
e fy
146.7 N
135 N
e fz ⌢ e fx ⌢ e fy ⌢ e fz
115.2 N
81 N
0.15
0.11
0.08
0.08
0.15
0.07
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Dynamics and Robust Control of Robot-Environment Interaction
Experiment 3.2: Contact transition trials with improved impedance control The aim of the experiments was to examine the contact transition behavior during an elemental contact task in which a nominal position behind a hindrance is commanded and a stable transition and equilibrium state should be achieved. For the tests, a Manutec r-3 robot has been used. The basic impedance control scheme (Fig. 3.3) has been implemented using the improved control law and integrated in the ARCOS control system [27]. The computational control cycle of T = 8 ms and a force sensing and computation time lag of τ ≈ 6 ms have been realized.
Fig. 3.17 Contact transition experiment
As a working object, i.e. the environment, a cantilever beam with variable cross-section, i.e. stiffness (Fig. 3.17-3.18), has been used. The stiffness of the beam was estimated at several locations from the clamped end in a relatively simple way using force control (see next section), by exerting a constant commanded force (30 N) upon the beam by the robot and measuring the robot position penetrationf into the environment. The accuracy of the stiffness estimation using this method is determined by the resolution of the used force sensor (0.05 N) and by the Cartesian resolution of the Manutec-r3 robot positioning sensors in joints (encoders with 2400 increments/rad), providing a Cartesian position measurement accuracy of about 10 µm (high due to very high gear ratios). That gives resolution of environment
f Relative position displacement of robot end-point after the robot hits the beam.
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303
stiffness estimates of about 5000 N/m, which is not so much, but this error is not critical since the environmental stiffness is not used in the control law. In addition, as demonstrated, a robust impedance control has to be designed in order to cope with uncertainties in the robot model and the environment. Due to the relatively slow penetration velocity caused by the relatively slow nominal approaching velocity, and foremost by compliance effects, the deformation of the beam, as will be demonstrated, could be considered as quasi-static with dominating stiffness effects, and negligible acceleration and velocity proportional to the environmental force components.
Fig. 3.18 Contact transition experiment
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Dynamics and Robust Control of Robot-Environment Interaction
By changing the contact point location with respect to the beam end we have varied the environmental stiffness in the interval 50000 to 200000 N/m. That represents a relatively quite stiff environment, covering most practical industrial cases. The target model parameters have been varied in a relatively large interval in order to prove their influence on the contact behavior. The second-order target impedance model (3.2) has been rearranged in the common normalized form
(
Gt (s ) = Zt (s ) = M t s 2 + Bt s + K t = M t s 2 + 2ξtωt s + ωt
2
)
(3.76)
with the target frequency and damping ratio respectively
ωt =
Kt Bt ; ξt = Mt 2 M t Kt
F ≈ K e p = K e ( x − xe )
(3.77) (3.78)
The target mass and stiffness have been varied within intervals: M t ∈[1, 40] (kg) and K t ∈[1000, 5000] (N/m). An important practical constraint in combining these parameters is imposed by the inner/outer loop architecture of the impedance control scheme which requires the target system frequency ωt to be less than the internal position control closed loop frequency (circa 60 rad/s i.e. 10 Hz). Considering the coupled impedance system sketched in (Fig. 3.7a) and expressed by (3.28), which describes a serial connection of the target system and environment (in an ideally realized target impedance)
F = Z c (s ) p0 Z c (s ) =
Gt (s )Ge (s ) Gt (s ) + Ge (s )
(3.79)
it is obvious from (3.76) that the damping ratio of the target system should be ξt >> 1 in order to achieve a well damped interaction, i.e. the above coupled system, taking into account that K t << K e . The selection of the target model parameters will be discussed in detail later. A second-order position control transfer function has been adopted in the form (3.75), providing the impedance compensator for the experiments according to the control law (3.68)
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Impedance Control
~ −1 −1 G f = G p (s ) G t (s ) =
s 2 + 75s + 3463 2
3463 M t ( s 2 + 2ξ t ωt s + ωt )
(3.80)
The experiment scenario was as follows. At the start of experiments the robot was located in the vicinity of the beam (see Fig. 3.17-3.18). Then the impedance control was activated and a linear motion was commanded along the surface normal, which coincided with the y-axis of the robot base frame (B). The goal point was located beyond the contact surface, defining the maximum nominal ∗ penetration p0 = 8 mm. To test robot contact behavior in specific experiments, a further relative linear displacement of 10 mm beyond the beam was commanded after the first nominal motion was realized. The maximum approaching velocity was varied within v0 ∈ [0.01, 0.05] m/s. The velocity profile was trapezoidal (see Fig. 3.18). After a short WAIT interval, the backward motion to the initial robot pose in the front of the beam was commanded and the impedance control was deactivated at the end of the motion. Typical contact transition experimental results are shown in (Fig. 3.19-3.21). Fig. 3.19 shows the contact force components (force exerted by the robot on the environment) w.r.t. the C-frame attached at the robot end-effector. The stiffness of the environment (beam) at the corresponding contact location in the experiment was considerably high, approximately K e ≈ 160000 N/m. The parameters of the selected target model were: M t = 20 (kg), Bt = 2771 (Ns/m) and K t = 1500 (N/m). The force components in the direction of the surface normal (Cartesian y direction in B, i.e. z axis of the C-frame) is the dominant force component for the considered contact process, while remaining components occur due to disturbance effects such as friction, geometric misalignments, control coupling effects etc. Fig. 3.20 shows the applied actuator forces and torques in Cartesian space (B frame) reduced at the robot end-effector. These components are computed using measured motor voltages according to
τ = J −T τ q where the joint actuator torques reduced at output shaft τ q are obtained as τ q = diag (ni ) ⋅ diag (k mui )u p , where u p is measured voltage vector, k mui is motor torque constant and n i is gear transmission ratio. One can see from (Fig. 3.20) that the largest actuation forces are in the B - y and z directions, in which the influence of the high power robot axes 2 and 3 is dominant in order to
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Dynamics and Robust Control of Robot-Environment Interaction
control the interaction and compensate for gravity and joint friction effects. The particular influence of interaction force (Fig. 3.19, component Fz ) on the actuating force is not so relevant due to the relatively small magnitude in comparison to other disturbance effects.
Fig. 3.19 Cartesian contact forces and torques
Let’s now examine the robot/environment interaction behavior in the surface direction. The robot transient behavior, including the robot motion x (t), as well as the impedance control position modification ∆x f (t ) and nominal position x0 (t ) , is presented in (Fig. 3.21) (the notation for characteristic motion quantities, i.e. positions, is undertaken from the impedance control scheme in Fig. 3.3). The interaction force is also added in order to compare the motion and force time history during contact. The most interesting contact transition interval is captured and presented separately in the diagram below.
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Fig. 3.20 Cartesian control forces
The following observation can be drawn from this experiment (Fig. 3.183.21). The contact transition was stable. After first impact the robot remains in contact until the backward motion is not commanded. The steady-state force corresponds to the expected one according to the coupled impedance model (3.31) ∗
Fz = Fz = t →∞ s →0
Kt Ke ∗ p0 Kt + Ke
(3.81)
which for the first penetration step yields ∗
Fz =
1500 ⋅ 160000 ⋅ 0.008 = 1486 ⋅ 0.008 = 11.88 1500 + 160000
in N. This value matches quite well with the experiment.
(3.82)
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Fig. 3.21 Characteristic impedance motion quantities in the contact direction: nominal position x0 , impedance control position correction ∆x f , reference position xr and actual position x (dashed)
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A force peak after the environment was hit corresponds to the impact between the end-effector and the environment. The time lag in the impedance control system (i.e. a delay between force sensing and impedance control reaction) causes this collision to be in a short time period (τ ≈ 8 ms) equivalent to the impact between a stiff robot and the environment. Due to discrete time control implementation, several samples (T = 8 ms) are required to achieve the desired impedance effect according to the target system dynamics and to reduce the initial impact force. To keep this force as small as possible, a considerable small approaching velocity and fast control computation are desirable. Following the position time history we can see that the robot actual position x tracks accurately the nominal one x0 in the free space, while immediately after the contact the impedance controller introduces the correction ∆x f so that the internal position controller is trying to track the reference input xr = x0 − ∆x f . After a very short transition interval, the robot position x remains very close to xr, which, due to a very stiff environment, corresponds to the stiffness surface location xe , i.e. x ≈ xe and p ≈ 0 (an infinitesimal penetration is required to produce the contact force shown in Fig. 3.21). The force transient behavior, however, is characterized by relatively high overshoots. That is expressed especially at the beginning of the contact where the force reaches about 250% of its steady-state values (3.82), whereby in the second step, the maximum force during transition comes to 170 % of the corresponding steady-state value (Fig. 3.21). There are two explanations for the overshoots during transition phase, which in essence have the same background. The first one can be drawn from control theory. The high force peaks are caused by the large target damping ratio used in the experiment
ξt =
2771 Bt = ≈8 2 M t K t 2 20 ⋅ 1500
which, as will be demonstrated later, represents a quite over-damped system providing more damping than the minimum needed to maintain stable contact. As well known, an over-damped system generally reacts sluggishly causing high transient forces during contact establishment. The second reason lies in the structure, performance and limitations of the impedance control concept itself. Practically, the aim of impedance control is to realize the robot target model reaction to the interaction force. The target model, in principle, defines a mechanical system shown in (Fig. 3.1) consisting of passive elements: mass, dashpot and spring, each of them producing a force
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component proportional to the deviation between nominal and actual robot motion during contact (3.1). The resultant force from these elements is equal to the contact force. A stiff stationary environment and the position modification by the impedance compensator cause the actual end-effector motion to be slowed done in a short time period. Practically a stable contact in a stiff environment (Fig. 3.21), with reliable contact forces and approaching velocity is characterized by
p ≈ 0 ⇒ x ≈ xe , xɺ << xɺ0 , ⇒ xɺ ≈ 0 ɺxɺ << ɺxɺ0 ⇒ ɺxɺ ≈ 0 during a greater part of the transition period. From (3.1) it follows
F = M t(ɺxɺ0 − ɺxɺ) + Bt(xɺ0 − xɺ) + K t(x0 − x) ≈ M t ɺxɺ0 + Bt xɺ0 + K t(x0 − x)
(3.83)
This means that the contact force is in a great measure shaped by the nominal motion generated by the interpolator. Since the target mass and the robot acceleration are relatively small, the inertial force component (first term in 3.83) is also small. Consequently, the remaining damping and stiffness components have a dominating influence on the contact force behavior. The stiffness force increases nearly linearly with augmentation of the nominal penetration into the surface, and reaches its maximum value which corresponds to the steady-state contact force
F ∗ = K t e∗ = K e p ∗ =
Kt Ke ∗ p0 Kt + Ke
(3.84)
where
e∗ = x0∗ − x∗ p ∗ = x∗ − xe p0∗ = x0∗ − xe It may be concluded that the damping component has the main influence on the force overshoots during transition. In a stiff stationary environment the impedance damping force component is approximately proportional to the nominal velocity. Consequently, in a commonly used velocity profile in industry (Fig. 3.6), the damping component has a trapezoidal form following the nominal
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velocity trajectory from the initial impact untill the end point, where it goes down to zero. To demonstrate this for the considered experiment, the inertia, damping and stiffness force components of the target model (3.1) have been computed using the measured robot position x(t ) and the nominal position x0 (t ) obtained from the interpolator, as well as the target model parameters. The result is presented in (Fig. 3.22). The nominal velocity and acceleration were obtained by numerical differentiation of x0 (t ) . The measured contact force is added in order to illustrate the relationship between the real and ideal target model force, i.e. its components. In order to prove the obtained contact force performance and suitability of the linear target impedance models used in control design, we have compared the contact force measured in the experiment with the model force computed using the following models: • exact target impedance model
F = M t(ɺxɺ0 − ɺxɺ) + Bt(xɺ0 − xɺ) + K t(x0 − x) • approximated target model
F = M t ɺxɺ0 + Bt xɺ0 + K t(x0 − x) • coupled impedance model
F=
(
)
K e M t s 2 + Bt s + K t ( x0 − x) M t s 2 + Bt s + K t + K e
• non-linear robot impedance control model
Λ ( x) ɺxɺ + Bxɺ + µ ( x, xɺ ) + p ( x) = PID( x0 − ∆x f ) + pˆ ( x) + F where the interaction force and the impedance control position correction are respectively obtained using stiffness model
F = K e ( x − xe ) • and discrete time control model
∆x f = G f ( z )z −1 F The first two models have been computed in the time domain based on the measured robot motion, the third model has been calculated in the s-domain using the MATLAB’s lsim function, and the last one is obtained using the realistic simulation in the ROBOTICS Toolbox [28]. This simulator utilizes the
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complete non-linear robot/environment model, also including various non-linear dynamic and control effects such as joint friction, discretization (e.g. in converters and encoders), control time lags etc. The results are presented in (Fig. 3.23) and (Fig. 3.24). A good matching between the forces obtained by using simple linear models and the measured force confirms the reliability of these models, as well as the good target model realization and correctness of the previous considerations concerning the contact system performance. That means that simple linear target and coupled impedance models, which are well suited for both the control design and system analysis, can accurately approximate the real system behavior.
Fig. 3.22 Nominal motion profile and target impedance model components
Better correspondence between measurement and simulation has been obtained using a non-linear model (Fig. 3.24). However, due to the very complex non-linear Cartesian stick-slip friction nature at low-velocities during contact, in the considered example the simulated limit cycle oscillations significantly differ from the measured performance. The effect of friction on the end-effector force
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depends on the robot configuration and stiction/friction behavior of separate joint axes. Practically, the joint stiction force is variable and depends on robot working conditions (e.g. temperature, initial control conditions etc.). If all axes are clamped, i.e. they are in the stiction area, it will produce a stable steady-state force without limit-cycles, such as is in the considered experiment (Fig. 3.213.24). However, if stable equilibrium conditions for all exes are not fulfilled, which is more common in practice, steady-state oscillations could occur. As will be shown in further experiments, these oscillations are quite similar to the simulated one in (Fig. 3.24). The role of friction effects in the impedance control, and generally in contact task control, will be discussed in more detail in the following sections.
Fig. 3.23 Measured vs. simulated contact force
There are several possibilities to decrease large force overshoots, i.e. to influence the contact transition performance. According to the above consideration, the simplest way could be to reduce the target damping. However, in general that can jeopardize stable contact transition. The influence of target
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impedance and environmental parameters on the contact performance will also be considered in the following sections.
Fig. 3.24 Non-linear model simulation
In order to illustrate the effects of contact parameters variations, the transition experiment has been performed using the same scenario but with different target models and contact point location (i.e. modified beam stiffness, see Fig. 3.17 and Fig. 3.18). The obtained contact forces are shown in (Fig. 3.25). In the first experiment (Fig. 3.25a), a contact point with K e ≈ 65000 N/m has been selected, while the target-damping ratio was settled at ξ t = 3 (target mass and stiffness remained the same as in the previous experiment M t = 20 kg and K t = 1500 N/m). As can been seen, the contact transition was unstable. If the damping ratio is increased to ξt = 5 (Fig. 3.25b) or the stiffness is reduced (i.e. contact location has been changed) to K e ≈ 25000 (Fig. 3.25c) the contact transition become stable. In the experiment with the environmental stiffness K e ≈ 160000 and ξt = 5 (Fig. 3.25d), the
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contact transition from free to constrained motion is nearly at stability limit. That means that the contact force is going down closely to zero after initial impact, or the force becomes zero within a short time interval. After stable contact has been realized in the first step, the next nominal penetration appears to be non-critical for the contact stability. As shown in (Fig. 3.21), a further increasing of the damping coefficient to ξ t = 8 provides a safe contact transition process.
Fig. 3.25 The effects of ξ t and K e variations on contact performance ( M t = 20 kg and
K t = 1500 N/m)
The effects of the target mass and target stiffness variations on the contact transition are illustrated in (Fig. 3.26). Generally, mass reduction causes a faster target system reaction (increases target frequency ωt) and up to some limitations it facilitates contact stabilization. Since the target damping Bt also becomes smaller with the decreasing mass (in the case where the damping ratio ξ t remains constant), the force overshoot is turned down (compare Fig. 3.26a and
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Dynamics and Robust Control of Robot-Environment Interaction
b). The effect of a target reduction of stiffness is similar. In general, the stiffer the environment and more elastic the target system, the larger the damping ratios must be to ensure a stable contact transition (see Fig. 3.26c and d). A quite elastic target system ( K t = 250 N/m), which provides small interaction forces even in the case of higher environment uncertainties, imposes a large damping ratio ( ξt ≈ 30 ) to ensure stable contact transition. These initial observations will be validated in the next section using mathematical formalisms.
Fig. 3.26 The effects of M t and K t variations on contact performance ( K e ≈ 60000 N/m)
The experiment in (Fig. 3.25a) clearly shows how dangerous a loss of contact during transition could be. Analyzing the force trajectory, it may be observed that after first impact the robot is moving backward losing the contact and pursuing the motion in the free space in the opposite direction to desired one. Thereby the interaction force tends to zero, causing the position modification ∆x f at the output of compensator G f ( s ) (see Fig. 3.4) also to
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reduce to zero. Moreover, during this period the nominal motion x0 generated by the interpolator is progressively increasing in surface direction, producing a significant position error relative to the actual robot position. The internal stiff position controller rapidly accelerates the robot in the surface direction in order to reduce this error, causing again a bump, however with higher velocity than the initial one. This generates an even higher force peak than in the first collision. A contact-unstable impedance controller reacts again dynamically with a large position modification producing a second lost of contact, and the entire cycle repeats. If during this period the nominal position continuously increases in the surface direction, each following bump is accompanied with a larger force and faster forward and backward motion, till a maximum allowed force at the sensor is achieved and the controller exception handling functions break the robot motion. If the nominal motion progression is stopped between two collisions, bump and lost of contact cycles can repeat periodically keeping a nearly constant force magnitude. The force behavior becomes thus a typical limit-cycle oscillation form with periodical impacts. These cyclical bumps, as well as robot shutdown due to maximum force overshoot, are quite dangerous and undesirable. Switching the controller off and activating motor brakes in the robot contact with a stiff environment generates in principle an uncontrollable non-linear transition process. During this procedure, incremental joint displacements are still possible, whereby the robot end-effector remains constrained by the environment. This may produce enormous forces at the end-effector that can cause damage to the sensor, robot or environment. Even when a crushing does not occur, it is difficult to resolve this situation and to restart the robot and continue the robot motion, either in position control mode or using the impedance control. The limit-cycle bump oscillations usually have small frequency and could excite the control and robot eigen oscillations or also cause damage. How to overcome these periodical bumps and continue the robot operation also is not clear. A very important note is that in the considered experiment the coupled robot/environment stability is ensured, while in spite of this the system exhibits the contact transition instability. That indicates that the contact transition stability is more critical implying stronger criteria on impedance control parameters than the previously considered coupled system stability. In some cases, when the contact is just narrowly lost in a very short time period and the end-effector remains closely to the contact surface, the contact could be established without repeating bumps and the negative consequence of the lost of contact may not be so critical as discussed above. For example, in repetitive trials of the transition experiment shown in (Fig. 3.25c), a stable
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contact transition (i.e. practically at the contact “stability limit”) has been achieved in some cases. An explanation for this is the influence of non-linear Coulomb friction effects, which practically supports a bounce-less establishment of the contact by motion resistant effects. However, the tuning of compliance controller based on experiment trials is quite risky. In spite of above-mentioned specific cases, in which the contact instability appears to be not so hazardous, this problem is fundamental and requires basic theoretical investigations. Moreover, it can be pointed out that a stable contact transition generally represents the most critical issue and primary practical control design requirement in almost all contact tasks.
Experiment 3.3:
Improved vs. conventional control law transition performance
It is interesting to compare the contact transition performance of the improved impedance control law (3.68) with the conventional one (3.19). In the experiment, a similar concept has been applied as in the above presented
Fig. 3.27 Conventional vs. improved impedance control: contact transition performance
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transition trials. The robot start position was about 6 mm in front of the surface, so that the contact with environment was realized in the deceleration motion phase (see Fig. 3.18), which, as will be demonstrated later, imposes worst contact transition conditions from the stability viewpoint. The following target parameters were chosen: M t = 20 (kg), ξt = 6 , K t = 1500 (N/m), while the environmental stiffness was K e ≈ 60000 (N/m) and the nominal penetration p0 ≈ 0.004 (m). The obtained contact forces with both laws are presented in (Fig. 3.27). Evidently, the conventional control law provides more oscillatory force behavior due to higher oscillating modes, partly due to a non-perfectly realized target model. Indeed, as discussed above, a stable contact transition with this control law is much more difficult to realize. Generally, this control requires a significantly larger amount of damping compared to the improved control law in order to stabilize contact with a stiff environment.
3.7 Contact Transition Stability In the previous section we have analyzed the coupled stability of the interconnected robot/environment system using linear interaction models. However, the coupled stability problem has been considered without regard to the nature of the contact, i.e. constraints. Basically it was assumed that the contact could not be lost. This assumption, however, is not reliable in unilateral, i.e. force constraints. Moreover, the loss of contact during transition from the free-space robot motion to the physical contact with the environment and vice versa appears to be the most critical interaction problem, even in bilateral i.e. geometric contacts. In effect, a geometric contact is commonly achieved by closing the gripper jaws, or by parts mating. Due to relative robot/environment position misalignment, the unilateral contact precedes the realization of a bilateral contact. The most critical issue in contact tasks is a stable contact establishment, i.e. a stable contact transition from and to the free space. The undesirable effects during transition, such as multiple impact (bouncing) and loss of contact, can cause very high interaction forces or a long transition time, respectively. Relatively little research has addressed the problem of contact transition stability. Coupled stability and contact transition stability were not clearly distinguished in the literature. The majority of theoretical contact stability investigations and developed transition control algorithms concern direct drive robots. The established transition stability criteria are either necessary or
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Dynamics and Robust Control of Robot-Environment Interaction
sufficient, and usually quite conservative. Moreover, these results are not applicable in industrial robots primarily due to their stiff position controllers and limited sensing and computation capabilities. In experiments with industrial robots [29] it was observed that an increased target damping and mass (“a dominant real pole” of the interaction model) support stable contact establishment. However, the high values of the target mass and damping needed to maintain a stable contact give the robot sluggish behavior, which is not desirable for handling both high impact forces and contact losses. The purpose of this section is to derive reliable contact transition stability criteria pertinent for industrial robots and real contact tasks. Therefore, the consideration is mainly concerned with position-based impedance control, but under some circumstances the results can be extended to various compliance and force control schemes. Our aim is to examine the effects of impedance control parameters on the transition stability, taking into account uncertainties and disturbances, as well retardations (control time lags), rather than to develop complex transition control algorithms. The main idea is to tune the target impedance parameters in order to meet both interaction performance and stability (contact transition and coupled stability). The validity of the derived conditions for contact transition stability will also be demonstrated in experiments with the Manutec-r3 robot.
3.7.1 Definition of the contact stability Based on the initial impedance control experiments (Experiment 3.2, Fig. 3.173.18) let us consider the transition of a robot under impedance control from the free space to contact with a stationary environment (e.g. a surface). Assume that the controlled robot-environment interaction can be described by the target impedance behavior
F = Gˆ t (s )( x0 − x ) = Gˆ t (s ) e Let consider the model of the environment in the form
F = Ge (s ) ( x − xe ) = Ge (s ) p = Ge (s ) ( p0 − e )
(3.85)
assuming that the environment is passive. That means that the environment does not involve an energy source and cannot produce mechanical work. A passive environment is characterized by the positive mechanical work during a stable contact
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t
∫F
T
pɺ dτ > 0, ∀t
(3.86)
0
The driving point impedance of an isolated passive environment is a strictly positive real (SPR) transfer function
pɺ (s ) s = F (s ) Ge (s ) jω Re > 0, ∀ω Ge ( jω )
(3.87)
To meet this condition the environmental transfer function Ge (s ) must be a Hurwitz polynomial of maximum second order, such as the linear environmental model (3.24). In order to ensure a stable contact transition with each passive environment it is efficient to utilize the most destabilizing environmental model (e.g. pure stiffness) in the control design. The passive environment, as was demonstrated, ensures the coupled stability in the contact with passive target impedance. Introduced assumptions will simplify the contact stability analysis, but not restrict the generality of the considerations. Under some circumstances, the contact stability investigations can be extended to nearly-passive and active environments, as well as on non-linear robotic systems and other compliance control schemes (e.g. force and impedance/force control). For the sake of simplicity let suppose the following initial condition at the time t = 0, when the robot hits environment: for the robot motion
p (0) = p0 (0) = 0; pɺ (0 ) = xɺ (0) = xɺ0 (0) = v0 ; D0 =: p (0) = ɺɺ x (0 ) = ɺɺ x0 (0) = a0 ɺɺ
(3.88)
The above conditions presume an ideal position servo in the free-space with zero initial errors immediately before the impact
e(0) = eɺ(0) = ɺeɺ(0) = 0 Assume further that the nominal trajectory x0 (t) is a continuous and bounded ∗ ∗ function and that p0 → p0 as t increases, where p0 is the maximum ∗ penetration corresponding to the maximum nominal goal position x0 (see Fig. 3.19) ∗
p0 = x0∗ − xe
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Dynamics and Robust Control of Robot-Environment Interaction
As was manifested in the initial transition experiments, in the robot contact with a stable passive environment the actual penetration p and position deviation e, as well as the interaction force F, reach the equilibriums when t → ∞ respectively (3.30)-(3.31)
p∗ =
Kt Ke p0∗;e∗ = p0∗ Kt + Ke Kt + Ke
F∗ =
Ke Kt ∗ p0 Kt + Ke
(3.89)
These assumptions allow us to consider the contact stability as a classical equilibrium stability problem [30]. However, the basic problems concern the definition of conditions for testing the stability of the contact transition, as well as the relationship between contact stability and previously considered coupled system stability. The transition examples depicted in (Fig. 3.28) illustrate several specific contact/coupled stability cases: a) The transition is contact unstable since the contact is lost after first impact. The bumps and contact lost are exchanged periodically. Probably the conditions for coupled system stability also are not filled. b) The transition is unstable, however the system meets the coupled stability conditions. After initial lost of contact a stable equilibrium force has been achieved in the second attempt. The contact establishment in this case relies on nominal motion parameters (e.g. maximum penetration, velocity profile, maximum velocity, duration of acceleration and deceleration phases in contact, etc.). Dependent on initial impact conditions and nominal motion time history, the contact can be lost and a similar situation as in the case a) could occur. c) The transition is stable. However, the coupled robot/environment system is at the stability limit. These samples demonstrate a non-transparent dependence between contact transition and coupled system stability. The relationship between these stability types depends on the impedance control (i.e. selection of target parameters), as well as parameters of the environment. In an elastic environment the contact stability appears not to be critical, since the end-effector penetrates the environment and the coupled system stability is essential for the connected system behavior (e.g. the sample c in Fig. 3.28). However, in a stiff environment
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the contact transition stability is crucial for the contact establishment, even when the coupled stability is guaranteed. In realistic stiff industrial environments the contact stability indices seem to be stronger than those of the coupled system stability.
Fig. 3.28 Specific contact transition cases
From a pragmatic viewpoint, the contact transition can be considered stable, if the contact is not lost after the manipulator hits the environment. A stable contact transition (e.g. in Fig 3.21) can be characterized by one of the following features:
● positive force (after contact is detected), ● positive penetration of the manipulator end point into the environment, ● nonappearance of bumps, etc. As demonstrated in the performed transition experiments, the most critical issue in transition control is the initial impact against a stiff environment. Obviously a stable controller should ensure the passage through the transition phase with maintaining contact until all impact energy has been absorbed. A typical transition behavior in the penetration and deviation phase planes, ( p,pɺ ) and (e,eɺ) respectively, is sketched in (Fig. 3.29). By the transformations
p = p − p∗
(3.90)
e = e − e∗ ∗
∗
the origins of these planes are shifted to the equilibrium points p = e = 0 . The phase planes provide useful information for the analysis of contact stability. Therefore, before defining indices for the stable contact transitions, in the following example some specific transition cases will be analyzed using transition phase planes diagrams.
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Example 3.1: Contact transition analysis in phase planes The experiments 3.1-3 have demonstrated the dependence of the contact transition stability on the impedance and environment parameters. This example illustrates the relationship between nominal motion and contact stability. The aim is to show the influence of the nominal motion phase in which the robot hits the environment (e.g. acceleration phase, constant velocity or deceleration motion phase) on the contact transition performance, as well as to demonstrate the contact transition effects in the phase plane. For this purposes a simple SISO simulation implemented in SIMULINK (i.e. ROBOTICS) has been applied. The Manutec r3 robot parameters [31] have been used in the simulation.
Fig. 3.29 Contact transition phase plane diagrams
As proven in (Fig. 3.23), the performance of the improved impedance control allows a simple impedance model to be applied in the simulation and analysis of relevant transition effects accurately. The advantage of simulation for the
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experiments is that the influence of specific effects, such as computation control delay, nominal motion, or Coulomb friction, can be tested separately and under various conditions. In the following simulation experiments the time delay and friction effects were switched off. The same scenario as in the previous experiments (Experiment 3.2) has been employed in the simulation. The following parameters have been selected: Mt = 10 (kg), ξ t = 2.2 , Kt = 1500 and Ke = 60000 (N/m). The location of the environment has been changed in the experiments; thus, the impact is achieved in different phases of the common trapezoidal velocity profile. The results of experiments, involving the nominal motion (with depicted contact point), contact force, as well as p ( pɺ ) , p0 ( pɺ 0 ) and e(eɺ) phase diagrams, are presented in (Fig. 3.30-3.32).
Fig. 3.30 Acceleration phase impact
In the first experiment (Fig. 3.30) the contact is achieved in the acceleration phase of the nominal velocity profile. The impact velocity was slightly smaller than the maximal velocity v0 max = 0.025 (m/s). The transition was stable. The contact force exhibits similar behavior as in real experiments (Fig. 3.26). After
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initial impact, the transition is characterized by damped oscillations that disappear in the constant velocity phase. In this period damping and stiffness components mainly determine the force shape. As already mentioned, due to the almost stopped actual robot penetration into the environment by the impedance control, these components are proportional to the nominal penetration and nominal velocity respectively. The deceleration phase initiates a second transition process, which is terminated by achieving the steady state force (3.89). In the p ( pɺ ) phase diagrams, both transitions can be observed following the phase trajectory. It focuses on two points (Fig. 3.30). The first one corresponds to the “quasiequilibrium” of the initial transition process after impact. This equilibrium is characterized by a nearly linear force rise. In the phase plane this is manifested by a slowly increasing penetration p with a negligible penetration velocity pɺ (in a stiff environment). The second transition starts with nominal deceleration phase and approaches the stable focus defined by steady-state values (3.89).
Fig. 3.31 Constant-velocity phase impact
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These transition processes are extremely critical from the contact lost viewpoint. It is especially apparent in the second example in (Fig. 3.31). In this experiment the impact is realized in the constant velocity phase with a higher initial velocity. The nominal acceleration/deceleration values are selected to be significantly higher than in the first example in (Fig. 3.30), in order to amplify the transition effects. The critical points for the loss of contact correspond to the local force (i.e. penetration) minima of both damped transition oscillation cycles: after the impact and at the start of deceleration nominal motion phase, respectively. In the phase diagrams these points correspond to the intersections between the phase trajectory and ordinate ( pɺ = 0 axis) (see Fig. 3.29).
Fig. 3.32 Deceleration phase impact
In the third example the robot collides with the environment during the nominal deceleration phase (Fig. 3.32). Since in that case both impact and deceleration transition effects coincide, it is the most contact destabilizing transition situation. As can been see, the contact is lost. However, since the nominal motion (i.e. nominal penetration) progression is decelerating and after a short time period is stopped, in the next bump the contact is again established.
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Therefore, the consequence of the contact lost is not as dangerous as in an unstable contact during the first transition phase after impact (e.g. Fig. 3.25), in which the nominal penetration is increasing. However, for practical reasons it is desirable to tune the impedance control parameters to ensure a stable contact transition even in this worst-case situation. An example of unstable contact transition is presented in (Fig. 3.33). In this experiment the same contact transition parameters (contact point, impact velocity, target mass and stiffness, stiffness of the environment) as in the case shown in (Fig. 3.30) have been applied, except for the target damping ratio which is reduced to ξ t = 1,5 . This example again demonstrates the negative contact stability consequence of a decreasing target damping. It should be remarked that in spite of the simplified linear simulation model, the obtained behavior is quite similar to the unstable contact experiment shown in (Fig. 3.25).
Fig. 3.33 Unstable contact transition
Based on the above contact transition experiment and simulation analysis, we can now define the contact stability. As already mentioned, the contact stability
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analysis considers a unilateral (i.e. force) contact representing most the general and critical contact case. The bilateral contact, i.e. geometric form contact, which is in robotics practice commonly established by closing the gripper (grasping an object), is due to uncertainties usually realized by an initial unilateral contact between the gripper jaws and the environment. Definition 3.4 (Contact Transition Stability) The contact establishment between a robot and a passive environment at an initial location xe is said to be stable under the initial conditions D0 (3.88) if and only if the robot/environment system is coupled stable and one of the following equivalent statements is ensured 1. The robot end-point position penetration remains positive after the initial contact
p(t,D0 ) = x − xe > 0, t ≥ t 0
(3.91)
2. The interaction force is non-negative
F (t,D0 ) ≥ 0, t > t0
(3.92)
3. The position deviation is less than the nominal penetration
e(t,D0 ) < p 0 (t ), t > t 0
(3.93)
i.e.
e(t,D0 ) p0 (t )
=
p 0 (t ) − p(t,D0 ) p0 (t )
=
x0 (t ) − x(t,D0 ) x0 (t ) − xe
<1
(3.94)
For sake of simplicity in the above inequalities a basic contact situation has been assumed. In practice, however, for testing the sign of the penetration or the force it is needed to consider the location of the contact point w.r.t. a reference frame, i.e. the direction of the normal vector to the environmental surface. For example, the real positive penetration condition requests ( x − xe ) sign ( xe ) ≥ 0 . The first two criteria in the above definition are obvious, while the third one requires additional explanations. This requirement presumes the position deviation to be smaller or equal to the nominal penetration. In other words, this relation implies the actual end-effector position during a stable contact transition to always be located between the position of environment and the nominal position (Fig. 3.34). Since this contact stability condition is based on a simple
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geometric consideration it will be referred to as the geometric criterion. This criterion theoretically can be applied in the cases when the actual position overshoots the nominal one, i.e. when p > p0, which provides a negative position error e. However, in a contact between a reliable industrial robotic system with realistic stiff environment, the actual motion is, as already mentioned, nearly stopped by the resistant force and impedance control effects. Hence this case does not have practical relevance. The advantage of the geometric criterion (3.93)-(3.94) over the previous ones (3.91)-(3.92) is that it compares two time signals. For this purposes various norms can be applied
e ≤ p0
(3.95)
such as the 1-norm (l1) ∞
e 1 = ∫ e(t ) dt t0
2-norm (square root of the signal energy) 1
∞ 2 2 e 2 = ∫ e(t ) dt t 0 or ∞-norm ( l∞ )
e
∞
:= sup e(t ) t
The norm comparison offers possibilities to apply relatively simple and efficient control techniques for the contact stability analysis. In that case, however, the criterion (3.95) only ensures sufficient contact stability conditions, but not the necessary ones. Obviously, even when e ≤ p0 is filled there may exist time intervals in which e(t ) > p0 (t ) (see the example in Fig. 3.31). Consequently, the obtained contact stability indices might be conservative. Definition 3.4 supposes that the robot/environment system still meets the coupled stability conditions. From this viewpoint, the contact transition stability implies stronger conditions than the coupled stability. As will be demonstrated that is a quite realistic postulate.
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Impedance Control
Fig. 3.34 Geometric contact stability
The stability of the contact transition can be reasonably defined using the practical stability concept [32]. From a practical view the contact transition can be considered stable if the deviations of the actual position remain within certain bounds determined by the environmental surface. Practical stability generally does not require a stable equilibrium in the sense of Lyapunov. For the test of contact stability one defines a region X0 (t) around the nominal trajectory x0 (t), i.e. relative to the contact point a pre-defined region E (t) around p0 (t ) (Fig. 3.35). To guarantee the contact this region should be selected to ensure
{
[
E (t ) ≤ p0 (t ) . A finite time period T = t , t ∈ t0 , t1
]} and admissible initial
values region DI should also be selected.
Definition 3.5 (Practical Contact Transition Stability) The contact transition is said to be stable if ∀p (0 ) ∈ D0 , D0 ⊂ DI and ∀t ∈ T the deviation between the nominal penetration p0 (t ) and the actual penetration p (t ) , e(t ) = p0 (t ) − p (t ) , is constrained by E (t ) around p0 (t ) , i.e. e(t ) ∈ E (t ) . Definition 3.6 (Domains of contact stability) The sets of all initial values D0 and parameters Dt for which the contact transition is stable define the domains of contact stability.
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Dynamics and Robust Control of Robot-Environment Interaction
Fig. 3.35 Practical contact stability region
For the developed position based impedance control law the domain of stability in the parametric space is determined by the target impedance and
{
}
environment parameters Dt = M t , Bt , K t , K e ,⋯ . It is customary to use a normalized set of parameters such as
Dt = {M t , ξ t , ωt , κ,⋯}
(3.96)
3.8 Contact Stability Conditions In this section reliable explicit conditions and domains D0 and Dt under which the proposed impedance control law fulfils the specified impedance control tasks and ensures a stable contact transition will be determined. The main goal is to examine the parametric domains of the contact stability, and to benefit from the simple impedance control law and design (practically based on the selection of target impedance parameters and approaching velocity), rather than to develop complex transition control algorithms. It is desirable to keep the domains D0 and Dt as large as possible in order to meet the requirements of a broad class of contact tasks and to ensure robustness against environmental and robot uncertainties. Contact stability tests and conditions may be used to either check the stability of the specified control laws or to define procedures for the synthesis of the parameters. As will be demonstrated, the established contact stability conditions can easily be extended to other compliance control laws as well as non-linear interaction models.
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Impedance Control
3.8.1 Time domain analysis A good place to begin the analysis is the ideal deviation impedance model (3.29) realized by the improved control law (3.68) in each constrained direction (for the sake of simplicity the environment is presented as a stiffness)
(
)
2 2 2 eɺɺ + 2ξtωt eɺ + ωt + ωe e = ωe p0
(3.97)
where analogous to the target frequency, ωe represents the equivalent frequency of the environment
ωe =
Ke Mt
(3.98)
The relationship between the frequencies is determined by the stiffness ratio
κ = ωe ωt >> 1 It is assumed that the nominal trajectory p0 (t ) is a continuous, monotone increasing and bounded function, with initial conditions D0 (3.88) and the ∗ constant maximum value (maximum penetration) p0 (t , t ≥ T ) = p0 ∗ corresponding to the stationary nominal goal position x 0 and ɺɺ0 (t, t ≥ T ) = 0 . The target and environmental parameters are pɺ 0 (t, t ≥ T ) = p assumed to be constant. Under these circumstances, the position deviation e(t ) and the actual penetration p (t ) reach equilibrium as t → ∞
1 p0∗ 1+ κ κ e∗ = p0∗ 1+ κ p∗ =
(3.99)
The equilibrium of the second-order perturbed system (3.97) is asymptotically stable for ξt > 0 , which ensures negative real parts for characteristic equation eigenvalues
(
)
λ2 + 2ξtωt λ + ωt 2 + ωe 2 = 0 λ1, 2 = −ξtωt ± jωt 1 + κ − ξt 2 = −ξtωt ± jωd where
(3.100)
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Dynamics and Robust Control of Robot-Environment Interaction
ωd = ωt 1 + κ − ξt 2 Thereby it is assumed that ξt < 1 + κ which corresponds to the undercritical damping case. Increasing target damping is a common practical approach in the impedance control to stabilize contact between the robot and the environment. Basically the strategy to maintain a stable contact is to enlarge ξ t to achieve a critical or overcritical coupled system (3.97) [29], i.e. a dominant real pole of the coupled target-model/environment transfer function. That implies ξ t ≥ 1 + κ . However, the high damping gives the robot sluggish behavior. That is not desirable during contact establishment, since it generally produces large forces and long transition times. Therefore in our consideration we are looking for the minimum damping needed to ensure both stable contact and satisfactory robot performance. The stability of the coupled system (3.97) can be easily proven using the Liapunov method. For this purpose we utilize an equivalent model obtained by shifting the origins to the equilibrium (3.97) yielding
(
)
(
2 2 2 ∗ eɺɺ + 2ξtωt eɺ + ωt + ωe e = ωe p0 − p0
)
(3.101)
Choose a Liapunov function such as
V=
1 ɺ2 2 2 e + ωt + ωe e 2 ≥ 0 2
[ (
) ]
(3.102)
which is a positive definite function (except at the origin) representing total position deviation energy. Then
(
)
2 ∗ Vɺ = −2ξtωt eɺ 2 − ωe p0 − p0 eɺ ≤ 0
(3.103)
Considering typical contact transition behavior sketched in (Fig. 3.29), we can see that the deviation velocity eɺ = eɺ remains positive as the nominal motion ∗ increases ( p0 → p0 ), and takes relatively small negative values after the goal nominal position is reached. Hence the above condition requires a minimum amount of damping ξt > 0 to ensure coupled system stability. However, as demonstrated in experiments, that is not valid for contact stability. To obtain conditions ensuring contact stability by the Liapunov method, it is necessary to find a more reliable Liapunov function, which is generally a difficult problem. Estimation of contact transition stability measures (3.91)-(3.94) based on the Liapunov function (3.102) might be useless for the transition stability analysis. For the considered simple linear system (3.97), more accurate bounds can be obtained by solving the differential equation taking into account the initial
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Impedance Control
conditions (3.88). For the considered under-critical damping case ( ξt < 1 + κ ) the solution has the form [33]
e(t ) =
ωe 2 t p0 (ζ ) e −ξ ω (t −ζ ) sin ωd (t − ζ ) dζ ∫ ωd t t
(3.104)
t
0
As mentioned, the nominal penetration function p0 (t ) represents a common nominal motion profile. In industrial robotic systems p0 (t ) is an at least C2continuous function, having continuous second (acceleration profiles) or higher derivatives. Thus, we can apply partial integration upon (3.104), taking into account the initial conditions (3.88), which yields
e(t ) = +
2ξ t v κ pɺ 0 (t ) + 0 e −ξ ω (t −t ) sin[ωd (t − t0 ) − 2θ ] p 0 (t ) − ωd 1+ κ (1 + κ )ωt t t
0
−ξtωt (t −ζ ) ɺ ɺ ( ) ( ) p ζ e sin [ ω t − ζ − 2 θ ] d ζ d ωd t∫0 0 1
t
(3.105) where
sin θ =
1 + κ − ξt 1+ κ
2
; cos θ = −
θ = atan2 (sin θ , cos θ );
π 2
ξt 1+ κ
(3.106)
≤θ ≤π
The above equations provide a starting point to analyze nominal system transition behavior dependent on a nominal motion phase in which the robots impacts the environment (see Fig. 3.30-3.32). 3.8.1.1 Constant velocity phase contact Consider firstly the constant velocity phase impact (Fig. 3.31). Since the nominal acceleration is zero, i.e.
ɺpɺ0 (t ) = 0; pɺ 0 (t ) = v 0 = const; p (t ) = v0 (t − t0 ) it follows from (3.105)
(3.107)
336
e(t ) =
Dynamics and Robust Control of Robot-Environment Interaction
2v0ξ t v κ + 0 e −ξ ω (t −t ) sin[ωd (t − t0 ) − 2θ ] v0 (t − t0 ) − (1 + κ )ωt ωd 1+ κ 0
t t
(3.108)
Time-differentiation yields the position-error velocity
eɺ(t ) =
1+ κ v0 1 + e −ξtωt (t −t0 ) sin[ωd (t − t0 ) − θ ] 2 1+ κ 1 + κ − ξt
κ
(3.109)
The most critical point for the lost of contact C on the phase diagrams (Fig. 3.29) corresponds to the minimum penetration. This point can be obtained from the condition
pɺ (tC ) = 0
(3.110)
which is equivalent to
eɺ(tC ) = pɺ 0 (tC )
(3.111)
Substituting (3.107) and (3.109) in (3.111) and assuming t 0 = 0 for the sake of simplicity yields the transcendental equation
e−ξ t ω t t sin (ωd t − θ ) =
1 + κ − ξt
κ 1+ κ
2
=
sin θ
κ
(3.112)
The solution of this equation t = tC depends on the target impedance parameters ξ t , ωt and environment/target system stiffness ratio κ . Generally we can apply some known numerical methods to compute this solution. However, taking into account that in real applications κ is large (usually κ ≈ 50 …100 ), i.e. the term on the right side is small, it is possible relatively simply to obtain an approximate solution. For this purpose we can utilize the diagram in (Fig. 3.36) illustrating the graphical solution of the transcendental equation (3.112) for a typical transition case. Apparently for small sin θ κ the time period tC needed to reach the minimum penetration after initial impact is quite close to the second zerog of sin (ωd t − θ ) corresponding to
g The first zero
t = θ ωd
correspond to maximum penetration after impact (see Fig. 3.31).
Impedance Control
tC ≈
π +θ ωd
337
(3.113)
i.e. tC ≈ [(2k + 1)π + θ ] ωd (k = 1,2,…) for following minima along phase trajectory. In the example in (Fig. 3.36), the numerical solution of (3.112) is tC = 0.09279 , while the approximate solution according to (3.113) provides tC ≈ 0.09505 (s).
Fig. 3.36 Graphical solution of minimum penetration transcendental equation for constant velocity phase impact ( ξ t = 2, ωt = 8.7rad s , κ = 40 )
The contact stability conditions imply the following equivalent inequalities to be fulfilled
p (tC ) ≥ 0,
i.e.
e (tC ) p0 (tC )
≤1
(3.114)
Substituting (3.107)-(3.108) and (3.113) in (3.114) yields π +θ
e tan θ ≤
π +θ − 2 cosθ κ sin θ
(3.115)
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Dynamics and Robust Control of Robot-Environment Interaction
For the known stiffness ratio κ we can numerically solve the above transcendental equation in terms of the angle θ in order to obtain the parametric stability domain. The contact transition is stable for
θ ≥ θC min
(3.116)
where θ C min is a solution of the inequality (3.115). The minimum amount of target damping needed to guarantee a stable contact transition is then obtained from (3.106)
ξt ≥
1+ κ 1 + tan 2 θC min
(3.117)
The following observations are indicative for constant velocity contact transition stability analysis. •
In the considered ideal transition case the contact stability does not depend on the initial impact velocity v 0 nor the target frequency ωt .
•
The relevant parameters are only the target damping ξ t and stiffness ratios
κ.
Fig. 3.37 Contact transition stability bounds in the parametric space Dt (ξ t , κ ) (constant velocity case)
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339
Fig. 3.37 presents the bound of contact stability in the parameter space (the stability domain Dt) for the considered constant velocity phase transition. Obviously, as the stiffness ratio increases, i.e. as the environment becomes stiffer and the target system more elastic, a larger amount of damping is needed to stabilize the contact transition. 3.8.1.2 Constant acceleration/deceleration phase contact We will omit the mathematical considerations for this transition case (for more details see [34]). It is worth mentioning that the obtained contact stability model is significantly more complex than the previous model for the constant velocity transition (3.107)-(3.117). In the considered case the contact stability depends not only on the impedance and environmental parameters, but also on the initial contact conditions (impact velocity and acceleration). In a case where nominal acceleration/deceleration a 0 is very small, the contact transition behavior is similar to the constant velocity case. However, dynamical nominal motion changes during contact cause transition behavior to be highly shaped by the impact velocity and acceleration ( v 0 and a 0 ), as well as by the nominal motion profile.
Fig. 3.38 Influence of impact acceleration sign on contact transition stability 2 ( M t = 20 kg, ξt = 1.05 , K t = 1500 N/m, κ = 40 , v0 = 0.005 m/s, a 0 = ±0.03 m/ s )
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Dynamics and Robust Control of Robot-Environment Interaction
Thereby, as it was verified in [34], in the considered trapezoidal nominal profile the acceleration in the contact direction ( a 0 > 0 ), i.e. increasing nominal velocity, improves the contact stability, while the nominal motion deceleration after impact destabilizes the contact. That is illustrated in the example in (Fig. 3.38), where a transition at the stability limit ( p0 = e , i.e. p = 0 ) with constant impact velocity ( a 0 = 0 ), grows stable ( p0 > e ) when a 0 > 0 , i.e. instable ( p0 < e ) for a 0 < 0 . Solving the contact transition model in the considered acceleration/deceleration contact transition phase in terms of the system parameters is considerably more complex and involves significant numerical stability and solution separation problems in transcendental equations. Therefore, this model is reliable to use as an analysis tool, for testing the contact stability for a set of selected impedance and initial impact parameters, however not for synthesis purposes. This example shows that the time domain contact stability analysis and synthesis are, even for the simplified ideal second-order target impedance model, extremely complex and only efficient in specific cases, such as a transition with constant nominal velocity. Therefore, it is practical to apply concepts that simplify the contact transition stability analysis, providing a usable frame for the development of impedance control design procedures. As a rule, the price which is paid is conservativeness.
3.8.2 Passivity-based contact transition stability analysis Passivity theory provides a common technique for the stability analysis of both linear and non-linear control systems. As demonstrated in [35], passivity theory is closely related to fundamental stability analysis techniques, such as the smallgain theorem, Liapunov stability theory, L2-gains and H ∞ techniques. In the prior analysis passivity methods have been used to establish conditions for the coupled stability of a robot contacting an arbitrary passive environment −1 (Theorem 3.2). These conditions imply the realized target admittance s Gˆ t (s ) , describing the input/output relationship between the interaction force F and motion deviation eɺ to be a positive real matrix. Unfortunately, this result is obtained considering the coupled stability around the equilibrium point ( pɺ 0 (t ) = 0 ), and cannot be directly applied for the analysis of contact transition stability.
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Impedance Control
Likewise, it is possible to apply the passivity theorem, i.e. the concept of positive dynamic systems, to obtain the conditions ensuring contact stability. To solve the contact stability using the passivity approach, first one must relate this problem to a feedback system and input/output relationships. Fig. 3.7 represents basic linearized feedback systems describing the interaction of a robot under impedance control and a passive environment. The input in these systems, the nominal penetration p0 (t ) , is a monotone-increasing positive function
p0 (t,D0 ) = x0 − xe ≥ 0,
t ≥ t0
The contact stability criteria indicate the output interaction force F(t), i.e. the penetration p (t ) to be non-negative during transition (3.91)-(3.92)
p(t, D0 ) = x − xe ≥ 0,
t ≥ t0
Theorem 3.6 (Passivity and contact stability): A sufficient condition to ensure contact stability of a linearized robotic control system under impedance control, during transition from the free space to a unilateral contact with any passive environment, is that the feedback system with the input-output pair satisfy the passivity condition
{p , p} 0
t
∫ p(τ )
T
p0 (τ )dτ ≥ 0
(3.118)
o
Proof : Substituting p(t ) = p0 (t ) − e(t ) in the above inequality yields t
t
t
o
0
0
2 T T ∫ p(τ ) p0 (τ )dτ = ∫ p0 (τ ) dτ − ∫ e(τ ) p0 (τ )dτ ≥ 0
(3.119)
This inequality implies that the “nominal penetration power” pushing the robot in the direction of the environment should be greater than the “position deviation power”, which draws the robot actual position away from the nominal one to the contact surface (see Fig. 3.34). Obviously, if during transition
e(t ) ≤ p0 (t ) for t ≥ 0
(3.120)
then the contact is stable and (3.118) is satisfied. This theorem, however, defines sufficient conditions in a stable contact, rather than necessary prerequisites ensuring stable contact transition. In practice this means that though the integral inequalities (3.118)-(3.119) are satisfied, the
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Dynamics and Robust Control of Robot-Environment Interaction
contact could be lost for a short instant. Despite this limitation, the above theorem provides a quite practical result. Theorem 3.7 (Passivity-based contact stability criterion): A necessary and sufficient condition for the input-output pair {p0 ,p} to satisfy passivity (3.118)
[
]
is that the transfer function matrix I + Gˆ −1 (s )G (s ) t e
−1
be positive real.
Proof : For the considered linear interaction system and initial conditions (3.88), the actual penetration is t
p(t ) = ∫ K (τ , t ) p0 (τ )dτ
(3.121)
o
A necessary and sufficient condition for K (τ, t) = K (τ - t) to be a positive definite kernel is that its Laplace transform
[
]
∞
−1 −1 I + Gˆ t (s )Ge (s ) = ∫ K(t) e −st dt 0
be a positive real matrix of rational functions of the complex variable s = jω . Definition 3.3 provides conditions for the transfer matrix
[
]
−1 −1 G ( s ) = I + Gˆ t (s )Ge (s )
(3.122)
to be positive real. For a SISO system with
G (s ) =
Gt(s) Ge(s) + Gt(s)
these conditions imply
Gt ( jω) Re ≥0 Ge ( jω) + Gt ( jω ) Substituting: Ge ( jω ) = K e , and ωt =
(3.123)
Gt ( jω) = K t − M t ω2 + jωωt ,
Bt = 2ξt M t K t
K t M t in the above inequality provides
(K(
)(
)
− M t ωt 2 K )( e +e Kt −t M t ωt 2 +) Bt tω2
t t
((
2
))
= M t 2 ω 4 + M t 4ξ t 2 K t − 2 K t − K e ω 2 + K t ( K e + K t ) ≥ 0
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Impedance Control
A necessary and sufficient condition ensuring that the obtained biquadratic (quartic) trinomial is non-negative for any real ω is that the discriminant satisfies
( ) (κ( + 2 − 4ξ )) − 4ω (1 + κ ) ≤ 0
2
2
2
2
D = M t K e + 2 K t − 4ξ t K t − 4M t K t (K e + K t ) = ωt
4
2 2
t
4
t
which yields the contact stability criterion
ξt ≥
1 1+ κ −1 2
(
)
(3.124)
That is a simple, nevertheless very usable outcome for the synthesis of impedance control law, i.e. selection of target impedance parameters ensuring stable contact transition. As in the “exact-solution” (3.115)-(3.117) derived for the constant velocity phase transition, in the condition (3.124), obtained using passivity-based contact stability, the relevant parameters for the contact maintenance are only the target damping ξ t and stiffness ratio κ . Moreover, the amount of damping needed to stabilize the contact transition is again increasing with rising environmental-stiffness/target-stiffness ratios. The obtained minimum damping values (Fig. 3.39) indeed are higher than in the “exact solution”, however, more than two times smaller than in the so-called “dominant real-pole” criterion proposed in [29]. This condition imposes a critically damped 2 2 total impedance s + 2ξ t ωt s + ωt (1 + κ ) , i.e.
ξt ≥ 1 + κ
(3.125)
The effectiveness of the obtained contact stability criteria can easily be proven by a simulation using the ideal interaction model (3.97) with variable initial conditions ( v0 , a 0 ) and environmental stiffness ( κ ). It can be remarked that in almost all transition tests with realistic impact velocities and accelerations 2
(e.g. v0 < 0.1 m/s, a 0 < 1 m s ) even the “exact solution” (3.115)-(3.117) ensures a stable transition. However, in critical deceleration-phase transitions with a very small nominal penetration p 0 (i.e. goal position close to the environmental surface), a slightly larger amount of damping than provided by the passivity-based criterion (3.124) is still needed to prevent the lost of contact. Therefore this criterion can be used as “absolutely safe” for the ideal system. In the ideal case, assuming a diagonal nominal target model and a decoupled environment, the criterion (3.124) can be applied for the target impedance
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Dynamics and Robust Control of Robot-Environment Interaction
synthesis in the real six-dimensional task space. In other words, the target impedance parameters can be tuned along each C-frame direction to meet (3.124). In a more general case, the impedance control synthesis should be based
[
]
on ensuring that the transfer matrix I + Gˆ −1 (s )G (s ) t e defined “worst case” environment.
−1
is positive real for a
Fig. 3.39 Contact transition stability bounds
3.8.3 Robust transition stability - generalized contact stability As already pointed out, the established geometric contact transition stability criteria (3.91)-(3.94) offer the possibility to apply different norms for contact signals: p (t ) , p0 (t ) , and e(t ) , in order to test their relationships, i.e. the performance of the transition process.
