ROMANSY 18 - Robot Design, Dynamics and Control

~ SpringerWienNewYork CISM COURSES AND LECTURES Series Editors: The Rectors Giulio Maier - Milan Franz G. Rammerstor...

Author: Werner Schiehlen | Vincenzo Parenti-Castelli

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~ SpringerWienNewYork

CISM COURSES AND LECTURES

Series Editors: The Rectors Giulio Maier - Milan Franz G. Rammerstorfer - Wien Jean Salençon - Palaiseau

The Secretary General Bernhard Schreﬂer - Padua

Executive Editor Paolo Seraﬁni - Udine

The series presents lecture notes, monographs, edited works and proceedings in the ﬁeld of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series is to make known in the international scientiﬁc and technical community results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences.

INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES - No. 524

ROMANSY 18 ROBOT DESIGN, DYNAMICS AND CONTROL PROCEEDINGS OF THE EIGHTEENTH CISM-IFToMM SYMPOSIUM

EDITED BY VINCENZO PARENTI CASTELLI UNIVERSITY OF BOLOGNA, ITALY WERNER SCHIEHLEN UNIVERSITY OF STUTTGART, GERMANY

This volume contains 291 illustrations

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned speciﬁcally those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 2010 by CISM, Udine Printed in Italy SPIN 80016144

All contributions have been typeset by the authors.

ISBN 978-3-7091-0276-3 SpringerWienNewYork

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Keynote Lectures

Where Future Robots should Go and should not Go *;EINI (;H?EI ;H> *CNMOLO %CA;MBCGILC *

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1

Introduction

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4

M. Kaneko and M. Higashimori

Figure 1. 'OAAFCHA ;H> G;HCJOF;NCHA ; J?H=CF

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Where Future Robots should Go and should not Go

5

LIC?L?HN @LIG ?;=B INB?L 1I >CM=OMM NB?M? CMMO?M Q? B;P? NI EHIQ NB? =OLL?HN MN;N? I@ NB? ;LN I@ LI?RN?LCNS ;H> MJ??> CH G;HCJOF;NCIH <;M?> IH IOL @ILG?L QILEM * (;H?EI ;H> 1MODC ; * %CA;MBCGILC ;H> (;H?EI ; < * (;H?EI ;H> 1MODC < * (;H?EI ;H> &MBCE;Q; * %CA;MBCGILC ;H> &MBCE;Q; * %CA;MBCGILC ;H> (;H?EI = ; < S =IGJ;LCHA NB?M? @OH=NCIHM @IL BOG;H ;H> LI ;N >?RN?LIOM G;HCJOF;NCIH QBCF? LI LI ;N MJ??> JIQ?L L?J?;N; MI @ILNB QBCF? BOG;H ;L? HIN CH L?F;NCP? M?HM? 1BLIOAB NB?M? L?PC?QM Q? >CM=OMM BIQ Q? MBIOF> =IFF; QB;N MECFF Q? MBIOF> ?RJ?=N ;H> HIN ?RJ?=N @IL LIO=? ; G?>C=;F LI; 3CH=C >?P?FIJ?> <S &HNOCNCP? 0OLAC=;F )N> 1BCM =IGJ;HS CM >ICHA ; L?;FFS AII> CH=FO>? MIG?NBCHA CGJILN;HN @OH=NCIH 4B;N @OH=NCIHM >I NB?S CH=FO>? ;H> =ON CHN?HNCIH;FFS 1BLIOAB ?RJFILCHA NB?M? KO?MNCIHM Q? ?P?HNO;FFS L?;=B ;HMQ?LM NB;N Q? ;L? FIIECHA @IL 1BCM J;J?L CM ILA;HCT?> ;M @IFFIQM &H 0?=NCIH Q? =IHMC>?L MJ?=C;F MECFFM @IL LI IH NBCM =IHMC>?L;NCIH Q? JLIJIM? ; =IFF; LIO=? ;H ?R;GJF? I@ LICL?=NCIH I@ @ONOL? LICHA L?G;LEM CH 0?=NCIH

2

The Special Skill for Human and Robot

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
6

M. Kaneko and M. Higashimori

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Where Future Robots should Go and should not Go

7

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8

M. Kaneko and M. Higashimori

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Where Future Robots should Go and should not Go

9

Figure 7. 1QI HA?L?> B;H> =;JNOL?M ;H I
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
10

M. Kaneko and M. Higashimori

Figure 8. MN?J ;H> MN?J MNL;N?AC?M

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Where Future Robots should Go and should not Go

11

Figure 9. *;HCJOF;NCHA ; JCTT; <S ; G;MN?L

Figure 10. H ?R;GJF? I@ JCTT; G;HCJOF;NCIH LI
; JCTT; G;MN?L %IQ?P?L NBCM ;JJLI;=B QCFF HIN QILE Q?FF MCH=? ; >?RN?LCNS QBC=B CM HIN AII> ;N @IL LI 1I =IJ? QCNB NBCM CMMO? Q? B;P? >?P?FIJ?> ; LI?AL??M I@ @L??>IG * %CA;MBCGILC ;H> (;H?EI = CHMN?;> I@ ; BOG;HIC> LI PCMCIH Q? B;P? ;=BC?P?> P;LCIOM ?RJ?LCG?HNM MO=B ;M LIN;NCIH;F ;M Q?FF ;M NL;HM F;NCIH;F GINCIHM I@ IIH? G;HCJOF;NCIH I@ NB? I
12

M. Kaneko and M. Higashimori

Figure 11. JIMMC BOG;H

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3

A Collaboration Style between Robots and Human

*;HS L?M?;L=B?LM B;P? MBIQH ; =IFF; ; LI IH NB? C>?; NB;N Q? ?RJ?=N ;H ?KO;F MECFF @IL LI CH NB? JL?PCIOM M?=NCIH NB?L? ;L? >C?L?HN MJ?=C;F MECFFM @IL BOG;H ;H> LI ;N >?RN?LCNS QBIM? MECFF CM ?P?H GIL? NB;H NB? =OLL?HN LI LI ;N ; MCGJF? N;ME QCNB BCAB MJ??> M @IL ; =IFF; BOG;H Q? MBIOF> N;E? NB?M? MJ?=C;F MECFFM CHNI =IHMC>?L;NCIH #CA ?RJF;CHM ; JIMMC?RN?LIOM GINCIHM @IL BOG;H ;H> BCAB MJ??> MCGJF? GINCIH @IL LI?L ;H ?R;GJF? I@ NBCM MNSF? 0OJJIM? NB;N ; G?>C=;F >I=NIL CM

Where Future Robots should Go and should not Go

13

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
4

A Success Example in Robot Business

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14

M. Kaneko and M. Higashimori

Figure 12. H ?R;GJF? I@ G?>C=;F LI
CH@ILG;NCIH I NBLIOAB ! MN?L?I CG;A? #CA MBIQM NQI JIMMC ??=NILM ;P;CF;C=;F LI < ;L? ; ALCJJ?L NSJ? B;H> ;H> ; GOFNC DICHN?> B;H> L?MJ?=NCP?FS 1B? B;H> CM NB? JIN?HNC;F =;J; QBCF? NBCM @OH=NCIH B;M H?P?L CH ?CNB?L CH>OMNLS IL G?>C=;F ?F> &H G;HCJOF;NCIH I@ I LI BOG;H ;M >CM=OMM?> CH 0?=NCIH && 1B? >? MCAH?LM CH &HNOCNCP? 0OLAC=;F =ON NBCM @OH=NCIH @LIG >; 3CH=C ;H> ;>IJN?> ; MCGJF? ALCJJ?L NSJ? B;H> ;M MBIQH CH #CA ; +?P?LNB?F?MM CN FIIEM NB;N >; 3CH=C CM ;=BC?PCHA >?RN?LIOM ;H> MECF@OF G;HCJOF;NCIH <S OMCHA NQI ;LGM 1B? E?S CM NB;N ?;=B ALCJJ?L MCGJFS AL;MJM NB? IF? ; NBL?;> IL CHN?LH;F ILA;H ;H> ; MOLA?IH JLI>O=?M ; ALCJJ?L F?P?F =IIL>CH; NCIH =IHNLIF NBLIOAB ; G;MN?L ;LG 1B? >?RN?LIOM GINCIHM ;L? NB?L?@IL? JLI>O=?> <S NB? >?AL??M I@ @L??>IG I@ ;LG CHMN?;> I@ HA?L ,H=? ;A;CH Q? MBIOF> HIN? NB;N NB? >?RN?LCNS I@ >; 3CH=C CM HIN JLI>O=?> <S NB? HA?L GINCIHM O=?> L?GIN?FS <S NB? MOLA?IH M @;L ;M Q? =;L?@OFFS ?R;GCH? >; 3CH=C NB?CL ;JJLI;=B CM KOCN? H;NOL;FFS ;==?JN?> CH N?LGM I@ NB? =IFF; LI =IGJF?N?FS ;AL?? QCNB NB?CL >?=CMCIH @IL =ONNCHA NB? @OH=NCIH I@ GOFNC DICHN?> HA?L

Where Future Robots should Go and should not Go

15

Figure 13. 1QI NSJ?M I@ B;H> @IL G?>C=;F ;JJFC=;NCIH

5

Where Future Robots should Go and should not Go

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6

Concluding Remarks

4? B;P? >CM=OMM?> NB? >CL?=NCIH I@ @ONOL? LI CM ; OHCP?LM;F NIIF @IL G;HCJOF;NCHA ; EHC@? ; MJIIH ; @IFE ; J?H=CF ;H> G;HS INB?LM 4CNB NBCM MCGCF;LCNS J?IJF? ;L? ?RJ?=NCHA NB;N GOFNCHA?L?> LIM QCFF ??=NIL @IL @ONOL? LIC=OFN NI >?MCAH ;H> >?P?FIJ ; GOFNCHA?L?> LI ;P;CF; NB? >C=OFNS I@ >?RN?LIOM G;HCJO F;NCIH <S =IGJ;LCHA NB? >C?L?H=? BOG;H "MJ?=C;FFS NB? >C=OFNS I@ G;HCJOF;NCIH I@ I =IG?M @LIG NB;N

16

M. Kaneko and M. Higashimori

Q? ;L? ;FFIQ?> NI OM? ; M?N I@ JOMBCHA @IL=?M IHFS 1B? MGIINB ;H> KOC=E GINCIH I@ I <S G;HS JBSMC=M MO=B ;M MFCJJCHA LIFFCHA ;H> CHMN;O? NI ; F;LA? JOMBCHA @IL=? IH I; 3CH=C CM M?H>CHA ; G?MM;A? NIQ;L> ; JL;=NC=;F MIFONCIH &HMN?;> I@ >?MCAHCHA ; =IGJFC=;N?> B;H> MSMN?G Q? MBIOF> >?MCAH ; B;H> ;M MCGJF? ;M JIMMC G;S QILE CH CN QCNB LG L?FC;?MCAHCHA ; LI FCPCHA NBCHAM ;L? @;L >C?L?HN @LIG ?;=B INB?L CH N?LGM I@ ;=NO;NIL M?HMIL ;H> ?P?H G?=B;HCMG %OG;H HA?LM B;P? =IHNCHOIOMFS ?PIFP?> NBLIOAB NB? JLI=?MM CH OMCHA P;LCIOM NIIFM @IL FIHA FIHA J?LCI>M MCH=? G;ECHA ;JJ?;L;H=? ;H> ?P?HNO;FFS HIN ?RJ?=N MO=B ; OHCP?LM;F B;H> @IL ;H ;LNC=C;F IH? ,P?L S?;LM B;P? J;MM?> ;FL?;>S MCH=? G;HS JLID?=NM IH GOFNC HA?L?> LI B;P? N F?;MN NB? L?M?;L=B L?MOFNM IH GOFNCHA?L?> LIM MI @;L B;P? MOJJILN?> IOL G?MM;A?M QCNBION ;HS =IOHN?L?R;GJF? 4? ;L? HIQ CH ; MN;N? QB?L? Q? MBIOF> M?H> G;HS LICHA Q? QIOF> HIN? IH=? ;A;CH NB;N Q? MBIOF> HIN >?PIN? IOL JIQ?L CH >?MCAHCHA ; BOG;H HA?L FCE? OHCP?LM;F B;H> @IL ; @ONOL? LI C@ Q? QIOF> M?? ; LI?MCAH ; B;H> CH NB? @ONOL? #CH;FFS NBCM QILE CM J;LNC;FFS MOJJILN?> <S +0# '01 DICHN L?M?;L=B JLI AL;G IH IHN;=N &HN?L@;=? *I>?FCHA ;H> 0NCH?MM <;M?> CIG?>C=;F !C;A HIMCM QCNB 0?HMCHA 1?=BHIFIAS 1IQ;L>M ; ?NN?L .O;FCNS I@ )C@?

Bibliography 0' IF;HIQMEC ;H> /1 3?LLCFFI 1?GJ?L;NOL? ;H> =LCN?LCIH ??=NM CH ; MIG;NIM?HMILS MO<MSMN?G H?OLIJBSMCIFIAC=;F ;H> JMS=BIJBSMC=;F MNO>S &H Journal of Neurophysiology PIFOG? J;A?M U 0' IF;HIQMEC ;H> '' 7QCMFI=EC &HN?HMCNS ;H> @L?KO?H=S =B;L;=N?LCMNC=M I@ J;=CHC;H =ILJOM=F?M C ;=NCIH JIN?HNC;FM &H Journal of Neurophysiology PIFOG? J;A?M U !)/ BNNJQQQ>FL>?LG?H>?MENIJ>?@;OFN;MJRN; & &MBCC ;H> * &MBCE;Q; %CAB MJ??> N;LA?N NL;=ECHA ;FAILCNBG @IL GM PCMO;F @??><;=E MSMN?G &H J. Robotics Soc. Japan 17 J;A?M U * (;H?EI ( %;L;>; ;H> 1 1MODC /IFFCHA <;M?> G;HCJOF;NCIH @IL GOFNCJF? I & &MBCC !?P?FIJG?HN I@ ; BCAB L?MIFONCIH BCAB MJ??> PCMCIH MSMN?G OMCHA =GIM CG;A? M?HMIL N?=BHIFIAS ?HB;H=?> <S

Where Future Robots should Go and should not Go

17

CHN?FFCA?HN JCR?F M?F?=NCIH N?=BHCKO? &H Optics East 2004 PIFOG? J;A?M U +;GCEC * %CA;MBCGILC * (;H?EI ;H> * &MBCE;Q; !?MCAH I@ NB? A =;JNOLCHA LI IH >SH;GC= JL?MB;JCHA &H The International Journal of Robotics Research PIFOG? J;A?M U & &MBCC * %CA;MBCGILC * (CGOL; ;H> * (;H?EI !SH;GC= =;JNOLCHA MNL;N?AS @IL ; > MNC=E MB;J?> I IH @LC=NCIH CH>?J?H>?HN =IF FCMCIH &H IEEE Transactions on Robotics PIFOG? J;A?M U 6 ,GINI * %CA;MBCGILC ;H> * (;H?EI +IH AL;MJ G;HCJOF;NCIH I@ >?@ILG;FCHA G?=B;HCMG &H Proc. of the IEEE Int. Conf. on Robotics and Automation J;A?M U ; 6 ,GINI * %CA;MBCGILC ;H> * (;H?EI "RJ?LCG?HN;F ;JJLI;=B IH >? @ILG; <S NB? B;H>FCHA I@ ; JCTT; J??F &H Proc. of 9th Int. IFAC Symp. on Robot Control J;A?M U < 6 ,GINI * %CA;MBCGILC ( 2NMOGC ;H> * (;H?EI !SH;GC= G;HCJOF; NCIH CHMJCL?> <S NB? B;H>FCHA I@ ; JCTT; J??F &H IEEE Transactions on Robotics PIFOG? J;A?M U = + 1B;CJL;M?LN * (;H?EI ;H> 1 1MODC "RJ?LCG?HN;F ;JJLI;=B IH ?H P?FIJCHA AL;MJ @IL =IFOGH I * &MBCE;Q; 1B? A =;JNOLCHA LI 1 1MODC 0=;F? >?J?H>?HN AL;MJ &H IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics PIFOG? J;A?M U * (;H?EI * 0PCHCH ;H> 1 1MODC &HN?LH;F @IL=?M ;H> MN; ( 1;DCG; !?P?FIJG?HN I@ ; G?A; JCR?F ;H> GCFFCM?=IH> PCMCIH MSMN?G OMCHA CHN?FFCA?HN JCR?F M?F?=NCIH &H TExCRA2004 J;A?M U * &MBCE;Q; 1 (IGOLI & &MBCC ;H> 6IMBC>; >CACN;F PCMCIH =BCJ MJ?=C;FCT?> @IL BCAB MJ??> N;LA?N NL;=ECHA &H IEEE Trans. Electron. Devices 50 J;A?M U /1 3?LLCFFI &HP?MNCA;NCIH I@ MIG? J;L;G?N?LM I@ NB? =ON;H?IOM NBL?MB IF> @IL PCS ; !; 3CH=C BNNJDJL?>E;L;IE?=IGMIHAMHI @ONOL? CH NB? J;MNPC>?I <

18

M. Kaneko and M. Higashimori

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Robust Drives for Parallel Robots

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22

J. Angeles

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Robust Drives for Parallel Robots

23

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24

J. Angeles

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26

J. Angeles

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28

J. Angeles

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30

A. Kecskeméthy

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Minimal-Form Multibody Dynamics for Embedded Applications

31

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32

A. Kecskeméthy

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Minimal-Form Multibody Dynamics for Embedded Applications

33

$ $ $ $

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34

A. Kecskeméthy

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Minimal-Form Multibody Dynamics for Embedded Applications

35

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36

A. Kecskeméthy

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Minimal-Form Multibody Dynamics for Embedded Applications

37

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0,0 ,0 " + > 0 '0, +" B &* ) '): G4G5CJL@MN: GAA9B

38

A. Kecskeméthy

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2 : .,+": > 0,: , JJ>JR JEGEB $ 1 "1+B

Chapter I

Novel Robot Design and Robot Modules/Components

Constructive Redesign of a Modular Metamorphic Microgripper Luca Bruzzone, Giorgio Bozzini Department of Mechanics and Design of Machines (DIMEC), University of Genova, Genova, Italy E-mail: {bruzzone, bozzini}@dimec.unige.it Abstract. The paper presents the constructive re-design of a modular microgripper equipped with metamorphic fingertips. The experimental tests on the first prototype of the microgripper has shown that the adoption of the metamorphic fingertips permits to increase remarkably the grasping efficiency; nevertheless, the tests have highlighted some operative drawbacks. The proposed constructive re-design is oriented to increase the reliability and to make easier the tuning and maintenance operations.

1 Introduction One of the most important paradigms of the present industrial scenario is miniaturisation. The market size of miniaturized products is constantly growing, due to the market expansion towards Developing Countries. Miniaturization can be achieved by means of the so-called mechatronic approach, that is the strict integration between mechanical, electrical and electronic components. The outlay of the assembly phases represents a large share of the overall production cost for a wide variety of microsystems, in particular in case of small batches (Koelemeijer Chollet et al., 2003). For all of these reasons, the design of flexible, effective and reliable micromanipulation devices is an important issue in robotics research; for example, significant application fields are: assembly of micromechanisms (Zubir and Shirinzadeh, 2008; Koelemeijer Chollet et al., 2002; Yesin and Nelson, 2005); assembly of micro electro-mechanical systems (MEMS) (Wang et al., 2008; Donald et al., 2008; Dechev et al., 2004; Nelson et al., 1998; Yang et al., 2003); manipulation of objects for microscopy observation (Bergander et al., 2000). Industrial microassembly systems are generally composed of a gripper and a machine for the positioning of the gripper within the workspace. As regards microgrippers, there are several techniques that can be used for payload grasping: mechanical, cryogenic, vacuum, magnetic; mechanical grasping is usually adopted in case of complex products because it is more general-purpose.

42

L. Bruzzone and G. Bozzini

Mechanical microgrippers can be divided into two categories: - microtweezers, characterized by two opposing flat tips, with central gripping point; - special-purpose mechanical microgrippers, designed to grasp specific micro-objects by means of specially shaped tips. To simplify the production cycle and consequently to shorten the cycle time, it is evidently useful to adopt general-purpose microgrippers to grasp different components; nevertheless, sometimes non-negligible assembly forces are required and the adoption of special-purpose grippers is necessary. A possible solution to combine operative flexibility and grasping effectiveness is the development of metamorphic fingertips. In the following of the paper the design and the experimental tests on the proposed microgripper, MorphoGrip, will be discussed; then, on the basis of the tests on the preliminary prototype, a partial redesign is proposed to improve the mechanical reliability and to simplify the tuning and maintenance operations.

Figure 1. Design of the metamorphic fingertip (a); grasping of a 1 mm cylinder (b)

2 The MorphoGrip Architecture The starting point of the design is the definition of the reference applications; we have considered the manipulation of cylindrical parts with diameter ranging from 0.5 to 2 mm, in presence of assembly forces up to 0.3 N; however, we tried to obtain a microgripper capable of grasping also payloads with different shapes. On these specifications, we have developed metamorphic fingertips characterized by a highly deformable contact zone. The fingertip plate (Figure 1a) is obtained by electro-discharge machining (EDM) from a Nitinol sheet; the adopted Nitinol alloy has superelastic behaviour with minimum residual strain (Young modulus 75 GPa, tensile strain up to 20% at 20°C). A thin protrusion with square wave shape enhances the contact compliance; the shape of this protrusion has been refined by means of FEM analysis to obtain a good compromise between high flexibility and structural resistance. Adopting the final protrusion design, the contact zone

Constructive Redesign of a Modular Metamorphic Microgripper

a

43

b

Figure 2. Architecture of the finger module (a); finger actuation (b)

Figure 3. Off-line reconfigurability of the modular gripper

maximum deformation is of 60 m while grasping a cylinder with 1 mm radius in presence of a closing force of 0.16 N (Bruzzone and Bozzini, 2007). The fingertip plate is carried by the finger module shown in Figure 2; it is based on a flexible four-bar mechanism (Figure 2b); it is realized in Nitinol alloy, with 2 mm thickness. The fingertip plate is connected to the four-bar plate by a slot and bolts, to tune their relative position. The actuation of the four-bar plate is performed by means of an actuation rod (Figure 2a) pushed along the actuation direction (Figure 2b) by a piezoelectric actuator. The actuator opens the gripper, whereas the closing force is provided by the elastic return force of the four-bar flexure joints; the closing force can be tuned by regulating the position of the fingertip plate in the slot.

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Figure 4. The MorphoGrip prototype

In the proposed gripper, the finger modules can be arranged in two different ways (off-line reconfigurability): a general-purpose two-fingered configuration (Figure 3, left), or a three-fingered configuration (Figure 3, right), with three fingers placed at 120°, to grasp components with axial symmetry.

3 Experimental Results A first prototype of the MorphoGrip has been built (Bruzzone and Bozzini, 2008) (Figure 4) and an experimental campaign has been carried out. The gripper was mounted on a static layout to compare the grasping performance of the metamorphic fingertips with respect to standard non-metamorphic fingertips, characterized by the same geometry, material and surface finishing but without deformable square wave contact zone. The tests were performed using cylindrical payloads with diameters between 1 and 2 mm. In this test layout the linear actuator is vertical, and a vertical force Fg is applied to the payload by means of a wire connected to a small vessel, filling slowly the vessel with liquid (Bruzzone and Bozzini, 2009). Some numerical results are summarized in Table 1. In this table: the closing force Fc (ranging from 0.1 to 0.25 N) is the force exerted on both sides of the payload by the fingertips; Fg,nm is the force Fg that is necessary to wrest the cylinder from the non-metamorphic gripper; the initial slipping force Fg,slip is the force Fg that is necessary to start the slipping of the cylinder in case of metamorphic gripper; the maximum grasping force Fg,max is the force Fg that is necessary to wrest the cylinder from the metamorphic gripper. The tests show that: 1) the grasping release with non-metamorphic fingertips is sudden when Fg reaches the value Fg,nm; 2) with metamorphic fingertips the grasping stability is higher, and the release gradual: when the traction force reaches the value Fg,slip the payload starts to move; a further increment of Fg to

Constructive Redesign of a Modular Metamorphic Microgripper

45

Fg,max is necessary to have the complete release of the grasping; 3) Fg,slip is much greater than Fg,nm . A detailed comparison of metamorphic and non-metamorphic fingertips in different operative conditions is discussed in (Bruzzone and Bozzini, 2009). Table 1. Experimental comparison of metamorphic and non-metamorphic fingertips.

payload type steel cylinder with 1 mm diameter

steel cylinder with 2 mm diameter

Fc [N]

Fg,nm/Fc

Fg,slip/Fc

Fg,max/Fc

0.10 0.15 0.20 0.25 0.10 0.15 0.20 0.25

0.53 0.38 0.24 0.20 0.93 0.66 0.47 0.38

1.40 1.03 0.78 0.69 1.68 1.20 0.98 0.86

1.84 1.50 1.33 1.14 2.12 1.70 1.44 1.32

4 Constructive Re-Design of the Finger Module The experimental results confirm the effectiveness of the design of the metamorphic fingertips. On the contrary, the tests have highlighted some operative drawbacks of the four-bar plate: 1) the alignment of the fingertip is critical: the four-bar plates are connected to the gripper base by means of slots and lateral bolts (Figure 3); small errors in the machining of these slots cause non-negligible misalignments of the fingertips; these misalignments can be corrected only by placing thin sheets in the slots; 2) the flexure hinges of the four bar plate, and in particular the ones of the actuation rod, are frail and have to be handled very carefully; moreover, a damage in one flexure hinge requires the expensive substitution of the whole four-bar plate. In order to solve these problems the finger modules have been re-designed as shown in Figure 5, maintaining the same operating principle. The four flexure hinges have been replaced by thin sheets of superelastic alloy with 1.5 mm thickness (Figure 5, a), connected by bolts; these flexure sheets are endowed with slots (Figure 5, b) to tune the joint lengths. The actuation rods have been replaced by a more robust flexible element (Figure 5, c). The four-bar links (Figure 5, d,e,f) are realized in steel. A screw (Figure 5, g) has been added to tune more precisely the position of the fingertip plate in its slot and consequently the elastic preload. The advantages of this re-design are the following: 1) the alignment of the fingertips can be easily tuned by means of the slots of the flexure sheets; 2) the break of a single flexure joint does not require the substitution of the whole finger module; 3) the elastic return force of the four-bar can be tuned by regulating the

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L. Bruzzone and G. Bozzini

lengths of the flexure joints lj (Figure 5) in a range between 5.5 and 8.5 mm; 4) the module construction is less expensive (lower amount of superelastic alloy).

Figure 5. Second version of the finger module

The structural features of the first and second versions of the finger module have been compared by means of FEM analysis. Let us define: the functional stiffness kf = Fa / dft , where Fa and dft are the actuation force and the corresponding translation of the end of the fingertip (Figure 2b); the lateral stiffness kl = Fl / dft,l , where Fl is a lateral force applied to the end of the fingertip (perpendicular to the four-bar plane) and dft,l is the corresponding lateral displacement of the end of the fingertip; the torsional stiffness kt = Mt / t , where Mt is a moment applied to the lower surface of the link f with direction parallel to the actuation direction and t is the corresponding rotation of the same link. In the estimation of these stiffness values, the model boundary conditions are: upper link of the four-bar fully constrained, upper end of the actuation rod constrained to translate along the actuation direction (Figure 2b). The results of the FEM simulations are collected in Table 2. Figure 6 shows the deformed shapes in the three load cases for the re-designed finger module (lj = 8.5 mm). While the functional stiffness has to be compatible with the maximum actuation force and the required closing force, high values of lateral and torsional stiffness allow a more precise grasping, without unwanted deformations. In the second version the functional stiffness is variable in a range between 0.525 N/mm (lj = 8.5 mm) and 1.083 N/mm (lj = 5.5 mm). This allows a wider regulation of the

Constructive Redesign of a Modular Metamorphic Microgripper

47

closing force: in the first version, on the contrary, it can be tuned only by moving the fingertip plates in their slots.

Figure 6. Displacements in mm (lj = 8.5 mm); load cases corresponding to: functional stiffness, Fa = 1 N (a); lateral stiffness, Fl = 1 N (b); torsional stiffness, Mt = 1 Nmm (c). Table 2. Functional, lateral and torsional stiffness of the different finger modules. Finger module first version second version (lj =8.5 mm) second version (lj =5.5 mm)

kf [N/mm] 0.823 0.525

kl [N/mm] 2.083 5.434

kt [Nmm/rad] 820.5 3097.3

1.083

8.696

6730.8

The increments in the values of lateral and torsional stiffness mean that the structural quality is better even if the flexure hinges (characterized by lumped section restriction) are replaced with flexure sheets (with distributed elasticity).

5 Conclusions MorphoGrip is a modular and reconfigurable microgripper designed to increase the level of operative flexibility with respect to the state-of-the-art. The experiments on the first prototype has shown that the adoption of the metamorphic fingertips allows to remarkably increase the strength and the stability of the grasping; nevertheless, the tests have highlighted some practical drawbacks. The second version is characterized by an easier tuning of the alignment of the fingertips, by an easier and more flexible regulation of the closing force and by lower costs of realization and maintenance. In the next future the second prototype will be realized; it can be

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considered as a pre-production development stage of a general-purpose industrial microgripper.

References Bergander, A., Breguet, J.M., Schmitt, C., and Clavel R. (2000). Micropositioners for Microscopy Applications based on the Stick-Slip Effect. In Proceedings of the IEEE Intl. Symposium on Micromechatronics and Human Science (MHS 2000), Nagoya, Japan, October 22-25, 2000, 213-216. Bruzzone, L., and Bozzini, G. (2007). Design of electro-discharge machined flexible fingertips for microassembly. In Proceedings XVIII Congresso Associazione Italiana di Meccanica Teorica e Applicata (AIMETA 2007), Brescia, Italy, September 11-14, 2007, CD. Bruzzone, L., and Bozzini, G. (2008). MorphoGrip: metamorphic gripper for miniaturized assembly. In Proceedings of the RAAD 2008, 17th International Workshop on Robotics in Alpe-Adria-Danube Region, Ancona, Italy, September 15-17, 2008, CD. Bruzzone, L., and Bozzini, G. (2009). A flexible joints microassembly robot with metamorphic gripper. Accepted for publication on Assembly Automation. Dechev, N., Cleghorn, W.L., and Mills, J.K. (2004). Microassembly of 3D Microstructures Using a Compliant, Passive Microgripper. IEEE Journal of Microelectromechanical Systems 2(13):176-189. Donald, B.R., Levey, C.G., and Paprotny I. (2008). Planar Microassembly by Parallel Actuation of MEMS Microrobots. IEEE Journal of Microelectromechanical Systems 4(17):789-808. Koelemeijer Chollet, S., Moll, B., Bourgeois, F., Wulliens, C. Benmayor, L., and Jacot, J. (2002). A Flexible Microassembly Cell for Small and Medium Sized Batches. In Proceedings of the Intl. Symposium on Robotics (ISR02), Stockholm, Sweden, October 7-11, 2002, CD. Koelemeijer Chollet, S., Bourgeois, F., and Jacot, J. (2003). Economical Justification of Flexible Microassembly Cells. In Proceedings of the 5th IEEE Intl. Symposium on Assembly and Task Planning, Besancon, France, July 10-11, 2003, 48-53. Nelson, B., Zhou, Y., and Vikramaditya, B. (1998). Sensor-based microassembly of hybrid MEMS devices. IEEE Control Systems Magazine 6(18):35-45. Wang, L., Mills, J.K., and Cleghorn, W.L. (2008). Automatic Microassembly Using Visual Servo Control. IEEE Tran. on Electronics Packaging Manufacturing 4(31):316-325. Yang, G., Gaines, J.A., and Nelson, B.J. (2003). A Supervisory Wafer-Level 3D Microassembly System for Hybrid MEMS Fabrication. Journal of Intelligent and Robotic Systems 1(37): 43 68. Yesin, K.B., and Nelson, B.J. (2005). A CAD model based tracking system for visually guided microassembly. Robotica, 4(23):409 418. Zubir, M.N.M., and Shirinzadeh, B. (2008). Development of a Compliant-Based Microgripper for Microassembly. In Proc. IEEE/ASME Intl. Conf. on Mechtronic and Embedded Systems and Applications (MESA 2008), Beijing, China, Oct. 12-15, 2008, 522 527.

A One-Motor Full-Mobility 6-PUS Manipulator Patrick Grosch(°), Raffaele Di Gregorio(°°), Federico Thomas(°)∗ (°)Institut de Robòtica i Informàtica Industrial (CSIC-UPC), Barcelona, Spain (°°)Department of Engineering – University of Ferrara, Ferrara, Italy

Abstract. This paper presents the feasibility study of an under-actuated parallel manipulator with 6-PUS topology, destined to handle work-tables in CNC machine tools. The proposed device exploits the fact that, in such an application, the path between the initial and final poses of the mobile platform is not assigned to reduce the number of actuators to only one.

1 Introduction The need of making an object move along an assigned path arises only in a limited number of applications. In most cases, the only initial and final poses of the object are assigned, whereas the path between them must just satisfy weak constraints (e.g., obstacle avoidance, preventing interferences among machine components, etc.) which leave the choice of the path practically free. Such a freedom can be exploited during design to simplify the machine architecture. Work-tables of machine tools usually either perform simple translations or just lock the workpiece during cutting. Thus every time the workpiece has to be reoriented or, in general, repositioned with respect to the spindle axis either manual operations or external devices must intervene. Repositioning workpieces is a manipulation task that involves small six-dimensional workspaces, good positioning precision and high stiffness in the final configuration; it does not impose any constraint to the path between the initial and final poses. Parallel manipulators can satisfy the requirements on positioning precision and stiffness; moreover, they are specially suitable for applications that involve small workspaces. Therefore, they are natural candidates to move the work-table during workpiece repositioning.

—————————— ∗

This work was supported by the Spanish Ministry of Science and Innovation, by the contribution of Regione Emilia Romagna (District Councillorship for Productive Assets, Economic Development, Telematic Plan), PRRIITT misura 3.4 azione A, to InterMech (Division Acoustics and Vibrations - LAV), and by UNIFE funds.

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How to exploit the free path for reducing the complexity of a manipulator destined to move the work-table during repositioning is an open problem. This paper proposes an under-actuated parallel manipulator that, by exploiting the free path, is able to control the mobile platform pose in a six-dimensional workspace by using only one motor. Section 2 describes the manipulator architecture and illustrates its operation. Section 3 addresses the kinetostatic analysis of the machine and gives conditions the path must satisfy to keep the mobile platform pose controllable during motion. Eventually, the conclusions are drawn in section 4.

2 Manipulator Architecture and Operation Parallel manipulators (PMs) with topology 6-PUS1 feature a mobile platform connected to a fixed base through six in-parallel kinematic chains (legs) of type PUS. Their architectures vary according to the relative disposition of the prismatic-pair sliding directions, the platform geometry, and the six fixed distances (leg lengths) between universal-joint center and spherical-pair center of each leg. By changing these geometric parameters, a number of 6-PUS PMs have been proposed in the literature (see Merlet (2006, chap. 2) for Refs.). Boye and Pritschow (2005) named them linapods. Honegger et al. (1997) proposed the Hexaglide that has six parallel and coplanar guides. Moreover, some of the proposed architectures (Bernier et al. 1995; Pritschow et al. 2002) exhibit coincident guides for couples of prismatic pairs, and, in particular, Nabla 6, proposed by Bernier et al. (1995), has three coplanar guides each carrying two sliders. The actuation of each prismatic pair is independent of the other actuations in all the linapods proposed in the literature. Figure 1 shows the proposed linapod. On the base, a single motor, through a transmission, actuates, one at a time, two racks that are constrained to slide along two mutually orthogonal guides forming a cross-shaped path. The transmission is able to actuate one or the other rack by using two clutches that also act as brakes for the non-actuated rack. The racks carry suitably shaped hooks which can firmly lock revolute-pair housings (the cubes attached to the racks in Fig. 1). In these revolute-pair housings, legs’ universal joints insert one pin of their cross link so that the resulting revolute pair has the axis perpendicular to the plane of the guides. In so doing, all the universal joints have the other revolute-pair axis parallel to the plane of the guides, and their centers are constrained to lie on T paths that are all parallel to the plane of the guides. The universal-joint centers slide on these T paths when the racks are moved. On the mobile platform, the housing of the spherical pairs, which join the leg endings to the platform, are embedded in the platform. —————————— 1

P, U, and S stand for prismatic pair, universal joint, and spherical pair, respectively; whereas, the underscoring indicates that the corresponding joint is actuated.

A One-Motor Full-Mobility 6-PUS Manipulator

51

mobile platform spherical-pair housing

base

rack II guide II

guide I revolute-pair housing

rack I

(a)

(b)

(c)

Figure 1. Under-actuated 6-PUS: (a) 3D CAD model, (b) detail of the racks, the guides, and some of the revolute-pair housings, (c) revolute-pair housing at the intersection between the two racks.

The hooking between rack and revolute-pair housing is managed by a purely mechanical device carried either on the revolute-pair housing or on the guides. This hooking device and the hooks on the racks are conceived so that the following functional requirements are satisfied:

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P. Grosch, R. Di Gregorio and F. Thomas

(i) if the revolute-pair housing is in contact with only one rack, the hooking device must provide a constraint that, combined with the constraint of rack’s hooks, firmly holds the housing fixed to the rack; (ii) if the revolute-pair housing is in contact with both racks (i.e., at the intersection of the T path), the hooking device must not provide any constraint, whereas the hooks of both racks must provide the constraints necessary to hold the housing; (iii) if the revolute-pair housing is at the intersection of the T path and one rack starts moving, the hooks of the moving rack must be able to tow the housing, whereas the hooks of the other rack must not forbid this towing and must be so shaped that, during the housing motion, make the hooking device intervene to provide its constraint.

rack II 4 5

6

rack II 4 5

3

6

rack II 4 5

3

2

(a)

3

(b)

2

6 2

(c)

rack I

rack I

1

1

rack I 1

rack I

rack I rack II 4 5

(d)

3

2

1

6

rack II 4 5

3

(e)

1

2

6

rack II 4 5

2

6

(f) 3 rack I 1

Figure 2. Sequence of rack motions (the numbered boxes represent the revolute-pair housings): (a) initial configuration, (b) rack I was moved to make housing 3 touch rack II, (c) rack II was moved to make the dotted box centered on rack I, (d) rack I was moved to make housing 2 touch rack II, (e) rack II was moved back to its initial position, (f) final configuration obtained by moving back rack I to its initial position.

A One-Motor Full-Mobility 6-PUS Manipulator

53

Many hooking devices and complementary hooks for the racks can be easily devised to satisfy the above technical requirements. For instance, in Fig. 1, the hooks of rack I are dovetail joints parallel to rack II, whereas the hooks of rack II are frontal teeth with rectangular cross section; moreover, the hooking device is constituted of two lateral stops fixed to guide I, and of dovetail joints, identical to the ones of rack I, fixed to guide II (see Figs. 1(b) and 1(c)). The many design alternatives for these equipments will not be discussed here, for the sake of conciseness. By exploiting the above-reported properties of hooking device and racks’ hooks the positions of the universal-joint centers on the T paths can be about freely changed. In fact, if, for instance, housing 2, on rack I, (Fig. 2(a)) must be moved to the dotted position on rack II and housing 3 must be moved to the actual position of housing 2, the sequence of operations shown in Fig. 2 can be implemented. In general, many different rack-motion sequences lead to the same final configuration, and the number of operations to implement may decrease when the number of housing permutations increases.

3 Kinetostatics and Constraints on the Path The inverse position analysis (IPA) that, for this linapod, means the determination of the housing positions on the racks for an assigned platform pose (position and orientation) must be solved every time the platform is repositioned. This determination is straightforward once the positions of the universal-joint centers have been computed. The assigned relative pose between base and platform involves that the six T paths (one per leg), the universal-joint centers must lie on, have assigned poses with respect to the spherical-pair centers embedded in the platform. Thus, for each leg, the determination of the universal-joint center’s position reduces itself to compute the intersection points between the T path the universal-joint center must lie on and a sphere, with center at the spherical-pair center and radius equal to the leg length. This geometric problem has at most four solutions: the two sets of intersections between the sphere and the two sides of the T. At most four solutions for each leg yields at most 46 (i.e., 4096) leg arrangements compatible with an assigned platform pose. Such a high number of IPA solutions is mainly theoretical. In fact, many line-sphere intersections will fall out of the line segments actually occupied by the T-path’s sides. Moreover, other solutions will be excluded by the fact that two or more housings cannot be located on the same position, and that, on each rack, the hooks’ sequence has a fixed pitch, which implies that the distance between couples of housings positioned on the same rack can only be multiples of the hooks’ pitch. Eventually, all the leg arrangements that give a singular configuration (see below) must be excluded. The direct position analysis (DPA) of the proposed linapod consists in the determination of the platform poses compatible with an assigned disposition of the

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revolute-pair housings on the two racks. If the positions of the revolute-pair housings are assigned, the positions of the universal-joint centers will be assigned, too. Thus this problem reduces itself to the determination of the assembly modes of the 6-US structure (i.e., two rigid bodies connected by six in-parallel US legs), which was broadly treated in the last two decades in connection with the DPA solution of the general Stewart platform (see Merlet (2006) for Refs.). The result of these studies is that the 6-US structure can have at most forty assembly modes which can be even analytically determined (Husty, 1996; Innocenti, 2001). The singularities of the forward instantaneous kinematics are, for this linapod, the configurations where the platform can perform instantaneous motions even though the racks are locked (i.e., they are uncertainty configurations of the 6-US structure). At a singularity of this type, the platform pose is not controllable, and the internal loads of one or more links of the legs are not able to equilibrate the external loads applied on the platform. Thus, they must be identified during design and avoided during operation. The uncertainty configurations of the 6-US structure have been studied by many authors, and, in the literature, both geometric and analytic conditions to identify them have been provided (see, for instance, Merlet (1989), McCarthy (2000), St-Onge and Gosselin (2000), Di Gregorio (2002)). The actual implementation of the proposed linapod requires that all this literature be exploited to correctly design and control it. In this paper, for the sake of brevity, the authors will only give the justification of some design choices due to the need of avoiding uncertainty configurations. From a static point of view, in 6-US structures, an uncertainty configuration occurs when the six forces applied to the platform through the spherical pairs are not able to equilibrate any system of external loads. The fact that these forces are aligned with the leg axes2 allowed the geometric classification of the singular configurations through particular arrangements of the six leg axes (McCarthy, 2000). All these singular arrangements satisfy at least one out of the following three geometric conditions: (a) the six axes either intersect or are parallel to a line, (b) the six axes are all parallel to a plane, and (c) the six axes are tangent to coaxial helices with the same pitch. For the linapod under study, the possibility of locating all the revolute-pair housings on one rack would greatly improve the path planning algorithms (see below). Therefore, making this housing arrangement non-singular is important. Once all the housings are located on the same rack all the universal-joint centers lie on the same plane. And, in order to avoid the geometric conditions (a) and (b), the universal-joint centers must not be located at the same height on the rack (i.e., the T paths must not coincide); whereas the spherical-pair centers must be suitably dis—————————— 2

In a US leg, the leg axis is the line passing through the centers of the universal joint and of the spherical pair.

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55

tributed on the platform. The manufacturing conditions that allow condition (c) to be avoided are much more difficult to be visualized and a careful numerical check is necessary. It is worth noting that the leg arrangements with all the housings on the same rack geometrically coincide with the Hexaglide architecture (Honegger et al., 1997), and the results obtained for the Hexaglide can also be usefully exploited. A path-planning algorithm for the proposed linapod has to take into account all the above-reported kinetostatic considerations. In addition, it needs the implementation of a motion strategy for choosing the sequence of rack motions able to move from the initial platform pose to the final one. Each step of this sequence finishes with a particular arrangement (state) of the revolute-pair housings on the racks that is reached when both the racks are at rest and the actuation is about to be switched from one rack to the other. Thus, a path-planning algorithm has to determine the states’ sequence by respecting the rule that the transition from one state to the successive one must be possible by moving only one rack. For instance, the motion described in Fig. 2 is characterized by six states and five transitions. Two different paths that have the same initial and final housing arrangements can be compared on the basis of the number of intermediate states, and, of course, the lower is the number the better is the path. The sequence that moves only one housing from any position to any other without changing the positions of the other housings, in the final state, can be easily automated. Thus, a simple path-planning algorithm could reduce itself to implement six separate sequences each of which brings only one housing from its initial to its final position and, in the final state, does not change the positions of the housings already brought to their final positions. Such a motion strategy employs a great number of intermediate states. For instance, it is easy to realize that, in Fig. 2, the motion of the only housing 2 without changing the position of housing 3 would have required nine states, whereas the strategy reported in Fig 2 uses only six states to move both housings 2 and 3 to their final positions. A much better motion strategy can be obtained by finding a state (parking state) from which any other state can be reached through a reduced number of intermediate states. The state with all the housings located on rack II and no housing at the rack intersection could be a parking state. In fact, from this parking state, a housing can be put on rack I, at any position, with a sequence involving only two intermediate states, whereas only four intermediate states are required to change the position of a housing on rack II. A path-planning algorithm based on this parking state, first, has to implement the sequences that bring all the housings on rack II (note that the only housings located on rack I in the initial state are involved in this phase); then, it has to move all the housings from the parking positions to their final positions.

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4 Conclusions The feasibility study of an under-actuated parallel manipulator with 6-PUS topology, destined to handle work-tables in CNC machine tools, has been presented. The proposed device exploits the fact that, in such an application, the path between the initial and final poses of the mobile platform is not assigned to reduce the number of actuators to only one. For the proposed manipulator, all the hardware critical points have been addressed. Its kinematic and static characteristics have been discussed, and the availability of the solutions to all the problems involved in its design and control has been verified. Two motion strategies that can be used in the path-planning algorithms have been proposed. A formalization of the allowed rearrangements using group theory will probably provide a deeper insight into this path planning problem (Joyner, 2002). This is certainly a point that deserves further attention References Bernier, D., Castelain, J.-M., and Li, X. (1995). A new parallel structure with six degrees of freedom. In Proceedings of the 9th World Congress on the Theory of Machines and Mechanisms, Milan (Italy), 8 - 12. Boye, T., and Pritschow, G. (2005). New transformation and analysis of a N-DOF LINAPOD with six struts for higher accuracy. Robotica 23: 555-560. Di Gregorio, R. (2002). Singularity-locus expression of a class of parallel mechanisms. Robotica 20: 323-328. Honegger, M., Codourey, A., and Burdet, E. (1997). Adaptive control of the Hexaglide a 6 dof parallel manipulator. In Proc. of the 1997 IEEE Int. Conf. on Robotics and Automation, Albuquerque, 543-548. Husty, M.L. (1996). An algorithm for solving the direct kinematics of the Stewart-Gough platform. Mechanism and Machine Theory 31(4): 365-380. Innocenti, C. (2001). Forward kinematics in polynomial form of the general Stewart platform. Journal of Mechanical Design 123(2): 254-260. McCarthy, J.M. (2000). Geometric Design of Linkages. Springer-Verlag, New York, chapter 12: 272-283. Merlet, J.-P. (1989). Singular configurations of parallel manipulators and Grassmann geometry. Int. J. of Robotics Research 8(5): 45-56. Merlet, J.-P. (2006). Parallel Robots. Dordrecht, The Netherlands: Springer, 2nd edition. Pritschow, G., Eppler, C., and Lehner, W.-D. (2002). Highly dynamic drives for parallel kinematic machines with constant arm length. In Proc. 1st Int. Colloq., Collaborative Res. Center, Vol. 562, Braunschweig, Germany, 199–211. St-Onge, B.M., and Gosselin, C.M. (2000). Singularity analysis and representation of the general Gough-Stewart platform. Int. J. of Robotics Research 19(3): 271-288. Joyner, D. (2002). Adventures in Group Theory. Rubik’s cube, Merlin’s machine and other mathematical toys. The Johns Hopkins University Press, Baltimore (USA).

Dynamic Hybrid Position/Force Control for Parallel Robot Manipulators Wenbin Deng, Hyuk-Jin Lee and Jeh-Won Lee Dept. of Mechanical Engineering, Yeungnam University, Gyeongsan, Korea Abstract It is necessary to control both the position of a manipulator and the force when a robot manipulator interacts with the environment. In this paper, a simple compliant motion control algorithm for parallel robot manipulators is presented. Based on the concept of hybrid position/force control scheme, the task space is split into position controlled and force controlled subspaces. We designed two controllers for position and force control to track the desired position trajectories and force in operational space. Experimental results obtained from a two degree of freedom parallel robot show and verify the effectiveness of the control method.

1

Introduction

Force control has been known to be one of the complicated control methods for robotic manipulator interacting with other objects. Specially, the desired force should be maintained while following the desired trajectories when the robot contact with the environment. So it is necessary to control both force and motion when a robot manipulator interacts with the environment. At the same time, force control of robot manipulators has been studied by many researchers (see Whitney, 1977, Khatib, 1985, Li et al., 2009 and so on). Although the demand of applying a force control technique is increasing, many problems still remain to be solved. Recently, parallel robot manipulators are being more and more used in industry. A parallel robot typically consists of a moving platform which is connected to a fixed base by several kinematic chains, also called “limbs” or “legs”. Typically, every limb is controlled by one actuator and all the actuators can be mounted at or near the fixed base. Comparing with a traditional serial manipulator that is constituted of rigid-body links and joints connected in series, a parallel robot manipulator has a more rigid structure and better payload-carrying ability. So it is more suitable for situations in high precision, stiffness, velocity, and heavy load-carrying suggested by Merlet (2001). In this paper, we adopt hybrid position/force control scheme because most machining tasks require precise positional control in some directions and force control in others. Based on the dynamics of parallel manipulators, two controllers were designed to achieve the desired force and position.

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Dynamic Modeling of Parallel Manipulator

The equation of motion for the constrained parallel robotic manipulator, considering the contact force and the constraints, is given (Tsai, 1999) in joint space as follows: M (q)q + h(q, q )q + G (q ) = u − J −T ( q) Fe

(1)

where: q , q , q denotes the joint angle, velocity, acceleration of the manipulator; M(q) is the robot inertia matrix which is symmetric and positive; h(q, q ) contains the Centrifugal terms and Coriolis terms; G(q) is the gravity matrix; u denotes the joint input control torques; Fe is the interaction forces associated with the constraints; J(q) is the Jacobian matrix of parallel manipulator. It is well known that when the end-effector contacts the environment, a task space coordinate system defined with reference to the environment is convenient for the study of contact motion. In the absence of friction, the dynamic equation of a parallel manipulator can be expressed by Lagrangian approach in the task space as

M ( X ) X + h( X , X ) = F − Fe

(2)

, F = JTu . where: M ( X ) = J T M (q) J , h( X , X ) = J T h(q ) J − J T M (q ) JJJ It is easy to see that the control F should be chosen according to F = Fe + M ( X ) X + h( X , X )

(3)

The generalized input force F is related to the input torque u by Jacobian matrix. u = J −T ( q ) F

3

(4)

Hybrid Position/Force Control Method

Precise control of manipulators in the face of uncertainties and variations in their environments is a prerequisite to feasible application of robot manipulators to complex handling and assembly problems. A pure position control of a manipulator is severely limit in a manufacturing environment where uncertainties are significant and sophisticated tasks such as polishing, deburring, grinding etc.. The necessity of force control along with position control was recognized and proposed by Raibert et al. in 1981. But it did not take into account the manipulator dynamics rigorously. Yoshikawa (1987) presented the dynamic hybrid control approach which is based on the dynamic equation of the robot manipulators and a description of the end effector constraints. However, he took dynamics of serial manipulator into consid-

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eration and adopted a SCARA robot for the application to test the control scheme. We adopted and developed this method for the parallel robots in this paper. To implement the hybrid position/force scheme, we design both a position control law along force constrained directions and a force control law along position constrained directions through the use of so-called selection matrices. Such an overall control scheme is shown in Figure 1. The inner loop control law consists of an inverse dynamics control computed in task space.

Figure 1. Hybrid control architecture.

The selection matrix S is diagonal with ones and zeros shown in Figure 1, which is used to decide that each DOF is uniquely specified as being either position controlled or force controlled. We derive the control law as follows. Considering the dynamics of a parallel robot as equation (2) and the hybrid position/force control scheme in Figure 1, we define the control force as F = M ( X ) SAX + h( X , X ) + ( I − S ) Af

(5)

where AX and Af are the outer loop control inputs for position control and force control respectively. The selection matrix S is diagonal with ones and zeros on the diagonal. When a one is present in S, position control is in effect. Otherwise, for force control, a zero is present. Substituting equation (5) into (2) yields M ( X )( X − SAX ) + [ Fe − ( I − S ) Af ] = 0

(6)

Since we desire to control position, we would like to use a servo-control law for AX discussed by Satya et al. (1995), and we add an integral term to the control law to eliminate steady-state error. Hence AX = ρd + K d ( ρ d − ρ ) + K Pp ( ρ d − ρ ) + K Pi ∫ ( ρ d − ρ )

(7)

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where Kd is the derivative gain, KPp is the proportional gain and KPi is the integral gain. and Kd, KPp and KPi are constant diagonal matrices, ρ d is the desired position. And we use a PI controller to achieve force tracking. t

Af = K Fp ( f d − f ) + K Fi ∫ ( f d (u ) − f (u ))du 0

(8)

where KFp, KFi are constant diagonal matrices, fd is the desired force. Now we can simultaneously track motion and force by choosing AX = [ a Xi ] , Af = ⎡⎣ a fi ⎤⎦

(9)

where ⎧a Xi = ρdi + K di ( ρ di − ρ i ) + K Ppi ( ρ di − ρi ) + K Pii ( ρ di − ρi ) ∫ ⎪ ⎨ t ⎪a fi = K Fpi ( f di − f i ) + K Fii ∫0 ( f di (u ) − f i (u ))du ⎩

(10)

afi = 0 if i belongs to the position controlled subspace; else i belongs to the force controlled subspace. Base on the above discussion, we can get the block diagram of the hybrid control scheme as Figure 2.

Figure 2. The control block scheme of the proposed hybrid control method.

4

Experiments

In this section, the proposed control method is applied to a new five-bar parallel manipulator and experimental results are presented.

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61

Experimental Setup

The robot used in this experiment is shown in Figure 3. This manipulator has two degrees of freedom as illustrated in the schematic diagram of Figure 4. Two AC servo motors (FMA- CN04A-AB00) are attached to the joint A of link AE and C of link CG respectively. Motor rotations are measured by means of two incremental encoders with 2000 pulses/rev resolution. Velocity is estimated through numerical differentiator. An ATI GammaSI-130-10 force/torque sensor with a resolution of 0.05N is mounted at the end-effector. A low-pass second order filter is used to process the signal from force/torque sensor. The contact is achieved by a bearing whose purpose is to reduce tangential friction force which may arise from contact with the environment. Communication with the manipulator is provided by a data acquisition card (Quanser Q8 H.I.L Control Board) mounted in PCI slot of a personal computer and terminal board.

Figure 3. The experimental setup.

Figure 4. The schematic diagram.

The system software core is WinCon, real-time Windows XP application. It allows running code generated from a Simulink diagram in real-time on the same PC. Acquisition and control were performed with sampling time of 0.001 s. Chung et al. (2001) investigated the kinematic analysis and performance of this parallel robot. We can get the Jacobian matrix and kinematic equations of the parallel manipulator from it. The dynamic model of this manipulator can be described by equation (1) as explained in section 2. The parameters of the robot can be detailed as

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⎡ m11 ⎢m M (q ) = ⎢ 21 ⎢ 0 ⎢ ⎣ 0

m12

0

m22

0 m33 m43

0 0

0 ⎤ ⎡0 ⎥ ⎢h 0 ⎥ , h(q, q ) = ⎢ 21 ⎢0 m34 ⎥ ⎥ ⎢ m44 ⎦ ⎣0

h12

0

0

0

0 0

0 h43

0⎤ ⎡0⎤ ⎥ ⎢u ⎥ 0⎥ , u = ⎢ 2 ⎥ (11) ⎢0⎥ h34 ⎥ ⎥ ⎢ ⎥ 0⎦ ⎣u4 ⎦

where m11 = m1lc12 + m3lc 3 2 + m4 l12 + I1 + I 3 , m12 = m21 = (m3l2 lc 3 + m4 l1lc 4 ) cos(θ 2 − θ1 ) , m22 = m2lc 2 2 + m3l2 2 + m4lc 4 2 + I 2 + I 4 , m33 = m5lc 5 2 + m7 lc 7 2 + m8 l5 2 + I 5 + I 7 , m34 = m43 = (m7 l6 lc 7 + m8l5lc 8 ) cos(θ 4 − θ 3 ) , m44 = m6 lc 6 2 + m7 l6 2 + m8 lc 82 + I 6 + I8 , h12 = −(m3l2 lc 3 + m4 l1lc 4 ) sin(θ 2 − θ1 )θ2 , h21 = (m3l2 lc 3 + m4 l1lc 4 ) sin(θ 2 − θ1 )θ1 , h34 = −(m7 l6 lc 7 + m8l5lc 8 ) sin(θ 4 − θ3 )θ4 , h43 = ( m7 l6 lc 7 + m8 l5lc 8 ) sin(θ 4 − θ 3 )θ3 . θi denotes the joint angle; mi, li and lci denote the mass, the length and the distance from the previous joint to the center of mass of link i respectively; Ii denotes the moment of inertia of link i about an axis coming out of the page, passing through the center of mass of link i. 4.2

Approaching and Contacting a Stiff Wall

The first experiment is to test the performance of the robot with the proposed control method when the end-effector approaches and contacts a stiff environment. A wooden wall was used as the environment object. For the 2-DOF manipulator, the selection matrix was S=diag(1,0). Along the position controlled direction, Xd = 0 , X d = 0 , X d = X 0 , X 0 is the initial position. For the force controlled direction, the desired force was 20N. The end-effector was controlled to approach and contact the wooden wall along the normal. Figure 5 is the force response. We can see that the system remains stable after the collision from it. Figure 6 shows the position response of the end-effector.

Figure 5. Force response.

Figure 6. Position response.

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Approaching objects is an important phase of automatic tasks with robots. This experiment shows that the proposed control method is competent to control the manipulator to approach and contact the stiff environment. And the distance between the end-effector and the object surface can be totally unknown. 4.3

Trajectory Tracking in Unconstrained Environment

A trapezoidal velocity profile is assigned in Y-axis, which imposes the resulting trajectory formed by a linear segment connected by two parabolic segments to the initial and final positions. The position of the start and end points of the trajectory is (0, 0.2) and (0, 0.3) respectively. The selection matrix was S=diag(1,1), both directions were position controlled. The end-effector moved along the trajectory in 10 seconds. Figure 7 shows position error of the end-effector.

Figure 7. Position error of the end-effector.

From Figures 7, we can see that the manipulator can follow the desired trajectories within small error during the whole tracking process. 4.4

Trajectory Tracking in Constrained Environment

The last experiment is designed to demonstrate the effective control of force and position in a constrained environment. The selection matrix also was S=diag(1,0). The X-axis was position controlled, and the Y-axis was force controlled. The position trajectory of the end-effector along X direction is illustrated in Figure 8. The external force along Y direction shown in Figure 9 was applied to the end-effector.

Figure 8. Position of the end-effector.

Figure 9. Force of the end-effector.

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From Figures 8-9, we can see that the hybrid position/force control method has the ability to control the position and force in the constrained environment. All the experimental results show that the proposed dynamics based hybrid control strategy can control the parallel manipulator working in both the unconstrained environment and the constrained environment.

5

Conclusions

Parallel robot manipulators are beginning to play an important role in industry, especially in process of assembly, grinding, deburring and so on. They need position/force control in unconstraint and constraint environment at different time. Based on the task of position/force control, we designed two controllers for position control and force control respectively. This work shows satisfactory performance when such control techniques were implemented on a two degree of freedom parallel robot. Several experiments have been done to test the control scheme. The results have confined the effectiveness of the proposed control approach.

Bibliography D. E. Whitney. Force feedback control of manipulator fine motions. Trans. ASME, J. DSMC, vol. 99, 91–97, June 1977. K. J. Salisbury. Active stiffness control of a manipulator in Cartesian coordinates. In Proc. 19th IEEE Conf. Decision and Control, 95–100, Nov. 1980. O. Khatib. The operational space formulation in the analysis, design, and control of robot manipulators. In Proc. 3rd Znt. Symp. Robotics Research, 103–110, Oct. 1985. M. Callegari, M. C. Palpacelli, M. Principi. Dynamics modelling and control of the 3-RCC translational platform. Mechatronics, vol. 16, 589–605, 2006. Y. M. Li, Q. S. Xu. Dynamic modeling and robust control of a 3-PRC translational parallel kinematic machine. Robotics and Computer-Integr. Manuf., vol. 25, 630–640, 2009. J. P. Merlet. A generic trajectory verifier for the motion planning of parallel robots. Trans. of the ASME, J. of Mech. Design, vol. 123, no. 4, 510–515, Des. 2001. L. W. Tsai. Robot Analysis: The Mechanics of Serial and Parallel Manipulators. New York: John Wiley & Sons, Inc., 1999. M. H. Raibert and J. J. Craig. Hybrid Position/Force Control of Manipulators. ASME J. of Dynamic System, Measurement and Control, Vol. 102, 126–133, 1981. T. Yoshikawa. Dynamic hybrid position/force control of robot manipulators-description of hand constraints and calculation of joint driving force. IEEE J. of Robotics and Automation, Vol. RA-3, No. 5, 386–392, Oct. 1987. S. M. Satya and M.W. Spong. Hybrid control of a planar 3-Dof parallel manipulator for machining operations”, Trans. of the NAMRI/SME, Vol. 23; 273–280, 1995. Y. H. Chung and J. W. Lee. SenSation: A new translational 2 DOF haptic device with parallel mechanism. Trans. on Control, Autom., and Syst. Eng., Vol. 3, 217–222, 2001.

Decoupled Parallel Manipulator with Universal Joints and Additional Constraints Leonid Tyves1, Victor Glazunov1, Pavel Danilin1, Nguyen Minh Thanh2 1

Mechanical Engineering Research Institute of RAS, Moscow, Russia HoChiMinh City University of Transport, HoChiMinh City, Vietnam

2

Abstract. Decoupled parallel manipulator with three parallel kinematic chains is considered. Each kinematic chain contains two universal joints with additional constraints. The synthesis of this manipulator is carried out by means of screw groups. This approach allows avoiding completed equations by synthesis and singularity analysis of mechanisms.

1 Introduction It is well-known that the closed-loops of parallel manipulators cause high stiffness and payload capacity (Merlet, 2000, Cecarelli, 2004, Kong and Gosselin, 2007). However, due to the coupling between kinematic chains, control of the motions of the moving platform (end-effector) becomes complicated. There exist different solutions of this problem. One of them is to arrange for coincidence of centers of the spherical kinematic pairs of three identical kinematic chains S-P-S of the Gough-Stewart platform (Koliskor, 1986, Innocenti and Parenti-Castelli, 1991). Another solution corresponds to the special architecture of the U-P-U kinematic chains of a 6 degrees of freedom (6-DOF) parallel manipulator in which three U-joints mounted on the moving platform are designed as a spherical mechanism (Mianovski, 2007). Another approach to a solution of this problem is applicable for a parallel manipulator with reduced degrees of freedom. For example, the well known Delta robot consists of three R-R-P-R kinematic chains causing translation motions of the moving platform and of one R-U-P-U kinematic chain, causing rotation about the vertical axis (Clavel, 1990). This robot corresponds to Schoenflies motions besides, in this robot, three translation motions and one rotation motion are decoupled. Another well known robot with Shoenflies motions of the platform is PAMINSA (Arakelyan et al., 2005). In this manipulator one vertical motion is decoupled from planar motions. Note that the translation kinematic pairs can be represented as planar four-bar parallelograms (Mianovski, 2007, Clavel, 1990, Gogu, 2009). By this approach

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numerous families of decoupled and translation parallel mechanisms are obtained. A 6-DOF manipulator with decoupled translation and rotation motions and with linear and rotating actuators is synthesized by means of geometrical constraints (Yan Jin et al., 2007). This manipulator consists of three kinematic R-P-P-P-R-R chains. In this paper, we use the approach to synthesis of decoupled parallel manipulators based on closed screw groups (Dimentberg, 1965) that include all the screw products of the main members of these groups. Similar screw groups are considered from different viewpoints (Sugimoto, 1990, Herve, 1991, Angeles, 2004, Glazunov, 1995). In this article a new decoupled 6-DOF parallel manipulator is proposed which consists of three kinematic R-P-P-P-R-R chains where two P-pairs correspond to two U-joints with additional constraints. Structural design is discussed with details but no performance characteristics are outlined. With regard to the determination of the singularity the Jacobian matrices can be applied (Gosselin and Angeles, 1990). Using this approach one needs to differentiate equations expressing the constraints. In this paper we use the screw groups approach to describe singularities which allows avoidance of mathematical equations.

2 The kinematic chains with universal joints and additional constraints and 4-DOF mechanisms Let us consider a kinematic chain with two universal joints (Figure 1, a). These U-joints can be represented by four rotation kinematic pairs. In particular, the Plücker coordinates of the unit screws E1 , E2 , E3 , E4 of the R-pairs can be expressed as E1 (1, 0, 0, 0, 0, 0), E2 (0, 1, 0, 0, 0, 0), E3 (1, 0, 0, 0, eo3 y , eo3 z), E4 (0, 1, 0, eo4x , 0, eo4 z). This chain can transfer rotations from the input axis to the output axis. These axes are drown vertical. E3 E4 E2

E1

E1 E2

E1

E2

a)

b)

c)

Figure 1. The kinematic chains with universal joints and additional constraints.

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In order to retain the constant orientation of the output axis relating to the input axis two additional constraints can be mounted between the U-joints (Figure 1, b, c). One of these constraints can be mounted by R-pairs (Figure 1, c). The kinematic chain with additional constraints can be represented by two P-pairs. The Plücker coordinates of the unit screws E1 , E2 of the P-pairs can be expressed as E1 (0, 0, 0, eo1x , eo1y , eo1z), E2 (0, 0, 0, eo2x , eo2y , eo2 z). The axes of the P-pairs are perpendicular each to another and to the link connecting the U-joints. This chain transfers rotations from the input axis to the output axis and retain orientation of the output axis. Now let us consider a parallel mechanism of Schoenflies motions (Figure 2, a). E14 ≡ E24 E12

E13 ≡ E23 E22

E11

E21 E36 E35

a)

Ω4

Ω2

Ω3 Ω1

z

R2 y

x O

E33

E34

R1

E31

Ω2

Ω3

R3

z

R2 y

x

Ω1 R1

O

b)

c)

Figure 2. The Schoenflies motion parallel mechanism.

The first and the second kinematic chains consist of one actuated rotation pair (rotating actuator) situated on the base, two rotation kinematic pairs and one prismatic kinematic pair (the axes of all the pairs of these two chains are parallel). The third kinematic chain consists of one actuated rotation pair (rotating actuator) situated on the base, one actuated prismatic pair (linear actuator) (the axes of rotating and linear actuators coincide) and two U-joints represented as two rotation kinematic pairs. The unit screws of the axes of these kinematic pairs can be represented by coordinates: E11 (0, 0, 1, eo11x , eo11y , 0), E12 (0, 0, 1, eo12x , eo12y , 0),

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E13 (0, 0, 1, eo13x , eo13y , 0), E14 (0, 0, 0, 0, 0, 1), E21 (0, 0, 1, eo21x , eo21y , 0), E22 (0, 0, 1, eo22x , eo22y , 0), E23 (0, 0, 1, eo23x , eo23y , 0), E24 (0, 0, 0, 0, 0, 1), E31 (0, 0, 1, eo31x , eo31y , 0), E32 (0, 0, 0, 0, 0, 1), E33(e33x , e33y , 0, eo33x , eo33 y , eo33 z ), E34(e34x , e34y , 0, eo34x , eo34 y , eo34 z ), E35(e35x , e35y , 0, eo35x , eo35 y , eo35 z ), E36 (e36x , e36y , 0, eo36x , eo36 y , eo36 z ). The screws E14, E24, E32, are of infinite pitch. All other screws are of zero pitch. The first and the second kinematic chains impose the same constraints corresponding to four-member closed screw group, so that one can insert other similar chains between the base and platform and the degree of freedom will remain equal to four. The third kinematic chain does not impose any constraints. The wrenches of the constraints have coordinates (Figure 2, b): R1 (0, 0, 0, 1, 0, 0), R2 (0, 0, 0, 0, 1, 0). All the twists of motions of the platform can be represented by the twists reciprocal to the wrenches of the constraints (Figure 2, b): Ω 1 (0, 0, 0, 1, 0, 0), Ω 2 (0, 0, 0, 0, 1, 0), Ω 3 (0, 0, 0, 0, 0, 1), Ω 4 (0, 0, 1, 0, 0, 0). In this mechanism singularities corresponding to loss of one degree of freedom exist if four screws Ei1 , Ei2 , Ei3 and Ei4 (i = 1, 2) or if six screws E31 , E32 , E33, E34, E35 and E36 are linearly dependent. It is possible if three axes Ei1 , Ei2 , Ei3 (i = 1, 2) are situated on the same plane or if the link connecting the U-joints in the third chain is horizontal. In particular if the axes E11 , E12 , E13 are situated on the plane parallel to the y axis then there exist three wrenches (Figure 2, c) of constraints imposed by the kinematic chains: R1 (0, 0, 0, 1, 0, 0), R2(0, 0, 0, 0, 1, 0) and R3 (0, 1, 0, 0, 0, 0) and only three twists of motion of the platform reciprocal to these wrenches Ω 1 (0, 0, 0, 1, 0, 0), Ω 2 (0, 0, 0, 0, 0, 1) and Ω 3 (0, 0, 1, 0, 0, 0). This mechanism is particularly decoupled. The rotating actuator drives the orientation of the end-effector. The linear actuators drive the position of the end-effector. Now let us consider a parallel mechanism of Schoenflies motions containing one kinematic chain with two universal joints and additional constraints (Figure 3, a). The first and the second kinematic chains consist of one actuated prismatic pair, one spherical pair and two rotation pairs with intersecting axes. The third kinematic chain consists of one actuated rotation pair, one actuated prismatic pair and two U-joints represented as two rotation kinematic pairs. The unit screws of the axes of these kinematic pairs can be represented by coordinates: E11 (0, 0, 0, 1, 0, 0), E12 (1, 0, 0, 0, eo12y , eo12z ), E13 (0, 1, 0, eo13x , 0, eo13z ), E14 (0, 0, 1, eo14x, eo14y , 0), E15(e15x , e15y , 0, eo15x , eo15 y , eo15 z ), E16 (0, 0, 1, eo16x , eo16 y , 0), E21 (0, 0, 0, 0, 1 , 0), E22 (1, 0, 0, 0, eo22y , eo22z ), E23 (0, 1, 0, eo23x , 0, eo23z ), E24 (0, 0, 1, eo24x, eo24y , 0), E25(e25x , e25y , 0, eo25x , eo25 y , eo25 z ), E26 (0, 0, 1, eo26x , eo26 y , 0), E31 (0, 0, 1, eo31x, eo31y , 0), E32 (0, 0, 0, 0, 0, 1), E33 (0, 0, 0, eo33x , eo33 y , eo33 z ), E34 (0, 0, 0, eo34x , eo34y , eo34z ). The screws E12, E13, E14, E15, E16, E21, E22, E23, E24, E25, E26 , E31 are of zero pitch. All other screws are of infinite pitch. The first and the second kinematic chains do not impose any constraints. The third kinematic chain imposes two constraints corresponding to four-member closed screw group. The

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wrenches of the constraints have coordinates (Figure 3, b): R1 (0, 0, 0, 1, 0, 0), R2 (0, 0, 0, 0, 1, 0). All the twists of motions of the platform can be represented by the twists reciprocal to the wrenches of the constraints (Figure 3, b): Ω 1 (0, 0, 0, 1, 0, 0), Ω 2 (0, 0, 0, 0, 1, 0), Ω 3 (0, 0, 0, 0, 0, 1), Ω 4 (0, 0, 1, 0, 0, 0). The twists Ω 1, Ω 2 and Ω 3 are of infinite pitch, the twist Ω 4 is of zero pitch. In this mechanism singularities corresponding to loss of one degree of freedom exist if six screws Ei1 , Ei2 , Ei3 , Ei4, Ei5 and Ei6 (i = 1, 2) or if four screws E31 , E32, E33, and E34 are linearly dependent. It is possible if one of the axes of the links connecting the axes Ei4 and Ei5 (i = 1, 2) is perpendicular to the axis Ei1 or if the axis of the link connecting the U-joints of the third chain is perpendicular to E32. E16 ≡ E26 E15 E24

E25 E13

E14

E11

E22

E23 E12

E21 E34

E33

Ω4

Ω2

Ω3

a)

R3

z R2

x

y

E31

O b)

Ω1

z

R2

R1

Ω2

Ω3

E32

Ω1

x

y

R1

O c)

Figure 3. The Schoenflies motion mechanism with U-joints and additional constraints.

In particular if the axis of the link connecting the U-joints of the third chain is parallel to the y axis then there exist three wrenches (Figure 3, c) of constraints imposed by the kinematic chains: R1 (0, 0, 0, 1, 0, 0), R2(0, 0, 0, 0, 1, 0) and R3 (0, 1, 0, 0, 0, 0) and only three twists of motion of the platform reciprocal to these wrenches Ω 1 (0, 0, 0, 1, 0, 0), Ω 2 (0, 0, 0, 0, 0, 1) and Ω 3 (0, 0, 1, 0, 0, 0).

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3 The design of 6-DOF decoupled parallel mechanism In this section, let us consider a 6-DOF decoupled parallel mechanism 3 R-P-P-P-R-R (Figure 4, a). Each kinematic chain consists of one rotating actuator situated on the base, one linear actuator, two prismatic kinematic pairs represented as two U-joints with additional constraints and two passive rotation pairs. The axes of all the passive rotation pairs intersect in the same point O which is the origin of the coordinate system. This point O is movable but the directions of the coordinate axes are constant. The unit screws of the axes of kinematic pairs have coordinates: E11 (1, 0, 0, 0, 0, 0), E12 (0, 0, 0, 1, 0, 0), E13 (0, 0, 0, eo13x , eo13 o o o o y , e 13z), E14 (0, 0, 0, e 14x , e 14 y , e 14z), E15 (e15x, e15y , e15z , 0, 0, 0), E16 (e16x, e16y, e16z , 0, 0, 0), E21 (0, 1, 0, 0, 0, 0), E22 (0, 0, 0, 0, 1, 0), E23 (0, 0, 0, eo23x , eo23 y, eo23z), E24 (0, 0, 0, eo24x , eo24 y , eo24z), E25 (e25x, e25y , e25z , 0, 0, 0), E26 (e26x, e26y , e26z, 0, 0, 0), E31 (0, 0, 1, 0, 0, 0), E32 (0, 0, 0, 0, 0, 1), E33(0, 0, 0, eo33x , eo33 y , eo33z), E34 (0, 0, 0, eo34x , eo34 y , eo34z), E35 (e35x, e35y , e35z , 0, 0, 0), E36 (e36x, e36y , e36z , 0, 0, 0). The screws Ei1, Ei5, Ei6 are of zero pitch, the screws Ei2, Ei3, Ei4 are of infinite pitch (i = 1, 2, 3). All six twists of motions of the platform can be represented as: Ω 1 (1, 0, 0, 0, 0, 0), Ω 2 (0, 1, 0, 0, 0, 0), Ω 3 (0, 0, 1, 0, 0, 0), Ω 4 (0, 0, 0, 1, 0, 0), Ω 5 (0, 0, 0, 0, 1, 0), Ω 6 (0, 0, 0, 0, 0, 1). If rotating actuators are fixed then the linear actuators drive translation motions of the end-effector. By this the kinematic pairs corresponding to the screws Ei2, Ei3, Ei4 are used. If linear actuators are fixed then the rotating actuators drive orientation motions of the end-effector. By this the kinematic pairs Ei1, Ei5, Ei6 are used and rotations are transferred by the U-joints. In considered mechanism (Figure 4, a) singularities expressed by loss of one or more degrees of freedom exist if any six screws Ei1 , Ei2 , Ei3 , Ei4 , Ei5 , Ei6 (i = 1, 2, 3) are linearly dependent. It is possible if at least one of the axes of the links connecting the U-joints is perpendicular to the axis Ei2. In particular if the axis of the link connecting the U-joints of the third chain is parallel to the y axis then there exists one wrench of the constraint imposed by this kinematic chain: R (0, 1, 0, 0, 0, 0) and only five twists of motion of the platform reciprocal to this wrench Ω 1 (1, 0, 0, 0, 0, 0), Ω 2 (0, 0, 1, 0, 0, 0), Ω 3 (0, 0, 1, 0, 0, 0), Ω 4 (0, 0, 0, 1, 0, 0) and Ω 5 (0, 0, 0, 0, 1, 0). If the unit screws E11 (1, 0, 0, 0, 0, 0), E15 (e15x, e15y , e15z , 0, 0, 0), E16 (e16x, e16y, e16z , 0, 0, 0) are coplanar then there exists one wrench of the constraint imposed by the first kinematic chain: R (0, 0, 0, 0, roy, roz) and only five twists of motion of the platform reciprocal to this wrench Ω 1 (1, 0, 0, 0, 0, 0), Ω 2 (ω2x, ω2y, ω2z , 0, 0, 0), Ω 3 (0, 0, 0, 1, 0, 0), Ω 4 (0, 0, 0, 0, 1, 0), Ω 5 (0, 0, 0, 0, 0, 1). The wrench R is of infinite pitch, it is perpendicular to the axes E11, E15, E16. If the actuators are fixed then the wrenches of the constraints have coordinates (Figure 4, b): R11 (r11x , r11y , r11z , 0, 0, 0), R12 (0, 0, 0, ro12x , ro12y , ro12z), R21 (r21x , r21y , r21z , 0, 0, 0), R22 (0, 0, 0, ro22x , ro22y , ro22z), R31 (r31x, r31y , r31z , 0, 0, 0), R32 (0,

Decoupled Parallel Manipulator…

71

0, 0, ro32x , ro32y , ro32z). The wrenches Ri1 and Ri2 are imposed by the i-th kinematic chain. The wrenches Ri1 are of zero pitch, the wrenches Ri2 are of infinite pitch (i = 1, 2, 3). The wrenches Ri1 are perpendicular to the links connecting the U-joints, the wrenches Ri2 are perpendicular to the screws Ei5 and Ei6. Singularities corresponding to non-controlled infinitesimal motion of the moving platform exist if the wrenches R11, R12, R21, R22 , R31, R32 are linearly dependent. It is possible if the wrenches R11 , R21, R31 or R12 , R22, R32 are coplanar. In particular if the wrenches R12 , R22, R32 are coplanar (Figure 4, c) then the twist Ω (ωx, ωy, ωz , 0, 0, 0) exists which is perpendicular to the axes of the wrenches R12 , R22, R32 and reciprocal to R11, R12, R21, R22, R31, R32 . E16

E35

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In this mechanism, also singularities exist corresponding both to loss of one degree of freedom and to non-controlled infinitesimal motion of the platform. The corresponding conditions are represented above. This mechanism is decoupled. The rotating actuators drive the orientation of the end-effector, the linear actuators drive the position of the end-effector.