Theorem 3.8 (Robust contact transition stability criterion): A sufficient condition for a stable contact transition of a linearized robotic control system under impedance control from the free space to a unilateral contact with any passive environment, is that the 2-norm/2-norm system gain of the feedback
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Impedance Control
{p0 ,e}, i.e. ∞ − norm of the corresponding −1 transfer function matrix [I + Ge −1 (s )Gt (s )] , is less than 1. system with the input-output pair
Proof : It is based on Theorem 3.1 (see Eq. 3.8) defining bounds for the 2-norm input/output gain. In the considered case the relationship between the nominal penetration and position deviation signals “energy” is limited by
e p0
[
2
≤ I + Ge
−1
(s )Gt (s )]
−1
2
(3.126)
∞
A stable contact transition is characterized by (3.94)
e(t ) < p 0 (t ) imposing ∞
∫ e (t )dt 2
e p0
2
= 2
0 ∞
[
≤ I + Ge
−1
(s )Gt (s )]
−1
2
∫ p (t )dt
<1
(3.127)
∞
0
0
Introducing the unstructured perturbations in the robot/environment interaction model (Fig. 3.10) and assuming the uncertainty in the form (3.42), the above condition becomes
e p0
[
≤ W (s ) I + G e
2 2
−1
(s )Gt (s )]
−1
≤1
(3.128)
∞
The obtained contact transition stability condition is the same as previously in the criterion for robust coupled stability (3.45)-(3.47) derived from Theorem 3.3. Practically the criterion (3.128) satisfies both robust coupled and contact transition stability.
Theorem 3.9 (Generalized contact stability): A sufficient condition for the stability of a linearized robotic control system under impedance control subject to contact with any passive environment, considering both the stability of the equilibrium and the stability of the contact transition process, is that the inputoutput pair {p0 ,e} transfer function satisfies the condition (3.128). Proof : The statement follows from Theorems 3.3 and 3.8.
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Dynamics and Robust Control of Robot-Environment Interaction
Generally in a MIMO interaction system the computation of the H ∞ operator norm of the {p0 ,e} system transfer function matrix (3.107) is complicated [36]. According to the definition this norm is obtained as the peak value of the principal gain σ {G ( jω )}
G
∞
:= sup σ {G (s )} = sup σ {G ( jω)} Re ( s )>0
ω
within a frequency interval of interest. The above relation represents a generalization of the maximum modulus theorem for matrix functions. In a SISO system, or in the case of an ideal diagonally decoupled interaction model, the principal gain is equal to the magnitude of the {p 0 , e} transfer function
σ {G ( jω)} = G ( jω) =
Ge ( jω) Ge ( jω) + Gt ( jω)
(3.129)
2 Substituting Ge ( jω ) = K e and Gt ( jω) = K t − M t ω + jωωt yields
Ke K e + Kt − M t ω2 + jωBt
G ( jω) = =
(K
Ke e + Kt − M t ω
)
2 2
+ ω2 Bt 2
(
)
K e + K t − M t ω2 − jωBt
Thus the robust contact stability criterion becomes
Ke
G ( jω ) =
(K
e
+ Kt − M t ω
Substituting Bt = 2ξt M t K t and ωt =
(
)
2 2
2
+ ω Bt
2
≤1
K t M t yields
)
ω 4 − 2ω t 1 + κ − 2ξ t ω 2 + ω t (2κ + 1) ≥ 0 2
2
(3.130)
4
(3.131)
The obtained biquadratic (quartic) trinomial will be non-negative for all real frequencies of interest ω ∈ (0, ∞] if the discriminant is non-positive 4
(
D = 4ω t 1 + κ − 2ξ t which implies
)
2 2
− 4ω t (2κ + 1) ≤ 0 4
Impedance Control
347
2
1 + κ − 2ξ t ≤ 1 + 2κ from which the robust contact stability condition
ξt ≥
1 2
( 1 + 2κ − 1)
(3.132)
follows. As in the previously derived contact stability conditions (time-domain “exact solution”, passivity-based positive real requirement (3.124) and dominant real-pole limit (3.125)), in the robust-stability infinity-norm criterion (3.132) the contact stability bound is only determined by the target damping and stiffness ratios. The minimum target damping values required by the bound (3.132) for various target stiffness ratios κ are compared in (Fig. 3.40) with the stability limits prescribed by prior contact stability criteria.
3.8.4 Equivalence of robust- and passivity-based contact stability As may be remarked from (Fig. 3.40), the infinity-norm boundary line is placed between the positive-real and dominant real-pole bounds. A further important observation concerns similar forms of the expressions for the contact stability conditions (3.124) and (3.132) obtained in the passivity-based and robuststability methods respectively. Practically, the positive-real stability limit provides the same stability bound as the robust-stability condition when the stiffness ratio, i.e. the stiffness of the environment, is taken to be twice the real stiffness.
Fig. 3.40 Comparison of contact transition stability bounds
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Dynamics and Robust Control of Robot-Environment Interaction
This relationship is the result of the equivalence between positive-realness, the small-gain theorem and the H ∞ norm. The principal basis for this correlation is the transformation referred to as the bilinear sector transform [37], or in a more general form as Cayley transformation [38]. This transformation can convert a positive real problem concerning a system within the sector [0, ∞ ) into small-gain i.e. H ∞ problem relating systems inside the sector [− 1,1] , or vice versa. Definition 3.7 (Sector transformation) Let T (s) and G (s) denote the systems lying within sectors [− 1,1] and [0, ∞ ) respectively. Then the bilinear sector transformation is defined by −1
T(s) = sec tf (G(s)): = (I − G(s))(I + G(s))
(3.133)
We define further
herm ( A(s)) =
1 A(s) + A∗ (s) ; 2
(
)
Im ( A(s)) =
1 A(s) − A∗ (s) ; 2
(
)
where the asterisk means conjugate. When A( s ) ∈ [− 1,1], then A( s )
(3.134) ∞
≤ 1 , and
when A(s ) ∈ sector [0, ∞ ) , then herm ( A ( s )) ≥ 0 for s = jω, ∀ω. Thus, using the above definitions and notation, we have the following theorem Theorem 3.10 (Passivity and H ∞ -norm) For the sytems related by (3.134)
G(s ) ∞ ≤ 1 if and only if herm (T ( s )) ≥ 0 . Proof : See [37]. Hence, the following theorem expresses the equivalence between passivitybased contact stability and the infinity norm condition used in the robust contact stability. Theorem 3.11 (Passivity-based and robust contact stability) The input-output pair {p0 ,p} will satisfy passivity, i.e. the corresponding transfer function matrix will satisfy −1 herm I + Gˆ t −1 ( s )Ge ( s ) ≥ 0
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Impedance Control
which ensures the stable contact transition, if and only if the equivalent system {p0 ,e } describing the interaction between the same target impedance and a “halved” environment satisfies the robust contact stability −1
I + 2Ge −1 ( s ) Gˆ t ( s ) ≤ 1 Proof : It is relatively easy to confirm that
I + Gˆ t −1 ( s ) Ge ( s )
−1
−1 = sec tf I + 2Ge −1 ( s ) Gˆ t ( s )
In the SISO contact stable transition case the ensures
{p0 , e}
(3.135)
input/output system
Ge ( jω) ≤1 Ge ( jω) + Gt ( jω)
(3.136)
for the target damping values
ξt ≥
1 1 + 2κ − 1 2
(
)
while the associated sector-transformed system T (s) ∈ [0, ∞ ) satisfies
Gt ( jω ) herm (T ( jω)) = herm Gt ( jω) + 2Ge ( jω)
(3.137)
The passivity (i.e. positivity) based contact stability Theorems 3.6 and 3.7 imply the transfer function relating {p0 , p} input/output system to fulfil
G t ( jω) herm ( ) ( ) G j ω + G j ω e t
⇔
G e ( jω) 2 ≤1 G e ( jω) 2 + G t ( jω)
(3.138)
In the parametric space this yields the condition (3.103)
ξt ≥
1 1+ κ −1 2
(
)
Basically Theorem 3.11 defines the equivalence between robust and passivity-based contact transition stability. The interaction system considered in the robust contact stability relates a sector-transformed passive system that has the same form as the coupled system considered in the passivity-based contact stability theory. However, this sector-transformed system involves the double
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Dynamics and Robust Control of Robot-Environment Interaction
size of the environmental parameters (3.138). That means that the robust contact stability and the passivity based stability are essentially correspondent and have the same system-theoretical background. However, the robust contact stability permits 100% of environmental uncertainties and therefore is more conservative than the passivity-based stability (see Fig. 3.40). Finally, it should be pointed out that derived robust and passivity-based contact stability conditions are based on the sufficient stability criteria, but not the necessary ones. That means that both stability conditions are satisfied in a stable contact transition; however, the accomplishment of these conditions does not guarantee a stable contact. According to (3.91)-(3.94) the contact stability is defined in terms of the relationships between instantaneous time signals characterizing the contact transition (p0, p and e). However, the established stability criteria in Theorems 3.6-9 are based on the relationships between the 2norms of these signals. The squares of these norms define the signals energy obtained by integrating the instantaneous signal power over time (see 3.95). Consequently, though the contact stability norm relationships have been fulfilled over the specified transition time, the contact could be lost in instantaneous interval when the stability relationship between direct signals is momentarily violated as demonstrated in (Fig. 3.33). Therefore, a practical way to test the applicability and reliability of sufficient contact stability criteria is to test them experimentally.
Experiment 3.4:
Preliminary tests of contact stability bounds
For the testing of the established contact transition stability conditions, initial impedance control trials with the improved impedance law, presented in the Experiment 3.2 (Fig. 3.25-3.26), have been used. In each experiment the transition has been characterized by analyzing the force component in the contact direction. Dependent on the minimum force peak, which otherwise occurs in the first phase immediately after impact (Fig. 3.25), the transition process has been evaluated as: instable (contact is lost, e.g. Fig. 3.25-a), at stability limit (e.g. Fig. 3.25-d), or stable (e.g. Fig. 3.25-b, c). Each experiment is presented as a point in (κ , ξt ) diagram specifying the contact transition of a SISO system. The results are presented in (Fig. 3.40), which also depicts theoretical stability bounds for different criteria. Thereby, the following symbols have been used to characterize the transition experiments in (Fig. 3.40): x
-unstable transition;
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o +
351
-at (closely to) stability limit; -stable transition;
Considering the fitting of the experimental results by theoretical stability criteria bound lines, it may be concluded that the most reliable contact stability limit provides an infinity-norm based criterion, i.e. robust contact stability (3.132).
3.9 Influence of Non-Linear Effects on Contact-Stability Indeed the result of the last experiment is surprising, since according to the above discussion robust contact stability limit is expected to be more conservative than passivity-based and exact solutions contact stability bounds. Obviously the experiment differs from the simulation tests based on the ideal target impedance model (Fig. 3.30-3.33), which have proved the theoretical transition stability results. Hence, it may be concluded that real systems involve additional effects that were not considered in the previous analysis. Both theoretical analysis and simple simulation tests of non-linear effects in the coupled impedance model indicate the relevance of Coulomb’s friction, quantization and roundoff, as well as control time lag effects.
3.9.1 Coulomb’s friction The influence of the Coulomb friction on the contact transition process can be analyzed by introducing the equivalent Cartesian friction in the considered impedance control models. For example, substituting the impedance control law (3.64) into the error-model (3.13) and introducing the friction disturbance force yields
e = Gt
−1
(s )F + S p (s )x0 + S p (s )Gs (s )(F + Fc )
(3.139)
where Fc is Cartesian Coulomb friction force. From the above equation we can see that the Coulomb friction has the equivalent effect on the transition process as the perturbation contact force. In industrial robotic systems with very stiff position controller, i.e. very small sensitivity transfer function matrix Sp(s) within the frequency interval of interest, the perturbation friction effect is significantly diminished. Moreover, considering the motion resistant effect of the Coulomb friction, it may be stated that Fc supports the end-effector to remain
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Dynamics and Robust Control of Robot-Environment Interaction
in contact, rather than to destabilize the contact establishment. Thus, the stability results obtained with the ideal model are on the safe side. The perturbation effects of non-linear Coulomb friction are crucial in the steady-state behavior causing limit cycle oscillations, which can be observed in the experiments (Fig. 3.25). The quantization and round-off phenomena have similar effects on the feedback system performance. These effects are caused by signal conversion in analog-to-digital and digital-to-analog converters and floating-point computations. Generally in modern control system these effects give raise to relative small errors [39]. However, in contact with a stiff environment, these effects may become significant and could jeopardize steady-state behavior. Therefore these effects will be considered separately below.
3.9.2 Control lags and sampling effects A control delay in real sampling-data control system exhibits quite the opposite effect on the transition process. In the considered interaction process and impedance control system the presence of a delay time mainly is caused by the retardation between input control signals and control reactions. Generally real digital control systems operate in a continuous framework, but some signals are sampled in certain time intervals in order to be processed by digital computers operating at fixed rates. The most critical delay from the view of the stability of contact transition process concerns the tardiness of the contact force information. An extensive force sensor processing computation is needed to obtain instrumental virtual contact force information in a selected compliance, i.e. task-frame, to be used in the impedance control. This processing includes: filtering, compensation for offset, coupling and payload, as well as transformation in different frames. The other major cause of the retarded contact force information is the structure of the selected implicit position control system (Fig. 3.4). Basically, in this scheme the sampling period of the outer impedance control is limited by the sampling rate of the internal position controller. The sampled-data (SD) model of the considered impedance controlled interaction system in (Fig. 3.41) involves the plants (the robot mechanical part and the environment, i.e. contact process) operating in continuous time and the controllers (position and impedance) operating in discrete time. The control model consists of samplers (A/D converters), discrete-time controllers implemented in a digital computer and holding elements, which support D/A
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353
conversion of control signals by holding them constant over the sampling intervals. The unilateral delay is presented with a nonlinear unilateral-limiter block. The symbol u ⊗ denotes the sampled representation of the signal u at the sampling instant. The blocks in this figure represent various control tasks and sampled-data computation effects, such as the reading and processing of force information, computation of position and impedance control laws, process and control monitoring, communication with the operator, etc. These functions are usually realized in several control tasks running at various sampling time intervals, including different control delays. Such complex multirate sampling systems, however, are time-varying even for a time-invariant controlled plant, and quite difficult to analyze. Therefore, usually various simplifications are needed to consider dominant sampling-time effects. Generally, the presence of a controldelay in feedback system may cause the self-exciting oscillations and even the instability of the system [40, 33, 41]. As already mentioned, in high dynamic processes, such as the contact transition process, the control delay appears to be especially critical.
Fig. 3.41 Sampled-time impedance control system
There are essentially three approaches to design a digital controller [42]: i) analog design and digital implementation,
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Dynamics and Robust Control of Robot-Environment Interaction
ii) iii)
plant discretization and design in discrete time, direct sampling-data control design.
Each of these approaches has specific advantages and limitations. The advantage of the first (most common) approach is that the design is performed in continuous time, where the performance specification is most obvious. However, this method provides good performance only for very small sampling periods. The problems with the second approach is related to discrete-time performance specifications, as well as control redesign when the sampling time has to be changed. The third approach considers accurately the performance of SD systems; however, the design is more complicated because SD systems are time varying. In the following sections the effects of a force delay and sampling-data in the impedance control contact transition will be analyzed. Thereby, the simple and reliable contact transition analysis/synthesis approach based on the passivity and robust control design will be pursued and extended to delayed and SD systems. 3.9.2.1 Ideal target system with force delay The simplest model of a time-delayed control system assumes that all control signals are continuous (i.e. very fast sampled signals, with a rate T < 1 ms) except for the delayed force information. The approximation of the force aftereffect in a pure stiff environment is based on the interaction force model that depends on penetration at some preceding time instance. Introduce the delayed contact force
F (t − τ ) = K e p (t − τ ) = K e [ p0 (t − τ ) − e(t − τ )]
(3.139a)
where τ is the time delay. Assuming p0 (t ) to be a monotone, relatively slowly increasing function during transition, we can write p0 (t − τ ) ≈ p0 (t ) without loss of generality and
F (t − τ ) ≈ K e [ p0 (t ) − e(t − τ )] Moreover, it is relatively easy to show that this assumption does not affect the passivity and infinity norm of the corresponding interaction transfer function, thus providing a suitable model for the comparison of the time-domain stability conditions with those obtained by passivity-based and robust control approaches. Substituting the above expression into the ideal target model system (3.17)
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355
yields the linear difference-differential equation (i.e. differential equation with retarded, or deviating argument)
ɺeɺ(t ) + 2ξ t ωt eɺ(t ) + ωt 2e(t ) + ωe 2e(t − τ ) = ωe 2 p0 (t )
(3.140)
with constant coefficients and constant deviating argument. Consider first the coupled stability problem around the equilibrium point. Similarly to the linear non-retarded systems, the coupled stability analysis is based on an investigation of the location of the roots of the corresponding characteristic quasipolynomial [33, 41]
φ ( z ) = z 2 + 2ξtωt z + ωt 2 + ωe 2e −τz = 0
(3.141)
A necessary and sufficient condition for the asymptotic stability of the solution of the linear stationary equation with deviating argument is negativity of the real parts of all the roots of the characteristic quasipolynomial (3.141). However, φ ( z ) has an infinite set of roots, the computation of which is relatively complex. Therefore, to check the stability several practical tests of negativity of the real part of all roots of the quasipolynomial have been developed (see [41]. The method of D-partitions determines the regions of asymptotic stability in the parametric space by separating the quasipolynomial coefficients values into intervals providing at least one zero on the imaginary axis. Since the case z = 0 does not provide a reliable parametric solution 2 2 ( ωt + ωe ≠ 0 ), we should look for pure imaginary roots (at the stability limit) by substituting z = iy ( y ∈ ℜ > 0 ) into (3.141), which yields 2
2
− y 2 + 2ξtωt iy + ωt + ωe (cos τy − i sin τy ) = 0 or the system of transcendental equations
a = 2ξtωt = 2
2
sin τy 2 ωe y
(3.142)
2
b = ωt = y − ωe cos τy The above equations define D-partitions boundaries in the parametric plane 2 2ξtωt , ωt . The typical form of this D-partition is sketched in (Fig. 3.42). A
(
)
reliable part of this boundary corresponding to the target frequency parameter ωt ≤ 100 (rad/s) is sketched by the bold line. It is relatively easy to prove that the first transcendental equation
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Dynamics and Robust Control of Robot-Environment Interaction 2
2ξtωt y = ωe sin τy does not provide real solutions for y representing the intersection between the 2
straight line 2ξtωt y crossing the origin and the sine functions ωe sin τy for
ξt >
τωe 2 2ωt
(3.143)
Fig. 3.42 D-partition boundaries ( τ = 0.008 sec, K e = 60000 N/m, M t = 10 kg) 2
As y → 0 the D-boundary line approaches the point (τωe ,0) . In the example in (Fig. 3.42) the above inequality defines the region of the parametric 2 plane on the right side from the line 2ξtωt = τωe = 48 As can been seen from (Fig. 3.42), within this region the quasipolynomial φ ( z ) has no pure imaginary roots, i.e. above this region only conjugate complex roots z = x + iy are possible. In order to check if this region is the region of asymptotic stability
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Impedance Control
( x < 0 ) we should check the sign of the real part of the root x in the neighborhood of the D-boundary. Based on (3.141)-(3.142) it can be written
∂φ 2 2 dz = 2 z + a − ω e τe −τz dz = 2 x + 2iy + a − ω e τ (cosτy − i sin τy ) ( dx + iy ) ∂z ∂φ ∂φ ∂φ dz = − da − db = − zda − db = − ( x + iy ) da − db ∂z ∂a ∂b
(
)
[
]
from which the differential of the real part of the root is
(x + iy ) da + db dx = − Re 2 2 x + 2iy + a − ω e τ (cos τy − i sin τy ) Usually it is sufficient to test the differential dx upon considering the variation of only one parameter whose change guarantees passage across the examined D-partition. Assuming across the D-boundary: x = 0 and b = const , yields
dx = −
2 y + ω e 2τ sin τ y
(
a − ω e 2τ cosτ y
2
) (
+ 2 y + ω e 2τ sin τ y 2
)
2
da
Considering that for y > 0 and also 2 y + ωe τ sin τy > 0 , it may be concluded that the sign of dx is opposite the sign of da . Hence, the directions of increasing value of the parameter a = 2ξ t ωt relative to the D-partition boundary determine the regions of asymptotic stability in the parametric plane (Fig. 3.42). Based on the above analysis it can also be concluded that the condition (3.143) (see Fig. 3.42) is necessary and sufficient to ensure the coupled stability of the system (3.140) describing the interaction between an ideal targetimpedance and a pure stiffness environment including delayed force feedback information. Unlike the interaction without retardation (3.103) in which ξ t > 0 was sufficient to ensure the coupled stability, the control lags require a much more complex stability condition (3.143) to be guaranteed. In this case the coupled stability depends on the time delay, environmental stiffness and target impedance frequency parameters. As with the time-domain contact transition stability analysis of an ideal coupled system without retardation (3.97), the goal is to establish the parametric
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Dynamics and Robust Control of Robot-Environment Interaction
{
}
domain of stability Dt = M t , ξ t , ωt , κ ,⋯ of the considered system (3.140) which ensures the geometric contact stability criteria (3.91)-(3.95). The stability analysis of differential equations with delayed argument generally is very complex (see [33, 40, 41]). However, in some specific second-order delayed systems [43] it is possible to establish the equivalence with the systems without time lags. That makes it possible to utilize previous contact stability results. Consider the initial value problem (IVP)
ɺeɺ(t ) + 2ξ t ωt eɺ(t ) + ωt 2e(t ) + ωe 2e(t − τ ) = ωe 2 p0 (t )
( )
( )
e 0 − = 0; eɺ 0 − = 0
(3.144)
In the general case for arbitrary initial functions e(t − τ ) ≡ φ (t − τ ) when t − τ < 0 , the solution space of the above type of equations is, according to the conditions for the existence and uniqueness of solutions [43], infinitedimensional. However, for a fixed initial function (3.144) the space of solutions is two-dimensional, which is also the case for equations without delay. Then the following theorems define the equivalence between differential equations of the second order with and without delayed argument [43]. Theorem 3.12 (Common fundamental systems of solutions of homogenous system) Let the functions u1 (t ) and u2 (t ) belong to the two-dimensional space
L2φ of linearly independent solutions of the homogenous equation with delayed argument (fundamental system of solutions) 2 2 eɺɺ(t ) + 2ξ t ωt eɺ(t ) + ωt e(t ) + ωe e(t − τ ) = 0
( )
( )
e 0 − = 0; eɺ 0 − = 0
(3.145)
Then there exists a unique differential equation without delay of the form
ɺeɺ(t ) + peɺ(t ) + qe(t ) = 0 for which u1 (t ) and u2 (t ) also form a fundamental system of solution. Theorem 3.13 (Common fundamental systems of solutions of non-homogenous 2 system) Let u1 (t ), u 2 (t ) ∈ Lφ be a fundamental system of solutions of the IVP
(3.145), and let v(t ) be a particular solution of (3.144). Then there exists a function
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Impedance Control
ψ (t ) = vɺɺ(t ) + pvɺ(t ) + qv(t ) such that the solution of non-homogenous equation without delay
eɺɺ(t ) + peɺ(t ) + qe(t ) = ψ (t )
( )
(3.146)
( )
e 0 − = 0; eɺ 0 − = 0
coincides with the general solution of the equivalent delayed system (3.144).
Proof : It is based on basic properties of second-order differential equations with delayed arguments (see [43]) and well-known theorems of fundamental systems for linear differential equations without delay. The equivalence between second-order systems with and without delay based on the above theorems provides useful tools to enlarge the result of the contact stability analysis to the case involving time lags. Using well-known formulas for representing the coefficient of a linear homogenous differential equation without z1,2 t
x ±iy
= e , see for delay in term of the fundamental system of solutions ( u1, 2 = e example [44]), we can obtain the relationship between the coefficients of the equivalent systems p = −(z1 + z 2 ) = 2ξ t ωt + ωe
2
D(u1,u 2 ,τ ) W (u1,u2 )
(3.147)
W (u1,u 2 ,τ ) q = z1 z2 = ωt + ωe W (u1,u 2 ) 2
2
where the Wronskian W and the determinant D have the forms
D(u1,u2 ,τ ) = W (u1,u 2 ) =
u1 (t ) u 2 (t ) u1 (t − τ ) u2 (t − τ )
u (t − τ ) u2 (t − τ ) u1 (t ) u2 (t ) ; W (u1,u2 ,τ ) = 1 ɺu1 (t ) uɺ2 (t ) uɺ1 (t ) uɺ2 (t )
Without loss of generality we can assume coefficients of the form 2
2
2
p = 2ξtωt ; q = ωt + ωe = ωt (1 + κ )
(3.148)
where ξ t and κ represent the damping and stiffness ratios of the equivalent system without delay. Substituting in the above equations:
u1,2 = e
z 1,2 t
− ξ ω ± iω 1+ κ − ξ 2 t t t t t
= e( x ± iy )t = e
= e −ξ t ω t t ± iω d t
(3.149)
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Dynamics and Robust Control of Robot-Environment Interaction
yielding 2
ω p = 2ξ t ωt − e e ξt ωt τ sin (ωd τ ) ωd 2
q = ωt + ω e
1+ κ
2
1 + κ − ξt
2
e ξt ωt τ sin (ωd τ + θ )
(3.150)
where
cosθ = −
ξt
;sin θ =
1+ κ
1 + κ − ξt
2
1+ κ
;ωd = ωt 1 + κ − ξt
2
From (3.148)-(3.150) it follows for the relationship between the target coefficients of the equivalent systems
ξt = ξ t − κ=
κ 2 1 + κ − ξt
κ 1+ κ 1 + κ − ξt
2
2
eξ t ωtτ sin (ωdτ )
eξ t ωtτ sin (ωdτ + θ )
or explicitly
ξt = ξt + κ =κ
sin (ω d τ ) 2 1 + κ sin (ω d τ + θ )
κ
sin θ
(3.151)
e −ξ tωtτ
sin (ω dτ + θ )
Assuming that the nonhomogenous equivalent systems with and without delay (3.145) and (3.146) respectively have the same particular solution, it is relatively easy to derive the relation between the nonhomogenous parts applying the method of variation of the constants
u 1 (ζ ) 2
ψ (t ) = ω e p 0 (t ) + ω e
2
t
∫τ
t−
t −τ ≤ ζ ≤ t
u 2 (ζ )
u 1 (t − τ ) u 2 (t − τ ) W (u 1 (ζ ), u 2 (ζ ))
ψ (ζ ) dζ
(3.152)
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Impedance Control
Substituting (3.149) in the above equation provides an integral equation of Volterra type of the second kind
ψ (t ) = ω e 2 p0 (t ) +
ωe 2 ωd
t
∫τe
−ξ t ωt (t −τ −ζ )
sin ω d (t − τ − ζ )ψ (ζ ) dζ
(3.153)
t−
For the continuous and bounded kernel (3.143) this equation has a unique and continuous solution that may be obtained by the method of iteration (successive approximations). Furthermore, it can be written
ψ (t ) = ωe 2 p0 (t ) where the equivalent nominal motion function p0 (t ) has the form
p0 (t ) = p0 (t ) +
1
ωd
t
∫τe
−ξtωt (t −τ −ζ
)
sin ωd (t − τ − ζ ) p0 (ζ )dζ
(3.154)
t−
The equivalence with the system without retardation is now completely defined. The established analogy allows us to obtain solutions of the considered IVP’s with retardation (3.144) and (3.146) using the previous results on the contact transition in the system without time lags (3.97) and (3.104)-(3.106) respectively. The solution of the equivalent IVP without retardation has the known form
ωe 2 t −ξ ω (t −ζ ) e(t ) = e sin ωd (t − ζ ) p0 (ζ )dζ ωd ∫0 t t
or
e(t ) = +
2ξt v κ pɺ 0 (t ) + 0 e −ξ ω (t −t ) sin [ωd (t − t0 ) − 2θ ] p0 (t ) − ωd 1+ κ (1 + κ )ωt t t
0
ɺpɺ0 (ζ )e −ξtωt (t −ζ ) sin [ωd (t − ζ ) − 2θ ] dζ ωd t∫0 1
t
(3.155) Together with the geometric contact transition stability condition
e(t ) ≤1 p0 (t )
(3.156)
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Dynamics and Robust Control of Robot-Environment Interaction
the derived model (3.150)-(3.155) defines a mathematical frame that can be applied for the impedance control synthesis in a system with retardation, i.e. for
{
}
computing the domain Dt = M t , ξ t , ωt , κ ,⋯ of target impedance parameters that ensures a stable contact transition. However, the control design algorithms based on iterative numerical solving of the systems of integral and transcendental equations appear to be extremely complex and time consuming. A practical solution can be obtained by neglecting the difference between nominal motion functions p0 (t ) and p0 (t ) , corresponding to the systems with and without retardation respectively. Usually this difference is for a considerable delay very small. In general, the difference between non-homogenous functions of the equivalent systems with and without retardation (3.144) and (3.146) is bounded by [43]
sup ψ (t ) − f (t ) ≤ f 0 K 0 τe K0τ [0,t )
where
(3.157)
f (t ) denotes the non-homogenous part of the retarded system 2
( f (t ) = ωe p0 (t ) ), f 0 = sup f (t ) , K (t , ζ ) is the kernel of the integral equation (3.152) and K 0 = sup K (t,ζ ) , Substituting (3.152)-(3.153) in (3.157) yields
K0 = ω e2
sin ω d τ
ωd
eξtωtτ
and
sup p0 (t ) − p0 (t ) ≤ [0,t )
1 ∗ p0 K 0 τe K0 τ 2 ωe
In a “worst-case” transition with: κ = 100 , τ = 0.01 (s), M t = 10 (kg) and ∗ K t = 1500 (N/m) the above bound is about 0.0005 p0 . This means that the nominal motion function can be applied in the system without time lag without loss of accuracy. Then, the algorithm runs as follows. Algorithm 3.1 Computation of the contact stability domain Dt (ξt , κ ) for a SISO delayed system Step 1: For known κ , ωt and τ , and assuming the most common constant velocity phase impact, one computes the equivalent stiffness ratio κ and damping ratio ξ t at the stability limit for the equivalent system without delay.
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363
These parameters are obtained by solving the system of transcendental equations describing the equivalence between delayed and non-delayed interaction systems (3.151) and contact stability bound for the constant velocity transition (3.115), respectively
κ
sin θ
e − ξ t ω tτ − κ = 0
sin (ω τ + θ ) d
e(π +θ ) tan θ + 2 cosθ −
π +θ =0 κ sin θ
Step 2: The damping ratio ξ t of the original system with delay, which ensures the same stable contact transition (i.e the same basis solution of IVP) as in the equivalent system without delay is then computed from (3.151)
ξt = ξt +
κ
sin (ω dτ )
2 1 + κ sin (ω dτ + θ )
The contact transition stability bounds obtained by the above algorithm for various values of the computation time lags τ are presented in (Fig. 3.43). The stability limit defined by the robust contact stability, which in the previous experiment was identified as the most reliable one (see Fig. 3.40), has also been sketched. Obviously, compared to the ideal case τ = 0 , also shown in the figure, the control delay requires significantly higher damping ratios in order to ensure a stable transition. These curve diagrams also give the explanation why the most reliable stability results in the experiments have been provided by the robust stability criterion that is more conservative than the other criteria. Namely, in the performed experiments the time delay in the implemented control system was approximately τ ≈ 0.008 (s). For the stiffness ratios κ applied in the Experiment 3.4 ( 40 ≤ κ ≤ 100 , see Fig. 3.40), the robust stability bound, computed for the system without delay, quite well matches the “exact” stability limit obtained for the ideal time-domain model with retarded force signal and corresponding to the time lag τ = 0.008 (s).
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Dynamics and Robust Control of Robot-Environment Interaction
Fig. 3.43 Contact stability bounds of the retarded systems: time-domain analysis (“exact solution”) ( M t = 10 kg, K t = 1500 N/m, ωt = 12.25 rad/ s )
Fig. 3.44 Dependence of minimum damping on target frequency – time domain delayed system analysis ( M t = 152 − 0.6 kg, K t = 1500 N/m, ωt = 3.14 − 50.26 rad/ s )
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Impedance Control
Practically this experiment also proves the stability analysis of the retarded system based on the linear impedance model (3.144) and its corresponding system without control delay (3.146). It is important to remark that unlike the contact stability of the ideal coupled system, the stability bounds of the delayed system depends on the target frequency ω t . An increasing target frequency for a constant time lag and stiffness ratio requires even higher values of the damping ratio to guarantee stable transition. This relationship using the considered transition experiment example is illustrated on (Fig. 3.44). Numerical determination of the stability limits based on the Algorithm 3.1, however, becomes very sensitive to the selection of the initial values for higher target frequencies. 3.9.2.2 Robust and passivity-based contact stability of discrete-time system In spite of the relative simple analytical approach and resulting reliable contact stability bounds, target impedance system synthesis based on the Algorithm 3.1 has two main limitations. The first one concerns the numerical problems in solving the system of transcendental equations describing the relationship between equivalent systems with and without delay. Second, it is difficult to generalize this closed-form model approach to MIMO or sampled-data impedance digital control systems (Fig. 3.41), as well as to include modeling uncertainties. As still demonstrated above, the H ∞ induced norm, describing the maximum energy gain measure, is quite useful in analyzing the performance and synthesis of stable interacting impedance control systems. In linear systems the result of H ∞ -norm based synthesis can be directly applied in both continuous and discrete time control. A common proximal method of converting a continuous (analog) system to a digital system with the same properties is based on the bilinear transform, a special case of which is the Tustin transform
s=
2 z −1 T z +1
(3.158) sT
where T denotes the sampling period and z = e is the delay operator for fixed sampling period. The bilinear transform has the advantage of mapping the left half s-plane into the unit disk in the z-plane, ensuring a stable continuous system to remain stable after discretization. For the contact stability analysis, a key property of the Tustin transform is relevant: it preserves the H ∞ norm.
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Dynamics and Robust Control of Robot-Environment Interaction
Considering the correspondence between the H ∞ norm and the passivity, Tustin’s method is also a passivity preserving discretization technique. Hence, the results of stable interacting system synthesis in the continuous-domain can be applied in the discrete-time and vice versa. If the Tustin transform is applied to a stable system considered in passivity-based or infinity-norm based contact transition stability theory, (3.122) and (3.128) respectively, then the resulting discrete system also is contact stable. Translating a linear continuous system, represented by the transfer function G (s), into the corresponding discrete-time transfer function H (z) using the Tustin transform is done by substituting (3.158)
G (s ) → H ( z ) = G (2 T z − 1 z + 1) For a stable discrete time interaction system matrix considered in the robust contact stability Theorem 3.8 (3.128), the induced infinity norm for testing the contact stability conditions is defined in terms of the frequency-dependent singular values −1 jωT 2 e jωT − 1 − 1 2 e jωT − 1 −1 2 e I + Ge Gt ≤ 1 (3.159) sup σ W jωT jωT jωT ω T e + 1 T e + 1 T e + 1
or for the SISO case, assuming unity weighting matrix and a pure stiffness environment,
Ge ( z ) Gt ( z ) + Ge ( z )
2
∞
ωe T 2(z + 1 )2 = 2 2 4( z − 1) + 4ξ t ωtT z 2 − 1 + (1 + κ ) ωt T
(
)
≤1
(3.160)
∞
jωT
Substituting z = e in (3.160), for which according to the maximum modulus principle the supreme norm (magnitude) is obtained, it is relatively easy to derive the contact stability condition in the parametric space. As expected, the obtained stability condition is the same as in the continuous system (3.132)
ξt ≥
1 2
( 1 + 2κ − 1)
Likewise, the passivity based contact stability analysis (see the Theorem 3.7) for the discretized system generated using the passivity preserving Tustin transform will provide the identical condition as in the continuous case (3.144). The robust and passivity-based contact transition stability theorems allow the force control delay to be considered in a relatively simple manner. Let us
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consider for this purpose the interaction model in (Fig. 3.45) with transport force delay modeled by the transfer function e-τs , where τ is the time delay. The infinity norm for testing the robust contact stability according to Theorem 3.8 for the interaction system with force retardation becomes
[G (s )e e
-τs
]
−1
+ Gt (s ) Ge (s ) e -τs
∞
[
]
−1
= Ge (s )e -τs + Gt (s ) Ge (s )
≤1
(3.161)
∞
Furthermore, adding the model uncertainties in the form (3.42) yields
[
]
−1
W (s ) Ge (s )e -τs + Gt (s ) Ge (s )
≤1
(3.162)
∞
where W (s ) is a stable transfer matrix describing the perturbations effects in the realized target model (i.e. robot) or in the environment. Based on the relationships between passivity-based and robust contact stability (3.138), the infinity norm for testing the passivity-based contact stability for delayed system has the form
[G (s )e e
-τs
]
−1
2 + Gt (s ) Ge (s )
≤2
(3.163)
∞
For a SISO interaction system without delay the passivity-based and robust contact stability provide the stability criteria in the simple explicit parametric forms (3.124) and (3.132), respectively. In the case with time delay, however, the determination of the target impedance parameters ensuring a stable transition based on criteria (3.161) and (3.163) becomes much more complex. For example, the robust stability implies
σ {G ( jω)} = G ( jω) = Substituting Ge ( jω ) = K e transformations
(
Ge ( jω) ≤1 Ge ( jω) e -jωτ + Gt ( jω)
and Gt ( jω ) = K t − M t ω 2 + jωωt
(3.164) yields after
)
ω4 − 2ωt 2 1 + κ − 2ξt 2 ω2 + ωt 4 (1 + 2κ ) ≥ −4ωt 2 κ sin 2 (
ωτ 2 ωτ ) ω + 2ξt ωt ctg ( )ω + ωt 2 2 2
(3.165)
The biquadratic parabola on the left side is identical to the non-delayed case (3.131). In the delayed system, however, the transcendental quadratic function on the right side of (3.165) complicates computation of ξ t which satisfies the
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robust stability inequality. The target system synthesis is based on the numerical solution of transcendental inequalities. An alternative general design approach is to select some of the target parameters based on the given task and environment knowledge. The remaining parameters (usually target damping ratio) should be varied (increased) and the corresponding infinity norm (i.e. principal gains of matrix transfer functions) should be computed using well-known numerical methods and functions (e.g. MATLAB’s sigma function) until contact stability inequalities are not fulfilled. Taking into account the preservation of the infinity norm by the bilinear transformation, the equivalent robust and passivity-based contact stability conditions in the time-discrete domain respectively have the form −1
2 z − 1 2 z − 1 -nT 2 z − 1 2 z −1 W Ge z + Gt Ge T z + 1 T z +1 T z +1 T z +1
≤1 ∞
(3.166) −1
2 z − 1 -nT 2 z − 1 2 z −1 Ge T z + 1 z 2 + Gt T z + 1 Ge T z + 1
≤2 ∞
where the delay time is assumed to be a multiple of the sample time τ = nT (commonly n = 1). The discrete time model is more suitable to describe transport delay effects, including also different sampling and delay times.
Fig. 3.45 Interaction system with force delay
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For a practical SISO interaction case, the obtained stability limits in the parametric domain are presented in (Fig. 3.46). It can be remarked that as with the results of the prior time domain analysis, the increasing delay time requires a larger amount of damping to stabilize the interaction. As expected, the robust stability criterion requires higher damping ratios at the stability limit (robust stability assumes a twice as stiff environment than in reality, i.e. in the passivitybased one). As pointed out, an important difference of stability analysis of a time delayed system compared to the ideal case without delay is that the minimum target-damping ratio stabilizing the contact does not depend only on the stiffness ratio and time delay, rather also on the target frequency ω t . The effects of the target frequency on the parametric contact stability limits are illustrated in (Fig. 3.47). Apparently, for a constant delay the increasing target frequency requires still higher damping to stabilize contact transition. 3.9.2.3 Contact stability of sampled-data system The previous analysis, however, does not cover the most common class of SD systems in which the plant operates in the continuous time while the controller operates in the discrete time. In the considered interaction systems (Fig. 3.4) the robot Gs (s ) and the environment Ge (s ) are continuous time components while the position Gr (z ) and impedance controllers G f (z ) , as well as the nominal motion interpolator, are discrete time components. As mentioned, the continuous and discrete parts are interfaced with each other using A/D and D/A converters, i.e. sampler S and hold ZOH operators (Fig. 3.41). Real A/D and D/A converters are electronic devices possessing some dynamic characteristics. However, for the sake of simplicity, we will assume ideal system elements: as samplers S that instantaneously retrieves input, and the hold operators H immediately providing signals at the sampling instant. Furthermore, assume that all digital control components are synchronized at a unique sample rate T (usually real controllers are realized as multitasking systems running at different sample frequencies). As the most critical control delay, the force signal retardation will be taken into account. We will assume this time delay in the form τ = nT , with the transfer function e-nTs , or z − n in the discrete-time domain, where n denotes the number of sampling intervals (usually n = 1 , though, in some implementations impedance control sample rate can be significantly larger than the position control rate, causing an even longer force delay).
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Fig. 3.46 Contact stability bounds of the delayed systems (s-domain analysis): passivity-based (above) and robust (below) contact stability ( M t = 10 kg, K t = 1500 N/m, ωt = 12.25 rad/ s )
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Fig. 3.47 Dependence of minimum damping on the target frequency (s-domain analysis): passivity-based (above) and robust (below) contact stability ( M t = 152 − 0.6 kg, K t = 1500 N/m, ωt = 3.14 − 50.26 rad/ s )
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The above idealization of the real SD interaction system is needed to simplify the system modeling facing the most relevant effects, and thus to make contact stability analysis and impedance control synthesis much easier. Consider the model of the SD interaction system in (Fig. 3.41). Assuming again that the internal position controller is compensated for using the outer loop impedance compensator (3.72), implemented in discrete form as
G f (z) = G p
−1
(z )Gt −1 (z )
the obtained SD transfer function of the interaction system {p0 , e} considered in the robust (i.e. generalized) contact stability theory and its limited stability norm condition have the form
[I + (G G (z)) s
e
−1
]
−1 G s ( z )Gˆ t ( z ) G p ( z )
[
]
−1 ≤ I + (G s Ge(z)) −1G s ( z )Gˆ t ( z ) ∞
<1 ∞
(3.167) In the preceding expression it was assumed that a robust position controller satisfies
G p (z ) ≤ 1 ∞
Introducing force signal delay in (3.167) yields
[I + z (G G (z)) G (z )Gˆ (z )]
−1
−1
n
s
e
s
<1
t
(3.168)
∞
Since the interaction between a robotic mechanism and the environment is in the continuous time domain, the discrete time transfer function has to be computed for the product of the continuous transfer functions
Gs Ge(z) = Z {Gs (s )Ge (s )} However, in specific cases (e.g. worst case stiffness environment) the separated discrete time transfer functions can be obtained yielding the robust stability condition
[I + z G (z) Gˆ (z )]
−1
−1
n
e
<1
t
(3.169)
∞
Based on the analogy between passivity-based and robust contact stability (3.138), we can directly write the condition for passive contact stability of SD interaction system
[I + 2 z G (z) Gˆ (z )]
−1
−1
n
e
<1
t
∞
(3.170)
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Fig. 3.48 Numerical computation of minimum damping coefficient satisfying robust (i.e. passivity based) contact stability
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Even in the SISO case, the ascertainment of the parametric domain of contact stability, based on the above criteria, is extremely complex. Therefore, the numerical algorithms are more reliable. For a practical interaction system the infinity norm (system gains) can be determined by computing singular values of the transfer function matrix (e.g. by using MATLAB’s sigma function) for the frequency spectrum within the system bandwidth and for given target impedance parameters. This indeed analysis-oriented approach, can be used for an iterative solution of the target impedance synthesis problem by defining some of the parameters from the given control task (commonly κ and ω t , i.e. K e , K t and
M t ), and varying the remaining ones ( ξ t ) untill conditions (3.169)-(3.170) are fulfilled. A corresponding computation algorithm, written in MATLAB, is illustrated in (Fig. 3.48). For the discretization of transfer functions the MATLAB c2d function (continuous-time model to discrete time conversion) has been utilized. As an illustrative example, the results of the synthesis of damping ratios at contact stability limits for varying time-delay and target frequency values are presented in (Fig. 3.49) and (Fig. 3.50) respectively. Obviously, considering the form of the stability bounds and their dependence on interaction system parameters, it may be concluded that in all above considered contact stability analysis cases the obtained results (stability limits) are quite similar. The main difference among the specific stability conditions concerns the numerical values of the stabilizing damping ratios bounds. However, focusing on the form of the stability boundaries, several qualitative rules describing the contact stability law can be identified. These rules are useful to define the stability criteria in a more general form, as well as to predict the needed stabilizing damping ratio level in each specific case, dependent on relevant effects, or applied stability criterion. The main set of stability rules includes:
● the contact stabilizing damping ratio grows with increasing stiffness ratio; ● the amount of damping in passivity-based stability is higher than in the ideal time domain analysis (“exact solution”), while the highest values require robust contact stability; ● significantly higher damping is needed to stabilize the interaction system with force delay, compared to the non-delayed or insignificantly delayed system (e.g. with τ < 0.001 s); ● in delayed systems, as target impedance becomes faster (i.e. higher target frequency ωt ), larger damping is needed to keep the contact transition stable;
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● the influence of the target frequency is insignificant in interaction cases with very small control delay; ● the damping ratio bounds obtained in the discrete-time analysis (SD system) are higher than in the continuous analysis (i.e. Tustin transformed system). To illustrate the above rules, the parametric stability bounds and damping ratio values obtained for relevant practical cases including: different contact stability analysis methods (e.g. “exact-solution”, passivity and robust stability), effects (non-delayed and retarded system), and analysis domains (continuoustime, discrete-time and sampled-data), are presented in (Fig. 3.51). The presented contact stability analysis techniques and cases differ not only from the view of the corresponding stability bounds values, but also considering the computation efficiency and applicability for the impedance control design. The explicit parametric stability expressions, which are quite desirable for the design, were only obtained in the ideal SISO interaction case. The derived explicit stability criteria are, however, applicable only for ideal, very fast control systems with high sampling rates. In real SD and MIMO systems the appropriate interaction control analysis and design are based on numerical methods for the computation of principal transfer matrix gains. For both analysis and synthesis of appropriate impedance control gains, i.e. target impedance parameters, the same procedure can in principle be applied in which the target parameters have to be shaped in order to satisfy the task requirements and the infinity norm, i.e. singular value loop shape conditions. Based on the developed contact stability criteria, the impedance control design problem can essentially be considered as a robust control design task for which numerous developed approaches and tools can be utilized (e.g. MATLAB’s Robust Control and µ –Analysis and Synthesis Toolboxes).
3.10 Evaluation of Contact Transition Stability Conditions The established contact transition stability bounds for various interaction systems have been partially evaluated by using corresponding simulation models and initial transition experiments. However, the unanswered practical problem is how to select the most appropriate transition model including specific relevant effects (e.g. time-delay, discrete-time control implementation etc.), as well as
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corresponding stability criterion (passivity-based or robust stability) for each real interaction system and case.
Fig. 3.49 Contact stability bounds of the retarded systems (SD system analysis): passivity-based (above) and robust contact (below) analysis ( M t = 10 kg, K t = 1500 N/m, ωt = 12.25 rad/ s )
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Fig. 3.50 Dependence of minimum damping on target frequency (SD system analysis): passivitybased (above) and robust contact (below) stability bounds ( M t = 152 − 0.6 kg, K t = 1500 N/m,
ωt = 3.14 − 50.26 rad/ s )
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Fig. 3.51 Comparison of contact stability bounds for the parameters. Interaction system parameters: M t = 1 kg, K t = 1500 N/m ( ω t = 38.73 rad/s) and τ = 0.008 s. Used symbols, lines and markers: t-continuous time analysis, τ -control delay effect (diamond marker), d-sampled-data system analysis (circle marker), “exact-solution” (solid line), passivity-based stability (dashed line) and robust contact stability (dash-dot line).
Various stability criteria define different parametric stability bounds (Fig. 3.51). These bounds are similar in form; however, they impose quite different amounts of damping needed to stabilize the interaction. A realistic design problem is to select the most appropriate model and criterion for a specific robotic system and environment. We may conclude that the main goal is to find a minimum amount of damping which ensures stable and safe contact, as well as provides fast transition with acceptable contact force magnitude. As demonstrated, the increasing damping supports the stabilization of contact transition. However, it causes a sluggish robot reaction with unacceptably high force overshoots (see Fig. 3.25-3.26). Similar to the damping, we found the requirements on further target impedance parameters also to be contradictory. The control design problem deals with finding the best compromise between these conflicting objectives.
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The specific problem is that real robotic systems and environments include effects that are quite complicated to model (e.g. friction, high dynamic effects etc.). These effects might also be relevant for the interaction control performance. Therefore, before developing impedance control design algorithms, it is relevant to obtain a proper view on the applicability of specific stability conditions in real robotic systems, as well as to get practical experience relevant for design. The practical way to achieve this goal is to perform transition experiments in real industrial robotic systems. Moreover, the obtained contact stability bounds are based on the sufficient stability criteria that may be conservative. The best way to evaluate their applicability is to test them experimentally. The experiments should also be used to improve the simulation to become more realistic by identifying models and parameters of relevant effects. The realistic simulation provides a powerful tool for examining the theoretical stability and impedance control design results. The simulation offers possibilities to add/exclude specific perturbation effects in simulation experiments and to test the reliability and robustness of the system stability. The main benefit of the simulation is that virtual transition experiments are easier and “safer” to realize. In real testing, contact instability effects could produce high interaction forces (e.g. bouncing) and may cause damage to robot, sensor and environment.
3.10.1 Contact transition performance indices Before evaluating the contact transition criteria it is important to define measures of impedance control performance during contact realization. Due to the specific impedance control goal, the contact-transition performance cannot be considered in terms of common transient response control indices [45]. It is therefore necessary to define characteristic impedance control measures of performance to indicate to the control designer the quality of the interaction control system. The relevant characteristics of the impedance control are realized target impedance and interaction force. As demonstrated, the examination of the target impedance can be performed using simple interaction tests (Fig. 3.13). The evaluation of the force history, however, is more complex since the interaction force is shaped not only by the target model, but also by the nominal motion and the environment. In a typical contact transition with a trapezoidal nominal motion profile and constant velocity impact (Fig. 3.52) two transient process can be distinguished: the first one immediately after impact, and the second after the
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interpolator reaches the maximum penetration. The following quantitative measures represent the performance of contact transition processes: ● “maximum impact force” Fi (Fig. 3.52). This force mainly depends on the approaching velocity and time-delay. Since the impact is not explicitly modeled, the initial impact force practically represents the force due to initial penetration during time delay, before the impedance control starts to reduce the interaction force, 0
Fi = K e δp = K e v0 τ where v0 is impact velocity. Discrete target impedance elements (mass, dashpot and spring) do not affect the interaction force during very short impact time intervals. After the initial impact, the interaction force increases further though the impedance effects and reaches the maximum Fi (Fig. 3.52); ● maximum force Fmax, which commonly occurs at the end of the nominal penetration phase (i.e. beginning of second transient process), or in some cases immediately after the impact; ● “minimum transition” force Fmin is critical for loss of the contact during transient periods after impact (commonly first minima) or at the end of the interpolation phase (Fig. 3.30-3.31); ● steady-state force F*, i.e. final force of a contact transition process, defined for a SISO system according to (3.31) by
1 1 ∗ p0 F ∗ = + K K e t
(3.171)
● settling time Ts, represents the time required for the system to settle within certain percentage of nominal force behavior (e.g. target force computed for the nominal impedance model and motion profile, or steady-state force). Furthermore the following relative contact transition measures will be used.
Definition 3.8 (Contact Stability Margin) The relationship between the minimum and maximum impact force is referred to as the contact transition stability margin (zero value denotes an instable contact transition)
ς=
Fmin × 100% Fi
(3.172)
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Definition 3.9 (Contact Overshoot) The relationship between maximum and steady state forces is referred to as the contact percent overshoot
ο=
Fmax − F ∗ × 100% F∗
(3.173)
It should be pointed out that the maximum and minimum impact forces depend not only on the target impedance and environmental parameters, but also on the impact velocity and nominal motion. However, impact commonly occurs in constant velocity phase. In this case, the ratio (3.172) is almost independent of the approaching velocity. Hence, this measure provides a practical index for evaluating the contact transition.
Fig. 3.52 Contact transition indices
3.10.2 Contact transition assessment The next experiments present the result of both practical and simulation transition trials with the Manutec-r3 robot. The same experimental configuration
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as in the initial impedance control experiments (Fig. 3.17-3.18) has been utilized.
Experiment 3.5:
Contact stability trials
To examine the reliability and applicability of established contact transition stability criteria and corresponding bounds, a set of interaction systems, i.e. test cases, have been designed with the target stiffness
K t = [250 500 1000 2000 4000 8000] ( N/m)
(3.174)
which in contact with the experimental environment K e = 56000 (N/m) provides the following stiffness ratios
κ = [224 112 56 28 14 7] These selected values cover almost the entire interval used in the contact stability analysis (Fig. 3.43-3.51). In order to define a set of possible (realizable) target frequencies within the bandwidth of the internal position controller (approximately 10 Hz), a set of target masses has been determined for each target stiffness value (3.174) to achieve
ωt = [0.5 1 3 6 8] (Hz)
(3.175)
A minimum realized target mass in the experiments was Mt ≈ 4 kg (for Kt=250 and ωt=8 Hz). The corresponding maximum reduction of the apparent Cartesian robot mass (approx. 300 kg in the contact direction) was 98%, which is significantly greater than the reduction of robot inertia recently achieved in [46]. The above intervals define 30 interaction cases. For each case three different target damping ratios have been computed based on the following criteria: “exact solution” (analytical bounds) obtained for the time-domain model with delay (Algorithm 3.1), passivity-based and robust contact stability for the sampled-data interaction systems with delay (3.170) and (3.169), respectively. By this means a total of 90 transition experiments was designed. After first tests the initial target damping ratios were slightly modified. It was observed that the damping ratio obtained by the “exact solution” in almost all cases is not sufficient to stabilize the contact between the robot and the environment. Therefore, these values were increased by multiplication with factors 1.3-2 (depending on ωt ). The passivitybased criterion provides the damping close to the practical contact stability. Therefore the corresponding theoretical target damping values were either kept
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or insignificantly increased (sometimes to 1.2-1.5 times). The contact transition with the robust-stability criterion damping was in all cases quite stable and safe. However, the numerical stability values were reduced by the factor 0.7-0.9. The aim of these modifications was to obtain safe experiments, with minimum undesired bouncing pertinent to damage the sensor, and from the other side, to make the target damping as close as possible to the practical contact stability limit (with the margin ς ≈ 0 ). Although the theoretical stability limits were not directly applied in the experiments, it is still straightforward to practically evaluate their reliability based on the performed trials by using above introduced contact transition measures. The specified transition experiments were realized and the force, motion, as well as control-output data were acquired. From the measured force data the relevant transition segments were cut off for the purpose of comparison and evaluation of contact stability measures. Due to limited space, only few of typical results will be presented. Figures 3.53-56 illustrate the transition force for the applied damping ratios according to the different target stiffness and frequency cases. The comparison of contact transition for two boundary target frequencies ωt and selected stiffness ratios κ is done in (Fig. 3.53-3.55). Each transition test consists of three experiments for the selected triplets of target damping ratios ξ t . The contact realization for intermediate frequency and target stiffness ratios is presented in (Fig. 3.56). The initial observation is that in the majority of experiments the desired goal was achieved to practically capture transition stability limits for various interaction system parameters. The obtained diagrams confirm the contact transition characteristics deduced from the theoretical analysis. Obviously, in real sampled data systems with control lags, faster target impedances (bottom diagrams in Fig. 3.53-3.55) impose significantly higher target damping ratio values to stabilize contact. However, in these systems the compliant reaction is more rapid, which is manifested by significantly lower force overshoots in comparison to the slower systems. In the slower systems the target damping ratio interval between unstable and quite safe transition is very small, especially for lower κ . Typical contact transitions at stability limit ( ς = 0 %) and for a 100% stable transition are presented in (Fig. 3.56) and (Fig. 3.57) respectively. In order to evaluate the reliability of established stability criteria, it is relevant to put the theoretical limits and experiments side by side. An approach to do this is presented in (Fig. 3.58-3.60). These diagrams show the damping ratios at stability limits computed from theoretical stability criteria (two different
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Fig. 3.53 Contact transition comparison: slow (top) and fast (bottom) systems (low stiffness ratio, i.e. stiff target system; delay τ =0.008 s, K e = 56000 N/m)
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Fig. 3.54 Contact transition comparison: slow (top) and fast (bottom) systems (medium stiffness ratio; delay τ =0.008 s, K e = 56000 N/m)
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Fig. 3.55 Contact transition comparison: slow (top) and fast (bottom) systems (high stiffness ratio, i.e. elastic target system, delay τ =0.008 s, K e = 56000 N/m)
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Fig. 3.56 Contact transition comparison: intermediary case (control delay τ =0.008 s, K e = 56000 N/m)
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Fig. 3.57 Typical transitions cases: at stability limit (above) and 100% stable contact transition (below). Experiment parameters: τ =0.008 s, K e = 56000 N/m
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Fig. 3.58 Theoretical (“exact-solution”-star, passivity-based limit-triangle, robust stability diamond) vs. experimental (unstable transition-x, transition at stability limit-circle, stable transition-plus) stability limits (captured stability limit-dashed, estimated-dotted) (delay τ =0.008 s, K e = 56000 N/m)
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Fig. 3.59 Theoretical (“exact-solution”-star, passivity-based limit-triangle, robust stability diamond) vs. experimental (unstable transition-x, transition at stability limit-circle, stable transition-plus) stability limits (captured stability limit-dashed, estimated-dotted) (T =0.008 s, τ =0.008 s, K e = 56000 N/m)
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Fig. 3.60 Target damping stability limits corresponding to the damping ratios from Fig. 3.58
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Fig. 3.61 Target damping stability limits corresponding to the damping ratios from Fig. 3.59
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ξ t -axes are used to distinguish different damping ratio scales for lower and higher κ ) versus the stiffness ratio values realized in experiments (a logarithmic scale is used for the κ -axis). In the consideration of the previous contact transition, the target damping ratio ξ t and target frequency ωt were mainly used to assess the contact performance. By this means the target damping parameter set is reduced only to two relative parameters. However, a second-order target system, i.e. impedance transfer function (3.1)-(3.2), is characterized by three parameters: mass M t , damping Bt and stiffness K t . Therefore, it is interesting to analyze the influence of total damping Bt on the contact stability. The diagrams in (Fig. 3.60-3.61) present the equivalent stability bounds on the target damping parameter
Bt = 2ξt M t K t
(3.176)
The following symbols are utilized to denote the theoretical stability limits * ∆ ◊
-“exact”, i.e. time-domain solution” (3.117); -passivity-based limit (3.170); -robust-stability limit (3.169);
The results of the transition experiments has been categorized by using the following markers: x o +
-unstable transition; -transition at (closely to) the stability limit; -stable transition;
For stable transitions the contact stability margins are given explicitly. The captured stability limit curvatures in experiments also are drawn (dashed line). For the experiments where the stability limit was not exactly achieved, it is roughly estimated (dotted line) by computing stability margins. The following conclusions can be drawn from the above comparison of contact stability criteria:
● in all experiments the most accurate contact stability result provides the passivity-based criterion (∆-marker). For lower κ values (7-28) the passivitybased contact stability gives the damping ratios that are on the safe side, relatively close to the limits obtained experimentally for the tested system ( τ =0.008 s). For the higher stiffness ratio values, however, the passivity-based theory results are slightly under the practical stability limits, while with
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increasing ωt this difference becomes smaller, and for a quite fast interaction system ( ωt =8 Hz) the results are on the safe-side.