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4 Conclusion In this article, new decoupled parallel manipulator consisting of three parallel kinematic chains containing U-joints with additional constraints is represented. The synthesis and the singularity analysis is carried out by using of Plücker coordinates of twists and wrenches corresponding to the kinematic chains. The application of closed screw groups allows obtaining all the twists of the platform and the wrenches of the constraints without any equations. References Merlet, J.-P. (2000). Parallel Robots. Kluwer Academic Publishers. Ceccarelli, M. (2004). Fundamentals of Mechanics of Robotic Manipulations. Kluwer Academic Publishers. Kong, X., Gosselin, C. (2007). Type Synthesis of Parallel Mechanisms. Springer. Koliskor, A. (1986). The l-coordinate approach to the industrial robot design. In V IFAC / IFIP / IMACS / IFORS Symposium, Suzdal, URSS, April 22-25: 108-115. Innocenti, C., Parenti-Castelli, V. (1991). Direct kinematics of the 6-4 fully parallel manipulator with position and orientation uncoupled. In European Robotics and Intelligent Systems Conf., Corfou, June 23-28. Mianovski, K. (2007). Singularity analysis of parallel manipulator POLMAN 3×2 with six degrees of freedm. In 12th IFToMM World Congress, Besançon (France), June 18-21. Clavel, R. (1987). Device for Displacing and Positioning an Element in Space. In Patent WO 87/03528. Arakelian, V., Guegan, S., Briot, S. (2005). Static and Dynamic Analysis of the PAMINSA. In ASME 2005. International Design Engineering Technical Conferences & Computers and Information in Engineering Conference. Long Beach. CA, USA: 24-28. Gogu, G. (2009). Structural Synthesis of Parallel Robots, Part 2: Translational Topologies with Two and Three Degrees of Freedom, Springer, Dordrecht. Yan Jin, I-Ming Chen, Guilin Yang. (2007). Structure Synthesis of 6-DOF 3-3 Decoupled Parallel Manipulators. In 12th IFToMM World Congress, Besançon, June 18-21. Dimentberg, F. (1965). The Screw Calculus and its Applications in Mechanics. Nauka. (Eng. trans.: AD680993, Clearinghouse for Federal Tech. and Sc. Inf., Virginia). Sugimoto, K. (1990). Existence Criteria for Overconstrained Mechanisms: An Extension of Motor Algebra. In Tr. ASME, J. of Mech. Design, 3: 295-298. Herve, J. (1991). The Lie group of rigid body displacements, a fundamental tool for mechanism design. In Mechanism and Machine theory, 34: 719-730. Angeles, J. (2004). The Qualitative Synthesis of Parallel Manipulators. In Journal of Mechanical Design, 126: 617-624. Glazunov, V. (1995). Principles of the construction and analysis of spatial parallel structure mechanisms. In Jour. of Mach. Manuf. and Reliab., Aller. Press Inc. 1: 10-15. Gosselin, C., and Angeles, J. (1990). Singularity Analysis of Closed Loop Kinematic Chains. In IEEE Trans. on Robotics and Automation, 6(3): 281-290.

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92

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Identification of Base Parameters for Large-scale Kinematic Chains…

93

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5? @41 ?;8A@5;: ;2 & )41>1 -:0

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94

K. Ayusawa and Y. Nakamura

Link shape

mass point

Condition of physical consistency k :condition of point

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Identification of Base Parameters for Large-scale Kinematic Chains…

95

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Modeling Effects on Free Vibration of a Two-Link Flexible Manipulator Carmelo di Castri† and Arcangelo Messina† †

Dipartimento di Ingegneria dell’Innovazione, Università del Salento, Lecce, Italia

Abstract In this article, the frequency characteristics of a two-link flexible manipulator in an arbitrary position are investigated by using two analytical models. In particular, two formulations based on Euler-Bernoulli beam theory are herein used to derive analytical modal data for different slenderness ratios of robotic manipulators. The comparison between the results stresses that, except for extremely flexible manipulators, appreciable discrepancies can occur; these latter can be attributed to axial vibrations which, are neglected in one model, but are instead introduced in the other as proposed by the present authors. A finite element analysis validates this latter model thus making the authors confident on the ability of the model to provide more accurate results for a larger class of robotic manipulators.

1 Introduction During recent decades, increasing demand for precise high speed operation has made flexible manipulators attractive and great attention has been dedicated to the modeling and vibration control of such structures, mainly in the area of space and industrial robots with flexible links. Many researchers have tried to develop dynamic models that describe the motion of flexible manipulators by taking into account the effects of links flexibility in order to have an explicit, complete and accurate dynamic model (see, e.g., Tomei and Tornambè, 1989; De Luca and Siciliano, 1991; Chen, 2001). Their dynamic formulations of the flexible multibody system (based on EulerBernoulli beam theory) led to infinite-dimensional models described by a set of partial differential equations that could not be solved analytically; then, approximate techniques were used to change these partial differential equations to a set of ordinary differential equations in order to obtain a finite-dimensional system that can easily be solved numerically. Milford and Asokanthan (1999) derived the exact partial differential equations governing the system modes of a two-link flexible manipulator within the framework of the Euler-Bernoulli beam theory; they used an exact boundary conditions formulation and showed that eigenfrequencies and eigenfunctions are a strong function of manipulator configuration. Here, based on the assumptions by Di Castri et al. (2009),

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C. di Castri and A. Messina

an exact model to determine modal data (i.e. natural frequencies and mode shapes) of a two-link manipulator based on Euler-Bernoulli beam theory is presented; in this work, the configuration-dependent differential eigenvalue problem is derived by also taking the axial dynamics into account. It is highlighted how the inclusion of the axial influence is not computationally expensive as also stressed by Di Castri et al. (2009); however, in this latter paper, a complete validation of the axial influence was not provided neither with respect to a frequency range on modal data nor within the frame of occurring configurations of robotic manipulators. In this paper, a comparison between the above mentioned two formulations is presented; it is herein shown how the results predicted by the two models can become very different when the slenderness ratio of the structure changes and these discrepancies can be attributed to the axial deformation effect, which is absent in Milford and Asokanthan (1999). The comparison is done firstly through simulations carried out for an extremely flexible two-link manipulator and subsequently for different values of the slenderness ratio. Natural frequencies are reported for different nominal configurations of the manipulator, chosen as significant cases. Finally, results of a finite element package are presented and are found to be in excellent agreement with those of the model here introduced.

2 Manipulator models The two-link manipulator model is illustrated in Figure 1 in its undeformed shape. It consists of two homogeneous, isotropic and elastic links with constant cross section Si and linear mass density μi (i=1,2), with each link having length Li and flexural rigidity EiIi. The flexible robot has a shoulder joint variable Q1 and an elbow joint variable Q2. Robot motion and vibration modes are planar with gravity being neglected. Mass Mi and inertia Ji take account for the presence of joint actuators, whereas MP and JP are referred to a possible payload handled by the manipulator end effector. Moreover, in order to express elastic deformations, two local reference frames (X1,Y1) and (X2,Y2) are defined, which are respectively attached to the shoulder and elbow hub as shown in Figure 1. X1 and X2 axes coincide with the undeformed neutral axes, which fully characterize the spatial shapes of the links. Henceforth, let us denote with “M model” the formulation presented by Milford and Asokanthan (1999), and with “D model” the one presented here. Both models present modal analysis of the two-link manipulator, based on small perturbation about a nominal configuration; in order to derive configuration-dependent modal data, rigid body motions of the system are prevented by blocking both joints. The main difference between analytical formulations is whether or not the axial deformations are considered in the system dynamic description. In particular, M model neglects longitudinal dynamics of the links, whereas D model takes into account the axial and transverse dynamics; this leads to consider, according to the exact model,

Modeling Effects on Free Vibration of a Two-Link Manipulator

101

Figure 1͘ Two-link flexible manipulator.

different arrangements of the local reference frames to represent the deformed shape of the manipulator. Firstly, referring to Figure 2, since the boundary-value problem does not depend on joint variable Q1, a simplifying cantilever boundary condition is considered in the shoulder section O1 for both models. In order to represent the ith mode shape of the manipulator, M model moves the local frame (X2,Y2) along X2 by a measure equal to the rigid displacement that the second link undergoes along that direction because of the bending of the first link. For this reason, the position of the local frame (X2,Y2) changes according to the represented mode shape. Moreover, if we consider the system at rest and a point of the ith link (i=1,2) which has coordinates (xi,0) in the reference system (Xi, Yi), being xi the distance from Oi measured on the neutral axis, its position after deformation will be (xi, vi(xi,t)), where vi(xi,t) is the field variable associated with the transverse displacement. Conversely, D model always represents all the mode shapes in the same coordinate systems, which are those initially associated with the undeformed shape of the system (Figure 2). In this case, the above considered point (xi,0) is located, in its actual state, by the position (xi+ ui(xi,t), vi(xi,t)), being ui(xi,t) the axial displacement. According to the Euler-Bernoulli beam theory, M model requires the following two partial differential equations (PDEs) to describe the transverse motion of the links:

(1)

102

C. di Castri and A. Messina

In addition to the PDEs in (1), D model introduces the following PDEs for the axial dynamics of the links:

(2)

In spite of the possible simplicity of equation (2), it is here shown that the relevant results can introduce appreciable variations with respect to its absence when coupling phenomena are involved through the boundary conditions. Now, choosing a value for Q2 (denoted by Q20) and using boundary conditions, it is possible to obtain the corresponding differential eigenvalue problem for each model, associated with the nominal configuration Q20; such a problem is expressed in compact form as A()x = 0

(3)

where A() is the characteristic matrix of the manipulator, with W natural frequency, and x is the vector of modal coefficients that characterize the axial eigenfunction Ui(xi) and the transverse eigenfunction Vi(xi), i=1,2. The appendix explicitly illustrates the relevant analytical quantities for the model here introduced.

Figure 2. Displacement functions for deformed shape.

Modeling Effects on Free Vibration of a Two-Link Manipulator

103

We observe that A() and x are quite different for both models (e.g. see Milford and Asokanthan (1999 pp.198-199) compared to the appendix); in fact, the characteristic matrix for M model is a square matrix of order 8 and x is 8×1, since boundary conditions for this model have been formulated from relations for transverse displacement, slope, bending moment and shear force, which are related to eight modal constants. D model is instead based on a 12×12 characteristic matrix with a modal vector x having 12 components, since, in addition to the aforementioned physical quantity, contributions due to axial displacements and axial forces have been taken into account in the boundary sections. To this end, we stress that even if PDEs for axial and transverse vibrations are always uncoupled, this uncoupling also depends on the boundary conditions; indeed, D model, with its axial-transverse formulation, couples the two vibratory behaviours through boundary conditions.

3 Simulations and discussion To demonstrate the effects of the modeling strategies described in the previous section, simulation results are presented in this section. The basic structural model adopted is the extremely flexible two-link manipulator used by Milford and Asokanthan (1999) to experimentally validate their analytical predictions. In order to derive modal data through both exact models, M model and D model are implemented in MATLAB® R2007a and a contextual finite element analysis is also carried out by using ABAQUS 6.6, with each link divided into fifty-five Euler-Bernoulli beam elements. Among all the possible choices, natural frequencies are calculated for three nominal configurations of the manipulator, which correspond to Q20 = –30°, Q20 = 0° and Q20 = 45°; for each configuration, a slenderness-dependent analysis is carried out. More precisely, a thickness factor tf is introduced and modal data are derived for the assigned configuration when tf is equal to 1, 25 and 50, so that thicknesses equal to tf ·hi (i=1,2) are respectively considered for each case (where hi is the original thickness value of the experimental manipulator originally used by Milford and Asokanthan (1999)). We observe that by varying tf from 1 to 50, local slenderness ratios range approximately between 3000 and 60; this considerable variation is chosen in order to allow a better investigation of the modeling effects on frequency characteristics (see also the mode shape change of Figure 3) of the two-link flexible manipulator if thicker links than those used by Milford and Asokanthan (1999) were considered. In fact, hi is really a small quantity (0.6 mm) whilst larger values can easily be encountered when treating with flexible manipulators.

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C. di Castri and A. Messina

Table 1. Natural frequencies [Hz] for Q20 = –30°. tf = 1

tf = 25

Mode M model D model

e%

M model D model

tf = 50 e%

M model D model

e%

1

0.2051

0.2051

0.00

11.253

11.253

0.00

23.454

23.454

0.00

2

0.8898

0.8898

0.00

57.199

57.189

0.02

121.88

121.79

0.07

3

4.8716

4.8716

0.00

188.34

188.26

0.04

390.49

389.87

0.16

4

9.0299

9.0299

0.00

336.89

335.95

0.28

699.08

687.48

1.7

5

14.478

14.478

0.00

594.38

592.59

0.3

1243.3

1204.0

3.3

6

20.610

20.610

0.00

854.83

844.05

1.3

1771.6

1439.4

23

7

36.366

36.366

0.00

1187.7

1162.3

2.2

2525.4

1902.9

33

8

36.749

36.749

0.00

1546.4

1396.0

11

3302.6

2576.1

28

9

62.809

62.809

0.00

1967.6

1621.9

21

4196.4

3314.0

27

10

70.890

70.890

0.00

2341.4

2004.1

17

5125.3

3692.1

39

Table 1 reports the first ten natural frequencies calculated through M model and D model when Q20 is equal to –30°. The occurring natural frequency changes are calculated as

(4)

Finite elements results are reported in Table 2 and correspond to results obtained after having submitted the relevant FE-analysis to a relevant convergence test (55 elements for each beam). As it can be observed from tables 1 and 2, both models agree with each other and with the finite element analysis only for tf =1, i.e. when an extremely high flexibility is taken into account. In this case, the assumption made by M model, which neglects the axial deformations, is validated and the model can be used to obtain accurate data. However, as tf increases, the above agreement ceases and appreciable discrepancies occur, up to 21% and 39% for tf equal to 25 and 50 respectively. Since finite element results fully agree with D model, we can state that this latter represents a more reliable way of deriving modal data, whereas M model becomes appreciably inaccurate; moreover, it can be expected that this inaccuracy is intended to grow for larger values of tf; however, in these latter cases the possibility of adopting a higher order model (uniform o higher order shear deformable theories) should also to be taken into account. Figure 4 again presents the same comparison between exact models by a graphical representation of the frequency changes occurred for Q20 = 0° and for Q20 = 45°. It is observed that in the second case (on the right of the figure), the same considerations of the previous comparison in Table 1 can be deduced;

Modeling Effects on Free Vibration of a Two-Link Manipulator

105

in particular, for tf =50, the error is again over 30%. Finite element analysis agrees again with D model . The only case for which M model and D model give the same results, for any thickness, is when Q20 = 0° (on the left of the figure); it can be explained by observing that, in this condition, axial and transverse vibrations are uncoupled, and the manipulator behaves like a cantilever beam. Table 2. Finite element results [Hz] for Q20 = –30° (ABAQUS 6.6, B23 element). Mode

tf = 1

tf = 25

tf = 50

1

0.2051

11.253

23.454

2

0.8898

57.189

121.79

3

4.8716

188.26

389.87

4

9.0299

335.95

687.48

5

14.478

592.59

1204.0

6

20.610

844.05

1439.4

7

36.366

1162.3

1902.9

8

36.749

1396.0

2576.1

9

62.809

1621.9

3314.0

10

70.890

2004.1

3692.1

Figura 3. Analytical modes (D model: 20.610, 1439.4 Hz).

4 Concluding remarks In this paper, two analytical models for investigating the modal analysis of a two-link flexible manipulator have been compared and the model here proposed has been numerically validated too. The two analytical descriptions differ for a different treatment of both the axial dynamics of the links and the local system of coordinates. Exact modal data have been derived through both formulations for different geometrical configurations of a manipulator having links modeled as Euler-Bernoulli beams. For each chosen configuration, the influence of the structural slenderness ratio on the predicted results by the two analytical models has been evaluated through a simulation study. Natural frequencies and data plots have been reported in detail. It has been shown that, except for extreme flexibility conditions (tenths of millimeter thicknesses), substantial discrepancies can arise between both models. A finite element analysis has validated all the results obtained through the analytical axial-transverse model here introduced.

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References Chen, W. (2001). Dynamic modeling of multi-link flexible robotic manipulators. Computers and Structures 79:183-195. De Luca, A., and Siciliano, B. (1991). Closed form dynamic model of planar multilink lightweight robots. IEEE Transactions on Systems, Man, and Cybernetics, JulyAugust 1991, vol. 21, no. 4, 826-839. Di Castri, C., Messina, A., and Reina, G. (2009). Modal Analysis of a Two Link Flexible Manipulator. Proceedings of the RAAD 2009 18TH International Workshop of Robotics in Alpe-Adria-Danube Region, Brasov, Romania, May 2009. Milford, R. I., and Asokanthan, S. F. (1999). Configuration dependent eigenfrequencies for a two-link flexible manipulator: experimental verification. Journal of Sound and Vibration 222:191-207. Simulia (2006). ABAQUS Version 6.6 HTML Documentation. The MathWorksTM (2007). MATLAB® Online Documentation. Tomei, P., and Tornambè, A. (1988). Approximate modeling of robots having elastic links. IEEE Transactions on Systems, Man, and Cybernetics, September-October 1988, vol. 18, no. 5, 831-840.

Figure 4. Natural frequency changes. * tf =(1,25,50); tf =1; tf =25; tf =50.

where:

! ! " "

"

! "

"

"

! "

"

"

!

!

"

! ! ! !

Characteristic matrix A() =

ch(·) = cosh(·)

!

!

transverse Vi(xi) = Ci1s(Aixi)+Ci2c(Aixi)+Ci3sh(Aixi)+Ci4ch(Aixi); axial Ui(xi) = Ci5c(W·xi/ci)+Ci6s(W·xi/ci) Ai = [MiW2/(EiIi)]1/4; ci = (EiSi/Mi)1/2 Vector of modal coefficients x = [C11 C12 C13 C14 C15 C16 C21 C22 C23 C24 C25 C26]T

sh(·) = sinh(·);

Eigenfunctions (i=1,2):

Parameters:

c(·) = cos(·);

s(·) = sin(·);

Abbreviations:

Appendix: mathematical formulation of the two-link manipulator with axial dynamics (D model)

Modeling Effects on Free Vibration of a Two-Link Manipulator 107

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06 ++, 6 1 6 &:6 $6 , 8 6 6 6 , &,,6 6 ,6806 &6*66+ 8+,6, 616",, O6,66 ,# 6*++6 &6*6 060,++61 6,1 "66+ 8+,76 "+66,6, 8,6 :6$6 6 , &,,666+ 6 6&,,6 6 0, 6 ",:6$6"6 6,6, 6+,66 6 1,'6+ 6"66 "++6 +"6 6 6 ++6 6 + 6 *6 6 ,# 6 ,6 06 6 + 6 ",76 , +6 6 6 6 6 6 "1, :6 6 " "6 , /0, 6 6 6 6 , 76 + 6 ,6 +6 6 ,+6 6 6

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= § 4 5 · 4 7 5 :% ¨ 6 ¸ 76666 % 4 5 ® 4 5 # © ¹ ¯ *6 6661 " ,8+06,1 "66*6",:66

4?56

Chatter Suppression in Sliding Mode Control

111

?56$6( **( 6,616" 661 6 % 6, 6%6+ 6 0 6 ++6F=76D76GI:6%6,66 06 6

4 7 5

4 5 4 5

6

4A56

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4 7 5 4 5 4 5 6

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++6 6+ 60 606*66+,&6 6,&6,+ 616

&+ 6FJI:6 $6#*$$ *FK76=
= 4 4 55 ? 4 4 55 6

4D56

*6 = 66, 6 ? 6,6 6,,06,6,6 = ? :66 $6 %$ * ), (%$* $$ 6F?76==I6"66/0,+6&,+"6 66 *6 &,,1+76 ::6 6 0,/0"06 , 6 00"06 &,+"6 6 6 *6 &,,1+76,6,6, ,&606*66"":6(+'66 6?8+ 60 6 ++76 60 ,6 , 66" 6 6 &6 4 5 :6$6"+66 ,60+6,+ 06*6+6"6 :6$6 ++6,66 06 6

= 4 4 5 ? 5 6

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112

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M. D O’Toole, K. Bouazza-Marouf and D. Kerr

!#!# '$% "#

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Chatter Suppression in Sliding Mode Control

113

4 5 4 5 4 5 4 4 57 4 55 < 6

64J56

6

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4 5 %4 5 4 5 6 4 5 % 4 5 4 5 4 5 4 5

6

4=<56

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4 4 57 4 55 4%7 7 5 4 5 =

= 4%7 7 5 4%7 7 5 4 5 %

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% 4 57 % 4 5 4 5 4 5 4 5 ³ = ³ % 4 57 % 4 5 4 5 4 5 4 5

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114

M. D O’Toole, K. Bouazza-Marouf and D. Kerr

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Chatter Suppression in Sliding Mode Control

115

=< ' ' 6

6

64=D56

6 6 6 "6 ,,06 6 ,6 6 6 + 6 0 6 +6 ,,06 ,6 6 6 &,&+6 ,6 6 ,1+6 6 6 ,1+6 =6 6 ,6 ,6 6 ",:6$6 "66 6A:?66 ++ * 6 6,6 66 ++6 6 6,6,0+" 6, 6+60:6$6,6,0+" 66,, 6 066 "6+ 6"66,,+60 6 1 :6 +6066 " 6 16 "6 ,6 6 ,+ 06 6 6 6 , 6 ,6 6 6 6 % 76 , 6 6 "0,+6, 6 6 6,6 +" 6 6", 64==5:6$6 6&,+"6,6 0, 6 *6 ,6 0"+, 76 0+0 6 "6 6 0"+'6 & 06 4,* '6%76( 5:6$6"+6,6 *66,1+6A:6 6,6 ++766 0 6 "+ 66 6A:?66,1+6 6+ +6 66,6, 6+606 660"+, 60:6

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# # ! # <:AG=A6

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# # !# <:?G6 <:A=6 <:?D6 <:??6 <:=D6 <:=D6

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116

M. D O’Toole, K. Bouazza-Marouf and D. Kerr

"+,6,8" 6,+ 066,

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# F=I6 :6 *, 6 , 6 :6 .:6 " 76 9 + 6 0 6 +M6 6 , 6 ,+, 7E6 M6$,+ 6>6,76=KKJ:66 F?I6 %:6 ' 76 :6 0,76 :6 , 76 , 6 :6 (,76 @,+6 6 ,6 6 06 *6 8 6+ 60 796 *%%**(%*$+* & +:6D?76:6?<JD8 ?=
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2 2?5 E96 86?6C2= 96I2A@5 #( H2D CDE D@=G65 H:E9 D 3J )@==2?5 2?5 92?5C2

H96C6 E96 A:G@E >FE2E:@? @A6C2E@C H2D :?

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?62=:?8 :>AC@G65 A6C7@C>2?46 ) 92?5C2 2?5 )@==2?5

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118

L. Rolland and R. Chandra

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C:E9>D *64E:@? AC6D6?ED E96 6IA6C:>6?ED E96 C6DF=ED 2?5 E96:C 2?2=JD:D *64E:@? 4@?4=F56D E96 A2A6C 2?5 DF886DED 5:C64E:@?D 7@C 7FEFC6 C6D62C49

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5:?2E6D :? E96 32D6 C676C6?46 7C2>6 2?5 E96 >@3:=6 A=2E7@C> ;@:?E 4@@C5:?2E6D :? E96 >@3:=6 A=2E7@C> C676C6?46 7C2>6 +96 <:?6>2E:4D >@56= G2C:23=6D 2C6 E96 24EF2E@C 4@@C5:?2E6D 2?5 6?5 664E@C 4@@C5:?2E6D E96 6?5 664E@C A@D:E:@? 2?5 @C:6?E2E:@?

)& +96 A2C2==6= >2?:AF=2E@C

($& ! $&")!( $# $ ( ' #"( ' $! @?E2:?:?8 2D >2?J 6BF2E:@?D 2D G2C:23=6D G64E@C:2= 7@C>F=2E:@? 4@?DECF4ED 2? 6BF2E:@? DJDE6> 2D 2 4=@D65 G64E@C 4J4=6 36EH66? E96 7@==@H:?8 A@:?ED 2?5 +96 5:DE2?46 36EH66? 2?5 :D D6E E@

On Solving the Forward Kinematics of the 6-6 Manipulator…

119

+96 >@3:=6 A=2E7@C> :D A@D:E:@?65 3J 5:DE:?4E A@:?ED D6=64E65 2D E96 ;@:?E 46?EC6D )@==2?5

+96 G2C:23=6D 2C6 D6E 2D 0 1 7@C '?6 C676C6?46 7C2>6 :D AC64:D6=J =@42E65 @? +96 F?:E G64E@CD 2?5 C6AC6D6?E E96 ?6H ?@? 2CE6D:2? 7C2>6 2I6D 56?65 3J

?J A=2E7@C> A@:?E 42? 36 6IAC6DD65 2D H96C6 2C6 4@?DE2?ED :? E6C>D @7 E96D6 A@:?ED 6?46 :? E96 42D6 @7 E96 !?G6CD6 <:?6>2E:4D AC@3=6> !#( E96 4@?DE2?ED 2C6 ?@E65

2?5 42? 6IA=:4:E=J 36 565F465 7C@> 3J D@=G:?8 E96 7@==@H:?8 =:?62C DJDE6> @7 6BF2E:@?D

H96C6 2?5 2C6 A2C2>6E6CD @?=J 56A6?5:?8 @7 A=2E7@C> 86@>6ECJ ,D:?8 E96 C6=2E:@?D :? BF2E:@? E96 5:DE2?46 4@?DEC2:?E 6BF2E:@?D 42? 36 6IAC6DD65 +96C67@C6 7@C E96 !#( :D @3E2:?65 3J :D@=2E:?8 E96 24EF2E@C G2C:23=6D :? E96 D:I 7@==@H:?8 6BF2E:@?D

+96 !#( 4@>AC:D6D E96 CDE 6BF2E:@?D :? E6C>D @7 A@:?E G2C:23=6D +96 >6E2 96FC:DE:4 E649?:BF6D ?665 @?6 @3;64E:G6 7F?4E:@? E@ 36 >:?

:>:D65 E96C63J C6BF:C:?8 4@?G6CD:@? @7 E96 ?@? =:?62C 6BF2E:@? AC@3=6> :?E@ 2? @AE:>:D2E:@? @?6 H96C6 E96 !#( :D :?E68C2E65 :?E@ @?6 D:?8=6 @3;64E:G6 7F?4E:@? :E?6DD 7F?4E:@? C6AC6D6?E:?8 E96 E@E2= 6CC@C @? 6249 <:?6>2E:4D 492:? =6?8E9D $6E 36 E96 =6?8E9 @7 <:?6>2E:4D 492:? 2?5 7C@> BF2E:@? E96 E?6DD 7F?4E:@? :D D6E E@ 4@?E2:?:?8 E96 <:?6>2E:4D 492:? =6?8E9 @3;64E:G6D (C6=:>:?2CJ E6DED H:E9 D =65 E@ D6G6C2= D@=FE:@?D H9:49 H6C6 &'+ :? 4@CC6DA@?56?46 H:E9 E96 6I24E AC@G6? @?6D )@==2?5 2?5 92?5C2

+9:D 7F?4E:@? :D E96? 2F8>6?E65 3J E96 4@?DE2?E 5:DE2?46D 36EH66? E96 5:DE:?4E A=2E7@C> A@:?ED 2?5

120

L. Rolland and R. Chandra

+96 ?2= E?6DD 7F?4E:@? :?4=F56D E96 D:?8=6 @3;64E:G6D @3E2:?65 7C@> E96 <:?6>2E:4D 492:? =6?8E9D 2?5 A=2E7@C> 5:DE2?46 4@?DEC2:?ED

"(! )& !& "$& %

)62= 4@565 86?6E:4 2=8@C:E9>D ) 2C6 4@5:?8 H:E9 C62= ?F>36CD >2:?

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6?ED *@>6 4@>>@? 4C@DD@G6C @A6C2E@CD 2C6 2E 4C@DD@G6C D:>A=6 4C@DD@G6C 2C:E9>6E:4 4C@DD@G6C 3=6?5 4C@DD@G6C =:?62C 3C6656C 4C@DD@G6C 2?5 .C:89ED 96FC:DE:4 4C@DD@G6C )(( $# $%&($& !E AC@G:56D C2?5@> 5:G6CD:EJ :? E96 A@AF=2E:@? *@>6 4@>>@? >FE2E:@? @A6C2E@CD 2C6 C2?5@> F?:7@C> >FE2E:@? 2?5 %:492=6H:4K ?@? F?:7@C> >FE2E:@? !? F?:7@C> >FE2E:@? 2 C2?5@> ?F>36C :? E96 C2?86 @7 0 1 :D 25565 E@ 2 D6=64E65 86?6 H96C6 2?5 2C6 E96 9:896DE 2?5 =@H

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Bibliography Huang, Z. and Ge Q. J. A Simple Method for Mobility Analysis Using Reciprocal Screws, ASME 2006 IDETC September 10-13, 2006, Philadelphia, Pennsylvania, USA. DETC2006-99677. Herv´ e, J.M., Analyse Structurelle des M´ ecanismes par Groupe des D´ eplacements, M echanism and Machine Theory, Vol. 13, pp. 437–450, 1978. Kong, X. and Gosselin, C. M. Type synthesis of 3-DOF spherical parallel manipulators based on screw theory, ASME Journal of Mechanical Design, Vol. 126, pages 101–108, 2004. Kong, X. and Gosselin, C. M. Type Synthesis of Parallel Mechanisms, Berlin: Springer Verlag, 2007. Murray, R. M., Li, Z. and Sastry, S. S. A Mathematical Introduction to Robotic Manipulation, Boca Raton: CRC Press, 1994. Rico, J. M. and Ravani, B. On Mobility Analysis of Linkages Using Group Theory, ASME Journal of Mechanical Design, Vol. 125, pages 70–80, 2003. Rico, J. M., Gallardo J. and Ravani, B. Lie Algebra and the Mobility of Kinematic Chains, J ournal of Robotic Systems, Vol. 20, pages 477–499, 2003. Rico, J. M., Cervantes, J. J., Tadeo, A., P´erez G. I. and J. Rocha, J. New Considerations on the Theory of Type Synthesis of Fully Parallel Platforms, ASME Journal of Mechanical Design, Vol. 130, 112302, 2008. Selig, J. M. Geometrical Foundations of Robotics, 2nd. Ed., New York: Springer Verlag, 2005.

Micro Hinges and their Application to Micro Robot Mechanisms Mikio Horie1 1

Precision and Intelligence Laboratory, Tokyo Institute of Technology, Japan, [email protected]

Abstract. In the present paper, two kinds of manipulators are shown and their characteristics are discussed. Their manipulators are as follows, (1) a micromanipulator with elastic hinges made by FAB(Fast Atom Beam) machine, and (2) a molding pantograph mechanism with large-deflective hinges, which is used as one component mechanism of a palmtop surface mount system using at the one room factory. In the case of the micromanipulator with elastic hinges, its micromanipulator’s relationship between input and output displacements in the experiments are coincident with theoretical results. In the case of the molding pantograph mechanism with large-deflective hinges whose materials are polypropylene, it was confirmed that the hinge in the mechanism had not fractured even if more than one million repeatable motion of the mechanism in the mechanism’s fatigue test. Moreover, in order to have a long life hinge, the fatigue process toward the fracture of the hinge specimen was clarified in experiments by hinge-specimen’s fatigue test.

1. Introduction In the case of micromechanisms, micromachines, MEMS, micro systems, instead of bearings at joints, the flexural hinges, large-deflective elastic hinges, and large-deflective hinges are used to avoid of friction effects at the bearing . Mechanisms with elastic elements, that is, flexural hinges, have been researched and developed in the world. For example, Kota, S.(1999) proposed a compliant mechanism which has single-piece flexible structures that deliver the desired motion by undergoing an elastic deformation, and Müglitz, J. and Schönherr, J.(1999) suggested a miniaturized planar mechanism for guiding a surgical instrument whose joints are designed as thin compliant segments acting like bending beams, that is, large-deflective elastic hinges, while the joints of this pantograph mechanism are large-deflective hinges whose aspect ratio of thickness to length is nearly 1 to reduce the displacement error by Mikio Horie et. al.(2001).

134

M. Horie

In the present paper, the feature of them and differences among them are explained, and their applications are introduced. The flexural hinge has not a large angular displacement. Therefore, the flexural hinges are used to have a large and precision output displacement by use of a lot of hinges and an enlargement mechanism. On the other hand, a large-deflective elastic hinge is used to obtain a large angular displacement described in Chapter 2. Moreover, a new concept, that is, a large-deflective hinges are used to obtain a large angular displacement and to have a precision output, because of a hinge center small motion described in Chapter 3. Then, the characteristics of the micromechanisms with large-deflective hinges which has a long life, are discussed in Chapter s 4 and 5.

2. Micro Motion Convert Mechanism made by FAB Machine Figure 1 shows a FAB(Fast Atom Beam) manufacturing machine. Figure 2 shows the manufacturing view by use of the mask and the FAB. A PRRR Mechanism, which consists of revolute pairs (R) and prismatic pairs (P), is shown in Fig. 3. In this mechanism, the Y-direction displacement of the central revolute pairs (R) is taken as the output when an X-direction displacement is given to the prismatic pairs (P) as the input. A micro motion convert mechanism based on the PRRR Mechanism is shown in Fig. 4. The micro motion convert mechanism consists of micro order hinges, which are substituted for the revolute pairs in the PRRR mechanism, and links. A micro motion convert mechanism was manufactured with polyimide for the structural material based on the manufacturing process of the micro motion convert mechanism as shown in Fig.5, and aluminum for the mask by semiconductor micromachining technology including FAB etching, which is shown in Fig. 6 (a) and (b). In Fig. 6 (b), the hinges of Type B are made in the same

FAB source Table Mask TMP

Figure 1. Fast Atom Beam(FAB) manufacturing machine

Specimen

Figure 2. Manufacturing view by use of mask and FAB

Micro Hinges and their Application to Micro Robot Mechanisms

135

Micro Hinges and their Application to Micro Robot Mechanisms

R2 Y L1

R1

L2

L3

1

R3

X

O

P

L4

Figure 3. PRRR mechanism ( R :Revolute pairs, P :Prismatic pairs) SiO2 (Sacrificial layer) Si substrate Al

Y R1 O

R2

R3

X

FAB

Polyimide Cr

P

Figure 4. Micro motion convert mechanism

(a) Type A

Figure 5. Manufacturing process of the micro motion convert mechanism

(b) Type B

Figure 6. Manufactured micro motion convert mechanisms one way manufacturing direction. Figures 7 (a) and (b) show the dimensions of the micro motion convert mechanisms Type A and Type B, respectively. Using the experimental apparatus, the input and output displacement characteristics of the micro motion convert mechanism were examined. In the results, the Type B has not the determined motion direction because of the same manufactured direction of the hinge. Figure 8 shows the relationship among output displacements Yout, 1, 2 and input displacement Xin of the Type A. In this experiment, when the X-direction displacement was given to the mechanism as the input, Y-direction displacement of the central hinge and the angles between the each rigid body part of the mechanism(link parts) were taken as the output. Through the use of a micromanipulator operated with the joystick, the tip of the needle was made to scrape the rigid body and move parallel to the mechanism so that it would produce the same effect with prismatic pairs, creating the mechanism to undergo a displacement. Also, when the X value or input displacement increased, it was taken as forward while when X value

136

M. Horie

decreased, it was taken as backward. The images of the process were grabbed and analyzed to measure the output displacement as well as the output angles relative to the input displacement. Output values relative to each input displacement, X, were presented in graphical forms: Figure 9 shows the output displacement, Y. The output angles 1 and 2, for X are obtained similarly. In the results, the input-output displacement relationship is investigated experimentally.

3. Palmtop Surface Mount System for Micro Devices by Use of a Pantograph Mechanism with Large-Deflective Hinges

20.8 52.5 (147.3)

1.4

10.9

17.0

11.4

1.4

When we opened the cellular phone case, we can look a lot of micro devices on the electric circuit board. In the case of conventional surface mount systems to make the electric circuit board with micro devices, the size is very large, for example, 2m(width)× 4m(length)× 2m(height). In this chapter, a new palmtop surface mount system which consists of groups of manipulators that have been

52.0

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(a) Type A A (a) Type

9.8

(147.3)

Y Yout

Xin 1 2

Y0 O

X

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Output displacement Y [μm]

(b)(b)Type Type B B Figure 7. Dimensions of the micro motion convert mechanisms [ Unit: μm] 30 forward 20 backward 10

0

10

20

Input displacement X [μm]

Figure 9. Output displacement Y

30

Micro Hinges and their Application to Micro Robot Mechanisms

137

Micro Hinges and their Application to Micro Robot Mechanisms minimized by an molding pantograph mechanism with hinges and links is proposed as shown in Figs. 10 and 11 by Mikio Horie et. al.(2001). Figure 10 shows the proposed new palmtop surface mount system. Figure 11 shows the molding pantograph mechanism with large-deflective hinges and links. The pantograph mechanism has linear input actuators in the experimental apparatus. The mechanism material is the Polypropylene. The hinge length of the pantograph mechanism used in the experiment was determined to be 200 μm, with the thickness being 180 μm. The width is 5 mm because of the section of the link 5*5mm. We confirmed that the mechanism has not fractured even if more than one million repeatable motion of the mechanism in the fatigue test shown by Mikio Horie et. al.(2006).

4. Fatigue Tests of an Molding Pantograph Mechanism with Large Deflective Hinges Large deflective hinges are known to be able to obtain large angular displacements by producing larger deformations on the materials than the elastic regions. Therefore, when they are used as hinge parts of a pantograph mechanism, it is thought that there might be some possibility that the materials could be destroyed by fatigue under the condition that repeated use creates large deformations. Consequently, these results led to conducting the fatigue test of the model- devised pantograph mechanism using the apparatus. The experimental apparatus consists of three parts: the vibrator based on the slider crank mechanism, the laser displacement sensors, and the oscilloscope. The test is conducted as follows: Using a vibrator, the sinusoidal displacement input at 7.5mm vibration is given to the input part(Point A in Fig. 11) in the horizontal direction of a pantographmechanism, and then the displacements of input-output ending points are measured by a laser displacement sensor. The relation between the input vibratory frequencies and the repeatable numbers until hinge parts has fractured is determined after the molding pantographs using hinges of different thickness lengths have been examined. The hinge thickness, the vibration frequency, the number of cycles, and failed hinge parts of the molding pantograph mechanisms in each conducted experiment, are shown in Table 1. A hyphen mark in Table 1 shows that a pantograph mechanism does not fracture even after one million-time repetition. The A, B, C, etc. in Table 1 mean a hinge name in Fig. 11.

5. Experimental Method of the Fatigue Test of One Hinge Model Two kinds of hinge specimens used in a fatigue test were manufactured by injection molding, whose shape and dimensions(Length*Thickness*Width) are

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C

A

B (F1)

Figure 10. A new palmtop surface mount system composed of parallel arrangement pantograph mechanisms

F

D

E (F2)100

mm

Figure 11. A pantograph mechanism with large-deflective hinges and links ( Material: Polypropylene)

respectively 0.2mm*0.2mm*5mm and 0.2mm*0.5mm*5mm. The material of the specimens is a widely used thermoplastic resin such as polypropylene(PP), polyethylene(PE), polyamide(PA) and polystyrene(PS). Regarding polypropylene(PP), Type A[PP(A)] and Type B[PP(B)] are used and are made by a different company with same materials. A fatigue test was conducted on hinge specimens using the test apparatus. A cyclic bending load with ±45o angular displacement, and a sinusoidal cyclic tensile-compressive load were applied on the specimens. Cyclic frequency of 10Hz was applied on hinges 200μm thick, while 20Hz was applied on hinges 500μm thick. The results are shown in Fig. 12. The horizontal axis represents thenumber of cycles when hinges were broken whilethe vertical axis represents cyclic tensile compressive load by the fatigue test apparatus. The results show that hinges 500μm thick are more durable than hinges 200μm thick. This is Table 1 Results of the fatigue test Group I Exp. No. 1 2 3 4 Hinge thickness [μm] 60 90 90 90 Input frequency [Hz] 19.4 15 15 20 Number of cycles100 - 153900 3140 Failed hinge D - C, D B,C,D Group Exp. No. Hinge thickness [μm] Input frequency [Hz] Number of cycles Failed hinge

II 5 6 7 8 100 110 110 115 22.2 520 18 100 - 3900 30600 F2, D - C, D D

9 120 15 -

II III 10 11 12 13 14 15 16 17 125 125 130 135 150 165 225 260 17 20 26 20 26.8 17 1020 - 6040 100 8600 100 - F2, D F2, D A F2, D -

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Attention: (1) Polypropylene(PP), (2) Polymerization blend PP (A1~A4) which we call PP+R-TPO, that is, the thermoplastic elastomer to which PP monomer and ethylene are polymerized by multistep, (3) Machine blend PP(B1~B3) called PP+EPDM that PP and Ethylene Propylene-Diene Monomer(EPDM) were mechanically mixed using a mixer, and (4) Another machine blend PP(B1~B3) (+MAH-PP) that we mixed maleic anhydride modified PP(MAH-PP) with the above (3) machine blend PP. Figure 12. Results of fatigue test because stress by the tensile-compressive load gets smalleras the cross section of hinge gets larger. However, the number of cycles of tensile-compressive load stayed at about one and a half times as many when the cross section of the hinge was two and a half times as large. It is conceivable that damages by bending load got severer as hinges were thicker. It was also confirmed that PP was the most durable among 8 kinds of specimens with 200μm-thick hinge and 4 kinds of specimens with 500μm-thick hinge, while PP was asdurable as A1 regarding 500μm-thick hinges. A1 which is one of the name of specimen has the bending ratio PP: Elastomer = 70: 30(R-TPO) and bending ratio of Mah-PP = 0 %. It is because 200μm-thick hinges with elastomer, such as A1, got significantly longer and buckling in the hinge part became prominent during the test. This had an adverse effect on hinge durability. It was also found that the durability of hinges with a thickness of 200μm or 500μm became less durable as the proportion of elastomer blended in the material became

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larger. But the result could be reversed when hinges are made thicker and the fatigue damage by bending load got severer. Moreover, it was found that Machine Blend PP was far less durable than Polymerized Blend PP.

6. Conclusions The characteristics of the two kinds of micro-manipulators with large-deflective elastic hinges and large-deflective hinges are discussed, and the following conclusions are obtained. (1) A micro motion convert mechanism was manufactured with polyimide for the structural material based on the FAB(Fast Atom Beam) manufacturing process. (2) A molding pantograph mechanism consisting of large deflective hinges with 200 μm of the length,180 μm of the thicknessan5 mm of the width did not fracture in displacement inputfatigue tests on the mechanism even after repeatedly used one million times. (3) This study performed cyclic-bending tests with a static tensile load on hinges made from polypropylene, polyethylene, polyamide, as well as polystyrene, and ascertained that polypropylene, and polyamide hinges have a relatively high fatigue resistanceamong them. Regarding the polypropylene hinge, the relationship between the tensile strength and the normal logarithm of the number of bending cycles to fracture was displayed in a straight line. (4) From the results of a fatigue test on the hinge specimens in the chapter 5, it was found that hinges 500μm thick were more durable than 200μm thick specimens, and that Machine Blend PP was less durable compared to Polymerized Blend PP.

References Kota, S., Design of compliant mechanisms with applications to MEMS and smart structures, Proceedings of the 10th World Congress on the Theory of Machines and Mechanisms, Oulu, Finland, 20 to 24, June 1999, Vol.7, pp. 2722-2728. Müglitz, J. and Schönherr, J., Miniaturized mechanism - joint design, modeling, example, Proceedings of the 10th World Congress on the Theory of Machines and Mechanisms, Oulu, Finland, 20 to 24, June 1999, Vol.2, pp. 848-855. Horie, M., Uchida, T., and Kamiya, D.(2001), A Pantograph Mechanism with Large-deflective Hinges for Miniature Surface Mount Systems, ROMANSY 13-THEORY AND PRACTICE OF ROBOTS AND MANIPULATORS, ISBN 3-211-83333-1,Springer-Verlag Wien New York, 93-102. Horie M., Okabe Y., Yamamoto M., and Kamiya D.(2006), Durability of Large-Deflective Hinges made of Blend Polypropylene used for Molded Pantograph Mechanisms, ROMANSY 16-ROBOT DESIGN, DYNAMICS, AND CONTROL, ISBN 3-211-36064-6, Springer-Verlag Wien New York, 355-362.

Parallel RPR--RPR Manipulator Generating 2-DOF Translation along One-Sheet Revolute Hyperboloid Chung-Ching Lee Professor, Dept. Tool & Die-Making Engineering, National Kaohsiung University of Applied Sciences, 415 Chien Kung Road, Kaohsiung, 80782, Taiwan R.O.C. E-mail:[email protected]

Jacques M. Hervé Honor Professor, Ecole Centrale Paris, Grande Voie des Vignes, F-92295, Chatenay-Malabry, France E-mail: [email protected]

Abstract. Based on the theorem that any subset of translations is normal or invariant by conjugation in the group of a spatial 3-DoF translation, a novel type of parallel manipulator RPR--RPR generating 2-DoF translation along a revolute hyperboloid of one sheet is synthesized. A special type of plane symmetric RPRP Delassus mechanism, which has two noncontiguous parallel revolute axes and two prismatic pairs bilaterally symmetric with respect to the plane containing the revolute axes, is further derived as being a special case of the previous manipulator with coincident R axes. Keywords: Group conjugation, 2-DoF translation, revolute hyperboloid of one sheet, parallel manipulator, doubly ruled surface, Delassus linkages

1. Introduction Parallel generation of 3-degree-of-freedom (3-DoF) translation has been attractive to researchers for a long time [Clavel, 1987; Hervé, 1991; Wenger and Chablat, 2000; Gao et al., 2002; Kong and Gosselin, 2002; Angeles, 2004, Lee and Hervé, 2006, 2007, 2009a]. However, a parallel manipulator generating 2DoF translation along a surface that is neither a plane nor a sphere is still a novel topic in robot kinematics. In [Hervé, 2009], a useful theorem derived from the group-algebraic properties of the 6-dimensional (6D) set of displacements is stated: any subset of translations is invariant by conjugation (or normal) in the

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commutative (or Abelian) Lie 3D group T of 3-DoF translations. By application of the foregoing theorem, any 2-DoF translation along a surface that may be, for instance, a revolute hyperboloid of one sheet is normal in the group T. Through this theorem, a novel parallel manipulator generating 2-DoF translation along a revolute hyperboloid is synthesized. Under a special geometric arrangement, a new derivation of a particular Delassus paradoxical mechanism, which is a plane symmetric RPRP chain, is also obtained. The first step is to introduce a single-closed-loop chain with an end effector moving by translation along the surface of a revolute hyperboloid of one sheet with respect to a fixed frame. This chain can be considered to be the parallel arrangement of a PPP generator of the group T with a RPR open chain with parallel revolute axes. Hereinafter, P and R are the usual notations of a prismatic pair and a revolute pair respectively and the underline indicates the parallelism of R axes. The RPR chain generates a special 3D submanifold of a 4D group X of Schoenflies motions. An explicit vector calculation is done for verification. The second step is the addition of another single closed-loop RPR--PPP chain, which is conjugate or congruent by translation to the first RPR--PPP chain. As for the third step, the rigid connection of the two end-effectors and the removal of the superfluous PPP subchains are made. From the plane symmetry of the hyperboloid, a novel manipulator architecture is derived. Finally, we also attain a plane symmetric RPRP mechanism with 1-DoF finite paradoxical mobility, which is a special Delassus four-bar linkage.

2. Single-Closed Loop Chains with 2-DoFs The parallel arrangement of a PPP generator of the group T with a RPR open chain having two parallel R axes produces a single-closed-loop RPR--PPP chain, which is depicted in Figure 1 where (i, j, k) indicates an orthonormal vector base. The points A and B are assumed to be located on the R axes, which are both parallel to k. R(A, k) and T(s) denote the 1D group of rotational motions of axis (A, k) and the 1D group of rectilinear translations parallel to s, respectively. Such a chain embodies R(A, k)T(s)R(B, k ) T(j)T(i)T(k) and is movable with two DoFs as being a trivial chain associated to the 4D Schoenflies group X(k) [Lee and Hervé, 2009b]. In a 2-DoF translation, any point of the translating body moves on a surface that is a 2D submanifold of the 3D Euclidean affine space. In RPR--PPP manipulator, let M be any point belonging to the moving end-effector. One limb is the RPR chain. The other limb is a PPP generator of T. The 3D group T is the set of point transformations

Parallel RPR-||-RPR Manipulator Generating 2-DOF Translation…

M M' =M+a1i+b1j+c1k

143

(1)

which is also expressed by (MM')= a1i+b1j+c1k where (MM') denotes the free vector obtained from the points M and M'. The parameters a1, b1 and c1 are the scalar values of the rectilinear translations produced by the P pairs from a given home configuration of the PPP chain. In Figure 1, the RPR limb is such as (AN) is perpendicular to (NB), which is parallel to the unit vector s; (AN)=ri, (NB)=bs, s=sin j+cosk where the angle is a given constant. To facilitate a clear understanting, the point position B does not coincide with N and b is not zero in the figure. Because displacements or rigid-body motions are the body-position changes, the geometric description of an initial home configuration of a mechanism shown in Figure 2 is a datum that is necessary for the characterization of all relative displacements. For simple calculation, at the home configuration, the position B0 of the end-effector point B is assumed to coincide with N. Hence, B0=N(B) in the following. Moreover, the position of M is M0 at the home configuration, as shown in Figure 2. The set of feasible displacements generated by the RPR limb can be attained by a sequence of three 1-DoF motions. In the first stage, the second R moves with an angle around the axis (B0, k) while the first R and the P keep their home postures (=0 and b=0); M0 becomes Mi , which is expressed by M Mi = B0 + exp(k)(B0M0) or (B0Mi) = exp(k)(B0M0)

(2)

In the second stage, the second R is locked (angle keeps its value), the first R keeps its home position (=0) and the P provides a translation parallel to s , with an amplitude b. Then, the point Mi becomes Mi' Mi Mi' = Mi +b s

(3)

With (B0M'i)=bs+exp(k)(B0M0) and (NM'i)=bs+exp(k)(B0M0) because of the choice B0=N, we have (NM'i)=b sin j+bcosk+ exp(k)(NM0)

(4)

In the third stage, the second R and P are locked and the parameters and b keep their values. The first R moves with angle around the fixed axis (A, k) and Mi' is transformed into M', Mi' M' = A+ exp(k)(AMi') or (AM') = exp(k)(AMi')

(5)

Using (AMi')=(AN)+(NMi')= ri+b sin j+bcos k+exp(k)(B0M0) yields (AM')= =bcosk+exp(k)(ri+bsin j)+exp[(+)k](B0M0)

(6)

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Figure 1. RPR--PPP generator of 2-DoF translation along a one-sheet hyperboloid: general configuration

Figure 2. RPR--PPP generator of 2-DoF translation along a one-sheet hyperboloid: initial posture of parametrization.

Notice that exp(k)(ri)=r(cos i+sin j) and exp(k)(b sin j)=b sin (sin i+cos j ) and we have (AM')=bcosk+(rcos -b sin sin)i+(rsin +b sin cos )j+exp[( +)k](B0M0) (7) This is an explicit expression of the product of transformations R(A, k;)T(s;b)R(B, k; ) belonging to the set product R(A, k )T(s)R(B, k). In the RPR--PPP parallel mechanism, for any point M, the feasible displacements of the moving platform have to satisfy Eq.(7) and have to be a translation, which is achieved for =-. As a matter of fact, by translation, (BM) must keep unchanged, that is, (B'M')=(B0M0). Then, Eq.(7) becomes

Parallel RPR-||-RPR Manipulator Generating 2-DOF Translation…

(AM')= bcosk+[r(cos -1)-bsin sin]i+(rsin+bsincos)j+(AM0)

145

(8)

It is straightforward to verify that the end-effector point B that coincides with B0 at the home configuration moves on the ruled hyperboloid with the axis (A, k), which has the parametric representation (AB')= r(cos i+sin j)+b[sin(- sin i+cos j)+cos]k =(AN)+bs

(9)

where N and s are respectively the transformation of N and s through a rotation of angle around the axis (A, k). Because of the end-effector translation, any point moves also on a ruled hyperboloid, which is congruent by translation with the orbit of B. In other words, the end-effector moves by a 2-DoF translation along a surface of revolute hyperboloid of one sheet. This hyperboloid is a doubly ruled surface resulting obviously from its plane symmetry.

3. Parallel Manipulator Generating 2-DoF Translation along a Ruled Hyperboloid Two 2-DoF translation manipulators that are congruent by any given translation produce the same translational motion of their end effector [Hervé, 2009]. In the two congruent manipulators, two end-effectors move with the same motion and consequently can be rigidly connected. Then, more than two congruent limbs can connect the fixed base with the end effector. In Figure 3, two RPR--PPP mechanisms produce the same ruledhyperboloid translation. That way, rigidly connecting the two end effectors and removing the PPP subchains lead to a parallel generator RPR--RPR of 2-DoF translation (2D submanifold of the group T ) along a hyperboloid of one sheet, as in Figure 4. This surface has a special property: it is doubly ruled and symmetric with respect to any plane containing its axis of revolution. In addition, based on this property, the double generation of the 2-DoF translation along a revolute hyperboloid of one sheet is derived and a new type of RPR--RPR parallel chain is synthesized as depicted in Figure 5.

4. Derivation of Delassus RPRP Paradoxical Chain The end-effector motion in the previous manipulator is 2-DoF translation along a revolute hyperboloid of one sheet. The foregoing surface is doubly ruled, which leads to not only a non-obvious parallel generator of 2-DoF translation but also a new demonstration of the Delassus plane symmetric RPRP mechanism. In

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Figure 6, actually, each of the two Rs is made up of two coaxial Rs, which are equivalent to one R when ignoring the internal self rotation of the intermediate body. This special geometric case is obtained when the distance d between the parallel R axes of the translating end-effector is equal to zero. Then the distance between the parallel R axes of the fixed base, which is also equal to d, becomes zero. In fact, d is the amplitude of the translation used in section 3 to introduce two conjugate RPR limbs.

Figure 3. Two mechanisms generating the same 2-DoF translation

Figure 4. An RPR--RPR parallel generator of 2-DoF translation along a one-sheet hyperboloid

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Figure 5. Another RPR--RPR manipulator of 2-DoF hyperboloidal translation

Figure 6. A plane symmetric RPRP Delassus chain

5. Conclusion A novel parallel RPR--RPR manipulator generating 2-DoF translation along the surface of a hyperboloid of one sheeet is synthesized via the invariance by conjugation of any submanifold of translations in the commutative (or Abelian) Lie group T. In addition, a special case of plane symmetric Delassus mechanism, namely the RPRP paradoxical mechanism with 1-DoF finite mobility is introduced through the double generation of the surface by the rotation of a straight line together with an adequate geometric arrangement of the RPR--RPR

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parallel mechanism. The simulation of the proposed manipulators has been verified by computer animation software. The new mechanism becomes a RR-RR hinged parallelogram when one P is locked and therefore it has the singular posture of the flattened parallelogram. In despite of its singularity, the hinged parallelogram has many practical applications. In further work, the mechanisms revealed here will be extended to mechanisms with screw pairs and will lead to a new derivation of the general Delassus paradoxical HPHP four-bar linkage. Last but not least, generators of 2-DoF translation can be employed as composite joints in the synthesis of robotic manipulators and further generalize the hinged parallelogram, which generates translation with only one degree of freedom. This is also the motivation of our work.

Acknowledgements The authors are very thankful to the National Science Council for supporting the research under grant NSC 98-2221-E-151-019.

References Angeles, J. (2004), The qualitative synthesis of parallel manipulators, ASME J. Mech. Des., 126: 617-624. Clavel, R. (1987), Dispositif pour le déplacement et le positionnement d'un éléments dans l'espace, Swiss patent no 672089A5, 1985; international patent (PCT) no WO 87/03538. Gao, F., Li, W., Zhao, X., Jin, Z. and Zhao, H. (2002), New kinematic structures for 2-, 3-, 4-, and 5-DOF parallel manipulator designs, Mech. Mach. Theory, 37:1395-1411. Hervé, J. M. and Sparacino, F. (1991), Structural synthesis of parallel robots generating spatial translation, Proc. 5th IEEE Int. Conf. Adva. Rob., Pisa, Italy, 808-813. Hervé, J. M. (2009), Conjugation in the displacement group and mobility in mechanisms, Trans. Canadian Society for Mechanical Engineering, 33: 3-14. Kong, X.-W. and Gosselin, C. (2002), Kinematics and singularity analysis of a novel type of 3-CRR 3-DOF translational parallel manipulator, Int. J. Robot. Res. 21:791-798. Lee, C.-C. and Hervé, J. M. (2006), Translational parallel manipulators with doubly planar limbs, Mech. Mach. Theory, 41: 433-455. Lee, C.-C. and Hervé, J. M. (2009a), On some applications of primitive Schoenfliesmotion generators, Mech. Mach. Theory, 44:2153-2163. Lee, C.-C. and Hervé, J. M. (2009b), Type synthesis of primitive Schoenflies-motion generators, Mech. Mach. Theory, 44:1980-1997. Lee, C.-C. and Hervé, J. M. (2007), Cartesian parallel manipulators with pseudoplanar limbs”, ASME J. Mech. Des., 129:1256-1264. Wenger, P. and Chablat, D. (2000), Kinematic anlaysis of a new parallel machine tool:The orthoglide, Advances in Robot Kinematics, J. Lenarcic and M. M. Stanisic eds., Kluwer Academic Publisher, Dordrecht, 305-314.

A Study on the Effects of Cable Mass and Elasticity in Cable-Based Parallel Manipulators Erika Ottaviano and Gianni Castelli LARM: Lab. of Robotics and Mechatronics, DiMSAT, University of Cassino, Italy Abstract. This paper presents numerical and experimental results for an analysis of the effects of cable’s mass and elasticity in cable-based parallel manipulators. It will be shown, through numerical simulation and experimental results, that effects of these characteristics cannot be always neglected, because they affect the pose capability of the end-effector. A planar 2 cable point-mass manipulator is considered as a case of study. The model and results can be further extended to all classes of cable-based manipulators, such as under or fully constrained types.

1 Introduction Cable-based manipulators belong to the class of parallel robots in which the fixed frame and platform are connected by several cables, which can be exerted or retracted by suitable actuation system. A cable-based manipulator has better characteristics than a classical parallel robot as reported in (Verhoeven, 2004). Although cable manipulators have a rather long history of industrial applications, many of their fundamental problems regarding the kinematics, dynamics, and control were addressed by researchers only in the recent years. The stiffness problem has been studied by (Verhoeven et al., 1998). To investigate the vibration effects some researchers modeled cables as linear springs (Behzadipour and Khajepour, 2006) or nonlinear springs (Kawamura et al., 2000). However, all of them assumed that a cable has axial flexibility only (i.e., ignoring the transversal flexibility), which was taken into account by (Diao and Ma, 2009). The cable’s mass is usually neglected in the modeling phase, considering the end-effector mass only. The effect of cable mass has been considered in few cases only, for which large manipulation operations are considered (Bouchard and Gosselin, 2006; Riehl et al., 2009). In this paper it will be shown that the key point for the choice of an appropriate model deals with the ratio between end-effector and cable masses or, more generally, between the end-effector wrenches and cables’ tensions. In particular, it will be shown that axial and transversal flexibility of a cable affect the position capability of the end-effector (Castelli, 2010). Cables with no negligible mass will tend to sag under their own weight and deviation from ideal mass less cable model can be significant and cannot be ignored when an accurate modeling is required.

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2 Cable Modeling for Cable-Based Manipulators In the following 4 different models of a cable will be considered for the analysis of cable-based manipulators: mass less inextensible cable, mass less cable with elasticity, cable with continuous mass and elasticity, and lumped mass cable with elasticity. In most of cases cables are modeled as mass less and inextensible. Some authors take into account cable’s elasticity only, but for some applications cable mass should be considered also to have an accurate model. Cable mass should be taken into account in general when cable sagging effects cannot be neglected. 2.1 Mass Less Inextensible Cable Model A model for the Kinematic analysis can be based on the assumption of mass less inextensible cables, with the hypothesis that they are always in tension and can thus be treated as line segments representing bilateral constraints. Therefore, the IK consists in computing the norm of the vector li connecting each cable attachment point, to the ending point of the cable attached to the mobile platform. In order to check the cables forces distribution, Statics or Dynamics analyses can be performed. 2.2 Elastic Mass Less Cable Model Cables can be supposed to be mass less, therefore, deformation due to gravity can be neglected. According to this model, cables behave as linear springs and their elasticity coefficient can be evaluated as function of the cable section area A, the Young Modulus E, and the cable length l. Therefore, the cable tension Fi can be evaluated as function of the elasticity coefficient ki and cable longitudinal deformation li. The passive stiffness (Verhoeven, 2004) describes the reaction of a mechanical system by small perturbation. For many practical applications, due to high elasticity of real cables (especially using plastic materials) the stiffness may be low in some regions of the workspace. This can affect the precision of the overall system, requiring appropriate control strategies or a suitable design. 2.3 Continuous Cable Model with Mass and Elasticity In a static or quasi-static application in gravitational field cables can be assumed to have homogeneous properties and negligible flexural stiffness. In the following cables are considered elastic, and cable’s attachment points are assumed as fixed. Let us consider the unstrained length of the cable given by l0, and the force applied to the end of the cable given by F, both of them are assumed as known. Figure 1a) shows the model of a sag cable under the gravity effect. Main parameters are , E, A, and l, which represent the unstrained linear density, the elastic modulus, the

A Study on the Effects of Cable Mass and Elasticity…

151

unstrained cross-sectional area, and the strain of the cable, respectively. Each point of the cable can be represented by Cartesian coordinates (x, z) and path variable denoted by s. The variable s represents the strained length of the cable segment as measured from the attachment point of the cable to the current point. A model has been used (Irvine, 1981; Kozac et al. 2006) to compute the static displacement of a cable and thus, its derivation will not be presented here. Coordinate of each cable point (x, z) can be defined by its curvilinear abscissa s as z(s)

Fz s EA

x( s )

%0 g s

!

EA # 2

Fx s EA

l0 s "

%0 g

2

(sinh

-1

*

Fx Fz % 0 gl0 2

2

%0 g

$

&

Fx

Fx Fz % 0 g s l0 2

2

Fz % 0 g s l0 ' -1 F % 0 gl0 - sinh ! z ! " " ) Fx Fx # $ # $,

(1)

Equations (1) describe the profile of an elastic cable under the action of gravity and forces at its free end. In these equations, the axial elasticity can easily be identified. An experimental validation of the above-mentioned model is given in (Kozac et al. 2006). Although the analysis provides a continuous model, it can be used for static applications only (Riehl et al., 2009). 2.4 Lumped Mass and Elastic Cable Model Each cable can be modeled as lumped masses connected in series by mass less springs, as shown in Fig. 1b). Therefore, considering a cable, it will be composed by N lumped masses mi, and N+1 linear springs connecting the adjacent masses with elasticity coefficient ki. In this context, it has been assumed that all the mi masses are equal, as well as the ki. The sum of the mi masses equals to the total cable mass, and ki is equal to (N+1)k, in which k is the overall cable elasticity. FO

FO

Z

Z

X

O

O l0a la

, E, A (x, z, s) l0

B

g

Fx l

a)

g

X ma l0b lb

mb mc l0c B lc l0d l

Fx

d

Fz

F

Fz

F

b)

Figure 1. A cable with mass and elasticity: a) continuous model; b) lumped mass model.

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3 Case of Study: Planar 2 Cables Point-Mass Robot In the following the above-mentioned cable models have been used to evaluate the effects of elasticity and cable mass for end-effector positioning for a planar 2 cables point-mass manipulator. In particular, the mass less inextensible cable, elastic cable and cable with mass and elasticity have been modeled in ADAMS environment for simulation purposes. Moreover, the latter one shown in Fig.2b) has been compared with the continuous cable model. Simulations and experimental tests have been carried out by varying the end-effector mass leaving the cables’ nominal lengths and, consequently, their masses unchanged. Let us consider a planar point mass cable-suspended robot shown in Fig. 2a). A fixed reference frame OXY is considered and attachment points are denoted as A1 and A2. According to the mass less inextensible cable model cables lengths can be readily computed as (for i = 1, 2)

li ( xG ) Ai xG

Aix x

2

A z 2 Aiz z ; i ( xG ) tan 1 ! iz " # Aix x $

(2)

where xG = [x, z]T are the coordinates of center of gravity of the end-effector expressed in the fixed frame and li and i are respectively the length and angle for the i-th cable. Vectors Fi and FG respectively identify the tension in the i-th cable and weight force of the end-effector under the gravity action. According to Fig. 2a) the static equilibrium can be expressed as F1cos(1 ) F2 cos(2 ) 0 ; A1

F1sin(1 ) F2 sin(2 ) mee g

(3)

A2 2

1

l1a Cable 1

Cable 2 l2

l1 F1

LZ

m1a

m2a l1b

F2

l2b m2b

m1b l1c

FG = mee g

l2c

m1c g

l2a

Planar constrain

m2c

l1d l2d

Z X

End-effector

O

Fixed frame

LX

a)

b)

Figure 2. A scheme for a planar 2 cables point-mass manipulator: a) with mass less cables; b) ADAMS model with cables’ mass and elasticity.

A Study on the Effects of Cable Mass and Elasticity…

153

3.1 Simulation Results

According to the scheme of Fig. 2a), parameters (expressed in mm) are: LX = 2000.0, LZ = 2000.0, l1=1788.85, l2=2000.0, the initial pose is x=800.0 and y=400.0. Cables have a nominal diameter of 0.8 mm, Young's modulus of 200 GPa, linear density of 0.003 kg/m and breaking load of 543 N. The first simulation deals with the mass less inextensible cable model, therefore it has been assumed k1= k2= N/m; m1= m2= 0 kg. A second simulation is related to a model with mass less elastic cables, and cables are modeled as linear springs with k1= 5.619E+005 N/m and k2= 5.026E+005 N/m with m1= m2= 0 kg. Third simulation deals with a model considering cables with mass and elasticity. Each cable has been modeled by lumped masses and linear springs, according to the scheme shown in Fig. 2b) such as: mia= mib = mic=ma/3; and lengths lia= l1ib = lic = lid =li/4. Cable elasticity is kia= kib = kic = kid =4ki with ki=EAi/li. Cables masses are m1= 6.45 g; m2= 7.21 g. Simulations have been carried out in ADAMS environment by varying the end-effector mass from 13.66 g (which represents the overall cables mass mtot for the given configuration) to 79.56 kg, which is 700 times mtot (Castelli, 2010). Figure 3 shows the effect of end-effector mass over cable mass on the end-effector positioning capability. Figure 4 shows a comparison between the lumped and continuous mass cable’s models. Results for the latter one are obtained by Eqs.(1). Figures 3a) and 3b) show respectively horizontal and vertical end-effector displacements, when the second and third models are considered. These two models give different displacements up to a threshold value for the ratio between end-effector and cables’ masses, that in this case is equal to 100. Over this value the two models have the same behaviour, and cable mass can be neglected. Therefore, the end-effector positioning capability depends on the ratio between end-effector and cables’ masses. In particular, for low ratios the cable’s sagging is quite relevant and determines a large deviation in end-effector positioning. 20.0

Only elasticity Elasticity and cable mass

End-effector vertical displacement [mm]

End-effector horizontal displacement [mm]

20.0

15.0

10.0

5.0

0.0

a) -5.0

Only elasticity Elasticity and cable mass

15.0 10.0 5.0 0.0 -5.0 -10.0

b)

-15.0 1

10

100

Mee/Mcs

1000

10000

1

10

100

1000

10000

Mee/Mcs

Figure 3. Comparison among the proposed cables models for the end-effector position: a) horizontal displacement; b) vertical displacement.

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2000

2000 Continuos model

Continuos model

Lumped mass model

Lumped mass model

1500

z [mm]

z [mm]

1500

1000

500

500

M end -effector M cables 0 0

1000

200

1

M end -effector

a) 400

600

800

x [mm]

1000

M cables 0 0

200

1000

b)

400

600

800

1000

x [mm]

Figure 4. A comparison between lumped and continuous mass cable’s models: a) low end-effector and cables masses ratio; b) high end-effector and cables masses ratio.

For the lumped mass model it has been chosen to use 3 masses, this value provides an optimal compromise between computational complexity and accuracy of the model (Castelli, 2010). A simulation test shown in Fig.4 has been carried out to compare results for the lumped and continuous mass models. 3.2 Experimental Characterization

A validation of numerical simulations has been carried out by considering a planar 2 cables point-mass manipulator. For the prototype under-study, according to the scheme of Fig. 2a) the pose is x=629.45 and y=1153.65. Plastic cables were chosen, with average nominal diameter of 4 mm and linear density of 10.94 g/m. The Young's modulus was determined experimentally by means of a stress-strain test on a test-piece of the cable and it was set equal to 50 MPa, neglecting the nonlinear relationship between stress and deformation of the cable. During the tests 5 different masses have been used as end-effector, while the total mass of cables was always equal to 29.88 g. Table I shows values of the lengths of the cables and end-effector coordinates collected for the 5 tests. Measurements of the manipulator’s configurations were performed by using an algorithm based on Image Processing and implemented in MATLAB environment (Castelli, 2010). Figure 5 shows the phases of the implemented program. In Table I deformation rates 1 and 2 are reported for the 2 cables and deviations of the end-effector coordinates x and z with respect to the initial conditions. As expected, according to numerical results in previous Section, by increasing the mass of the end-effector a greater axial deformation of the cable will occur because of its elasticity together with lower longitudinal strain due to the mass of cables. It is worth noting that deviations of the end-effector position from the initial configuration are due to both elasticity of the cables (on a priority basis for tests M4 and M5) and mass of cables (on a priority basis for tests M1, M2 and M3). Indeed, tests M1 and M2 show

A Study on the Effects of Cable Mass and Elasticity…

155

significant vertical deviations of the end-effector along Z+ direction due to the mass of cables, while the vertical deviation of the end-effector along the Z- direction is due to the elongation of the cables. The ADAMS model in Fig. 2b) has been used for a comparative analysis (Castelli, 2010). As for experimental tests, during the simulation the end-effector weight was increased continuously from the value of 98.1 mN (10 g) to a value of 49.05 N (5 kg), while nominal cables lengths were kept constant. Figure 6 shows a comparison for end-effector position during the experimental tests and numerical simulation. The maximum deviation between experimental and numerical values do not exceed the 8% of the average of the cables’ lengths. Table I. Experimental results for the tests varying the end-effector masses. test

mass [g]

M1 M2 M3 M4 M5

53.26 101.34 676.47 1305.40 3048.30

l1 [mm] 1052.1 1054.8 1080.6 1109.8 1117.6

l2 [mm] 1690.9 1695.4 1721.6 1777.2 1779.8

1 [%] 0.10 0.36 2.81 5.55 6.28

2 [%] 0.47 0.74 2.29 5.60 5.75

x [mm] x [mm] z [mm]

z [mm]

672.2 667.6 651.4 630.7 625.9

74.6 66.8 17.9 -41.9 -45.1

42.7 38.1 21.9 1.2 -3.5

1227.3 1219.5 1170.6 1110.7 1107.6

(1) Image acquisition and identification of the manipulator’s area

(2) Image processing and recognition of the cables

(3) Coordinate transformation from pixels to mm for the cables

Figure 5. The 3 phases of the Matlab program based on Image Processing.

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750.0

1250.0 Simulation data Experimental results

Simulation data Experimental results

Vertical position [mm]

Horizontal position [mm]

1200.0 700.0

650.0

1100.0

1050.0

a) 600.0 10

1150.0

100

1000

End-effector mass [g]

10000

1000.0 10

b) 100

1000

10000

End-effector mass [g]

Figure 6. Comparison between the end-effector position obtained by experimental tests and numerical simulation: a) x coordinate, b) z coordinate.

Conclusions In this paper modeling and experimental results are presented for the analysis of the effects of cable’s mass and elasticity. It has been shown that they affect pose capability of cable-based parallel manipulators. These effects must be analyzed through the ratio of end-effector and cables masses. Observing the results it is evident that the possibility to neglect the cables’ masses is related to the end-effector and cables and do not depend by the manipulator dimensions. Further extension of this work is related to vibration and dynamic problems considering the cables’ masses.

Bibliography Behzadipour, S., Khajepour, A. (2006). Stiffness of Cable-Based Parallel Manipulators with Application to Stability Analysis. Jnl. Mech. Des. 128(1), 303–310. Bouchard, S. Gosselin, C. M. (2006). Kinematic Sensitivity of a Very Large Cable-Driven Parallel Mechanism”, ASME DETC '06, Philadelphia, paper n. DETC2006-99222. Castelli. G. (2010). Analysis, Modeling and Experimentation of Cable-Based Manipulators. PhD thesis, University of Cassino. Diao. X. and Ma. O. (2009). Vibration Analysis of Cable-Driven Parallel Manipulators. Multibody System Dyamics. 21, 347–360. Irvine, H. (1981). Cable Structures. Cambridge, MA: MIT Press. Kawamura, S., Kino, H., Won, C. (2000). High-Speed Manipulation by Using a Parallel Wire-Driven Robots. Robotica, 18, 13–21. Kozak, K., Zhou, Q., Wang, J. (2006). Static Analysis of Cable-Driven Manipulators with Non-Negligible Cable Mass. IEEE Trans. on Robotics, 22 (3), 425-433. Riehl, N., Gouttefarde, M., Krut, S., Baradat, C., Pierrot, F., (2009). Effects on Non-Negligible Cable Mass on the Behavior of Large Workspace Cable-Driven Parallel Mechanism. IEEE Int. Conf. Robotics and Autom., Kobe, pp. 2193-2198. Verhoeven, R. (2004). Analysis of the Workspace of Tendon-Based Stewart Platforms. PhD thesis, University of Duisburg-Essen.

Chapter II

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Mechanism Design Improvements of the Airway Management Training System WKA-3 Y. Noha, K.Satoa,A. Shimomuraa, M. Segawaa H. Ishiib, J. Solisbc, A. Takanishibc, K. Hataked a

Graduate School of Advanced Science and Engineering, Waseda University, Tokyo, Japan Department of Modern Mechanical Engineering, Waseda University, Tokyo, Japan c Department of Modern Mechanical Engineering and the Humanoid Robotics Institute (HRI), Waseda University, Tokyo, Japan d Kyoto Kagaku Co. Ltd., Kyoto, Japan b

E-mail: [email protected] Abstract. This paper reports on the development of an Airway Management Training System. Airway management is a standard operation executed in emergency and operating rooms. However, medical accidents occur due to unskilled operations. In order to avoid accidents, medical doctors undergo to mannequin-based training. However, traditional training techniques do not provide quantitative information on the trainee's performance and are not capable to assess the trainee's performance based on the quantitative information. In this context, we proposed a novel airway management training system, dubbed WKA-3. This work presents both the electromechanical and the control sub-systems of the WKA-3. The results of preliminary experiments carried out by medical doctors to doctors to verify the training system are also provided.

1 Introduction Airway management is a standard operation executed in emergency or operating rooms. However, medical accidents occur due to the unskilled operations. In order to avoid these accidents, medical doctors undergo to mannequin-based training (Satake et al. 1999). However, traditional training techniques i) fail to simulate the real conditions of the task, ii) do not provide quantitative information on trainee's performance and iii) do not assess the trainee's performance based on the collected quantitative information. In this context, using Robot Technology (RT), we would propose more efficient training system in the medical field. The advantage of using RT is that we could conceive advanced medical training systems designed to reproduce the conditions of the task and provide information of the task performance of the trainee. In particular; in Solis et al., 2007, the concept of an active training system has been introduced based on robotic platforms. Such training system should fulfills at least three conditions: i) reproduce the real-world condition of the task, ii) provide objective assessments of the training progress, and iii) provide useful feedback to

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trainees. As a result; since 2004, we propose a robotic patient as a long-term project. The robotic patient is capable: to serve i) as an active training device and ii) as an advanced evaluation tool. In order to fulfill these tworequirements, the robotic patient should i) be designed to emulate the human body (both anatomically and physiologically), ii) have sensors embedded into the simulator, iii) have actuators capable of reproducing the real conditions of the task, iv) provide feedback information to the trainee, and v) assess the trainee's performance objectively (Solis et al., 2007). For the purpose of these, we are proposing an innovative effective medical training system for the Airway management training. In this paper, we propose the WKA-3, which provides quantitative information on the training progress and simulates the real conditions of the task. The WKA-3 features the same basic functionalities of the previous prototypes WKA-1R (Noh et al., 2008) and WKA-2 (Noh et al., 2009) but with improved performance both at the mechanical and at the control level. In this work, we present the electro-mechanical design and the control system of the WKA-3. The design of the WKA-3 is focused on integrating sensors and actuators into both the link drive mechanism and the wire drive mechanism. From a control point of view, the WKA-3 is capable to support both position control and Virtual Compliance Control. Finally, we provide the experimental results on the effectiveness of the WKA-3 using the feedback from medical doctors.