● the time-domain solution (Algorithm 3.1) in all cases provide instable transition. Relative deviation from the practical stability limit is smaller for the stiffer target systems (smaller κ ). ● the robust contact stability always ensures a safe transition. For small ωt the obtained transition results are quite conservative. For higher ωt and smaller κ the safety margin becomes tolerable (about 30%-60%), while for higher κ it is further reduced, approaching the practical stability limit. ● an interesting observation can be made by comparing the stability limits on the damping ratio ξ t and effective damping parameters Bt (Fig. 3.58-3.61). As mentioned, the higher target frequency requires higher damping ratio to ensure stable contact transition. However, the effective damping Bt (3.176) becomes smaller (Fig. 3.60-3.61). The reason for this is the reduction of the target mass needed to obtain higher target frequency. Moreover, contrary to the damping ratio which grows large with the increasing stiffness ratio κ , the effective damping appears to be independent of κ (for higher target frequencies, see Fig. 3.61), or even is slightly decreasing with higher κ (Fig. 3.60). These results are quite in agreement with the reducing force overshoots with rising target frequency observed both in experiments and simulations. As demonstrated in (Fig. 3.22), the force overshoots are proportional to the target damping parameter Bt .
● to achieve better contact transition performance, it is desired to achieve the highest possible target frequency. The internal position controller bandwidth imposes the practical upper limit on the target impedance frequency. These conclusion, as well as theoretical and experimental contact transition stability investigations, provides a background for reliable design and an impedance control tuning algorithm for industrial robotic system, which will be presented in the following sections.
3.10.3 Upper limits on target impedance frequency Based on the theoretical contact stability analysis and the above experiments, it may be concluded that increasing the target model frequency as much as possible
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can significantly enhance the performance of the impedance control contact transition process. The improvement of transition performance is mainly characterized by smaller force overshoots and faster reaction to force signals. As will be demonstrated below, the increasing target frequency also contributes to the better quality of the steady-state behavior. Therefore, it is interesting to determine upper limits on the realizable target frequency. In principal, by cancelling the position control loop (3.68), the external impedance controller theoretically may be just as rapid as or even faster than the position loop. Practically, however, the structural oscillatory modes, the excitation of which is avoided by limiting the position control bandwidth, may now be affected by the impedance control loop. A realistic solution to avoid excitation of the frequency associated with the robot structural resonance could be to design a notch filter [39] and/or to apply active damping, in order to reduce resonance effects. However, in uncertain interaction systems these approaches may not be reliable. The bandwidth of the internal position feedback loop ωb is commonly defined [47] as the lowest frequency such that
G p ( jωb ) =
G p (0 ) 2
The bandwidth indicates the speeds of disturbance elimination and control reaction. A proper tuning of the internal position control loop in the industrial practice provides the bandwidth that is large enough to ensure a fast speed of response to the reference signal, however, also to provide sufficient attenuation of disturbances (e.g. high oscillation modes, sensor noise etc.). As well known, to avoid excitation and resonant oscillation of the robot structure, which varies with the robot configuration, it is practical to set the undamped natural frequency of the position controller to no more than one half of the structural frequency estimate
ω p ≈ 0 .5 ω 0 The position loop bandwidth is usually very close to the cross-over frequency ωc ( ωc ≈ ωp ≈ ωb ) at which the open loop gain is equal to one. Since the main purpose of the position feedback system is to combat the uncertainties in the robot system, the target system bandwidth must be limited by the position control bandwidth ωb in order to achieve acceptable performance of the impedance control. Even more, to avoid dynamic interaction at higher frequency
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ranges within ωb , in which the position control disturbances in conjunction with a stiff environment could cause undesired oscillation upon contact, it is recommended to limit the target frequency to
ω t ≤ 0 .5 ⋯ 0 .8 ω b
(3.177)
Moreover, since in contact tasks the robot mechanically interacts with the environment, in some specific cases the target frequency can also be limited by the structural frequency of the environment
ω t ≤ 0.5 ω se Those are rather heuristic rules. Indeed, these limits of the target frequency are due to engineering rather than mathematical reasons. Experiment 3.6:
Contact transition experiment vs. simulation
The performed experiments have also been used to improve the simulation models (ROBOTICS blockset). Since the dynamic and control parameters of the Manutec-r3 robot and ARCOS control system were accurately known [27, 31], especial efforts were made to improve the model and parameters of the robot joint friction at lower velocity during contact [25]. As already pointed out, to obtain a realistic simulation, various non-linear effects, such as control delay, quantization and roundoff etc., were included in the simulation. Using these parameters, as well as the applied impedance control gains, the contact transition was simulated and compared with the transition experiments (Fig. 3.17-3.18). A typical example is presented in (Fig. 3.62). At first glance, a very good matching between experiment and simulation was achieved. However, a detailed analysis of transient interval shows the slight differences in the amplitude and frequency of force oscillations. The reason for this is the lack of precise knowledge of several effects, such as impact, which play an important role in contact establishment and which are not considered in simulation. In contact with a very stiff environment, the interaction force appears to be significantly sensitive to various non-modeled effects. Despite the difficulty to exactly simulate the contact transition, the obtained result can be estimated as quite satisfactory for the investigation of contact transition in industrial robotic systems. As still mentioned, the main benefit of simulation is the possibility to test non-linear effects separately and estimate their specific influence on contact transition stability. The ROBOTICS
Impedance Control
Fig. 3.62 Contact transition experiment vs. simulation (Mt = 1 kg, Kt = 1500 N/m, ξ t = 5, Ke = 157000 N/m, τ = 0.008 s)
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simulation system allows relatively accurate testing of the impedance control design outcomes. Knowing relatively precise robot control and environment parameters, this system provides a correct representation of the contact transition and steady-state performance of real robotic systems. Concerning the valuation of contact stability results, there are fairly small deviations between experiments and simulation which could be critical for the judgment of transitions close to the stability limits (with small ς margin). However, generally the simulation transition results are on the safe side. This means that an unstable transition in simulation is almost unstable in the experiments.
3.11 Conclusion This chapter focused on the problems which address the following phase of the impedance control design:
● specification of the control objectives and performance, ● selection of the control scheme, and ● preliminary testing. The main impedance control performance specification addresses the capability to achieve the target model. In order to assess how well designed the impedance controller meets the control objective, position- and force-model based performance criteria have been introduced. For the sake of simplicity, the impedance control design problem is split into two subproblems concerning the realization of the target impedance, and the selection of target impedance parameters which ensure specific desired task performance, as well as common control design requirements, such as stability, fast reaction and robustness. The elementary position based impedance control schemes that on one side meet the impedance control performance criteria, and on other side are suitable for implementation in convenient industrial robotic control systems have been proposed and discussed. The position model-error scheme has been selected as the most appropriate for practical implementation. Two control laws with this scheme are considered. The convenient control utilizes the target admittance as the impedance compensator. Though quite simple and obvious, this control law exhibits limited performance in realizing the target impedance and stabilizing the interaction. To overcome these limitations a new control law has been proposed. The improved position based impedance control includes the compensation of the internal position control loop. Considering the diagonal dominancy of the
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Cartesian position controller in industrial robotic systems, this control law is very simple and reliable for implementation. The stability of interaction between the robot and environment in contact, which is essential for the impedance control synthesis, is defined by means of the coupled stability. For the examination of stability we have applied a common approach utilizing the properties of the system at equilibrium, and various modern control techniques (e.g. positivity and H ∞ control concepts etc.). The system passivity concept provides a relatively simple test for the assessment of coupled system stability. In this test only passivity of the environment should be proven, without an accurate knowledge of the parameters. However, in real control systems the realization of a passive target system is disturbed by various practical implementation effects, such as control delay or data sampling. Robust control provides an efficient framework for the synthesis of the impedance control capable to cope with uncertainties occurring in the robot model and the environment. For the testing of the coupled stability, the worst or most destabilizing environment, consisting of a series of elemental springs or masses, has to be used in order to guarantee stability with an arbitrary passive environment. The initial interaction and contact transition experiments with industrial robots have proven the theoretical design results and the expected performance of the proposed new impedance control law. The problem of contact transition stability has been identified as essential for contact realization. We have further considered the stability of the contact transition process. Although recognized to be most fundamental in contact tasks control synthesis, this problem was not appropriately investigated untill now. Especially the contact stability in industrial robotic systems has not been explored adequately. Several practical contact stability definitions are proposed in order to clearly distinguish contact and coupled stability, often mistaken in the literature. Based on these definitions new stability conditions are proposed. For an ideal secondorder interaction model the contact stability has been analyzed in the time domain. This analysis in conjunction with initial experiments has further been applied to derive more general and reliable contact stability criteria based on common system techniques for the stability analysis of both linear and nonlinear control systems. The concept of dynamic systems passivity and robust stability analysis are applied to obtain reliable conditions ensuring contact stability. In the considerations of contact stability the “most destabilizing” environmental model has been applied.
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The established stability criteria provide the basis for examining the impedance control parameters effects on the transition process stability. The analysis/synthesis oriented stability examinations allow the tuning of target impedance parameters in order to meet both interaction performance and stability. In SISO interaction systems the contact stability conditions are derived in closed forms relating the target impedance and environment parameters. The target damping and stiffness ratios are essential parameters for the stabilization of the contact transition process. The established criteria determine a minimum amount of damping that guarantees the stable contact and suitable transition performance. MIMO stability is expressed in terms of weighted infinity-norms of linearized interaction system transfer functions. Within the robust stability analysis framework, the generalized contact stability condition also is derived ensuring both contact and coupled system stability. With the aim to obtain more realistic contact stability bounds matching the initial transition experiments with industrial robots, the non-linear effects of Coulomb friction and control delay have been analyzed. On the basis of these considerations it was concluded that control lags and sampling effects are essential for the precise determination of practical contact stability limits in real sampled data robotic systems. Therefore, the stability analysis is pursued in discrete-time and sampled-data interaction systems. Utilizing the Tustin transform which preserves the infinity system norm, the results of stable interacting system synthesis in the continuous domain are applied for discrete systems. For idealized sampled data interaction systems with dominant force lags, the passivity-based and robust contact stability also are derived. Several numerical algorithms for the computation of contact stability limits in parametric space are proposed. As the result of the performed contact stability analysis, a set of different contact stability criteria and conditions for various classes of interaction systems has been obtained. Various stability criteria define different parametric stability bounds. These bounds are similar in form; however, they impose quite different amounts of damping to stabilize the interaction. The realistic design problem is to select the most appropriate model and criterion for the specific robotic system and environment. The main goal thereby is to find the minimum amount of damping that on one side should ensure stable and safe contact, and on the other side, will provide a fast transition with acceptable force magnitude. The derived contact stability conditions are based on the sufficient stability criteria, but not the necessary ones. That means that the conditions are satisfied in a stable contact transition; however, the accomplishment of these conditions
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does not guarantee a stable contact. Therefore, a practical way to test the applicability and reliability of the sufficient contact stability criteria was to test them experimentally. In order to evaluate the derived contact stability criteria a large set of several hundred contact transition experiments was designed and realized using the Manutec-r3 robot. Relevant contact stability parameters (target frequency, damping and stiffness ratios) have been varied. The aim was to capture a practical parametric contact stability limit in a typical real robot/environment interaction system. In order to compare the performance of different transition processes new measures for the assessment of the impedance control performance during contact realization are proposed. The experimental results finally are compared with theoretical limits prescribed by different stability criteria. On the basis of tests performed it was concluded that the most accurate contact stability results (the parametric limits closest to the experimental bounds) provide the passivitybased criterion for the sampled-data system. Robust contact stability always ensures a safe transition and appears to be very practical for the control synthesis in an uncertain robot/environment interaction system. The experiments have proven the theoretical contact stability analysis.
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Chapter 4
Practical Synthesis of Impedance Control
4.1 Introduction A widespread application of impedance control in industrial robotic systems is still a challenging problem. One of the limitations is the absence of a widely accepted framework for the synthesis of the impedance control parameters that ensure the stability of both the contact transition and interaction processes and guarantee the desired contact performance. The existing design procedures based on robot passivity appear to be exceedingly conservative in applications in which the interaction between an industrial robot and a stiff environment should be controlled [1]. The proposed new interaction stability paradigm ensures contact stability during all phases of interaction. Moreover, the new design framework realizes low-impedance performance allowing considerable reduction of high apparent industrial robot inertia and stiffness. The novel stability criteria are established based on robust control theory and take into account estimates of the environmental stiffness, tolerating thereby large uncertainties and variations in industrial environments. These criteria are proved by extensively testing in industrial and space robots and have been recently extended to the control synthesis of human robot interaction systems (haptic admittance displays and rehabilitation robots). The analyses of coupled and contact stability, as well as the practical evaluation of established stability criteria provide a theoretical basis for the synthesis of impedance control in industrial robotic systems. Splitting the impedance design problem into two independent steps focused on the selection and realization of a target impedance system respectively, has significantly simplified the impedance control synthesis problem. As demonstrated, in industrial robotic systems with robust position servo systems, the improved position-based impedance control accurately realizes the selected target impedance even in the presence of model uncertainties and disturbances. The main problem of the design is, then, to adjust the target impedance to the given task, i.e. to select a target impedance model and parameters to meet basic contact task requirements. 405
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This chapter addresses the synthesis of the adopted second-order target impedance model for a generic contact task. The contact task consists of the realization and maintenance of a stable contact with the environment. The interaction force should be kept within the prescribed limits, dependent on position tolerances and the environmental stiffness. This assignment is intrinsically involved in almost all robot interaction tasks. The control synthesis to be considered mainly concerns basic problems of the impedance control design at the servo-control level. The impedance control synthesis at higher control layers (e.g. motion planning, programming and interpolation levels), which reflect more specific task and control system requirements, will also be considered in this chapter. 4.2 Influence of the Target Parameters on the Impedance Control Performance 4.2.1 Influence of target frequency and mass on contact transition As demonstrated in the experiments in the previous chapter, an increase in the target impedance frequency leads to the improvement of control performance. This reduces force overshoots in the transition phase, as well as limit-cycle amplitudes in the steady-state phase. Experiment 4.1: Effects of target frequency on contact performance To prove the effects of the target frequency on contact performance, we have performed similar experiments as in the contact transition trial (Experiment 3.2, Fig. 3.17), but with variable target masses, i.e. target frequencies. The experimental results (Fig. 4.1) clearly confirm the above statement. In a real industrial robot system, reducing the limit-cycle amplitude caused limit-cycles to vanish (e.g. see Fig. 3.26 a and b). In the last experiment, with ωt ≈ 6.2 Hz (i.e. target mass M t = 1 kg), as can be remarked, the contact was at the stability limit. Obviously the higher target frequencies (i.e. the lower target masses) produce lower force overshoots during contact transition and reduce limit-cycle
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Fig. 4.1 Experimental results of force-overshoots and limit-cycle reductions by increasing target frequency ωt (in Hz); ( ξ t = 8 , Kt = 1500 N/m, Ke = 160000 N/m) (above: sequences after the initial impact; below: complete transition experiments)
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amplitudes, but the contact transition requires a higher damping ratio to ensure a stable transition. The initial sampled force is given by
F o = Ke po where p o = p (t = 0 ) is the start value of penetration. This gives
∆x f =
ΛK e o p MtKp
Assuming that the position error at the beginning of contact is zero, i.e. x ≈ x0 , as a rule of thumb it can be inferred that the position modification is lower than the initial penetration
∆x f ≤ p o which implies that
Mt ≥
Ke Λ Kp
(4.1)
Once the impact between the end-effector and the environment is detected, the above condition presumes that the initial value of the modified nominal o position xr should not abandon the connected surface. This could be considered as an additional measure to prevent the loss of contact. This condition defines the lower limit on the target mass parameter by considering the initial position correction bound. The minimum target mass depends on the robot and environmental parameters. To some extent the condition (4.1) is heuristic and may be quite conservative. For example for K p = 1450000 (N/m), Λ = 300 (kg) and K e = 60000 (N/m), it is implied that M t ≥ 12.5 (kg). Assuming K t = 1500 (N/m), this provides
ωt < 2 (Hz). As demonstrated in contact transition experiments, depending on the position control bandwidth, sampling rate and nominal motion, the contact must not be lost though these conditions are not fulfilled. It is more reliable to obtain the target mass, i.e. frequency, from the condition (3.177), defining realizable target systems. It is worth pointing out that a high target mass causes higher impact forces, while a low mass assumes a large target bandwidth, which cannot be realized by industrial robots position control servo systems. As a result of the target mass-
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409
damper-spring behavior, the transition force becomes significantly higher than the steady-state interaction force component (spring effect). As was shown in (Fig. 3.22), inertial and damping force components contribute to the total force magnitude during the contact transition phase. Especially the influence of the damping force component on force overshoots is significant. The simplest strategy to reduce the contact overshoots, i.e. the index ο (3.173) (see also Fig. 3.52), is to slow down the approach velocity. When taking into account the task execution time, however, this solution is often not appropriate. Another strategy results from the consideration of the expression for the force overshoot index (3.173)
ο=
∗ ∗ Fmax − F ∗ FB max + FK − FK B eɺ Bt xɺ0 ≈ ≈ t ≈ ∗ ∗ F K t e K t ( x0 − x ) FK
(4.2)
It might be remarked that this index can be reduced without changing the nominal motion by decreasing the target damping/stiffness ratio. This ratio can be written as
Bt 2ξ t M t K t Mt ξ = = 2ξ t =2 t Kt Kt Kt ωt
(4.3)
This relationship is not easy to analyze because in real control systems both ξ t and ω t affect the transition stability limit. It is relatively easy to demonstrate that the ratio (3.183), and according to (4.2) also the force overshoot ο , can be reduced for the constant nominal motion parameters by increasing the target frequency, i.e. by reducing the target mass and thereby keeping constant target stiffness (i.e. κ ). This effect of reduction is significant in the lower frequency interval, while in the higher one a saturating effect occurs which limits the potential to diminish the force overshoot index. 4.2.2 Reduction of force overshoots – Hogan’s target impedance model (generalized stiffness control) Due to the considerably slow movement of real robots during interaction, the damping interaction force component of the target impedance model (3.1) can be approximated by the product of the target damping and the nominal velocity. Relatively high damping is required to stabilize the contact transition. Then the elimination of the nominal velocity from the basic target impedance control law (3.1) theoretically should significantly reduce the damping force that mainly
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contributes to interaction force overshoots. The corresponding second-order target impedance model was proposed by Hogan [2] and has the form
F = M t ɺxɺ + Bt xɺ + K t (x − x0 )
(4.4)
Obviously, the interaction in this model is mainly characterized by the linear target stiffness behavior, while the damping and acceleration effects are negligible due to slow real motion (penetration) during contact with a stiff environment. Therefore, the target model (4.4) and the corresponding control scheme can be referred to as generalized stiffness control. In industrial robot systems it is relatively easy to show that the following control law realizes Hogan’s target model (4.4) using the selected impedance control structure (Fig. 3.3)
(s ){[Gt −1 (s ) − S p (s )Gs (s )]F + [G p (s ) − Gt −1 (s ) K t ]x0 } −1 −1 −1 −1 ≈ G p (s )Gt (s ) F + [I − G p (s )Gt (s ) K t ]x0
∆x f = G p
−1
(4.5)
The corresponding control scheme (Fig. 4.2) includes the same compensator as in the improved impedance control (3.68)
~ −1 −1 G f (s ) = G p (s )Gt (s )
as well as an additional nominal position feed-forward filter
~ −1 −1 G0 (s ) = I − G p (s )Gt (s ) K t
(4.6)
The effect of this filter is that the nominal motion will be cancelled and the effective filtered nominal motion
~ −1 −1 −1 x0 ef = G p (s )Gt (s ) K t x0 ≈ Gt (s ) K t x0
(4.7)
will be pursued to the internal position controller. When taking into account that the second-order target impedance must be quite over-damped in order to ensure stable contact with a stiff environment in the presence of the force control delay, it is clear that the filter (4.7) has low-pass behavior with a very small cut-off frequency (commonly < 0.1 Hz). As a consequence, the filtered nominal motion would slow down considerably. Substituting the position deviation (4.5) in the position control model (3.11) and neglecting the contact force disturbance in the position control yields the desired interaction model
Gt (s ) x = F + K t x0
(4.8)
The following example illustrates a typical performance of the generalized stiffness control realized in an industrial robot control system (Fig. 4.3).
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Fig. 4.2 Control scheme realizing Hogan’s target model
Example 4.1: Performance of the generalized stiffness control In order to illustrate the basic performance of the generalized stiffness control a simple contact transition in a SISO interaction system will be considered by simulation. The task configuration is the same as in the previous experiments and simulations (Fig. 3.17-3.18). The control scheme in (Fig. 4.2) for a SISO system is implemented in SIMULINK using the Manutec r3 robot sampled-time Cartesian control model and data in the considered configuration. The example in (Fig. 4.3) presents the nominal x0 (t ) and effective filtered
x0ef (t ) command motion for a characteristic contact transition case. As can be observed, due to the feed-forward filter effect (Fig. 4.3) the effective command motion is slowed down, so that the robot hits the surface with a lower impact velocity. The effective motion achieves the final position
∗
x0 = x0ef
∗
considerably later. The filter smoothens the path and velocity profiles by removing quick accelerations and jerks typical for the convenient interpolation. Such nominal motion produces a feasible contact force with reduced impact and overshoots caused by the damping force component (Fig. 4.4). Obviously, this interaction force profile is more suitable and desirable in practical contact tasks. In both cases, the steady state force has the same value. It should be noted that for the sake of simplicity the Coulomb friction effects are neglected in the simulation.
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Fig. 4.3 Nominal x0 (t ) and effective filtered x0 ef (t ) command motion ( x0 = max ( x0 ) = 0.03 m, xe = 0.005 m, v0 = 0.01 m/s, a 0 = 0.05 m/s2) ∗
Fig. 4.4 Interaction force in the improved impedance F (t ) and generalized stiffness F f (t ) control ( M t =20 kg, ξ t =6, K t =1500 N/m, K e =100000 N/m, T = 0.008 s, τ =0.008 s)
Let us consider now the contact transition and coupled stability of the generalized stiffness control. The basic interaction models (3.21)-(3.29) used in the contact stability analysis define relationships between the actual and nominal penetration (penetration model), i.e. between the motion deviation and nominal penetration (deviation model). To obtain these models for the generalized stiffness, one begins from (4.7) defining the relation between the effective and nominal command penetrations
p0ef = Gt
−1
(s ) K t p0
(4.9)
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Substituting the environment model − F = G p (s ) p into the interaction equation (4.8) and assuming a stationary environment (contact point), as well as an ideal position control servo system, yields
[(
p = I + Gt
−1
(s ) Ge (s ))]
−1
Gt
−1
(s ) K t p0 = [(I + Gt −1 (s ) Ge (s ))]
−1
p0 ef
(4.10)
The effective position deviation in generalized stiffness control (4.11)
eef = p0ef − p is then directly derived from (4.10)
eef = Gt
−1
(s )Ge (s )[I + Gt −1 (s )Ge (s )]
−1
p0 ef
(4.12)
Comparing the above expressions with the corresponding relations for the impedance control (3.12)-(3.15), it may be concluded that identical relations describe the interaction in the impedance and generalized stiffness control. Consequently, the previously presented contact and coupled system analysis, as well as the established stability limits for various classes of impedance control interaction systems (e.g. continuous, sampled-data, with control delay etc.), are valid for Hogan’s target impedance model and can directly be applied in the control design. At first glance, the more desirable performance of the interaction force represents an obvious advantage of the stiffness target impedance model (4.4) over the convenient second-order impedance model (3.1). However, when analyzing the control realization several practical constraints may be identified, that cause the implementation and integration in the industrial robotic systems to be quite difficult. The main problem concerns the modification of the nominal motion by the feed-forward filter (Fig. 4.2). As demonstrated (Fig. 4.3), this filter provides a more suited command motion profile, when considering the interaction force performance, however, from another point of view, it produces several undesirable effects. In particular, reducing the speed of the nominal motion is not always allowed in the practice. As already pointed out, the interaction in the impedance control does not represent the task goals. Rather it is an inevitable consequence of the robot/environment mispositioning and uncertainties. According to the control scheme in (Fig. 4.2), the feed-forward motion filter should be realized at the impedance control servo layer. Thus, as opposed to the impedance control compensator, the motion filter is active while still in the free space, independent of the contact establishment. By these means, the motion
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could be slowed down even if the robot does not contact the environment. In almost all industrial applications this is quite disagreeable. Since the filter continues to propagate the motion noticeably after the interpolator finishes the motion sequence (Fig. 4.3), an additional synchronization between the interpolator and servo layer is required in order to manage the planed program motion sequences properly. Such synchronization is not easy to realize in existing industrial robot controllers. Though demonstrating advantages, the above-mentioned difficulties cause the generalized stiffness control scheme (Fig. 4.2) to be very complex to implement and integrate in commercial industrial robotic control systems. This control approach is more suitable for new controllers and non-time-critical applications, such as in space robotics. 4.3 Selection of Target Impedance Parameters – Impedance Control Design at Lower Control Layer In the adopted impedance control approach, the control design is split into two subtasks: one concerned with the realization of the target impedance model and the other with the design of the target parameters. The realization of the target impedance model using improved impedance control (3.68) was examined in the previous chapter (Section 3.5.3). The selection of the target impedance parameters, i.e. the matrices M t , Bt and K t , for a specific contact task will be addressed in this section. The theoretical and experimental assessment of the interaction system behavior during transition and the steady state, presented in previous sections with the main emphasis to highlight the specific influence of the target impedance parameters on the control performance, provides the basis for the design. However, before presenting the design, an algorithm for the generic impedance control task will be specified. As already mentioned, in principle the impedance control does not require any modification of the conventional nominal motion planning concepts. This means that a constrained motion can be planned analogously to the free-space motion tasks. When assuming ideal and known constraints, a compliant task can be specified in terms of nominal-motion sequences. The specific consideration relevant for the control design concerns the analysis of inaccuracies and deviations from the nominal task geometry. In addition, acceptable interaction force and torque magnitudes should be defined. For example, consider the
Practical Synthesis of Impedance Control
415
HINGE of a rotating knob (Fig. 4.5). At the motion planning layer (left figure) an ideal rotation around the vertical axis defines the nominal path (circle). At the servo impedance control layer, however, the designer should analyze worst-case deviations of the rotation axis position and orientation, in conjunction with admissible forces, which could be produced by these errors. These parameters represent the input to the impedance control design algorithms. The impedance control design procedure considers a common and idealized interaction task configuration. The task arrangement will be represented by a single-point and frictionless contact between a simple shaped end-effector and the environment surfaces. The contact surface, along which the ideal effector moves, is assumed to be smooth and continuous. The adopted task geometry allows intuitive reasoning and specification of contact task parameters to be applied for the target impedance selection. Taking into account that, even for a simple task geometry, the design of the impedance control for a real sampled data system still could be very complex, additional simplifications, such as neglecting the non-linear roundoff and quantization effects and considering dominant system delays only, are needed to obtain a reliable design problem.
Fig. 4.5 Impedance control task planning
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Dynamics and Robust Control of Robot-Environment Interaction
4.3.1 Specification of impedance control geometry The geometric model of the robot/environment interaction provides the basis for the analysis and synthesis of compliance tasks. Mason [3] has established the most common geometrical compliance model. This model defines the orthogonal compliance C-frame in Cartesian space in which the contact task is specified, as well as the constraint C-surface that separates free and constrained motion DOFs. This model, however, was mainly utilized in hybrid position/force control [4]. Lozano-Perez [5] has applied the constraint frame model to specify compliant motion using decoupled stiffness for each DOF relative to the Cframe. The decoupled compliance model has been primarily applied for describing compliant behavior of the RCC [6]. As mentioned, in this device the associated C-frame and his origin, referred to as the compliance center, exhibit decoupled compliant behavior manifested by a pure translation displacement if an interaction force is applied through the center, and analogously by a pure rotation if the pure torque (couple) has been applied. Since the directions of compliant displacements coincide with the force/torque acting lines, such behavior is termed directionally decoupled compliance [7]. The authors have generalized the decoupled stiffness concept by introducing accommodation center and mobility center respectively as the origins of frames in which directionally decoupled damping and inertial behavior have been realized. When the compliance, accommodation and mobility centers coincide in one coordinate frame, they build the admittance center. Goldenberg and Shimoga [8] have demonstrated that the common second-order spatial impedance system should meet the following conditions in order to have an admittance center: (i) The mass, damping and stiffness matrices are diagonal
M t = diag (M ti ); Bt = diag (Bti ); K t = diag (K ti ), i = 1… 6
(4.13)
(ii) The damping ratios and eigen-frequencies of the decoupled second-order impedance models are equal.
ξ ti =
Bti = ξ tj ; ω ti = 2 M ti K ti
K ti = ωtj M ti
i,j = 1… 6
(4.14)
A compliance model commonly used in robotics for modeling the interaction tasks [9] describes an ideal elastic interconnection (e.g. spring, elastic beam) between two rigid bodies representing the environment and the robot
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Practical Synthesis of Impedance Control
respectively (Fig. 4.6). It is well known that in an ideal elastic medium the force/displacement relationship for small displacements has a linear form expressed by Hook’s law (1676) of springs end elasticity [10]. Generally a 6x6 spatial stiffness matrix relates infinitesimal elastic displacements and corresponding rates of change of the interaction force. A linear and angular velocity pair {v ; ω} with respect to a frame of reference represents the spatial motion of rigid bodies base. The corresponding infinitesimal displacements pair is referred to as the twist. Accordingly, the {δr ; δθ } according to screw theory force/torque vector pair {F; M } referred to as the wrench gives the spatial representation of the load. The linear theory of elasticity usually considers infinitesimal displacements of a compliance counterpart from an unloaded (zero external force) equilibrium. In a reference coordinate system the following 6x1 spatial vectors represent the twist and wrench respectively
w = F ; δt = δr M δθ
(4.15)
Then the spatial stiffness relationship can be represented in the form
w = Kδt
(4.16)
In the following text we will consider the wrenches as generalized 6x1 spatial forces and according to the previous notation also use the symbol F. Hence, the generalized 6x1 spatial displacement ∆x will also be used for the twist.
Fig. 4.6 Elastically coupled rigid bodies: undeformed configuration (left) and deformed, strained configuration (right)
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Dynamics and Robust Control of Robot-Environment Interaction
The spatial stiffness matrix describes the compliance of a coupled system for three-dimensional displacements about an unloaded equilibrium. This matrix can be represented in the following general form [11]
K K = TT KC
KC K R
(4.17)
where 3x3 K T and K R are stiffness submatrices describing the pure translation and rotation respectively (i.e. the relationship between forces/torques and elastic displacements/rotations), while K C describes the coupling between translation/rotation DOFs (i.e. the relationship between forces/torques and elastic rotations/displacements respectively). In a zero-load state, the spatial stiffness matrix is necessarily symmetric. The submatrices K T , K R and K C characterize the elastic behavior in a manner that is analogous to the expression of the inertia dynamic effects expressed by the mass, inertia and linear-mass-moment matrices [12]. However, it is generally not possible to define the center of stiffness analogously to the mass center relative to which the inertia can be decoupled. By means of Lie algebra, Loncaric [13] has demonstrated that, for a compliantly supported spatial rigid body, there does not always exist a point at which the translational and rotational elasticity are decoupled, but that there exists a point at which they are “maximally decoupled”. The author referred to this point as the center of stiffness and the corresponding form of the stiffness matrix, with diagonal K C , is termed the normal-form. Loncaric’s center of stiffness is obtained by translating the reference coordinate frame to a position where K C is symmetric [14], provided det (K R − tr (K R )I ) ≠ 0 . Then a rotation can be found which diagonalizes K C . Selig [15] has defined different possible transformations to normalize the stiffness matrix. Howard et al. [11] have demonstrated that the form of the spatial Cartesian stiffness matrix, associated with the linear elastic coupling between two rigid bodies, depends on the reference frame. The stiffness matrix in the moving (i.e. body-fixed) frame is the transpose of the stiffness matrix in the reference inertial frame. In general, the spatial stiffness matrix is asymmetric if the resulting equilibrium wrench is not zero (loaded state). It is possible to find a basis frame in which it becomes symmetric. The resulting symmetric stiffness frame is the average of the stiffness matrix in the moving frame and that in the fixed frame.
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Practical Synthesis of Impedance Control
The stiffness matrix can be, as second-order tensors, decomposed into symmetric and skew-symmetric counterparts [14]. The stiffness and its inverse compliance matrix can be normalized relative to two distinctive points referred to as the center-of-stiffness and the center-ofcompliance, respectively. Normally these two centers are distinct. Ciblak and Lipkin [16] introduced a third special point referred to as the center-of-elasticity representing the intersection point of the eigenwrenches, i.e. eigentwist threes (reciprocal three-systems). They show that whenever these three points coincide, the stiffness matrix becomes diagonal. All lines through these centers are compliant axes. Several authors have considered the realization of a stiffness matrix by linear springs [17, 18]. This problem is relevant for designing RCC devices. Ciblak and Lipkin [19] have demonstrated that a stiffness model of range r is realizable with at least r linear springs. Let us consider the well-known transformations specifying the propagation of the velocities (twists) and forces (wrenches) between two distinct points A and B of a rigid body (Fig. 4.6). Generally, different frames can be associated to these points. The velocity (twist) propagation is defined by:
xɺB = BJ A xɺ A
(4.18)
where the Jacobian matrix has the form B
B RA JA = 0
B − r AB B RA B RA
(4.19)
B
where r AB is a skew-symmetric matrix (used to present the vector product in
matrix form) associated with the position vector between A and B ( rAB ) and expressed in the frame B; B RA is the rotation matrix between frames A and B; and 0 denotes a 3x3 zero submatrix. The following transformation relates skewsymmetric matrices expressed in the two frames B
A
r AB B RA = BRA r AB The work on the virtual elastic displacement must be the same during the rigid body displacement, independent of the reference frame T
T
FB δxB = FA δx A Hence the propagation of forces (wrenches) is expressed by
(4.20)
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Dynamics and Robust Control of Robot-Environment Interaction −T
FB = BJ A FA B
JA
B RA = B B − r AB RA
−T
B
0 RA
(4.21)
The relation between the stiffness matrices describing the elastic deformation of the attached compliant element at A and B, respectively (Fig. 4.6), can be derived based on the virtual work principle. Substituting (4.18) and (4.21) into (4.20) and considering (4.16)-(4.17), we obtain −T
K B = BJ A K A B J A
−1
(4.22) Assuming an unloaded equilibrium, i.e. the symmetric stiffness matrix form (4.17), we can write the relations for the stiffness matrix component transformations T
K BT = BRA K AT B RA
T
T
B
K B C = BRA K AC B RA + BRA K AT B RA r AB T
B
T
B
K B R = BRA K A R B RA − r AB B RA K AT B RA r AB + BRA K AC B
TB
T
B
RA r AB
(4.23)
T
− r AB B RA K AC B RA
The inverse transformation from B to A provides T
K A = BJ A K B B J A
(4.24)
i.e. T
K AT = BRA K BT B RA T
T
B
K AC = BRA K B C B RA + BRA K BT r AB B RA T
T
B
B
T
T
B
K A R = BRA K A R B RA − BRA r AB K AT r AB B RA − BRA K AC r AB B RA T
(4.25)
B
+ BRA r AB K AC B RA Obviously the stiffness matrix depends on the rigid body transformation, i.e. on the selection of the frame of reference. Hence, the specification and understanding of the target compliance becomes quite complicated. In the common linear second-order impedance target system (3.1) the mass and damping elements conveniently specify the system dynamics (i.e. transition processes), while the stiffness apparently provides a simple relationship between
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Practical Synthesis of Impedance Control
the displacements and forces (steady-state behavior). However, due to the configuration dependence of the spatial stiffness, it is difficult to specify and select the compliance parameters as well as to understand the spatial compliance relationships. The reasoning about admissible or realizable wrenches during design is not a simple task, and the practical goal is to explore means of facilitating it. In some compliance analysis and design cases it is simpler to consider the compliance instead of the stiffness matrix. This matrix provides the displacement reaction to the applied load. δx = CF (4.26) Assume again an unloaded configuration with the symmetric compliance matrix having the form
C C = K −1 = TT CC
CC CR
(4.27)
where, analogous to (4.17), the submatrices CT, CR and CC describe the translational, rotational and coupling effects respectively. The rigid body transformation (4.18)-(4.20) causes the following changes in the compliance matrix
C B = B J AC A B J A −1
T
C A = BJ A CB B J A
(4.28)
−T
i.e. in the developed form T
B
T
T
B
B
CBT = BRAC AT B RA − r AB B RAC A R B RA r AB + BRAC AC B RA r AB B
− r AB B RAC AC T
TB
T
RA B
(4.29)
T
CB C = BRAC AC B RA − r AB B RAC A R B RA T
CB R = BRAC A R B RA and T
T
B
B
T
B
C AT = BRA CBT B RA − BRA r ABCB R r AB B RA + BRA r ABCB C T
RA
B
− BRA CB C r AB B RA T
T
B
C AC = BRA CB C B RA + BRA CB R r AB B RA T
TB
C A R = B RA C B R B RA
(4.30)
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Dynamics and Robust Control of Robot-Environment Interaction
The design problem would be simplified if the compliance parameters could be chosen independently of the interaction system configuration (e.g. reference frame, contact point etc.). In the compliance control design, it is quite desirable to specify the target stiffness matrix independently of the robot/environment configuration. Considering the equations describing the transformation of stiffness and compliance submatrices dependent on the rigid-body change of the coordinates (4.23)-(4.30), it appears that the application of principal (eigen), i.e. characteristic (singular) values is quite suitable for the specification of the stiffness (compliance) submatrices. Since the stiffness submatrices are positive semi-definite, the eigenvalues are identical to the singular ones. The characteristic values are independent on the rigid-body rotation occurring in the transformations. The “cross-product” multiplication in the above expressions can be interpreted by means of the displacements, i.e. the force transformations from one to another point of a rigid body. This multiplication reduces the rank of the corresponding matrix elements in expressions. Hence, the appropriate parameterization of the stiffness and compliance matrices involves the following parameter sets: i) The set of non-spatial parameters consisting of the principal values of the submatrices KT, KC and KR, i.e. CT, CC and CR. Based on the principal value transformation (i.e. singular value decomposition for symmetric positive semidefinite matrices)
K T = U T S KTU T
T
(4.31)
where UT is an orthonormal matrix with the principal axes of the translational stiffness submatrix, while S KT = diag ktx , kty , ktz is the matrix of the principal
(
)
translational stiffnesses. By analogy, the principal rotational and coupling stiffness and compliances will be respectively denoted as:
S KR = diag (kϕx , kϕy , kϕz ), S KC = diag (ktϕx , ktϕy , ktϕz ) SCT = diag (ctx , cty , ctz ) , SCR = diag (cϕx , cϕy , cϕz ) , SCC = diag (ctϕx , ctϕy , ctϕz ) ii) The set of spatial dependent parameters describing the rigid body B
transformation: B RA and r AB .
Practical Synthesis of Impedance Control
423
The above transformations and parameterization provide the basis for the computation of Loncaric’s centers of stiffness and compliance [15] for the given stiffness matrix and rigid body configuration. Commonly the center of compliance is defined as the point at which a pure force produces a pure translation in the same direction. This can be written as
CT C T C
CC F F 0 = α 0 CR
(4.32)
i.e.
CT F = αF
(4.33) which, for an arbitrary applied force, requires the translational compliance to be proportional to the identity matrix at the compliance center
CT = αI
(4.34) Likewise for the torques and rotations, if the compliance center is defined as the point where a pure torque results in a pure rotation about the same axis, we obtain
CR = β I
(4.35) This means that the diagonal compliance matrix with uniform translational and rotational compliance elements
CC = αI 0
0 βI
(4.36)
provides a trivial solution for the center of compliance. As demonstrated in [15] such a compliance matrix can be realized using stretched springs. In practice, we can weaken the definition of the center of compliance by permitting the applied force/torque to produce pure translations/rotations but not in the same direction [20]. This case is convenient when it is needed to eliminate the compliance in some directions. In this case, the appropriate compliance matrix is also diagonal but has mutually different elements
0 diag (ctx , cty , ctz ) CC = 0 diag (cϕx , cϕy , cϕz )
(4.37)
The goal of the compliance control is to realize the target impedance (i.e. compliance) by means of the appropriate control algorithm. For a given interaction task, we can put the center of compliance in a desired location, then select the target compliance values at the center and realize the specification by
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Dynamics and Robust Control of Robot-Environment Interaction
the control. The goal is thereby to compensate for possible model uncertainties. Although this is a simplification, the selection of the compliance center C and the compliance elements is still quite a difficult problem that will be discussed in the following sections. One of the most common approaches to word modeling (specification of robot and object positions) in robot programs is based on coordinate frames. Beside coordinate frames that are convenient for the programming of the robot motion in the free space (e.g. robot base B-, end-point E-, tool T-frame etc.), the compliance control system includes two new frames specific to the compliant motion programming: force sensing S and compliance frame C. The S-frame is a force-sensor specific frame in which the forces and torques are measured. This frame is commonly defined relative to the robot end-point E. With respect to the C-frame, the target impedance behavior (robot impedance reaction) is specified and controlled. Since the location of the C-frame depends on the current task, we have chosen, as most convenient, to specify the C-frame relative to the task Tframe (see the next section) while taking into account that the T-frame is also a variable frame selected to meet a specific task motion requirement. Usually, the desired robot position specifies the location of the T-frame (specified relatively to the E-frame, e.g. tip of the tool) with respect to an object frame. The selection of C- frame for different tasks will be considered in the next section. A specific problem in the selection of the target compliance parameters concerns the fact that the center of compliance and the actual interaction point between the robot and the environment must not coincide. As will be shown, the center C has to be selected to meet various task specifications and requirements. A quite important requirement is that the interaction system must reach a stable equilibrium around C, while considering all interaction forces and torques. The actual interaction point, in general, should not fulfill this condition. Specific interaction tasks involve several interaction points that are switched during execution. That additionally complicates the specification and handling of the compliance parameters. The relations (4.23)-(4.25) and (4.29)-(4.31) describing the rigid body transformation facilitate the selection of the compliance parameters. Consider, for example, the interaction case presented in (Fig. 4.7) with the interaction point in A and the compliance center in B≡C. If in the interaction point A the target compliance is selected to allow only the displacement in the vertical direction, the transformation (4.29)-(4.31) provides the same compliance in the point. However, if a simple rotational compliance is needed in B≡C only around y-axis (Fig. 4.8), the equivalent compliance matrix in A becomes quite complex.
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Practical Synthesis of Impedance Control
This compliance matrix includes coupling elements, which ensures that a force/torque applied in A produces also rotations/displacements around B. In practice, the center C and target compliance should be selected to describe given tasks as simply as possible.
0 C C C A = AT ⇒ C B = AT 0 0 0 C AT = diag(0, 0, ktz )
0 0
Fig. 4.7 Pure translation compliance
− rC BR r CA = T − C BR r
rC BR ⇐ C B = 0 0 0 C BR C BR CBR = diag (0, kϕy , 0)
Fig. 4.8 Pure rotational compliance
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Dynamics and Robust Control of Robot-Environment Interaction
The above considerations can likewise be applied for the specification of the target damping and inertia matrices. In addition, to well-known mass-center and principal inertia paradigms [12] we can in the same manner also introduce the center of damping and principal damping coefficients. All the above-mentioned centers, as well as the non-spatial and spatial parameter sets specifying stiffness (compliance), damping and inertia properties, can be independently selected and transformed into a point of interest. However, for the sake of simplicity, it is convenient to assume a decoupled diagonal target model at the unique center C with coinciding principal axes, without losing generality or performance of the impedance control. The magnitude of the principal stiffness, damping and mass thereby can be selected arbitrarily in each of the directions corresponding to the specific task. Thus, the impedance control synthesis algorithm will concern the design of the common second-order target systems for the translational and rotational C-frame directions (see Fig. 3.2), respectively.
Gti (s ) = M ti s 2 + Bti s + K ti
(i = x,y,z )
Gtj ( s ) = J tj s 2 + Βtj s + Κ tj
(j =ϕ
x
,ϕ y ,ϕ z
(4.38)
)
(4.39)
where Jt, Bt and Kt are the equivalent angular inertia, damping and stiffness coefficients, respectively. In some practical cases, the angular deviation and torque could not be directly specified from the given task, since they are not relevant for a particular interaction problem for which force and position displacements are specified. A typical configuration for this case is presented in (Fig. 4.7-8), where the C-frame is shifted relative to the force interaction direction for the offset r. However, the corresponding rotational parameters can easily be obtained based on the virtual work principles, i.e. rigid-body transformations (4.23)-(4.31) derived for the target stiffness and compliance parameters. In the considered case, we will have
J tj = M ti r 2 Btj = Bti r 2
(i = x, y, z;
j = ϕ x ,ϕ y ,ϕ z )
(4.40)
K tj = K ti r 2 4.3.2 Specification of input design parameters - user interface Taking into account the above considerations and assumptions, we can now specify the following typical impedance control tasks.
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Practical Synthesis of Impedance Control
Definition 4.1 (Impedance control basic design task specification) Given is the following information: a nominal path of the robot end-effector in contact with a nominal constrained surface, worst case estimates of the robot and environmental position tolerances, as well as the maximum acceptable interaction force and torque magnitudes. Design the second-order impedance control model parameters (4.38-39) that will meet the task constraints and ensure a stable contact transition and interaction in spite of the environmental and robot parameter inaccuracies. Definition 4.2 (User interface parameters) According to the above tasks specification the following sets of input parameters are needed: ● Maximum geometrical deviation (tolerances) of the constraint specified in terms of maximum displacement and/or rotation deviation magnitudes
∆pmax , ∆ϕ max In the case of a complex task geometry (Fig. 4.5), the user should compute the worst-case composite position deviation involving the displacement components due to the translational and rotational errors, which assuming small rotation, can be expressed as
∆pmax ≈ ∆pmax + R∆ϕ max
(4.41)
where R represents the position offset between the end-effector contact point and the task specific (e.g. C-frame) rotation axis (Fig. 4.7-4.8). Considering the nominal constraints, we can define the maximum nominal penetration vector in the C-frame
[
∗
p 0 max = ∆p xmax
∆p ymax
∆ϕ xmax
∆p zmax
∆ϕ ymax
∆ϕ zmax
]
T
(4.42)
● Maximum admissible interaction steady-state force and torque magnitudes along each C-frame direction Fmax components. ∗
[
Fmax = Fxmax
Fymax
Fzmax
*
Μ xmax
involving three force and three torque
Μ ymax
Μ zmax
]
T
(4.43)
● Maximum target frequency ωt . Generally it should satisfy the feasiblity condition (3.177) of the external impedance control around the internal position control loop, and specifically, it should be less than the critical structural frequency of the environment in order to avoid resonant interaction effects.
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Dynamics and Robust Control of Robot-Environment Interaction
● The stiffness of the worst-case environment, with respect to the C-frame, should be roughly estimated and expressed via a diagonal stiffness matrix
K e = diag (K ex ,K ey ,K ez )
(4.44)
In general by considering different C- and interaction-frame (contact point) locations (Fig. 4.7-4.8), we can transform the environmental stiffness based on (4.23)-(4.25). The environmental stiffness then becomes non-diagonal. However, the assumption (4.44) is common in basic design. Additionally, the rotational stiffness can also be specified directly or by using the equivalent of the translational stiffness (4.40). Since generally the estimation of the environmental stiffness could be very complex and uncertain, the value for the Ke components can be specified using a linguistic (“fuzzy”) description, such as very elastic, elastic, medium stiff, stiff and very stiff. Based on the impedance and force control practice with various passive engineering environments, the corresponding heuristic values for the stiffness coefficients, including environment types as examples, are given in Table 4.1. It should be noted that the given values express the order of the stiffness magnitude and are very roughly estimated. Using the robust impedance control design, the uncertainties of the environmental stiffness can be compensated for. Using the experiments with the predesigned impedance controller, the initial environmental stiffness data can be improved. The stiffness estimation experiments using the position controller and position and force sensor could be very dangerous and should be avoided. Table 4.1 Linguistic (fuzzy) environmental stiffness specifications and corresponding numerical values Environment
K e (N/m)
Example
very elastic elastic medium stiff stiff very stiff
<5000 10000 25000 50000 >100000
non-metallic surfaces springs thin plates thick plates metallic block
● Sampling time period T for the implementation of the external impedance control and estimate of the maximum impedance control system delay τ max . The selection of T concerns the fundamental problem of the discrete-time control
Practical Synthesis of Impedance Control
429
system design [39]. Generally, T should fulfill the sampling theorem (it should be less than the smallest time constants of the impedance control compensator), when considering the outer/inner-loop control structure (Fig. 3.3). It should be equal to or higher than the sampling time of the internal position control loop. This condition is in agreement with higher computation efforts needed to process the contact force information and compute the impedance control law. The time delay usually can be expressed as τ max = nT (commonly n = 1). The following sets of parameters should also be specified. Indeed these parameters are not explicitly applied in the target impedance design procedure, but are very useful to test the design outcomes by simulation and, if needed, to improve the design by repeating some steps and changing the input parameters. These parameters involves: ●
{x
0s
, x0e , v0max, a 0max , aɺ 0max }: start and end position, maximum velocity,
acceleration and jerk, specifying the robot nominal path x0 (t) and the robot endeffector motion. ● Considering the dynamical behavior of the force transition process with relatively high force overshoots, a tolerable overshot ο max index should also be specified. Last but not least, a very important precondition for the design and implementation of the impedance controller is knowledge of the internal position control performance. More precisely, the estimate of the continuous or discrete time transfer function of the closed position control loop is required. The favorable performance of the industrial position control systems in joint- and Cartesian-space analyzed in [22], manifested by diagonal dominance, normality and spatial roundness, significantly simplifies the determination of the needed transfer function. Essentially, only one transfer function should be estimated, representing the closed loop performance independent of the considered motion space, C-frame location and motion direction. This transfer function has the form, in the s- and z-domains, respectively
~ Num_s (s ) G p (s ) = Den_s (s ) ~ Num_z ( z ) G p (z ) = Den_z ( z )
(4.45)
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Dynamics and Robust Control of Robot-Environment Interaction
where Num and Den denotes numerator and denominator polynomials in descending powers of s and z, generally having the form 0
0
i= N
i=D
0
0
i=Z
i =Z
Num_s (s ) = ∑ ni s i ; Den_s (s ) = ∑ d i s i Num_z ( z ) = ∑ bi z i ; Den_z ( z ) = ∑ ai z i ~ Since G p (s ) is the proper transfer function, the order of the denominator is greater than, or equal to, the order of the numerator N ≤ D . Furthermore, ~ assume that G p (s ) is a stable minimum-phase transfer function, so that all its poles and zeros are in the open left half-plane. This condition allows the use of
~
the inverse transfer function G p (s ) in the impedance control law (3.68). To obtain a realizable compensator Gf (s) for the adopted second order target impedance, it is further required that D − N ≤ 2 . Usually the industrial position controller is tuned to obtain a well-damped aperiodic response [22]. Therefore, the solution that is almost universally applicable is to adopt a second-order [22], or even first-order, closed loop position control estimate in the forms
~ G p (s ) = ~ G p (s ) =
ωp
2
s 2 + 2ξ p ω p s + ω p
2
(4.46)
1 = s + ω p Tp s + 1 ωp
~
Since the impedance control law (3.68) utilizes the inverse of G p (s ) , it is favorable to select the first-order estimate of the closed loop position control transfer function in order to avoid possible excitations of high-order system oscillating modes caused by differentiation of the force signals. There are several relatively simple ways to obtain the coefficients of the above transfer functions, such as using step-responses of the individual axes, common tracking experiments in joint or Cartesian space, or dynamic modeling of the robot position control system. Various control tools, such as the MATLAB Identification Toolbox or IDCON (ExpertControl), are available to support the estimation of transfer function parameters in discrete or continuous
Practical Synthesis of Impedance Control
431
time. The bandwidth ωb , which is required as an upper limit of the target frequency in the design algorithm, can thereby also be estimated. Assuming the position control transfer function to satisfy the robust stability
~
condition G p (s ) < 1 , the impedance control design that ensures stable and robust interaction is significantly simplified. Essentially, the position control parameters are not explicitly required in the design. However, they are crucial for the realization and implementation of the impedance control compensator. In principle, it is possible to design the control parameters for each task separately, and thus to obtain optimal control performance in each practical case. However, this is not a practical approach. A large and complicated database of impedance control parameters may be required to integrate and distinguish all relevant cases in an industrial robot controller. Furthermore, the outcomes of the control adjustment for each specific task could be insignificant. Therefore, it is more reliable to perform the design for a group of similar tasks to be realized under similar environmental conditions. In this case the input parameters should reflect the worst-case conditions for the considered group of tasks. Taking into account the contact stability and impedance control performance, the worst-case conditions are characterized by: ● maximum environmental stiffness coefficient, ● maximum position deviation, and ● minimum allowable interaction force and torque, for the considered group of tasks. 4.3.3 Design algorithm The synthesis of the impedance control for a practical contact task is based on the following: the theoretical framework elaborated in the previous chapter, focusing on the performance of industrial robots, improved impedance control law, and coupled and contact stability respectively, as well as the considerations of the impedance control task specification presented in this chapter. In spite of this framework, however, it is still difficult to define a general control synthesis algorithm that can be applied to each practical case. A real design procedure, in great measure, depends on the specific task conditions and the designer’s experience. In effect, the control design is a very creative job, which is difficult to completely automate or schematize.