2

Design of the WKA-3

The WKA-3 consists of a standard medical mannequin integrated with sensors and actuators. The mannequin consists of six parts such head, neck, mandible, tongue, vocal cord and trachea, as reported in Figure 1. The WKA-3 features seven degrees of freedom (six active and one passive). Tongue and mandible actuation is achieved by three mechanisms integrated into the WKA-3, namely a link drive mechanism, wire drive mechanism and under-actuated mechanism. Both the wire drive mechanism and the under-actuated mechanism are driven by Wire Drive Unit. Airway management challenges, such as restricted mouth opening, restricted cervical range can be reproduced by synchronization of WKA-3 mandible and tongue motions. Force control strategies (based on torque and force sensor readouts) are adopted to reproduce the stiffness of the muscular system and to response to operator's forces. In order to register the quantitative information of operator's performance, several sensors were integrated in redesigned WKA components, such as incisor teeth, vocal cord and trachea.

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Figure 1. WKA-3 Waseda Kyotokagaku Airway management training system 2.1 Wire Drive Unit To improve the efficiency of the torque transmission to the wire a novel design, named Wire Drive Unit, is proposed. The latter consists of a rotary stage, a force sensor, a wire tube and a pulley, as described in Fig. 1. The pulley and wire are connected through the wire tube, while the pulley and the rotary stage are assembled in order to measure the torque applied on the wire. These units can be controlled either via position or torque control. 2.2 Mandible The mandible can be model via a 3-DOF (two rotational and one translational) system when simulating mouth opening motions (Ishihara et al., 1975). However, while performing the airway management, the motion can be simplified to a 2-DOF system. For this reason, we implemented a 2-DOF system (Figure 2), which represents the rotational motion (Jaw) and the translational one (dJawAxis). In order to control both rotational and translational motions of the mandible, we have attached a four-bar mechanism and crank mechanism to the actuators. For the dJawAxis, both the four-bar and the crank mechanisms are driven. For the Jaw, only the four-bar is driven. Upon solving the inverse kinematics, both JawRot and JawTrans angles can be computed, as shown in Eqs.1-4; JawRot is the angle describing the rotation of the jaw; JawTrans is the angle associated to the translation of the jaw. Force/torque sensors were embedded both in the four-bar and in the crank

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mechanism.

Figure 2. Operation of the mechanism of the mandible Pulley

Wire

Compression spring

Fixed end

Motor and torque sensor compression spring block

F 㱂

wire

a)

b)

Figure 3. Tongue mechanisms: a) linear guide mechanisms; b) under-actuated mechanism. dt L1sin JawTrans L12 sin 2 JawTrans L12 L2 2 1

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2

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(4)

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2.3 Tongue We focused on two motions of the tongue, which are typical in airway management tasks. During unconsciousness, all the muscles are relaxed; as a result, the tongue falls down into the pharynx, thereby blocking the airway. This medical condition is named tongue swallowing. To treat this medical condition, the tongue needs to be lifted up in order to observe the vocal cords and insert a tube into the vocal cords to provide oxygen to the lungs. With reference to the tongue, we

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simulate the translational motion and the shape change (Turkstra et al., 2007). This is achieved via a linear guide mechanism and an under-actuated mechanism. The linear guide mechanism, which consists of a linear guide, tensile spring and wires, reproduces the translational motion of the tongue (Figure 3). The under-actuated mechanism consists of blocks, wire, and compression spring. The foregoing blocks are disposed along an array. This array is in charge of changing the shape of tongue. The compression spring is responsible for restoring the original shape of the tongue upon applying tension via the wire. Each wire is connected to the Wire Drive Unit, thus allowing for position and force control.

3

Force and Position Control of the WKA-3

The muscular system is distributed over all the bones. When arms, legs and neck are moved, all the muscles associated with these elements undergo contraction or relaxation. In order to simulate the muscles, we implemented a virtual compliance control for the WKA-3. This control schemes adjusts the stiffness of the end-effector by modifying the virtual spring coefficient (K) and the virtual damper coefficient (C). The virtual compliance control is applied on each of the joints such as the neck, mandible, and tongue. A block diagram of the control system of the WKA-3 is shown in Fig. 4. Fext (t ) Human Operator (Medical doctor)

K, C

Force and Torque Sensors

Virtual compliance calculator

xint (t )

xcc (t ) + +

xtar (t )

Inverse Kinematics calculator

Controller

Actuators

Figure 4. Block diagram of the control system of the WKA-3 The first step consists in selecting the patient's medical condition xint(t), such as restricted mouth opening, restricted cervical range, and shapes of the tongue. When this selection is made the WKA-3 reproduces the desired medical condition. As soon as an external force Fext(t) is applied on the mandible, the neck and the tongue, the WKA-3 computes the virtual compliance xcc(t) by adjusting Virtual Spring Compliance Coefficient K and Virtual Damper Coefficient C according to Eqs. 5-7. The controller compensates the virtual compliance so as to respect the selected medical condition. This attempts to simulate the stiffness of the actual muscles.

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Experimental results

Based on medical literature, we can define the distance between the minimum and maximum tongue-pharynx distance. Tongue swallowing condition feature a trivial distance 0 mm. Distance being different from zero and smaller than 20 mm is associated to a certain degree of the consciousness (Turkstra et al., 2007). Tongue swallowing condition can be reproduced using the WKA-3 as shown in Fig. 5a. Protrusion is the medical condition where the upper jaw protrudes from the mandible; the maximum value of the protrusion is 18.7 mm. Retrusion defines, instead, the mandible retruding from the upper jaw; the maximum value of the retrusion is 7.4 mm. The protrusion (retrusion) can be reproduced using the WKA-3 from a distance of 0 mm up to a distance of 22mm (-15mm), as shown in Fig. 5b. On the other hand, several medical conditions, such as restricted cervical range, restricted mouth opening and responses to specific external forces can be reproduced by adjusting the virtual compliance coefficients, such as spring and damper characteristics. Restricted mouth opening takes place when the distance between the incisor teeth of the upper jaw and the mandible is less than 30-40 mm (American Society of Anesthesiologists, 2003). By setting the compliance coefficients to high (low), WKA-3 can reproduce normal (restricted) mouth opening, as it is shown in Figure 5c. WKA-3 can reproduce normal (restricted) cervical range (Greenland et al., 2008), by setting the compliance coefficients to high (low), as it is shown in Fig. 5d. Finally, the WKA-3 can also simulate the stiffness of the tongue by adjusting the compliance coefficients, as depicted in Fig. 5e. Finally, in order to verify the WKA-3 system, the system was tested by medical doctors (MDs) and a survey was conducted aimed at gathering the feedback from the MDs. Eight MDs participated to the experiments. For each of the medical conditions reproduced by the WKA-3, MDs touched and operated the airway management system in order to make an assessment. The results are summarized in Fig. 6. According to the MDs, the reproduction of protrusion, retrusion, and restricted cervical range is acceptable. However, the tongue swallowing and the restricted mouth opening need major revisions.

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0 mm

20 mm

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b) Abnormal

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Figure 5. Airway-related medical conditions reproduced using the WKA-3: a) normal tongue position (left) tongue swallowing (right); b) protrusion (left) retrusion (right); c) normal mouth opening (left) restricted mouth opening (right); d) normal cervical range (left) restricted cervical range (right); e) tongue response to external forces applied by a laryngoscope.

Tongue Restricted Restrict Protrusion Retrusion Swallowing Opening mouth

Restricted Cervical range

Figure 6. Results of medical doctor feedback on the WKA-3 system.

5 Conclusions & Future Work In this paper, we have presented the detailed design of the Waseda-KyotoKagaku Airway No. 3 (WKA-3) system. The improvements with respect to previous designs are the easiness of maintenance, the mechanism compliance, and the

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reduction of actuators. A set of experiments was conducted to assess the performance of the WKA-3.㩷The Virtual Compliance Control was tested for each of the following test cases: normal mouth opening restricted mouth opening restricted cervical range and tongue lift-up. Finally, a survey, aimed at verifying the medical performance, was conducted among eight medical doctors. As a future work, we will focus on additional experiments aimed at validating the Virtual Compliance Control. Moreover, the design mechanism to control the shape of the artificial tongue will be proposed.

Acknowledgment The authors would like to express gratitude to G-COE Global Robot Academia, Waseda University. Finally, we would like to thank the doctors from the Department of Anesthesiology at the Tokyo Women’s Medical University and the Otsuka Hospital for their valuable time.

Bibliography Satake, J.(1999), Is it possible to prevent medical accidents? Practice of activity in the nursing care safety measure committee. Starting at analysis of near-miss cases,in Japanese Journal of Nursing Administration, vol. 9(8), pp.599-602. Solis, J., Marcheshi, S., Frisoli, A., Avizzano, C.A., Bergamasco, M. (2007), Reactive Robots System: An active human/robot interaction for transferring skills from robot to unskilled persons, Advanced Robotics Journal, vol. 21(3/4), pp. 267-291. Noh, Y., Segawa, M., Shimomura, A., Ishii, H., Solis, J., Hatake, K., Takanishi, A. (2008), Development of the Evaluation System for the Airway Management Training System WKA-1R, in Proc. of Int. Conference on Biomedical Robotics and Biomechatronics, Tokyo, Japan, pp. 574-579. Noh, Y., Segawa, M., Shimomura, A., Ishii, H., Solis, J., Takanishi, A., Hatake, K. (2009), Development of the Airway Management Training System WKA-2 Designed to Reproduce Different Cases of Difficult Airway, in Proc. of Int. Conference on Robotics and Automation, Kobe, Japan, pp. 3833-3838. Ishihara T, Hasegawa S, AI M. (1975), Mandibular Movement and Articulators (in Japanese). Tokyo, Japan,. HYORON, p 103. Turkstra TP, Pelz DM, Shaikh AA, Craen RA. (2007), Cervical spine motion: a fluoroscopic comparison of Shikani Optical Stylet vs Macintosh laryngoscope, Can J Anaesth. Jun;54(6):441-7. Greenland KB. (2008), The sniffing and extension-extension position: the need to develop the clinical relevance, Anaesthesia; 63(9):1013-4. American Society of Anesthesiologists, Inc. Lippincott Williams & Wilkins, Inc. (2003), Management of the Difficult Airway Practice Guidelines for Management of the Difficult Airway- An Updated Report by the American Society of Anesthesiologists Task Force on Management of the Difficult Airway- Anesthesiology; 98(5):1269–77

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Mechanical Design of a Novel Biped Climbing and Walking Robot Giorgio Figliolini, Pierluigi Rea and Marco Conte DiMSAT, University of Cassino, Italy Abstract. The present paper deals with the mechanical design of a novel biped climbing and walking robot, which is provided of a 2 (3-RPS) leg mechanism with 6 d.o.f.s. and which makes use of suction-cups for climbing on flat and rigid vertical surfaces. The serial-parallel kinematic structure of each leg mechanism, along with the climbing and walking motions of the proposed biped robot, are analyzed.

1 Introduction In the last years, the mechatronic design of a low-cost biped robot was carried out in Cassino and several prototypes of the EP-WAR (Electro-Pneumatic-Walking-Robot) were designed, built and tested since 1995, as reported in (Figliolini and Ceccarelli, 1997; 1999; 2004). In particular, the last prototype, named EP-WAR3, was provided of a binary pneumatic actuation in order to be controlled in on/off environment by a PLC (Programmable-Logic-Controller) and good performances were obtained to walk along a straight line, turn right and left, climb and descend stairs. The equilibrium of the EP-WAR prototypes was obtained by means of two suction-cups, which were installed on the underside of each foot. Later, the attention was focused on the gait analysis and mechanical design of six-legged walking robots, as shown in (Figliolini and Rea, 2007; Figliolini et al., 2009). Currently, the mechatronic design of a novel biped climbing and walking robot is in progress at the University of Cassino. Several prototypes of walking and climbing robots were designed, built and tested around the world, but only some of these have inspired the current research project. In fact, the most significant and pertinent prototypes are described in (Nishi, 1992; Hirose and Kawabe, 1998; Bahr et al., 1996; Minor and Mukherjee, 2003; Balaguer et al., 2005). In particular, referring to (Nishi, 1992; Hirose and Kawabe, 1998), a biped robot capable of moving on wall surfaces with irregular shapes and the quadruped robot NINJA, were developed in Japan at the Miyazaki University and Tokyo Institute of Technology, respectively. Similarly, a quadruped climbing robot for the aircraft maintenance, named ROSTAM, was developed at Wichita State University (U.S.A.), before to design a new light-weight biped robot, as reported in (Bahr et al., 1996). Two biped designs for miniature climbing robots were proposed in (Minor

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and Mukherjee, 2003), as based on under-actuated kinematic structures. Finally, an interesting overview on the main features of non-conventional climbing robots mobility on complex 3D environments is reported in (Balaguer et al., 2005). Therefore, at moment, our attention has been focused on the overall mechanical design of a novel biped climbing and walking robot, which is provided of a 2 (3-RPS) leg mechanism with 6 d.o.f.s. and which makes use of suction-cups for climbing on flat and rigid vertical surfaces. The type and dimensional synthesis of the serial-parallel kinematic structure of each leg mechanism, along with the climbing and walking motion analysis, are presented in this paper by referring to (Kim and Tsai, 2003; Mattiazzo et al., 2005; Lukanin, 2005; Di Gregorio and Parenti-Castelli, 2006) and as first step of the current research project.

2 The 2 (3-RPS) Leg Mechanism The idea of the proposed 2 (3-RPS) leg mechanism comes from the design specifications, which are aimed to obtain wide workspace and good mobility, high rotation and stiffness, along with a good integration of the linear actuators in the overall kinematic structure. In particular, the foot is represented by the second moving platform of the serial-parallel kinematic structure, which can be rotated up to 90° with respect to the robot body, as required to climb vertical walls by starting from an horizontal walking. Referring to Fig.1, the proposed leg mechanism shows 6 d.o.f.s since composed by two modules of 3-RPS (Revolute-Prismatic-Spherical) parallel mechanisms, which are connected in series from the robot body to the foot. Each 3-RPS parallel mechanism shows 3 d.o.f.s of the moving platform with respect to the reference platform, i.e. one translation along the normal (Z-axis) to the reference platform and two rotations across the X and Y axes of the O X Y Z fixed frame. In particular, referring to the kinematic sketch of Fig. 1a, each 3-RPS parallel mechanism is composed by a moving platform, which is connected to the fixed platform through 3-RPS legs. z2 x2

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P Z

R

Z O O

X Y

a)

X

Y

b)

Figure 1. The proposed 2 (3-RPS) leg mechanism: a) kinematic sketch; b) 3D model.

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Each of these legs is composed by a prismatic pair P, which is connected to the moving platform through a spherical joint S and to the fixed platform through a revolute joint R. These R and S joints are installed on the fixed and moving platforms at 120° among them in the order to allow only the radial rotation of each RPS leg. Figure 1b shows a 3-D view of the proposed leg mechanism, where the foot is parallel to the platform of the robot body. Therefore, the proposed 2 (3-RPS) leg mechanism has been assembled in order to have the equilateral triangle among the three spherical joints S of the upper 3-RPS parallel mechanism in opposite position with respect to the antagonist leg mechanism. In particular, Figs. 2a and 2b show the leg mechanism configurations to perform the long and short steps, with step sizes p1 and p2, respectively. Figure 2c shows the starting configuration to climb a vertical wall, before to in stroke completely the left leg to perform the next climbing step, as shown in Fig. 2d.

a)

b)

1

2

p1

2

1

p1

p2

c)

p2

d) 2

2

1 1

Figure 2. Starting configurations of both 2 (3-RPS) leg mechanisms (left and right legs are indicated with 1 and 2, respectively): a) long step with step size p1; b) short step with step size p2; c) vertical wall climbing; d) first configuration on the vertical wall.

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3 The 3-RPS Parallel Mechanism: Kinematic Synthesis The kinematic synthesis of one-module of the proposed 2 (3-RPS) leg mechanism has been formulated by referring to the kinematic sketches of Fig. 3, where Figs. 3a and 3b show the top and front views of one 3-RPS parallel mechanism for the maximum clockwise and counter-clockwise rotations, M and M , respectively, of the moving platform (link AB). Figure 3a shows the case for which the linear actuator that is represented through the link CB, moves outstroke to perform the maximum clockwise rotation M of the link AB. Similarly, Fig. 3b shows the case for which the two linear actuators that are represented through the link DA, move outstroke to perform the maximum counter-clockwise rotation M of the link AB. The proposed algorithm has been formulated as function of the following design parameters, the side size L of the equilateral triangle made by the three R joints and the lenght l0 of the fully instroke RPS leg, while α is equal to 60°. The input data are L, l 0 and = 60°, while the main design specification is M = 45°. Thus, referring to Fig. 3a, one has d = L cos (α 2 ) , xB ' = d (1 − cos β M )

(1)

yB ' = d sin β M ,

and

d

d

α B

α

G1

L

A

R

G1

B

R

B yB B

(2)

A

L

A

βM

xB

A B yB

xB

gM

A

B L0

L0 l0

C

a)

C

D

b)

Figure 3. Kinematic sketches of one-module for two configurations: a) max clockwise rotation through M of AB; b) max counter-clockwise rotation through M of AB.

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203

which give xB ' 2 + ( l 0 + y B ' )

L0 =

2

s = L 0 − l0 ,

and

(3)

where s is the stroke of the linear actuator of link CB. Likewise, referring to Fig. 3b, one has xB '' = d (1 − cos γ M )

yB '' = l 0 − L 0 + d sin γ M ,

and

(4)

which give l0 =

(l

− y B '' ) + x 2B '' 2

0

(5)

.

Thus, substituting the Eqs. (4) in Eq. (5) and developing, it yields 2 L 0 sin γ M + 2 d 2 cos γ M + l 20 − L20 − 2 d 2 = 0

(6)

that for sin γ M =

2t 1+ t 2

and

cos γ M =

1− t 2 , 1+ t 2

(7)

gives the following second order algebraic equation

(L

2 0

− l 02 + 4 d 2 ) t 2 − 4 L 0 d t + L 02 −l 02 = 0 .

(8)

Both solutions of Eq. (8) are given by

γM

1,2

= 2 tan −1 t 1,2

for t 1,2 =

2 L0 d ±

2 l 02 L 02 − l 04 − L 04 + 4 d 2 l 02 L 02 − l 02 + 4 d 2

.

(9)

Thus, for L = 300 mm, l 0 = 545 mm, α = 60° and M = 45°, this algorithm gives : d = 259,81 mm, d0 = 183,71 mm, L0 = 732,67 mm, s = 187,67 mm and M = 48,58°. Two configurations of a prototype with pnuematic actuation is shown in Fig. 4.

a)

b)

Figure 4. Two configurations of a built prototype: a) fully ahead to perform a short step, as leg 1 of Fig.2b; b) fully ahead to perform a vertical wall climbing, as leg 2 of Figs.2c and 2d.

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4 The 3-RPS Parallel Mechanism: Workspace Analysis The workspace for the 3-RPS parallel mechanism has been carried out by using a numerical procedure for the Forward Kinematics (FK), as proposed in (Lukanin, 2005) and then extended in (Figliolini et al., 2009). In particular, the Newton-Kantorovich Method has been applied in order to obtain a numerical solution for the FK and workspace analysis of the 3-RPS parallel mechanism. Referring to Fig. 5, a set of geometrical constraints can be expressed as | A1 B1 | = | B1 C1 | = | C1 A1 | = 3 R or in the Cartesian form as ⎧ ( X − X ) 2 + ( Y − Y ) 2 + ( Z − Z ) 2 = 3R 2 A1 B1 A1 B1 A1 ⎪ B1 2 2 2 ⎪ 2 , ⎨( X C 1 − X B 1 ) + (YC 1 − YB 1 ) + ( Z C 1 − Z B 1 ) = 3R ⎪ 2 2 2 ⎪ ( X A 1 − X C 1 ) + (YA 1 − YC 1 ) + ( Z A 1 − Z C 1 ) = 3R 2 ⎩

(10)

where the Cartesian coordinates of the points A1, B1 and C1 are given by ⎧ X A1 = R − l 1 cos α ⎪ ⎨YA1 = 0 ⎪ ⎩ Z A1 = l 1 sin α

⎧ X B1 = −1 2 ( R − l 2 cos β ) ⎧ X C 1 = −1 2 ( R − l 3 cos γ ) ⎪ ⎪ , ⎪⎨YB1 = − 3 2 ( R − l 2 cos β ) ⎪⎨YC 1 = 3 2 ( R − l 3 cos γ ) (11) , . ⎪ ⎪ ⎪⎩ Z B1 = l 2 sin β ⎪⎩ Z C 1 = l 3 sin γ

Thus, substituting these Cartesian coordinates into Eqs. (10), a system of three non-linear equations is obtained in the unknown , and for the input parameters l1, l2 and l3, which can be expressed in compact form as f i (α , β , γ ) = 0

x1

i = 1, 2, 3.

for

z1

A1

B1 G1= O1

R

y1

l1

l2 Z α

β

O X

C1 l3

γ

R Y

Figure 5. K

the 3-RPS parallel mechanism.

(11)

Mechanical Design of a Novel Biped Climbing and Walking Robot

205

This system of three non-linear equations has been solved numerically by using the Newton-Kantorovich Method, which is based on the linearization of each equation through the Taylor Series. In particular, a system of three linear equations has been obtained and solved by applying the common Cramer’s determinant method, where the 3 x 3 matrix of the coefficients is given by the Jacobi matrix, which elements are obtained as partial derivatives of the functions f i for i = 1, 2 and 3 of the Eqs. (11) with respect to , and . This algorithm for the FK analysis of the 3-RPS parallel mechanism has been implemented in a MatLab code and the workspace of point G 1 of the moving platform has been obtained by varying the lengths l1, l2 and l3 within the range of values to perform the maximum stroke s of Eq. (3). In fact, this stroke can give the maximum rotation M = 45° (G1m moves up to GA1, GB1 or GC1) or the maximum translation s (G1m moves up to G1M) of the moving platform, when only one ( l1, l2 or l3) or all three ( l1, l2 and l3) linear actuators are moved at end outstroke, respectively, as shown in the axonometric projections of Fig. 6.

A1

B1

GC1

G1M

GB1

M A1

A1 C1

C1

GB1

GA1

B1

GA1 GC1

M

M B1 G1m

G1M

C1

A1 C 1

a)

c)

G1m

B1

b)

d)

Figure 6. Workspace of the 3-RPS parallel mechanism: a) X Z front view; b) Y Z lateral view; c) X Y top view; d) X Y Z axonometric view.

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Conclusions The mechanical design of a novel biped climbing and walking robot with a 2 (3-RPS) leg mechanism has been described. In particular, the type synthesis of the leg mechanism along with the analysis of the climbing and walking motions, the dimensional synthesis and the workspace analysis of the 3-RPS parallel mechanism, have been carried out. Moreover, a first prototype of the 2 (3-RPS) leg mechanism with pneumatic actuation has been built and shown in the paper. The dimensional synthesis and the workspace analysis of the whole leg mechanism is in progress to formulate a general algorithm for the required walking and climbing performances.

Bibliography Figliolini, G. and Ceccarelli, M. (1997). Mechanical design of an anthropomorphic electropneumatic walking robot. ROMANSY 11: robot design, dynamics, and control. Eds. A. Morecki, C. Rzmkowski, G. Bianchi, Springer, Vienna. 189−196. Figliolini, G. and Ceccarelli, M. (1999). Walking programming for an electropneumatic biped robot, Mechatronics, 9 (8). 941−964. Figliolini, G. and Ceccarelli, M. (2004). EP-WAR3 biped robot for climbing and descending stairs, Robotica, 22 (4). 405−417. Figliolini, G. and Rea, P. (2007). Mechanics and simulation of six-legged walking robots. Climbing & Walking Robots, Towards New Applications, Ed. H. Zhang, I-Tech Education and Publishing, Vienna. 1−22. Figliolini, G., Rea, P. and Conte, M. (2009). Mechanical design of a six-legged walking robot with Sarrus-2-(3-RPS) leg mechanism. Mobile Robotics: Solutions and Chal, World Scientific Company, Singapore. 487−494. lenges. Eds. Nishi, A., (1992). A biped walking robot capable of moving on a vertical wall, Mechatronics, 2 (6). 543−554. Hirose, S. and Kawabe, K. (1998). Ceiling walk of quadruped wall climbing robot NINJA-II. 1st Int. Symp. on Mobile, Climbing and Walking Robots, Brussels. 143−147. Bahr, B., Li, Y. and Najafi, M. (1996). Design and suction cup analysis of a wall climbing robot, Computers & Electrical Engineering, 22 (3). 193−209. Minor, M.A. and Mukherjee, R., (2003). Under-actuated kinematic structures for miniature climbing robots, ASME J. of Mechanical Design, 125 (2). 281−291. Balaguer, C., Gimenez, A. and Jardon, A. (2005). Climbing robots’ mobility for inspection and maintenance of 3D complex environments, Autonomous Robots, 18 (2). 157−169. Kim, H. S. and Tsai, L. W. (2003). Kinematic synthesis of a spatial 3-RPS parallel manipulator, ASME J. of Mechanical Design, 125 (2). 92−97. Mattiazzo, G., Pastorelli, S. and Sorli, M. (2005). Motion simulator with 3 d.o.f.s pneumatically actuated, Bath Workshop on Power Transmission & Motion Control. Eds. D.N. Johnston, C.R. Burrows and K. Edge, John Wiley & Sons, London. 395−406. Lukanin, V. (2005). Inverse kinematics, forward kinematics and working space determination of 3 d.o.f. parallel manipulator with S-P-R joint structure, Periodica Polytechnica, Ser. Mech. Eng., 49 (1). 39−61. Di Gregorio, R. and Parenti-Castelli, V. (2006). A new approach for the evaluation of kinematic and static performances of a family of 3-UPU translational manipulators. ROMANSY 16: robot design, dynamics, and control. Springer, Vienna. 47−54.

Chapter III

Humanoid Robots, Bio-Robotics, Multi-Robot, Multi-Agent Systems

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2.3

Experiments and Results

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Development of a Visual Interface for Sound Parameter Calibration of the Waseda Flutist Robot WF-4RIV Klaus Petersen 1 , Jorge Solis 1

2,3

and Atsuo Takanishi

2,3

Graduate School of Advanced Sci. and Eng., Waseda University, Tokyo, Japan 2 Faculty of Science and Engineering, Waseda University, Tokyo, Japan 2 Humanoid Robotics Institute, Waseda University, Tokyo, Japan E-mail: [email protected], [email protected], [email protected] Abstract The Waseda Flutist Robot WF-4RIV is a humanoid robot that is able to imitate a human ﬂute performance. In the recent years the mechanical construction of the robot has been improved, so that the robot’s playing capabilities have reached the level of an intermediate human instrument player. To make the robot able to play at this performance level, careful calibration of its musical parameters is necessary. Using the ﬂutist robot’s core control software, this procedure is very complicated to perform. In this paper we present the implementation of a visual control interface that allows also non-technical users to calibrate the sound settings. The newly developed control interface enables a musician to adjust certain parameters of the robot performance while the robot is playing. Experiments in which we verify the functionality of this sound calibration system are presented. We examine the visual processing system’s perception of the instrument movements of a human performer and analyze their eﬀect on the performance of the robot.

1

Introduction

Thanks to recent advances in computer science, electronics, sound processing and artiﬁcial intelligence, musical performance robot research has not been limited to only developing sound-making devices that automatically play musical instruments. In fact, several researchers have been interested in developing musical robots that are able to display human-like dexterity as well as an amount of intelligence required for playing musical instruments. From this point of view, the development of musical performance

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robots provides opportunity to study human physiology and behavior from diﬀerent approaches such as: Human-Robot Interaction (Solis et al., 2007); Human Motor Control (Solis et al., 2006b); Art/Entertainment (Weinberg and Driscoll, 2007) and Education (Solis et al., 2004). The authors have been developing an anthropomorphic ﬂutist robot to study the methodology of human motor control, to introduce novel ways of man-machine interaction and to propose interesting applications for humanoid robots (i.e. musical tutor, etc.). After initial stages of research the ﬂutist robot has been capable of basic playing techniques, the sound production quality (ﬁngering, breath control etc.) being similar to a ﬂutist beginner’s. After further developments, extended technical skills (Solis et al., 2006c), which are typically practiced by intermediate level ﬂutists (i.e. vibrato, etc.), have been successfully implemented. More recently, in (Solis et al., 2006a), the authors have presented the Waseda Flutist Robot No.4 Reﬁned III (WF-4RIII). In this paper we present results on the development of a visual tracking system that enhances the possibility to calibrate performance parameters of the robot. The purpose of this interface is to allow a human without special technical skills to inﬂuence sound parameters of the robot (e.g. vibrato amplitude) in an intuitive fashion. As most skilled musicians that might use the ﬂutist robot as a musical tool do not have the necessary technical skills to operate the core control software of the robot, improvement of the procedure of adjusting these settings is necessary to give more users the possibility to operate robot.

2

Waseda Flutist Robot No.4 Reﬁned IV

The Waseda Flutist Robot No.4 Reﬁned IV (WF-4RIV) has a total of 41DOFs (Figure 1a), which mechanically emulate the human organs involved in ﬂute playing (Solis et al., 2008). The robot has humanoid arms and ﬁngers that strike the keys of the ﬂute. The air pressure to produce the ﬂute sound is generated using a lung mechanism. The lung compressor has two (right and left) parts similar to the human lung shape. The air is transferred into the oral cavity of the robot through a tube system. There is an artiﬁcial vocal cord that is used to regulate the strength of the air stream coming from the lung. During a performance, periodic (opening / closing) movement of the vocal cord applies a vibrato eﬀect the sound of the ﬂute play. The oral cavity contains a tonguing mechanism to make more precise note-to-note transitions possible. When the air stream leaves the oral cavity it is shaped using an artiﬁcial lips mechanism that generates the characteristic beam to produce the ﬂute

Development of a Visual Interface for Sound Calibration…

235

Figure 1. The Waseda Flutist No.4 Reﬁned IV (WF-4RIV) and its newly developed lips, tonguing and oral cavity mechanisms.

sound. In recent research the design of the lips and the tonguing mechanism (2-DOFs) has been improved: Due to the use of a new material with very realistic shape and elasticity properties for the lips, a more natural sound can be generated. The improved tonguing mechanism allows accurate control of note attack times and enhances the double tonguing (extended technical skill) capabilities of the robot. In addition, the implementation of an auditory feedback control system to enable the robot to improve its own performance has been proposed. Building upon this achievement, we focused our latest research on improving the capabilities to control the robot’s sound calibration parameters more comfortably. Up to now the procedure of setting these parameters could only be achieved by an expert in controlling the robot, as understanding of the usage of the robot’s control-software was necessary. Using the visual control system, a musician without technical skills can adjust the robot’s performance parameters with instrument gestures. In this case the movements of the musician’s interface are translated into values that adjust

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K. Petersen, J. Solis and A. Takanishi

the robot’s sound setting. In the experimental section we show how a musician can control the vibrato amplitude of the robot’s ﬂute performance. The vibrato is generated by movement of the artiﬁcial vocal chord of the robot. By moving the instrument the user of the robot can eﬀectively control the amplitude of the opening-and-closing motion of the vocal chord.

3

Instrument Position Mapping

In the ﬁrst stages of our research, we have concentrated on creating a visual interface, that enables a human to control the robot through gestures with his musical instrument. Our emphasis here is to create an interface, that allows to control the sound parameters of the robot accurately, but also robustly and computationally eﬃciently (at a later stage we would like among other things to perform audio processing on the same hardware). In order to satisfy those requirements, we propose to implement color histogram matching (Saxe and Foulds, 1996) and particle tracking (Arulampalam et al., 2002) to follow the movement of a musical instrument, while satisfying the previously introduced requirements. We model the shape of the saxophone as a line. The hands of the player are located on two spots on that line. The average of the position of the hands is recorded as the center position of the saxophone. Similarly we deal with the orientation: We consider a line drawn from the center of one hand to the center of the other. The inclination of the line is the orientation of the instrument. ¿From the 2D coordinates of the four hand particles we calculate the relative position, inclination and rotational angle of the instrument. To compute the depth values of both hands we use a z-transformation: Δxp = abs(xpl − xpr ), z =

1 α Δxp

(1)

Δxp is the distance between the X-coordinate of the patch in the left camera image (xpl ) and the right camera image (xpr ). Accordingly z denominates the Z-coordinate of the patch. We use α as a constant to adjust the value of Δz for further calculations. We use a Cartesian coordinate system parallel to the view plane of the robot to calculate roll (φ), pitch (θ) and yaw (ρ) inclination of the instrument. Δy Δy Δx φ = arctan( ), θ = arctan( ), ρ = arctan( ) (2) Δx Δz Δz A particle ﬁlter is a Bayesian ﬁlter method that uses randomly generated samples to approximate an unknown state conﬁguration. Similar to a

Development of a Visual Interface for Sound Calibration…

237

Kalman ﬁlter it works in prediction-update cycles. It is considered to be superior to other ﬁltering schemes used in object tracking (Hess and Fern, 2007). A Bayesian ﬁlter represents the PDF (Probability Density Function) p(xk |z k−1 ) of state xk given observation z k−1 were k is the discretized time. Speciﬁcally for the particle ﬁlter this PDF is deﬁned through a set of Ns random measurements sik with weights πki . In this case, the current observation X k is given by Ns πki sik (3) Xk = i=1

and the PDF p(xk |z k−1 ) can be approximated as (Arulampalam et al., 2002) p(xk |z k−1 ) ≈

Ns

πki δ(xk − sik )

(4)

i=1

with

Ns

πki = 1

(5)

i=1

δ denotes the Dirac delta function. δ(xk − xik ) represents the deterministic relationship between the random samples and the actual state. Both of the above equations show, how we gather random samples, assign them with weights and construct an approximation of our actual state. The random samples (or particles) are our sensor input, that we use to predict a new location of the object of interest. We restrict the algorithm to a limited set of samples Ns to keep the computational eﬀort manageable / real-time.

4

Experiments and Results

We performed two types of experiments: In each of the experiments the vibrato amplitude of the ﬂute player is manipulated by changing the saxophone orientation. We considered that the absolute positioning of the saxophone relative to the robot does not have to be evaluated: the orientation of the saxophone is calculated from the absolute position values of the hands. Thus, is the absolute position of the instrument calculated as the mean of the hand positions. The saxophone can be rotated by the player, in terminology changing pitch, roll and yaw. We did not assign any function to the yaw, as normally a saxophone player will not move the instrument in this way. Modes of individual experiments as follows: 1. Variation of the roll changes the vibrato amplitude of one tone, that the robot is playing.

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Figure 2. a) The recorded roll inclination data of the musical instrument. In the experiment, a musician performed a continuous regular up-and-down motion. b) The volume plot of the tone played by the ﬂutist robot. Representative areas for strong Vibrato and Vibrato with low amplitude are indicated. We notice a signiﬁcant volume drop (at 6s), when the artiﬁcial lung of the robot is deﬂated and needs to be reﬁlled with air.

2. Changing the pitch (orientation) manipulates the vibrato amplitude of the tone, that the robot is playing. As described in the previous section, we have requirements for our method to fulﬁll certain standards in recognition quality and ease of use. In our experiments we tried to give rather a qualitative than quantitative examination of the characteristics of our method. As performance environments may strongly vary, we tested our system with diﬀerent backgrounds and varying lighting conditions. Fig. 2 and Fig. 3 show that the visual input to our system does closely relate to its acoustic output. In the graphs areas of high and low vibrato amplitudes are marked. These areas match with the controller data peaks and troughs, respectively.

Development of a Visual Interface for Sound Calibration…

239

Figure 3. a) The recorded pitch inclination data of the musical instrument. Again, the musician performed a continuous regular up-and-down motion. The recorded information is notably more noisy than in the case of the previous experiment (Figure 2). b) Again, the volume plot is shown, with indicators for strong and weak vibrato. The breathing point occurs at 2.5s.

5

Conclusion and Future Works

In this paper, we have displayed our current research on the development of the Waseda Flutist Robot No.4 Reﬁned IV (WF-4RIV). Besides presenting signiﬁcant improvements of the actual sound production mechanisms, we have shown our preliminary results in enhancing the calibration system of the WF-4RIV. Performance sound parameters can be manipulated visually in real-time by instrument gestures of the musician partner of the robot. Through this a part of the calibration procedure necessary for the operation of the robot can be facilitated by non-technical users. We plan to integrate the pressure sensors built into the lip mechanism of the robot with the overall motor control system. The goal is to dynamically readjust the ﬂute position during a performance in order to provide consistent sound results. To achieve this we currently perform experiments to determine the relation between sensor output, ﬂute position and sound quality. In future works we will feed the sensor output to the robot’s motor control software to adjust the position parameters of the arms and hands.

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Bibliography M. Arulampalam, S. Maskell, N. Gordon, and T. Clapp. A tutorial on particle ﬁlters for online nonlinear / non-gaussianbayesian tracking. In IEEE Transactions on Signal Processing, pages 174–188, 2002. R. Hess and A. Fern. Improved video registration using non-distinctive local image features. In Proceedings IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 127–151, 2007. D. Saxe and R. Foulds. Toward robust skin identiﬁcation in video images. In 2nd International Conference on Automatic Face and Gesture Recognition, pages 379–384, 1996. J. Solis, S. Isoda, K. Chida, A. Takanishi, and K. Wakamatsu. An anthropomorphic ﬂutist robot for teaching ﬂute playing to beginning students. In Proceedings of the IEEE International Conference on Robotics and Automation, pages 146–150, 2004. J. Solis, K. Chida, K. Suefuji, and A. Takanishi. The development of the anthropomorphic ﬂutist robot at waseda university. International Journal of Humanoid Robots, 3:127–151, 2006a. J. Solis, K. Suefuji, K. Chida, K. Taniguchi, and A. Takanishi. The mechanical improvements of the anthropomorphic ﬂutist robot wf-4rii to increase the sound clarity and to enhance the interactivity with humans. In Proceedings of the Sixteenth CISM-IFToMM Symposium on Robot Design, Dynamics, and Control, pages 247–254, 2006b. J. Solis, K. Suefuji, K. Taniguchi, T. Ninomiya, M. Maeda, and A. Takanishi. Implementation of expressive performance rules on the wf-4riii by modeling a professional ﬂutist performance using nn. In Proceedings of the IEEE International Conference on Automation and Robotics, pages 312–317, 2006c. J. Solis, K. Suefuji, K. Taniguchi, T. Ninomiya, M. Maeda, and A. Takanishi. Implementation of expressive performance rules on the wf-4riii by modeling a professional ﬂutist performance using nn. In Proceedings of the IEEE International Conference on Robotics and Automation, pages 2552–2557, 2007. J. Solis, K. Taniguchi, T. Ninomiya, and A. Takanishi. Understanding the mechanisms of the human motor control by imitating ﬂute playing with the waseda ﬂutist robot wf-4riv. Mechanism and Machine Theory, 44(3): 527–540, 2008. G. Weinberg and S. Driscoll. The perceptual robotic percussionist: New developments in form, mechanics, perception and interaction design. In Proceedings of the ACM/IEEE International Conference on HumanRobot Interaction, pages 97–104, 2007.

Simulation of a Humanoid Robot walking gait on Moon Surface Aiman Musa M. Omer1, Hun-ok Lim2, and Atsuo Takanishi3 1

Graduate School of Science and Engineering, Waseda University, Tokyo, Japan Department of Mechanical Engineering, Kanagawa University, Yokohama, Japan 3 Department of Mechanical Engineering / Humanoid Robotics Institute, Waseda University, Tokyo, Japan 2

Abstract. Humanoid robots are developed to be used in many areas in the human environment, especially area where it is highly dangerous for human to work or live in. Outer space or even the moon surface is one of the areas where are people trying to explore, but it is highly risky and costly. Sending humanoid robot to the moon could one of the solution to explore the moon surface and research on the real effect of the moon gravity on human walking motion. Using dynamic simulation we can have an approach to imitate human walking gait and analyze the effect of the low gravity which could result on the walking velocity.

1. Introduction The explosion of the outer space has been a great interest for human begins since centuries ago. Physical exploration of space began in the mid of the 20th century by robotics spacecraft before human could successfully go to space in a Human spaceflight in late fifties. Successful exploration of the space had extended the human interests to exit the Earth orbit and reach fare planets. The challenge of exploring the moon and land on it surface has driven the research and development for the technology to reach the goal. The first robotics spacecraft to reach the surface of the moon with an impact was Luna 2 in 1959. Further attempts had been made to make a soft landing on the surface until it was successfully made in 1966 by Luna 9. Human spacecrafts were not developed to operate out of the Earth orbit until the 1968 when Apollo 8 carrying three astronauts flew around the moon and returned safely to earth. This had lead the first human walk on the moon by Neil Armstrong in 1969, a part of Apollo 11 mission which was the first manned spacecraft to the moon. The NASA`s Apollo program had made nine missions to the moon all occurred between 1968 and 1972. The high cost of sending human to the moon had stopped the any planned mission after 1972. Moreover, great risk of the mission could be also considered. After more 30 years from the last manned mission to the moon, many countries

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are now developing projects and research to send human beings back to the moon. Currently, EU, India, China, Japan, USA, and even Russia are under developing manned missions to the moon by 2020 to 2030. Recently, there are several robotics spacecrafts impacted on moon surface or flying around the moon orbit. The future planes are to construct bases on the moon surface for several purposes including scientific research. However, this will required many aspects concerning the payload work, safety, and energy. Therefore, the use of space robotics could reduce the load work on astronauts by providing assistance, and reduce cost for sending large number in a group or life support system. The Japan Aerospace Exploration Agency (JAXA) has proposed a plane to explore the moon using bipedal walking robots. The proposal is being considered as a part of Japan’s new space strategy. Sending bipedal robot to the moon to on its surface has never been achieved. Success in operating a bipedal walking robot could extend for further robotics mission in the moon. Japan advanced robotics technology could be used to develop and build a moon base and work with astronauts in many operates needed be conducted during moon exploration. In Japan many bipedal walking robots had been developed by different research groups and institutes. For examples, ASIMO a humanoid robotthat is developed by HONDA. The Japanese National Institute of Advanced Industrial Science and Technology (AIST) and Kawada Industries, Inc. have developed HRP-2P.These robots are highly advanced and can perform bipedal walking motion. In Waseda University a long research has been going in developing a bipedal walking robot. The latest achievement made is the development of WABIAN-2R (Waseda Bipedal Humanoid), which is capable of simulating human walking motion. The major difference that was achieved by WABIAN-2R is the ability to perform a human like walking motion with stretched knees where other bipedal robots walk with bend knees stable walking motion. The use of WABIAN-2R on the moon surface could help for the research and understanding the effect of moon gravity on human walking motion. The concern about walking on the moon is the low gravity. It may case a high velocity of movements which will make the walking motion unstable. Therefore, the astronauts whom had landed on the moon were in fact jumping or even running. In order to perform a walking gait motion, the velocity should decrease to have a stable walking. Simulating walking gait using WABIAN-2R, which can walking much similar to human than other robots, could be tested and checked through a dynamic simulation. The performance in the dynamic simulation could give an approach to word the real walking robot on the moon.

2. Walking Behavior on Moon Gravity The ability of walking on moon surface could face many difficulties due to the low

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gravity which is one sixth of the earth gravity. This will have a great impact in the dynamical behavior due reduction in the potential energy store in each step. Therefore, the amount of kinetic energy that is transform from potential energy will be decrease. Less amount of kinetic energy will result in low velocity for a walking gait. The increase of the mechanical energy will either be by increasing the vertical displacement which will result in switching to running gait instead of walking gait, or increase the mass of the body which is not probably accepted.

2.1. Human body modeling The human body could be modeled as a spring damper system. The leg could act as a spring that store potential energy and released it during walking, as shown in figure 1. A low stiffness elastic leg model is the form of running gait simulation where the higher stiffness will result in a walking gait. In the running gait the vertical displacement with respect to the step is slightly high compare to the walking gait. In the case of low gravity, the vertical displacement increases which could decrease the stability of movement. Therefore, high stiff leg could increase the motion stability in a walking gait.

Figure 1. Modeling of the leg in terms of spring system

The walking gait for locomotion could be simulated in terms of an inverse pendulum. The body is supported by the leg and rotating around the ankle joint, as shown in figure 2. In order to perform the walking gait properly the rotational velocity has to be quite enough to complete the step cycle. Moreover, the velocity must be too large otherwise the normal acceleration will be larger than the gravitational acceleration which is in the opposite direction. The period of the step cycle which is defined by the law of the pendulum: P = 2 (l / g) Where l is the leg length and g is the gravity. Since the moon gravity is lower than the earth gravity, the step cycle period will increase which means long time to complete one step on the moon surface than on earth.

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2l

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Figure 2. Human Walking gait as Inverse Pendulum system

2.2. Froude number Froude number is a dimensionless ratio of centripetal force to gravitational force. It could set as: Fr = v2/gl where v is the forward velocity of the body, g is the gravitational acceleration, and l is the length of the leg. Froude number is used to determine the walking velocity according to the gravity. Walking gait for locomotion is defined as a movement with keeping a least one foot connected to the ground. On the other hand, running gait could make by lifting both feet for ground at some point, it could for a long or a short period of time depending on leg length or velocity. The maximum walking velocity is approached when the double phase, both feet are touching the ground, is equally zero. This is classified as walk-run transition where it is transfer from walking gait to running gait. Scientific researches found that the Froude number for maximum velocity is closely to 0.5. The walking velocity could be a function of gravity for a walking style by setting the Froude number as a constant value. However, since the leg length is constant value as the Froude number, the velocity is the square root of the gravitational acceleration. v = (g l Fr) For compression between the velocity on earth and the velocity on the moon both could be divided as vEarth / vMoon = (gEarth l Fr) / (gMoon l Fr) by solving the above equation, the velocity on the moon will be vMoon = vEarth (gMoon / gEarth) since the moon gravity is one sixth of the earth gravity, the velocity on the moon surface with respect to velocity on earth is as follow

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vMoon 0.4 vEarth If the same walking gait performed on the earth and is needed to be performed on the moon, the velocity should be reduced to almost 40 per cent. Some researches and studies show that the maximum possible walking velocity for a 0.9 m leg long is close to 3 m/s. Therefore, the maximum velocity on the moon could be about 1.2 m/s.

3. Control Method of the Robot WABIAN-2R control system is trajectory positioning control base. It reads the trajectory planning for waist, foot, and hand from a pattern file previously generated. And through inverse kinematics method the controller calculate the each joint position.The walking pattern is generated base on ZMP (Zero Moment Point) method which is a general theory for bipedal walking robot. WABIAN-2R control loop is run every 30 ms. The low level control of the system control the actuator velocity by checking the difference between the current and target position with respect to the control loop timing. When the simulation of the walking on low gravity the forward velocity must be decreased in order to have a stable walking gait. The velocity of the robot on the moon surface will have to be reduced to about 40%. The waist and feet of the robot has a trajectory planned based on position and each phase has to be operated each 30 ms. However, since the positions will not change if the same pattern of walking gait is used on the moon, the control loop timing has to be increase in order to slow the motion of legs. Therefore, the step time will be increase to about 250% to have the reduction on the velocity. The controller of the robot run with 75 ms for one control step to perform the slow motion need on the moon.

4. Simulation Result of Walking Motion To simulation a real physical environment of the moon is highly difficult due to low gravity simulation on the earth. Therefore, computer simulation could give an easy approach for simulating robot walking in low gravity. In our research we use Webots which is a highly advanced and fast prototyping and Simulation used for mobile robots. We modeled the structure of our robot WABIAN-2R and the simulation environment of the moon. The simulation can show the behavior of the robot walking in such a low gravity. We run the simulation of the robot by setting the control step to 30 ms which is the same that is used for the real walking simulation. The robot slipped of the ground due to the high velocity of the leg motion and fell down. However, when the control step is increased to 75 ms, the walking motion was going smoothly and stable (see

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figure 3). The walking motion with high control step (75 ms or higher) could result on better stability of walking motion but it requires a longer time due to the slow motion and high time to complete one step. The walking pattern took 38.4 second to have a stable walking on the moon gravity where it takes 15.36 second to complete the same walking pattern when it is on earth gravity. The velocity data shows a little bite of disturbance due to the angular momentum which could increase in low gravity satiation (see figure 4).

Figure 3. Simulation of walking gait on Moon Surface

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Figure 4. Forward Velocity (X axis) Data

5. Conclusion & Future Work The walking of a bipedal humanoid robot is be tested and checked through dynamic simulation. For achieving high stability for walking gait on moon gravity, long time is needed to complete one step cycle. If higher velocity of forward movement on the moon surface is needed, transferring to running gait could be the only solution for stable motion. The future work in this research involves development of walking and running method on low gravity. Successful achievement in this filed will help to design a real prototype the can be send to the moon in future mission for moon exploration. References Alexander, R. M. (1996). Walking and running. The Mathematical Gazette, Vol. 80, No. 488, pp. 262-266. Breniere, Y. (2001). Simulation of gait and gait initiation associated with body oscillating behavior in the gravity environment on the Moon, Mars and Phobos. Biological Cybernetics, Vol. 84, No. 4, pp. 261-267. Donelan, JM., Kram, R. (1997). The effect of reduced gravity on the kinematics of human walking: a test of the dynamic similarity hypothesis for locomotion. Journal of Experimental Biology, Vol 200, Issue 24, pp. 3193-3201. Donelan, JM., Kram, R. (2000). Exploring dynamic similarity in human running using simulated reduced gravity. Journal of Experimental Biology, Vol 203, Issue 16, pp. 2405-2415.

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Kram, R., Domingo, A., Ferris, DP. (1997). Effect of reduced gravity on the preferred walk-run transition speed. Journal of Experimental Biology, Vol 200, Issue 4, pp. 821-826. Margaria, R. (1976). Biomechanics and energetics of muscular exercise, Oxford University Press. Oda, M. (2008). JAXA’s Space Robotics Road Map. In Proceedings of International Symposium on Artificial Intelligence, Robotics and Automation in Space. Hollywood, USA, February 26 - 29, 2008. Vaughan, C., O`Malley, M.(2005). Froude and the contribution of naval architecture to our understanding of bipedal locomotion. Gait & Posture, Vol. 21, Issue 3, pp. 350-362.

Development of New Biped Foot Mechanism Mimicking Human’s Foot Arch Structure Kenji Hashimoto*, Yuki Takezaki†, Kentaro Hattori†, Hideki Kondo†, Takamichi Takashima‡, Hun-ok Lim¶, § and Atsuo Takanishi*, § * Department of Modern Mechanical Engineering, Waseda University, Japan † Graduate School of Science and Engineering, Waseda University, Japan ‡ College, National Rehabilitation Center for Persons with Disabilities, Japan ¶ School of Engineering, Kanagawa University, Japan § Humanoid Robotics Institute, Waseda University, Japan E-mail: [email protected] Abstract. The humanoid robot, WABIAN-2R, has achieved human-like walking with heel contact and toe off motions with a foot mechanism with a passive toe joint. However, the foot structure is different from a human’s. In this paper, we describe a new foot mechanism mimicking the human’s foot arch structure to figure out its function. Especially, the developed foot mimics the elastic properties of the arch and the change of the arch height during walking. We conducted several walking experiments by using WABIAN-2R, and we confirmed that the arch elasticity could absorb a foot-landing impact force at the plantar contact phase and the change of the arch height contributed to a strong thrust at the push-off phase.

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Introduction

A human foot is anatomically composed of 26 bones excluding sesamoid bone. Considering that an adult human body has 206 bones, a quarter of the bones concentrate in the feet. A human foot has a complex structure, and few mammals, including humans, have a foot arch structure. There are several theories about the reasons for the existence of the human’s foot arch structure. The development of the medial longitudinal arch was a primary step in the evolution of bipedal human gait (Day and Napier, 1965; Morton, 1924). This terrestrial modification to the arboreal foot had two major effects. First, it provided the plantarflexors enough mechanical advantage to lift the weight of the body during the stance phase (Morton, 1924; Schultz, 1963). Second, it provided the foot with the capacity to absorb some of the increased shock caused by upright striding (Ker et al., 1987). The transformation of the grasping function of the medial rays of our primate ancestors into the plantigrade propulsive feet of modern human was clearly a seminal event in prehuman evolution (Harris, 1990). But few theories verified the

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significance of arch structure quantitatively. The reasons for the existence of the arch structure are still remain to be clarified. Meanwhile, we have developed a biped humanoid robot named WABIAN-2R (Waseda Bipedal humanoid - No.2 Refined) to simulate human motion (Ogura et al., 2006) (see Figure 1). WABIAN-2R has a foot with a passive toe joint and achieves a human-like walk with the knees stretched, heel-contact and toe-off motion. However, the foot of WABIAN-2R is different from the human’s on the point that the foot sole is flat and the foot breadth is wide. In this paper, we aim to clarify the function of human foot in gait. We develop a new biped foot mechanism mimicking a human’s foot arch structure and verify the significance of arch structure quantitatively through walking experiments by WABIAN-2R mounted on the new feet.

(a) (b) Figure 1. WABIAN-2R: (a) Overview; (b) DOF Configuration.

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Development of Human-like Foot Mechanism

2.1. Modeling of Human’s Foot Arch Structure The human foot has three arches; the medial longitudinal arch, the lateral longitudinal arch, and the transversal arch (Kapandji, 1988). The chief characteristic of the medial longitudinal arch is its elasticity, due to its height and to the number of small joints between its component parts. The lateral longitudinal arch allows only a limited movement. The most marked features of this arch are its solidity and its slight elevation compared with the medial longitudinal arch. The transversal arch is elastic to follow the unevenness on a ground and cannot keep the arch structure when a large load is applied during walking. The stiffness of the arch was identified at the following four phases in a gait cycle (see Figure 2); I. Heel contact phase II. Plantar contact phase III. Push-off phase IV. Swing phase

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In the phase I, the medial longitudinal arch is elastic because a load isn’t applied. In the phase II, it provided that the foot can absorb a foot-landing impact force (Ker et al., 1987). In the phase III, the toes are extended, and they pull the plantar pads and hence the aponeurosis forward around the heads of the metatarsals, like a cable being wound on to a windlass (Hicks, 1954) (see Figure 3 (a)). As a result, the arch is caused to rise because the distance between the metatarsal heads and the calcaneum is thereby shortened. The arch becomes the stiffest in a gait cycle. In the phase IV, the arch becomes elastic again because a foot is unloaded. The characteristics of the human’s foot are the changes of the elasticity of the medial longitudinal arch with the change of the arch height during walking. In this research, we approximate the arch structure by a rotational spring and damper as shown in Figure 3 (b), referring to the Takashima’s method (Takashima et al., 2002). The identified viscoelasticities of k Nm/rad and c Nm/rad2 show different values in a gait cycle as shown in Table 1. In this paper, we focus on only the medial longitudinal arch because the change of its elasticity is much larger than the lateral longitudinal arch and the transversal arch. The foot is classified into the following four parts; the medial toe, the lateral toe, the medial plantar and the ankle-lateral plantar (see Figure 4 (a) and (b)).

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Figure 3. Modeling of human’s foot arch structure: (a) Windlass mechanism (Hicks, 1954); (b) Foot model with a rotational spring-damper.

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We set the targets of arch viscoelasticity to the elasticity in the phase II and the phase III. We designed a new foot mechanism as shown in Figure 5 to mimic the change of the elasticity with the change of the arch height. It consists of a toe axis, an arch axis and a wire that mimics the plantar aponeurosis to realize the windlass mechanism. It has two soft materials to limit the arch movement, and it can realize the change of the arch elasticity during walking. Table 1. Viscoelasticity at each phase (Takashima et al., 2003).

I 19 17

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Figure 4. Foot skeleton model (left foot): (a) Parts; (b) 3-DOF model; (c) Dimension of human-like foot mechanism.

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2.2. Human-like Foot Mechanism Design Fugure 6 shows the human-like foot mechanism developed. Its weight is 1.1 kg equal to a human’s foot, and its size and the position of each joint axis are determined based on the converted size by the mean size of adult women’s feet (Kouchi and Mochimaru, 2005) using the WABIAN-2R’s height (see Figure 4 (c)). It has a medial toe axis, a lateral toe axis and an arch axis. A medial toe axis and an arch axis are connected by a wire which mimics the plantar aponeurosis, and we selected a stainless wire which has a high mechanical strength and a high stiffness. The toes are passive joint as the previous research (Ogura et al., 2006). The developed foot also mimics the change of the arch height in the phase II and the phase III. The developed foot mechanism mimics the rotational spring constant by attaching a soft material between the foot frame and the stopper. The soft material is selected through compression tests using a universal testing machine as shown in Figure 7. A compressive load is applied to the developed foot with the universal testing machine, and a rotational spring constant is calculated from the moment about the arch axis.

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Figure 6. Human-like foot mechanism (left foot): (a) Photograph; (b) DOF configuration.

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After conducting experiments repeatedly by changing kinds and thickness of soft material, we selected a polyurethane rubber whose Shore hardness is Shore A 30 with 10 mm thickness as the soft material in the phase II. On the other hand, the rotational spring constants had a constant value as 60 Nm/rad even if we changed kinds and thickness of soft material, and we could not find significant differences of rotational spring constants among soft materials. This is because a stainless wire mimicking the plantar aponeurosis is “a stranded wire” and it can be stretched at a low-load condition. The soft material for the phase III cannot be compressed enough due to the stretch of the stainless wire. To mimic the rotational spring constant k III , a wire which is hard to be stretched should be selected, or the foot mechanism itself should be reconsidered. However, the developed foot mimics the rotational spring constant at the phase II and the change of the arch height during walking. It is possible to verify the shock absorbing function at the phase II which is explained as a hypothesis about the reason for the existence of the arch structure in ergonomics field (Ker et al., 1987). It is also possible to verify the function of the windlass mechanism related to the change of the arch height. So we conducted some experiments described in the next chapter.

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Experimental Tests and Consideration

3.1. Verification of Walking with Human-like Foot Mechanism Because the breadth of the developed foot is as narrow as 90 mm, we must confirm whether a humanoid biped robot, WABIAN-2R can walk with the knees stretched, heel-contact and toe-off motion. The human-like foot mechanism was attached to WABIAN-2R, and forward walking experiments were conducted with a step length of 450 mm/step and a walking cycle of 1.0 s/step. Although we applied a polyurethane rubber with the hardness of Shore A 30 and the thickness of 10 mm to the soft material in the phase II, we tentatively applied a polyurethane rubber with the hardness of Shore A 70 and the thickness of 5 mm to the soft material in the phase III, and the medial longitudinal arch is transformed by the stainless wire. Then the rotational spring constant in the phase III is 60 Nm/rad. As shown in Figure 8 (a), WABIAN-2R mounted on the human-like foot mechanism realized a stable walk with the knees stretched, heel-contact and toe-off motion. 3.2. Consideration of Function of Foot Arch Structure We compared the vertical ground reaction forces when a polyurethane rubber (Shore A 30 and 10 mm thickness) is attached to the arch and when a fixation part is attached to the arch.

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Figure 8 (b) shows the ground reaction force in a gait cycle. At the beginning of the phase II, the foot-landing force with fixed-arch condition was larger than that with movable-arch condition. So, we confirmed that the foot can absorb a foot-landing impact force by the elasticity of the medial longitudinal arch. We can also find that the ground reaction force with movable-arch is larger than that with fixed-arch at the end of the phase II. It is considered that the windlass mechanism which causes the change of the arch height contributes to this strong thrust at the push-off phase. If the heel point is fixed in Figure 3, the toe position moves to the left with elevation of the arch height, and this movement contributes to the strong thrust at the push-off phase. I

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Figure 8. Walking experiment with human-like foot mechanism developed. The walking cycle is 1.0 s/step and the step length is 450 mm/step: (a) Sequential photographs; (b) Vertical ground reaction force in a gait cycle. GRF with movable-arch is smaller than GRF with fixed-arch at the beginning of the phase II. GRF with movable-arch is larger than that with fixed-arch at the end of the phase II.

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Conclusions and Future Work

To clarify the function of the foot arch structure, we developed a new human-like foot mechanism that mimics the changes of the elasticity of the medial longitudinal arch with the change of the arch height during walking. As experimental results using a biped humanoid robot, WABIAN-2R mounted on the new human-like foot mechanism, we confirmed that the arch elasticity could absorb a foot-landing impact force at the plantar contact phase and the windlass mechanism which caused change of the arch height contributed to a strong thrust at the push-off phase. At the present stage, the mimesis of the elasticity at the phase III has not realized. Therefore, we will improve the human-like foot mechanism and continue to research to clarify the function of the arch structure quantitatively.

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Acknowledgment This study was conducted as part of the Advanced Research Institute for Science and Engineering, Waseda University, and as part of the humanoid project at the Humanoid Robotics Institute, Waseda University. It was also supported in part by Global COE Program “Global Robot Academia” from the Ministry of Education, Culture, Sports, Science and Technology of Japan, Grant-in-Aid for Scientific Research (B) (18360126), SolidWorks Japan K.K., QNX Software Systems, and DYDEN Corporation, whom we thank for their financial and technical support.

Bibliography M. H. Day and J. R. Napier. Fossil foot bones. Current Anthropology, 6(4): 419-420, 1965. M. Harris. Our Kind: Who We Are, Where We Came From, Where We Are Going. Herper Perennial, 1990. J. H. Hicks. The mechanics of the foot: II. The plantar aponeurosis and the arch. Journal of Anatomy, 88:25-30, 1954. I. A. Kapandji. The Physiology of the joints: Lower limb, Volume 2. Churchill Livingstone, 1988. R. F. Ker, M. B. Bennett, S. R. Bibby, R. C. Kester, and R. McN. Alexander. The spring in the arch of the human foot. Nature, 325:147–149, 1987. M. Kouchi and M. Mochimaru. AIST Research Information Database (http://riodb.ibase. aist.go.jp/riohomee.html). H16PRO287, 2005. D.J. Morton. Evolution of the longitudinal arch of the human foot. Journal of Bone Joint Surgery, 6:56-90, 1924. Y. Ogura, K. Shimomura, H. Kondo, A. Morishima, T. Okubo, and S. Momoki. Human-like walking with knee stretched, heel-contact and toe-off motion by a humanoid robot. In Proceedings of the 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems, pages 3976-3981, 2006. A. H. Schultz. Relatives between the lengths of the main parts of the foot skeleton in primates. International Journal of Primatology, 1:150–171, 1963. T. Takashima, H. Fujimoto, and A. Takanishi. Determination of the longitudinal arch moment of the human foot complex during gait: inverse dynamics model analysis. Transactions of the Japan Society of Mechanical Engineers. C, 68(672): 2425-2430, 2002. (in Japanese) T. Takashima, H. Fujimoto, and A. Takanishi. Analysis of the human foot arch viscoelasticity using the simple model of the arch support elements. Transactions of the Japan Society of Mechanical Engineers. C, 69(685), 2003. (in Japanese)

Design and Development of Bio-mimetic Quadruped Robot Hiroyuki ISHII, Qing SHI, Yuichi MASUDA, Syunsuke MIYAGISHIMA, Syogo FUJINO, Atsuo TAKANISHI Faculty of Science and Engineering, Waseda University Satoshi OKABAYAHI, Naritoshi IIDA, Hiroshi KIMURA Faculty of Letters, Arts and Science, Waseda University Abstract Robot Technology should contribute to the progress of other research fields. We’d like to propose one example of it. In psychology, basic medicine and biological science, several experiments on animal behavior have been performed to clarify brain functions or mechanisms of learning. These experiments have been playing very important role to understand human mind. Rats and mice have been well used in these experiments. Therefore, we have been developing quadruped robots that reproduce animal behavior as new experimental tools. We developed a quadruped robot that has 4 legs and reproduces some social behaviors. Each leg has 3 DOFs. The robot also has 2 DOF at the neck and 2 DOF at the waist.

Introduction Robot Technology (RT) should contribute to the progress of other research fields. The application of RT to other fields is also very important for the progress of itself. Based on this idea, we’d like to propose an example. In psychology, brain science and basic medicine, several experiments on animal behavior have been performed to clarify brain functions or mechanisms of learning. These experiments have been playing very important role to understand human mind. In these experiments, rats and mice have been well used [1][2], and several experiments on social interactions have been performed to study mental disorders or social learning of human beings [3][4]. However, some researchers mention lack of reproducibility in these experiments. We consider it’s very difficult to improve reproducibility of these experiments because the social interactions of animals are not reproducible. On the other hand, some computer scientists and psychologists have been developing intelligence models of rodents, such as navigation or sensory-motor coordination [5]-[7]. Some researchers developed rodent-like robots to verify their intelligence model [8][9]. However, none of them has legs hence some researchers require new robots that have fore legs like a real rat. In robotics, several quadruped robots have been developed. One of the most

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famous quadruped robots is AIBO [10]. Big Dog and Little Dog are also well known [11][12]. We then considered that reproducibility of the experiments on social interactions can be improved by using a biomimetic quadruped robot as an agent that interacts with an animal subject. In addition, a quadruped robot can be used as a platform to verily the intelligence model developed by computer-scientist. We had developed two different quadruped robots. The first one had four 2-DOFs legs [13]. However, this robot doesn’t reproduce rat’s social behavior such as rearing, sniffing or mounting. The second one had four 2-DOFs legs, and a 1-DOF waist yaw joint [14]. However, this robot doesn’t reproduce rat’s social behavior either. Thus, we developed a new biomimetic quadruped robot WR-2 (Waseda Rat No. 2). WR-2 is original in respect of reproducing not only walking behavior but also other behaviors such as rearing, grooming and mounting of a rat. Because of this characteristic, WR-2 can be used as an agent that interacts with a rat in the experiments on social interactions and also as a robotic platform for computer-scientists. Before designing its hardware, we analyzed rat’s proportion and motion. We then determined DOF arrangement and proportion of WR-2. DOF arrangement of WR-2 is unique in respect of having 2 DOF at the waist. This robot also has 1-DOF at the neck. It also has a microcontroller, a wireless communication module and a battery. In this paper, we describe mechanical design, control system and specification of WR-2.

z x

y

Fig. 1 Real rat (left) and WR-2 (right)

Analysis of Rat’s Motion and Proportion To determine the specification and DOF configuration of the new quadruped robot, we analyzed rat’s motion and muscle skeleton. Size and Proportion. An x-ray picture of a mature rat and the proportion of each part are shown in figure 2. Comparing to other quadruped animals such as dogs or horses, the proportion of rat’s spine to its front and hind legs is longer. As shown in the figure, the lower hind leg is longer than the upper hind leg while the upper

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fore leg and the lower fore leg is almost same.

0.24

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Fig. 2 Proportion of a rat obtained from X-ray picture

t=0.7s

t=0s

Fig. 3 Stick diagram of walking rat, drawn in ever 0.03 [ms] DOF arrangement. The DOF arrangement model of a mature rat is experimentally obtained by a motion analysis. Walking motion of a mature rat is recorded by a video camera, and a stick diagram is obtained by observation as shown in figure 3. We analyzed the motion of both the fore and hind legs in not only x-z plane but also y-z plane. From the result of this analysis, we consider that each fore/hind legs can be represented as a 3-DOF manipulator, a roll and pitch at the shoulder/hip, a pitch at the elbow/knee. In addition, rats have a flexible spine, and they can bend their body when they rear or groom. We consider these kinds of motions can be reproduced by a 2-DOF waist joint, a pitch and a yaw. Through this kind of motion and kinematical analysis, we consider that the muscle skeleton structure of the mature rat can be represented by the DOF arrangement model as shown in figure 4.

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Fig. 4 DOF arrangement model of rats

Hardware Design Hardware of WR-2 was designed based on the model as shown in figure 4 and the studies in anatomy [15]. Dimensions and weight are shown in table 2. Table 2 Dimensions and weight of WR-2 Width mm 70 Height mm 90 Length mm 240 Weight g 850 Fore legs. Each fore leg consists of 3 DOF, a roll and pitch at the shoulder and a pitch at the elbow. The proportion between the upper and the lower leg is determined based on the model as shown in figure 2. As shown in figure 2, the legs of a rat are very short and small. Therefore, we consider that actuators should be implemented in the body, not in the legs. Each joint is actuated by a DC motor with a rotary encoder. Specifications of the motors are shown in table 1. Table 1 Specifications of the motors on WR-2 Encoder resolution Joint Power W deg/pulse Shoulder/Hip roll 2.5 8.8 x 10-2 Shoulder/Hip pitch 0.75 3.5 x 10-1 Elbow/Knee pitch 0.75 3.5 x 10-1

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As shown in figure 5, only the shoulder roll is directly driven by the DC motor. The shoulder pitch and the elbow pitch are driven by the DC motors via the wire and outer-tube connection mechanisms. The motors to drive these joint are implemented in the body. The problems of this kind of wire and outer-tube mechanisms are buckling of the tube and friction between the wire and the tube. To solve these problems, the outer-tubes are made from fluorine resin. Hind legs. Design of each hind leg is almost same as that of the fore leg except for position of the motor to drive the knee pitch joint as shown in figure 5. Waist. The waist consists of 2-DOF joint, a yaw and pitch. Each axis is actuated by a servo motor for hobby use.

Shoul _ P _ M

Hip _ P _ M

Elb _ M Shoul _ R

Knee _ M

Shoul _ P _ J Elb _ J x

Hip _ P _ J

Hip _ R

z y

(a) Fore leg (left)

Knee_ J x

(b) Hind leg (left) Fig. 5 Design of legs

Control of WR-2 A microcontroller, wireless communication module are implemented in WR-2. This robot is controlled by the PC via wireless connection. Controller circuit. A controller circuit that is originally designed for WR-2 is implemented. A microcontroller and 6 motor drivers (each drives two motors) are implemented in it. All the motors, not only the DC motors but also the servo motors are connected with this circuit and controlled. Some motion patterns are implemented in the microcontroller as time-line data of each joint angle. The microcontroller controls the angle of each joint according to the pattern when it receives the instruction from the PC. Walking pattern generator. Walking motion patterns are computed using the walking pattern generator as shown in figure 6 and recorded in the controller. This generator was developed for WR-2 based on the theory of ZMP stability criterion

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[16]. Kinematic model with mass distribution as shown in figure 6 (b) is implemented in this generator. It computes time-line data of each joint angle based on the procedure as shown in figure 6 (a). Start

Computing inertial force generated by motion of legs Computing body trajectory to make ZMP trajectory on target ZMP trajectory Computing motion of each leg using inverse kinematics Computing ZMP

Computing error moment eM

eM is acceptable?

(b) Kinematics model Motion pattern of each joint is obtained.

(a) Algorithm Fig. 6 Walking pattern generator for WR-2

Performance Verification We performed a performance verification test of WR-2. In this test, WR-2 walked 5 trials and we measured the walking speed. The result of the test, average of the speed and its standard deviation of 5 trials are shown in table 3 and the sequence photographs of a trial are shown in figure 7. Table 3 Result of performance verification test Average of 5 trials SD Walking speed mm/s 20.7 5.9

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Summary and Future works We developed a new quadruped robot WR-2. This robot was designed to reproduce not only walking but also other behaviors of a rat. Its DOF arrangement and proportion were determined based on the motion and proportion analysis of a mature rat. In verification test, we confirmed walking performance of WR-2. It walks like rat. However, the walking speed was slower than that of a real rat. We are now tuning some mechanical parameters such as tension of each wire and material of the feet. After tuning, we believe the walking speed of WR-2 become faster. We are also now, making motion pattern generator for social behaviors such as rearing, grooming and mounting. 1

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Fig. 7 Sequence photographs of walking of WR-2 References [1] DS. Woodruff-Pak, “Animal models of Alzheimer's disease: therapeutic implications,” Journal of Alzheimer’s disease, Vol. 15(4), pp. 507-521, 2008. [2] N. Hiroi, et al., “A 200-kb region of human chromo-some 22q11.2 confers antipsychotic-responsive be-havioral abnormalities in mice,” Proceedings of the National Academy of Science of U.S.A., 102(52), 19132-7, 2005. [3] M. Tonissaar, L. Herm, M. Eller, K. Koiv, A. Rinken, J. Harro, “Rats with high or low sociability are differently affected by chronic variable stress,” Neuroscience. Vol. 152(4), pp.867-76. 2008.

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[4] L. Jans, G. Korte-Bouws, S. Korte, A. Blokland. “The effects of acute tryptophan depletion on affective behaviour and cognition in Brown Norway and Sprague Dawley rats,” Journal of Psychopharmacology. 2008. [5] A. Barrera and A. Weitzenfeld, Biologically-inspired robot spatial cognition based on rat neurophysiological studies, Autonomous Robot Vol. 25(1-2), pp. 147-169, 2008. [6] M.D. Humphries, Is There an Integrative Center in the Vertebrate Brain-Stem? A Robotic Evaluation of a Model of the Reticular Formation Viewed as an Action Selection Device, Journal of Adaptive Behavior, Vol. 13(2), 2005. [7] V. Hafner, Cognitive Maps in Rats and Robots, Journal of Adaptive Behavior, Vol. 13(2), 2005. [8] JA. Meyer, et al., The Psikharpax project : towards building an artifical rat, Robotics and Autonomous Systems, Vol 50. 2005. [9] S. Elfwing, E. Uchibe, K. Doya, and H. I. Christensen. Evolutionary Development of Hierarchical Learning Structures. IEEE Transactions on Evolutionary Computations, Vol. 11, Issue 2, pages 249--264, 2007. [10] Sony AIBO Europe; http://support.sony-europe.com/aibo/ [11] R. Playter, M. Buehler, M. Raibert, “BigDog,” Proceedings of SPIE, Vol. 6230, 2006. [12] L.D. Jackel, et al., “How DARPA structures its robotics programs to improve locomotion and navigation,” Communications of the ACM, Volume 50 (11), 2007. [13] C. Laschi, B. Mazzolai, F. Patane, V. Mattoli, P. Dario, H. Ishii, M. Ogura, S, Kurisu, A. Komura, A. Takanishi, “Design and Development of a Legged Rat Robot for Studying Animal-Robot Interaction,” Proceedings of BioRob2006, 2006. [14] F. Patane, V. Mattoli, C. Laschi, B. Mazzolai, P. Dario, H. Ishii, S. Kurisu, A. Koumura, A. Takanishi, “Biomechatronic Design and Development of a Legged Rat Robot,” Proceedings of ROBIO 2007, 2007. [15] F.W. Warren, G. H. Dominique, “Anatomy and Dissection of the Rat,” W. H. Freeman, 1997. [16] H. Nishizawa, A. Takanishi., H. Lin, “Realization of Quadruped Walk with ZMP stable (in Japanese),” Proceedings of RSJ2003, 2003.

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Control, Modelling, and Analysis of Robotic Systems

Kinematic Calibration of Small Robotic Work Spaces Using Fringe Projection Dipl.-Ing. Klaus Haskamp

*‡

and Prof. Dr.-Ing. E. Reithmeier

†‡

*

Research Associate in the work group Production Measurement and Technology † Head of the Institute and head of the working group Control Engineering ‡ Institute of Measurement and Control Engineering, Leibinz Universität Hannover, Hannover, Germany Abstract In recent years a number of medical therapy concepts have taken hold in the ﬁeld of microsurgery. These concepts require measurement accuracies below . The positioning accuracy needed in surgical applications is higher than what surgeons usually are able to achieve. In this case robot manipulators can be employed to support surgical skills. The robotic movement has to be suﬃciently reliable and has to incorporate safety procedures like fast collision detection and avoidance. Furthermore, important premisses to the technical system were given by the absolute and the relative accuracy. In industrial applications the absolute accuracy is enhanced by calibrating the kinematic parameters and compensating manufacturing errors. The achieved accuracies are less than and do not comply with the actual medical standard. In this article a new method for modeling and calibrating the kinematics of robots with the aim of achieving precisions less than respectively in a work space is presented. The used mathematical description of the kinematics and the calibration strategy is explained in detail.

1

Introduction

Robots are applied in industrial applications as manipulator systems for tasks like pick and place, path-welding, bonding or milling. Medical engineering is a new application ﬁeld. For example a robot can be equipped with a milling cutter or a burr and be used in the precision engineering from hard tissue like bones or tooth enamel. Thereby, new problems that refer to positioning- and the motion-behavior have to be solved. For example, a very high absolute positioning accuracy and an excellent tracking accuracy

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have to be realized to remove only material, which is planned to carry oﬀ. Furthermore, the whole system has to be insensitive with respect to exterior disturbances, which are induced through the material removal. Recent research works deal with the calibration of industrial robots, which is described in Wiest (2001), Atkinson (1996), Everett (1988). The kinematics is usually described using the modiﬁed Denavit-Hartenberg- Parameters to consider geometric errors like home position errors. Thereby, absolute accuracies of less than were achieved. Medical applications have higher requirements relating to the absolute positioning accuracy and due to this the kinematic model have to be extended to include additional parameters. The problem is that the parameters are correlated and therewith cannot be identiﬁed seperately. In this work, a new calibration strategy for identifying the whole kinematics is presented. The kinematic is described using polynomial functions and due to this each geometrical and non-geometical error, like oﬀset angles and elasticities, are regarded. Additionall to the description of the kinematics the measurement procedure of the pose of the endeﬀector is explained in detail. The paper is concluded presenting a validation of the determined forward and inverse kinematic functions.

2 2.1

Methods for the Modeling of the Kinematics Conventional Modeling

Generally, the forward kinematics is described using the Denavit-Hartenberg-Notation or the modiﬁed Denavit-Hartenberg-Notation, as speciﬁed in Wiest (2001). denotes the pose of the robot endeﬀector (TCP) with respect to an inertial world coordinate system and represents the joint angles of the robot. The inverse kinematics can be obtained from the solution . Usually, this cannot be done analytically due to the fact that has a nonlinear character. Normally, nonlinear optimization algorithms like the NelderMead-Method or the Rosenbrock-Method have to be implemented in order to calculate the joint angles for a given pose , as written in Logt (1998). This modeling method implicates some advantages and disadvantages. The main beneﬁt is given by the fast setup of the forward kinematic functions through easy geometrical relationships. On the other hand, the description is not clear if consecutive axis are parallel. Furthermore, singularities appear if the robot is in an inappropriate angle conﬁguration. Another disadvantage is that mechanical components like elasticity and friction cannot be considered in this model so that an error remains between the virtual and the real kinematics.