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In spite of this, we will try here to define an impedance control design procedure for a generalized contact task. The design algorithm is, to a great extent, based on our own long-years practice in integrating the impedance control into real industrial robotic systems. This algorithm does not include the identification of the robot position control system, which is assumed to be known. A solution to this problem is illustrated in a practical example concerning the position control loop identification of a new-generation industrial robot. Algorithm 4.1 Design of the impedance control parameters for typical impedance interaction tasks and input set of parameters. Step 1: Computation of the target stiffness parameter The first step in the impedance control design procedure is to select a target stiffness parameter Kt along each C-frame Cartesian direction (Fig. 3.2) (4.38)(4.39). Based on the steady-state condition (3.170), maximum target stiffness is defined by maximal object position tolerances, i.e. maximal penetration p0max, and force limits Fmax. Using a simple linear spring model, for the displacement DOF’s it can be written
K ti ≤ K timax =
1 p0imax 1 − Fimax K e
(i = x,y,z )
(4.47)
or assuming a stiff environment
K ti ≤ K timax ≈
Fimax p0imax
(i = x,y,z )
N m
(4.48)
Likewise, for the rotational DOF’s the target stiffness is obtained as
Κ tj ≤ Κ tjmax ≈
M jmax ∆ϕ 0jmax
(j =ϕ
x
,ϕ y ,ϕ z
)
Nm rad
(4.49)
The angular stiffness can also be computed based on (4.40) using translational parameters and the worst-case offset-estimation between the interaction point and C (Fig. 4.7-4.8)
Κ tj ≤ Κ tjmax = K timax R 2 ≈
Fimax R 2 p0imax
(i = x, y,z;
j = ϕ x ,ϕ y ,ϕ z
)
Nm rad (4.50)
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The above formulas define upper limits of the target stiffness. The choice of smaller target stiffness leads to even smaller forces than requested. However, elastic target systems are quite sensitive to disturbances and noise in the force sensor and the position control system. Zero target stiffness leads to the pure damping impedance behavior (integral reaction) that tries to reduce the steadystate force to zero. Generally, very elastic systems require high relative damping to stabilize interaction, which causes a sluggish system reaction. By selecting the target stiffness, the target stiffness ratio κ = K e K t has also been computed for further usage in the design procedure. Step 2: Computation of target mass parameters In the next step, the target mass has to be determined. In almost all interaction tasks with a passive environment, the target mass usually is not directly specified. In this case, the minimum target mass is determined by a given maximum target frequency and by the maximum stiffness parameters computed in the previous step
M ti ≥ M timin = J tj ≥ J tjmin =
K timax
ωt 2
Κ tjmax
ωt2
(i = x, y, z ) [ kg ]
(j =ϕ
x
,ϕ y ,ϕ z
)
(4.51)
kgm 2
For the task class illustrated in (Fig. 4.8), the rotational inertia can be computed based on the translational/rotational systems analogy (4.40). In specific applications, such as actuating an object or passive mechanism, the target mass affects the interaction system dynamics and may be specified as the task parameter. In this situation, it should be tested if the target stiffness and mass satisfy the target system feasibility condition (3.177)
ωt =
Kt ≤ 0.5⋯0.8 ωb Mt
ωt =
Kt ≤ ωse Mt
(4.52)
If these inequalities are not fulfilled, the target mass and stiffness values should then be adjusted. The choice of larger target masses is coupled with higher impact forces during transition and lower target frequencies, while the
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Dynamics and Robust Control of Robot-Environment Interaction
mass reduction is limited by the restricted target frequency. The target stiffness is bounded by the maximum steady-state force for the predicted position deviation. Step 3: Computation of target damping ratio The last parameter that should be determined is the target-damping ratio ξ t , which is essential to ensure stable contact transition and interaction. The selection of this parameter is based on the contact transition stability analysis. Concerning uncertain environmental stiffness, the most reliable criterion for computation of ξ t is the robust stability criterion (3.169). For the real sampleddata delayed interaction system this condition imposes the condition
[I + z G (z) Gˆ (z )]
−1
−1
n
e
<1
t
(4.53)
∞
The flow-chart of a practical algorithm for the computation of ξt written in the MATLAB syntax is presented in (Fig. 3.48). The start value of ξt in this algorithm is obtained using the robust stability criterion for the ideal non-delayed system. Then, in each step the initial damping ratio is increased for a constant rate of change and the linear discrete interaction model is computed. This procedure is repeated untill condition (4.53) is fulfilled. The last iteration provides the desired target damping ratio at robust contact and the coupled stability limit. As demonstrated in the experiments (Fig. 3.53-3.61), this solution almost ensures a safe stable transition, even in an uncertain environment. The obtained contact stability margin and force overshoots index, i.e. the entire contact transition and interaction behavior (e.g. higher force oscillations), however, may be non-optimal (i.e. too conservative) with the obtained solution. In order to obtain a more suitable performance it is reasonable to allow the designer to correct the ξˆt_rd value obtained in robust design. By this means the designer’s own experience and knowledge can be efficiently applied in order to meet specific task goals and to take into account particular environment performance. For this purpose, the algorithm also computes the smaller damping ratio satisfying the passivity-based contact stability ξˆt_pb . The passivity-based contact stability criterion (3.170), written in the equivalent infinity–norm form, imposes
Practical Synthesis of Impedance Control
[I + 2 z G (z) Gˆ (z )]
−1
−1
n
e
<1
t
435
(4.54)
∞
This stability criterion was clearly identified in experiments to provide the most accurate practical contact stability limit. The damping limit ξˆt_pb can be utilized as the lower limit for the reduction of the predesigned damping coefficient computed by robust design. This reduction is relevant, for example, in the case when a faster force reaction is most important for the practical contact task, whereby the force oscillations are not critical. However, if the force overshoots and reaction are not significant, rather a smooth force performance is crucial for the given task, the initial damping can be increased, even over the theoretical robust stability limit. Thus practically, the damping ratio is obtained as
ξt = wξ ξˆt_rd
(4.55)
where the damping weighting factor wξ has the value
ξ t_pb ξ t_rd
≤ wξ ≤ n
(4.56)
where ξt_rd and ξt_pb denote damping ratios at stability limits obtained by passivity-based and robust contact stability criteria respectively, while n is commonly n = 2…3 . Indeed, the above-proposed procedure for target damping selection is based on a sufficient stability condition. As was demonstrated, in common contact transition problems the robust control design almost provides the target parameters “on the safe side”, ensuring stable transition and interaction in spite of environmental uncertainties. The proposed impedance control synthesis appears to be very efficient when the designer has gained considerable experience with impedance control and contact tasks in a specific environment. Then, this procedure will allow the control designer to optimally utilize his own practice. However, we cannot exclude possibilities that in particular interaction cases, mostly in contact with a non-linear or highly variable environment, the design can be very conservative, or even more the contact stability may be jeopardized by specific environmental effects (e.g. elastic oscillations, etc.). Robust control design provides a systematic approach to synthesize a reliable impedance control capable of interacting with an insufficiently known, variable or slightly non-linear environment, and thereby retaining a very practical contact
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stability design based on the linearized interaction model. This approach allows the integration of a simple model of the perturbation effects that influence the contact transition and stability. A simple mathematical formalism to consider perturbations is provided by the multiplicative model (Fig. 3.10) defined by (3.41)-(3.45). In fact, this model is not optimal for every application. However, in conjunction with developed robust contact stability theory it provides very practical results. The robust and passivity-based contact stability conditions (3.169-3.170) taking into account the multiplicative perturbations upon an interaction system impose, respectively
[ W ( z )[I + 2 z G (z)
]
−1 W ( z ) I + z n Ge (z) −1Gˆ t ( z )
n
e
−1
]
<1 ∞
−1 Gˆ t ( z )
(4.57)
<1 ∞
where W (z) is a stable weighting transfer function used for the design, or used to reflect the specific design objective. This function describes uncertainties, i.e. deviations between the actual and nominal model. For example, W (z) may be used to model sensor noise or environmental oscillations, or to reflect the force perturbation effects due to the nominal motion or interaction force [23]. The appropriate choice of weights for a specific contact tasks design problem is not trivial. If the weights are selected without any physical base, so that these weights may be considered as design parameters, the control synthesis based on (4.57) become equivalent to the practical weighting of the target damping parameter (4.55)-(4.56). In numerous impedance control implementations in industrial robot systems interacting with various technical environments we have achieved quite satisfactory results using a basic robust stability design, without considering the perturbation effects, e.g. due to friction. However, the practical selection of the weighting functions for specific interaction problems represents an attractive future research topic. Step 4: Computation of impedance control compensator After all of the target impedance parameters have been selected, the next step is to obtain the sampled-time impedance control compensator transfer function to be implemented in the control system. For the estimated position control closed loop and target impedance transfer functions, the compensator Gf (z) is directly computed based on
Practical Synthesis of Impedance Control
~ −1 −1 G f ( z ) = G p ( z )Gt ( z )
437
(4.58)
Generally, this compensator is computed for each C-frame direction separately if various performances are required, or at least for translational and rotational DOF’s. Step 5: Simulation tests of the target model performance For the final selection of the impedance control parameters it is useful to test the system performance by simulation using a simple SISO interaction model (Fig. 4.9). If some of the performance indices (e.g. stability margin, maximum force or force overshoot) do not satisfy the specification, the designer can repeat one or several of above steps untill satisfactory interaction behavior is reached. The simulation model should include all effects relevant for the preliminary system tests and assessment of the impedance control design outcomes. The following modules and effects are relevant: position interpolator, discrete impedance and position controllers, robot plant model, worst-case stiff environmental model, variable contact point location, etc. Various non-linear effects such as the control lag, quantization etc., are useful to obtain more realistic performance in the real time control system. Force signal filtering as well as dynamic perturbations are involved to test the robustness. The impact effect can be tested using different approaching velocities. A more complex simulation model of a typical non-linear multivariable industrial robotic system (ARCOS controller and Manutec r3 robot) under impedance control is presented in (Fig. 4.10). This model utilizes the Simulink Toolset ROBOTICS [24]. As mentioned, this Toolset uses an identified MIMO model of the Manutec r3 robotic system and can accurately be applied for the testing of the impedance control performance (Fig. 3.24). Step 6: Experimental design evaluation After performing satisfactory simulation tests, it is recommended to evaluate the results of the design by simple experiments before starting the complex task applications. The elementary practical experiments include: ● testing of the desired target model realization in the free-space (Fig. 3.133.16), ● simple contact transition, i.e. impact test (Fig. 3.17-3.20). Again, if test results are not satisfactory, it is necessary to redesign control parameters and/or, if permitted to change the requirements.
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Dynamics and Robust Control of Robot-Environment Interaction
Fig. 4.9 SISO interaction model for tests of synthesized control performance
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439
Fig. 4.10 MIMO impedance control simulation model implemented using the ROBOTICS tools
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Dynamics and Robust Control of Robot-Environment Interaction
The developed impedance control design algorithms are implemented in MATLAB. The particular functions for the automatic computation of impedance control gains for the SPARCO control systems [25, 26] are also implemented and integrated in the MATLAB Impedance Control Design Toolbox [27]. In conjunction with general MATLAB functions and specific MATLAB control design toolboxes, this toolbox provides all the necessary functions to synthesize and test the impedance control in an industrial robot environment. The design steps described above are illustrated in the following examples of the impedance control design and evaluation experiments in commercial robotic systems that belong to the latest generation of industrial robots. Example 4.2: Impedance control design for Comau SMART-S4 industrial-robot The aim of this example is to illustrate the above presented impedance control design algorithm. In this example we will consider a general contact transition and interaction task. The basic contact task consists of the realization of a stable contact with an object or surface, similar to in the configuration presented in (Fig. 3.17). The impedance control is synthesized for the Comau SMART-S4 robot intended to be implemented in a quite complex cooperating robot assembly task [28]. The standard control system COMAU-C3G, including the SPARCO project development and extension SW and HW [26] has been applied to the experiments. It is worth mentioning that the design steps are commonly repeated depending on the evaluation experiments in order to improve the design results. The experiments on SMART S4 that will be used to illustrate the design outcomes are presented in [29]. According to the design procedure, before starting with the target impedance design algorithm, the input data that include the task, environment and robot control parameters, should be specified, i.e. identified. Initial Step: Specification of task parameters and identification of robot position controller Potentially there are several strategies to identify the position control closed loop transfer functions of the industrial S4-robot. The Comau C3G robot control system provides a simple specific solution based on the open test-bed controller C3G-OPEN and so-called PC-C3LINK [30]. This system allows the user to
Practical Synthesis of Impedance Control
441
execute a control algorithm in the robot control system using a PC with the BIT3 communication adapter. This system also supports acquiring joint positions with a high sample rate (1 ms). By these means a simple step response can be implemented and realized at the joint control level. Considering the usually high gains of the industrial position controller, a relatively small joint axis step (e.g. < 0.2 grad) should thereby be given. This test, however, is specific for the COMAU robots and cannot be applied in common control systems that usually do not permit the realization of step signals by bypassing the motion interpolator. It should be mentioned that recently other industrial robot producers have also tried to offer similar open interfaces mainly for research purposes. In convenient industrial robot controllers, a “step response” can be realized only approximately by selecting the maximum accelerated nominal motion. Considering the performance of industrial robotic systems any tracking experiment of relatively fast dynamic trajectories in joint or Cartesian space can also be used for the identification. The step response of the Comau SMART-S4-robot, realized using the C3GOPEN system is presented in (Fig. 4.11). This example is again the proof of practical controller tuning in conventional industrial robot control systems to achieve quite similar local axes (denoted by 1, 2, …,6) performance. Indeed, the more marked non-linear effects in joints 1 and 2 (Fig. 4.11) (e.g. due to higher friction and oscillatory effects) are not relevant for the system performance, especially at relatively low interaction velocities. The independent joint control tracking of the nominal trajectories is presented in (Fig. 4.12), while (Fig. 4.134.14) show a simple Cartesian trajectory tracking experiments for relatively slow and fast linear motion, respectively. In these experiments the nominal trajectories are given in the internal joint coordinate space. Again, similar control performance could be observed. Obviously, the tracking of fast trajectories (Fig. 4.13) is poorer than that of the slow ones (Fig. 4.14). The similarity of the local joint controller (i.e. the correspondence of the industrial robot S4 position control performance in various Cartesian directions) becomes explicit after the identification of the linear transfer functions based on the above experiments. The identification results for each axis controller in the sdomain are obtained using the IDCON toolbox. The results specify the natural frequencies and damping factors of the first and second order estimates of the position control transfer functions (4.46) for various identification experiments. The following transfer function estimates representing all axes are obtained by combining the experiments:
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Dynamics and Robust Control of Robot-Environment Interaction
11.3 ~ G p (s ) = s + 11.3 (4.59) 17.52 ~ G p (s ) = 2 s + 2 * 0.86 *17.5s + 17.52 where ω p = 11.3 (rad/s) and ω p = 17.5 (rad/s) are frequencies of the firstand second-order transfer function, and ξ p = 0.86 is the damping ratio. Compared to the previous generation of industrial robots (e.g. Manutec r3 and Comau SMART-3 6.12R), the obtained frequencies are appreciably lower. This result is reasonable when considering that the mechanical structures of the preceding robots were significantly stiffer and heavier. By optimizing the dimension of the new robots in order to reduce the mass, the elastic structural effects become dominant. Consequently, the bandwidth of the position controller is reduced. The validation of the identified transfer functions for the step response experiment is presented in (Fig. 4.15). As already mentioned, the first-order position transfer function providing a lower frequency is advantageous for computing the impedance control compensator (4.58). The remaining design parameters forming the impedance control design “user interface” have also to be specified. These parameters specify bounds on the interaction force and the position errors (nominal penetration), as well as the “worst-case” task environment and approach motion parameters. In this example we will undertake the task requirements specific to the space robotics and internal automation established in the Space Robot Controller SPARCO development project [25] SPARCO Control algorithms. The seminal SPARCO development involved: INSTALL/REMOVE, OPEN/CLOSE and ACTUATE tasks. In terrestrial applications such tasks are characteristic for laboratory automation or for specific assembly applications in industry. In comparison to the industrial applications, however, the space task performance requirements can be considered to be very strong. The SPARCO task requirements distinguish directions in which the interaction is required for the task execution (e.g. door or drawer axes) from those directions where the interaction occurs due to positioning errors (e.g. lateral hole directions during insertion). The lateral directions are referred to as parasitic directions and the corresponding forces as parasitic forces. The maximum allowed steady-state parasitic forces and torques were settled to 3 N and 0.45 Nm respectively. Higher interaction forces/torques are allowed in the
Practical Synthesis of Impedance Control
443
interaction direction that by convention coincides with the z-direction of the C-frame. Therefore, these forces are not considered here for the exemplar impedance control design. Thus, we have
Fmax = [3 3 ∗ 0.45 0.45 ∗] ∗
T
Fig. 4.11 SMART S-4 step responses
Fig. 4.12 Independent joint trajectories tracking
(4.60)
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Dynamics and Robust Control of Robot-Environment Interaction
Fig. 4.13 Tracking of joint fast trajectories
Fig. 4.14 Tracking of Cartesian trajectories
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Practical Synthesis of Impedance Control
Fig. 4.15 Validation of the estimated position control transfer function (dashed: first-order, dash-dot: second-order transfer function)
The corresponding position error was assumed to be 2-3 mm. In practice higher positioning errors have been compensated for, while keeping the maximum steady-state force within the above limit. This is achieved by means of the relax strategy (see the next section), which reduces the interaction forces to zero using the damping control. The rotation error in industrial robots usually is quite small and more difficult to predict for a robot/environment interaction case. Therefore, we estimate
p0 max = [0.002 0.002 0.002 ∗ ∗ ∗] ∗
T
(4.61)
As the worst-case pure stiffness environment we will select a “stiff environment” (Table 4.1) with
K e ≈ 50000
[N/m]
The rotation interaction compliance effects will be considered using equivalency with the translation compliance effects (Fig. 4.8) based on the rigid-body transformations (4.23)-(4.25) and (4.29)-(4.30). To utilize this analogy it is necessary to define a maximum distance, i.e. offset (Fig. 4.8), at which the equivalence will be established. In practice the offset is defined by the task geometry, for example by the dimensions of the end-effector (e.g. which should
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Dynamics and Robust Control of Robot-Environment Interaction
contact a surface), by the radius of mating parts, door or drawer dimensions, etc. In our example the offset at which the equivalent linear stiffness (i.e. force and linear penetration) will be used to determine the rotational compliance (i.e. torques and angular displacements relationship) is settled to
R = 0.020 (m) Considering other input parameters specifying the approach motion, a usually slow transition velocity with V0 ≈ 0.01 (m/s) is assumed. As mentioned, these parameters are used in the design procedure to test the design by simulation before performing the first transition tests. The defined set of parameters provides almost all the data needed to design the impedance controller at the servo-control layer. Some additional parameters will be adopted in the algorithm. Step 1: Determination of target stiffness parameter Based on the input parameters one computes the target impedance stiffness. As defined in the design algorithm, the target stiffness parameters mostly are determined directly from the given task requirements specifying the limits on the acceptable interaction force and robot/environment inaccuracy based on (4.48)(4.49) and the adopted limits (4. 60)-(4.61). For the translation stiffness, that provides
K ti ≤ K ti max ≈
3 = 1500 0.002
(i = x,y,z )
N m
(4. 62)
The rotational stiffness is determined based on the translation/rotation stiffness equivalence (4.40). For the adopted input parameters that gives
Κ tj ≤ Κ tj max = K ti max R 2 ≈ 1500 * 0.202 = 60
(i = x, y, z;
j = ϕ x ,ϕ y ,ϕ z
)
Nm rad (4.63)
Step 2: Computation of target mass parameters For the computation of the target mass parameters it is necessary to select the target system frequency. The achievable target frequency is bounded by socalled impedance control realizability conditions (3.177), (4.52), i.e. by the internal position control bandwidth and structural frequency of the sensoreffector system (as pointed out, the position control bandwidth is commonly properly tuned to avoid robot structural resonant effects).
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Practical Synthesis of Impedance Control
For the estimated first-order closed loop position control function (3.239) the position bandwidth is approximately
ωb ≈ 3π Accordingly to (3.177) it is recommended to select the target frequency as
ωt ≈ 1,5π In a spatial application such a relatively low target frequency (0.75 Hz) is not critical due to the commonly low operating speeds, however, for faster industrial tasks it appears ti be very restrictive. A reasonably low, reachable target frequency in the SMART-S4 robot is a consequence of low structural frequency that is, by increasing the inertia of the third arm, significantly reduced in comparison to similar robots (e.g. SMART-S2). Even more, by mounting a relatively heavy gripper and force sensor, this structural frequency is further reduced causing the initially estimated target frequency to also be high. Undesired interaction effects among the target, position control and mechanical robot systems, were observed in initial experiments where even in the free space the lateral arm oscillations in the impedance control mode occurred. These vibrations in contact with the environment caused unwanted resonant effects. Therefore, the target frequency was reduced in the next design step the 0.5 Hz (4.64)
ωt ≈ π
The minimum target mass parameters were then obtained based on (4.51)
1500 = 151.98 (i = x, y, z ) [ kg ] π2 60 = 2 = 6.08 j = ϕ x ,ϕ y ,ϕ z kgm 2 π
M ti ≥ M ti min = J tj ≥ J tj min
(
)
(4.65)
Step 3: Computation of target damping ratio The last and crucial impedance control parameter for the contact stability is the damping ratio ξ t . This parameter should be selected to ensure the contact and coupled system stability based on the established stability criteria (4.57) taking into account the real sampled-data system performance. The impedance control loop in the SPARCO (i.e. Comau) control system is implemented with a sampling rate T = 0.020 (s). We assume a maximum force signal retardation of τ = 0.020 (s). Since there are commonly no additional relevant non-linear effects in the considered industrial robot control system, the weighting function is selected as
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Dynamics and Robust Control of Robot-Environment Interaction
W (s ) = 1 . As a result of the damping ratio design procedure (Fig. 3.48), one obtains:
ξt_pb = 3.75
(4.66)
ξt_rd = 6.2
As recommended we will adopt the more conservative and robust value obtained from the robust contact stability ξt_rd = 6.2 . Thus, the synthesized target models for the translational and rotational DOFs in the C-frame, respectively, are
Gt (s )( x,y,z ) = M t s 2 + 2ξ t M t K t s + K t = 151.98 s 2 + 5920.56 s + 150 Gt (s )(ϕ x ,ϕ y ,ϕ z ) = J t s 2 + 2ξ t J t Κ t s + Κ t = 6.08 s 2 + 236.82 s + 60
(4.67)
Step 4: Computation of impedance control compensator Now we can compute the impedance control compensators according to the control law (3.68) in the s-domain
~ −1 −1 G f (s )( x,y,z ) = G p (s )Gt (s )( x,y,z ) =
s + 11.3 11.3 151.98 s + 5920.56 s + 1500 ~ −1 s + 11.3 −1 G f (s )(ϕ x ,ϕ y ,ϕz ) = G p (s )Gt (s )(ϕ x ,ϕ y ,ϕ z ) = 2 11.3 6.08 s + 236.82 s + 60
(
2
(
)
(4.68)
)
These transfer functions are converted into the discrete-time form using the bilinear Tustin approximation
G f ( z )( x,y,z ) =
4.661 z 2 + 0.9464 z − 3.714 −6 ⋅10 z 2 − 1.437 z + 0.4397
11.65 z 2 + 2.366 z − 9.286 G f ( z )(ϕ x,ϕ y ,ϕz ) = ⋅10 −5 z 2 − 1.437 z + 0.4397
(4.69)
or zero-order hold discretization method
9.115 z − 7.288 ⋅10 −6 z − 1.456 z + 0.4588 22.79 z − 18.22 G f ( z )(ϕ x,ϕ y ,ϕz ) = 2 ⋅10 −5 z − 1.456 z + 0.4588 G f ( z )( x,y,z ) =
2
(4.70)
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449
Step 5: Simulation tests of the target model performance Before starting the experiments on the synthesized impedance controller, it is recommended to test the design outcomes by simulation using the simplified interaction models (Fig. 3.7, 4.9). For the considered example, i.e. the estimated position controller and synthesized impedance compensator, the simulation model is presented in (Fig. 4.16). The main emphasis in the basic simulation test is to examine the contact transition. The robot end-effector is located closely to the environment and a small approaching velocity (0.005 m/s) realizes the contact. Such a slow approaching velocity is typically used in space automation experiments. The simulated interaction force is presented in (Fig. 4.17). The obtained contact transition is characterized by a relatively high contact stability margin ς ≈ 75% and contact overshoot ο ≈ 131% . This relatively conservative behavior is considered suitably robust for experiments concerning an uncertain robot environment and relatively high force delay. Step 6: Experimental design evaluation The target model realization example is presented in (Fig. 4.18). This figure presents the measured and modeled interaction force components in the Cartesian z-direction. A relatively good match between model and experiment can be observed. The measured interaction force is presented in (Fig. 4.19). It is quite similar to the experiment (Fig. 4.17). The real behavior is characterized by relatively higher oscillations during transition (e.g. due to friction effects).
Fig. 4.16 Simulation of the synthesized impedance control for the SMART-S4
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Dynamics and Robust Control of Robot-Environment Interaction
Final Step: Task performance evaluation The positive transition experiment provides the background for starting more complex task trials. For these purposes the complete set of SPARCO target system gains (for various impedance and damping cases) is synthesized and implemented. As an example, the OPEN-drawer task is selected (Fig. 4.20). The figure presents the interaction force components in the lateral directions (x, y) and opening direction (z) respectively. The lateral forces are within the desired limits. The z-force component mainly represents the friction force during opening, except for an impact that occurs at the drawer motion limit. Obviously the obtained interaction behavior corresponds to the above-specified task requirements. This is crucial for evaluating and accepting the design results, which are satisfactory.
Fig. 4.17 Simulated contact transition
Fig. 4.18 Measured (solid) and target model force (dashed) in z-direction
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Fig. 4.19 Measured interaction force during contact transition experiment
Fig. 4.20 Open-drawer task execution
4.4 Synthesis of Impedance Control at Higher Layers Despite the numerous sophisticated strategies and schemes elaborated for compliance and interaction control, advanced compliance control methods are still not implemented in the commercial robotic control systems available on the
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market. There are several reasons for this. First, the majority of compliance control concepts are concerned with particular problems, mainly at the lower control layers. Second, combining various algorithms and control concepts and integrating them into conventional robot position control systems is complex and tedious. Third, most of the studies on impedance control have been primarily focused with the servo control algorithms. Except for the initial works on compliance control [31], research studies have generally neglected the contact task planning and programming issues. In practice, however, these subjects are essential for compliance control applications. This section examines control problems at the planning and programming layers of impedance control. Solving these problems is crucial for the development of an integrated impedance control system that includes control, programming and monitoring functions. The problems of impedance control integration were comprehensively considered during the development of the Space robot control system - SPARCO [26]. The goal of this development was to design a reliable and integrated compliance control framework, based on the technology and standards of industrial robots. The SPARCO system includes commands and ready-to-use programming functions that allow the programmer to implement relatively complex high-level compliance control algorithms. The SPARCO system is capable of automatically executing various compliance tasks, under the safety and reliability requirements of space robotics and automation. The following considerations address the issues and algorithms of high-level impedance control, regardless of the specific control system. 4.4.1 World model for the impedance control In robot programs, one of the most common approaches to representing robot and object positions is based on coordinate frames. Although coordinate frames have several drawbacks (for example, they have non-unique specification of the robot configuration, as well as an over-specification of some tasks, such as grasping a symmetrical object), they are the best suited and established representation of the robot position and orientation in robot programming languages. In this section we will define specific frames needed for the integration of the impedance control in common industrial robot control systems. Modern industrial robot control programming systems (e.g. ABB-S4, KukaKRL, Comau-PDL2, Fanuc-KAREL etc.) provide a customary specification of robotic tasks in different relative reference frames in which the applications can
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easily be described. The utilization of relative frames has the following main objectives: ● flexible environmental object description (changes in the environment require only updating the immediately involved objects), and ● efficient robot motion specification (by using a variable gripper tool-centerpoint TCP-frame the desired motion can be specified in terms of the frame for which it actually holds). A common robot world model with absolute and relative coordinate frames associated with objects and features of interest is shown in Fig. 4.21 (robot and environment frames) and Fig. 4.22 (gripper frames). The robotic system presented in (Fig. 4.21) represents the CAT-arm (Columbus A&R Testbed) developed at the European Space Agency (ESA). The CAT system consists of an articulated robotic arm mounted on a rail-gantry system with 2 DOF’s. The robot is designed to support scientific experiments in the space laboratory. In the SPARCO project the following tasks were considered: open/close, actuate, and install/remove. The meaning of the presented coordinate frames is as follows: ● The reference world frame (W) is fixed relative to the robotic system; ● The robot-base (B) coordinate system is located relative to the robot pedestal. The transform W TB denotes the homogeneous transformation from B in W (in general Y TX describes a homogeneous transform from X frame into Y); ● The end-point coordinate frame (E) is located at the wrist flange. The location of E w.r.t. B is described by a non-linear direct kinematic model in terms of generalized robot coordinates q (joint angles); ● The robot gripper is assigned the end-effector (i.e. Gripper) coordinate frame G, relative to E. The gripper features (e.g. location of gravity center, gripping force, etc.) can be expressed with respect to G. The above-mentioned frames represent a convenient set of frames of a robotic system. These frames are independent of a specific task. The following sets of frames are related to specific tasks: ● A generalized variable gripper coordinate frame, referred to as the roolcenter-point T-frame, is associated with the robot end-effector to specify the robot motion needed to obtain the desired object motion directly. Any
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temporarily movable frame, i.e. any frame that belongs to the gripped objects can be declared to T; ● The facility (F) frame is associated with each element (i.e. facility) of a workcell; ● We can describe the location of associated objects relative to F by means of object O-frames; ● To facilitate the relationship of the robot position to the objects, as well as to easily define the motion of their parts, several work-piece frames (P) may be introduced. These frames are affixed to the important features of objects. The motion of the selected points is clearly defined in relation to the corresponding object frames. The impedance control system includes two new frames, which are specific for the compliant motion programming: ● The force-sensing S-frame is a force-sensor specific frame in which the forces and torques are measured. This frame is defined relative to the robot endpoint E (Fig. 4.22); ● The compliance C-frame in which the target impedance behavior (robot impedance reaction) is specified and controlled. Since the location of the C-frame depends on the current task, we have chosen the most convenient way to specify the C-frame relative to the task Tframe. As mentioned above, the T-frame is a variable frame selected to meet specific task motion requirements. Usually the desired robot position specifies the location of the T-frame with respect to an object frame. However, in the contact tasks the C-frame generally differs from the T-frame. The C-frame’s main concern is the robot/environment interaction and the compliance behavior, while the T-frame should be selected to easily describe and program the robot motion during interaction. A topological presentation of the above-mentioned basic robot frames is sketched in (Fig. 4.23). The basic PDL2 language predefined variables that represent the robotic frames are denoted in this figure as well. The variable $BASE specifies the robot base frame relative to the world frame. The T- frame is defined in the PDL2 by the variable $TOOL relative to the end-point E (faceplate of the arm). The last PDL2 modifications realized to support SPARCO development includes the following two new system variables:
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● $UFRAME – “user frame”, which specifies the object O-frame (Fig. 4.23) relative to the world frame, and ● $SFRAME specifies the general “sensor-frame” relative to the tool frame.
Fig. 4.21 Impedance control world model (CAT arm)
Fig. 4.22 End-effector frames
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$SFRAME has been applied in the SPARCO project to define the compliance C-frame. The end-effector E-frame is specified with a predefined internal variable for each particular robotic system. The remaining frames (S, G, F and P) can easily be defined using the PDL2 POSITION data type, which is used in robot programs to represent relative Cartesian frames of reference.
Fig. 4.23 World model: frames relations
Example 4.3: Robot frames specification in PDL2-extended Programming the robot motion involves specifying successive T-frames. Thereby, the absolute transformation of a frame can be expressed as the product of transformations between relative frames (a so-called compound transformation), which is usually supported by high-level industrial robot languages (in the PDL2 the “:” command denotes the compound transformation). The following program fragments illustrate the specification of the basic frames in the PDL2 (note that “--“ denotes comments in the program): -- declaration of variables
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VAR default_tool : POSITION -- w.r.t E-frame grip_pose : POSITION -- w.r.t P cat_jnt : JOINTPOS --robot joint pose default_tool : POSITION -- w.r.t E-frame compliance_frame : POSITION -- w.r.t T … BEGIN -- read actual joint position cat_jnt := ARM_JNTP … -- set robot base frame frame in the actual gantry position $BASE := POS(-cat_jnt[7],-cat_jnt[8],0,0,0,0,'') $UFRAME := POS(0, 0, 0, 0, 0, 0, '') -- user frame = world … -- set tool in default position (G-frame) $TOOL := sp_robot.default_tool … -- after grasping an object shift T-frame in P $TOOL := $TOOL : POS_INV(subj.grip_pose) -- Compliance Frame = Tool Frame $SFRAME := POS(0, 0, 0, 0, 0, 0, '') To facilitate the programming and specification of complex tasks a structured database was implemented in the SPARCO system. This database involves a set of predefined relative poses required to realize specific tasks (e.g. approach-, grasp-, on- and off-poses etc.). For more details about the SPARCO programming approach see [22]. 4.4.2 Impedance control integration into forward industrial robot position control This section addresses the specification of control functions needed to implement the impedance control. Let us consider the adopted position based impedance control scheme (Fig. 4.24). The output from the position interpolator (standard robot control element) is the nominal position, defined by the position vector x0 or transformation matrix T0, which is specified with respect to a Cartesian frame of reference given by the user (commonly in an object O-frame that is in the PDL2 defined using the system variable $UFRAME). The output from the impedance controller is a
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Fig. 4.24 Impedance control schemes including coordinate frames specification
force-dependent position modification, expressed by the vector ∆xF or corresponding transformation matrix ∆TF, which is defined with respect to the Cframe, and is also specified by the user (e.g. using $SFRAME variable). The nominal position in the scheme is modified by the incremental adjustments (∆xF,∆TF) providing the so-called reference position of the tool-center point, which is pursued to the standard position controller. Commonly, the Cartesian position set point is converted into joint coordinates using an inverse kinematic transformation and provides a joint set point that represents the input to the local joint position controller. Taking into account the different time slices between the motion planning (e.g. 10 ms in the Comau C3G control system) and position control (1ms), the joint set points are linearly interpolated to achieve the required short distances between the set points of the motion control. As can be seen, the impedance control scheme involves several functions that perform computations on various parameters and variables defined in different coordinate frames. Therefore, the control functions have to include transformation control variables between different frames using transformation matrices (for position variables) or Jacobian matrices (for forces and incremental displacements).
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In principle the following groups of data are required to implement the above sketched specific impedance control blocks and interfaces: i) Robot and task (action) parameters (from robot machine data or application program): • ETT - homogenous matrix specifying the tool T-frame location w.r.t. the endpoint E-frame ($TOOL); • T TC - homogenous matrix of C-frame location w.r.t. the T-frame ($SENSOR); • ETG - homogenous matrix of G-frame location w.r.t. the robot end point frame; • ETS - homogenous matrix of S-frame location w.r.t. the end-point E-frame; • Set of impedance control gains. In the SPARCO control systems the gain specification is realized in the program language using descriptive linguistic variables, such as HIGH-, MEDIUM- and SMAL-IMPEDANCE, i.e. DAMPING for the damping control. In the PDL2 these commands are realized using the system variable $SENSOR_GAIN. Based on the gain setting, low layer control functions copy the appropriate gains from an impedance control look-up table, which involves the synthesized gains for all DOFs (translational and rotational) and all impedance cases, into a local control function buffer (shared memory). The SPARCO controller applies the discrete impedance compensator GF (Fig. 3.41) of order 3, which in the standard normalized form can be represented as
GF ( z ) = K t
−1
B3 + B2 z −1 + B1 z −2 + B0 z −3 1 + A2 z −1 + A1 z −2 + A0 z −3
(4.71)
where Ai and Bi (i=0, 1, 2, 3) are coefficients of the denominator and numerator −1 polynomials, respectively, while K t is the inverse of the target stiffness. Hence, the set of gain parameters in the SPARCO system includes 36 polynomial coefficients (4.71), for 6 impedance cases and 6 DOFs, a total of 288 parameters. • Fmax - the maximum allowable force/moment (force limits) used in monitoring algorithms. Two sets of force limits must be distinguished: F S max the maximum force/moment in the sensor S-frame that specifies the physical robot, i.e. sensor limitations, and the task-specific maximum force/moment F C max defined in the C-frame for each task;
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• FSP - the force set point vector (nominal force and moment F C 0 ) in the Cframe which is used in the damping control mode; • F S Czero - the zero contact force vector in the S-frame which specifies the minimum contact force/moment components that can be measured (i.e. estimated) depending on the performance of the applied force sensor and contact process (e.g. taking into account offsets, bias, etc.). This force is used in the algorithms that compute contact force and provide contact flag; • ε F - epsilon environment used to check when the force set point is reached. ii) Robot system state data: • 0TT - the nominal end-point robot position (w.r.t. O-frame), which is obtained from interpolator; • W TE - the actual end-point robot position w.r.t. W, which is computed in the common robotic control system; iii) Gripper and end-effector inertial parameters used to compensate for payload gravitational and inertial effects in order to compute the pure contact force based on force/moment sensing: • M GE – the total mass of the gripper and end-effector; G • rmc - the position vector of the gripper/end-effector mass center point w.r.t. the gripper G-frame (in [m]); • g – the gravitation acceleration vector in W, which is specifically used in the SPARCO project to distinguish terrestrial tests and space applications.
The impedance control functions block feed the following information back to the position robot control kernel (Fig. 4.24): i) Robot system state data: • ∆T C imp the force dependent impedance correction specifying the new Cframe location w.r.t.the initial one. ii) Status information: • “force-setpoint reached” signal (in SPARCO this is utilized to start clock measurement of the needed time for apply-force); • “maximum-force reached” signals (in the C- and S-frame) used for task and system force limits monitoring purposes; • contact flag.
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The above minimal set of data needed to implement the impedance control, as well as the corresponding control functions are presented in (Fig. 4.25). In the following sections the basic impedance control functions will be specified. 4.4.3 Impedance control functions and algorithms 4.4.3.1 End-effector position modification The end-effector position modification function provides the functional interface between the external impedance control and the internal position controller (Fig. 4.25). The inputs to these control functions are transformation matrices that define the position modification ∆T C imp and the nominal position of the tool frame 0TT respectively.
Fig. 4.25 Impedance control schemes including coordinate frames specification
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The basic principle of the position correction algorithm is illustrated in (Fig. 4.26). The impedance controller provides the relative position modification ∆T C imp in the C-frame. This modification is proportional to the external forces and moments (reduced at C) and selected impedance control gains that are also specified in the C-frame. The position correction defines the modified location of the compliance frame Cmod relative to the nominal C-frame pose (Fig. 4.26). The position correction algorithm calculates the corresponding modification of the nominal tool frame T0 , which is defined by the transformation matrix 0TT (“tool in object frame”) generated by the position interpolator. The modified frame T0 mod is obtained so that the relative arrangement between T- and Cframes defined by T TC remains unchanged after the position shift (Fig. 4.26). In mathematical form this condition can be expressed as T
TC ∆T
C imp
= ∆T
T imp
T
(4.72)
TC
so that the modified T-frame location w.r.t. the nominal location is defined by the transformation matrix
∆T
T
imp
=T TC ∆T
C imp
T
TC
−1
(4.73)
and in the selected object frame is given by
∆T
O imp
= OTT T TC ∆T
C imp
T
TC
−1
(4.74)
This T-matrix represents the output from the position modification function, i.e. the reference position input to the internal position controller at the actual time step.
Fig. 4.26 Impedance control schemes including coordinate frames specification
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4.4.3.2 Selection and initialization of impedance gain parameters This function performs the following operations: • receive a command from the command handler; • select the gains parameters from the impedance control look-up table based on the user selection (e.g. $SENSOR_GAIN from the PDL2 level) for each C-frame DOF; • copy the gains into the local impedance control buffer. The gain initialization is done during system setup. For this purpose a default impedance case has to be selected in the control system (in SPARCO it is HIGHIMPEDANCE, denoting “high stiffness”). The change of the impedance gains during contact may be hazardous causing control chattering and impacts. In order to ensure a continuous adjustment of the impedance control gains, a specific control algorithm should be implemented at the higher control layer. This algorithm as well as the preconditions for a smooth gain variation will be described below. 4.4.3.3 Computation of path correction This function implements the impedance control compensator. It computes the actual position modification in the C-frame based on the selected gains and nominal interaction force (defined in the application program), as well as the measured actual contact force/moment in the C-frame, using the discrete impedance control law:
(
∆x C = G f ( z ) F
C
0
−F
C c
)
(4.75)
Where F C 0 and F C c are desired and actual contact force/moment vectors in C, G f (z ) is the vector of discrete impedance control transfer functions (4.71) along each C-frame DOF. The gains are stored in a local shared memory. In order to implement the above control law, the interaction force, i.e. force control error history should also be recorded. The position modification vector is computed based on the adopted target impedance model in the C-frame consisting of six independent mass-damperspring systems along each C-frame axis and DOF (Fig. 3.2).
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4.4.3.4 Delta x to delta T transformation (deltax_2_deltaT) Since industrial robot control systems commonly utilize the homogenous transformation matrices for internal computations, this control function transforms the position modification vector (4.75) into a corresponding T-matrix form. Generally the input position modification is not an infinitesimal vector. For example, when LOW-IMPEDANCE (low target stiffness), and especially LOWDAMPING (zero target stiffness) are selected, as well as when higher interaction forces and moments are exerted, this vector may involve finite relative displacement and rotation components. Thereby the specific problems are related to the selection of an appropriate form for the representation of relative finite orientations, as well as to the dependence of the rotation matrix on order of successive elemental rotations around the principle coordinate axes in threedimensional space. The most customary way to resolve these problems is to assume the angular displacement in the impedance control to be represented by an equivalent rotation vector (so-called angle-axis representation). The components of the equivalent orientation vector, i.e. their projections on to Cframe axes, are computed independently in the path-correction control function using a selected impedance relationship between the external moments and angular displacements. The transformation matrix that represents the position modification has the form
∆T
C imp
∆R(∆xi=4 ,…,6 ) ∆p (∆xi=1,…,3 ) = 0 1
(4.76)
where the vector of force proportional linear displacement is
∆p = [∆x1 ∆x2
∆x3 ]
T
(4.77)
Based on the adopted angle-axis representation (Craig, 1986), the displacement angle and the rotation axis can be presented in the form 2
2
2
θ = ∆x4 + ∆x5 + ∆x6 ;θ ≠ 0 ∆x k= 4 θ k 2 =1
∆x5 θ
T
∆x6 = kx θ
[
ky
providing the equivalent incremental rotation matrix
kz
]
T
(4.78)
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k x 2vθ + cθ ∆R = k x k y vθ + k z sθ k k v − k s y θ x z θ
k x k y vθ − k z sθ 2
k y vθ + cθ k y k z vθ + k x sθ
k x k z vθ + k y sθ k y k z vθ − k x sθ 2 k z vθ + cθ
465
(4.79)
where
s = sin θ, cθ = cos θ v = vers θ = 1 − cos θ 4.4.3.5 Compensation for payload and inertial effects This function extracts the real contact force vector from the measured forces/moments in the S-frame by compensating for components due to the endeffector gravitation and inertia forces and moments. The compensation for inertial forces and moments requires the estimation or measurement of the endeffector’s linear and angular accelerations (e.g. using a navigation sensor). In the SPARCO system, due to the very slow contact motions, the inertial terms are neglected. In this case, considering that the end-effector’s inertia is defined in a G-frame, we have
F
S C
= − F S + SJ G
−T
−T M gG J Gmc GE 03 x1
G
S
(4.80)
where F C is the contact force vector (exerted by the robot upon the environment) in the compliance C-frame, F S is the measured force vector in Sframe, S J G is the Jacobian matrix (see 4.18-4.19) that defines the velocity (twist) propagation from the G into the S-frame, G J mc is the Jacobian matrix from the end-effector mass-center point to the G-frame origin, and g G is the gravitation vector in the G-frame. When taking into account (4.18)-(4.19), the above Jacobian matrices have the forms
S R − S RG r GGS JG = G S RG 03×3 G I − r mc G J mc = 3×3 03×3 I 3×3 S
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where the orientation matrix
S
RG , of G-frame relative to the S, and the distance
G vector rGS , from the G to the S origin, are taken from the corresponding
homogenous matrix S TG that is computed based on S
where
E
TG = E TS
−1 E
SR TG = G 03×1
G rGS 1
(4.81) G
TS and ETG belong to the machine data, while r GS is a 3x3 skew-
symmetric matrix formed from the corresponding vector in order to present vector multiplication in matrix form. The gravitation acceleration vector in G is computed according to
g G = ERG
T W
g W = [0 0 − 9.81]
T
T
RE g W ;
(4.82)
The actual robot position matrix W TE is obtained from the direct kinematics control module. Unlike the computation (4.81), which can be performed one time during thecontrol system setup, the transformation (4.82) must be calculated in each of the control steps. Finally, this control function provides a dead zone output of the contact force vector signals in the S-frame. Using the dead zone effect the force offsets and bias are cut according to the following algorithm (written in a MATLAB-like coding style)
for i = 1:6
(
if abs(F
S C
(i)) <= F
S Czero
(i)
)
F
S C
(i) = 0.0;
(4.83)
S
where F Czero is selected according to the performance of the force sensor and the interaction process. 4.4.3.6 Sensor frame to compliance frame transformation (Sf2cf) This function transforms the interaction force vector from the S- into the Cframe in which the impedance control is realized. This transformation is computed based on the formula
F S
C C
= CJ S C
−T
F
S C
(4.84)
where F C and F C are the contact force/moment vectors in the S- and Cframe respectively, and C J S is the Jacobian matrix from S to C
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C
03×3 I 3×3 C RS 03×3
CR JS = S 03×3
467
S − r SC I 3×3
S (from S to C, where the orientation matrix C RS and the distance vector rSC defined by projections in S) are obtained from the homogenous matrices CTS , computed using the system variables T TC ($SENSOR in PDL2) and E TT ($TOOL), as well as the machine data ETS , according to C
TS =T TC
−1 E
TT
−1 E
CR TS = S 01×3
− C RS rSCS 1
Since the above variables must remain constant during impedance control execution (they can be changed in the program only when the impedance control is inactive in order to avoid discontinuities in the control), the Jacobian matrix can be calculated once in the impedance control initialization phase, and kept constant during transformation. 4.4.3.7 Force setpoint check This function monitors the realization of the force setpoint and captures the time instance when the force setpoint is reached. It is used in SPARCO mainly within relax and apply-force control functions that apply a steady state force and keep it constant during a given time period established by the programming layer. The function checks the condition
−F
C
0
+F
C C
< εF
(4.85)
and provides the force set point reached flag to higher control layers. In SPARCO this flag is used at the programming layer to start a common robot control condition handler that deactivates the external impedance control after set time period. The usage of control flags at programming layer is realized utilizing additional buffer providing selectable floating and integer vectors and introduced in the PDL2 to support sensor applications. 4.4.3.8 Force sensor data preprocessing This function processes the raw force sensor data. It effectively obtains the force information through the use of filtering, scaling, offsets and coupling compensation, etc.
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In the SPARCO control system this function is realized within the sensor functional module. SPARCO uses the ROTEX [72] gripper that was still being used in space experiments. 4.4.3.9 Contact check This function examines if the robot is in contact with the environment based on the actual effective force contact value.
for i = 1 : 6
(
if abs( F
S C
(i)) >= F
S Czero
(i )
)
contact _ flag = TRUE;
(4.86)
In most cases it is enough to detect that some of force components are outside of the dead zone in order to prove contact. If these forces are zero, the torques will be also tested.
4.4.4 Higher control layers algorithms 4.4.4.1 Impedance control operating modes Although the impedance control feedback can be activated or deactivated by the physical contact with the environment (Fig. 4.24), the impedance control system requires mechanisms to functionally manage the activation of the above described control modules and functions. It is convenient to introduce the following impedance control operating modes: • Stopped mode, with no impedance control functions performed. This mode corresponds to the conventional position control systems; • Monitoring mode, which performs monitoring functions in real control time such as examination of sensor and process force/torque limits, checking position correction bounds, contact check, collision detection, end-effector observation functions, etc.; • Running mode, in which monitoring and compliance control functions (computation of impedance control loop) are executed at each sampling interval. Switching between operating modes can be managed by the new commands set_imco_status(mode), which is implemented in the robot language at the programming layer.
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In the initial stopped status, the impedance control is initialized by selecting the sensor type, as well as by transferring the control and machine data to the local control functions. Before activating the impedance control functions, the status is changed to the monitoring mode. This mode allows changing taskdependent control parameters, such as the T- and the C-frame locations, impedance gains, contact force limits, desired force, etc. This can be done during contact with the environment, without deactivation of the external control. An algorithm, referred to as the relax control function, has to be used to realize the conditions for a continuous change of the impedance control parameters. Once the running mode has been started, the robot positions in all subsequent robot motions are automatically modified by the corrections corresponding to the force and selected impedance target gains. Any transition to the monitoring or stopped modes automatically resets the position correction offset by replacing the nominal robot position in the interpolator by the actual robot position (so-called position synchronization). By this mean, the robot motion can be pursued in the position control mode starting from the actual position. 4.4.4.2 Change of impedance gains (relax) Since the impedance control parameters define a desired target mechanical system, they should often be changed in order to meet task-specific requirements (some actions, such as INSERT, require the target impedance to be changed several times during the execution). In addition to the control gains, the location of the compliance C-frame, in which the impedance is controlled, must also be varied. In a control system the main problem is to achieve a continuous transient switching of the parameters (bumpless parameter changes). When the robot is in contact with the environment, any discontinuous parameter change can cause control chattering due to an abrupt change of the impedance force/motion relationship. This can be caused by an alteration of impedance gains (i.e. variation of stiffness), or by a modification of the location of the compliance frame (i.e. change of force and torque components). Particularly the control gain switching can be critical in directions in which a high stiffness is replaced by a low one and vice versa. In the free space, however, impedance parameter switching is not critical since the impedance feedback loop is inactive due to a zero contact force. According to the above consideration, a logical way to obtain bumpless parameter changes is to achieve contact conditions very similar to the free space
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conditions. This can be achieved by reducing the interaction force to a minimum level while maintaining the contact with the environment. This is realized by the relax algorithm, and corresponding built-in program language functions. The relax algorithm is realized through the following steps (Fig. 4.27): Algorithm 4.2 Relax impedance control algorithm Step 1: Select monitoring mode Switch the impedance control to the monitoring mode and reset position correction offset; Step 2: Input damping gains Select the damping control gains (target stiffness is zero) in all C frame directions and set the desired force F0 at zero. For relax, the SPARCO system uses LOW_DAMPING gains synthesized for the ESA-CAT arm; Step 3: Running mode Switch the impedance control to running mode; Step 4: Relaxing Due to contact forces and moments, the robot moves until the given small force threshold has been reached in all directions during a selected time period; Step 5: Monitoring mode Switch the impedance control again to the monitoring and reset offset, as well as initial impedance gains. For a proper execution of the relax strategy it is crucial to select the C-frame location to be compatible with real task constraints. This topic will be discussed for each action separately. The relax algorithm steps are illustrated in (Fig. 4.27). In the SPARCO system a relatively large time period was applied for the relaxing in order to reduce reaction forces as much as possible, and thus to meet the high requirements on the space task execution performance. For the sake of simplicity, in SPARCO the same relax time was selected for each C-frame direction. In industrial applications, however, the execution time is relevant and the relax algorithms must be designed to satisfy fast task execution, as well as specific task requirements. The SPARCO system (i.e. its first realized version) covers the following tasks: OPEN (CLOSE), ACTUATE and INSTALL (REMOVE). The accomplishment of the above tasks requires the following actions: APPROACH
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(RETRACT), ATTACH (DETACH), EXTRACT (INSERT), HINGE, SLIDE and PUSH. These actions can be realized using standard robot motion instructions, as well as the new compliance control commands. Using the FRM methodology, each robotic task can be decomposed in a logical sequence of control actions. In the SPARCO control system the action commands are implemented as the PDL2 language routines. A task is described as a simple sequence of action calling commands.
Fig. 4.27 Relax steps
4.4.5 Actions and tasks control algorithms As examples, the main SPARCO actions are used to illustrate the solving of practical contact control tasks based on the new impedance control functions and commands. The SPARCO tasks and actions were extensively tested using the experimental facility (Fig. 4.28) that was designed by Fokker Space & Systems. This facility includes the following objects: door, drawer, and insertion port, grasping knob, linear and rotation actuating mechanism, push knob and switch
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mechanism. These objects were designed to support efficient testing of the above-mentioned tasks and actions under various initial and termination conditions. About 70 different experiments were specified and realized in the SPARCO acceptance test. In the experiments a successful completion was not only assessed by the formal task achievements, but rather the performance of interaction processes was quantitatively evaluated and compared with strong requirements. It is worth mentioning that experiments were performed based on the CAD models of the CAT environment, without teaching any position. The objective of compliance control algorithms was to compensate all uncertainties (e.g. robot and environments position inaccuracies, unknown and variable stiffness, friction effects, etc.) and thus to provide the high disposability and robustness required in the robot space-applications. Moreover, in some experiments the subjects were intentionally designed with relatively large deviations (e.g. sliding mechanism in Fig. 4.28), in order to demonstrate the robustness of task execution.
Fig. 4.28 Experimental facility
To achieve the demanding task goals, reliable and robust algorithms at high control levels were developed and implemented in the new programming environment. In the following, the underlying idea of these algorithms will be
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briefly presented, including typical experimental results. Finally, it is worth mention that these algorithms can be applied in several tasks in industry and service fields without any significant modification. 4.4.5.1 Grasping/detach action In order to grasp a fixed object properly (Fig. 4.29), a nominal relative configuration (position) between the end-effector and the object needs to be achieved with the required relative accuracy before the gripper is closed. Grasping is performed by means of closing the gripper jaws with active impedance robot control. The implemented grasping algorithm allows the gripper to self align to the fixed grasping objects during closing, thus compensating for inaccuracies in the environment and the robot control. The specific design of gripper jaws and grapple fixtures (CAT grasping interface, Fig. 4.30) provides a dependable hardware setting for precise and repeatable grasping. The CAT grasping interface is designed as a cube with grooves supporting the alignment of jaws, which includes compatible hemispheres (Fig. 4.30). The SPARCO grasping elements allow the compensation of positioning errors of ±3 mm and angular alignment up to ±10 degrees.
Fig. 4.29 Grasping a fixed knob
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A basic specification of compliance control algorithms generally involves the following conditions: • location of the C frame and the tool frame, • selection of control gains for the specific task, and • set of commands to be performed by the robot. To meet the grasping requirements, the following specification is used: • The compliance C-Frame is approximately located in the gripper middle point, between the hemispheres (Fig. 4.31), which are used to support the gripper self-alignment along grooves based on the impedance control effects; • In each direction LOW_IMPEDANCE is selected.
Fig. 4.30 Gripper jaws and grasping interface design
Fig. 4.31 Grasping using impedance effect
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The following grasping situations occur commonly. In a relatively good centralized grasp (the compliance center is located near to the grapple fixture axis), internal grasp forces are approximately balanced, while the resulting torque mainly causes a rotation about the center of compliance (Fig. 4.31). In an opposite case, lateral misplacements are dominant, causing one-side contact. The lateral force produced by closing the gripper, as well as corresponding moments around the compliance center, cause the robot to correct the initial position and orientation as far as both jaws grasp the object.
Fig. 4.32 Grasping experiment sequences: start of grasping (above left), unilateral contact (above right), rotation around C (lower left), grasping end – relaxing (lower right)
Due to the unbalanced contact forces/moments and the impedance effects, the robot moves until an internally stable desired grasp is achieved (the form of the jaws, hemispheres and grapple fixture notches in conjunction with the contact friction prevents further motion). Figure 4.32 illustrates these sequences in a
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typical grasping experiment. The grasping is completed when the gripper jaws are properly closed (in the SPARCO gripper it is pinpointed using the measurements of the distance between the jaws). However, the interaction forces and moments, produced by linear and angular misalignments and spring-like impedance control effects, produce a residual strain after attachment. In order to reduce the strain (interaction forces), the relax action has to be used. The Cframe location is not changed during the relax action. Due to the fact that the gripper is tightly connected to the environment, the robot-arm can modify the position and rotation (by utilizing the damping effects of the relax) until the interaction forces/moments are reduced nearly to zero. Thus an ideal grasping is achieved by correctly adjusting the robot/gripper position to the real environment. The grasping is involved in all tasks requiring bilateral contact between the robot and subjects (e.g. in INSTALL; OPEN, ACTUATE). The detach algorithm is mainly inverse to the attach action. Detaching starts with relaxing forces/torques occurring in a previous action (e.g. insertion, closing, etc.), and after an unstrained ideal position has been achieved, it is simply completed by opening the gripper jaws. 4.4.5.2 Insertion/extraction Industrial-robots programmers generally have experience with RCC passive compliance. The impedance control provides essentially a similar, however more flexible approach. The control system capabilities to change impedance control gains or compliance frames in various task phases provide a free programmable “compliance device”. The SPARCO insertion task mainly relates a circular container (rod) and an insertion port (Fig. 4.33). Under some circumstances, the developed robust insertion algorithm can be applied for installing rectangular objects (Fig. 4.34). Similar to the RCC insertion, the key condition for compliance control based insertion provides the presence of chamfers on the container and the port (Fig. 4.33). The chamfers support a contact-driven alignment of the installation counterparts. The proper dimensions of the chamfers depend on the peg and hole geometry, frictions, tolerances, and misalignments. The common compliance control based INSTALL algorithm consists of the following three procedures: engagement, insertion and termination. The
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principal selection of the impedance control gains in these phases is specified in (Fig. 4.35).
Fig. 4.33 SPARCO insertion experiment
Fig. 4.34 Insertion of rectangular container in CAT facilities using impedance control
The engagement phase consists of the initial contact and relative sliding between the part chamfers (Fig. 4.36). The following impedance control parameters are specified for the engagement phase • The C frame should be located near the position of the interacting forces (on the peg top);
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• Impedance gains along the C-frame directions, respectively
Fig. 4.35 Impedance gains selection for insertion phases
During the engagement a medium stiffness in the z-direction (Fig. 4.35) is selected taking into account that the robot motion is, in fact, unconstrained in this direction. In order to get the peg to slide along the hole chamfers, the stiffness in the lateral directions (x, y) must be less than the axial one, causing a faster lateral alignment than that encountered by the surface (the lateral force/stiffness behavior serves as a cue to the desired corrective motion). This strategy reduces contact forces in the insertion direction allowing the peg to cope with the friction (Fig. 4.36). A high rotational stiffness is required taking into account the engagement goal to compensate only for lateral misalignment, without producing unwanted rotations. At the end of the engagement (after a specified engagement depth is achieved), the insertion strategy is started. At first the contact forces are relaxed
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using the relax algorithm. Thereby the location of the C-frame remains close to the peg top. The insertion phase is characterized by the following settings: •
The C frame remains located near the peg top;
• The lateral rotation impedances (stiffness) around the x and y C-frame directions are switched to LOW in order to compensate for rotational errors. The impedance in the insertion direction (z) along the hole is set to MEDIUM since the robot motion is unconstrained in this direction and has to compensate for disturbing friction forces between the peg and the hole during insertion. In specific cases LOW impedance in lateral directions can also be applied; • The insertion motion consists of a linear displacement in the positive z-direction along the hole axis.
Fig. 4.36 Engagement phase
The insertion phase is terminated when a termination pose in front of the peg bottom is reached (Fig. 4.35). This pose is selected depending on the estimated position tolerances to avoid contact with the hole bottom. Analogously to the end of engagement, residual steady-state interaction force/torque accompanies the end of insertion. These forces occur due to misalignment between the peg
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insertion direction and the hole axis, which is corrected using the impedance spring-effects. In order to pursue with the termination phase and to change impedance gains according to the termination strategy, these residual forces must be relaxed. Specific for the insertion relaxing is that the location of the C-frame should be changed near the middle point of hole in order to achieve a consistent condition for the relaxing of both the residual force and moment components.
Fig. 4.37 C-frame selection for relaxing at the end of insertion
Generally the compliance frame for relaxing should be selected as closely as possible to an object’s geometric center, i.e. to the middle point or object’s “barycenter”, which is crossed by the resulting reaction force direction. By this mean, the relaxing of both external forces and moments with respect to the compliance frame becomes feasible, i.e. compatible with the physical constraints. For example, if the C-frame is selected at the grasping fixture (Fig. 4.37 a, b) or top of the peg, it is possible to reduce the forces but not synchronously the moments due to the opposite requirements on the force equilibrium (F1 = F2) and the moment equilibrium (F1 > F2). This
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incompatibility causes oscillations during relax (similar to “limit cycles”). In this case, the force and moment equilibriums can never be achieved simultaneously; thus, the relax cannot be completed. However, in the compatible case (Fig. 4.37 c), both force and moment equilibrium can be realized, and relax may be successfully performed. The selection of appropriate force/torque zerothresholds and the relax-time is also important for a robust relaxing. For time non-critical space-applications, a relatively longer time period can be chosen to reduce the residual forces. The termination algorithms depend on the termination strategy. The peg position, an external signal, or the contact force can govern the termination in the SPARCO system. In each case the LOW impedance is selected in the insertion direction to avoid large contact forces due to inaccuracies. The following set of gains is commonly used: along the C-frames directions, respectively. The C-frame is switched again close to the peg top. As mentioned above, after relaxing the contact forces and moments this change of the C-frame location is not critical. Depending on the termination strategy, the interaction forces can again be relaxed at the end of termination. Finally, it is worth discussing the effects of the payload changes at the end of the INSTALL task (after installing the container in the port), or at the start of the extraction action (after grasping) in the REMOVE task. In terrestrial applications in both cases, due to the attaching/detaching payload (as described above both actions are performed with active compliance control), gravitational effects suddenly change the interaction force. In order to compute pure contact forces for the impedance control it is necessary to compensate for these gravitational effects. As specified above, the control system includes functions for setting the payload inertia parameters and compensating for gravitational effects. In the SPARCO system this setting was done with the programming language using the set_payload command. Thereby, the compliance control is deactivated and put into MONITORING mode (after relaxing contact). Since the change of payload parameters acts as an abrupt virtual alteration of the contact force, which is computed by the control system, the collision detection must be deactivated. Indeed, in this case the collision monitoring is not relevant, since the robot is intentionally in contact with the environment. In order to achieve conformance between computation and real contact forces, the relax action can be applied again.
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Experiment 4.2: SPARCO INSTALL task experiment The insertion algorithms implemented in the above PDL2 program are realized in the CAT system to perform various INSTALL/REMOVE tasks. This experiment demonstrates a typical insertion task in the SPARCO port (Fig. 4.28). In the case considered here, the insertion is terminated when a nominal insertion depth (325 mm) has been reached.
Fig. 4.38 Insertion sequences: initial contact (above left), engagment (above right), insertion (below left and right)
Characteristic insertion sequences are presented in (Fig. 4.38). The interaction forces in the C-frame and the linear motion components with respect to the port O-frame are presented in (Fig. 4.39) and (Fig. 4.40) respectively. In order to distinguish the task execution phases from each other, the flags, which are characterized by the relevant actions, are presented in (Fig. 4.41).
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Fig. 4.39 Interaction force and torque in C-frame during installation
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Fig. 4.40 End-effector displacements
Fig. 4.41 Action phases flags: -2 -relax, -1 –detaching, 1-approaching and engagement, 2 –insertion, 6 –termination
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From the above diagrams the following observations may be drawn: • The beginning of the engagement phase is characterized by small impact forces in all directions (a time interval of 5-10 s). • Relatively high lateral position misalignments (ca 4 mm) are compensated for, while maintaining quite small forces (less than 4 N). At the end of the engagement these forces are relaxed. • At the start of the insertion phase (at the approximate time interval of 45-140 s), the interaction force components increase again. However, with the growing depth of insertion, the z-component in particular becomes smaller due to the peg and hole axes’ being relatively well centralized after the engagement. During this phase the lateral displacements remains insignificant since they have been mainly compensated for during the engagement. • When an insertion depth of ca. 270 mm has been reached, the insertion phase is completed and the residual forces are relaxed to near zero. • The final termination phase is then realized through a further insertion with a velocity of approximately 2 cm/s, until the given nominal depth has been reached. • Since for testing purposes a relatively high imprecision was introduced (ca. 10 mm), the peg tip impacted the bottom of the hole, which produced, in this experiment, a contact force of approximately 20 N. This force is proportional to the position error and the selected impedance gains (stiffness of ca. 2000 N/m). At the end of the sequence, this force is relaxed and a proper unstrained insertion is achieved by adjusting the peg to the real hole constraint. • Detaching the peg and moving the robot back to the standby pose terminates the insertion. 4.5 Conclusion This chapter considers algorithms for the practical impedance control design in industrial robotic systems. The developed algorithms integrate the theoretical and practical stability results investigated in the previous chapter. The considered impedance control synthesis addresses basic control design problems at the servo-control layer. The impedance control design has been established for a reliable decoupled compliance geometric model that allows a relatively simple
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parameterization of the target impedance behavior. For fundamental and common interaction tasks the compliance parameters can be chosen independently of the interaction system configuration. More complex robot/environment interaction was also considered based on the spatial compliance model. The control synthesis consists of the straightforward steps computing the target impedance parameters and impedance compensator gains. All input parameters to the design algorithm have been explicitly specified. The impedance control design functions are implemented in MATLAB. The feasibility of the developed algorithms was demonstrated using experiments with two modern industrial robot systems. However, it should be noted that control design is essentially a very creative job, which is quite difficult to completely automate. The impedance control design toolbox provides rather practical design functions; however, the designer’s creativity, expertise and experience are crucial for the final success. Finally, a reliable geometric and control framework for the implementation of compliance control in industrial and other advanced robotic systems has been developed and presented. Several practical and robust control algorithms at higher planning and programming control layers were designed and tested. The essential algorithms support a setting of the compliance parameters, such as the C-frame location and impedance gains, as well as a continuous switching of compliance control and variation of parameters. These features are proven to be essential for a stable and robust execution of the compliance control tasks. Powerful sets of control functions, also presented in this chapter, integrate the basic compliance control algorithms in the forward robot control. These functions perform all of the computations and management of the parameters between the convenient robot position control system and the impedance control kernel. The basic structure and implemented functions of the compliance control module are presented and specified. Finally, the new commands providing a flexible user-interface are designed and implemented in the PDL2 environment. The new programming language commands are illustrated by means of several examples. An essential design requirement was to combine the user’s experience with robot motion programming and simple understandable physical behavior of the impedance control, which mimics a variable spatial mass-damper-spring system. The new set of commands implemented in the PDL2 provides a useful framework for the implementation and programming of general compliance tasks.
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The experimental testing within the SPARCO control system has clearly proven the reliability and robustness of the presented high-layer compliance control algorithms. Certainly, a basic precondition for the implementation of compliant motion control is the design of a robust servo impedance controller, which ensures stable transition and coupling with the environment. However, the control integration and programming issues which are often underestimated in the literature are essential for a customary and efficient application of impedance control in practical contact tasks. Proper selections of the C-frame location and target impedance gains are crucial for a successful execution of the impedance control tasks. This selection should be compatible with the very nature of the motion constraints, i.e. contact task geometry and physical task characteristics (e.g. force/motion relationships). Consequently, the C-frame and the gains should be selected in such a manner that the contact reaction forces can move this frame to an equilibrium position similar to that of a passive mass-spring-damper mechanical system. The experience gained in performing the compliance tasks presented here is essential for compliance control design and implementation in a wide range of tasks in industry and service robotics.
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Buerger S. P. and Hogan N., “Relaxing Passivity for Human-Robot Interaction”, Proceedings of the Int. Conference on Intelligent Robot and Systems, Beijing, pp. 45704575, 2006. Hogan N., 1987, “Stable Execution of Contact Tasks Using Impedance Control”, Proceedings of IEEE International Conference on Robotics and Automation, Raleigh, North Carolina, pp. 1047-1054. Mason M.T., 1981, “Compliance and Force Control for Computer Controlled Manipulators”, IEEE Transaction of System, Man and Cybernetics, SMC-11, pp. 418-432. Raibert M.H., Craig J.J., 1981, “Hybrid Position/Force Control of Manipulators”, ASME Journal of Dynamic Systems, Measurement and Control, Vol. 102, pp. 126-133. Lozano-Perez T., 1983, “Robot Programming”, Proceedings of IEEE, Volume 71, pp. 821-841. Whitney D.E., Nevins J.L., 1986, “What is Remote Center of Compliance (RCC) and What Can it Do”, Robot Sensors Volume 2 - Tactile and Non-Vision, IFS Publication, pp. 3-17, Springer-Verlag.
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Surdilovic D., 2002, Synthesis of Robust Compliance Control Algorithms for Industrial Robots and Advanced Interaction Systems, Ph:D. Thesis, Mechanical Engineering Faculty of University in Nis, Serbia and Montenegro. Doyle J.C., Francis B., Tannenbaum A.R., 1992, Feedback Control Theory, Macmillan Publishing Surdilovic D., Lizama E., Kirchhof J., 1995, “A Toolbox for Simulation if Robotic Systems”, Proceeding Simulation Congress Eurosim ’95 Vienna, Elsevier, Amsterdam, pp. 693-699. Nicolodi S., Surdilovic D., Schott J., Boumanns J., Putz P., 1995, “Development of a Space Controller with Advanced Sensor-Based Control Capabilities”, Proceedings 7th International Conference on Advanced Robotics ICAR’95, Sant Feliu de Guixols, Spain. Surdilovic D., Bernhardt R., Colombina G., Grassini F., 1996, “A Space Robot Control with Advanced Sensor-Based Capabilities: Terrestrial Spin-off”, 27th ISIR, Milan, pp. 243-248. Surdilovic D., 2000, Impedance Control Design Toolbox – User’s Guide, IPK-Berlin. De Bartolomei M., Frassini F, S. Losito, 2000, Robotic arm co-operation for assembling a reticular structure, Proceedings Astra 2000, ESA/ESTEC. Surdilovic D., De Bartolomeo M., Grassini F., “Synthesis of Impedance Control for Complex Co-operating Robot Assembly Tasks”, Proceedings AIM’01 Conference, Como, Italy, 2001, pp. 1181-1187. Dogliani F., Magnani G., Sciavicco L., 1993, “An Open Architecture Industrial Robot Controller”, IEEE Robotics and Automation Magazine, Vol. 1, No. 2., pp.19-21. Mason M.T., 1982, “Compliant Motion”, Robot Motion: Planning and Control, Ed. by Brady M. et al., MIT Press Series in Artificial Intelligence, pp. 305-322.
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Chapter 5
Robust Control of Human-Robot Interaction in Haptic Systems
5.1 Introduction The new robust control design framework established in this book for the control synthesis of the interaction between an impedance-controlled robot and a passive or active environment could also be effectively applied in other interactive systems with physical or virtual interfaces. Recently, outstanding research interest addresses new interactive systems designed for the interaction between a human being and a robotic device, as well as with physical or virtual dynamic environments (Fig. 5.1). To the novel interactive systems belong kinaesthetic displays and haptic interfaces, teleoperation systems, human enhancers and augmentation devices, rehabilitation robots, robot assistants and collaborative robots, etc. These systems are designed to produce/receive kinaesthetic stimuli for/from human movements, as well as to provide the user with a realistic feeling of contact and dynamic interaction with nearby, remote or virtual environments. The advanced interaction systems have recently found very attractive applications in surgical and rehabilitation robotics, power assist-devices, training simulation systems, etc. The most critical issue in these systems is to ensure stable and safe interaction with a high fidelity of reproduction of a virtual environment. This is a challenging task, when taking into account serious problems such as unknown and variable human dynamics, commonly non-linear environmental characteristics, as well as various disturbances in computer-controlled systems. This chapter considers the stability of the interaction of a human-robotenvironment (real or virtual) system based on the robust control design approach. The proposed new interaction stability paradigm ensures contact stability during all phases of interaction. Moreover, the new design framework realizes lowimpedance performance allowing considerable reduction of high apparent industrial robot inertia and stiffness. The defined stability indices take into account the relevant effects in robot control systems, such as time lags and sampling data effects, as well as uncertainties in the environment and realized target admittance models. The synthesis of robust control laws has been 491
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confirmed to be very efficient for the stabilization of the interaction between a robot and a stiff and force-delayed environment taking into account the desired interaction performance. Testing this approach in various robotic systems has demonstrated the feasibility and reliability of the interaction control approach even for relative higher control rates and lags. Robust stability provides useful design tools for control synthesis for linear and non-linear systems. Therefore, it is promising to apply established robust control for haptic system design.
Fig. 5.1 Structure of the human-robot interaction with a real or virtual environment
5.2 Haptic System Structures Haptic technology, or haptics, refers to the devices capable of providing the user with a sensation of touch and interacting with objects in a virtual world. Critical performance considerations for haptic interface devices concern the degrees of freedom, bandwidth, maximum stiffness that can be rendered and transparency.
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The notion of haptic transparency is used to quantify the fidelity with which virtual object properties are presented to, and perceived by, the human operator. The idiom haptic rendering was initially introduced by Salisbury et al. [1] to denote algorithms for programming and controlling the force of interaction with a virtual environment. Essentially, the basic interaction chain in a haptic display consists of three principal elements (Fig. 5.2): the human operator (H), haptic device (D) and virtual environment (VE). A common model of a haptic interface is presented in (Fig. 5.2). The middle element in this model is a haptic device, which is, based on analogy with electrical network circuits, represented as a so-called two-port network. A haptic device interconnects the human being with the virtual environments (both linked as one-port networks) via force and velocity I/O signal pairs, describing the exchange of energy between blocks This representation has been demonstrated to be very useful in the analysis of teleoperation and haptic systems [2]. Since the haptic device is computer controlled, critical SD effects on the interaction system stability (identified in the previous chapter to be control delay and sample-and-hold effects) must be also introduced in the interaction model. These effects are involved in the digital control block (DC) presented in (Fig. 5.2). The main role of the SD control system is to measure and render I/O signals via the haptic interface, and thus to provide the operator with an enforced sense of haptic (or kinesthetic) presence in a virtual environment.
Fig. 5.2 Elementary network model of a haptic system
Depending on the signals measured in a haptic interface, two system classes may be distinguished: impedance and admittance displays. In the impedance display the velocity (position) of the haptic mechanism is measured and a command force is rendered. The admittance display ensures the tracking of a command position using interaction force/moment measurements in the handle. Impedance displays are commonly lightweight, back-drivable active mechanisms (e.g. Phantom in Fig. 5.3), while position-controlled high-inertia manipulators
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have often been used as admittance displays. In the target impedance realization experiment (Fig. 3.13), the impedance-controlled industrial robot could be considered as an admittance display rendering a target admittance. Although the study and modeling of human motor control and spatial limbs dynamics are fundamental challenges in biomechanics and neuroscience, the understanding of the human interaction with a dynamic environment is still insufficient. The key quantity describing human arm dynamic interaction is the end-point impedance [3]. Numerous studies have recently demonstrated surprising human capabilities to adapt the arm impedance to variable interaction conditions and perturbations, even so to perform mechanically unstable tasks [4]. The Cartesian end-point arm impedance is a non-linear and non-symmetric spatial impedance combining passive and active components [3]. However, in the control analysis the human impedance is commonly, for the sake of simplicity, considered as a linear variable impedance, often with one or two DOF’s. Likewise, the haptic display dynamics can be considered as a linear admittance, while the environment generally can be represented by non-linear impedance (Fig. 5.2).
Fig. 5.3 Haptic interaction system (impedance display Phantom 3.0)
However, the performance obtained with the direct rendering approach sketched in (Fig. 5.2) is commonly poor, and therefore such an interaction structure is not feasible. The essential interaction problems cannot be successfully resolved with this interaction system architecture. Such a haptic
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system could interact well with a particular environment and for a specific human behavior, but in other cases performance might be bad; moreover, the interaction could be unstable (especially upon contact with a stiff virtual wall). Generally it is not possible to guarantee the stability of the interaction with the simple haptic interface control system presented in (Fig. 5.2). Obviously, a more sophisticated controller is required to take into account the human and environmental dynamics. Synthesis of such a controller is, however, extremely complex. In order to simplify design and to improve the stability of the haptic interaction system, Colgate et al. [5] have proposed to couple an additional block, referred to as virtual compliance or virtual coupling (Fig. 5.4), between the haptic device and the virtual environment. The virtual coupling is commonly selected as impedance, i.e. admittance. The virtual coupling provides a simple, nevertheless stable and robust haptic controller. For a particular haptic device a corresponding virtual coupling might be designed regardless of simulated virtual worlds and real human behavior. The main design goal is to ensure the passive behavior of the coupled subsystem consisting of the virtual coupling and the haptic display, thereby also taking into account critical SD effects. By these means, when taking into account that the human produces an almost passive and stable interaction with a passive system, the stability of the entire haptic system may be ensured under all operating conditions if the virtual environment is passive. A haptic interface operates as a connection between human and virtual environments. The transparency and the Z-width are the main measures of performance of a haptic display. Transparency represents the degree to which velocities and forces (on the human and environmental sides) match each other. The Z-width [6] of a haptic interface can be defined as the achievable range of impedance that can stably be presented to the operator. An ideal haptic interface could simulate free motion without inertia or friction, as well as render infinitely rigid and massive virtual objects.
Fig. 5.4 Haptic interface with virtual coupling (admittance)
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5.3 Haptic Rendering The primary concern in haptic systems is to achieve stable interaction under any operating conditions and for all simulated virtual environments, without unwanted oscillations that degrade virtual surface rendering. However, that is a challenging goal, because several destabilizing effects tend to jeopardize interaction stability. In order to ensure stability almost all modern haptic systems utilize the advantages of the virtual coupling concept and implement the control architectures similar to the one sketched in (Fig. 5.4). Numerous experiments clearly demonstrate that the contact stabilization with stiff, delayed and non-linear environments still represents the crucial problem in haptic interfaces. The specific problem in haptic interfaces is lack of objective stability testing. A human being exhibits a good capability to stabilize (damp) the interaction with a slightly oscillating environment. Therefore, the lost of contact and bouncing in haptic interfaces appear to be less critical [7], compared with contact stability problems in industrial robots investigated in the previous chapters. However, these oscillations can jeopardize interaction fidelity. In the majority of experiments the increase in sampling rates and reduction of force magnitudes have been recognized as promising measures to reduce bouncing. The stability of the haptic interaction system is commonly considered based on passivity theory. Colgate and Hogan [8] have defined necessary and sufficient conditions to ensure the stability of a linear robotic system coupled to a linear environment. The authors argue that if the environmental impedance is positive real, representing any passive Hamiltonian environment, then a necessary and sufficient condition to ensure stability of the linearized robotic control system is that the realized admittance be positive real. In other words, it should represent the driving point impedance of a passive network. The system passivity concept provides a relatively simple test for the assessment of the coupled system stability. In this test only the passivity of the environment should be proven, without an accurate knowledge of the parameters. Assuming that an ideal second-order target impedance response is realized, the passivity of the target admittance implies positive definite inertiastiffness-damping matrices. The attractiveness of passivity theory is that it guarantees the phase lead or lag of any passive block in an interaction system (Fig. 3.7, 3.9) to be no more than 90 deg. Thus, when two passive blocks are connected together as shown in (Fig. 3.9), the maximum phase lead or lag of the overall open-loop system will be limited to no more than 180 deg. Therefore, it guarantees stability regardless of the overall gain. The problem with any
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practical application such as a robot performing impedance control or a haptic display is that additional phase lags caused by cut-off (analogue and digital) filters and sampling delays make the overall phase lag to be more than 180 deg. at high frequencies. If the overall gain of the open-loop system is too high (such as in contact with a stiff environment), instability will occur. The realization of a passive system in a real digital robot controller imposes a fundamental limit on the reduction of apparent robot inertia to a maximum of 50% [9]. In industrial robots with apparent Cartesian endpoint masses of several hundred kilograms, that is an exceedingly conservative condition. Newman [10] proposes a natural admittance control (NAC) approach that provides considerable compensation for friction, however, without significant improvement of achievable target admittance reduction which does not violate the passivity constraints. To solve this critical limitation in implementation, the author has proposed in [11] the insertion a mechanical filter between the manipulator and the environment in order to achieve low-impedance performance. However, this is a specific solution, rather than a general approach. The reduction of apparent inertia appears also to be essential in human-robot systems. In order to achieve valuable performance of the human-robot interaction, Buerger and Hogan have suggested [12] relaxing the restrictive passivity condition and designing the interaction system either by taking into account the limited knowledge of the particular environment or by lowering the target inertia beyond the passivity threshold. Colgate et al. [13] have derived explicit conditions for the passivity of a haptic systems including a linear haptic device, a virtual coupling and a virtual environment, taking into account sampling and computation delay effects. The authors argued the essential relevance of physical damping parameters for enhancement of system passivity and interaction stability. For a simple SISO coupling system consisting of the haptic-device, i.e. admittance Z D = 1 (ms + b ) , and the virtual-coupling impedance Z V = Bs + K , the stability (passivity) condition imposes
b>
KT +B 2
(5.1)
where T is the control sampling time. In this elementary case of a haptic interface, the virtual coupling represents a virtual wall which consists of the parallel connection of the virtual stiffness K and the virtual damping B, to be rendered to the human. Hence, the condition (5.1) means that physical damping must be involved in the system in order to ensure a stable interaction with the
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virtual wall. Higher sampling rates (i.e. smaller T) facilitate the implementation of stiffer walls. Brown and Colgate [14] have derived similar expressions for the minimum mass of the virtual wall that can be simulated passively. However, the stability conditions that were obtained appear to be quite conservative. Moreover, these criteria imply a physic-based approach to system design that is not always reliable. For example, higher additional damping (5.1) allows higher virtual impedance to be realized, but thereby the impedance of the haptic device must also be increased. Adams [15] has proposed an approach for a virtual coupling design based on the network stability which appears to be less conservative than the passivity based synthesis. The stability of the two-port network consisting of the haptic device and the virtual coupling guarantees the stability of a haptic interface when coupled with any passive virtual environment and human operator. Miller et al. [16] have extended the passivity based approach to haptic systems involving non-linear and time-delayed virtual environments. Hannaford and Ryu [17] have applied time-domain passivity analysis in order to improve the system performance in contact with a very stiff and delayed environment. Their approach utilizes a passivity observer to monitor the system energy and a passivity controller to modulate virtual damping in order to maintain a passivity constraint. Rendering stiff and delayed virtual environments remains a central challenge in the field of haptics. Recent research efforts address the effects of haptic device dynamics (inertia, natural damping and Coulomb friction) and SD control system non-linearities (sampling rate, quantization, computational delay) in the stability and performance of interaction [18, 19]. Physical damping and friction were recognized as key elements to dissipate energy and preserve system stability [20]. It was remarked that non-linear friction effects can create distinct stability regions ensuring that the haptic devices operate stably despite of a violation of the passivity criteria. However, it is very difficult to employ these effects for the control design. The robust control framework, however, has been proven to be a practical synthesis oriented approach ensuring stable interaction in the presence of system non-linearities [21]. 5.4 Robust Control of Haptic Systems Interaction The application of robust contact control will be considered for the two basic haptic interaction systems: admittance and impedance displays. Admittance displays measure the forces exerted by the human operator using a force sensor
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and generate the corresponding displacements. Conventional non-backdrivable and position controlled industrial robots can usually be utilized to realize the interaction based on the admittance model. Impedance displays are mechanical devices configured to render the interaction force while providing measurements of joint positions and/or velocities. Impedance controlled systems detect the motion provided by the operator and control the force exerted by a haptic device. Impedance displays have typically low inertia and are highly backdrivable (Fig. 5.3). 5.4.1 Admittance display control The principal scheme of the interaction control for an admittance display based on the robust control method developed in the previous sections is sketched in (Fig. 5.5). The display is presented as a closed loop position system Gp designed to accurately track the reference position xr received from the haptic control system. The reference position is obtained as the deviation ∆xF from an initial position x0 based on the proposed impedance compensator law (3.68) for conventional industrial robot systems
~ −1 −1 G f (s ) = G p (s )Gt (s )
(5.2)
~ where G p is a minimum phase estimate of the position control transfer matrix and Gt (s ) is the target impedance matrix. The main difference from the impedance control (Fig. 3.4) is that the input to the haptic compensator is the difference between the human and virtual environment forces. In an ideal positional servo x = xr , the computed position is rendered to the operator. The applied impedance controller (5.2) realizes the target admittance in such a way −1 that the display and the impedance controller can be replaced by Z t (Fig. 5.5). Hence, the interaction between the human operator, virtual-coupling (targetimpedance) and virtual-environment can be considered based on a simplified model, similar to the preceding robot/environment interaction (Fig. 3.7). In a simplified analysis, the human behavior can be considered as a passive variable impedance [22]. Thus, the contact and coupled stability of admittance displays will be ensured for the robustly stable interaction system presented in (Fig 5.6). Based on the established analogy between the impedance robot control systems in (Fig.3.7) and haptic admittance control system in (Fig. 5.6), we can apply the derived stability conditions (3.169, 3.170), as well as the target impedance design algorithms (Fig. 3.48) and tools developed for the robot/environment
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interaction, now for the design of admittance haptic systems. According to (Fig. 5.6), the relationship between the human and environmental force is defined by
[
F = Z e ( p0 − e) = Z e p0 − Z t
−1
(F − Fh )]
(5.3)
providing for a SISO system
F=
Ze Z Z Fh + e t p0 Zt + Ze Zt + Ze
Fig. 5.5 Admittance display control scheme
Fig. 5.6 Interaction model
(5.4)
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When considering the systems in (Fig. 3.7) and (Fig. 5.6), as well as the generalized stability conditions defined in the Theorem 3.9 and (3.128), the equivalence of the transfer function ( Fh → F ), which describes the force interaction in an admittance display, and the function ( p 0 → e ), which describes the robot/environment contact behaviour, can be established. Assuming that the operator intends to exert forces upon the virtual wall, the hand force can be, as with the nominal penetration p 0 , considered to have constant direction towards the virtual obstacle during contact establishment. Then, based on Theorem 3.9, which defines a sufficient condition for robustly stable interaction in the (p 0 , e ) subsystem, we can write the stability criterion for the considered equivalent (Fh , F ) subsystem regarding (5.4)
Ze Zt + Ze
≤1
(5.5)
∞
which ensures
F Fh
2
≤1
(5.6)
2
These relations define sufficient conditions for both stable contact transition and the coupled system interaction of an admittance display. Let us consider the role of p 0 in the model (5.3-5.4) and schemes sketched in (Fig. 5.5) and (Fig. 5.6). In the haptic system the initial position x0 (i.e. penetration p 0 ) is constant and has a different meaning from the nominal position (penetration) in the robot/environment interaction. In effect, in a haptic interface the hand force Fh directs the system motion, while x0 defines the start location. Conveniently, x0 is selected in front of the virtual hindrance. Therefore, p 0 (“negative penetration”) has no sense in the contact model (5.3) and can be neglected ( p 0 = 0 ). When the complete second-order target impedance is selected Z t (s ) = M t s 2 + B t s + K t , and assuming a stiff virtual environment Z e = K e >> K t , the interaction system (5.4) provides the following steadystate performance
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Dynamics and Robust Control of Robot-Environment Interaction
F∗ =
Ke ∗ ∗ Fh ⇒ F ∗ ≈ Fh Kt + Ke
(5.7)
However, in the free space the applied control (Fig. 5.5), like the robot −1 impedance control, realizes the target admittance (virtual coupling) Z t that is rendered to the operator. Since the general admittance exhibits a spring-like behavior, the operator should exert greater force than (5.7) in order to bring the virtual coupling system into contact with the virtual environment. Consequently the entire steady-state force becomes
(
∗
)
Fh = K h p * = K t x e - x 0 + p * + K e p * ∗
F = K ep
*
(5.8)
where p * denotes the equilibrium penetration and K h is the total stiffness rendered to the operator. From this it is obvious that
Kh ≥ Kt + Ke
(5.9)
In order to improve transparency, a target-damping virtual coupling (with zero stiffness) can be applied
Z td (s ) = M t s 2 + B t s
(5.10)
In free space with this virtual coupling the human operator feels only the target inertia and damping during motion, while the equilibrium hand force becomes zero. At contact (5.4), transparency is characterized by
F=
Ke ∗ Fh ⇒ F ∗ = Fh M t s + Bt s + K e 2
(5.11)
The virtual coupling target systems, which can take the form of a general second-order impedance or damping (5.10), determines the lower impedance bound of the Z-width achievable with the control system (Fig. 5.5). Theoretically the maximum bound might be infinite for K e → ∞ . As demonstrated, by means of the developed robust interaction control design, we can synthesize the target systems (virtual couplings) ensuring the contact and coupled stabilities. Practical limitations on the upper Z-width bound govern the control lags, which in haptic systems can be considerably larger in comparison to the robot impedance
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control. Geometrically and physically, complex environments and contact interaction situations might require significant computation efforts to determine the forces, causing relatively large delays in a reliable computer control environment. This delay must be considered in the design (Fig. 5.7). As mentioned above, the new control design approach provides a unified and efficient framework for continuous as well as SD delayed system synthesis based on the conditions for the robust or passivity based contact stability (3.169-170)
[I + z G (z) Gˆ (z )] [I + 2 z G (z) Gˆ (z )]
−1
−1
n
e
<1
t
e
(5.12)
−1
−1
n
∞
<1
t
∞
In particular, this is advantageous for haptic systems. Even more important, by means of weighting functions describing model uncertainties (3.166), nonlinear environment effects can also be effectively considered in the design.
Fig. 5.7 SD interaction control
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Dynamics and Robust Control of Robot-Environment Interaction
However, in specific cases the effect of a simplified design could be conservativeness. In other words, the synthesized virtual coupling for a nonlinear delayed environment may become quite over-damped, causing slow response and sluggish behaviour, which significantly reduces the system transparency. This is particularly critical in free-space. In order to overcome this problem, we can apply adaptive virtual coupling with different target systems appropriate for free-space handling, stable transition and interaction with a delayed environment. However, the adaptive compliance control may become very complex and difficult to implement. A reliable approach towards adaptive interaction control based on fuzzy logic, promising for both robotic and haptic systems, has been proposed in [23]. The following initial experiment illustrates the benefits of the new robust compliance control design for haptic systems. Experiment 5.1: 1-DOF admittance display interaction control The main issue of this experiment is to demonstrate the design and performance of the new haptic controller on a simple SISO admittance display (Fig. 5.8). The experimental system consists of a single linear axis with a direct-drive actuator consisting of a linear hybrid (variable reluctance) stepper motor with air bearings. A sophisticated digital control drives the actuator precisely and smoothly in microstep mode and compensates for resonant effects at low velocities. Similar to DC-motors, a “voltage interface” is realized between the internal stepper control and external feedback control. This interface allows servo-loops around the position sensor (high-resolution linear encoder strip) and force sensor (six-DOF Shunk/ATI sensor) to be closed. The position and impedance control (i.e. haptic admittance control) is implemented according to (Fig. 5.5-5.8) in the SIMULINK and realized using the Real Time Workshop and a dSPACE rapid control prototyping system. In the experiments the linear drive is located in front of a virtual wall ( K e = 60000 N/m). The coupling impedance is selected in the form (5.10) with the selected target mass M t = 10 (kg). The target damping is computed based on the robust stability condition (5.12) taking into account the sampling
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505
time T=0.001 (s) and control lag τ = 0.001 (s). The target damping needed to ensure robustly stable interaction is computed using the Impedance Control Design Toolbox [24]. For the adopted parameters the required minimum damping is B t = 1224 (Ns/m). The experiments consist of pushing the actuator by hand in the direction of the virtual wall (Fig. 5.8) until contact is achieved, pressing on the wall and pulling back. This procedure was repeated several times. The measured hand force and simulated wall force are presented in (Fig. 5.9). Obviously, the interaction was stable, and both contact transition stability and coupled stability were reached. Transition sequences to- and from-contact, as well as coupled interaction performance are presented in (Fig. 5.10). The matching of the applied hand Fh force and the environment force Fe can be considered as a measure of the transparency [25]. The experiment in (Fig. 5.11) illustrates the robustness of the applied control design method. In this trial the same coupling impedance synthesized for K e = 60000 is applied for interaction with a significantly stiffer wall K e = 100000 (N/m) without affecting the performance. However, if the stiffness is further increased until K e = 150000 , the interaction becomes unstable (Fig. 5.12). Nevertheless, after redesign of the coupling impedance for the actual environment the contact is again stabilized (Fig. 5.13).
Fig. 5.8 Experimental set-up
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Dynamics and Robust Control of Robot-Environment Interaction
Fig. 5.9 Hand Fh (dashed line) and virtual wall F (solid line) forces during haptic interaction
Fig. 5.10 Contact transition sequences
Robust Control of Human-Robot Interaction in Haptic Systems
Fig. 5.11 Robustness of interaction control design
Fig. 5.12 Contact unstable interaction
507
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Dynamics and Robust Control of Robot-Environment Interaction
Fig. 5.13 Performance of redesigned controller
Experiment 5.2: 2-DOF collaborative robot based on an advanced handling system The developed algorithm was implemented to control a 2-DOF a hand-driven handling manipulator (x-y railway crane) with a 1000 kg payload capacity (Fig. 5.14). Each DOF is actuated by a friction drive actuator in order to achieve power assistance. The position of the hub is measured using laser distance sensors. Chip x-y force sensors were integrated in the hand grippers to measure hand control forces. Although referred to as a “KOBOT” (i.e. Cobot – collaborative robot), this system indeed represents an admittance display. The developed algorithms were applied to display different admittances as well as the virtual obstacles (walls) introduced in the working space in order to constrain the motion or to guide (slide) the hub to a target position. In order to manipulate heavy payloads ergonomically with minimum strain exerted upon operator, a hand force amplification is introduced. Taking into account the relatively high perturbations of the force and position measurements, the admittance control was implemented
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509
in the open loop. To synthesize target impedances ensuring stable interaction with a stiff environment (Ke=5000 N/m), the stability conditions (5.12) were 2 applied to synthesize a mass-damper coupled impedance Z t (s ) = 10 s + 140 s which ensures robust stability of interaction with the selected virtual wall.
Fig. 5.14 First “cobot” prototype for industry
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Dynamics and Robust Control of Robot-Environment Interaction
Promising initial experiments (Fig. 5.15) with polygonal virtual walls, modelled as a pure stiffness environment, have clearly demonstrated the robust contact stability, even in an open loop control structure (in spite of sensing and control perturbations). This demonstrates the practical applicability of the novel haptic stability criterion. A stable interaction without bouncing is very important for this application in order to achieve an efficient and ergonomically smooth guidance of high-inertia payloads along virtual walls. The first implementation in industry has found a considerable acceptance for the efficient and ergonomic handling of heavy loads. 5.4.2 Impedance display control The impedance display control scheme is presented in (Fig. 5.16). The system −1 consists of a lightweight haptic device, presented by the admittance Z D , which is shifted by the human operator. The measured position of the display is, similar to the impedance control (Fig. 3.7), considered as the “nominal penetration” p 0 , which is modified proportionally to the interaction force F and the selected coupling admittance Z t −1 . By this mean the effective penetration p in the virtual-wall is computed. The effect of a virtual force is rendered to the human using the haptic device control system and actuators. The difference between the virtual, i.e. haptic device active force F and the hand force Fh provides effective force Feff which produces the haptic device motion. Based on (Fig. 5.16) the relationship between the hand and interaction force is expressed by
[
F = Z e ( p0 − e ) = Z e Z D
−1
(Fh − F ) − Z t −1 F ]
(5.13)
providing in the SISO case
F=
ZeZt Z Fh = H Fh Ze Z t + Z t Z D + Ze Z D ZD
(5.14)
where Z H denotes the equivalent impedance of the haptic interface consisting of the serial connection of the haptic device, virtual coupling and virtual environment impedances
Robust Control of Human-Robot Interaction in Haptic Systems
1 1 1 1 = + + ZH ZD Z t Ze
511
(5.15)
where ZD = ZDs . Commonly, the impedance of the haptic device has the form ZD = M D s 2 + B D s .
Fig. 5.15 Amplified hand force (solid line) and virtual interaction force (dashed line) during contact with a virtual wall (for ergonomic reasons, the hand force is scaled by a factor of 10)
Fig. 5.16 Impedance display control
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Dynamics and Robust Control of Robot-Environment Interaction
In the case where the effect of virtual coupling is excluded, i.e. Z t → ∞ , the relationship between interaction and hand force becomes
F=
Ze Fh Ze + ZD
(5.16)
which is similar to the relation (5.3) describing the force interaction in an admittance display (the difference is that in (5.16) the virtual coupling Z t is replaced by the haptic display impedance ZD ) Analogously, the robust contact stability requires
Ze Ze + ZD
≤1
(5.17)
∞
For a SISO system and a “worst-case virtual environment” Z e = K e the above condition will be satisfied at all frequencies when
BD ≥ M DK e
(5.18)
Based on this criterion it possible to design a haptic interface ZD ensuring stable interaction with a passive virtual environment. Since usually the virtual environment can significantly vary, the design should be performed for “the worst case environment”. Thereby main SD system control effects should also be taken into account (Fig. 5.17). The designed system, however, usually requires a significant amount of damping (5.18) to stabilize the worst-case interaction. Since commonly interaction with a stiff virtual environment is characterized by very low velocities, it is very difficult to realize the required damping using either a mechanical or control approach. Moreover, high damping causes sluggish display reaction, which reduces the transparency. Therefore this approach is mostly not reliable. A more promising strategy is to synthesize a lightweight mechanical display and to stabilize the interaction utilizing virtual coupling. Unlike previous works, the new control synthesis approach includes the virtual environment in the design. In order to ensure stable interaction based on (5.14) the following must be satisfied:
Robust Control of Human-Robot Interaction in Haptic Systems
ZeZ t ZeZ t + Z t ZD + ZeZD
≤1
513
(5.19)
∞
Simple virtual coupling impedance that can stabilize the above system and * ensure F ∗ = Fh can be adopted in the form
Zt = Bt s + K t The proposed approach allows the haptic device to be designed independently of the virtual coupling. Since the virtual coupling is realized in a virtual simulation environment, it can be synthesized for various virtual environments. Furthermore, as demonstrated above, robust control design tolerates considerable variations in virtual environment parameters without affecting the contact stability. Thus a virtual coupling may be applied for a wide range of virtual environments. The lower bound on the Z-width of the impedance display in the free virtual space ( Z E = 0) is ZD , while the upper bound for
Z E → ∞ also tends to infinity. A practically achievable bound depends on several factors, such as the interaction task, resolution of position sensors, realizable virtual coupling, etc.
Fig. 5.17 SD control scheme of impedance display
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Example 5.1: One-DOF impedance display interaction control This simulation example demonstrates the performance of the impedance display synthesis in the case of a simple one-DOF interaction system (Fig. 5.18). The simulation scenario is equivalent to the above experiment. The virtual environment is assumed in the form of a pure stiffness Z e = K e , while the haptic display is adopted as a SISO mechanical system ZD = M D s 2 + B D s . For a very stiff virtual environment K e = 60000 (N/m) and M D = 2 (kg) we can compute the damping needed to ensure robust stability based on (5.18) providing B D = 490 (Ns/m). The simulation of contact transition with a constant hand force Fh = 15 (N) is presented in (Fig. 5.19). Similar to the robot/environment interaction examined in (Fig. 3.55-59), in the considered example, robust design provides safe and a bit conservative interaction. The required amount of physical damping of the haptic device appears to be very high.
Fig. 5.18 Impedance display system
In the next simulation experiment a more feasible haptic system is adopted with considerably smaller damping parameter ZD = 2s 2 + 20s . A virtual coupling Z t = B t s + K t is added in order to stabilize the interaction with the same virtual environment as in the previous case. The parameters of the virtual coupling are determined based on (5.19) is such a way that damping was settled to B t = 50 (Ns/m) and the stiffness parameter was computed to satisfy the condition (5.19). The obtained value was K t = 500 (N). The interaction performance is presented in (Fig. 5.20). Evidently, the interaction is stable in
Robust Control of Human-Robot Interaction in Haptic Systems
515
spite of a significantly higher approaching velocity than in the previous experiment (due to lower mechanical damping of haptic device), which causes a higher impact force. As expected, the interaction remains stable if the virtual wall stiffness is increased to K e = 90000 (N/m), however, with worse performance (Fig. 5.21).
Fig. 5.19 Performance of haptic device design for stable interaction
Fig. 5.20 Test with virtual coupling and higher approaching velocity
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Fig. 5.21 Robustness test
5.5 Conclusion The robust control framework and the new contact stability theory established for the control synthesis of the interaction between an impedance-controlled robot and a passive environment has been expanded to the control and synthesis of haptic interfaces interacting with a virtual environment. This rapidly emerging technology imposes high requirements on the interaction stability and robustness of the control system in spite of considerable control computation efforts and time lags. Simple experiments and simulation results have demonstrated the advantages, high performance and reliability of the new algorithms. The advantage of robust stability was especially demonstrated in the interaction control of novel intelligent power-assist handling systems with significant perturbations in the force and position measurements. This shows the practical applicability of the novel stability criteria for haptic systems. A stable interaction without bouncing is very important for such applications in order to achieve an efficient and ergonomically smooth guidance of high-inertia payloads along virtual walls.
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Salisbury K.J., Brock D., Massie T., Swarup N. And Zilles C., “Haptic Rendering: Programming Touch Interaction with Virtual Objects”, in Proceedings Symposium on Interactive 3D Graphics, New York, pp. 123-130, 1995. Adams R., Hannaford B, “Stable Haptic Interaction with Virtual Environments”, IEEE Trans. on Robotics and Automation, Vol. 15, No. 3, pp. 465-474, 1999. Hogan N., “Controlling Impedance at the Man/Machine Interface”, Proceedings of IEEE International Conference on Robotics and Automation, Scottsdale, Arizona, pp. 1626-1631, 1989. Burdet E., Osu R., Franklin D., Millner T., Kawato M., “The central nervous system stabilizes unstable dynamics by learning optimal impedance”, Nature, Vol. 414, pp.446449, 2001. Colgate J. E., Stanley M.C., Brown J.M., “Issues in the Haptic Display of Tool Use”, Proceedings IEEE/RSJ International Conference on Intelligent Robots and Systems, Pitsburgh, PA, pp. 140-145, 1995. Colgate J. E., Brown J.M., “Factors Affecting the Z-Width of a Haptic Display”, Proceedings of IEEE International Conference on Robotics and Automation, San Diego, pp. 3205-3210, 1994. Miller B., Colgate E., Freeman R., “Guaranteed Stability of Haptic Systems with Nonlinear Virtual Environments”, IEEE Transaction on Robotics and Automation, Vol. 16, Nr. 3, pp.712-719, 2000. Colgate J. E., Hogan N., , “Robust Control of Dynamically Interacting Systems”, International Jour. of Control, Vol. 48, No. 1, pp.65-88, 1988. Colgate J. E., Hogan N., “An Analysis of Contact Instability in Terms of Passive Physical Equivalents”, Proceedings of IEEE Int. Conference on Robotics and Automation, pp. 404-409, 1989. Newman W. S., “Stability and Performance Limit of Interaction Controllers”, Trans. of the AMSE Journal of Dynamic Systems, 114, No. 4., pp. 65-88, 1992. Dohring M. and Newman W., 2002, “Admittance Enhancement in Force Feedback of Dynamic Systems”, Proc. of ICRA, Washington DC, pp. 638-643. Buerger S. P. and Hogan N., “Relaxing Passivity for Human-Robot Interaction”, Proceedings of the Int. Conference on Intelligent Robot and Systems, Beijing, pp. 45704575, 2006.
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Chapter 6
Intelligent Control Techniques for Robotic Contact Tasks
6.1 Introduction In view of the high and complex demands they have to meet as important integral parts of flexible manufacturing systems, robots are ideal objects for the application of intelligent control. Intelligent robots are functionally oriented devices built to perform sets of tasks instead of humans, as autonomous systems capable of extracting information from environment using knowledge about world and intelligence of their duties and proper governing capabilities. Intelligent robots should be autonomous to move safely in a meaningful and purposive manner, i.e. to accept high-level descriptions of tasks (specifying what the user wants to be done, rather than how to do it) and should execute them without further human intervention. Also, intelligent robots are autonomous robots with versatile intelligent capabilities, which can perform various anthropomorphic tasks in a familiar or unfamiliar (structured or unstructured) working environment. They have to be intelligent to determine all the possible actions in an unpredictable dynamic environment using information from various sensors. Human operator can transfer knowledge, experience and skill in advance to the robot to make it capable of solving complex tasks. However, in the case when the robot performs in a unknown environment, the knowledge may not be sufficient. Hence, the robot has to adapt to the environment to be able to acquire new knowledge through the process of learning. The learning properties of intelligent control algorithms in robotics are very important as they ensure the achievement of high-quality robot performance. Our knowledge of robotic systems is in many cases incomplete, because it is not possible to describe the system’s behaviour in a rigorous mathematical way. Hence, use is made of the learning of active compensation for uncertainties, which results in continuous improvement of robotic performance. It is well known that conventional adaptive and non-adaptive robot control algorithms 519
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Dynamics and Robust Control of Robot-Environment Interaction
comprise the problem of robot control during execution of particular robot trajectories without considering repetitive motion. Hence, in terms of learning, almost all manipulation robots have no memory. Thus, previously acquired experience about the dynamic robot model and control algorithms is not applied in the robot control synthesis. It is expected that the use of a training process by repeating the control task and recording the results accumulated in the entire process will steadily improve performance. The state variables-dependency of robot dynamics may also be solved by learning and storing solution, while timedependency of robot parameters requires an on-line learning approach. If the learning control algorithm once learns a particular movement, it will be able to control quite a different and faster movement using the generalization properties of the learning algorithm. It can be assumed that the characteristics of the manipulation robot environment are unknown and changing significantly in dependence of the given task. In addition to environmental uncertainties, various system uncertainties, being the result of imprecise position of workpiece, varying stiffness of the environment, robot tool and robot itself, etc., have an essential influence on the system behavior. For example, in the case of using fixed position-force gains for conventional compliance control tasks, these controllers perform satisfactorily when the environment parameters such as stiffness are known. However, the same controllers typically exhibit sluggish response in the case of contact with a softer environment, and become unstable in the case of a stiffer environment. Thus, one of the most delicate problems in compliant motion control of the robots interacting with dynamic environment is the stability of both the desired motion and interaction forces. A multitude of various conventional control approaches such as hybrid control, stiffness control, impedance control, damping control, etc. [1], point to the stability in control tasks as a problem which has not been satisfactorily solved yet, either from the theoretical and practical standpoint. It is well known that without knowing the environment model with sufficient accuracy it is not possible to determine, for instance, the nominal (desired) contact force. For example, in robot contact operations, such as grinding, deburring and polishing, it is essential to control the tangential velocity of the tool along the workpiece and the force normal to the work surface. It is a very difficult task to achieve an acceptable system performance because of a high level of unpredictable
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interactions between the robot and the environment. Also, various system uncertainties must be taken into account when considering the system behavior. For example, in the procedure of controller design, we have to cope with structured uncertainties (inaccuracies of the robot mechanism parameters, imprecise position of the workpiece, varying degrees of stiffness of the environment, robot tool and robot itself, etc.), unstructured uncertainties (unmodelled high-frequency dynamics as structural resonant modes, actuator dynamics, sampling effects) and measurement noise. Besides, the time-varying nature of the robot parameters and variability of the robot tasks can cause a high level of interaction force or the loss of the contact. Besides, the insufficiently accurate environment dynamics model can significantly influence the contact task execution. It is evident that in order to overcome these problems, the controller must be capable of adapting its parameters, and possibly its structure, to the changes in the environment parameters and environment model structure. Also, in the case of an unknown environment, it is very difficult to determine the maximum boundary of the position-force feedback gains. In this case, the conventional non-adaptive algorithms are not robust enough, because they can compensate only for a small part of the above uncertainties. Hence, a more appropriate approach would be the adaptive control techniques [2]. The conventional adaptive control techniques in robotics can tolerate wider ranges of uncertainties, but in the presence of sensor data overload, heuristic information, limits on real-time applicability and very wide interval of system uncertainties, the application of adaptive control cannot ensure a high-quality performance. As a result of the mentioned facts, efficient compliance control algorithms must include new intelligent features, which are necessary for active compensation of system uncertainties and for determination of optimal control parameters. In the case when the robot performs in an unknown environment, the previously acquired knowledge may not be sufficient. Hence, the robot has to adapt to the environment and be capable of acquiring new knowledge through the process of learning. There are several intelligent paradigms that are capable of solving intelligent control problems for robotic contact tasks. Beside symbolic knowledge-based systems (AI - expert systems), connectionist theory (NN - neural networks), fuzzy logic (FL), and theory of evolutionary computation (GA - genetic algorithms), are of great importance in the development of intelligent robot
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control algorithms. Also, great importance in the development of efficient algorithms has the hybrid techniques based on integration of particular techniques such as neuro-fuzzy networks, neuro-genetic algorithms and fuzzygenetic algorithms. Intelligent control systems can benefit from the advances in artificial neural networks as a tool for on-line learning optimization, and optimal policy making. The connectionist systems (neural networks) represent massively parallel distributed networks with the ability to serve in advanced robot control loops as learning and compensation elements using nonlinear mapping, learning capability, parallel processing ability, self-organizing ability and generalization ability. The fuzzy control systems based on mathematical formulation of fuzzy logic have a characteristic to represent human knowledge or experience as a set of fuzzy rules. Fuzzy robot controllers use human know-how or heuristic rules in the form of linguistic if-then rules, while fuzzy inference engine computes the efficient control action for a given purpose. The theory of evolutionary computation with genetic algorithms represents an approach to global optimization search, which is based on the mechanics of natural selection and natural genetics. It combines survival of the fittest among string structures with a structured yet randomized information exchange to form a search algorithm with expected ever-improving performance. Each of the proposed paradigms has its own merits and drawbacks. To overcome the drawbacks, certain integration and synthesis of hybrid techniques are important for efficient application in robotic contact tasks. Symbiotic intelligence incorporates a new type of robotic system having many degrees of freedom (DOFs) and multi-modal sensory inputs. The underlying idea is that the richness of inputs to and outputs from the system, along with co-evolving complexity of the environment mixed with various intelligent control paradigms, is the key to the emergence of intelligence. For example, neuro-fuzzy networks represent a combined tool by which a human operator can import his knowledge by means of membership functions. On the other hand, membership functions are modified through the learning process in the same way as fine tuning by neural networks. After learning, the human operator can understand the acquired rules in the network. Neuro-fuzzy networks are faster than conventional neural networks in terms of convergence of learning. On the other hand, fuzzy logic and
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neural networks can serve as evaluation functions for genetic algorithms. At the same time, genetic algorithms can be structure optimizers for fuzzy and neural algorithms. Computational intelligence techniques map well onto nonlinear problems and are better at handling uncertainties. 6.2 The Role of Learning in Intelligent Control Algorithms for Compliant Tasks Learning is the gathering of knowledge as well as of mental and physical abilities. Learning can be seen as a systematic changing of behaviour due to attained and elaborated information by examining changes in environment. To enable autonomous robots to operate in unknown environments, they have to be able to adapt their behaviour their own – they have to learn. So learning and adaptation are important paradigms in the field of current robotics research. Especially when a system cannot be completely modelled, data-driven methods enhance model-based approaches. Robot learning refers to the process of acquiring a sensory-motor control strategy for a particular movement task through a training process by trial and error. The goal of learning control can be defined as the need to acquire a task dependent control algorithm that maps the state variables of the system and its environment in an appropriate control signal (action). The linear or nonlinear control variable depends also on the system parameters that need to be adjusted by the learning system. The aim of robot learning control is to find a function that is adequate for a given desired behaviour and robotic system. Generally, robot learning can be considered through three main sub-branches: supervised, unsupervised and reinforcement learning. Supervised learning is needed to form nonlinear coordinate transformations and internal models of the robot environment. For autonomous movement systems, supervised learning has to proceed incrementally and in real time. It is possible to learn the control algorithm directly, where desired behaviour needs to be expressed as an optimization criterion that must be optimized over a certain temporal horizon, resulting in an appropriate cost function. In the cost function, a discount factor is used, that reduces the influence of the criterion in the far future. Such optimization problems have been solved in the framework of dynamic programming and its recent derivative, reinforcement learning. Reinforcement learning is used to allow the robot system to learn from scalar evaluation
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(punishment or reward) only, instead of precise teacher’s information (supervised learning). For reinforcement learning, the optimization criterion corresponds to the ‘immediate reward’, while the cost function is called the ‘long-term reward’. The control algorithm is acquired with reinforcement learning by first learning the optimal cost function for every state variable, usually by a technique called temporal difference learning, and then deducing the control algorithm as the control variable in every state variable that leads to the best future performance. Many variations of reinforcement learning exist, including methods that avoid estimating the optimization cost function. However, the main disadvantages of reinforcement learning are a large amount of exploration of all actions and states for proper convergence of learning and the lack of generalization among continuous variables. A possible way to reduce the computational complexity of the learning of a control algorithm comes from modularizing the control algorithm. Instead of learning the entire control algorithm in one big representation, one could try to learn sub-algorithms that have reduced the complexity and, subsequently, build the complete algorithm out of such sub-algorithms. This approach is also appealing from the viewpoint of learning multiple tasks: some of the subalgorithms may be re-used in another task such that learning new tasks should be strongly facilitated. Robot learning with such modular control systems, however, is still in its preliminary phase. A topic in robot learning that has recently received increasing attention is that of imitation learning. The idea of imitation learning is intuitively simple: a student watches the performance of a teacher, and, subsequently, uses the demonstrated movement as a seed to start his/her own movement. Imitation involves the interaction of perception, memory, and motor control, subsystems that typically utilize very different representations, and must interact to produce and learn novel behaviour patterns. The ability to learn by imitation has a profound impact on how quickly new skills can be acquired. From the viewpoint of learning theory, imitation can be conceived as a method to bias learning towards a particular solution, i.e., that of the teacher. However, not every representation for motor learning is equally suited to be biased by imitation. Imitation learning thus imposes interesting constraints on the structure of a learning system for motor learning. Imitation learning forces us to deal with perceptual and action uncertainty, non-stationarity, and real-time constraints. As
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a result, imitation approaches to learning, inspired by learning in biology, strive for efficient solutions that can cope with the challenges of real-world domains. Whether direct or indirect control is employed in a motor learning problem, the core of the learning system usually requires methods of function approximation in the neural network and statistical learning literature. Function approximation is concerned with approximating a nonlinear function (model of the robotic system with its environment) from noisy data. Many different methods of function approximation exist in the literature. These methods can be classified into two categories, local and global algorithms. The power of learning in neural networks comes from the nonlinear activation functions that are employed in the hidden units of the neural network. Global algorithms use nonlinear activation functions (sigmoid function) that are not limited to a finite domain in the input space of the function. In contrast, local algorithms make use of nonlinear activation functions that differ from zero only in a restricted input domain (Gaussian function). Despite both local and global algorithms being theoretically capable of approximating arbitrarily complex nonlinear functions, the learning speed, convergence and applicability to high-dimensional learning problems differ significantly. Global learning algorithms can work quite well for problems with many input dimensions, since their non-local activation function can span even huge spaces quite efficiently. However, global algorithms usually require very careful training procedures such that the hidden units learn how to stretch appropriately into all directions. Along with the problem how to select the right number of hidden units, it becomes quite complicated to train global algorithms for high dimensional robot learning problems. Local learning algorithms approximate the complex regression surface with the help of small local patches, for instance locally constant or locally linear functions [3]. The problem of local algorithms is the exponential explosion of the number of patches that are needed in high dimensional input spaces. The only way to avoid this problem is to make the patches larger, but then the quality of function approximation becomes unacceptably inaccurate. There is theoretically no way out of the curse of dimensionality - but empirically, it turns out not to be a problem. Local learning can exploit this property by using the techniques of local dimensionality reduction and can thus learn efficiently, even in very high dimensional spaces.