Kinematic Calibration of Small Robotic Work Spaces… 2.2

277

Kinematic Modeling with respect to Polynomial Functions

In our application a 6-axis-robot as shown in ﬁgure 1 is moved in a small work space of about . Due to the assumption that the joint movements are small the behavior of the forward and the inverse kinematics is approximately linear and both can be described using polygons:

(1)

(2)

(3)

with and as the polynomial coeﬃcients. The polynomial functions present a black-box-method. Because of this, each physical eﬀect like deﬂection of the axis or home position errors are considered which is an advantage compared to the conventional model. However, the dimension of the working area restricts the applicability from describing the kinematics through polynomials. In the case that the work space is too large the kinematics and the polynomials become nonlinear and the functions and oscillate in the boundary area.

Figure 1. -KRoS-Robot

278

3 3.1

K. Haskamp and E. Reithmeier

Identiﬁcation of the Kinematics Calibration Functional

The mathematical model of the kinematics deviates from the real kinematic structure of the robot with the consequence that positioning errors appear when a pose is approached. The errors can be divided in pure geometrical errors and non-geometrical errors. Examples for geometrical errors are home position errors or deviations from the orthogonality of the axis. Gear elasticities, friction or temperature inﬂuence are examples for non-geometrical errors. In order to enhance the absolute accuracy the geometric and non-geometric eﬀects have to be taken into account within the model and to be identiﬁed through a calibration process. In Wiest (2001) null position errors for example can be considered in the Denavit Hartenberg Matrix as an oﬀset to the joint angle . In order to identify the model parameter, a functional has to be deﬁned which combines a set of measured poses with the modeled poses , as written in Wiest (2001):

(4)

with as the parameter vector. Usually, is expressed through the forward kinematics and the joint angles are given by the angle encoders. In this case we use the polynomial functions described in chapter 2.2. To estimate capable values for the parameters, the minimum of has to be determined using numerical methods. Wiest (2001) points out that the measured pose should have 6 dimensions to get a high information content. Using laser tracker, a 3D-position , as decan be captured with a very high accuracy of about scribed in Illemann (2007). By measuring diﬀerent targets, which are ﬁxed at the robot endeﬀector, a 6D-position measurement can be realized. Another opportunity for 6D-position measurement is given by fringe projection systems. This will be explained in detail in chapter 3.2. 3.2

Experimental Setup

Figure 2 shows the experimental setup. The measurement is accomplished by three spheres which are ﬁxed at the endeﬀector and positioned in the working area. As the result of the measurement using fringe projection, data points are ﬁtted to spheres and the center points , and are available for further analysis. A fringe projection system, as shown in ﬁgure 2, consists

Kinematic Calibration of Small Robotic Work Spaces…

279

of a beamer and one or more cameras. Common fringe projection systems use straight fringe patterns which are projected from the beamer onto an object, as speciﬁed in Valkenburg, Mc Ivor (1998). Afterwards the deformed patterns on the surface of the measurement object are recorded by the camera. The relationship between a camera pixel and the beamer phase is calculated using image processing and is used to reconstruct the object surface with triangulation. Furthermore, in ﬁgure 2 the principle of the construction of the endeﬀector coordinate system is illustrated. To start with the analysis the distances between the center points of the three spheres , and have to be calculated, whereas all lengths are diﬀerent. The center of the coordinate system is given by the center point . The axis is created from the diﬀerence vector of and : The -axis is # $

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(5)

Furthermore, the coordinate system is deﬁned in the coordinate system of the fringe projection system. To get the measured pose a basis

280

K. Haskamp and E. Reithmeier

coordinate system has to be determined:

(6)

This coordinate system is the new inertial coordinate system and is given by the joint angle conﬁguration , which is absolute with respect to the home position of the robot. Through a movement of the robot endeﬀector the coordinate system, represented by the spheres, is displaced and tilted with respect to the coordinate system of the fringe projection system and . This coordinate system is named :

(7)

The displacement and the tilting of with respect to result in the pose . Thereby, the diﬀerence vector

(8)

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(9) (10) (11)

whereas is the 3x3 rotation matrix deﬁned by the roll-pitch-yaw-angles (RPY).

4

Determination of the Kinematics

For the calculation of the polynomial coeﬃcients the relative positioning accuracy from the movement to the joint angle conﬁguration has to be estimated. Therefore, multiple measurements of the basis coordinate system have to be accomplished whereas the following standard deviations could be speciﬁed: Table 1. Standard deviation of the pose Position

8 8 11

Rotation

0.01 0.009 0.007

Kinematic Calibration of Small Robotic Work Spaces…

281

To determine the polynomial coeﬃcients and , 500 measurements were done to get a high resolution of the work space. Each measurement consists of the measurement of the joint angles through the angle encoders and the measurement of the spheres with the fringe projection system. After that the coordinate systems and the measured pose were calculated whereby corresponding pairs of values are available for the estimation of and . Finally, and were calculated using nonlinear optimization algorithms. For convenience the values of the calculated polynomial coeﬃcients are not shown.

5

Validation of the Determined Kinematic Functions

To specify the quality of the estimated kinematic polynoms the forward and the inverse kinematics have to be tested. To test the forward kinematics the robot was moved to a position and the pose was calculated with the forward kinematic functions and the joint angles from the angle encoders. Furthermore, the three spheres, which are ﬁxed at the robot endeﬀector, were measured through the method described in chapter 3.2 with the result . Comparing and shows that the deviation is quite small:

(12)

The values showed above are the maximum measured values from 20 difTable 2. Deviation between and

Position

87 40 56

Rotation

0.08 0.05 0.09

ferent poses in the new working room of the robot. This implies that the forward kinematic functions coincide with the real kinematics of the robot. To test the inverse kinematics the joint angles for a given pose were calculated and the robot was moved to the conﬁguration . After that, the fringe projection system was used to measure the pose of the endeﬀector . The comparison of and

shows that there is just a small diﬀerence:

(13)

282

K. Haskamp and E. Reithmeier Table 3. Deviation between and

Position

87 39 55

Rotation

0.06 0.08 0.06

The values shown above are the maximum measured values from 20 diﬀerent poses in the new working room of the robot. This implies that the inverse kinematic functions projects the real kinematics of the robot.

6

Conclusion

In this paper a new method for the identiﬁcation of the kinematics was described. Beside the description of the kinematic functions the measurement procedure was explained in detail. To verify the forward and inverse kinematic functions the deviation for given poses were determined and the maximum occured deviation is less than respectively .

Bibliography U. Wiest, Kinematische Kalibrierung von Industrierobotern, Dissertation Technische Universität Karlsruhe, 2001. G. van de Logt, Entwicklung von numerischen Verfahren zur Planung von Trajektorien fuer Roboter, Dissertation Universität des Saarlandes, 1998. L.J. Everett, T.W. Hsu, The Theory of Kinematic Parameter Identiﬁcation for Industrial Robots, Trans. ASME, Journal of Dynamic Systems, Measurement and Control 110, pp. 96-100, 1988. K.B. Atkinson, Close Range Photogrammetry and Machine Vision, Whittles Publishing, UK, 1996. J. Illemann, Aufbau, Funktionsweise Lasertracker und Produktübersicht, in: Schriftenreihe des Geodätischen Instituts, Heft 4. Dresdner Ingenieurgeodäsietag - Industriemesstechnik", Geodätisches Institut der Technischen Universität Dresden, 2007. R.J. Valkenburg, A.M. Mc Ivor, Accurare 3D Measurement using a structured light system, in: Image and Vision Computing, pp. 99 - 110, 1998.

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Shaking Forces Minimization of High-Speed Robots via an Optimal Motion Planning S. Briot1, V. Arakelian2, N. Sauvestre2 and J.-P. Le Baron2 1

Institut de Recherches en Communications et Cybernétique de Nantes (IRCCyN), Nantes, FRANCE [email protected]

2

Institut National des Sciences Appliquées (INSA), Rennes, FRANCE [email protected] [email protected] [email protected]

Abstract. This paper deals with the problem of shaking force balancing of high-speed robots based on a new optimal trajectory planning approach. The aim of the new approach is the optimal path planning of the robot links centre of masses, which allows a considerable reduction of the variable inertia forces transmitted to the robot frame. The efficiency of the suggested method is illustrated by a numerical simulation of a planar two links 2R serial robot, in which reductions in the shaking force of 63 % and in input torque of 84 % are achieved.

1

Introduction

A primary objective of linkage balancing is to cancel or reduce the variable dynamic loads transmitted to the frame and surrounding structures. Different approaches and solutions devoted to this problem have been developed and documented for one degree of freedom mechanisms (Lowen et al., 1983), (Arakelian et al., 2000), (Arakelian and Smith, 2005). A new field for their applications is the design of mechanical systems for fast manipulation, which is a topical problem in advanced robotics. The balancing of a mechanism is generally carried out by two steps: (i) the cancellation (or reduction) of the shaking force and (ii) the cancellation (or reduction) of the shaking moment. Traditionally, the cancellation of the shaking force transmitted to the robot frame can be achieved via adding counterweights in order to keep the total centre of mass of moving links stationary (Lowen et al., 1983), (Arakelian et

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al., 2000). However, this approach leads to the increase in the total mass of the mechanical systems and consequently the increase in input torques. With regard to the shaking moment balancing of robots, the following approaches were developed: (i) balancing by counter-rotations (Berkof, 1973), (Dresig et al., 1994), (Arakelian and Smith, 1999), (Herder and Gosselin, 2004), (ii) balancing by adding four-bar linkages (Gosselin et al., 2004), (Ricard and Gosselin, 2000), (iii) balancing by optimal trajectory planning (Papadopoulos and Abu-Abed, 1994), (Fattah and Agrawal, 2006), (Arakelian and Briot, 2008) and (iv) balancing by adding an inertia flywheel rotating with a prescribed angular velocity (Arakelian and Smith, 2008). It should be noted that the complete dynamic balancing can only be reached by a considerably complicated design of initial robot mechanisms and by unavoidable increase in the total mass. This is the raison why we focused our research studies on the development of robot balancing methods via optimal motion planning approaches, i.e. without modification of the initial mechanical structure and without any adding masses. The paper is organized as follows. In the next part, the suggested optimal motion planning is described. Then, for illustration of the efficiency of this approach, simulations carried out using ADAMS software for a planar two links 2R serial robot are presented. Finally, conclusions are drawn in the last section.

2

Minimization of the Shaking Forces via an Optimal Motion Planning of the Total Mass Centre of Moving Links

The shaking forces fsh of a robot can written in the form: f sh = mtot x S

(1)

where mtot is the total mass of the moving links of a robot and x S the acceleration of the total mass centre. The classical balancing approach consists in adding counterweights in order to keep the total mass centre of moving links stationary. In this case, x S = 0 for any configuration of the mechanical system. But, as a consequence, the total mass of the robot is considerably increased. Thus, in order to avoid this drawback, in the present study, a new approach is proposed, which consists of the optimal control of the total mass centre of moving links. Such an optimal motion planning allows the reduction of the total mass centre acceleration and, consequently, the reduction of the shaking force. Classically, robot displacements are defined considering either articular coordinates q or Cartesian variables x. Knowing the initial and final robot configurations at time t0 and tf, denoted as q0 = q(t0) and qf = q(tf), or x0 = x(t0) and xf = x(tf) , in the

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case of the control of the Cartesian variables, the classical displacement law may be written in the form:

or

q(t ) = sq (t ) (q f − q 0 ) + q 0

(2a)

x(t ) = s x (t ) (x f − x 0 ) + x 0

(2b)

where sq(t) and sx(t) may be polynomial (of orders 3, 5 and higher), sinusoidal, bang-bang, etc. laws (Khalil and Dombre, 2002). From expression (1), we can see that the shaking force, in terms of norm, is minimized if the norm x S of the masses centre acceleration is minimized along the trajectory. This means that if the displacement xS of the robot centre of masses is optimally controlled, the shaking force will be minimized. Let us consider a robot composed of n links. The mass of the link i is denoted as mi (i = 1, …, n) and the position of its centre of masses as xSi. Once the articular coordinates q or Cartesian variables x are known, the values of xSi may easily be obtained using the robot kinematics relationships. As a result, the position of the robot centre of masses, defined as xS =

1 mtot

n

∑m x i

Si

(3)

i =1

may be expressed as a function of x or q. It may also be shown that, if dim(xS) = dim(q) (= dim(x) in the case of non redundant robots), x and q may be expressed as a function of xS. We would like to mention that the case where dim(xS) < dim(q) will not be considered in this paper. It will be the topic of our further research works. Thus, taking into account that q = f(xS) (and as a consequence x = g(xS)), the problem remains to optimally define the trajectory xS(t). For this reason, let us consider the displacement xS of a point S in the Cartesian space. We assume that, at any moment during the displacement, the norm of the maximal admissible acceleration the point S can reach is constant and denoted as xSmax . Taking that into consideration, the displacement law that minimize the time interval (t0, tf) for going from position xS0 = xS(t0) to position xSf = xS(tf) is the “bang-bang” law (Khalil and Dombre, 2002), given by (Fig. 1) ⎧x S (t ) = s (t ) (x Sf − x S 0 ) + x S 0 ⎪ (4) ⎨x S (t ) = s(t ) (x Sf − x S 0 ) ⎪x (t ) = s(t ) (x − x ) Sf S0 ⎩ S with ⎧⎪ xSmax for t ≤ (t f − t 0 ) / 2 1 s(t ) = (5) ⎨ x Sf − x S 0 ⎪− xSmax for t ≥ (t f − t 0 ) / 2 ⎩

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Figure 1. “Bang-bang” displacement law.

Consequently, if the time interval (t0, tf) for the displacement between positions xS0 and xSf is fixed, the “bang-bang” law is the trajectory that minimizes the value of the maximal acceleration xSmax (Khalil and Dombre, 2002). Thus, in order to minimize x S for a displacement during the fixed time interval (t0, tf), the “bang-bang” law has to be applied on the displacement xS on the robot total mass centre.

3

Illustrative Example

For illustration of the efficiency of the suggested solution, let us consider a planar 2R serial robot (Fig. 2b) with following parameters: - lOA = 0.5 m, lAB = 0.3 m, where lOA and lAB are the lengths of segments OA and AB, respectively; - r1 = 0.289, where lOS1 = r1 lOA and r2 = 0.098, where lAS2 = r2 lAB, lOS1 and lAS2 being the lengths of segments OS1 and AS2, respectively. Its mass and inertia parameters are:

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m1 = 24.4 kg and m2 = 8.3 kg, where mi is the mass of element i (i = 1, 2); mtool = 5 kg, where mtool is the payload; I1 = 1.246 kg.m² and I2 = 0.057 kg.m², where Ii is the axial moment of inertia of element i. We would like to note that the mentioned parameters correspond to the IRCCyN’s robot parameters (Fig. 2), which will be used in future for experimental tests and validation of the force minimization approach developed in the present work. In order to have the possibility to control the robot, let us express the articulated joint positions q = [q1, q2]T as a function of the position xS of the robot centre of masses. From (3), we obtain: -

⎡ x ⎤ m r l ⎡cos q1 ⎤ m2 x S = ⎢ S ⎥ = 1 1 OA ⎢ ⎥+ mtot ⎣ sin q1 ⎦ mtot ⎣ yS ⎦

⎛ ⎡cos q1 ⎤ ⎡cos( q1 + q 2 )⎤ ⎞ ⎜ lOA ⎢ + r2 l AB ⎢ ⎥ ⎥ ⎟⎟ ⎜ sin sin( ) q q q + 1 1 2 ⎣ ⎦ ⎣ ⎦⎠ ⎝

(6)

This expression leads to: ( xS − leq1 cos q1 ) 2 + ( y S − leq1 sin q1 ) 2 − leq2 2 = 0

(7)

where leq1 = (m1r1 + m2 )lOA / mtot and leq 2 = m2 r2l AB / mtot . Replacing cos q1 and sin q1 by (1 − t12 ) /(1 + t12 ) and 2t1 /(1 + t12 ) (t1 = tan(q1/2)), respectively, and developing (7), we obtain: ⎛ − b ± b2 − c 2 + a2 q1 = 2 tan −1 ⎜ ⎜ c−a ⎝

⎞ ⎟ ⎟ ⎠

(8)

where a = −2 leq1 xS , b = −2 leq1 y S and c = xS2 + y S2 + leq2 1 − leq2 2 .

(9)

In expression (8), the sign ± stands for the two possible working modes of the robot (for simulations, the working mode with the “+” sign is used). Once q1 is known, q2 may easily be found from (6): ⎛ y S − leq1 sin q1 ⎞ ⎟−q . (10) q2 = tan −1 ⎜ ⎜ x S − leq1 cos q1 ⎟ 1 ⎝ ⎠ Now, let us consider a pick-and-place displacement between two positions of the end-effector x0 = [0.3 m, 0.7 m]T and xf = [0.3 m, 0.4 m]T, beginning at t0 = 0 s and finishing at tf = 0.25 s. Two cases will be simulated: 1. a fifth order polynomial law is applied on the displacement of the robot’s end-effector (Eq. (2b));

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(a)

(b)

Figure 2. The 2R serial robot: (a) prototype; (b) schematics.

Figure 3. Displacement of the end-effector for two examined cases: 1) fifth order polynomial law applied on the displacement of the end-effector (full line) 2) “bang-bang” law applied on the displacement of the robot centre of masses (dotted line).

Figure 4. Shaking forces for two examined cases: 1) fifth order polynomial law applied on the displacement of the end-effector (full line) 2) “bang-bang” law applied on the displacement of the robot centre of masses (dotted line).

Shaking Forces Minimization…

(a) actuator q1

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Figure 5. Input torques for two examined cases: 1) fifth order polynomial law applied on the displacement of the end-effector (full line) 2) “bang-bang” law applied on the displacement of the robot centre of masses (dotted line).

2.

a “bang-bang” law is applied on the displacement of the robot centre of masses (Eq. (5)); The resulting displacements of the end-effector are shown in Fig. 3. These trajectory parameters are implemented into ADAMS software and we obtain the variations of shaking forces (Fig. 4). The obtained results shown that the optimal trajectory planning (case 2) allows the reduction of the shaking forces up to 63 %. It should be noted that the optimal trajectory planning (case 2) has a good influence also on the input torque reduction (Fig.5). For example, in the case of the first actuator, the reduction in the input torque is 84 %.

4

Conclusions

In this paper, we have presented a new approach, based on an optimal trajectory planning, which allows the considerable reduction of the shaking forces transmitted to the robot frame. The aim of the suggested method is to optimally control the acceleration of the robot centre of masses using “bang-bang” displacement law. In other words, in the suggested approach, the robot is controlled not by applying end-effector trajectories but by planning the displacements of the total mass centre of moving links. The trajectory of the total mass centre of moving links is defined as straight line and it is parameterized with “bang-bang” displacement law. Such a control approach allows the reduction of the acceleration of the total mass centre of moving links and, consequently, the reduction in the shaking forces. It should be noted that such a solution is also very favourable for reduction of input torques. Numerical simulations carried out using ADAMS software for a planar 2R serial

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manipulator have shown that reductions in the shaking force of 63 % and in input torque of 84 % have been achieved. References Arakelian, V., and Briot, S. (2008). Dynamic balancing of the SCARA robot. In Proceedings of the 17th CISM-IFToMM Symposium on Robot Design, Dynamics, and Control (Romansy 2008), Tokyo, Japan, July 5-9. Arakelian, V, Dahan, M., and Smith, M.R. (2000). A historical review of the evolution of the theory on balancing of mechanisms. In Proceedings of the International symposium on history of machines and mechanisms. Dordrecht: Kluwer Academic Publishers. 291-300. Arakelian, V, and Smith, M.R. (1999). Complete shaking force and shaking moment balancing of linkages. Mechanism and Machine Theory. 34(8). 1141-1153. Arakelian, V, and Smith, M.R. (2005). Shaking force and shaking moment balancing of mechanisms: a historical review with new examples. Transactions of the ASME, Journal of Mechanical Design. 127. 334-339. Arakelian, V. and Smith, M.R. (2008). Design of planar 3-DOF 3-RRR reactionless parallel manipulators. Mechatronics, 18, 601-606. Berkof, R.S. (1973). Complete fore and moment balancing of inline four-bar linkages. Mechanism and Machine Theory. 8(3). 397-410. Dresig, H., Naake, S., and Rockausen L. (1994). Vollständiger und harmonischer Ausgleich ebener Mechanismen.VDI Verlag, Düsseldorf, 73p. Fattah, A., and Agrawal, S.K. (2006). On the design of reactionless 3-DOF planar parallel mechanisms. Mechanism and Machine Theory. 41(1). 70-82. Gosselin, C.M., Cote, G., and Wu, Y. (2004). Synthesis and design of reactionless tree-degree-of-freedom parallel mechanisms. IEEE Transactions on Robotics and Automation. 20(2). 191-199. Herder, J.L., and Gosselin, C.M. (2004). A counter-rotary counterweight for light-weight dynamic balancing. In Proceedings of ASME 2004 DETC/CIEC Conference, September 28 – October 2, Salt Lake City, Utah, USA. 659-667. Khalil, W., and Dombre, E. (2002). Modeling, identification and control of robots. Hermes Sciences Europe. Lowen, G.G., Tepper, F.R., and Berkof, R.S. (1983). Balancing of linkages – an Update. Mechanism and Machine Theory. 18(3). 213-230. Papadopoulos, E., and Abu-Abed, A. (1994). Design and motion planning for a zero-reaction manipulator. In Proceedings of the IEEE International Conference on Robotics and Automation, San Diego, CA. 1554-1559. Ricard, R., and Gosselin, C.M. (2000). On the design of reactionless parallel manipulators. In Proceedings of the ASME 2000 DETC/CIEC Conference, Baltimore, Maryland, September 10-13. 1-12.

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Economical Control of Robot Systems Using Potential Energy Werner Schiehlen* and Makoto Iwamura *

Institute of Engineering and Computational Mechanics, University of Stuttgart, Stuttgart, Germany Department of Mechanical Engineering, Fukuoka University, Fukuoka, Japan Abstract This paper considers the problem to minimize the energy consumption of horizontal robot systems utilizing passive storage elements. We propose a simultaneous optimization method for trajectory planning and power saving based on the optimal control theory. The eﬀectiveness of the design method is demonstrated for a 2DOF planar manipulator.

1

Introduction

The energy problem has captured the attention of people worldwide in recent years. In the manufacturing industry, a lot of robot systems controlled by actuators are utilized and these actuators are consuming a great amount of energy. Therefore, saving the energy of such systems is a very important problem. As a fundamental study toward the energy conservation for mechanical systems, Schiehlen and Guse (2001) analyze the power aspects of an actively controlled oscillator and point out that a large amount of energy can be saved by using the proper linear and nonlinear springs. Furthermore, Schiehlen and Guse (2005) discuss the energy conservation problem for mechanical systems with prescribed trajectories. They show that adapting the periodic trajectories as closely as possible to the limit cycle of the underlying mechanical system and combining with the inverse dynamics control can reduce the energy consumption dramatically. On the other hand, the problem to ﬁnd the minimum energy trajectories between two given points is also actively studied mainly for manipulators, see e.g. Izumi et al. (1997). However, in the area of trajectory planning, utilization of storage elements for energy conservation is not discussed yet. In this paper, we consider the simultaneous optimization problem of storage elements and trajectories with respect to the energy consumption for horizontal robot systems as proposed by Schiehlen and Iwamura (2009). Firstly, based on the linearized equations of motion, we analyze the relationship between the consumed energy and the operating time, and the optimal trajectory using optimal control

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theory. Then, we verify the analytical solution by comparing with the numerical one computed considering the full nonlinear dynamics. After that we derive a condition for the operating time to be optimal, and propose the optimal design method for springs. Finally, we show the eﬀectiveness of the method by applying it to a robot arm modeled as a 2DOF planar manipulator.

2

Optimal Control of Robot Systems

The equations of motion for a robot system in the horizontal plane with degrees of freedom and parameters read as ˙ = ( ) + ()¨ + ( )

(1)

where is the vector of generalized coordinates, is the inertia matrix, is the vector of centrifugal and Coriolis forces, is the vector of applied forces/torques, is the vector of parameters, and is the vector of actuator driving forces/torques. In this paper, we consider only the applied forces/torques by passive storage elements that can be expressed in the form ( ) = ( ) where is the stiﬀness matrix, is the vector of spring mounting positions. We deﬁne = [1 2 ] as the vector of independent spring stiﬀnesses that form . Then, the vector of design variables can be expressed as = [ ] . Here, we consider motions that move between two points and rest at both ˙ ˙ ) = 0. The energy consumed by this = ( ends, i.e., (0) = 0 ( ) = (0) motion can be estimated by the following cost function 1 (2)

= 2 0 where is a positive deﬁnite symmetric matrix. The problem considered in this paper can be formulated as follows: Find (), (), and = [ ] by minimizing the energy consumption (2) subject to the initial and ﬁnal conditions for the system (1).

3 3.1

Linear Analysis of Optimal Solution Linearization of the Equations of Motion

We analyze the optimization problem approximately using the linearized equations of motion based on the optimal control theory. Firstly, to make the later calculations easy, we shift the reference point to the middle point of the initial and ﬁnal points. To this end, we deﬁne and as = ( +0 )2 = ( 0 )2 ˜ = () , ˜ = . This transforms the and shift the coordinates as () ˜ ˜ ) = ˜˙ (0) = initial and ﬁnal conditions to the symmetric form (0) = ( ˜˙ ( ) = 0. And the equations of motion (1) can be rewritten as ˜ ˜ () ¨ ˜ ˜ + ( ˜ ˜˙ ) + ˜ = + ˜

(3)

Economical Control of Robot Systems Using Potential Energy

325

˜ ˜ ˜˙ ) 0. Here we assume that the centrifugal and Coriolis forces are negligible ( ˆ = ˜ (0) = ( ) = const. In addition, we approximate the inertia matrix as The validity of these approximations will be examined numerically in Section 3.3. These assumptions lead the following linearized equations ˆ ¨ ˜ + ˜ = + ˜

(4)

Let us consider the free vibration system corresponding to (4) and compute the ˆ Φ = and Φ Φ = Ω2 , where modal matrix Φ that satisﬁes Φ is the identity matrix, Ω = diag[1 2 ], is the -th natural frequency arranged as 1 2 where some of the lowest may vanish. We make the coordinate transformation ˜ = Φ substituted into (4). Premultiplying the equation by Φ yields

¨ + Ω2 = Φ + Ω2 (5) ˙ where = Φ1 ˜ . We deﬁne the state vectors = [1 2 ] as 1 = 2 = . Then, the state equations are written as ˙ 1 = 2

(6)

˙ 2 = Ω2 1 + Φ + Ω2

(7)

The initial and ﬁnal conditions are expressed as 1 (0) = 1 ( ) = 2 (0) = 2 ( ) = 0, where = Φ1 .

3.2

Analysis Using Optimal Control Theory

We introduce an adjoint vector = [1 2 ] and deﬁne the Hamiltonian as =

1 + 1 2 + 2 (Ω2 1 + Φ + Ω2 ) 2

(8)

Then, the optimal control can be derived from = 0 as = 1 Φ2

(9)

Substituting (9) into (8) yields = 1 2 2 Ω2 1

1 1 Φ Φ2 + 2 Ω2 2 2

(10)

From (10), the canonical equations of Hamilton are derived as follows.

=

0 Ω2

0

= + + = = 0 0 0 = = Ω2 0 Φ 1 Φ

˙ =

(11)

˙

(12)

(13)

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By solving these diﬀerential equations under the two-point boundary conditions, we can obtain the optimal solution that minimizes the energy consumption. However, it is diﬃcult to obtain the closed-form solution of these equations since they are coupled due to the term Φ 1 Φ in . On the other hand, selecting the ˆ 1 results in the cost function of mechanical power weighting matrix as = ˆ Φ = . Hence, in the following, and allows to decouple these equations by Φ we analyze this case. In this case, we can get the closed-form solution of 2 () as 2 () = [21 () 22 () 2 ()] 2 () =

(14)

22 sin (

) sin + sin

22 sin (

) + sin sin +

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=

1 2

0

2 2 =

23 (1 + cos ) 2 23 (1 cos ) 2 + (15) sin + sin =1 =1 2 (1cos )

In the above equation, it is easily conﬁrmed that +sin 0. Therefore, it is understood that the optimal spring mounting position that minimizes is always = 0, i.e., = . Hence, in the following, we analyze the problem by setting = 0. In this case, the relationship between the minimum value of the consumed energy and the operating time can be expressed as

( ) =

( )

(16)

=1

3 2 (1 + cos ) 2 sin

( ) = 24 2 3

( if = 0 )

(17)

( if = 0 )

where we use L’Hospital’s theorem to derive the equation for the singular point = 0 that is lim 0

2 (1+cos ) sin

=

24 .

Equations (16) and (17) show that

we can reasonably understand the consumed energy of robot systems with storage elements as the sum of energy corresponding to the mode.

3.3

Comparison with Simulation Results

The analytical solution obtained in the previous section is derived based on the linearized equations of motion (4). Therefore, we should check the inﬂuence of the nonlinearity ignored in the analysis. Hence, in this section we compare the analytical solution with the numerical one considering the full nonlinear dynamics (1) by a general purpose optimal trajectory planning algorithm for multibody

Economical Control of Robot Systems Using Potential Energy

327

Figure 1. 2-DOF manipulator.

Figure 2. Power of 2-DOF manipulator with spring.

systems (Iwamura et al., 2009). Here we consider the horizontal robot manipulator shown in Fig. 1. In this ﬁgure, is the angle, is the torque, is the spring stiﬀness, is length of the link, is the distance between the joint and the center of mass, is the mass, and is the moment of inertia about the center of mass. The parameters used for the simulation are 1 = 025m, 2 = 016m, 1 = 0125m, 2 = 008m, 1 = 1425kg, 2 = 100kg, 1 = 043kgm2 , 2 = 0244kgm2 .

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As an example, we consider the motion under the initial and ﬁnal conditions 0 = [30 30] deg and = [30 30] deg. Figure 2 shows the comparison between the analytical solution (16) and the numerical one of for the cases of = [1 2 ] = [30 0] [0 30] [30 30] . We can observe that both results are well coinciding everywhere. Since the analytical solution is well approximating the characteristics of the exact one, it is proposed that the analytical solution can be used for analysis and design of robot systems with potential energy.

4 4.1

Optimal Design Method Optimal Operating Time

We ﬁrst derive a condition for to be optimal. Let us consider the case that all are non-zero. If we consider as the free quantity that takes any value, the transversality condition ( ) = 0 should be satisﬁed. Since (10) does not contain explicitly, = 0, i.e., = const. is hold. Hence, the condition (0) = 0 should be satisﬁed. By substituting initial and ﬁnal conditions, = 0 and (0) into (10), we get (0) = 2 (0)Ω2

25 sin 2 1 2 (0)2 (0) = 2 ( sin )2 =1

(18)

From (18), it is understood that (0) = 0 is satisﬁed if sin = 0 ( = 1 2 ), or equivalently = ( = 1 2 ) where is an integer. By substituting this into (16), the cost function can be expressed as

=

23 (1 + cos ) =1

2

(19)

2 From (19), we can understand that takes the maximum 43 if is an even number and vanishes if is an odd number. Therefore, takes the global minimum if all are odd number resulting in the minimum value of zero.

4.2

Optimal Design of Springs

In this section, we consider the problem to design the spring parameters = [ ] to make the consumed energy minimum at the speciﬁed time . As for , it is already proven that = is always the best choice, hence in the following we explain the optimal design method for . We assume that the number of independent springs is . Firstly, from the optimal condition = , we can determine as = ( = 1 2 ). All should select to be odd number so that all takes the minimum. Moreover, should satisfy 1 2 since we assumed that 1 2 . It is noted that as becomes larger, the mode shape becomes more complicated. Therefore, in the practical sense, it is reasonable to choose as = 2 1 so that it becomes the smallest possible integer.

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The spring stiﬀnesses = [1 2 ] contained in the stiﬀness matrix ˆ]= should be determined as they satisfy the characteristic equation det[ 2 ˆ ] and the 0 ( = 1 2 ). Let us deﬁne the error function = det[ 2 error vector = [1 2 ] . Then, the problem here is to ﬁnd that satisﬁes () = 0. By solving this nonlinear equation, we can obtain the optimal spring stiﬀness . For more details see Schiehlen and Iwamura (2009).

4.3

Application Example

The optimal design method is demonstrated for the 2DOF manipulator shown in Fig. 1. As in the previous chapter, we consider the motion under the initial and ﬁnal conditions 0 = [30 30] deg and = [30 30] deg. Here we assume that the operating time is speciﬁed as = 1 s. Firstly, we choose as 1 = 1 2 = 3 , then the natural frequencies become 1 = , 2 = 3. We set the initial guess of the spring stiﬀnesses as (0) = [1 2 ] = [30 30] . Though the consumed energy corresponding to this spring stiﬀnesses was already shown in Fig. 2, we show it again on a diﬀerent scale with 1 and 2 in Fig. 3. Since the minimum of 1 and 2 do not coincide, is not vanishing. Hence, we optimize the spring stiﬀnesses. By solving () = 0, we obtain = [1 2 ] = [21854 14182] . The consumed energy corresponding to this spring stiﬀnesses is shown in Fig. 4. We can observe that the minimum of 1 and 2 coincides at = 1 s and therefore

also becomes zero at = 1 s. In Fig. 4, the minimal value of corresponding to the case without spring, i.e. a conventional robot manipulator, is also shown by a dotted line. We can conﬁrm that the consumed energy is strongly reduced at the design point = 1 s. This proves the eﬀectiveness of the proposed method.

5

Conclusion

In this paper, we considered the problem to minimize the energy consumption of robot systems in the horizontal plane utilizing storage elements and proposed a simultaneous optimization method for storage elements and trajectory. The eﬀectiveness of the design method was shown by applying it to a 2DOF manipulator.

Bibliography M. Iwamura, P. Eberhard, W. Schiehlen, and R. Seifried. A general purpose optimal trajectory planning algorithm for multibody systems. In Proceedings of the 28th IASTED International Conference on Modeling, Identiﬁcation, and Control, K.M., Hangos (ed.), pages 55–62, 2009. T. Izumi, P. Boyagoda, and T. Satoh. Optimal trajectory for minimizing dissipative energy of a horizontal articulated manipulator with energy regeneration and the eﬀect of operating time. Journal of the Robotics Society of Japan, 15(4):601–607, 1997. W. Schiehlen and N. Guse. Power demand of actively controlled multibody systems. In Proceedings of DETC, ASME paper DETC/VIB 21343, 2001.

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Figure 3. Power before optimization.

Figure 4. Power after optimization.

W. Schiehlen and N. Guse. Control of limit cycle oscillations. In Proceedings of the IUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes in Mechanics, G. Rega, and F. Vestroni (eds.), pages 429–439, 2005. W. Schiehlen and M. Iwamura. Minimum energy control of multibody systems utilizing storage elements. In Proceedings of DETC, ASME paper DETC2009 86327, 2009.

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Dept. of Mechanical Engineering, University of Bologna, Italy

Abstract This work presents a 3D kinematic model for the passive ﬂexion motion of the tibio-talar joint. With only knowledge of the articular surface shapes, the spatial trajectory of passive motion is obtained as the envelop of joint conﬁgurations that maximize the surfaces congruence. An increase in joint congruence corresponds to an improved capability of distributing an applied load, allowing the joint to attain better strength with less material. Thus, joint congruence maximization is a simple geometric way to capture the idea of joint energy minimization. The results obtained are compared with in vitro measured trajectories. Preliminary experimental data provide strong support for the predictions of the theoretical model.

1

Introduction

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349

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2 2.1

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3

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351

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4

Result and Discussion

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;64E@CJ 7@C E96 A2DD:G6 >@E:@? @7 E96 E9C66 DA64:>6?D +96 CDE A=@E @7 6249 :8FC6 D9@HD E96 AC@8C6DD:@? @7 E96 AC@?@ DFA:?2E:@? 2?5 :?EC2 6IEC2 C@E2E:@? 2?8=6D H9:=6 E96 D64@?5 D9@HD E96 AC@8C6DD:@? @7 E96 I J 2?5 K 5:DA=246>6?ED 2== A=@EE65 G6CDFD E96 5@CD: A=2?E2C 6I:@? 2?8=6 )6DF=ED 7@C E96 CDE 2?5 E96 D64@?5 DA64:>6?D D66>65 E@ 36 264E65 3J 2 4@?DE2?E 6CC@C :? E96 56?:E:@? @7 E96 2?2E@>:42= 7C2>6 @? E96 3@?6D +@ E6DE E9:D :562 E96 EC2?D7@C>2E:@? 3C:?8:?8 4@>AFE65 2?5 6IA6C:>6?E2= EC2;64E@C:6D E@ 4@:?4:56 7@C 2 AC6D4C:365 6I:@? 2?8=6 92D 366? 42=4F=2E65 2?5 E92? 2AA=:65 E@ E96 H9@=6 4@>AFE65 EC2;64E@CJ +96 D:>:=2C:EJ 36EH66? E96 EH@ EC2;64E@C:6D 27E6C E9:D @A6C2E:@? D66>D E@ G2=:52E6 E96 :562 E92E E96 >@56= AC65:4E:@?D 5:6C 7@C> 6IA6C:>6?E2= 52E2 @?=J 3J 2 4@?DE2?E C@E@EC2?D=2E:@? E9:D 36:?8 2D4C:323=6 E@ 2 4@?DE2?E 6CC@C :? E96 >62DFC6

M. Conconi and V. Parenti Castelli

x, y and z translations, mm

prono−supination, intra−extra rotation, degrees

352

comp. ps exp. ps comp. ie exp. ie

10 5 0 −5 −20

−15

−10

−5

0

5

10

15

20

25 comp. x exp. x comp. y exp. y comp. z exp. z

30 20 10 0 −10 −20

−15

−10

−5

0 5 10 dorsi−plantar flexion, degrees

15

20

25

prono−supination, intra−extra rotation, degrees

a)

15

10 25

x, y and z translations, mm

comp. ps exp. ps comp. ie exp. ie

20

−25

−20

−15

−10

−5

0

5

10

15 comp. x exp. x comp. y exp. y comp. z exp. z

20 15 10 5

−25

−20

−15

−10 −5 0 dorsi−plantar flexion, degrees

5

10

15

b) prono−supination, intra−extra rotation, degrees

30

comp. ps exp. ps comp. ie exp. ie

20 10 0 −10 −20

x, y and z translations, mm

−40

−30

−20

−10

0

10

20

30 comp. x exp. x comp. y exp. y comp. z exp. z

30 20 10 0 −10 −40

−30

−20

−10 0 10 dorsi−plantar flexion, degrees

20

30

c)

Figure 2. @>AFE65 4@>A 2?5 6IA6C:>6?E2= 6IA AC@?@ DFA:?2E:@? AD :?EC2 6IEC2 C@E2E:@? :6 2?5 I J 2?5 K 5:DA=246>6?ED G6CDFD 5@CD: A=2?E2C 6I:@? 7@C E96 2 CDE 3 D64@?5 2?5 4 E9:C5 DA64:>6? C6DA64E:G6=J

A Kinematic Model of the Tibio-Talar Joint…

353

Table 1. %62? 3D@=FE6 CC@C % 7@C AC@?@ DFA:?2E:@? (* :?EC2

6IEC2 C@E2E:@? ! / 0 2?5 1 5:DA=246>6?E PS (degrees)

IE (degrees)

X (mm)

Y (mm)

Z (mm)

leg

st leg

st leg

st

:8FC6D 2 2?5 3 D9@H E96 C6DF=ED 27E6C E9:D 4@CC64E:@? ?@E ?66565 7@C E96 E9:C5 DA64:>6? +23=6 D9@HD E96 >62? 23D@=FE6 6CC@C % 36EH66? 6249 4@FA=65 4@>A@?6?E @7 E96 4@>AFE65 2?5 >62DFC65 A2DD:G6 >@E:@? @E9 :?DA64E:@? @7 E96 A=@ED 2?5 ?F>6C:42= C6DF=ED D9@HD 2 8@@5 28C66>6?E 36EH66? AC65:4E65 2?5 >62DFC65 >@E:@?