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Applying function approximation to the problems of robot learning requires a few more considerations. The easiest applications are those of straightforward supervised learning, i.e., where a teacher signal is directly available for every training point (state variables, output variables and control variables can be taken directly from sensors). Learning becomes more challenging when instead of the teacher signal only an error signal is provided, and the error signal is just an approximation. Thus, learning proceeds with “moving targets”, which is called a nonstationary learning problem. For such learning tasks, neural networks need to have an appropriate amount of plasticity in order to keep on changing until the targets become correct. On the other hand, it is also important that the network converges at some point and averages out the noise in the data, i.e., the network does not have too much plasticity. Finding appropriate neural networks that have the right amount of plasticity-stability tradeoff is a non-trivial problem, and so far, there are many heuristic solutions. Nonstationary learning problems are quite common in robotic learning. Learning the cost function in reinforcement learning (temporal difference algorithm) can only provide approximate errors. Other examples include feedback error and learning with distal teachers. Feedback error learning creates an approximate motor command error by using the output of a linear feedback controller as the error signal. Learning with distal teachers essentially accomplishes the same goal, except that it employs a model taught in advance to map an error in sensory space to an approximate motor error. In contemporary technological systems, robotics and automation technology have an important role in a variety of manufacturing tasks. These manufacturing operations can be categorized in two classes, related to the nature of the interaction between the robot and its environment. The first one is concerned with non-contact, i.e. unconstrained motion in a free workspace. In non-contact tasks the robot’s own dynamics has a crucial influence upon its performance. A limited number of simple and most frequently performed robotic tasks in practice, such as pick-and-place, spray painting, gluing or welding, belong to this group. In contrast to these tasks, many complex advanced robotic applications such as assembly and machining operations (deburring, grinding, polishing, etc.) require the manipulator be mechanically coupled to the other objects. These tasks are denoted as essentially contact tasks, because they include the phases where the robot end-effector must come into contact with objects in its
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environment, produce certain forces upon them, and/or move along their surfaces. Inherently, each manipulation task involves contact with the object being manipulated. The terms constrained or compliant-motion are usually applied to the contact tasks. The objective of using adaptive and learning capabilities for the above mentioned operations is to simplify the implementation process, to improve the system’s reliability and thus achieve a tremendous practical impact. Another important characteristic of learning techniques may be the enhanced capability to design robust hierarchical robotic systems. Robotic learning is the ability of a robot to adjust to its dynamic environment and changing task conditions. It can increase flexibility by enabling the robot to deal with different and unexpected situations. On the strategic and tactical (learning) hierarchical control levels, there are many opportunities for the application of learning systems to task-planning and task-reasoning problems, particularly those that confront the issue of uncertainty in the task environment. Also, a clear opportunity for the impact of learning systems is in the use of sensing and inspection technology for industrial applications. It is important to notice that for these control levels, specific methods from machine learning, inductive learning, case-based learning, explanation-based learning, evolutionary learning, reinforcement learning are the dominant learning tools for problems on these levels. Robots will have to be autonomous in unmodeled dynamic environments while behaving in ways useful to humans. The question is therefore how can robots acquire these behaviours? Some behaviours can be explicitly programmed, but this requires an explicit description of the tasks and a model of the environment. Some behaviours can be learned using methods such as reinforcement learning, or genetic algorithms. This again requires, the need to explicitly define the behaviours by the intermediary of a reward or fitness function and the use a trial-and error scheme that is impossible to achieve in most environments. From the human user point of view, a good way to define a behavior is to interact directly with the robot in the destination environment. A set of methods that could be grouped under the name of Empirical Learning are gaining interest in the literature. These methods are Learning by demonstrations or from examples, or more classically Supervised Learning.
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Our main concern will be the integration of learning techniques for the control of manufacturing operations on the executive (skill and adaptation) hierarchical level. The influence of couplings between the subsystems and interaction with the environment is substantial, so that we should include a “dynamic” control \cite, which uses the dynamic model of the robot mechanism and the model of the robot environment in the process of control synthesis. However, a common problem, especially in manufacturing operations, is how to describe correctly the robot-environment dynamics and how to synthesize control laws that simultaneously stabilize both the desired position and interactive force. For example, in robot contact operations, such as grinding, deburring and polishing, it is essential to control the tangential velocity of the tool along the workpiece and the force normal to the work surface. It is a very difficult task to achieve an acceptable system performance because of a high level of unpredictable interactions between the robot and the environment. Also, various system uncertainties must be taken into account when considering system behaviour. For example, in the procedure of controller design, we have to cope with structured uncertainties (inaccuracies of the robot mechanism parameters, imprecise position of the workpiece, varying degrees of stiffness of the environment, robot tool and robot itself, etc.), unstructured uncertainties (unmodelled high-frequency dynamics as structural resonant modes, actuator dynamics and sampling effects) and measurement noise. Moreover, the timevarying nature of the robot parameters and variability of the robot tasks can cause a high level of interaction force or the loss of contact. In this case, the conventional non-adaptive algorithms are not robust enough, because they can compensate only for a small part of the above-mentioned uncertainties. Hence, a more suitable approach would be the one using adaptive control techniques. The adaptive control technique in robotics was applied as a parameter adaptation technique with the possibility of adaptation in feedforward or feedback loops. All such methods usually comprise on-line schemes with recursive least-squares (RLS) optimization criterion and short-term learning in which past experience is completely forgotten. To summarize, the conventional adaptive control techniques in robotics can tolerate wider ranges of uncertainties, but in the presence of sensor data overload, heuristic information, limits on real-time applicability and a very wide interval of system uncertainties, the application of adaptive control cannot ensure a high-quality performance.
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Therefore, to achieve best performance of the robotic system, a solution of the robot control problem would probably require a combination of conventional approaches with new learning techniques. The application of learning techniques in the field of robot control on the lowest hierarchical level is very important because these techniques can significantly enhance the robotic performance with a priori low level of information about the model of the manipulation robot and the environment. Another important characteristic of learning control in contact tasks is its repetitive nature, which is very important for the process of learning by trial-and-error. Also, the inclusion of the learning concept in robotic control algorithms ensures some specific features as generalizations from multiple examples and the reuse of past experience. There are many possibilities in the development of more robust robot controllers by utilizing learning systems to identify more accurately the robot kinematics and dynamics, to more efficiently adapt dynamic control parameters to particular tasks, and to more effectively integrate sensory information into the control process. Another possibility for such an application is the field of kinematic calibration of robot arms, utilizing sensing systems to measure positions of the arm end-effector, as the learning system can identify a complex nonlinear model of the robot arm kinematics, which could be used to improve the positioning accuracy of the robot arm itself. The learning systems may also improve the capabilities to execute fine motion operations, such as force control and grasping. Robotic systems offer a promising domain for experimental exploration of learning systems in manufacturing operations, since the practical application of complex robotic systems may require adaptive and learning behaviour in order to achieve the desired functionality. The problem of control of manipulation robots is considered through a single execution of some technological operation. However, in most cases in industry, robotic operations are inherently of a repetitive nature, where the influence of the working environment is time-varying while the parameters of the robotic system vary in unpredictable ways. In this case, poor performance of the whole system can arise. This is a consequence of the fact that the robot controllers did not use experience acquired through previous repetitive trials, i.e., the robot controllers were based on “short-term memory”. The compensation of unmodelled dynamics was not achieved because of the lack of improvement mechanisms of control laws based on repetitive control trials. Hence, important characteristics of new learning control algorithms must be the use of skill
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refinement, i.e., the ability to gradually improve performance on the same or similar robotic tasks by repeated practice over time. In this way it is possible to achieve autonomous acquisition of the model of the robotic system and active reduction of the system’s uncertainties based on “long-term memory”. Another important characteristic of new learning laws should be the ability to execute old or new robotic tasks with higher quality (with greater speed, skill and accuracy) based on the generalization of the acquired knowledge. The robot controllers with the mentioned two properties are called repetitive controllers or training controllers and they involve two essential phases of work. The first phase is the training phase based on repetitive execution of the technological operation with a constant process of knowledge acquisition. In the second phase, called phase of knowledge association, the learning process is stopped and, using efficient association from memory, efficient control execution is performed. In this case, the learning control problem is considered as a pattern recognition problem with the input-output mapping based on associative memory data. For robot learning control on the executive hierarchical level, we can identify four main paradigms: 1. Iterative - analytical methods. These methods are based on successive attempts at following the same trajectory with betterment properties. Typically, control input values for each time instant in the trajectory are adjusted iteratively on the basis of the observed trajectory errors at similar times during the previous attempts. These iterative learning algorithms may be very precise and exhibit rapid convergence. On the other hand, a drawback of such control techniques is that they are applicable only to operations that are repetitive. These algorithms have no capability of generalization on quite different movements. 2. Tabular methods. These methods are related to the use of the local, memorybased, nonparametric function approximators , associative content - addressable memories, statistical locally weighted regression or kd-trees. In the case of addressable memory, the robot model is taught by storing experience about command signals and current state coordinates in the memory. Each time a particular set of robot positions, velocities and accelerations is requested, the entire memory has to be searched for the closest experience. In this approach, the problems are extended search time due to the great amount of stored experience, ways to measure similarity, and methods of efficient generalization.
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3. Reactive learning methods. These are numeric, inductive and continuous methods based on the psychological concept that is necessary to apply a reward immediately after the occurrence of a response in order to increase its probability of reoccurring, while providing punishment if the response decreases the probability. These methods include a component critic that is capable of evaluating the response and sending the necessary reinforcement signal to the robotic control. The problems with this type of learning are the credit assignment and finding optimal decision policy. A robotic agent may have many possible actions it can take in response to a stimulus, and the policy determines which of the available actions the robot should undertake. Two types of reinforcement learning algorithms predominate: the first Adaptive, heuristic critic learning (the process of learning the decision policy for action is separated from learning the utility function the critic uses for state evaluation) and second, Q-learning (a single utility Q-function is learned to evaluate both actions and states). 4. Connectionist methods. These represent the neural network approach based on distributed processing, where learning is the result of alterations in synaptic weights. Neural networks show great potential for learning the robot structure model and the model of the robotic system together with a great ability for knowledge association and knowledge generalization. They have the capability of being a general approximation tool for complex nonlinear systems. The connectionist approach does not require explicit programming because of general input-output mapping based on fast parallel architecture and sophisticated learning rules. In the context of robot control on the lowest hierarchical level, the primary aim of neural networks is the implementation of complex input-output kinematic and dynamic relations, i.e., the learning of inverse kinematic and dynamic robot models as parts of robot control algorithms. The important feature of neural networks is the ability to approximate the complete model of the control system, which is important for the compensation of a wide range of the system’s uncertainties. By using the properties of association and generalization, neural networks utilize the knowledge acquired on one set of test trajectories to efficiently control a quite different set of working trajectories. Another important role of neural networks in robotics is connected to the robot sensors, the process of sensor fusion and robot perception in order to present important information from the robot, needed for control decisions and algorithms.
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The issue of using neural networks in robotics represents a field of intensive research aimed at overcoming some difficulties connected with practical applications. These difficulties are related to the synthesis of practical topologies for complex mappings, forming of fast learning rules, optimal definition of network structures, accuracy of approximation, choice of “good” supervisor for learning, etc. 6.3 A Survey of Intelligent Control Techniques for Robotic Contact Tasks Along with recent extensive research in the area of intelligent robot control for non-contact tasks, various intelligent algorithms for compliance tasks have been proposed using different approaches such as connectionist, fuzzy, genetic and hybrid intelligent methods. Two essentially different approaches can be distinguished: one, whose aim is the transfer of human manipulation skills to robot controllers and the other, in which manipulation robot is examined as an independent dynamic system in which learning is achieved through repetition of the working task. The principle of transferring human manipulation skill is essentially connected with learning by imitation approach. Instead of rethinking and programming every nifty bit to solve a problem, it is try to reuse knowledge from nature – especially from us, the humans. Humans are still better than every robot, as soon as complexity rises over simple pick-and-place tasks. With learning by imitation, the robot is taught with trajectories relevant to solve a specific task, and the robot uses this information to solve similar tasks. However, many problems arise: How could the robot distinguish between relevant and irrelevant actions? How to segment the demonstrated action in reusable action-primitives? How to make use of the gathered information? Adaptation to new tasks is the goal. The principle of transferring human manipulation skill has been developed in the papers of Asada and co-workers [4-8]. The approach is based on acquisition of manipulation skills and strategies from human experts and subsequent transfer of these skills to robot controllers. It is essentially a playback approach, where the robot tries to accomplish the working task in the same way as an experienced worker. Various methods and techniques have been evaluated for acquisition and transfer of human skills to robot controllers.
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The transfer of human skill to the robot controllers, described in [7] was accomplished through specialized knowledge that was integrated in weighting factors of the neural network (Fig. 6.1). Teaching of neural network is based on an off-line procedure using standard back-propagation algorithm. The proposed neural controller was analyzed for robot deburring operation, whereby the input to the three-layer perceptron represents the characteristics of deburring process (characteristics of scraping and tool), while the output of the neural network is the cutting force in normal directions and damping and stiffness system gain. However, no experimental analysis was given, nor direct connections with the real-time control action.
Fig. 6.1 Transfer of human skills to robot controllers by neural network approach
This approach is very interesting and important, although there are some critical issues related to explicit mathematical description of human manipulation skill because of the presence of subconscious knowledge, inconsistent, contradictory, and insufficient data. These data may cause system instability and wrong behavior of the robotic system. As is known, the dynamics of the human arm and robot arm are essentially different, and therefore it is not possible to apply in the same way human skill onto the robot controllers. The sensor system for data acquisition of human skill may be insufficient for extracting the complete set of information necessary for transfer to robot controllers. Also, this method is inherently an off-line learning method, whereas for robot contact tasks on-line learning is a very important process because of the
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high-level robot interaction with the environment and because of unpredictable situations that were not captured in the skill acquisition process. Using similar approach, Skubic and Volz [9-12] have focused on the problems of learning low-level force-based assembly skills from human demonstrations. To avoid position dependencies, force-based discrete states are used to describe qualitatively how contact is being made with the environment. Sensorimotor skills are modeled using a hybrid control method, which provides a mechanism for combining continuous low-level force control with higher-level discrete event control. The human teacher demonstrates each single-ended contact formation while force data from force sensors are collected, and these data are used to train a state classifier (fuzzy or neural network classifier). The human teacher then demonstrates a skill, and the classifier is used to extract the sequence of single-ended contact formation and transition velocities, which comprise the rest of the skill. A second group of learning methods, based on autonomous on-line learning procedures with repetition of the working task, is also evaluated through several algorithms [13-18]. The main distinction between these algorithms is in the goal of learning, which is in the first case direct adjustment of control signals or parameters, while in the second case the goal of learning is the building of internal model of robotic system with compensation of the system uncertainties. The algorithms with on-line modification of control signal are basically related to automated contact-operation (more precisely, the assembly process). The control goal in the assembly process is to accomplish the whole set of corrective movements using learning rules in order to achieve valid realization of the working task. Asada [19-20], considered the problem of nonlinear “compliance” in the process of peg-in-hole insertion. The compliance task is defined as a nonlinear mapping of the measured force and moments on the robot end-effector into corrective velocity movements defined through linear and angular velocity of the robot end-effector. The nonlinear mapping was represented by multilayer perceptron with learning rules defined on the basis of back-propagation method. This is a very interesting approach, but the learning analysis was realized off-line, using the recorded input/output patterns without experimental verification. A very interesting approach belonging to this group is the one by Gullapali and coworkers [18]. The authors use reactive admittance control for
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compensating system’s uncertainties by on-line control modification and sensor information. Using a similar approach, the realization of active compliance in [19] is based on nonlinear mapping of the admittance from sensors of position and force in commanding velocity movement. The robotic controller learns this mapping through repetitive trial of peg-in-hole insertion. The learning rules are based on method of associative reinforcement learning [18], [21]. This method can be characterized as random search and reinforcement learning. Mostly we learn by interacting with our environment and experiencing success or failure within our intended task – which in turn either reinforces the successful behavior or drives us to try something else. This simple principle (also called learning-by-doing or trial-and-error) is also useful for computers and especially for robotics. Often we don’t exactly know how to solve a problem, so a robot that solves it by just trying out various things would be a great leap. The robot learns from a reward (“good” or “bad”), which is often only available after a sequence of actions has been performed. The reward is often simple to formulate – e.g. in a grasping task a simple rule would be: Object stable grasped is good, everything else is bad. Whereas it is not given to the robot what actions to choose to achieve the goal – it has to learn it on its own. To reduce the complexity, learning-by-imitation might be incorporated, as well as the abstracting features of neural networks might prove helpful. In contrast to the supervised learning paradigm, the role of the teacher in reinforcement learning is more evaluative than instructional. The teacher provides input to the learning system with an evaluation of the system performance of the robot task according to a certain criterion. Based on both the input to the learning system and the action it generated for that input, the environment computes and returns an evaluation “reinforcement”. Over time, the learning system has to learn to respond each with the action that has highest expected evaluation. In order to iteratively improve the evaluation obtained for the action associated with each input, the learning system has to determine how the modifying action affects the ensuing evaluation, for example, by estimating the gradient of the evaluation with respect to its actions. The goal of direct reinforcement learning algorithm is to compensate for the system uncertainties and sensor noise using learning by position-force feedback. Structure of the neural network used is given in Fig. 6.2. The neural network has 6 inputs from the robot sensor system (the peg position X ,Y , θ from the sensed joint position and force, and the moments Fx , F y , M z from the robot end-effector). In addition to two hidden layers with 15 neurons,
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the output layer contains 3 stochastic real-valued (SRV) neural units [18] for generation of the linear and angular velocities vx, vy and ωz. These units generate real-valued output stochastically and use the ensuing evaluation to adjust the output so as to maximize the expected evaluation over time [18]. The units do this by estimating the local gradient of the evaluation with respect to its output and using this estimate to perform gradient ascent. Using the SRV units in the output layer enabled the network to conduct a search in the space of control actions in order to discover appropriate compliant behavior. The authors have experimentally verified this approach in robot control with satisfactory results even in the presence of a high degree of noise and uncertainties. From the domain where a primary goal is the learning of internal robot models with compensation of the system uncertainties, we can mention some connectionist algorithms for learning control of contact tasks [13-16], [22-28].
Fig. 6.2 Neural network used for peg-in-hole insertion
In paper [29], a new neural network controllers for the constrained robot manipulators in task space is presented. The neural network will be used for adaptive compensation of the structured and unstructured uncertainties. The controller consisted of a model-based term and a neural network on-line adaptive compensation term. It is shown that the neural network adaptive compensation is
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universally able to cope with totally different classes of system uncertainties. Novel adaptive learning algorithms for tuning the weights of neural network are proposed. A suitable error filtered signal for training the neural network can be easily obtained from the controller design without using any model knowledge of the robot manipulator itself. The closed-loop system with neural network adaptation on line is guaranteed to be stable in the Lyapunov sense. Detailed simulation results are given to show the effectiveness of the proposed controller. Fukuda and his co-workers [13], proposed a complex control scheme, which uses centralized connectionist structure in the frame of explicit hybrid control laws [14,30]. Tao and Luh [14,30] used a technique of model reference control as basic control algorithm and neural network as robust controller in order to compensate for the uncertainties of the dynamic model of multiple robots with redundancy. Kiguchi and Fukuda [23] have proposed an adaptive neural controller for hybrid position-force control of robot in contact with an unknown environment. They used a new type of artificial neurons, which possess visco-elastic properties. The use of the proposed visco-elastic neurons enables the damping of unexpected overshooting and oscillation caused by unknown unmodeled robot or environment dynamics. In the papers [24,26,28] the authors have presented a position-based neural force control algorithm for solving a wide range of surface tracking tasks for robot manipulators involving the defined contact with moving rigid objects. This approach comprises a hybrid position-force controller where neural control algorithm performs force control by modifying the desired joint angles in the force direction. These are fed into a computed torque controller, where the inverse dynamics of manipulator is represented by neural networks. In paper [31] a new approach of controller to treat the problem of the force control of active compliance robots witch are in interaction with their environment. In the goal to get a behavior of the system that either satisfactory in different situations, non linear adaptive controller was observed. The connectionist approach that is proposed rests on a methodology of control while considering the treatment of the task according to aspects off line and on line. The external force control structure that uses the reinforcement paradigm for the on-line adaptation of the controller’s parameters to variations of parameters of the task was proposed. To put in evidence the interest of the proposed approach, the proposed structures to perform a peg in hole insertion of complex shapes with different tolerances and speeds of insertion was
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considered. The results prove the efficiency of the approach and show that the on-line training improves the controller’s performances by off line learning appreciably. In order to satisfy all these constrains a control structure based on an external force control scheme is proposed (Fig. 6.3). Indeed, this structure of control consists in a hierarchy of the force control loop with regard to the position control loop. The law of force control that intervenes in the diagram works out, from the force error (∆F) between the desired contact force torsor Fd and the measured one Fcm , an increment of position ∆X that modifies the task vector Xd according to the following relationship: ∆X = Cf ( ∆F )
(6.1)
where C f represents the forces control law. At the time of a assembly task the interaction robot-environment constitute a non-linear process coupled and variable in the time, a satisfactory behavior during this final phase of the assembly requires a regulating of C f control parameters. The ability of neural networks to approximate a large class of non-linear functions with sufficient accuracy incited us to use neural network to compute the function C f . So at the time of a constrained displacement, a neural controller is used to achieve a compliant control. The implementation of this control takes place according to the following step: First the FFNNC is initialized off-line from the identification of a classical force control. Then an on line paradigm of adaptation of neural controller parameters is implemented in order to refine the regulating of neural networks parameters according to the real behavior of the system. This structure of control uses on-line the learning paradigm called associative reinforcement learning. For the implementation of the different neural networks, multilayers networks including each a hidden layer of 20 neurons were used. Due to the location of the force sensors the measured forces in free space are not equal to zero. So in order to estimate the contact forces from the measured ones in a compliant motion, an identification model was proposed (Fig. 6.4) based on a multilayer neural network due to its ability to model a non-linear behavior. In our approach, the task is realized with constant velocity. This FFNNM is trained off-line using the data obtained from the robot’s displacements in the free space.
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The learning process is performed off-line using Quasi-Newton method combined with a one dimensional search. This method consists in optimizing the step of training to increase the speed of convergence of the global criteria. While achieving a control of the system in the task space with a conventional controller of PID type, a basis of training is constructed (Fig. 6.5). This one is constituted of measured forces torsor Fcm , and wanted ones Fd and control vector U fs generated by the conventional controller. Values of mistakes to the previous sampling instants are taken in amount in the learning. Parameters of the controller PID have been adjusted experimentally for a speed of insertion of 5 mm/s of way to get a satisfactory behavior of the point of view of the minimization of contact forces. The period of sampling of the control is 5 ms. The input vector Ic of neural controller is constituted by force error vector e = Fd - Fcm .
Fig. 6.3 Adaptive controller using associative reinforcement paradigm
Where: • • • • • • • •
DGM: the direct geometric model of parallel robot. IGM: the inverse geometric model of parallel robot. DFM: direct force model. X d : the desired position trajectory. Fd : the desired forces trajectory. Fg : the estimate free motion force torsor. Fm : the measured forces torsor. Fcm : the estimate contact force torsor.
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Fig. 6.4 FFNNM learning a free motion forces
In this step the FFNNC is trained off-line using the data obtained from the robot’s displacements in the constrained space under classical external force control. The learning structure is presented Fig. 4.5. The Quasi-Newton method is used for the FFNNC learning.
Fig. 6.5 Initialisation of neural networks controller (FFNNC) scheme
To improve performances of neural controller with regard to the real behavior of the system, a simple retropropagation algorithm with minimization of a partial gradient to every instant of sampling is implemented. The objective of this algorithm is to adjust on-line the network parameters after every sampling of input and output. To-date relatively few attempts have been made at combining intelligent control methodologies with practical real-time robotic force control. In the case
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of fuzzy control, an explanation for this is that the derivation of fuzzy rules for non-trivial, real-world tasks is often not intuitively obvious. In addition, the approach often leads to fuzzy systems with huge numbers of highly coupled rules, which are difficult to interpret, and time-consuming, if not impossible, to tune manually. For this reason, fuzzy control is often employed where a ‘good enough’ solution is required, which does not necessarily have to be an optimum one, or meet a particularly tight specification. Where attempts have been made to employ fuzzy logic in explicit robot force controllers, the main goal has been to improve the performance when the robot is in contact with an environment whose parameters are either unknown or rapidly changing. Simulation studies on adaptable fuzzy force controllers have demonstrated good tracking performance and effectiveness despite wide variations in environment stiffness, and for specific contact situations, e.g. deburring. Improved performance using a hierarchical fuzzy force control strategy has also been demonstrated for various contact situations, such as pegin-hole insertion. However, most have employed one-off fuzzy solutions, designed for specific robots and applications. The paper [32] describes an ongoing research programme at the Universities of Sunderland and Newcastle to investigate the application of fuzzy and neurofuzzy techniques applied to robotic force control. Initial investigations have concentrated on the application of fuzzy logic to improve the performance of force controllers operating in environments where Ke is variable, and two unique methods for fuzzy force controller design are presented. These were initially developed using simulation, before being implemented on a robotic test facility. In the first approach, a structured fuzzy design method is outlined that uses experimental data, obtained from a system employing a conventional controller, as the basis for the construction of an initial fuzzy system that emulates it. Additional membership functions (MFs) and rules are then applied in order to minimize the effect of one or more varying parameters (in this case Ke) by attempting to maintain an ideal force error (fe), and force rate error (∆fe), profile. The second method is also based around a conventional controller, but employs a secondary fuzzy system that calculates factors by which the conventional controller gains are multiplied as the task progresses. An analysis of both controllers is undertaken and the performance of each system is evaluated on the experimental robot compared to the conventional solution in contact situations where there exists a wide range of Ke.
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A fuzzy inference system (FIS) can be considered as a rule-based expert system employing linguistic rules, and in control can facilitate a mathematical formulation of the uncertainty and imprecision associated with certain processes. This enables non-linear controllers to be devised that would be difficult to design using conventional methods. The two most common forms of the fuzzy inference process are usually referred to as the Mamdani method, and the Sugeno (or Takagi-Sugeno-Kang) method. To summarise, Mamdani-style systems employ output fuzzy sets, and generally require more complex defuzzification techniques. In contrast, firstorder Sugeno-style systems have rules of the form: If x is A and y is B then z=p*x + q*y + r (6.2) where A and B are fuzzy sets in the antecedent (the initial ‘if’ part of the fuzzy rule), x, y and z are inputs and the output, respectively, and p, q and r are constants. As already described, attempts have been made to combine fuzzy logic and explicit robot force control. However, no general and analytically based design strategy has evolved, for example to enhance the performance of conventional controllers using fuzzy techniques. To address this, two fuzzy control techniques have been presented at paper [32], described in the following text. A first method for designing Sugeno-style fuzzy controllers has been developed that effectively produces a PV controller with variable gains, capable of maintaining acceptable performance irrespective of Ke. A block diagram of the arrangement is shown in Fig. 6.6, and the design method can be summarized as follows. Firstly, a FIS is created to emulate a conventional PD controller, tuned for a high Ke environment. The FIS is assigned three inputs (force error, force rate error and velocity: fe, ∆fe, ∆x), and one output (controller output: u), where the input ranges can be obtained mathematically or measured from conventional system data. In order to create a linear system, initially only a single MF for each input and output is required. A linear MF is chosen for the output, of the form given in eq. (6.2). By assigning names normal to the input MFs, and u1 to the output MF, the following rule produces the desired linear control surface. Note that a consequence of employing only one rule is that no defuzzification algorithm is required: if (fe is normal) AND (∆fe is normal) AND (∆x is normal) then u is u1
(6.3)
Output u1 is defined by u1 = K1*fe + K2 * ∆fe + K3 *∆x + K4
(6.4)
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where K1 is a positive constant (equal to the forward gain Kp of a conventional PV controller) and K3 a negative constant (equal to the velocity feedback gain Kv). K2 and K4 in this case are set to zero.
Fig. 6.6 Fuzzy Control using SFAC
Since the single rule system emulates a conventional PD controller it suffers the same disadvantages when Ke is unknown or variable. However, having created the initial FIS, it is now possible to tune the controller using a combination of analytical and intuitive methods. With the system tuned for high Ke, during low Ke contact the maximum value of ∆fe is reduced. This reflects the over-damped response of the system at low Ke, in this case an undesirable effect that can be minimized by increasing the proportional gain component of the controller output given by eq. (6.4) if lower ∆fe is ‘detected’ by the FIS. This is achieved firstly by adding a second Gaussian input MF to the ∆fe input set, with a smaller standard deviation (or ‘width’) than σ fe, and named low. A second rule is then added to take into account this decrease in ∆fe relative to the ‘normal’ (desired) profile and the knowledge that during the dynamic response to a step input, ∆x is inversely proportional to Ke. In the case of a system with varying gain Kp, assuming it is initially at a lower value and increases if Ke is less than maximum (one of the fundamental aims of this system), then ∆x will also begin to increase. Therefore, the following rule is added:
if (∆fe is low) AND (∆x is high) then u is u2
(6.5)
where u2 has the same form as u1, eq. (6.3), but with a modified forward gain
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component, K1a, such that K1a>K1, and σ∆x high > σ∆x normal. During initial testing it was found that the system performed well over a wide range of Ke, although had a slight tendency to overshoot at some intermediate values. To eliminate this it was found necessary to modify σ∆fe low and slightly reduce the width. This had the effect of reducing the firing strength of the second rule (eq. (6.5)) in intermediate regions of Ke. An optimum value was found by trial and error. The Second method - Fuzzy Model Reference Adaptive Control (FMRAC) scheme is illustrated in Fig. 6.7, and is based upon a classical MRAC method to adjust the loop gain of a feedback control system. Note that Gd(s) is the ideal closed loop transfer function (i.e. the desired response) for a unit step force demand.
Fig. 6.7 Fuzzy Control using FMRAC
A typical MRAC method consists of a ‘parameter adjustment’ algorithm acting on the model error ε. The output of this algorithm is then integrated to give the loop gain adaptation parameter θ. The output of the parameter adjustment algorithm goes to zero as the estimate of the parameter tends to the desired value. The problem with the classical approach is finding a suitable algorithm, many of which have been proposed over the years, including ‘gradient algorithms’ that attempt to minimize some cost function J(ε).
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In the fuzzy approach, the non-linearity required of the parameter adjustment algorithm is again achieved via the rule base. In this case the fuzzy block has two inputs; the magnitude of the model error |ε|, similar to the traditional approach, and the magnitude of the adaptation parameter |θ|. The output is parameter factor, which is integrated to give θ. The rule base consists of only four rules, and is tuned to ensure factor varies between zero and an upper limit. The semirecursive use of θ as an input to the fuzzy block prevents large swings in factor, effectively enabling the rule base to ‘remember’ the most recent value of θ. The relative widths of the MFs and the input range for |θ| (i.e. ‘universe of discourse’ in fuzzy terminology) were selected for each axis initially using simulation, before being fine tuned on the actual system. Two constants were chosen as output MFs (parameter r, eq. (6.2)): Big=100 and Zero=0 and a rule base created as follows: If (|ε| is Poor) AND (|θ| is Big) THEN (factor is Big) If (|ε| is Good) AND (|θ| is Small) THEN (factor is Zero) If (|ε| is Poor) AND (|θ| is Small) THEN (factor is Big) (6.6) If (|ε| is Good) AND (|θ| is Big) THEN (factor is Zero) Two distinct methods for fuzzy logic-based force control have been compared using an experimental test robot. Both methods tested have been shown to improve system performance where a high degree of environmental uncertainty exists, without the need for a stiffness detection routine, and the relative advantages and disadvantages of each technique have been briefly discussed. The practical realization of robotic force control remains a problematic area of research. However, the potential of using fuzzy logic to overcome fundamental difficulties associated with applications where environmental uncertainty exists has been demonstrated, using techniques derived from conventional controller design theory. Thus, an important aspect of the work has involved bridging the gap between traditional automatic control theory and so-called soft computing, which aims to replace a highly analytical and mathematical approach with a behavioural and experimental one. The SFAC controller is particularly promising in this respect, since although the results would appear to indicate a slightly inferior performance compared to the FMRAC method, it should be remembered that the latter controller must be reset prior to each contact operation. The next stage in the validation of the methods will be to implement a similar controller on a 6-DOF industrial robot, and then perform a range of comparative tests with
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conventional solutions. This work is currently underway using a PUMA 762 industrial robot employing a six-axis, wrist-mounted F/T sensor to measure Cartesian forces and torques. In particular, they are being applied to gear deburring and extensive work is being carried out to validate their efficiency in this, and similar, industrial applications. This paper [33] presents an integrated fuzzy approach to recover the performance in impedance control, reducing the errors in position and force, considering uncertainties in the parameters of the manipulator model and contact surface model. This integrated strategy considers a fuzzy adaptive compensator in the outer control loop that adjusts the manipulator tip position to compensate for uncertainties present in the environment. In the inner loop, a fuzzy sliding mode-based impedance controller compensates for uncertainties in the manipulator model, based on an inverse dynamics control law. The system error defines the sliding surfaces of the fuzzy sliding controller as the difference between the desired and the actual impedances. In order to evaluate the force/position tracking performance and to validate the proposed control structure, simulations results are presented with a three-degree-of freedom (3DOF) PUMA robot. The proposed control structure can be defined as a position based explicit force control, such that in the inner loop a position based controller is feeded, in part, with position errors by a controller in the external loop (Fig. 6.8). In the external loop, the controller is of fuzzy adaptive type, which transforms the force error in a position adjustment. The main objective of the fuzzy adaptive controller is to compensate environment uncertainties, like stiffness and geometric location. As referred above, the fuzzy adaptive controller feeds in part the inner position controller, because it corrects only the manipulator tip position x1in the perpendicular direction to the contact surface, based on the force error ef. Moreover, the inner position controller will compensate for uncertainties in the manipulator dynamic model. The inner position controller is a fuzzy sliding mode-based impedance algorithm, with an inverse dynamics control law. As the dynamic model of the manipulator is not exactly known, an impedance error will appear. This impedance error defines a sliding surface, which, when the system is in sliding mode, after it has reached the surface, the state trajectory continues to be on it.
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Fig. 6.8 Block diagram of the overall control structure
The overall fuzzy adaptive control scheme (FAC) presented in this paper, based on the Fuzzy Model Reference Learning Controller is shown in Fig. 6.9.
Fig. 6.9 Fuzzy adaptive control scheme (FAC)
In this article, an integration of fuzzy adaptive and fuzzy robust force/position control to compensate for overall uncertainties, considering an impedance control formulation, is presented. The tip position adjusted by the fuzzy adaptive controller in the perpendicular direction to the surface, to compensate for uncertainties in the environment is feeded to an inner position controller. This controller is a fuzzy sliding mode-based impedance control that compensate for uncertainties in the manipulator dynamic model. The adjustment in position done
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by the fuzzy adaptive controller, based on the force error and the effort done by the fuzzy sliding mode-based impedance controller in the inner loop, recovers the impedance control performance and reduces the errors in position and force. This control structure also exhibits a good performance considering nonrigid materials with uncertainties.
Fig. 6.10 Fuzzy scheme of force control
The basic principle of application of fuzzy logic for force control and hybrid position-force control [34-40] in robotics is the same as in the case of position tracking control, except for the input level, where the space of input variables is expanded by including force tracking errors. Conventional force controllers give satisfactory results in the case when parameters of the robot environment are known and fixed. However, in most cases, this assumption is not fulfilled, so that the control structure and parameters of robot controllers have to adapt to the changing environment. One of techniques for solving these problems is the application of fuzzy logic [35,41-44]. In the process of control synthesis, there are many restrictive assumptions connected with the availability of force derivation and possibility to measure the velocity of the robot end-effector and these assumptions are effective only when the estimations of environment parameters are close to their real values. However, in practical application,
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information about force derivation contains the measured noise, and the accuracy of position and velocity of the end-effector is not sufficient for a satisfactory force control. Also, the range of parameters of the working environment can be very wide because, for example, the stiffness coefficient can vary in a large span, depending on whether the environment is soft or hard. The purpose of the fuzzy scheme of force control [44] (Fig. 6.10) is to achieve exact force regulation when the stiffness of the working environment is unknown and exhibits great variations, using no restrictive assumptions. The proposed adaptive fuzzy scheme for control of the force F has three main components: force reference model which describes the desired behavior of the control scheme according to the force component Fm , main force fuzzy controller which calculates the adaptation U in the position control loop, and the algorithm of fuzzy learning and adaptation (command δ ) which modifies the main fuzzy force controller based on the difference between the real and desired force. A common way for defining the force reference model is the synthesis of a second-order linear system in state space with specified damping factor and natural frequency of the system. The main fuzzy controller is a fuzzy PI controller with two input values (force error and integral force error) and one output value, which represents the adaptation of position control loop. Structure of the fuzzy controller is defined by simple fuzzy rules with the membership functions of 5 fuzzy sets. The task of the fuzzy mechanism of adaptation and learning is to learn parameters of the working environment and modify the fuzzy force controller to ensure the system tracks the output of force reference model. In this case, inputs to the fuzzy mechanism are force error and integral force error, while output is a modification of the output of the basic fuzzy force controller. This modification is represented by a change of the membership function of the output fuzzy set with respect to the center and width of the membership function. The efficiency of this approach is presented through simulation studies with different parameters of the working environment. Neuro-fuzzy networks can be efficiently applied for dynamic learning control of position-force in robot machining operations [45-49]. In this type of operation, the problem is to determine an efficient position-force controller, together with obtaining the desired force and desired tool feedrate. In practice, it is hard to obtain these values, especially for the deburring process, because the human expert realizes effectively the desired force and toll feedrate according to the environment characteristics and burr height. Also, it is very hard to model the
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expert knowledge, hence it is better to extract flexible linguistic rules based on his knowledge, skill and experience. The inclusion of adaptive and intelligent control properties is necessary too because of the complexity of modelling the robot, tool, and time-varying characteristics of the working environment. Because of these facts Kiguchi and Fukuda [48] proposed an intelligent controller (Fig. 6.11), which consists of two parts: intelligent planner and neurofuzzy controller of position and force. This intelligent method, without experimental work and previous learning process, enables the acquisition of expert knowledge for deburring process. The intelligent planner consists of a neuro-fuzzy planner and a fuzzy estimator of characteristics. The neuro-fuzzy planner, which is tuned in real time, determines the desired tool feedrate. The fuzzy estimator of unknown characteristics of the working environment generates the desired force and input coefficients for the adaptive neuro-fuzzy force controller. The intelligent planner transmits the generated signals to the neuro-fuzzy controller of position and force. According to the principle of hybrid position-force control, the two neuro-fuzzy networks exist based on TakagiSugeno architecture, one for position control and the other for force control. The inputs to the neuro-fuzzy position controller are position and velocity errors in external coordinates E and Eɺ , while the inputs to the neuro-fuzzy force controllers are the position errors in external coordinates E and robot moment in the force control direction M 0. The training process for these neuro-fuzzy networks is realized by supervised back-propagation method, using measurement of position and force. The fuzzy estimator of characteristics based on measuring position and force evaluates stiffness of the working environment as fuzzy variable and, using expert knowledge expressed through fuzzy rules, determines the desired force and desired tool feedrate. The second role of the fuzzy estimator of characteristics is to avoid great force overshoot based on modified coefficient of input variables for the fuzzy force controller. The neuro-fuzzy network is included in the neuro-fuzzy planner for determining the desired tool feedrate using information about the burr height given by a camera or laser sensor. A further extension of the proposed approach, to deal with the unknown or unexpected environment, is given in the paper [49]. A suitable neuro-fuzzy force controller is selected immediately from the initially prepared multiple neurofuzzy controllers for various kinds of environments, and then harmonized with a
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proper ratio using fuzzy reasoning according to the dynamic properties of the unknown environment. In this method, on-line classification of the unknown environment is carried out with an off-line trained neural network [50], and then fuzzy reasoning for controller selection and harmonization is performed on the basis of the classification information. Consequently, exactly suitable neurofuzzy controller is selected for an initially expected environment, and some suitable neuro-fuzzy controllers are selected and harmonized with a proper ratio for an unexpected environment.
Fig. 6.11 Intelligent planner for neuro-fuzzy control of robot machining operations
In paper [51] an adaptive neuro-fuzzy controller with an adaptive neuro-fuzzy friction compensator is proposed for hybrid position-force control of robot manipulators. Thanks to the adaptive neuro-fuzzy modelling, both for the controller and the friction compensator, the proposed method is independent of the robot dynamics as well as the conditions of the environment. The main advantage of the proposed control method is that the adaptation law is based on the Lyapunov stability theory, which guarantees the stability of the controller. Although the simulations are performed on a robot manipulator with three degrees of freedom with revolute joints, the proposed controller can be extended to robot manipulators with more degrees of freedom and different kind of joints. Moreover, the structure of the controller and the compensator is very simple, making it a very fast and appropriate method for different applications of robot manipulators. The simulation results show good performance of the proposed method as compared with other conventional control methods such as computed torque method.
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In the study [45], GA is applied in combination with neuro-fuzzy force controllers for robotic contact tasks. In this case, multiple genetic neuro-fuzzy force controllers are suitably combined with a proper rate in accordance with the unknown dynamics of an environment. In order to carry out the proposed force control method, several kinds of neuro-fuzzy controllers designed for different kinds of environment are prepared. The optimal combination rate of the prepared neuro-fuzzy force controllers according to the environment dynamics is defined on-line by a neural network that is off-line trained using genetic algorithms. Exactly, optimal weights of this Controller Combining Neural Network are defined by using GA to output proper weight values for each neuro-fuzzy force controller in accordance with dynamic property of the environment. In the subsections to follow, special attention will be paid to the new connectionist and hybrid connectionist methods in robotics as the basic research paradigm for robotic intelligent control. 6.4 The Synthesis of New Connectionist Learning Control Algorithms for Robotic Contact Tasks 6.4.1 The background of the new connectionist control synthesis The main intention of the control synthesis was to compensate for the system uncertainties by connectionist structures in a global fashion, without separately considering the problems of environment modeling, identification, and uncertainties. The previous research works in most cases, however, did not consider these uncertainties and nonlinearities of environment and, thus, the above approaches are limited to specified working conditions that satisfy only certain assumptions. Namely, when considering specific contact tasks, simplifications in the modeling of robot and environment dynamics are introduced in almost all control approaches, in order to obtain simpler control algorithms. Hence, the uncertain nonlinearities and other characteristics of environment models still remain a critical issue in robotic contact task research. As a result of the mentioned facts, efficient compliance control algorithms, and especially learning compliance control laws, must include new comprehensive learning features, which are necessary for active compensation of the
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environment uncertainties and for the determination of optimal control parameters. In this case, the algorithms that identify the type of environment models on-line, could significantly improve the performance of contact task control schemes. The main idea in the proposed new comprehensive intelligent control strategy for robotic contact task is the inclusion of connectionist structures with broader sense of application [50,52]. Beside the “black-box” and hybrid learning of robot and environment uncertainties, the new role of connectionist structures is related to sufficiently exact classification of dynamic environments in order to achieve highly reliable behavior of the whole robot-environment system. This classification is necessary in order to calculate control parameters of the learning control algorithms that achieve best performance of the robotic systems. Because of a new role of connectionist structures, the main feature of the proposed hybrid learning control algorithms will be the integration of two connectionist structures into some typical non-learning control laws (control laws based on the stabilization of robot motion and interaction force and impedance control). The first neural network (neural classifier) is capable to perform the classification of unknown characteristics of the environment needed for selecting the appropriate control parameters of basic non-learning compliance control algorithm. In this way, learning compliance control algorithm will significantly reduce the influence of robotic system uncertainties. It is important to notice that the neural classifier objective is to classify the model profile and parameters of the environment in an on-line manner. The classification capability of the neural classifier will be realized by efficient off-line training process. This off-line training process is defined by single execution of the same robotic contact tasks but with different dynamic environments and different features of these environments. The classification knowledge of the first neural network is the base for excellent generalization in the on-line procedure through the process of pattern association. As classification tool, wavelet network was chosen. The wavelet network classifier is chosen in order to enhance the pattern recognition properties in comparison with pure multilayer perceptron approach. The wavelet preprocessing includes the necessary feature extraction capability, together with the classification capability of multilayer perceptrons. On the other hand, the wavelet transformation is very convenient for representing nonstationary signals with brief, high-frequency components as are the signals from the robotic force sensors. In on-line operational mode, after the classification process, the basic
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learning control algorithm utilizes the information about the type of environment from the first neural network to determine control parameters and parameters of the environment model. The determination of these parameters can be achieved by the values corresponding to the appropriate type of particular environment that is defined in advance, or they can be obtained by the process of linear interpolation. The second proposed neural network (neural compensator) plays the general role of a robust learning controller needed to compensate for uncertainties of the dynamic model of the manipulation robot in contact with a dynamic environment. Hence, the main objective of this approach is to employ the coordination of the well-trained neural classifier (first network) with the neural compensator (second network), to significantly improve the robot’s performance in the contact tasks involving uncertain environment. 6.4.2 Model of robot interacting with dynamic environment – task setting In the case of contact robot tasks, the dynamics of the robot mechanism interacting with its dynamic environment has a crucial influence on the system performance. Model of the robot dynamics interacting with the environment is described by the vector differential eqs. in the form:
τ = H (q)qɺɺ + h(q, qɺ ) + J T (q) F
(6.7)
where τ ∈ R n is the vector of driving torques or forces; H (q ) : R n → R n×n is the inertia matrix of the system; h(q, qɺ ) : R n × R n → R n is the Jacobian vector which includes centrifugal, Coriolis and gravitational effects; J (q ) : R n → R n×m is the Jacobian matrix connecting velocities of the robot end-effector and the velocities of robot generalized coordinates; F ∈ R m is the vector of generalized forces or of generalized forces and moments from the environment acting on the end-effector; q ∈ R n is an n -dimensional vector of the robot generalized coordinates; n is the number of DOFs; m is the number of interacting force components. Here, it will be assumed that n = m (in general n ≥ m ). The dynamic model (6.7) can be transformed into an equivalent form in the operational space. This form of the model describes the motion of the endeffector in the Cartesian space:
Φ = Λ ( x) ɺxɺ + µ ( x, xɺ ) + F
(6.8)
where the relationships between the corresponding matrices and vectors from
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eqs. (6.7) and (6.8) are given as:
x = f (q ) xɺ =
where
(6.9)
∂f (q ) qɺ = J (q ) qɺ ∂q
(6.10)
Λ ( x) = J −T (q ) H (q ) J −1 (q )
(6.11)
µ ( x, xɺ ) = J −T (q) h(q, qɺ ) − Λ ( x) Jɺ (q)qɺ
(6.12)
Φ = J −T (q ) τ
(6.13) T
J −T (q ) = ( J −1 ) (q ) .
The end-effector position and orientation are represented by the vector of external coordinates: T
x = [ xk , yk , zk , ϕ , θ , ψ ]
(6.14)
where ( x k , y k , z k ) are the Cartesian coordinates of the reference coordinate frame attached to the manipulator base. Orientation of the end-effector with respect to the base frame is described in terms of the Euler angles (ϕ , θ ,ψ ). The working environment model represents one of the most complex and least investigated issues in the robot contact tasks. In the case when the environment does not exhibit the displacements that are independent of the robot motion, mathematical model of the environment can be described by the nonlinear differential eqs. [2]:
M ( s ) ɺsɺ + L ( s, sɺ) = F ,
s = ϕ (q)
(6.15)
where s is a vector of the environment coordinates (displacements); ϕ (q ) is a vector function connecting the two coordinate frames. In the frame of robot joint coordinates, the model of environment dynamics can be presented in the form: M (q ) qɺɺ + L(q, qɺ ) = S T (q ) F (6.16) where M ( q ) ∈ R n× n is a nonsingular matrix; L( q, qɺ ) ∈ R n is a nonlinear vector function; S T ( q ) ∈ R n×n is the matrix with rank ( S ) = n . In the case of contact, we shall also assume that all the mentioned matrices and vectors are continuous functions of their arguments.
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The presented forms of dynamics models of the robot and environment that can be used for learning control synthesis, have some important features that are given in general nonlinear form of generalized coordinates, although the mathematical models that are commonly used for contact tasks are based on linearized models and external coordinates. Hence, it is convenient to adopt in practice a simplified model of the environment, taking into account the dominant effects, such as stiffness:
F = K '( x − x0 )
(6.17)
or the environment damping during the tool motion:
F = B ' xɺ
(6.18)
where K '∈ R n×n , B '∈ R n×n are the semi-definite matrices describing the environment stiffness and damping respectively, and x0 ∈ R n denotes the coordinate vector of the point of impact between the end-effector (tool) and a constraint surface. However, it is more correct to adopt the relationship defined by specification of the target impedance [1]:
F = M ' ∆ɺxɺ + B ' ∆xɺ + K ' ∆x
(6.19)
where ∆x = x − x0 and M ' is a positive definite inertia matrix. The matrices M ' , B ' , K ' define the target impedance which can be selected to correspond to various objectives of the given manipulation task [53]. A general formulation of robot control task can be given as the robot motion along the desired trajectory q p (t ) while the desired force Fp (t ) is acting between the robot and the environment. In this case, it is important to notice that the desired robot motion q p (t ) and the desired interaction force Fp (t ) can not be arbitrary. These two functions must satisfy the following relation:
Fp (t ) = f (q p (t ), qɺ p (t ), qɺɺp (t ))
(6.20)
The goal of robot learning control in contact tasks can be formulated by the following goal conditions: q k (t ) → q p (t ) F k (t ) → Fp (t ) (6.21) where k is the number of learning epochs, while the quality of transient response is specified in advance.
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6.4.3 Factors affecting task performance and stability in robotic compliance control To emphasize the importance of connectionist learning approach and comprehensive control strategy we will analyze the factors affecting task performance and stability in compliance control algorithms. As typical compliance control algorithms we consider in particular special control algorithms for stabilizing position and force based on the quality of transient processes [54], along with impedance control algorithms. The main idea of using neural networks for learning the system uncertainties and classification of unknown robot dynamic environment can be efficiently applied onto other types of robot contact control algorithms, too. As a first example we consider the control algorithm based on stabilization of the robot motion with a preset quality of transient responses, and this has the following form [54]:
τ = H (q ) qɺɺp − K pη − K vηɺ + h(q, qɺ ) − J T (q ) F
(6.22)
The family of desired transient responses is specified by the vector differential eqs:
ηɺɺ = − K pη − K vηɺ
(6.23)
η (t ) = q (t ) − q p (t )
(6.24)
where K p ∈ R n×n is the diagonal matrix of position feedback gains; K v ∈ R n×n is the diagonal matrix of velocity feedback gains. The right side of eq. (6.17), i.e. the PD-regulator is chosen so that the system defined by (6.23) has a property of asymptotically global stability. The values of the matrices K p and K v can be selected according to the algebraic stability conditions. The proposed control law represents a version of the well-known computed torque method including the force term, which uses dynamic robot model and the available on-line information from the position, velocity and force sensors. In this case, the robot dynamics model (exactly, the uncertainties of the matrix H (q ) and vector H (q, qɺ ) has explicit influence on the performance of contact control algorithm. Another important characteristic of this control algorithm is that the model of robot environment does not have any influence on the performance of the control algorithm. Hence, the influence of different robot environments is expressed through the different values of initial force at the
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Dynamics and Robust Control of Robot-Environment Interaction
robot tip, i.e. through the different parameters of the environment model. In the initial contact, there are different values of initial force for various robot environments, while all the other model and control parameters are equal. These different initial conditions cause different force transient responses. Also, the inclusion of noise from the robot sensors cause different force steady-state responses for various robot environments. Hence, if the control goal is to achieve the same quality of force steady-state responses for different environments, the same force performance can be achieved only with the different values of PD gains. In this case, learning of the matrices of robot dynamic models by the neural network and the determination of PD local gains based on neural classification of robot environment will significantly influence the task performance. As a second example we consider a control algorithm based on stabilization of the interaction force with a preset quality of transient responses, which has the following form [54]:
τ = H ( q ) M −1 ( q ) − L ( q, qɺ ) + S T ( q ) F + h( q, qɺ ) − t ω − J (q) Fp − ∫ K f p µ (ω ) + K fi ∫ µ (ω1 )dω1 dω t0 t0 T
(6.25)
where µ (t ) = F (t ) − Fp (t ); K f p ∈ R n×n is the matrix of proportional force feedback gains and K fi ∈ R n×n is the matrix of integral force feedback gains. Here, it has been assumed that the interaction force in the transient process should behave according to the following differential eq.:
µɺ (t ) = Q( µ )
(6.26) t
Q ( µ ) = − K f p µ − K fi ∫ µ d ω
(6.27)
t0
whereby the PI-force regulator (continuous vector function Q ) is chosen such that the system defined by (6..26) is asymptotically stable as a whole. It is evident that the uncertainties of the robot dynamics model have explicit influence on the performance of contact control algorithm. However, the environment dynamics model has explicit influence on the performance of contact control algorithm, also influences the PI force local gains. It is clear that
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if the knowledge of the environment model (parameters of the matrices M (q ), L(q, qɺ ), S (q )) is not accurate enough, it is not possible to determine the nominal contact force F p (t ) . Besides, an inexact model of environment dynamics can significantly influence the contact task performance. Hence, if the control goal is to obtain the same quality of force steady-state processes for different environments, the same force performance can be achieved only by determining the environment model profile and/or identifying parameters of the robot environment model, and using equal fixed PI force local gains. The unknown model profile and/or parameters of robot environment model have a greater significance for the system performance in comparison with the PI force local gains. In this case, the robot environment model profile and parameters, based on classification of robot environment, influence the robot task performance, as also do the uncertainties of the robot dynamics model. 6.4.4 The comprehensive connectionist control algorithm based on learning and classification for compliance robotic tasks The main objective in the compliance control strategy is the application of connectionist learning structures used as part of stabilizing control algorithms. The role of the proposed connectionist structure is to compensate for the possible uncertainties and differences between the real robot dynamics and the assumed dynamics defined by the user in the process of control synthesis. But, the proposed learning control algorithms do not work in a satisfactory way if there is no sufficiently accurate information about the type of robot environment model and parameters of the environment model. Hence, in order to enhance connectionist learning of the general robot-environment model, a new connectionist control strategy is proposed based on learning and classification properties. The main idea of the proposed strategy is the use of neural network approach as classification and learning tool. Hence, the application of connectionist approach to this type of problems is divided into two phases: first, related to the acquisition process and off-line training of the proposed neural network and, second, association phase, where on-line control algorithms based on excellent generalization properties of the wavelet network classifier and learning properties of the neural compensator, should ensure the necessary quality of the system performance.
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Dynamics and Robust Control of Robot-Environment Interaction
Acquisition process for classification - the first phase In the acquisition process of the first phase, based on repeated realization of the proposed contact control algorithms using a previously chosen set of different working environments, force data from force sensors are observed and stored in specific data files. For each robot environment and for the same chosen contact control algorithm, values of normal force and error of normal force are measured, calculated, and stored as special input patterns for training the neural classifier. Generally, six contact force components can be collected during the task realization, but the attention will be focused only on the normal force, as one of the most interesting components, which is sufficient to classify the unknown environment characteristics. It is assumed that the normal force component can be obtained from the force sensor, because normal and tangential directions of force components are defined when considering machining operations. On the other hand, the acquisition process must be accomplished using various robot environments, starting with the environment with a low level of system characteristics (for example, with a low-level of environment stiffness) and ending with the environment having a high level of system characteristic (with a high-level of environment stiffness). As other important characteristic in the acquisition process, different model profiles of environment are used based on additional damping and stiffness members that are added to the basic general impedance model. It is important to notice that the main idea is the classification of environment type, not only environment parameter identification. This approach represents good foundation for encompassing a wide range of unknown robot environment characteristics. Wavelet network classifier In order to enhance the connectionist learning of a general robot-environment model, wavelet network classification of robot dynamic environment is proposed as another solution. Namely, in the pattern recognition tasks it is important to use feature extraction from the occurring patterns, which is the conversion of patterns to features that are regarded as a condensed representation, ideally containing all important information. Hence, the proposal of the new classifier represents a combination of the advantages of a multilayer perceptron with the
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appropriate feature-extraction methods by designing a classification procedure which consists of two parts: a) small number of features are calculated by wavelet transformation from high-dimensional input patterns (signals from force sensors) and b) the calculated features are regarded as inputs to a simple multilayer perceptron which is used as final classifier. To summarize the essential components of classifying procedure we will describe the process of feature extraction by wavelet transformation [55]. The signals from the robot force sensors are frequently characterized by a nonstationary time behavior. For this type of signals, time-frequency representation is highly desirable because of deriving important features. From a variety of different approaches to the theory of wavelets and signal processing [22,56-57] we will present one specific method for feature extraction based on the linear one-dimensional wavelet transformation ω (α, τ). In mathematical terms, the wavelet transformation is expressed as the inner product of a signal s (t ) with a basic wavelet function ha (t ) :
ω ( a , τ ) = s; h a = where
a
1
∫ s (t ) h a
∗ a
(
t −τ ) dt a
(6.28)
is the scale parameter, τ is the shift parameter of the wavelet
function; ∗ denotes the complex conjugate. The weighting factor
1 a
normalizes the wavelet function for all values of a to constant energy. The wavelet transformation ω (a, τ) can be considered as the correlation function between the signal s (t ) and the basic wavelet ha . Typically, basic wavelets are modulated window, i.e. they are composed as product of two functions. There are various basic wavelet families such as Mexican hat wavelet, Meyer wavelet and Morlet wavelet. As a typical example we can use the complex Morlet wavelet:
hMa =
1 t t exp − 0.5 ( ) 2 + j 2π f o a a a
(6.29)
The analysis by wavelet transformations can be generally considered as a filter bank comprised of bandpass filters with the bandwidths proportional to frequency. In this case, the filter bank is characterized by the particular basic wavelet function and the parameter f o . Depending on the scale factor a , the time course of the wavelet transformation can be changed. If a is large, the
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wavelet is a dilated low-frequency function, while for small values of a , the wavelet is constricted, corresponding to a high-frequency function. Important features of the extraction that are suited for distinguishing different patterns can be defined as particular values of ω (a, τ), at specific times τ k , and scale factors
a k , respectively, for the frequency values f k =
fo . ak
The concept of the wavelet network [58] for classification purposes is represented by an expanded perceptron for decision purposes together with the so-called wavelet nodes as preprocessing units for feature extraction. It is important to notice that during the learning phase, the wavelet network not only learns the appropriate decision functions and complex decision regions defined by the weight coefficients, but also searches for those parameter spaces that are suited for a reliable categorization of the input force signals. In Fig. 6.12 is shown a basic version of the wavelet network structure. The wavelet nodes, which are adjusted during the learning phase, are a modified version h(t − τ k / a k ) of a basic wavelet h(t ) . The nodes are described by a time shift parameter τ k , and a scale parameter a k , which is inversely related to the node frequency ( τ k and a k are the parameters of wavelet transformation). The input to the wavelet network, i.e. the output of the wavelet node φ ik represents the inner product of the node hk and the signal from the robotic force sensor Fi (the index i denotes the signal samples,
i = 1, ..., N ):
φik = hk ; Fi = ∫ hk* ( t
t −τ k ) Fi (t ) dt ak
(6.30)
The upper part of the wavelet network is represented by the topology of a 3layer perceptron, which bases its classification decision on the wavelet node output. Training of the perceptron can be achieved by using minimization of the least-squares error ( M = 1 in this case): N
E = ∑ (d i − y i ) 2 = min
(6.31)
i =1
where d i is the desired output vector which belongs to an appropriate class of training patterns. It is important to notice that not only the weights of the
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563
network are thus adjusted, but also the parameters of the wavelet nodes. The parameters of the wavelet transformation, the scale parameter a k and the shift parameter τ k , depend on the basic wavelet chosen, and they are determined using, for example, the well-known back-propagation algorithm:
Fig. 6.12 Wavelet network classifier
N ∂ φ ik ∂E = − ∑∑ δ ij ω kj ∂τ k ∂τ k i =1 j
(6.32)
564
Dynamics and Robust Control of Robot-Environment Interaction N ∂E =− ∂ ak i= j
∑∑ δ ω ij
kj
j
∂ φik ∂ ak
δ ij = (di − yi ) yi (1 − yi )
(6.33) for output layer
hid δ ij = O hid )ω out j (1 − O j ) j ( d i − yi ) yi (1 − yi )
for hidden layer ( j = 1,..., H ) O hid = f( j
∑ω
φ + bj )
in kj ik
(6.34) (6.35) (6.36)
k
where f is a sigmoid or other activation function; b j is a bias member. In our example, complex Morlet wavelet is applied as basic wavelet. Hence the wavelet node can be calculated in the following way:
φik = ∫ hk* ( t
t −τ k ) Fi (t ) dt = ( ∫ u1 dt )2 + ( ∫ u2 dt )2 ak t t
(6.37)
u1 = Fi (t ) cos (ω k
t −τk t −τk 2 ) exp (−0.5 ( ) ) ak ak
(6.38)
u 2 = Fi (t ) sin (ω k
t −τk t −τk 2 ) exp (−0.5 ( ) ) ak ak
(6.39)
where the frequency parameter ω k = 2π f k which, in contrast to the assumption of a fixed value f o , results in greater flexibility. Using summation for integration in discrete processing, and according to (6.32)–(6.39), the adjustment of parameters of the wavelet transformation is described by the following eqs.: ∂E = ∂τ k
N
∑∑δ ijω kj i =1
j
1
Tm
∑ F (t
φik m =1
i
m ) exp( −0.5(
tm − τ k 2 1 ) ) ak ak
t −τ k t −τ k t −τ k × ( u1dt ω k sin(ω k m )+ m cos(ω k ( m ) ak ak ak t
∫
+
∑ u dt −ω 2
t
k
cos(ω k
tm − τ k t −τ k t −τ k )+ m sin(ω k ( m )) ak ak ak
(6.40)
Intelligent Control Techniques for Robotic Contact Tasks
∂ E t −τk ∂ E = ak ∂ τ k ∂ ak ∂E = ∂ω k
N
∑∑δ ijω kj i =1
j
1
φik
Tm
∑ F (t i
m =1
m ) exp( −0.5(
(6.41)
tm − τ k 2 t m − τ k ) ) ak ak
t −τ k t −τ k − u1dt sin(ω k m ) + u2 dt cos(ω k m ) ak a j k t
∫
565
(6.42)
∫
where Tm is the number of signal sampling points. After the acquisition process, during the extensive off-line training process, in the same way as the perceptron, the wavelet network receives a set of inputoutput patterns, where input variables are the wavelet transformation of force signals, while the desired network output has a value between zero and unity, which exactly defines the type of training the robot environment. On-line Compliance Control Algorithms - the Second Phase Based on the first phase, related to the acquisition process and off-line training process of wavelet classifier, it is possible to determine the whole structure of compliance control algorithms, including the neural compensator and wavelet classifier. After the off-line training process with different working environments and different environment model profiles, the wavelet classifier is included in the on-line version of control algorithm, to produce some value between 0 and 1 at the network output. Based on this value, through the process of linear interpolation, the environment parameters M (q ), L( q, qɺ ) , along with the environment model structure, are effectively determined. This interpolation process is driven by the stored parameters of the dynamic models of different chosen environments and different chosen environment model structures. In the first example considered, the following stiffness model of robot environment is chosen for the control algorithm based on the stabilization of the robot motion with a preset quality of transient process:
F = K '( x − x0 )
(6.43)
After the off-line training process with different working environments (different environment stiffness), the neural classifier with fixed weighting factors is included in the on-line version of the control algorithm (6.22) to produce some value y at the network output being between 0 and 1, based on
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the on-line force inputs defined in the previous section:
τ = Hˆ (q) qɺɺp − Kˆ pη − Kˆ vηɺ + hˆ(q, qɺ ) − J T (q) F + P N N
(6.44)
Kˆ p = f k p ( y )
(6.45)
Kˆ v = f kv ( y )
(6.46)
where f k p , f kv are the linear interpolation functions for the position and velocity feedback gains; y is the output of the wavelet classifier. If we adopt the same force steady-state process for all different robot environments as performance criterion, then, using algebraic stability conditions, we can a priori choose the set of PD local gains for the previously defined set of known robot environments (in our case there are 5 different environments) that will satisfy this requirement. Hence, in the case of unknown environment type, the information from the wavelet classifier output can be efficiently utilized to calculate the necessary PD local gains by linear interpolation procedures. It is also assumed that the network output for the given environment varies in a small range. In this way, the local PD gains are relatively fixed during the operations. They are chosen for presetting stability conditions for each environment type. In the second example, for the control algorithm based on stabilization of the interaction force with a preset quality of transient process, the general impedance model of robot environment is chosen:
F = M ' ∆ɺxɺ + B ' ∆xɺ + K ' ∆x
(6.47)
Hence, after the off-line training process, the on-line version of compliance control algorithm with the wavelet classifier having fixed weighting factors based on the on-line force and force errors inputs for a specified environment model is given by the following relations:
τ = − H (q) Mˆ '−1 (q) Bˆ ' qɺ + Kˆ ' q + h(q, qɺ ) − ( J T (q) − t ω −1 ˆ − H (q ) M ' ) Fp − ∫ K f p µ (ω ) + K fi ∫ µ (ω1 )d ω1 d ω + P N N t0 t0 Mˆ ' = f M ' ( y ) Bˆ ' = f ( y )
B'
Kˆ ' = f K ' ( y )
(6.48)
(6.49) (6.50) (6.51)
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567
where f M ' , f B ' , f K ' are the linear interpolation functions for parameters of the matrices M ' , B ' , K '.
Fig. 6.13 Scheme of learning control law with wavelet network classifier
According to a similar principle, the same condition for control law and all different robot environments assumes the same local PI force gains. In our case, the parameters of dynamic models of the different chosen environments M ' , B' , K ' are stored as the information needed to calculate the basic control algorithm. In the case of an unknown environment, the information from the neural classifier output can be efficiently utilized to calculate the necessary environment parameters M ' , B ' , K ' by linear interpolation procedures. In the same way, the wavelet network classifier can be included in the comprehensive compliance learning control law. The overall scheme of the control algorithm is shown in Fig. 6.13 for the case of stabilization of interactive force.
6.4.5 The genetic-connectionist algorithm for compliant robotic tasks The main idea is to enhance the capabilities of the proposed connectionist control algorithm for contact robot tasks [50,52] by synthesizing new control
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Dynamics and Robust Control of Robot-Environment Interaction
algorithms based on genetic tuning of nonlearning control part (conventional force PI controller) and learning control part (neural “off-line” classifier and neural “on-line” controller). In the proposed nonlearning control part we use the control algorithms for stabilizing interaction force based on the quality of transient processes [54]. These stabilizing control laws ensure exponential stability of the closed-loop system. However, the property of closed-loop stability is a very basic requirement, but it is never the only one. In addition to stability, the user is usually very interested in manipulating the performance of the system in terms of overshoot, oscillation and settling time. For many such controllers, there are currently no systematic approaches to choose the control parameters yielding the desired performance. The controller parameters are usually determined by trial-and-error through simulation and long-lasting experimental tests. In such cases, the paradigm of GAs appears to offer an effective way of automatic and efficient searching for a set of the controller parameters yielding better performance. There are several efficient GA methods proposed, intended for various special purposes in robotics [59-63]. If compared with conventional optimization methods, GAs possesses many advantages (global, data-independent and robust method). Further, GA can be directly applied to solve an optimization problem with a certain fitness function without reformulating the problem into a suitable form. Here, a simple GA variant which works directly on real (decimal) parameters is used. Decimal-type GAs are equivalent to the traditionally used binary-type GAs in optimization. The real-type GAs for computer-based numerical simulations lead to high computational efficiency, smaller computer memory requirements with no reduction of precision, and to greater freedom in selecting genetic operators. A systematic approach to design of controllers for both closed-loop stability and desired performance by using GA to tune the position and force feedback gains is proposed. The GA utilized is selected to be of the decimal real number type, to achieve simple and efficient computation. Two types of fitness functions are considered for optimization of the controller performance: integral of squared errors (ISE) and integral time-multiplied absolute value of errors (ITAE). In order to improve the convergence process (time-consuming process of learning because nonsystematic approach to the determination of network structure), an efficient GA is proposed to choose the appropriate topology of the multi-layer perceptron that performs neural classification. Also, in order to improve the learning process, GA optimization is used to determine the weighting factors for neural compensation of the robot dynamic model in the on-
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line control algorithm. In both previous cases, binary representation of the problem is used to achieve good results for the robot compliance task based on the suitable fitness functions. The control algorithm based on stabilization of the interaction force with a preset quality of transient responses is considered, which has the following form [54]:
τ = H (q) M −1 (q) − L(q, qɺ ) + S T (q) F + h(q, qɺ ) − t ω − J T (q ) Fp − ∫ K f p µ (ω ) + K fi ∫ µ (ω1 ) d ω1 d ω t0 t0
where
µ (t ) = F (t ) − Fp (t ); K f ∈ R n×n = diag K f
p
( ) p
is the matrix of
ii
proportional force feedback gains; K fi ∈ R n×n = diag K fi
(6.52)
( )
ii
is the matrix of
integral force feedback gains. Here, it has been assumed that the interaction force in the transient process should behave according to the following differential eq.:
µɺ (t ) = Q( µ )
(6.53) t
Q( µ ) = − K f p µ − K fi ∫ µ dt
(6.54)
t0
The PI- force regulator (continuous vector function Q ) is chosen so that the system defined by (6.47) be asymptotically stable as a whole.
6.4.6 GA tuning of PI force feedback gains In order to further simplify the genetic process, the set of tuning force gains K f p and K fi is reduced to the single parameter ω n , where ω n is the natural frequency of the second-order linear system defined by the characteristic eq.
µɺ i (t ) + ∫ K f p to t
( )
ii
µi (ω ) + ( K f
ω
i
) ∫ µ (ω )dω dω = 0 i
ii
to
1
1
(6.55)
Previous forms of the characteristic eqs. are equivalent to the following eq.:
ηɺɺi + 2 ςω nηɺ i + ω n2 = 0
(6.56)
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Dynamics and Robust Control of Robot-Environment Interaction
If we assume for the second-order system the critical damping ς = 1, the feedback gains are given by
(K ) fp
(K ) fi
= 2ωn
(6.57)
= ωn2
(6.58)
ii
ii
In this way, only natural frequency is chosen for genetic tuning. The initial population of the size N is generated randomly to start the optimization process. The total population of each generation is evaluated using a suitably chosen performance criterion (ISE or ITAE). Reproduction as primary genetic operator is based on using the best N / 2 individuals of the current generation to be parents for producing the next generation. Weighted-average crossover genetic operator based on decimal numbers is applied [62]. From the parents ω n 1 and ω n 2 , two new offsprings are generated by the following terms:
ω n1 = r ∗ ω n 1 + (1 − r ) ∗ ω n 2
(6.59)
ω n2 = (1 − r ) ∗ ω n 1 + r ∗ ω n 2
(6.60)
where r ∈ (0, 1) is a random number. Mutations are based on the following changes of natural frequency:
ω n1 = ω n + (r − 0.5) ∗ 2 ∗ ∆ω nmax
(6.61)
where ∆ω nmax is the maximum change of natural frequency. The objective of the GA optimization is to obtain better end-effector performance, i.e. to find PI force feedback gains as fast as possible, with the minimal oscillation and overshoot. The fitness functions are defined according to the following eqs.: T
ISE = ∫ µ 2 (t ) dt
(6.62)
0 T
ITAE = ∫ t µ 2 (t ) dt
(6.63)
0
6.4.7 Case studies To demonstrate the performance of contact control schemes with the connectionist classifier and compensator, compliance control implementations
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are simulated using the PUMA 560 robot for the circular writing task on the various robot environments. The robot end-effector exerts the force that is perpendicular to the y − z plane, performing circular path with a diameter
d = 12 [cm] . The task of the robot is to carry out the writing process on the work surface along the prescribed trajectory with a desired contact force FN0 = 5 [N ] . As example, stabilizing interaction force algorithm is chosen, while general model of impedance as model of environment is used. The parameters of the environment model in the form of diagonal members of appropriate matrices for all different chosen environments and for both control algorithms are given in Tables 6.1 – 6.3. Table 6.1 The stiffness parameters of robot environment models Environment
K '11
K ' 22
K ' 33
K ' 44
K ' 55
K ' 66
Styrofoam Silicon Rubber Plastic Steel
660. 2000. 3300. 6000. 24000.
6600. 20000. 33000. 60000. 240000.
660. 2000. 3300. 6000. 24000.
0.0007 0.002 0.003 0.006 0.02
0.0007 0.002 0.003 0.006 0.02
0.0007 0.002 0.003 0.006 0.02
Table 6.2 The damping parameters of robot environment models Environment
B'11
B' 22
B ' 33
B' 44
B ' 55
B ' 66
Styrofoam Silicon Rubber Plastic Steel
69. 210. 346. 629. 2638.
277. 839. 1385 2517. 10704.
69. 210. 346. 629. 2638.
0.007 0.02 0.035 0.06 0.27
0.007 0.02 0.035 0.06 0.27
0.007 0.02 0.035 0.06 0.27
Table 6.3 The inertia parameters of robot environment models Environment
M '11
M ' 22
M ' 33
M ' 44
M ' 55
M ' 66
Styrofoam Silicon Rubber Plastic Steel
1.88 5.7 9.4 17. 68.15
6.7 20.3 33.5 60.9 243.41
1.88 5.7 9.4 17. 68.15
0.007 0.02 0.03 0.06 0.2
0.007 0.02 0.03 0.06 0.2
0.007 0.02 0.03 0.06 0.2
For the application of stabilizing force interaction control algorithm, the performance criterion based on selection of the same force PI gains is chosen.
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Dynamics and Robust Control of Robot-Environment Interaction
These PI force gains are synthesized using the same system frequencies for all different working environments (ω n = 2 H z ). The transient processes of internal coordinates error and force error are given in Figs. 6.14-6.15. We can notice the influence of different working environment.
Fig. 6.14 Internal error for stabilizing interaction force control algorithm
Fig. 6.15 Force error for stabilizing interaction force control algorithm
Intelligent Control Techniques for Robotic Contact Tasks
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After the acquisition process, the off-line wavelet network training is performed. The following network topology was chosen: 7 (number of wavelons in the wavelet node) - 42 -24 -1. The uncertainties of the robot dynamics model and dynamic environment model are defined by the parametric disturbances and additional white noise. In the generalization test, the “off-line learned” wavelet neural classifier with fixed weighting factors was included in the control algorithm for the recognition of unknown robot environment. The second neural network for uncertainty compensation utilized the same learning rules and parameters but had a different network topology (31-69-37-6). The profile model of environment using general impedance model with additional stiffness members was adopted. In this case, the robot environment with the dominant stiffness K = 65000 N / m was selected. The wavelet neural classifier based on input force data generates the appropriate value at the network output. For comparison, the learning control laws with and without exact information of environment stiffness are represented in Figs. 6.16 and 6.17. It is evident that in the case when there are no exact information about robot environment, the quality of position tracking performance is poor.
Fig. 6.16 Circular tracking – comparison with and without neural classifier
Fig. 6.17 Comparison with and without classifier
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Dynamics and Robust Control of Robot-Environment Interaction
In this section we presented a new comprehensive method for selecting the appropriate compliance control parameters for robotic contact tasks. The method classifies the type of environment by using in the first phase acquisition process of force sensor data and off-line training process by multilayer perceptrons of the wavelet networks. An important feature is that the process of pattern association can work in an on-line mode as a part of the selected compliance control algorithm. Simulation experiments showed that a wavelet classifier, together with the neural compensator, could significantly improve the robot performance in the contact tasks involving environment uncertainties. To investigate the effect of GA optimization procedure for tuning the PI local force gains, simulation experiments were conducted with the appropriate initial set of PI local force gains. It is necessary to specify the range of controller parameter (natural frequency). Including the maximal torque value given by the actuator limits, we obtain the range of the natural frequency:
0 < ω n ≤ 32
(6.64)
In the simulation, the population size of each generation was set to be N = 40 ; the maximum mutation values for ∆ω nmax = 0.5. The evaluation process is terminated when the change of fitness function is small in a certain number of successive generations. The results of GA optimization procedure are shown in Figs. 6.18 and 6.19.
Fig. 6.18 Best force feedback gains KFP and KFI according to
ISE
and
ITAE
criterions
It is obvious that better performance (corresponding to smaller values of the fitness functions) is obtained with the progression of the GA process.
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Based on the previous GA optimization, PI force gains are synthesized using the same system frequencies for all different working environments (ω N = 11.86 H z ) . In the phase of connectionist off-line training, an efficient GA is used in order to select optimal topology of the neural network. Using the adopted GA network topology and the learning process, the training is carried out with the stored weighting factors. In the generalization test, the “off-line learned” and GA tuned neural classifier is included into the control algorithm for recognition of the unknown robot environment. In a similar way, the GA approach based on binary representation is applied to determine the weights of the second neural network for compensating the robot uncertainties.
Fig. 6.19 Best values of ISE and ITAE criterion during evaluation process
6.5 Connectionist Reactive Control for Robotic Assembly Tasks by Soft Sensored Grippers The main purpose of this section is to present s one interesting example of connectionist approaches for special robotic application, in order to verify the efficiency of contemporary intelligent control methods. As example, connectionist reactive control for a robotic assembly task by soft sensored grippers is presented [64]. In this example, the problem of efficient control in assembly tasks realized by industrial robots with the gripper having soft and sensored fingers was considered. Instead of using the well known methods of reactive control based on force sensors at the manipulator wrist, a new method based on learning strategy by neural networks and pressure sensors on the soft fingers was proposed. The problem of learning reactive control was exemplified by the “peg-in-hole” task. Experiments were carried out using the industrial
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robot ASEA Irb-L6 with the Belgrade-USC-IIS Robot hand Model II whose fingertips are equipped with pressure sensors In spite of the fact that the problem of assembly is one of the most important and most investigated working operations in the industrial application, it has not yet been generally resolved. One of the reasons is that the tolerances of the parts to be assembled are in many cases smaller than the accuracy of the contemporary robots. Because the robotic grippers in the manufacturing industry grip the object in a stiff manner, no relative motion of the object in the gripper is allowed. The tolerances of the assembled parts are the same tolerances that limit the motion of the gripper to achieve a desired position and orientation. Small deviations in the object orientation and position can cause significant reaction forces, especially if narrow tolerances are involved. Various approaches have been employed to overcome this problem. One of the approaches which uses the benefits of passive compliance was proposed in [64]. The gripper fingers are equipped with pressure sensors and covered with soft material, allowing object motion while it is being grasped. Changes of the pressure on the contact surfaces are sensed and used as basic information for corrective action during the task realization. The basic principles of the peg-in-hole task realization involving a cylindrical peg are very simple, and the direction of corrective action can be successfully derived from the information obtained from the pressure sensors. Neural networks are very convenient for such tasks, and their use in the realization of the peg-in-hole assembly task is proposed. Practical working conditions of the peg-in-hole realization involve various sources of uncertainty and noise, so that they can substantially degrade the performance of traditional (off-line planning) control methods. Hence, reactive control methods try to cancel the effects of uncertainty and noise by using autonomous on-line learning procedures based on the repetition of the working task. The control goal of learning is an on-line modification of the control signal yielding a whole set of corrective movements to achieve a successful task realization. In the process of reactive control, compliant behaviour of the system is accomplished either actively (using contact forces or tactile stimuli) or passively, by the inherent physical characteristics of the robot. Hence, an approach to learning reactive control strategy for peg-in-hole insertion task using a manipulation robot with soft sensored fingers is proposed. The use of soft fingers in robotized assembly tasks enables self-adaptation of the manipulated object during the assembly operation. The main idea of the proposed method is the use of an array of separate pressure sensors on the surface of the robot’s fingers, which give a reduced set of information needed for the choice of
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direction of corrective movements in the assembly task. The new method of corrective actions is based on representation and learning of nonlinear compliance laws. In this case, the nonlinear compliance is treated as a nonlinear mapping from the pressure sensors to the corrected motion. This nonlinear mapping is accomplished by the learning process based on the application of multi-layer neural networks.
6.5.1 Analysis of the assembly process with soft fingers In contrast to the present practice, where the gripper’s surface in contact with the object is hard and allows no relative motion of the object in the gripper, it is proposed to use a soft material between the hard object and the gripper’s “skeleton”. Such a gripper with “soft” fingers can be used as a device with the characteristics of passive compliance. A change in the position of the object within the gripper causes a change of the finger elastic material stress state. Instead of measuring the reaction force at the robot wrist, the information is obtained from the stress state on the contact surface between the object and the gripper elastic pad. The force acting on each soft element is resolved in three mutually orthogonal directions. Two of them are parallel to the soft pad plane, while the third one is orthogonal to it. Thus, the force components are the normal component acting in the orthogonal direction with respect to the contact surface and two tangential components. One of the tangential components is selected to act along the gripper longitudinal axis, while the other is orthogonal to it. The normal component produces pressure between the object and the gripper, while the first tangential force in most cases corresponds to the insertion force, i.e., the force which has to be applied by the gripper to effect execution of the peg-in-hole assembly task. The other component of tangential force is produced by object rotation around the gripper roll axis. The influence of the normal force component represented by the pressure between the object and the gripper is sensed by a set of pressure sensors. They do not provide information about the exact intensity and direction of the force applied but on the pressure profile shape and its change, which have a very rich information content. The two other tangential stress components have the character of force and torque, and they may be sensed in some other way, if necessary. The basic information is the pressure portrait on the gripper fingers. The pressure portrait is established immediately when the object is gripped, even before the task execution starts. Let this portrait be called the initial pressure portrait. It is easy to see that any contact involving the free object will cause a change in this portrait. If the contact occurs as a consequence of a desired action, the pressure portrait
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obtained will be nominal. The whole control strategy is based on the information about pressure profiles and their dynamic behaviour, knowledge of task realization and previous experience. For a known strategy and for a particular task realization, a direct relationship between the sensed stress state and the compensating action performed has to be established.
6.5.2 Assembly process The most important problem in assembly processes is the forces, which may arise at the contact points. Let the “peg-in-hole” task be adopted as a general representation of assembly tasks, and suppose that the object has already been gripped and brought close to the desired (nominal) position. Practically, this means that the object has been brought somewhere into the chamfering cone. Then in the initial movement of joining, two types of deviation may simultaneously occur: a) Radial displacement of the object (peg) tip with respect to its nominal (desired) position at the beginning of insertion. b) The peg angular deviation with respect to the hole axis. Let us further define the nominal object position in the initial moment as the position, which enables the realization of the assembly task only by moving the object in the direction of its axis, without any additional correction. This situation occurs when the axes of the peg and the hole coincide. If nominal conditions are preserved during the insertion, the assembly may be realized without any resistant force. Because it is hard to realize such perfect positioning, the object and the hole come in contact. A compensating (corrective) action of the robot has to be performed in such a manners to eliminate or minimize deviations and enable successful completion of the assembly task. The first contact between the object and the hole may occur in the cone formed by chamfering. Due to the very high stiffness of the arm of the industrial robot, small positional inaccuracies can produce very large reaction and large friction forces. It is not reasonable to suppose that the object tip will “naturally” slip along the surface of the chamfering cone. The main requirement imposed on compensating action should be to prevent the occurrence of large friction forces. This will be the case if no high insertion force is applied. Thus, the proposed action, which has to enable progression of the object into the hole, is the application of very high pressure in the direction of insertion, combined with the appropriate compensation action. The rotation of the peg around the contact point toward the cone surface until the reaction force comes out of the friction
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cone must be realized and sliding along the surface of the chamfering cone continued. The phase is over when the object tip enters the hole. To continue insertion, a different compensation strategy has to be applied. Due to imperfections in the relative positioning, narrow tolerances, and especially due to the gripper’s soft contact surfaces, two-point contact between the peg and the hole will always occur. In this phase, when the object tip progresses into the hole and the two-point contact is established, the compensating movement is aimed at reducing or eliminating the friction force at the contact points. A deblocking movement has to be performed in the direction of motion generated by the reaction forces. The movement is limited by the size of the possible peg inclination with respect to the hole axis. When the object is inserted to its full depth, the motion of the object’s gripped end is reduced to the level of manufacturing tolerances. Thus, the gripper compensation motion is not a function of time, but of the depth of the realized insertion, and this is extremely inconvenient for on-line measurement and for use as feedback information.
6.5.3 Learning compliance methodology by neural networks The robotized assembly task can be considered as a combination of programmed motion and corrective actions in accordance with the information from the array of pressure sensors on the robot fingers. In this approach, corrective actions are based on representation and learning of the nonlinear compliance strategy. This nonlinear compliance is treated as a nonlinear mapping from the pressure sensors to a corrected motion. The nonlinear mapping is accomplished by the learning process, based on the application of a multilayer perceptron. The type of desired compensation motion is defined in advance, on the basis of studying the process to be realized or based on experience. The primary aim of the compensating motion is to enable slip of the object tip along the cone surface toward the end of the hole. The newly-proposed method allows a fast and smooth control action without unwanted intermittent actions. A real peg-in-hole task also involves great sources of uncertainty and noise, which can be efficiently compensated for through the neural network model. Synthesis of efficient learning reactive control is a multi-stage process, where a knowledge acquisition process is necessary as a preliminary phase. In this phase, the assembly task is realized by the operator’s commands, based on visual information. In teaching compliance, the operator demonstrates how the motion should be corrected in each training situation. During this phase, the intensive acquisition process of input network data (pressure information) and output network data (corrective actions) is realized. The training samples can be acquired using some other methods, such
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as measurement of human motions, recording of operations of a master-slave manipulator, or by simulation and off-line programming. After that, in the process of off-line learning using the previously collected input-output pairs, the three-layer perceptron must find a generic form of association of pressure with the a priori defined corrective actions. In order to enhance speed of the learning process, special rules, based on the recursive least squares approach, are used [66]. In the final stage, the robot works in an automatic mode, without interventions of the human operator and with the corrective strategy, which the neural network gives as a result of the on-line pressure sensor information. This corrective strategy is defined by the generic knowledge incorporated through the fixed values of weighting factors of the proposed three-layer perceptron. The neural network must be capable of high level generalization for input-output pairs which are not captured in the process of operator training. Also, by this online process, updating and improving the nonlinear mapping represented by the nonlinear network could likely be performed by the robot hand.
6.5.4 Experimental results The connectionist approach was experimentally verified on the example of an assembly task of the peg-in-hole type. Two basic actions had to be realized simultaneously: a small force acting constantly in the direction of the object axis, and adequate compensating movements performed according to the actual pressure profile. A Belgrade-USC-IIS multifingered hand Model II (Fig. 6.20) with three sensored fingers was used as gripper. The thumb and finger were equipped with a single tactile sensor covering the whole fingertip, while the tip of the middle finger was sensored with six separate sensors placed as strips orthogonal to the finger axis. Such a design of pressure sensors enables the pressure profile on the fingertip to be sensed and recorded. It is clear that it would be more desirable to sensor all fingertips in the same way as the middle finger, but this was not possible due to the limited number of sensor inputs available. As the sensor’s presence should not affect the finger’s local shape adaptability. Use was made of FSR tactile sensors that consist of two thin foils. On one of them, conductive pattern is printed, while the second one is covered with a thin film of FSR material, which changes its resistance in response to the applied force. It is clear that such a sensor enables only the normal component of the contact force to be measured, while the force in the direction of the object axis cannot be controlled in this way.
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Fig. 6.20 Belgrade-USC-IIS multifingered hand Model II
To ensure constant action of the force in the direction of insertion, a specially designed device, schematically shown in Fig. 6.21, was realized. Instead of utilizing the gripper with the grasped peg to ensure pressure in the insertion direction, this task was assigned to the object with the hole in it. Hence, this object was fixed to the end of a metal rod, which can slide along the fixture in a vertical direction. The other end of the metal rod has connected to the rubber strips. When an external vertical force is applied onto the object with the hole, it moves downward until the force is balanced by elastic forces generated by the rubber strips. When the peg is in the contact with the hole, and the robot is performing a compensating action to eliminate relative displacement in the peg position (while the peg axis is in the cone), the rod will move upward, and partial insertion will be automatically performed. In this way, full depth insertion may be realized. The robot hand was fixed to an industrial robot ASEA Irb-L6 with standard industrial controller.
Fig. 6.21 Experimental fixture
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The robot hand controller was a PC computer with two additional boards. Communication between these two controllers was realized via an ASEA standard Computer Link Board. Because direct access to the robot controller was not possible, but only via the Computer Link Board, a special technique was developed to accomplish compensation movement in the desired direction. Any inclination of the peg produces a unique pressure pattern on the fingertips, which requires hand movement in the proper direction. The object with hole was placed in the centre of the circle (Fig. 6.22). In the general case, an arbitrary direction of the peg relative to the hole axis will cause a unique change of the pressure on the contact surfaces of the fingers. Because of the specific hand design, the fingers are not able to grip the peg at equidistant points of its cross-section. As for the hand, two fingers are parallel, with the thumb being on the opposite side of the peg. Thus only some limited angular deviation relative to the direction defined by points 2 and 5 in Fig. 6.22 may be successfully handled. Then, instead of covering the whole circle, only six target points (1-6) are defined in advance. In the teaching phase, the operator, with the aid of joystick, brings the peg and the hole into contact. Instead of calculating the angle of direction in which the hand is to move, the operator on the basis of visual inspection, defines one of the points (1-6) and the robot moves in this direction. During this motion, partial (or full) insertion is realized. When jamming occurs, the motion is stopped. The operator now eliminates the direction of hand motion that caused jamming, and the procedure is repeated. Each time, all the pressures from the sensors are recorded, as well as the corresponding action chosen by the operator. This set of data is used to train the neural network.
Fig. 6.22 Selected compensation directions
The overall structure of the multi-layer perceptron for learning the pressure direction relations is presented in Fig. 6.23. The input layer of the network consists of normalized signals from the pressure sensors (eight pressure sensors on the three soft robot fingers plus the bias member). The hidden layer has 32 neurons plus the bias member. The network output produces 6 different control
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signals for the a priori defined compensation actions. The logistic sigmoid function was used as activation function. The process of learning has accomplished through iterative presentations of the training patterns (71 samples of input-output pairs) acquired in the preliminary phase. During learning, connection weights (matrices W1 and W2) are adjusted until the appropriate level of output error is achieved. The process begins with random connection weights. For the process of learning, rules based on the recursive least squares learning procedure [66] are chosen because of their fast convergence properties. The results of learning (square error criterion) are presented in Fig. 6.24. After learning, the values of the connection weights are stored in the robot controller memory for use in the on-line control.
Fig. 6.23 Structure of multilayer perception used in off-line learning procedure
During process of on-line control, the robot controller uses the stored connection weights, takes a forward pass through the neural network and makes decisions about compensatory action based on the value of the network output. The results obtained during the on- line control show that the neural network is capable of accurately performing the working task, using high-level generalization for the input-output pairs, not captured in the process of the operator training. Finally, a gripper with soft fingers with the characteristics of a passive compliance is used for the realization of the peg-in-hole assembly task. The contact force acting on the object’s free end causes object displacement within the gripper, resulting in stress changes at the elastic contact surface. The
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pressure profile is sensed by a set of pressure sensors placed under the soft finger tissue. A compliance control strategy for the assembly task, using robot hands with soft fingers and the fast learning capabilities of connectionist structures, are developed. The new method of corrective actions is based on representation and the multi-stage learning of nonlinear compliance between the information from the pressure sensors and the corrective motion, using the multi-layer neural network. The major contribution is a new learning compliance strategy, based on the relations between the pressure sensors on the robot hands and the direction of the corrective actions.
Fig. 6.24 Square error criterion during process of off-line learning
6.6 Intelligent Control of Contact Tasks in Humanoid Robotics 6.6.1 Introduction Many aspects of modern life involve the use of intelligent machines capable of operating under dynamic interaction with their environment. In view of this, the field of biped locomotion is of special interest when human-like robots are concerned. Studies in the area of humanoid robotics, as one of the most exciting topics in the field of robotics, have recently made a tremendous progress, especially visible in the last decade. The reason for increasing research interest in this domain is that major application areas have become self-evident. Humanoid robots are expected to be servants and maintenance machines with the main task to assist human activities in our daily life and to replace humans in hazardous operations. It is as obvious as interesting that anthropomorphic biped robots are potentially capable to effectively move in all unstructured environments where humans do. Hence, particularly, the fields of service robotics, medical applications, and operation in hazardous environments are of
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primary importance. Another important reason for the growth of humanoid robots research represents the development of advanced technologies in design and production of robot sensors, actuators and computing units. The significant progress has been made in the design of a hardware platform for humanoid robots and control of humanoid robots, particularly in the realization of dynamic walking in several full-body humanoids [67]. It should be emphasized that many humanoid projects [67] are currently being in progress worldwide, so that the number of representative humanoids will certainly grow in the near future. Having in mind the very high requirements to be met by humanoid robots, it is necessary to point out the need for increasing the number of degrees of freedom (DOFs) of their mechanical configuration and studying in detail some previously unconsidered phenomena pertaining to the stage of forming the corresponding dynamic models. Besides, one should emphasize the need for developing appropriate controller software that would be capable of meeting the most complex requirements of accurate trajectory tracking and maintaining dynamic balance during regular (stationary) gait in the presence of small perturbations, as well as preserving robot’s posture in the case of large perturbations. Finally, one ought to point out that the problem of motion of humanoid robots is a very complex control task, especially when the real environment is taken into account, requiring as a minimum, its integration with the robot’s dynamic model. Numerous studies of anthropomorphic walking robots and methods of biped walk control have been conducted in the recent years. As a result, general stationary walking, dynamic walking in the presence of unknown external force, and dynamic walking adapted to an unknown irregular ground, etc., were realized. The strategies for the walk control can be mainly classified [68] as the walking pattern generation related to the motion planning and on-line balance control. At the other side, humanoid robot applications usually demand the robot be highly intelligent. Naturally, the first approach to making humanoid robots more intelligent was the integration of sophisticated sensor systems. However, today’s sensor products are still very limited in interactivity and adaptability to changing environments. As the technology and algorithms for real-time 3D vision and tactile sensing improve, humanoid robots will be able to perform tasks that involve complex interaction with the environment (e.g. grasping and manipulating the objects). A major reason is that uncertainty and dynamic changes make the development of reliable artificial systems particularly challenging. On the other hand, to design robots and systems that best adapt to
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their environment, the necessary research includes investigations in the field of mechanical robot design (intelligent mechanics), environment perception systems and embedded intelligent control that ought to cope with the task complexity, multi-objective decision making, large volume of perception data and substantial amount of heuristic information. Also, in the case when the robot performs in an unknown environment, the knowledge may not be sufficient. Hence, the robot has to adapt to the environment and to be capable of acquiring new knowledge through the process of learning. The robot learning is essentially concerned with equipping robots with the capacity of improving their behavior over time, based on their incoming experiences. Although there have been a large number of the control methods used to solve the problem of humanoid robot walking, it is difficult to detect a specific trend. Classical robotics and also the more recent wave of humanoid and service robots still rely heavily on teleoperation or fixed behaviour-based control with very little autonomous ability to react to the environment. Among the key missing elements is the ability to create control systems that can deal with a large movement repertoire, variable speeds, constraints and most importantly, uncertainty in the real-world environment in a fast, reactive manner. We can however detect a major breakthrough that is definitely setting a new control direction based on introduction of learning and appropriate soft-computing paradigms to biped locomotion [69]. These methods have shown better results in more cases than conventional control methods. One approach of departing from teleoperation and manual ‘hard coding’ of behaviours is by learning from experience and creating appropriate adaptive control systems. A rather general approach to learning control is the framework of ‘reinforcement learning’. In this section, a novel integrated hybrid dynamic control structure for the humanoid robots is proposed, using the off-line line and on-line calculated complete model of robot mechanism. Our approach consists in departing from complete conventional control techniques by using hybrid control strategy based on model-based approach and learning by experience and creating the appropriate adaptive control systems. Hence, the first part of control algorithm represents some kind of computed torque control method as basic dynamic control method, while the second part of algorithm is modified GARIC reinforcement learning architecture [70-72] for dynamic compensation of ZMP (Zero-Moment-Point) error. The goal of this section is to propose the usage of reinforcement learning for humanoid robotics. Reinforcement learning offers one of the most general frameworks to take traditional robotics towards true autonomy and versatility.
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The reinforcement learning method proposed in this paper is based on the ActorCritic architecture. Actor-Critic methods are the natural extension of the idea of reinforcement comparison methods to Temporal Difference (TD) learning [7374]. The Actor network can be thought of as the control agent, because it implements a policy. The Actor network is part of the dynamic system as it interacts directly with the system by providing control signals for the plant. The Critic network implements the reinforcement learning part of the control system as it provides policy evaluation and can be used to perform policy improvement. This learning agent architecture has the advantage of implementing both a reinforcement learning mechanism as well as a control mechanism. For the Actor, we selected the two-layer, feedforward neural network with sigmoid hidden units and linear output units. For the Critic, neuro-fuzzy ANFIS network is proposed. The critic is trained to produce the expected sum of future reinforcement that will be observed given the current values of deviation of dynamic reactions and action. The Actor network receives the position and velocity tracking error from the biped system. It is trained via Back propagation (gradient descent) algorithm and training example provided by Critic net. The implemented algorithm was base on modified version of GARIC approach presented in paper [70]. In this paper, the external reinforcement signal was simply defined to be measure of ZMP error. Internal reinforcement signal is generated using external reinforcement signal and appropriate policy.
6.6.2 Definition of control problem and advanced control methods for humanoid robots There are various sources of control problems and various tasks and criteria that must be solved and fulfilled in order to create valid walking and other functions of humanoid robots. Previous studies of biological nature, theoretical and computer simulation have focused on the structure and selection of control algorithms according to different criteria such as energy efficiency, energy distribution along the time cycle, stability, velocity, comfort, mobility, and environment impact. Nevertheless, in addition to these aspects, it is also necessary to consider some other issues: capability of mechanical implementation due to the physical limitations of joint actuators, coping with complex highly-nonlinear dynamics and uncertainties in the model-based approach, complex nature of periodic and rhythmic gait, inclusion of learning and adaptation capabilities, computation issues, etc. The major problems associated with the analysis and control of bipedal systems is the high-order highly-coupled nonlinear dynamics and furthermore,
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the discrete changes in the dynamic phenomena due to the nature of the gait. Irrespective of the humanoid robot structure and complexity, the basic characteristic of all bipedal systems are: a) the DOF formed between the foot and the ground is unilateral and underactuated; b) the gait repeatability (symmetry) and regular interchangeability of the number of legs that are simultaneously in contact with the ground. During the walk, two different situations arise in sequence: the statically stable double-support phase in which the mechanism is supported on both feet simultaneously, and statically unstable single-support phase when only one foot of the mechanism is in contact with the ground. Thus, the locomotion mechanism changes its structure during a single walking cycle from an open to a closed kinematic chain. Also, it is well known that through the process of running the robot can be most of the time in nosupport phase. In this case, the control schemes that are successful for walking problem are not necessarily successful for the running problem. All the mentioned characteristics have to be taken into account in the synthesis of advanced control algorithms that accomplish stable, fast and reliable performance of humanoid robots. The stability issues of humanoid robot walking are the crucial point in the process of control synthesis. In view of this humanoid walking robots can be classified in three different categories. First category represents static walkers, whose motion is very slow so that the system’s stability is completely described by the normal projection of the Centre of Gravity, which only depends on the joint’s position. Second category represents dynamic walkers, biped robots with feet and actuated ankles. Postural stability of dynamic walkers depends on joint’s velocities and acceleration too. These walkers are potentially able to move in a static way provided they have large enough feet and the motion is slow. The third category represents purely dynamic walkers, robots without feet. Dynamic walkers can achieve faster walking speeds, running, stair climbing, execution of successive flips, and even walking with no actuators. In this case the support polygon during the single-support phase is reduced to a point, so that static walking is not possible. In the walk with dynamic balance, the projected centre of mass is allowed outside of the area inscribed by the feet, and the walker may essentially fall during parts of the walking gait. The control problems of dynamic walking are more complicated than in walking with static balance, but dynamic walking patterns provide higher walking speed and greater efficiency, along with more versatile walking structures. For all the mentioned categories of walking robots, the issue of stable and reliable bipedal walk is the most fundamental and yet unsolved with a high
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degree of reliability. This subject has been studied mainly through the following two classes of walking pattern generators and robot controllers. The first approach is to generate a dynamically consistent periodic walking pattern offline. It is done assuming that the models of robot and environment are available, and the kinematic and dynamic parameters of the robot model are precisely defined. On the other hand, the second approach uses limited or simplified knowledge of the system’s dynamics. However, in this case, the control relies much on the feedback control, and it is necessary to develop methods without high computation resources for real-time implementation. The rotational equilibrium of the foot is the major factor of postural instability with legged robots. The question has motivated the definition of several dynamic-based criteria for the evaluation and control of balance in biped locomotion. The most common criteria are the centre of pressure (CoP), the zero-moment point (ZMP) [75] and the foot-rotation indicator (FRI) from these criteria, the ZMP concept has gained widest acceptance and played a crucial role in solving the biped robot stability and periodic walking pattern synthesis [75]. The ZMP is defined as the point on the ground about which the sum of all the moments of the active forces equals zero. If the ZMP is within the convex hull of all contact points between the foot and the ground, the biped robot can walk. Even if the stability based on ZMP only describes contact condition between foot and the ground, ZMP based controller is mainly used in humanoid robot communities because it is known to work well experimentally. In most of cases, the trajectories of humanoid robot are designed off-line, and then controller is designed to track these trajectories. A walking pattern is generated to ensure the robot’s ZMP is all the time within the supporting foot projection. This is necessary for the biped robot to maintain dynamic balance during the walk. However, the desired ZMP of the walking pattern is different from the actual ZMP of the biped robot in a concrete walk. In order to compensate for the ZMP error, it is necessary to implement the balance control using force/torque (FIT) sensor or inclination sensor. Most research works dealing with balance control have focused on the compensation of ZMP error because the ZMP is the essential criterion of dynamic balance. In many cases, it has been assumed that the main source of errors represents the unexpected ground condition or model inaccuracies. As a matter of course, the balance controller should be robust against the inclination and model inaccuracy. For a legged robot walking on complex terrain, such as a ground consisting of soft and hard uneven parts, a statically stable walking manner is recommended. However, in the cases of soft terrain, up and down slopes or unknown environment, the walking machine may lose its stability because of the position
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planning errors and unbalanced foot forces. Hence, position control alone is not sufficient for practical walking, position-force control being thus necessary. Foot force control [76] can overcome these problems, so that foot force control is one of the ways to improve the terrain adaptability of walking robots. For example, in the direction normal to the ground, foot force has to be controlled to ensure firm foot support and uniform foot force distribution among all supporting legs; foot force in the tangential direction has to be monitored to avoid slippage. On the other hand, biological investigations suggest that human’s rhythmic walking is a consequence of combined inherent patterns and reflexive actions. The inherent dynamic pattern is rhythmic and periodic. It is considered as an optimal feedforward motion pattern acquired through the development in the typical walk environments without disturbances. The reflexive action is a rapid response due to the feedback control using sensory information. The reflexive action determines stability against unexpected events such as external disturbances or ground irregularity. A biped humanoid should not only perform rhythmic walk in a known environment but also adapt itself to real world uncertainties. In this case, the adaptability and ability of compensating external disturbances must be included in advanced control algorithms. A practical biped needs to be more like a human - capable of switching between different known gaits on familiar terrain and learning new gaits when presented with unknown terrain. In this case, even if stable trajectories are used, the existence of impulse disturbances on foot’s sole can make robot to tumble. In this sense, it seems essential to combine force control techniques with more advanced algorithms such as adaptive and learning strategies. Inherent walking patterns must be acquired through the development and refinement by repeated learning and practice as one of important properties of intelligent control of humanoid robots. Learning enables the robot to adapt to the changing conditions and is critical to achieving autonomous behavior of the robot. When the ground conditions and stability constraint are satisfied, it is desirable to select a walking pattern that requires small torque and velocity of the joint actuators. Humanoid robots are inevitably restricted to a limited amount of energy supply. It would therefore be advantageous to consider the minimum energy consumption, when cyclic movements like walking are involved. With this in mind, an important approach in research is to optimize simultaneously both the humanoid robot morphology and control, so that the walking behaviour is optimized instead of optimizing walking behavior for the given structure of humanoid robot. Optimum structures can be designed when the suitable
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components and locomotion for the robot are selected appropriately through evolution. In summary, conventional control algorithms for humanoid robots can run into some problems related to mathematical tractability, optimization, limited extendability and limited biological plausibility. The presented intelligent control techniques have a potential to overcome the mentioned constraints. There are several intelligent paradigms that are capable of solving intelligent control problems in humanoid robotics [69]. Due to their strong learning and cognitive abilities and good tolerance of uncertainty and imprecision, intelligent techniques have found wide applications in the area of advanced control of humanoid robots. Connectionist approach makes possible the learning of new gaits that are not weighted combinations of predefined biped gaits. Various types of neural networks are used for gait synthesis and control design of humanoid robots such as multilayer perceptrons, CMAC (Cerebellar Model Arithmetic Controller) networks, recurrent neural network, RBF (Radial Basis Function) networks or Hopfield networks, which are trained by supervised or unsupervised (reinforced) learning methods. The majority of the proposed control algorithms have been verified by simulation, while there were few experimental verifications on real biped and humanoid robots. Neural networks have been used as efficient tools for the synthesis and off-line and on-line adaptation of biped gait. Another important role of connectionist systems in control of humanoid robots has been the solving of static and dynamic balance during the process of walking and running on terrain with different environment characteristics. Tools and toys are often used in a manner that is composed of some repeated motion – consider hammers, bells, saws, rattles, drummers, brushes, etc. Therefore, strategies for the oscillatory control of movements of a humanoid robot are imperative, especially if they result on natural movements, which is the case of neural oscillators, since they track the natural frequency of the dynamic system to which they are coupled. All these techniques may be incorporated in advanced and sophisticated control systems of humanoid robots that were inspired in general by biological designs and neurobiological principles [69]. Some researchers used the fuzzy logic [69] as the methodology for biped gait synthesis and control of biped walking. Fuzzy logic was used mainly as part of control systems on the executive control level, for generating and tuning PID gains, fuzzy control supervising, direct fuzzy control by supervised and reinforcement error signals.
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It is considered that GA can be efficiently applied for trajectory generation of the biped natural motion on the basis of energy optimization, as well as for walking control of biped robots and for generation of behaviour-based control of these systems [69]. The main problem of GA application in humanoid robotics represents the coping with the reduction of GA optimization process in real time.. Recently, reinforcement learning has attracted attention as a learning method for studying movement planning and control [77-79]. Reinforcement learning is a kind of learning algorithm between supervised and unsupervised learning algorithms that is based on Markov decision process (MDP). Reinforcement learning concept that is based on trial and error methodology and constant evaluation of performance in constant interaction with environment. Reinforcement learning typically requires an unambiguous representation of states and actions and the existence of a scalar reward function. For a given state, the most traditional of these implementations would take an action, observe a reward (positive reinforcement), or punishment (negative reinforcement), update the value function, and select as the new control output the action with the highest expected value in each state (for a greedy policy evaluation). Updating of value function and controls is repeated until convergence of the value function and/or the policy. This procedure is usually summarized under “value update – policy improvement” iterations. In many situations the success or failure of the controller is determined not only by one action but by a succession of actions. The learning algorithm must thus reward each action accordingly. This is referred to as the problem of delayed reward. There are two basic methods that are very successful in solving this problem, TD learning and Q learning. Both methods build a state space value function that determines how close each state is to success or failure. Whenever the controller outputs an action, the system moves from one state to another. The controller parameters are then updated in the direction that increases the state value function. For the solution of large-scale MDPs or continuous state and action spaces, it’s impossible for reinforcement learning agent to go through all the states and actions. In order to realize the optimal approximation for value functions of continuous states and actions respectively, therefore, learning agent must have generalization ability. In other words, such an agent should be able to utilize finite learning experience to acquire and express a good knowledge of a largescale space effectively. How to design a function approximator with abilities of high generalization and computation efficiency has become a key problem for
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the research field of reinforcement learning. Using prior knowledge about the desired motion can greatly simplify controller synthesis. Imitation-based learning or learning from demonstration allow for policy search to focus only on the areas of the search space that pertain to the task at hand. In area of humanoid robotics, there are several approaches of reinforcement learning [71-72], [80-86] with additional demands and requirements because high dimensionality of the control problem. Furthermore, Benbrahim and Franklin showed the potential of these methods to scale into the domain of humanoid robotics [71] There are direct (model-free) and model-based reinforcement learning. The direct approach to RL is to apply policy search directly without learning a model of the system. It was shown that model-based reinforcement learning is more data efficient than direct reinforcement learning. Also, it was concluded that model-based reinforcement learning finds better trajectories, plans and policies, and handles changing goals more efficiently. On the other hand, reported that a model-based approach to reinforcement learning is able to accomplish given tasks much faster than without using knowledge of the environment. In paper [82] a model-based reinforcement learning algorithm for biped walking in which the robot learns to appropriately place the swing leg was proposed. This decision is based on a learned model of the Poincare map of the periodic walking pattern. The model maps from a state at the middle of a step and foot placement to a state at next middle of a step. Actor-Critic algorithms of reinforcement learning has a great potential in control of biped robots. This algorithm converges to the nearest local minimum of the cost function with respect to the Fisher in formation metric under suitable conditions. It offers a promising route for the development of reinforcement learning for truly high-dimensionally continuous state-action systems.
6.6.3 The model of the system 6.6.3.1 Model of the robot’s mechanism Biped locomotion mechanisms represent generally branched kinematic chains interconnected with spherical or cylindrical joints. During the motion, some kinematic chains in their interaction with the environment transform from open to closed type of chain. The kinematic scheme of the biped locomotion mechanism whose spatial model will be considered in this paper, is shown in Fig. 6.25. The model will be used to synthesize dynamic control of the
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locomotion mechanism and verify the research results obtained in simulation experiments. The mechanism possesses 18 powered DOFs, designated by the numbers 1-18, and two unpowered DOFs (1’ and 2’) for the footpad rotation about the axes passing through the instantaneous ZMP position. Thus, the mechanism considered has in total n=20 DOFs of motion. Dynamic model of the mechanism has been formed using the relations known from Newton’s rigid body dynamics. For this purpose, the first link in the branched chain, representing the supporting foot, is adopted as the basic link of the mechanism. Taking into account dynamic coupling between particular parts (branches) of the mechanism chain, one can derive a relation that describes the overall dynamic model of the locomotion mechanism in a vector form [75]: P + J T (q ) ⋅ F = H (q) qɺɺ + h(q, qɺ ).
(6.65)
where: P ∈ R n ×1 is the vector of driving torques at the humanoid robot joints; F ∈ R6 ×1 is the vector of external forces and moments acting at the particular points of the mechanism; H ∈ R n × n is the square matrix that describes ‘full’ inertia matrix of the mechanism shown in Fig. 25; h ∈ R n ×1 is the vector of gravitational, centrifugal and Coriolis moments acting at n mechanism joints; J ∈ R 6×n is the corresponding Jacobi matrix of the system; n = 20 , is the total number of DOFs (Fig. 6.25). The vector F is of special importance in the calculation of the model (6.65). It represents the vector of forces and moments of ground reaction at the moment when the foot of the free (unconstrained) leg is forming contact with the ground, i.e. at the moment when the weight is transferred from one foot to the other. It is necessary to make difference between the so-called supporting (constrained) and free (unconstrained) foot, the latter becoming constrained at the moment of making contact with the ground. 6.6.3.2 Definition of control criteria In the synthesis of control for biped mechanism gait it is necessary to satisfy certain natural principles. The control ought to satisfy the following criteria: (i) accuracy of tracking the desired trajectories of the mechanism joints, (ii) maintenance of dynamic balance of the mechanism during the motion, (iii) minimization of the impact arising at the moment of contact of the free foot and
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Fig. 6.25 Model of the biped locomotion mechanism with 18 active and 2 passive DOFs (a) kinematical scheme of the mechanism, (b) dynamic model of the environment
the ground during the gait, (iv) minimization of dynamic loads at the robot joints, and (v) realization of anthropomorphic characteristics of the gait. Fulfillment of criterion (i) enables the realization of a desired mode of motion, walk repeatability and avoidance of potential obstacles. To satisfy criterion (ii) it means to have a dynamically balanced walk. Fulfillment of criterion (iii) decreases the impact effects to the overall system at the moment when the unconstrained leg foot strikes the ground. Fulfillment of criterion (iv) is needed
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for the purpose of minimizing dynamic loads at the robotic joints, which is especially important for the joints bearing the highest load during the walk, e.g. the hip. Criterion (v) is related to the quality of artificial (human like) gait. Walk of a physically healthy human represents a balanced and harmonious sequence of movements, with minimal displacements of the position of its mass center about an imaginary central position corresponding to the human’s posture at rest. 6.6.3.3 Gait phases and indicator of dynamic balance The robot’s bipedal gait consists of several phases that are periodically repeated [75]. Hence, depending on whether the system is supported on one or both legs, two macro-phases can be distinguished, viz.: (i) single-support phase (SSP) and (ii) double-support phase (DSP). Double-support phase has two micro-phases: (i) weight acceptance phase (WAP) or heel strike, and (ii) weight support phase (WSP). Fig. 6.26 illustrates these gait phases, with the projections of the contours of the right (RF) and left (LF) robot foot on the ground surface, whereby the shaded areas represent the zones of direct contact with the ground surface. While walking, the biped is constantly in the state of dynamic balance. In the literature from this branch the terms “dynamically balanced walk” and “gait stability” are used very often in the same context. In this paper, we have used the term “dynamic balance” of robot mechanism in the sense that does not mean the stable system. Dynamic balance of biped mechanism represents its dynamic state in which there is no overturning around the edge of the constrained foot during the walk. Because of that, the term stable gait is used unjustified very often for the gait in which the system is attempting to avoid overturning by preserving its dynamic balance. The indicator of the degree of dynamic balance is the ZMP, i.e. its relative position with respect to the footprint of the supporting foot of the locomotion mechanism. The ZMP is defined [75] as the specific point under the robotic mechanism foot at which the effect of all the forces acting on the mechanism chain can be replaced by a unique force and all the rotation moments about the x and y axes are equal zero. Figs. 27a and 28b show details related to the determination of ZMP position and its motion in a dynamically balanced gait. The ZMP position is calculated based on measuring reaction forces [75] under the robot foot. Force sensors are usually placed on the foot sole in the arrangement shown in Fig. 6.27a. Sensors’ positions are defined by the geometric quantities l1, l2 and l3. If the point 0zmp is assumed as the nominal
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ZMP position (Fig. 6.27a), then one can use the following eqs. to determine the relative ZMP position with respect to its nominal:
Fig. 6.26 Phases of biped gait
l3 l ( F2 + F4 ) − ( F20 + F40 ) − 3 ( F1 + F3 ) − ( F10 + F30 ) , 2 2 = l2 ( F3 + F4 ) − ( F30 + F40 ) − l1 ( F1 + F2 ) − ( F10 + F20 ) ,
∆M x( zmp ) = ∆M y( zmp ) 4
Fr( z ) =
∑ i =1
Fi , ∆x ( zmp ) =
−∆M y( zmp ) Fr( z )
, ∆y ( zmp ) =
(6.66)
∆M x( zmp ) Fr( z )
where Fi and Fi 0 , i=1,...,4, are the measured and nominal values of the ground reaction force; ∆M x( zmp ) and ∆M y( zmp ) are deviations of the moments of ground reaction forces around the axes passed through the 0zmp ; Fr ( z ) is the resultant force of ground reaction in the vertical z-direction, while ∆x ( zmp ) and ∆y ( zmp ) are the displacements of ZMP position from its nominal 0zmp . The
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deviations ∆x ( zmp ) and ∆y ( zmp ) of the ZMP position from its nominal position in x- and y-direction are calculated by (6.66). The instantaneous position of ZMP is the best indicator of dynamic balance of the robot mechanism. In Fig. 6.27b are illustrated certain areas (Z0, Z1 and Z2), the so-called safe zones of dynamic balance of the locomotion mechanism. The ZMP position inside these “safety areas” ensures a dynamically balanced gait of the mechanism whereas its position outside these zones indicates the state of loosing the balance of the overall mechanism, and the possibility of its overturning. The quality of robot balance control can be measured by the success of keeping the ZMP trajectory within the mechanism support polygon (Fig. 6.27b). 6.6.4 Hybrid integrated dynamic control algorithm with reinforcement structure Humans, on the other hand, use a variety of complex strategies in order to maintain balance according to the amount of force applied to the body. For example, for relatively weak perturbations, the impact is absorbed by the ankle joint; the posture of the upper part of the body remains unchanged. When the impact is larger, the hip and knee joints are used, and the whole body is used to absorb the impact. If the impact is even stronger, the human will alter their gait and step out one or two steps to counteract the additional linear and angular momentum. Evidence from biomechanical studies indicates that a number of different strategies are prepared in advance, and the most appropriate response motion is launched when the perturbation occurs. This means that overall balance is preserved not only through feedback control, but the feedforward motion is also changed according to the current state of the body. These two balance compensation strategies vastly increase the flexibility and robustness of the human gait. In order to enable biped humanoids to approach the same level of performance and stability as humans, a balancing controller that is capable of “retuning” the upcoming balancing motion in realtime is needed. Biped locomotion mechanism represents a nonlinear multivariable system with several inputs and several outputs. Bearing in mind the control criteria, it is necessary to control the following variables: positions and velocities of the robot joints, ZMP position, contact force at the instant of foot striking the ground (i.e. at the moment of weight acceptance by the other leg). In accordance with the control task, we propose the application of the algorithm of the so-called
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integrated dynamic control based on the dynamic model of the complete system. Here we assume the following: (i) the model (6.65) describes sufficiently well the behaviour of the system presented in Fig. 6.25; (ii) desired (nominal) trajectory of the mechanism performing a dynamically balanced gait is known. (It is determined off-line by some of the known mathematical methods or calculated in real time at some of the higher robot control levels.); (iii) geometric and dynamic parameters of the mechanism and driving units are known and constant. These assumptions can be taken as conditionally valid, the rationale being as follows: As the system elements are rigid bodies of unchangeable geometrical shape, the parameters of the mechanism can be determined with a satisfactory accuracy.
Fig. 6.27 Zero-Moment Point: a) Legs of “Toyota” humanoid robot; General arrangement of force sensors in determining the ZMP position; b) Zones of possible positions of ZMP when the robot is in the state of dynamic balance
Based on the above assumptions we present in Fig. 6.28 a block-diagram of the dynamic controller for biped locomotion mechanism proposed in this work. It involves three feedback loops: (i) basic dynamic controller for trajectory tracking, (ii) dynamic reaction feedback at the ZMP based on reinforcement learning structure, (iii) impact-force feedback at the foot of the free (unconstrained) leg. The synthesized dynamic controller was designed on the basis of the centralized model. The vector of driving moments Pˆ represents the
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sum of the driving moments, Pˆ1 , Pˆ2 and Pˆ3 . The torque Pˆ1 is determined so to ensure precise tracking of the robot’s position and velocity in the space of joints coordinates. The driving torque Pˆ2 is calculated with the aim of correcting the current ZMP position with respect to its nominal. The torques Pˆ3 are determined with the objective of diminishing the amplitude of the force F that arises at the instant when the free foot touches the ground. The vector of driving torques Pˆ represents the output control vector. In the text below we describe in detail the control algorithms based on the mentioned feedbacks (i) – (iii).
Fig. 6.28 Block-scheme of the integrated dynamic control of biped with three feedback loops
6.6.4.1 Dynamic controller of trajectory tracking The controller of tracking nominal trajectory of the locomotion mechanism has to ensure the realization of a desired motion of the humanoid robot and avoiding fixed obstacles on its way. In [75], it has been demonstrated how local PD or PID controllers of biped locomotion robots are being designed. The proposed dynamic control law ha the following form:
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Pˆ1 = Hˆ (q, xcf ) ⋅ [ qɺɺ0 − KV (qɺ − qɺ0 ) − K P (q − q0 ) ] + hˆ(q, xcf , qɺ , xɺcf , ɺɺ xcf ) − Jˆ T (q, xcf ) F .
(6.67)
where Hˆ , hˆ and Jˆ a are the corresponding estimated values of the inertia matrix, vector of gravitational, centrifugal and Coriolis forces and moments and Jacobian matrix from the model (6.65); xcf ∈ R 6 ×1 , xɺcf and ɺxɺcf are the vectors of position, velocity and acceleration of the constrained foot. The matrices K P ∈ R n × n and KV ∈ R n × n are the corresponding matrices of position and velocity gains of the controller. The gain matrices Kp and Kv can be chosen in the diagonal form by which the system is decoupled into n independent subsystems. This control model is based on centralized dynamic model of biped mechanism. 6.6.4.2 Compensator of dynamic reactions based on reinforcement learning structure In the sense of mechanics, biped locomotion mechanism represents an inverted multilink pendulum. In the presence of elasticity in the system and external environment factors, the mechanism’s motion causes dynamic reactions at the robot supporting foot. Thus, the state of dynamic balance of the locomotion mechanism changes accordingly. For this reason it is essential to introduce in the control synthesis the dynamic reaction feedback at ZMP. Relations (6.66) define the relationship between the deviations of ZMP position ( ∆x ( zmp ) , ∆y ( zmp ) ) from the nominal position 0 zmp in the motion directions x and y and the corresponding dynamic reactions M x( zmp ) and M y( zmp ) acting about the mutually orthogonal axes that pass through the point 0 zmp (Fig. 6.27a). M x( zmp ) ∈ R1×1 and M y( zmp ) ∈ R1×1 represent the moments that tend to overturn the robotic mechanism, i.e. to produce its rotation about the mentioned rotation axes (axes of the joints 1’ and 2’ in Fig. 6.25) On the basis of the above the reinforcement control algorithm is defined with respect to the dynamic reaction of the support at ZMP. Reinforcement Actor-Critic Learning Structure
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This subsection describes the learning architecture that was developed to enable biped walking. A powerful learning architecture should be able to take advantage of any available knowledge. The proposed Reinforcement learning structure is based on Actor-Critic Methods [71]. Actor-Critic methods are temporal difference (TD) methods that have a separate memory structure to explicitly represent the control policy independent of the value function. In this case, control policy represents policy structure known as Actor with aim to select the best control actions. Exactly, the control policy in this case, represents the set of control algorithms with different control parameters. The input to control policy is state of the system, while the output is control action (signal). It searches the action space using a Stochastic Real Valued (SRV) unit at the output. The unit’s action uses a Gaussian random number generator. The estimated value function represents a Critic, because it criticizes the control actions made by the actor. Typically, the critic is a statevalue function that takes the form of TD error necessary for learning. TD error depends also from reward signal, obtained from environment as result of control action. The TD Error is scalar signal that drives all learning in both actor and critic (Fig. 6.29).
Fig. 6.29 The Actor-Critic Architecture
Practically, in proposed humanoid robot control design, it is synthesized the new modified version of GARIC reinforcement learning structure [70]. The reinforcement control algorithm is defined with respect to the dynamic reaction of the support at ZMP, not with respect to the state of the system. In this case external reinforcement signal (reward) R is defined according to values of ZMP error. If ZMP error is greater then chosen limit, external reinforcement signal is set to value 1, otherwise this signal is set to zero. Proposed learning structure is based on two networks: AEN (Action Evaluation Network) - CRITIC and ASN(Action Selection Network) - ACTOR.
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AEN network maps position and velocity tracking errors and external reinforcement signal R in scalar value which represent the quality of given control task. The output scalar value of AEN is important for calculation of internal reinforcement signal Rˆ . AEN constantly estimate internal reinforcement based on tracking errors and value of reward. AEN is standard 2layer feedforward neural network (perceptron) with one hidden layer. The activation function in hidden layer is sigmoid, while in the output layer there is only one neuron with linear function. The input layer has a bias neuron. The output scalar value v is calculated based on product of set C of weighting factors and values of neurons in hidden later plus product of set A of weighting factors and input values and bias member. There are also one more set of weighting factors B between input layer and hidden layer. The number of neurons on hidden later is determined as 5. The most important function of AEN is evaluation of TD error, exactly internal reinforcement. The internal reinforcement is defined as TD(0) error defined by the following eq.:
Rˆ (t+1)=R(t)+ γ v(t+1)-v(t)
(6.68)
where γ is a discount coefficient between 0 and 1 (in this case γ is set to 0.9). ASN (action selection network) maps the deviation of dynamic reactions in recommended control torque. The structure of ASN is represented by The ANFIS - Sugeno-type adaptive neural fuzzy inference systems. There are five layers: input layer. Antecedent part with fuzzification, rule layer, consequent layer, output layer with defuzzification. This system is based on fuzzy rule base generated by expert kno0wledge with 25 rules. The partition of input variables (deviation of dynamic reactions) are defined by 5 linguistic variables: NEGATIVE BIG, NEGATIVE SMALL, ZERO, POSITIVE SMALL and POSITIVE BIG. The member functions are chosen as triangular forms. SAM (Stochastic action modifier) uses the recommended control torque from ASN and internal reinforcement signal to produce final commanded control torque Pdr. It is defined by Gaussian random function where recommended control torque is mean, while standard deviation is defined by following eq.: σ ( Rˆ (t+1))=1-exp-| Rˆ (t+1)| (6.69) Once the system has learned an optimal policy, the standard deviation of the Gaussian converges toward zero, thus eliminating the randomness of the output. The learning process for AEN (tuning of three set of weighting factors A, B, C) is accomplished by step changes calculated by products of internal
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reinforcement, learning constant and appropriate input values from previous layers. The learning process for ASN (tuning of antecedent and consequent layers of ANFIS) is accomplished by gradient step changes (back propagation algorithms) defined by scalar output values of AEN, internal reinforcement signal, learning constants and current recommended control torques. The control torques Pdr obtained as output of actor structure cannot be generated at the joints 1’ and 2’ since they are unpowered (passive) joints. Hence, the control action has to be ‘displaced’ to the other (powered) joints of the mechanism chain. Since the vector of deviation of dynamic reactions has two components about the mutually orthogonal axes x and y, at least two different active joints have to be used to compensate for these dynamic reactions. Considering the model of locomotion mechanism presented in Fig. 25, the compensation was carried out using the following mechanism joints: 1, 6 and 14 to compensate for the dynamic reactions about the x-axis, and 2, 4 and 13 to compensate for the moments about the y-axis. Thus, the joints of ankle, hip and waist were taken into consideration. Finally, the vector of compensation torques Pˆ2 (Fig. 6.28) was calculated on the basis of the vector of the moments Pdr whereby it has to be borne in mind how many ‘compensational joints’ have really been engaged. If only two joints (e.g. the ankle joints 1 and 2) are engaged, then: Pˆ2 (1) = Pdr (1), Pˆ2 (2) = Pdr (2).
(6.70)
In the case when compensation of ground dynamic reactions is performed using all six proposed joints, the compensation torques Pdr are uniformly distributed over all of the selected joints, to load uniformly the actuators. Then the following relations hold: 1 Pˆ2 (1) = Pˆ2 (6) = Pˆ2 (14) = Pdr (1), 3 1 Pˆ2 (2) = Pˆ2 (4) = Pˆ2 (13) = Pdr (2). 3
(6.71)
In nature, biological systems use simultaneously a large number of joints for correcting their balance. In this work, for the purpose of verifying the control algorithm, the choice was restricted to the mentioned six joints: 1, 2, 4, 6, 13 and 14 (Fig. 6.25). Compensation of ground dynamic reactions is always carried out at the supporting leg when the locomotion mechanism is in the swing phase,
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whereas in the double-support phase it is necessary to engage the corresponding pairs of joints (ankle, knee, hip) of both legs. 6.6.4.3 Impact-force controller The role of impact-force controller in the proposed control structure (Fig. 6.28) is to counteract the ground reaction force that appears when the locomotion mechanism foot strikes the ground (heel strike). At that moment, the forces (and moments) that destabilize the system are transferred onto all of its links. Hence, a common practice is to introduce passive and/or active compliance into the system. Elastic footpad attached to the rigid foot sole is commonly applied as a passive compliance. Active compliance assumes the involvement of an active force or moment at the mechanism joint, generated by the contact-force controller. With a human being, the load forces arising at the joints of the legs and hip during a walk are absorbed by the appropriate contraction of leg muscles. With robots, the role of muscles is played by the actuators at the mechanism joints. In this work we propose the introduction of a force controller to control in a direct way the contact forces and moments at the landing (freeleg) foot. The magnitude of a ground reaction force is equal to the magnitude of a strike force. In the sequel we will assume that the vector F ∈ R 6 ×1 represents both the forces and moments of ground reaction in the case when the foot is in contact with environment. Thus the control law can be defined in the form of a PI-regulator as: t
∫
Fc = F0 − Q (∆F ) ⋅ dt , t ∈ (0, T ], ∆F = F − F0 , 0 T
F0 = F0 x F0 y F0 z M 0 x M 0 y M 0 z , T
F = Fx Fy Fz M x M y M z ,
(6.72)
t
∫
Q(∆F ) = K PF ∆F + K IF ∆F ⋅ dt , t ∈ (0, T ]. 0
where Fc ∈ R 6 ×1 is a vector of the so called generalized forces. In order to compensate deviations of ground reactions ∆F the generalized forces Fc should to act at the mass center of the last link of the swinging leg (i.e. at the mass center of the landing foot). In practice, this control action cannot be realized
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physically in the mass center of the constrained foot but indirectly using the actuators of robot’s mechanism; F0 ∈ R 6 ×1 and F ∈ R 6 ×1 represent the respective vectors of nominal and measured values of forces and moments of the ground reaction along, i.e. about three coordinate axes (x, y and z); ∆F ∈ R 6 ×1 is the vector of deviation of forces (moments) F of the ground reactions from their nominal values F0 calculated for the nominal robot’s motion q0 ; K PF ∈ R 6 × 6 and K IF ∈ R 6 × 6 are the square matrices of proportional and integral gains of the designed PI-regulator. If the gain matrices are adopted in the diagonal form K PF = diag{k pfj }, K IF = diag{kifj }, j = 1,6 , k pfj , kifj > 0 the characteristic
polynomial is obtained in the form f j ( s ) = s 2 + k pfj s + kifj . Finally, since the generalized forces Fc cannot be realized in a direct way, the control torques Pˆ3 (Fig. 6.28) are determined from the relation: Pˆ3 = − Jˆ T (q, xcf ) ⋅ Fc .
(6.73)
where Jˆ (q, xcf ) is the Jacobian matrix. It should be pointed out that the contact force feedback serves to correct dynamic performance of the robotic system, that is to ensure a smoother walk and minimize the effect of ground reaction of pulsed character. 6.6.4.4 Conflict between controllers The conflict between the compensation actions at the particular joints as a control problem with biped robotic systems has been known for many years , whereby the control task considered was divided into two subtasks: (i) tracking nominal trajectories of the mechanism joints and (ii) maintaining the mechanism dynamic balance. The subtask (i) performed all the mechanism joints, while subtask (ii) was allocated to only one (e.g. ankle) joint In the instance of endangered dynamic balance, the joint reacted so to return the ZMP to its nominal position. It is obvious that the additional requirement imposed on the ankle joint spoiled the quality of itsnominal trajectory tracking, but the maintaining of dynamic balance (preventing mechanism’s falling) is a priority task. The solving of this problem will be briefly explained in this subsection.
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The proposed controllers (Fig.6.28), operate independently from each other. These controllers generate the corresponding compensation torques Pˆ1 , Pˆ2 , Pˆ3 to be realized by the actuators at the particular joints of the mechanism. In a general case, in the presence of large disturbances (unpredictable changes of the environment, sudden impacts, model inaccuracies, etc.) acting upon the system, a conflict can arise between the calculated compensation torques. This conflict exerts a negative effect on the state of the system. There are various methods for overcoming the problem of possible conflict between the particular controllers implemented with robotic biped systems. It is especially worth mentioning the novel whole-body cooperative balancing method with modification of predesigned motion trajectories [86]. By implementing this advance method, the authors showed through a simulation example the way how to control simultaneously the joints’ positions and dynamic balance of a system in real-time with no conflict between the applied controllers. The essence of the applied method is in determining and manipulating the position of the center of gravity (COG). The manipulation was performed so to ensure the system’s balance in an anthropomorphic way, by calculating new reference positions qref of the mechanism’s joint angles instead of their pre-defined values. The important factors to ‘balance’ legged motions are [86]: (i) modification of the pre-designed posture trajectory in order to conserve the contact condition, and (ii) robust compensation of the deviation of the real posture from the pre-defined one. The considered method can also be used to calculate the control torques determined by relation (6.67). In the case of large perturbations acting on the robotic system, the proposed compensator of dynamic reactions Pdr can produce significant magnitudes of the compensating torques Pˆ2 , to ensure desired dynamic reactions on the biped locomotion mechanism. As a consequence, these torques can spoil the accuracy of the joints’ positions. In order to overcome this control conflict, the modified values of the reference positions qref should be used in eq. (6.67) instead of the pre-designed values of the nominal trajectory q0 . By the real-time modification of the predefined trajectory in the course of the mechanism’s motion, a possible conflict between the controllers Pˆ1 and Pˆ2 is avoided. The way of a suitable modification was explained in detail in the paper by Sugihara and Nakamura [86]. Using the reference trajectory qref , instead of the predefined nominal one
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q0 , less energy is consumed by the robot actuators to realize a desired motion, whereby the dynamic performance of the system is ensured.
6.6.5 Simulation studies The proposed control methods presented in Section 6.6.4 were analyzed on the basis of numerical data obtained by simulation of the closed-loop model of the locomotion mechanism. Total mass of the mechanism was m=[70 kg]. Its geometric and dynamic parameters were taken from the literature [87]. Simulation examples concerned characteristic patterns of artificial gait in which the mechanism makes a half-step of the length l=[0.40 m] in the time period T=[0.75 s]. Nominal motion of the robot mechanism (defined by its kinematic scheme in Fig. 6.25) walking on the ideally flat, horizontal surface during a halfstep phase of the gait is shown in Fig. 6.30.
Fig. 6.30 Computer animation of the stick-model of biped locomotion mechanism during the half-step
The half-step is repeated with this period T, whereby the gait phases presented in Fig. 6.26 alternate regularly. Nominal trajectories at robot joints were synthesized for the gait on the horizontal ground surface. Nominal angles at the mechanism joints (Fig. 6.31) and the corresponding angular velocities and accelerations were determined by the semi-inverse method [88] Initial (starting) conditions of the simulation examples (initial deviations of joints’ angles and
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joints’ angular velocities) are chosen to be the same in all considered simulation experiments. They were imposed by a definition of the following deviation vectors: ∆q = [0 0 0 0.051 0 − 0.023 0.034 0 − 0.02 0 0 0 0 0 0 0 0 0] [ rad ] T
and
∆qɺ = [0 0 ... 0] [rad / s ]. T
Simulation results were analyzed on the time interval T corresponding to the duration of one half-step of the locomotion mechanism in the swing phase (Fig. 6.26). The analysis of the efficiency of the developed dynamic controllers in the realization of dynamically balanced motion shows that the most delicate instances are the single-support phase (swing phase) and the free foot striking the ground. As the robot’s dynamic behavior in these time intervals is of special importance for control, simulation examples were selected so to encompass these critical phases.
Fig. 6.31 Nominal trajectory of robot mechanism – joint angles and joint angular velocities: a) small constrained leg joints 1-6, b) swing leg joints 7-12 and robot trunk joints 13-14
In the simulation example it was shown how the Basic dynamic controller is able to compensate the deviations of the dynamic reactions even in the presence of uncertainty of the ground surface inclination. Finally, the one more example proves that the impact controller together with the compensator of dynamic
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reactions successfully controls the impact force at the landing foot as well as keeps well the dynamic balance of the robotic system. In the first simulation example, the assigned initial deviations of particular angles ∆q at the mechanism joints were assumed to be as large as it was previously emphasized. Constant, small inclinations of the ground surface in the sagittal plane g1=3° as well as in the frontal plane g2=2° were introduced as an additional perturbation, too. Thus the simulation experiment dealt with the real case of walking on a quasi-horizontal ground. The issue of interest was the robot’s behaviour in the swing phase (Fig. 6.26), when the robot relies upon the ground by its rigid foot. The other one (free or swinging foot) was above the ground. Here, the control was realized in two cases, using: (i) only the basic dynamic controller described by relation (6.67) to track the desired trajectory with position-velocity feedback (Fig. 6.28), and (ii) hybrid control, consisting of the trajectory tracking and reinforcement learning compensator of dynamic reactions of the ground at the ZMP. In Figs. 6.32 and 6.33 are compared the results obtained by applying the controllers in cases (i) (without learning) and (ii) (with learning). It is evident that the ZMP relative positions (with respect to the coordinate system attached to the point 0zmp, Fig. 6.27a) are at the edges of supporting foot projection, and even outside it, whereas in the case (ii) they are mainly within the ‘safety zone’ Z0 (shown in Fig. 6.27b). Thus, it can be concluded that in the absence of the reinforcement learning feedback with respect to the ground reactions at the ZMP it is not generally possible to ensure (guarantee) dynamic balance of a locomotion mechanism during its motion. This comes out from the fact that the pre-designed (nominal) trajectory was synthesized without taking into account possible deviations of the surface inclination on which biped walks from an ideally horizontal plane. Therefore, the ground surface inclination influences the system’s balance as an external stochastic perturbation. The corresponding deviations (errors) ∆qi and ∆qɺi of the actual angles and angular velocities at the robot joints from their reference values, in the case when the controller of tracking desired trajectory was applied, are presented in Figs. 6.34 and 6.35. The deviations ∆qi and ∆qɺi converge to zero values in the given time interval T=0.75 [s]. It means that the controller employed ensures good tracking of the desired trajectory. Besides, the corresponding control torques at the robot joints Pˆ1 and Pˆ2 (Fig. 6.28), are presented in Fig. 6.36. The application of Pˆ and Pˆ ensures a desired convergence of joint positions and 1
2
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ZMP position to the corresponding reference positions as well as a dynamic balance of the locomotion mechanism as it is illustrated in Figs. 6.34 and 6.35. Simulation results shown in Figs. 6.34 and 6.35 confirm the efficiency of the proposed basic dynamic controller applied in controlling robot’s gait on the ground surface whose inclination deviates from an ideal horizontal plane by up to g1 (g2) ≤ 10 .
Fig. 6.32 Error of ZMP in x-direction
Fig. 6.33 Error of ZMP in y-direction
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Fig. 6.34 Convergence of the errors of tracking nominal angles
Fig. 6.35 Convergence of the errors of tracking nominal angular velocities
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Fig. 6.36 Control torques
In Fig. 6.37 value of internal reinforcement through process of walking is presented. It is clear that task of walking within desired ZMP tracking error limits are achieved.
Fig. 6.37 Internal reinforcement through process of walking
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The last simulation example was concerned with the quality of controlling impact forces in the course of robot motion. The attention was focused on the instant when the mechanism’s free (landing) foot comes in contact with its dynamic environment, whereby it is necessary to minimize the effect of the impact forces on the overall mechanism. Thus, the so-called weight-acceptance phase of the gait (Fig. 6.26) is of interest. It is the instant when the load is transferred from the supporting leg to the other leg. To realize control action, additional contact-force feedback was introduced. Having in mind that both feet are equipped with force sensors (Fig. 6.27), to measure force components in all three directions of robot motion (x, y, z), it is also possible to calculate the corresponding reaction moments acting on the foot about these axes. The additional controller was synthesized according to the relations (6.72)-(6.73).
Fig. 6.38 Comparative indicators of controlling the contact force between the free foot and the ground. Forces acting between the ground and a) foot of the mechanism’s supporting leg with as well without contact-force control in three coordinate directions
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At the end of the swing phase, the free foot strikes the ground at a certain rate and acceleration, which is accompanied by the vertical impact presented in Fig. 6.38. By using the contact-force controller it was possible to reduce significantly the amplitude of this force. In Fig. 6.39 are also presented the considered pressure forces at the supporting foot on the ground, with and without contactforce control. Fig. 6.39 shows the indicators of the state of the mechanism’s
Fig. 6.39 Deviations of the ZMP position caused by the contact force arising at the moment of free-foot striking the ground (case of existence of the impact-force feedback). Corresponding driving torques
dynamic balance before and after the moment when the free foot strikes the ground, using the contact-force control. At the moment of strike, there appears a very pronounced discontinuity in the ZMP position. However, the ZMP comes back very quickly, and after only several tens of millisecond it is at the border of
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the safety zone. By an appropriate choice of nominal motion it is possible to reduce significantly the nominal of the ground impact. Thanks to the action of the ‘Basic dynamic controller’ (Fig. 6.28), the mechanism will preserve dynamic balance at the moment of foot striking the ground. Fig. 6.39 shows also the values of control torques Pˆ (Fig. 6.28) that were used to obtain results presented in Fig. 6.38.
6.7 Conclusion The key features of this chapter dedicated to intelligent control algorithms for robotic contact tasks are represented through several issues. The first one is the theoretical background of intelligent control synthesis and a comprehensive survey of most up-to-date developments in the area of intelligent control of robotic contact tasks. Special attention is paid to the role of learning as one of the main intelligent capabilities of the control algorithms for robotic contact tasks. Robot learning refers to the process of acquiring a sensory-motor control strategy for a particular contact task through a training process by trial and error. The goal of learning control can be defined as the need to acquire a taskdependent control algorithm that maps the state variables of the system and its environment into an appropriate control signal (action). Hence the aim is to find a function that is appropriate for a given desired behavior and robotic system. In the second key focus is the learning principle and it is shown through the application of new, advanced learning algorithms for robotic contact tasks using nonrecurrent and connectionist structures. The main concern is the development of learning control algorithms as an upgrade of conventional non-learning control laws for robotic compliance tasks (algorithms for stabilization of robot motion and stabilization of robot interaction force). In view of the important influence of robot environment, a new, comprehensive learning approach is based on the simultaneous classification of robot environment and learning of robot uncertainties. The proposed comprehensive algorithms include the synthesis and application of two newly proposed classifiers: pure perceptron classifier and wavelet network classifier. Simulation studies validate the efficiency of the proposed connectionist control algorithms. In order to enhance capabilities of learning control algorithm, new hybrid intelligent (geneticconnectionist) control algorithms are synthesized based on genetic tuning of local feedback gains. The third focus is on the special application of connectionist reactive control for a robotic assembly task by soft sensored grippers. In this example, the
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problem of efficient control in assembly tasks is handled by industrial robots with the gripper having soft and sensored fingers instead of using the wellknown methods of reactive control based on force sensors at the manipulator wrist. A new method based on learning strategy by neural networks and pressure sensors on the soft fingers was proposed. The problem of learning reactive control was exemplified by the “peg-in-hole” task. Experiments were carried out using the industrial robot ASEA Irb-L6 with the Belgrade-USC-IIS Robot hand Model II whose fingertips are equipped with pressure sensors. Finally, the fourth focus is on the application of intelligent control techniques for humanoid walking problems. In this sense, the special hybrid integrated dynamic control of humanoid locomotion mechanisms are synthesized. The proposed structure of control involves two feedback loops: model-based dynamic controller including impact-force controller of the robotic mechanism joints and reinforcement learning feedback around Zero-Moment Point. The proposed reinforcement learning structure based on two networks: AEN (neural network) and ASN (neuro-fuzzy network) represent an efficient learning tool for compensation of reactions at the ZMP. The feedback loops were formed with respect to position and velocity of the mechanism joints. Basic dynamic controller was designed with the aim of ensuring precise tracking of the given motion and maintaining dynamic balance of the humanoid mechanism. Its application ensures the desired precision of robot’s motion, motion stability in the sense of maintaining dynamic balance of the locomotion mechanism. The primary control structure was supplemented with two additional feedbacks involving ground reaction force. The function of the impact-force controller is to stabilize the forces around the nominal values, determined for the nominal conditions of motion. The proposed hybrid control scheme fulfills the preset control criteria. Its application ensures the desired precision of robot’s motion, maintaining dynamic balance of the locomotion mechanism during the motion, and minimization of the impulse action of load forces at the joints, as well as of the forces arising at the moment of foot striking the ground during the motion. Performance of the control system was analyzed in a number of simulation experiments in the presence of different types of external and internal perturbations acting on the system. The obtained numerical results served as the basis for evaluation of the proposed controller performance that validated the efficiency of using intelligent control techniques for robotic contact tasks. In summary, the pure intelligent and complementary hybrid intelligent control algorithms represent a powerful tool for solution of robot compliance problems.
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Instead of Conclusion
Each chapter of this monograph has provided concluding remarks highlighting specifically considered topics. Here we would like to take a short retrospective view of the entire book, especially considering our initial goals and aims. Our aim was to consider interaction control problems from different viewpoints, primarily taking into account practical problems and needs. Basic strategies for controlling the interaction of a robot with the environment are the subject of the monograph. The book provides a historical perspective on interaction control, summarizing the major achievements in the field in the last twenty years. We have tried to identify open problems and insufficiently investigated issues which require additional attention in order to realize a wider application of interactive robotic systems in the industry and service fields. This book attempts to provide unified theoretical force and position control paradigms considering basic control issues: stability, performance and robustness. This framework assumes a general dynamic environment and uses an inverse dynamic control strategy to design various controllers for specific force and position stabilization tasks. We have focused our attention on the problems of simultaneous asymptotic stabilization of the robot motion and contact force of interaction of a robot with its dynamic environment. The unified approach to solving these problems has been considered. In the scope of the approach the control laws ensuring predetermined quality of transient processes with respect to position or force, depending on a specific-purpose technological task, have been synthesized. For a certain class of models of environment dynamics the generalization of this approach to position-force control is obtained. It has been proven the robustness of the offered control laws to every possible errors and perturbations arising in contact tasks including errors of feedback, inaccuracies of models of the robot and environment, and also external perturbations. In case of large-scale parametric perturbations the adaptive control laws have been proposed solving the set problems even when model of dynamics of the robot is
625
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nonstationary in a wide class of functions including quickly varying processes. It is shown, that these control laws can cope with essential inadequacy of models of the robot and environment dynamics. In the scope of the unified approach to control law synthesis for a robot manipulator in contact with a dynamic environment when the environment dynamics can be approximated sufficiently well by a linear time-invariant model in the Cartesian space, the task of stabilizing both the robot motion and interaction force simultaneously is solved under less restrictive conditions imposed on the environment dynamics than when some particular environment properties are required to ensure the overall system stability. The one-to-one correspondence between the closed-loop motion and force dynamics equations is obtained, a unique control law, ensuring the system’s stability and preset either motion or force transient response, is proposed. Advantages of the unified approach to robot’s control in contact tasks as compared with known approaches in their classical representation as hybrid and impedance control have been shown. These advantages are based on complete dynamic models of the robot and environment, and give the capability to synthesize more effective control laws. The same advantages are used in the synthesis of control laws demanding less restrictive stability conditions when the problem of so-called practical stability of the system is under consideration. The elaborated stability test may be used either to check the stability of the specified control laws, or to establish procedures for the synthesis of parameters of different control laws. By this, the control synthesis becomes much more accurate and effective. The impedance control has been selected as a special case of general active compliant control techniques, mainly taking into account the applicability in existing industrial robotic systems. The monograph proposes a new interaction stability paradigm for the design of impedance control which ensures contact stability during all phases of interaction. Moreover, the new design framework realizes low-impedance performance allowing considerable reduction of high apparent industrial robot inertia and stiffness. The novel stability criteria are established based on robust control theory into account estimates of the environmental stiffness, tolerating thereby large uncertainties and variations in industrial environments. The defined stability indices take into account the relevant effects in robot control systems, such as time lags and sampling data effects, as well as uncertainties in the environment and realized target admittance models. The theoretical robust stability framework has been used to develop algorithms for the practical impedance control design in industrial robotic
Instead of Conclusion
627
systems. The considered impedance control synthesis addresses basic control design problems at the servo-control layer. The impedance control design has been established for a reliable decoupled compliance geometric model that allows a relatively simple parameterization of the target impedance behavior. For fundamental and common interaction tasks the compliance parameters can be chosen independently of the interaction system configuration. More complex robot-environment interaction was also considered based on the spatial compliance model. This book also provides a reliable geometric and control framework for the implementation of compliance control in industrial and other advanced robotic systems. Several practical and robust control algorithms at higher planning and programming control layers were presented as well. The essential algorithms support a setting of the compliance parameters, such as the location of the compliance frame and impedance gains, as well as a continuous switching of compliance control and variation of parameters. These features are proven to be essential for a stable and robust execution of the compliance control tasks. Powerful sets of control functions, also presented in the monograph, integrate the basic compliance control algorithms in the forward robot control. These functions perform all of the computations and management of the parameters between the convenient robot position control system and the impedance control kernel. The basic structure and implemented functions of the compliance control module are presented and specified. Finally, the new commands providing a flexible user-interface are designed and implemented in an industrial robot control environment. The new programming language commands are illustrated by means of several examples. The novel interaction stability criteria are proven by extensive testing in industrial and space robotic systems. The execution of numerous complex robotic tasks illustrates the feasibility of the proposed interaction control techniques. The established contact stability theory has been expanded to the control and synthesis of haptic interfaces interacting with a virtual environment. This rapidly emerging technology imposes high requirements on interaction stability and robustness of the control system. New control approaches for admittance and impedance haptic display control based on the developed impedance control and contact stability results were proposed for the first time in the monograph. Simple experiments and simulation results have demonstrated the advantages, high performance and reliability of the new algorithms. The first implementations of these results in more complex haptic interfaces in the industry have also been briefly presented.
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Considering more complex tasks, the application of force sensors and interaction control techniques is certainly not sufficient to provide the robot with a required degree of autonomy and intelligence. The monograph is also focused on the learning principle, and this is shown through the application of new, advanced learning algorithms for robotic contact tasks. In view of the important influence of the robot environment, a new, comprehensive learning approach is based on the simultaneous classification of the robot environment and learning of robot uncertainties. It should be pointed out that the integration of different exteroceptive sensors, such as vision, is a big challenge for researchers today and will considerably improve the capabilities of robotic systems to interact with a complex and variable environment. Practical knowledge and experience gained in developing and implementing various interactive robotic systems is reflected in this book. In our opinion, this places a specific potential value on addressing the needs of a wide range of readers, including technical specialists in industry. The authors would foremost like to thank the researchers at the Fraunhofer Institute IPK in Berlin and the Mihajlo Pupin Institute in Belgrade for the fruitful cooperation in investigating interaction control problems focused on in the monograph. Warm thanks are due to the Robotic Division of the European Space Agency (ESA), the Robotics Laboratory at the Fraunhofer IPK and Tecnospazio SpA for providing exciting facilities in joint research projects for testing the research developments presented in this book and for invaluable support in experimental activities. The authors are very grateful to Mrs. S. Frankenstein for helping us to bring this work into its present form. Thanks should also go to Prof. Luka Bjelica and Mr. Le-Minh Ho for providing us essential English editorial help. A last but not least word of appreciation of thanks should be directed to MSc Milos Jovanovic and Mr. Svemir Popic for their technical editorial assistance.
Index
A
artificial constraints 29, 30
acceleration control approach 57 accommodation 2, 14, 21 active compliance 3, 7, 9, 18 active compliance control 3, 7, 9, 18 active damping 36 action evaluation network 602 action selection network 602 actor network 587 actor-critic architecture 587 actor-critic methods 587, 602 adaptable compliant devices 15 adaptable passive compliance 3 adaptive impedance control 27, 28 admittance center 416 admittance control 4, 48 admittance display 493, 498, 499, 504, 508, 512 algorithm adaptation algorithm processing speed 79, 129 finite-convergence adaptation algorithm 128, 137, 432 allocation of force sensor 6 angle-axis representation 464 angular stiffness 432 apply-force function 467 approximation first approximation 96, 114, 158, 193, 214, 237
B Back propagation 587 bandwidth 260, 268, 271, 299, 374, 382, 394, 408, 431, 442, 446 behaviour-based control 586, 592 Bihari lemma 213, 217, 240 bilateral i.e. geometric contacts 319 bilinear sector transform 348 bilinear transform 365, 368 biped locomotion 584, 586, 589, 593, 595, 598, 600, 607 C C3G-OPEN 440, 458 CAT-arm 453 Cayley transformation 348 center of damping 426 center of gravity 607 center of stiffness 418 C-frame 416, 424, 432, 437, 443, 448, 454, 458, 462, 474, 479, 486 cerebellar model arithmetic controller 591 chamfers 476 characteristic indices 97, 100, 115, 155, 212, 237 classifier 534, 553, 559, 563, 570, 616 629
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Dynamics and Robust Control of Robot-Environment Interaction
closed-loop 267, 271 closed-loop system 80, 90, 106, 121, 152, 157, 160, 169, 183, 192, 201, 204 cobot 508 compensation for inertia 465 compliance 1, 15, 21, 38, 42, 50, 57, 61 compliance center 416, 423 compliance control 520, 552, 557, 565, 570, 584 compliance frame 7, 37, 58 compliance matrix 419, 421, 423, 424 compliance or C-frame 19, 256, 416, 454, 456, 465, 469 compliance or constraint frame 5 compliance planning 56 compliance selection matrix 30 compliant environment 13 compliant motion 1, 6, 9, 18, 46, 65 computed torque method 22, 23, 557 connectionist reactive control 575, 616 control 552, 559, 616 configuration space 11 constrained motion control 2, 8, 13, 39 constraint-frame formalism 8 contact task 1, 7, 10, 18, 29, 40, 45, 49, 55, 65, 80, 84, 96, 101, 119, 129, 134, 145, 155, 166, 175, 180, 185, 191 constraints dynamic constraints 80, 82 kinematic constraints 81, 82 contact overshoot 381 contact stability domain 362 contact stability margin 380, 393 contact transition stability 46, 50, 317, 323, 330, 334, 338, 344, 350, 357, 361, 366, 375, 380, 394
control adaptive control laws 80, 101, 129, 135, 146, 191, 193 adaptive control scheme 79, 128, 145 admissible control 104, 135, 216 closed-loop control system 92, 108, 123, 126, 139, 182, 188, laws 78, 84, 91, 94, 98, 04, 110, 118, 121, 128, 135, 146, 149, 157, 161, 169, 173, 180, 190, 207, 220 conventional hybrid control 78, 173, 179 delay 266, 325, 352, 363, 369, 375, 387, 396, 400 dynamic control 78, 101, 158, 180, 193, 208 flags 467 force control 77, 173, 180, 191, 205 hybrid control 77, 170, 173, 175, 179, 190, 199, 207 hybrid position/force control 77, 179 impedance control 146, 182, 190 interaction force control 77 motion control 77, 170, 182, 191 position/force control 77, 179 position-force control 146, 158, 177, 190, 195, 198, 205 system delay 428 controller gain adjustment method 16 convenient impedance control law 287 Coulomb friction 318, 325, 351, 400 coupled or contact impedance 271 stability 46 system stability 272, 279, 317, 322, 334, 399
Index
coupling admittance 510 Critic network 587 crossover 570 D damping control 6, 20, 27, 42 damping matrix 19, 36 damping or accommodation control 21 damping ratio 304, 309, 314, 328, 362, 369, 374, 382, 391 damping weighting factor 435 design evaluation 437, 449 deviation model 272, 284 deviation-coupled model 281 direct rendering 494 disk uncertainty 282 disturbances 104, 192 domains of contact stability 331 double-support phase 588, 596, 605 D-partitions 355 dynamic balance 585, 594, 598, 606, 615 environment 8, 13, 43, 54, 60, 66 hybrid control 8, 39 impedance control 24, 57 interaction control 5, 64 position/force control 8 reaction feedback 599, 601 stability 35 dynamics nonstationary dynamics 79, 128, 145 stationary dynamics 129 system’s dynamics 80 E effective filtered nominal motion 410
631
end-effector motion 81, 83 coordinates 81 end-point 424, 453, 459 engagement 476, 484 environment dynamics 81, 85, 91, 95, 101, 108, 116, 124, 132, 146, 158, 168, 174, 183, 226, 232 coordinates 81, 83 parameters 165, 194, 204 external environment 88, 104 frictionless environment 81 passive dynamic environment 82 environmental stiffness 405, 428, 434 epsilon environment 460 equivalent haptic subsystem 501 equivalent impedance 510 essential contact tasks 4 example 324, 411, 428, 440, 456, 514 experiment 294, 302, 313, 318, 350, 382, 396, 406, 482, 504 expert systems 521 explicit force control 30, 36, 537, 540 external force control 37 explicit hybrid control laws 537 F feature extraction 553, 560 fixed passive compliance 3 foot force control 590 force -based assembly skills 534 -based impedance control 22, 37, 42, 261 -dependent position modification 458
632
Dynamics and Robust Control of Robot-Environment Interaction
class of programmed interaction forces 112 contact force 77, 82, 95, 100, 120, 128, 149, 155, 177, 187 desired contact force 77, 180, 189 control 77, 173, 180, 191, 205 mode 20, 52 model error 26, 259 model-error control 264 -mode or inner-loop control 6 interaction force desired interaction force 78, 150, 158 possible interaction forces 78 programmed interaction force 79, 104, 108, 112 generalized forces 81, 129, 146, 232 nominal force 122, 155, 189, 196, 200 overshoots 406, 429, 434, 466 -sensing S-frame 454 -setpoint reached 460 force of interaction perturbed robot’s force of interaction 88 programmed force of interaction 88, 90, 132, 158, 183 function approximation 525 function of uniformly bounded variation 139 fuzzy-genetic algorithms 522 fuzzy logic 101, 193, 521, 541, 545, 548, 591 fuzzy adaptive control 546 force controllers 541, 550 inference system 542, 603 sliding controller 546
model reference adaptive control 544 G general impedance control 6, 20 generalized contact stability 344, 400 coordinates 84, 129, 147, 158, 232 stiffness control 409, 411 stiffness formulation 21 genetic tuning 568, 570, 616 algorithms 521, 527, 552 geometric criterion 51, 330 grasping 452, 457, 471, 473, 476, 480 gripper frame 453 gross motion 46 H haptic interfaces 491, 496, 516 rendering 493, 496 transparency 493 haptics 492, 498 humanoid robotics 584, 592 hybrid control 5, 7, 30, 34, 39, 52, 66 dynamic control 586 position/force control 5, 28, 33, 42, 56, 65 position-force control 537, 548, 551 -impedance control 42 I impact-force controller 605, 617
Index
impedance 4, 19, 40, 46, 49, 57, 61 conventional impedance 189, 190 compensator law 499 control 4, 9, 18, 46, 52, 58 146, 182, 185, 189 control compensator 413, 429, 436, 449, 463 control design 406, 414, 428, 432, 440, 493 control error 20, 259 control gains 459, 476 control operating modes 447, 468 display 493, 508, 510, 514 equation 184, 190 improved impedance control law 301 implicit or position-based control 6, 37 inertia matrix 12, 19 infinity norm 348, 354, 366, 374, 380 insertion 442, 471, 476, 481 intelligent control 519, 532, 540, 550, 576, 584, 590, 616 internal reinforcement 587, 603, 613 stability 274 inverse control technique 23 IRCC, “Instrumented Remote Center Compliance” 15 J joint space control 7 K kinematic instability 33 model-based algorithms 7
633
L learning adaptive algorithms 521, 528 control 520, 529, 536, 549, 552, 567, 573, 586 global algorithms 525 imitation 524, 532, 535, 587, 593 local algorithms 525 Q 531 reinforcement 531, 539, 586, 592, 598, 613, 617 supervised 523, 527, 535, 550, 591 TD 587, 592, 602 unsupervised 523, 591 by demonstrations 527 empirical 9 limit-cycle 406 linear elastic environment 14 linguistic gains variables 459 Lyapunov generalized theorem 98 characteristic index 97 function 334 M manipulation mechanism 77, 84, 129, 170 matrix Cauchy matrix 116, 212, 238 fundamental matrix 116, 212, 238 Gramm matrix 86 Jacobian matrix 81, 182, 195 “maximum impact force” 380 “minimum transition” force 380 maximum modulus theorem 346, 366
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Dynamics and Robust Control of Robot-Environment Interaction
mechanical compliance devices 15 memory long-term 530 short-term 530 method computed torque method 149, 207 decomposition-aggregation method 194 connectionist 521, 531, 536, 552, 559, 567, 575, 580, 591, 616 iterative - analytical 530 reactive learning 531 tabular 530 model dynamic model 77, 158, 174, 199 environment dynamics model 79, 83, 89, 92, 101, 117, 132, 153, 158, 174, 183, 191 levels of inaccuracy of the models 116 linear impedance model 83 linear time-invariant model 80, 158 -matching 259 -based dynamic control 21 of the external environment dynamics 88 uncertainties 80, 194, 207 inaccuracy 103 nonlinear model 99 Moor-Penrose pseudo-inverse matrix 34 most destabilizing environment 48, 280, 321, 399 motion class of programmed motions 112 constrained motion 78, 182, 191 desired motions 78, 191 nominal motion 78, 120, 155, 166, 183
perturbed motion 96, 114, 183 possible motions 78 programmed motion 79, 86, 90, 100, 112, 139, 147, 158, 180, 183 real motion 86, 94, 104, 115, 152, 218 unconstrained motion 78 unperturbed motion 183 multilayer perceptron 534, 553, 561, 574, 579, 591 mutation 570, 574 N narrowing ε-narrowing 143 natural admittance control 497 constraints 29 neural compensator 554, 559, 565, 574 neural networks 101, 193, 521, 531, 537, 557, 575, 591, 617 neuro -fuzzy networks 522, 549 -genetic algorithms 522 nominal coupled system performance 285 penetration 501, 510 non-adaptable methods 3, 14 non-adaptive compliance control 7 non-backdriveable 17 non-contact tasks 1 nonlinear decoupling 23 O object frame 425, 454, 462 one-port network 493
Index
operational space control 7 space formulation 12 orthogonal complements 78, 173, 179 outer/inner loop stiffness control 25, 263 P parallel control 5, 42 position/force control 5 parameters actuator parameters 131, 210 damping and stiffness parameters 178 manipulation mechanism parameters 130 parameters’ drift 79, 129 uncertainties of parameters 80 passive compliance 3, 9, 14, 21, 52, 576, 583, 605 passivity 277, 279, 288 and contact stability 341, 265 and coupled stability 278 and -norm 348 -based and robust contact stability 348, 367, 382, 400 concept 496 peg-in-hole 534, 575, 583, 617 penetration 270, 284, 305, 323 model 270, 295 perturbation external perturbations 80, 101, 109, 112, 128, 145, 193 levels of external perturbation 119, 133 initial perturbations 87, 98, 105, 109, 117, 132, 155, 171, 210
635
initial perturbation levels 116 function 124 levels 108, 116, 119 uncontrollable perturbations 132 phase planes 323 PI-force regulator 558, 569 planning and programming layers 452, 486 planning of impedance control tasks 57 Poincare map 593 position control mode 53 correction 408, 462, 469 correction offset 470 deviation 259, 272, 284, 322, 333, 345 measure 4, 20, 39 model-error 259, 398 model error control 20, 25, 263 selection matrix 31 /force control 4, 8, 28, 42, 56 -based (implicit) force control 37 -based impedance control 24, 27, 41, 52, 59, 262, 320 -mode or outer-loop control 6, 24 positive system 277 potential contact tasks 3 practical stability 45, 56 contact transition stability 331 pressure sensors 575, 582, 617 R radial basis function 591 reactive admittance control 534 reference target model 256
636
Dynamics and Robust Control of Robot-Environment Interaction
reinforcement learning 523, 531, 538, 586, 592, 599, 610, 617 GARIC 586, 602 Remote Center Compliance\-RCC 3, 15 RCC passive compliance 476 reference world frame 453 relax algorithm 470, 479 function 467 reproduction 570 resolved acceleration control 8, 35, 57 robots structural elasticity 14 robot world model 453 -base frame 453 robotic assembly task 575, 616 robust contact stability 51 contact transition stability criterion 344 control 7, 28, 50, 55, 64 control design 435 coupled performance 285, 289 coupled stability 282, 284, 295, 345 running mode 468 S sampled-data (SD) model 352 sensitivity transfer functions 267, 271, 280, 293, 351 sensor errors 121, 123 inaccuracies 128 settling time 380 simulation model 437, 449
single-support phase 588, 596, 609 small gain theorem 275 SMART-S4 440, 447, 449 soft sensored grippers 575, 616 soft-servo or soft-float 18 SPARCO control systems 440, 449, 459, 468, 487 spatially round systems 285, 291 spatial stiffness matrix 417 stability asymptotic stability 87, 95, 114, 149, 158, 193 exponential stability 79, 98, 149, 153, 193 exponential stability of motion 79, 158 exponential stability of the contact force 79 practical stability 80, 101, 193, 199, 207 practical stability tests 80 conditions 80, 101, 192, 203, 206 (passivity) condition 498, 501, 504, 509 criterion 434, 447 structured robustness of coupled stability 287 stabilization adaptive stabilization 128, 139 exponential stabilization 79 position/force stabilization 78, 173 simultaneous stabilization 77, 104 stabilization accuracy 79 stabilization of interaction force 91, 172 stabilization problem 79 steady-state force 307, 313, 380
Index
stiffness control 6, 17, 22, 25 matrix 16, 19, 21, 57, 416, 428 -like behavior 16 ratio 273, 333, 336, 343, 359, 365, 374, 382, 400 stochastic action modifier 603 real valued unit 602 emporal difference algorithm 526 structured singular value 286 surface sliding contact task 29
637
transferring human manipulation skill 532 transient motion 46 motion transient processes 104, 148, 160 quality of transient processes 104, 110, 123, 148, 158 transparency 492, 495, 502, 505, 512 twist 417, 465 two-port network 493, 498 U
T target admittance 491, 494, 496, 499, 502 damping 313, 320, 334, 338, 343, 349, 391, 400, 409, 434, 437 damping ratio 309, 328, 368, 382, 393 impedance 494, 496, 499, 501, 509 impedance frequency 406 mass 304, 310, 314, 320, 328, 382, 394, 407, 433, 426 model realization 294, 298, 312 robot impedance 18 stiffness 258, 273, 297, 315, 347, 382, 409, 422, 426, 432, 446, 459, 464, 470 task - or compliance-frame 29 termination 472, 476, 480, 484 T-frame 424, 453, 459 time-domain contact transition stability 357 tool-center-point 453, 458
uncertainties model inaccuracies 80, 194, 207 parameter uncertainties 101, 191, 206 unified approach 43, 45 position-force control 5, 43 unilateral, i.e. force constraints 319 user parameters 427 V virtual compliance 61, 495 coupling 61 W wavelet classifier 534, 553, 563, 616 transformation 553, 561 wavelet network 553, 565, 573 weighting function 285 transfer function 436
638
Dynamics and Robust Control of Robot-Environment Interaction
worst destabilizing enviroment 48 passive environment 48 wrench 417, 421
Z zero-moment point 586, 599, 617