5

Algorithm Sensitivity

+96 C6DF=ED @7 E96 2=8@C:E9> >2J 36 :?F6?465 3J E96 C2?5@> 4@>A@?6?E :? E96 86?6C2E:@? @7 ?6H A@D6D H:E9 86?6E:4 >6E9@5D 3J E96 E9C6D9@=5 56?:?8 E96 4@?EC@= G@=F>6 3J E96 5:DE2?46 >2A 56?D:EJ 2?5 @7 4@FCD6 3J E96 :?:E:2= 8F6DD +@ 2DD6DD E96 D6?D:E:G:EJ @7 E96 2=8@C:E9> E@ E96D6 A2C2>6E6CD E96 E9:C5 DA64:>6? 92D 366? 49@D6? 36:?8 E96 @?6 E92E D9@HD =6DD 5:6C6?46 36EH66? 4@>AFE65 2?5 >62DFC65 A2DD:G6 >@E:@? 6G6? H:E9@FE 4@CC64E:@?D AC6=:>:?2CJ 2?2=JD:D 92D D9@H? E92E 56DA:E6 E96 C2?5@> 4@>A@?6?E E96 2=8@C:E9> :D 9:89=J C6A62E23=6 %F=E:A=6 =2F?496D AC@5F465 H6== 8C@FA65 EC2;64E@C:6D +23=6 2 D9@HD E96 >62? DE2?52C5 56G:2E:@? %* 7@C 6249 4@>AFE65 4@>A@?6?E @7 E96 A2DD:G6 >@E:@? E@86E96C H:E9 E96 % C6DA64E E96 >62DFC65 @?6 +96 D>2== %* G2=F6 F?56C=:?6D E96 C6A62E23:=:EJ @7 E96 2=8@C:E9> E96 5:D4C6A2?4J 7C@> 6IA6C:>6?E2= >62DFC6 36:?8 4@?D:DE6?E H:E9 E96 52E2 @7 +23=6 +96 E9C6D9@=5 56?:?8 E96 4@?EC@= G@=F>6 92D 366? >2?F2==J 49@D6? 6BF2= E@ >> D@ E@ >2I:>:K6 E96 A@CE:@? @7 E96 2CE:4F=2C DFC7246 =2J:?8 :?E@ :E +96 2?2=JD:D @7 E96 2=8@C:E9> D6?D:E:G:EJ E@ E9:D A2C2>6E6C 92D D9@H? E92E E96 4@?8CF6?46 >62DFC6 72:=D E@ 5C:G6 E96 D:>F=2E:@? 7@C 2 G2=F6 @7 >> :E AC@5F46D ?@E:4623=6 5:D4C6A2?4J 7C@> E96 6IA6C:>6?E2= A2E9 7@C 2 E9C6D9@=5 @7 >> 3FE E96 %* 2?5 % =2J :? E96 C6A62E23:=:EJ C2?86 7C@> E@ >> +23 3

354

M. Conconi and V. Parenti Castelli

Table 2. -2C:2E:@? @7 E96 %62? *E2?52C5 6G:2E:@? %* 2?5 %62? 3D@

=FE6 CC@C % H:E9 5:6C6?E 2=8@C:E9> A2C2>6E6CD 7@C AC@?@ DFA:?2E:@? (* :?EC2 6IEC2 C@E2E:@? ! / 0 2?5 1 5:DA=246>6?E PS (degrees)

IE (degrees)

X (mm)

Y (mm)

Z (mm)

a) Multiple Launches (repeatability) MSD

MAE

b) Control Volume Threshold: 4.5÷11 mm MSD

MAE

c) Distance Map Smaller Cube Side: 0.5÷2.5 mm MSD

MAE

d) Initial Guess Variation: -5÷+4 mm MSD

MAE

+@ DEF5J E96 :?F6?46 @7 E96 5:DE2?46 >2A C6?6>6?E E96 G2=F6 @7 :ED D>2==6C 4F36 D:56 H2D :?4C6>6?E65 7C@> E@ >> D9@H:?8 ?@ D:8?:

42?E G2C:2E:@? 2>@?8 E96 4@>AFE65 EC2;64E@C:6D +23 4 :6C6?E :?:E:2= 8F6DD6D H6C6 4@?D:56C65 3J G2CJ:?8 @7 2 D2>6 2>@F?E 2== E96 4@@C5:?2E6 @7 E96 @C:8:?2= @?6 3FE E96 6I:@? 2?8=6 +23 5 !? E96 C2?86 − ÷ E96 EC2;64E@CJ G2C:2E:@?D 2C6 DE:== 2D4C:323=6 E@ E96 2=8@C:E9> C6A62E23:=:EJ H9:=6 :? E96 C2?86 ÷ E96J 364@>6 >@C6 D6?D:3=6 E@ E96 :?:E:2= 8F6DD 49@:46 6DA64:2==J 7@C E96 :?EC2 6IEC2 C@E2E:@? H9:49 :? D@>6 A@:?E 5:6CD 7C@> 6IA6C:>6?E2= 52E2 FA E@ 568C66D ?JH2J 27E6C 2 CDE 2DD6DD>6?E E96 EC2;64E@C:6D DE:== E6?5 E@ 4@?G6C86 E@ E96 6IA6C:>6?E2= @?6D !E :D H@CE9 ?@E:?8 E92E E96 2=8@C:E9> AC6D6?ED 2 ?@? :D@EC@A:4 3692G:@C

A Kinematic Model of the Tibio-Talar Joint…

355

(C65:4E:@?D @3E2:?65 7@C E96 5@CD: 6I:@? 5:6C 7C@> E9@D6 7@C E96 A=2?E2C 6I:@? 2?5 2C6 :? 86?6C2= =6DD AC64:D6 2?5 DE23=6 +9:D 42? 36 5F6 E@ E96 ?@? DJ>>6ECJ :?E@ E96 E2=@ 2CE:4F=2C DFC7246 H9:49 2E E96 368:??:?8 @7 E96 5@CD: 6I:@? @6C 2 >F49 D>2==6C 2C62 E9FD =6DD DF:E23=6 7@C E96 86?6C2E:@? @7 >@E:@? 8F:52?46 +9:D 2DA64E DFC=J 56D6CG6D 7FCE96C DEF5:6D

6

Conclusion

+96 4@>A2C:D@? 36EH66? 6IA6C:>6?E2= 2?5 4@>AFE65 EC2;64E@C:6D D66>D E@ G2=:52E6 E96 >@56= 7@C E96 E:3:@ E2=2C ;@:?E @? 2 C2?86 @7 >@C6 E92? 568C66D @7 A=2?E2C 6I:@? AC6=:>:?2C:=J 2?2=JD:D D9@HD E96 2=8@C:E9> E@ 36 C6A62E23=6 2?5 C@3FDE !E :D H@CE9 E@ F?56C=:?6 E96 AC65:4E:G6 ?2EFC6 @7 E96 >@56= :E 5@6D ?@E C6=J @? 2 >62DFC6 @7 E96 EC2;64E@CJ :E 2:>D E@ >2E49 =D@ F?=6DD G6CJ 86?6C2= E9:D 2AAC@249 >2<6 A@DD:3=6 E@ 86?6C2E6 DF3;64E DA64:4 >@56=D C6BF:C:?8 C6=2E:G6=J D:>A=6 2?2E@>:42= >62DFC6D :?2==J E96 >@56= AC@G:56D 2 E96@C6E:42= C6AC6D6?E2E:@? 7@C A2DD:G6 >@E:@? 2D 2 >:?:>F> 6?6C8J A2E9 2?5 :E D9@HD E96 664E:G6?6DD @7 2 D6BF6?E:2= 2AAC@249 7@C E96 >@56=:?8 @7 9F>2? 5:2CE9C@5:2= ;@:?ED (@DD:3=6 2AA=:42E:@? @7 DF49 2 >@56= >2J 36 7@F?5 :? AC6 DFC8:42= A=2?

?:?8 :? E96 DEF5J @7 E96 664E @7 2CE:4F=2C DFC7246 52>286D @? E96 @G6C2== ;@:?E 3692G:@C 2?5 :? E96 AC@DE96D:D 56D:8? FEFC6 H@C E@ 7FCE96C G2=:52E6 E9:D >@56= @? E96 2?<=6 H:E9 >@C6 6IA6C:>6?E2= 2?2=JD:D 2?5 E@ E6DE :ED 2AA=:423:=:EJ @? @E96C ;@:?ED DF49 E96
Bibliography $ =2?<6G@@CE " #F:A6C ) F:D<6D 2?5 C@@E6?3@6CE CE:4F=2C 4@?E24E :? 2 E9C66 5:>6?D:@?2= >@56= @7 E96 @C2= ;@:?E Journal of Biomechanical Engineering

) : C68@C:@ - (2C6?E: 2DE6==: "" '@??@C 2?5 $62C5:?: %2E96

>2E:42= >@56=D @7 A2DD:G6 >@E:@? 2E E96 9F>2? 2?<=6 ;@:?E 3J 6BF:G2=6?E DA2E:2= A2C2==6= >6492?:D>D Medical and Biological Engineering and Computing L

) C2?4: 2?5 - (2C6?E: 2DE6==: @?6 568C66 @7 7C665@> 7F==J A2C2==6= >6492?:D> 7@C E96 >@56=:?8 @7 A2DD:G6 >@E:@? 2E E96 9F>2? 2?<=6 ;@:?E !? DETC2007 G@=F>6 ()+ A286D L $2D -682D &- ,?:E65 DE2E6D

356

M. Conconi and V. Parenti Castelli

) C2?4: 2?5 - (2C6?E: 2DE6==: @?6 568C66 @7 7C665@> DA96C:42= HC:DE 7@C E96 >@56==:?8 @7 A2DD:G6 >@E:@? @7 E96 9F>2? 2?<=6 ;@:?E !? Proceedings of IAK 2008 A286D L $:>2 (6CF "2?F2CJ

) C2?4: & *2?4:D: 2?5 - (2C6?E: 2DE6==: E9C66 DE6A AC@465FC6 7@C E96 >@56==:?8 @7 9F>2? 5:2CE9C@5:2= ;@:?ED !? Proceedings of the RAAD 2008 A286D L ?4@?2 !E2=J *6AE6>36C

% C@DE

FA52E6 @7 3@?6 A9JD:@=@8J 2?5 H@=D =2H 7@C 4=:?:4:2?D Angle Orthodontist L

* C@@5 2?5 . " *F?E2J ;@:?E 4@@C5:?2E6 DJDE6?7 7@C E96 4=:?:42= 56

D4C:AE:@? @7 E9C66 5:>6?D:@?2= >@E:@?D AA=:42E:@? E@ E96 2E:4D @7 E96 9F>2? 2?<=6 4@>A=6I :? A2DD:G6 6I:@? 2 D:?8=6 568C66 @7 7C665@> DJDE6> Journal of Biomechanics L %2C2: Data-Driven Predictive Modeling of Diarthrodial Joints (9 E96D:D C@H? ,?:G6CD:EJ

%2C2: $2:5=2H 6>:C2=A * ?5C6HD % C:>> 2?5 "" C:D4@ DE:>2E:?8 ;@:?E 4@?E24E 2C62D 2?5 =:82>6?E =6?8E9D 7C@> 3@?6 <:?6>2E:4D 2?5 DFC7246D IEEE Transactions on Biomedical Engineering L

- (2C6?E: 2DE6==: 2?5 ) : C68@C:@ Parallel mechanisms applied to the human knee passive motion simulation A286D L #=FH6C 4256>:4 (F3=:D96CD (:C2?@ (@CE@C@K *=@G6?:2 "F?6

& *2?4:D: 2?5 - (2C6?E: 2DE6==: 5@7 A2C2==6= DA96C:42= HC:DE 7@C E96 >@56==:?8 @7 E96 @E:@? !? Proceedings of IFToMM 2007 A286D L (2A6C ?@ 6D2?4@? C2?46 "F?6

. *:>@? * C:656?D36C8 2?5 * ):492C5D@? "@:?E 4@?8CF6?46 2 4@CC6=2E:@? @7 ;@:?E 4@?8CF6?46 2?5 E9:2E:4 >@56= @7 E96 HC:DE 32D65 @? >2I:>:K2E:@? @7 ;@:?E 4@?E24E 2C62 Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine L

) .:=D@? 2?5 "" '@??@C E9C66 5:>6?D:@?2= 86@>6EC:4 >@56= @7 E96 A@

?6?ED @7 A2DD:G6 @G6>6?E 2C6 4@FA=65 E@ 6I:@? 2?8=6 Journal of Biomechanics L

" .@= The Law of Bone Remodelling EC2?D=2E65 :?E@ ?8=:D9 3J ( %2BF6E 2?5 ) FC=@?8

Chapter V

Innovations in Sensor Systems for Robots and Perception

Development of the Ultrasound Probe Holding Robot WTA-1RII and an Automated Scanning Algorithm based on Ultrasound Image Feedback )KG &3=363F71 "AD97 *A>;E2,3 FEGA +3=3@;E:;2,3 %AFA3=; *G93I3D34 #;KA?; &;=;5 3@6 ;;5:; %;@393I36 1

,/. "))& ) 0( #( ( (!#(,#(! - (#0,-#.3 )%3) *( 2 /&.3 ) #( ( (!#(,#(! - (#0,-#.3 )%3) *( 3 /'()# )).#- (-.#./. - (#0,-#.3 )%3) *( 4

*,.'(. ) #& (!#(,#(! #'$# )%%3) (#0,-#.3 #'$# *( 5

*,.'(. ) #)'#& (!#(,#(! )%3) #.3 (#0,-#.3 )%3) *( 6

&)% ) . )%3) *( '#& ,(%. .%(#-"#'"1-$* -)&#- #),! )(.. .%(#-"#'"1-$* Abstract. ( ."#- **, 1 *,-(. )/, ,-," )( ." 0&)* '(. ) ( /.)'. '#& /&.,-)/( -((#(! -3-.' ), ." ,).# ,.,3 /-#(! *,) -/**),.#(! '(#*/&.), *,&&& &#(% '"(#-' ( *,.#/&, 1 .#& ." '"(#-' -#!( ) ." '(#*/&.), ( ,&.#' #'! *,)--#(! &!),#."'- .) .. ." ,).# ,.,3 ( .#--/ &3, ) #.- 1&&- #( ." ') ) /& .,-)/( #'! 0,& -+/(.#& *..,(- ) /&.,-)/( *,) .,$.),3 1, #'*&'(. #( ),, .) (& ." '(#*/&.), .) -( ." -/, ) (% .#0&3 -. ) 2*,#'(.- "- ( ,,# )/. .) 0,# 3 ." .#0(-- ) ." *,)*)- -3-.'

1

Introduction

+:7 BD7H7@F;H7 ?76;5;@7 FA 67F75F 3DF7D;AE5>7DAE;E 3@6 ?KA53D6;3> 6KE8G@5

F;A@ ;E :;9:>K ;?BADF3@F 3E F:7K 3D7 F:7 ?3;@ 3@6 ;@5D73E;@9 53GE7E A8 673F: ;@ F:7 D757@F IAD>6 '@7 A8 F:7 @A@ ;@H3E;H7 ?7F:A6E FA 7H3>G3F7 F:7 53D6;A

H3E5G>3D 8G@5F;A@ ;E F:7 ?73EGD7?7@F A8 .3H7 !@F7@E;FK .! "A@7E 7F 3> +:;E :7?A6K@3?;5 ;@67J ;E 53>5G>3F76 4K ?73EGD;@9 F:7 4>AA6 BD7E

EGD7 3@6 H7>A5;FK 3@6 :3E 67?A@EFD3F76 FA 47 775F;H7 ;@ 67F75F;@9 53D6;A

H3E5G>3D 5:3@97E +:7 .! ;E ?73EGD76 3F F:7 5A??A@ 53DAF;6 3DF7DK GE;@9 3@ G>FD3EAG@6 6;39@AEF;5 EKEF7? 3D363 7F 3>

360

R. Nakadate et al.

+:7 BD35F;53> BDA4>7? ;@ F:7 .! ?73EGD7?7@F ;E D7>3F76 FA F:7 BAE;F;A@

;@9 A8 F:7 G>FD3EAG@6 BDA47 *A@A9D3B:7DE 3D7 D7CG;D76 FA :A>6 F:7 BDA47 3F F:7 B3F;7@F @75= 3@6 A4F3;@ 3 5>73D ;?397 A8 >A@9;FG6;@3> E75F;A@ A8 F:7 +:7 G>FD3EAG@6 BDA47 E:AG>6 47 7J35F>K B>3576 3F F:7 57@F7D A8 F:7 3DF7DK ;@ AD67D FA ?73EGD7 F:7 5:3@97 ;@ F:7 3DF7DK 6;3?7F7D 4K :73DF 473F !F ;E 3 6;5G>F F3E= 8AD EA@A9D3B:7D 4753GE7 F:7 3>>AI34>7 7DDAD A8 BDA47 BAE;F;A@ ;E ?? DA4AF 3EE;EF76 EKEF7? 8AD F:7 BDA47 BAE;F;A@;@9 ?3K EA>H7 F:;E BDA4>7? +:7D7 3D7 ?3@K B3EF D7E73D5:7E A@ H;EG3> 8776435= 5A@

FDA> A8 ?76;53> G>FD3EAG@6 ;?397 4A>?37EG?; 7F 3>

#DGB3 7F 3>

%AEF A8 F:7? 3D7 8AD FD35=;@9 F:7 ;?397 873FGD7 :AI7H7D F:7D7 3D7 87I D7E73D5:7E A@ 3GFA?3F76 E53@@;@9 3@6 67F75F;A@ A8 3@K AD93@ 4K ?73@E A8 F:7 DA4AF;5 EKEF7? !@ F:;E B3B7D I7 DEF>K BD7E7@F 3 67H7>AB?7@F A8 G>FD3EAG@6 BDA47 ?3

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2 Development of the Probe Holding Robot WTA-1RII +:7 67F3;>E A8 F:7 BDA47 :A>6;@9 DA4AF .+ )!! ;E E:AI@ ;@ ;9GD7 !F 5A@F3;@E ' B3D3>>7> >;@= ?3@;BG>3FAD I:;5: :A>6E F:7 G>FD3EAG@6 BDA47 ' E7D;3> >;@= B3EE;H7 3D? I:;5: :A>6E I:A>7 ?3@;BG>3FAD 3@6 3 ?3EF7D 5A@FDA>>7D +:7 E7D;3> >;@= B3EE;H7 3D? ;E GE76 FA >A53F7 F:7 ?3@;BG>3FAD 5>AE7 FA F:7 @75= >> B3EE;H7 >;@= 3D? ;@5>G67 ?39@7F;5 4D3=7E I:;5: 3D7 GE76 FA >A5= G@>A5= F:7 >76 4K 8AAF EI;F5: +:7 ?39@7F;5 4D3=7E G@>A5= F:7 K ;E FGD@76 A@ EA F:3F 3@K BAEE;4>7 D;E= A8 ;@;?;@3F76 78AD7 I7 67E;9@76 F:7 ?3@;BG>3FAD A8 F:7 .+ )!! F:7 E7D;3> 3@6 B3D3>>7> >;@= ?75:3@;E?E I7D7 5A?B3D76 )793D6;@9 F:7 E7D;3> >;@= ?75:3

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3

Automated Scanning Method

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Figure 2. >A5= 6;39D3? A8 F:7 BDABAE76 3GFA?3F76 E53@@;@9 EKEF7? 67

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Image processing

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Development of the Ultrasound Probe Holding Robot…

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4

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3

4

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Probe Path Planning

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364

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4 4.1

Experiments and Results Manipulator Positioning Accuracy

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Development of the Ultrasound Probe Holding Robot…

365

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Conclusion and Future Work

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Bibliography ( 4A>?37EG?; * *3>5G673@ . 0:G % ) *;DAGEBAGD 3@6 * ( ;%3;A !?397 9G;676 5A@FDA> A8 3 DA4AF 8AD ?76;53> G>FD3EAG@6 IEEE Transaction on Robotics and Automation M

3D363 + '=363 % *G93I3D3 3@6 # &;=; 7H7>AB?7@F A8 3 @A@

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386

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388

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Extraction of Semantic Information from the 3D Laser…

389

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390

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392

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The Application of Icp and Sift Algorithms…

393

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3

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394

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The Application of Icp and Sift Algorithms…

395

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Chapter VI

Recent Advances in Robotics

Modeling and Motion Planning for a Population of Mobile Robots Qirong Tang and Peter Eberhard Institute of Engineering and Computational Mechanics, University of Stuttgart, Stuttgart, Germany [tang, eberhard]@itm.uni-stuttgart.de Abstract This paper investigates modeling and motion planing for a population of mobile robots. A mechanical model is built for the robot motion planning which is inspired by particle swarm optimization and combined with multibody system dynamics. It uses the augmented Lagrangian multiplier method to treat the constraints and an independent module to handle obstacle avoidance. Simulations show that the robots moving in the environment display the desired behavior well, and so this model and related algorithms will next be transfered to a large scale mobile robotic system.

1

Introduction

Nowadays, controlling of a multi-robot or so called swarm robots is still a challenge in the robotics area despite of its fast development. Thus, several researchers worked on ﬁnding methods of modeling and motion planning for such swarm robotic systems. In recent years, the comparatively new stochastic particle swarm optimization (PSO) algorithms have been applied to many engineering areas, among them also the robotics area. (Doctor et al., 2004) discussed using PSO for multi-robot searching. Their focus was on optimizing the parameters and they did not consider the scalability of the standard PSO for a large numbers of robots. (Hereford and Siebold, 2008) developed a distributed PSO but with shortcomings of nonconsideration of obstacles or restriction to just static simple obstacles in the environment and the real robots could only rotate at a speciﬁed angle range. (Pugh et al., 2006) contributed a simple PSO version in multi-robot searching and they mainly focused on how to model the biological algorithm. In contrast to the mentioned publications this paper builds a model and does the motion planning for a population of mobile robots based on PSO combined with multibody systems. It is expected that after some adaption, such models and methods can be adjusted for real mobile robots.

410

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Q. Tang and P. Eberhard

Modeling for a Population of Mobile Robots

A population of robots is not simply a group of individuals. The behavior shown by a population of robots should be collective, coordinated and requires information exchange. So far, many algorithms and methods are used in this area, among them traditional and recursive biology inspired methods, see (Tang and Eberhard, 2009). The PSO algorithm is very appealing due to its clear ideas, simple iteration equations, and the possibility to be mapped onto several robots or even swarm robots. 2.1 Basic Particle Swarm Optimization The original model of PSO, see (Kennedy and Eberhart, 1995), uses the vectors $xx and x to denote the particle’s ‘velocity’ and actual position, respectively. The so called ‘velocity’ of the i-th particle at the k 1-th iteration can be described with the equation k k k k xbestk $xxk1 W $xxki c1 ri1 xx bestk swarm x i i isel f x i c2 ri2 x

(1)

and the position update is done in the traditional PSO algorithm by x k1 x ki $xxk1 i i

(2)

If consider in a general way and for all n particles, the basic PSO model can be foumulated by

k1 k x˙ k1 x x (3) k k bestk W x˙ k x˙ k1 c1 r k1 xxbestk x swarm x k sel f x c2 r 2 ˆ Detailed interpretation for this structure and related parameters please see (Tang and Eberhard, 2009). 2.2 Build Mechanical Model for Practical Use in a Population of Mobile Robots This study wants to interpret the PSO algorithm as providing the required forces in the view of multibody system dynamics rather than a mathematical optimization tool as usual since the application purpose is swarm mobile robots motion planning. Each particle (robot) is considered as one body in a multibody system which is inﬂuenced by forces and torques from other bodies in the system but without direct mechanical constraints between them. The forces are artiﬁcially created by corresponding drive controllers. From another point of view, based on the Newton-Euler equations, the general form of equation of motion for swarm

Modeling and Motion Planning for a Population of Mobile Robots

411

mobile robots can be formulated as M x¨ k q

or

x¨ M 1 qq k M 1 F

(4)

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(6)

Here I 3 is a 3 3 unit matrix, 0 3n is a 3n 3n zero matrix. The forces f k1 and f k2 are attraction forces from the last best position of the robot itself and the last swarm best robot position and are proportional to their distances. The vector f k3 represents the force which is proportional to the last velocity and is a kind of inertia which counteracts a change in direction. One can also write Eq. (4) as a state equation with the state vector y xx x˙ T , where x and x˙ are the translational and rotational position and velocity of the robots. Together with the initial conditions, ﬁrst order of differential equation, the motion of the swarm robots over time can be computed, e.g., by the simple Euler forward integration formula, which yields the mechanical PSO model

k1 xk x M 1 hkf3 x˙ k I 6n $tM x˙ k1 (7)

k x ˙ $t k M 1 h k xˆ bestk x k M 1 h kf1 x bestk swarm f2 sel f x Comparing Eq. (7) to Eq. (3), one can see that they are quite similar and the corresponding relationships are introduced in (8), in (Tang and Eberhard, 2009) we give a very sound and useful interpretation, please see the related reference for details. M 1 h kf1 c1 r k1 $tM M 1 h kf2 c2 r k2 I 6n $tM M 1 h kf3 W $tM

3

(8)

Design a Control Scheme for a ROBOTINO

3.1 Constraint Handling Engineering optimization problems usually have constraints, e.g., if the particles (robots) are searching in the environment with some limitations, like inter-

412

Q. Tang and P. Eberhard

ference districts or several obstacles. In this study, we also take into account the treatment of constraints. So, here the general optimization problem with an objective function and constraints is minimize 9xx (9) g xx 0 me equality constraints subject to h xx 0 mi inequality constraints where x is the position of the particle bounded additionally by x min x xmax . For such an optimization problem, the augmented Lagrangian multiplier method can be used where each constraint violation is penalized separately by using ﬁnite penalty factors r p . Thus, the minimization problem with constraints in Eq. (9) can be transformed into an unconstrained minimization problem minimize LA xx L r p

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subject to

1 E x1 xm1 2 x2 xm2 2 f xx

(11)

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If the center of the source x m is infeasible, then the robots should at least get as close as possible. In this example, we choose x m 3 10 and the constrained minimum is x opt x m with 9xxopt 0. We try to use our model and related algorithms in the autonomous robot ROBOTINO (see Festo Home Website, 2009) for future work, so here we extract some main components of ROBOTINO and present the control scheme for simulation and also later for practical use, see Figs. 1 and 2.

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4

Simulation and Results

Some assumptions for the simulation experiments must be made: 1. the static obstacles are represented by polygons, 2. all simulations treat only the planar case, and 3. during the avoidance of obstacles, the robots can rotate full 360 degrees. This study classiﬁes the simulation experiments as shown in Table 1. The symbol of a circle ‘ ’ is used for a volume robot, and a ‘*’ for the target. The objective function and constraints are as same as described in (11). The trajectories of ﬁve searching robots of experiment no. 1 are shown in Fig. 3, where obstacle avoidance and target searching can be seen clearly. Experiment no. 2 uses external forces to steer a robot in a speciﬁed direction, see Fig. 4. The red robot e.g. will turn left when it meets the ﬁrst obstacle, see Fig. 3, but with inﬂuence from an external guiding force which is in right direction it turns to the opposite side. Compared to experiment no. 1 and no. 2, in experiment no. 3 we use more robots and try to show the ability of mutual avoidance, see Fig. 5. In experiment no. 4, the target is within an obstacle, see Fig. 6. The initial and ﬁnal status are shown. As the algorithm and model are designed, the 50 robots get as close as they can to the target and siege it. In summary, all above four experiments are obtaining the correct results from our swarm model and the designed algorithms, strategies, and methods.

5

Conclusion and Future Work

The application of PSO to our swarm model requires some extensions for the use of this scheme for the motion planning and control of large scale mobile robots. The results show that the algorithm and model used in our experiment are simple,

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4 15

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Figure 6. Experiment four, (a) t=0 s, (b) t=350.48 s.

Acknowledgements The authors want to thank Prof. W. Schiehlen for his encouragement and detail discussion, as well as the China Scholarship Council for providing a Ph.D. fellowship to Qirong Tang to study in Germany. Parts of this research are done in the framework of SimTech, the Stuttgart Research Center for Simulation Technology. All this support is highly appreciated.

Bibliography S. Doctor, G. K. Venayagamoorthy, and V. G. Gudise. Optimal PSO for collective robotic search applications. In IEEE Congress on Evolutionary Computation, pages 1390–1395, 2004. Festo Home Website. http://www.festo-didactic.com, 2009. J. M. Hereford and M. Siebold. Multi-robot search using a physically-embedded particle swarm optimization. International Journal of Computational Intelligence Research, 4(2):197–209, 2008. J. Kennedy and R. C. Eberhart. Particle swarm optimization. In Proceedings of the international conference on neural networks, volume 4, pages 1942–1948, Perth, Australia, November 1995. J. Pugh, L. Segapelli, and A. Martinoli. Applying aspects of multi-robot search to particle swarm optimization. In Proceedings of the 5th International Workshop on Ant Colony Optimization and Swarm Intelligence, pages 506–507, Brussels, Belgium, 2006. K. Sedlaczek and P. Eberhard. Using augmented Lagrangian particle swarm optimization for constrained problems in engineering. Structural and Multidisciplinary Optimization, 32(4):277–286, 2006. Q. R. Tang and P. Eberhard. A PSO-based algorithm designed for a swarm of mobile robots. Structural and Multidisciplinary Optimization, 2009. (submitted).

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Multi-Objective Trajectory Planning in Wire-Actuated Parallel Manipulators Maryam Agahi and Leila Notash Department of Mechanical and Materials Engineering, Queen’s University, Kingston, ON, Canada In the work presented, the redundancy resolution of planar wire-actuated parallel manipulators is investigated at the torque level in order to perform desirable tasks while maintaining positive tension in each wire. A local optimization routine is used in order to minimize the norm of actuator torques, maintain positive tension in each wire, reduce the effect of impact and avoid collision while modifying the trajectory of the mobile platform. The effectiveness of the presented approach is studied through a simulation of an example planar wire-actuated manipulator.

1. Introduction In closed-loop manipulators, if the number of actuators is greater than the degrees of freedom of the manipulator, the manipulator is redundantly actuated and for a given end effector trajectory and external forces/moments, an infinite number of actuator torques/forces exists. Redundant manipulators can use their degree(s) of redundancy to satisfy additional desirable task(s), e.g., to reduce actuators forces/torques. In wire-actuated parallel manipulators, wires connect the mobile platform to the base and have the advantage of being light weight, reconfigurable and transportable, and allowing high speed motion and a larger workspace compared to conventional manipulators. Wires can only apply force in the form of tension. Therefore, to design a fully controllable wire-actuated parallel manipulator, the manipulator has to be redundantly actuated. Thus, to keep positive tension in all wires, at least n + 1 wires are required for a manipulator with n degrees of freedom (DOF) (Kawamura et al. 2000). When resolving redundancy at the torque level, the null term of actuator forces/torques is interpreted as portions of actuator forces/torques that result in zero forces/moments at the end effector. Because of the existence of infinite solutions for the actuator forces/torques, the proper use of the null term is of great importance in redundancy resolution. The kinematic, dynamic and stiffness analyses, as well as, the design of wireactuated manipulators have been investigated, e.g., (Kawamura et al. 2000, Williams and Gallina 2003, Oh and Agrawal 2005, Notash and Kamalzadeh 2007, McColl and Notash 2009, Agahi and Notash 2009). Oh and Agrawal (2005), and Agahi and Notash (2009) investigated how to design positive tension controllers for wire-actuated manipulators with redundant wires to follow prescribed

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trajectories. For kinematically redundant manipulators, obstacle avoidance approaches have been proposed, e.g., (Maciejewski and Klein 1985). In the presented work, redundancy resolution of wire-actuated parallel manipulators during collision is investigated. The collision is detected considering the minimum distance between the mobile platform and a specified obstacle. Some of the challenges associated with the redundancy resolution of wire-actuated parallel manipulators, including positive tension in each wire and infinite inverse dynamic solutions, are addressed. The effect of impact during collision is reduced and the error in the trajectory of the mobile platform after collision is minimized. The kinematic and dynamic modelling of an example planar wire-actuated parallel manipulator is presented in Section 2. The collision detection method is explained in Section 3. The multi-objective redundancy resolution scheme is given in Section 4. Simulation results are discussed in Section 5 in order to verify the effectiveness of the redundancy resolution technique. The conclusion of the article is in Section 6.

2. Modelling The kinematics and dynamics of a planar wire-actuated parallel manipulator shown in Figure 1(a) are investigated for redundancy resolution. Moreover, the dynamical problem of collision with stationary obstacles is discussed. Figure 1 shows the coordinates and parameters used for the analysis of a planar wireactuated parallel manipulator. The fixed coordinate system ?X, Y, located at O, is attached to the base, while the moving coordinate system, @XA, YA, is attached to the mobile platform at its centre of mass P with position vector of x = [x, y]T in ?X, Y. All vector expressions will be in the base reference frame unless otherwise stated. The position of base attachment point of each wire (anchor) Ai is ai = [aix, aiy]T, the position of attachment point on the mobile platform Bi is rBi/P = [rBi/Pcosi, rBi/Psini]T in @XA, YA, and li = [licosi, lisini]T is the vector of the magnitude and direction of each wire length. When collision occurs, the mobile platform becomes in contact with the obstacle at point C, as shown in Figure 1(a). The position vector of contact point C is rC/P = [rC/Pcosnc, rC/Psinnc]T in @XA, YA. As shown in Figure 1(a), the position of the centre of the circular obstacle Q with respect to O and P are rQ/O = [rQ/Ox, rQ/Oy]T and rQ/P = [rQ/Px, rQ/Py]T, respectively. Fnc is the normal contact force acting towards the center of the mobile platform (due to the circular shape of the mobile platform) and Ff is the friction force acting on the mobile platform in the opposite direction of the motion of the mobile platform.

Multi-Objective Trajectory Planning in Wire-Actuated Manipulators

(a)

427

(b)

Figure 1. (a) Coordinates and variables for planar parallel manipulators during collision, (b) free body diagram of the mobile platform during collision.

The inverse velocity solution that relates the mobile platform velocity

T T x = [ x , y , ] to the wire length rates l = [ l1 , B, ln ] is given by

& l1 ' & c1 ( ) ( ( ) ( ( ) (c * n *ln ,

s1 sn

rB1 / P s ( 1 1 ) ' & x ' & x ' )( ) ( ) ) ( y ) J ( y ) (* ), rBn / P s ( n n )), (* ),

(1)

where ci, si and s( + i i) stand for cosi, sini and sin( + i i), respectively, is the orientation of the mobile platform and n (n 4) is the number of wires. From dynamic force and moment balances & Fext ' &C 1 ' &m 0 0 ' & x ' & 0 ' &( Fextx ') & x ' ( x ) ( ) ( ) ( ) ( ) ( ) ( J ( ) ( 0 m 0 ) ( y ) ( mg ) Fext y ) M ( y) g ( Fext y ) ( ) (*C n ), (* 0 0 I z ), (*), (* 0 ), ((M )) (*), (*M extz ), * extz , T

(2)

where m and Iz are the mass and moment of inertia of the mobile platform respectively, x and y are the components of linear accelerations and is the angular acceleration of the mobile platform, [C1, B, Cn]T is the vector of dynamic wire forces, Fext x and Fext y are the components of the external force, M extz is the external moment acting on the mobile platform, M is the inertia matrix, and g is the vector of gravitational force with g = 9.81 m/s2. Considering Figure 1(b), in the absence of any other external force/moment, Fext and M extz can be written as Fext Fnc F f M ext z Fnc rC/Psin ( nc nc ) F f rC/P cos ( nc nc ) sgn( )

(3)

where sgn( ) = 1 if > 0, sgn( ) = 0 if = 0, and sgn( ) = 1 if < 0, and nc defines the orientation of the normal contact force Fnc. Considering Ff = DkFnc, where Dk is the coefficient of kinetic friction, equation (2) is simplified as

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&C 1 ' ( ) J ( ) (*C n ), T

E & ' (cos nc D cos( nc 2 sgn( )) ) & x ' ( ) E ( ) ) M ( y) g Fnc ((sin nc D sin( nc sgn( )) ) 2 ( ) *(,) sin ( ) cos ( ) sgn( ) r D r nc nc C/P nc nc ( C/P ) (* ),

The solution of equation (4) is given by

FC1

(4)

T T C n G JT # MFx y G g Fncc (I JT # JT )k p h

(5) where c is the vector of coefficients of the normal contact force Fnc in equation (4). Cp = JT#(M[ x y ]T g – Fncc) is the minimum norm (particular) solution of equation (4) and JT# = J (JT J) 1 is the generalized inverse of matrix JT, and Ch = (I JT#JT)k is the homogeneous solution that maps the free vector k to the null space of JT. The homogeneous solution could be written as Ch = NH, where N is a matrix that its columns correspond to the orthonormal basis for the null space of JT and may be determined using the singular value decomposition, and H is an arbitrary vector. To resolve the redundancy at the torque level, for given trajectories of the platform, a H is identified (if it exists) at each instant such that a desirable task is achieved avoiding negative tension in each wire. It should be noted that throughout this paper, the term “norm” stands for the 2-norm, and all minimizations correspond to the 2-norm of the relevant vector. The constraint tension function, which is the solution to equation (2), is p N I min (6) T T# If J in equation (4) has full row-rank, then J is invariant to the choice of weighing metric, e.g., Doty et al. (1995). Thus, using a weighing metric is not required.

3. Collision Detection To detect collision, the minimum distance between the mobile platform and the obstacle is monitored at each time instant. When collision occurs, the given trajectory of the mobile platform is modified such that the minimum distance between the mobile platform and the obstacle is greater than a threshold. When the mobile platform and the obstacle are circular, the minimum distance d is calculated as d = PQ (rC/P + rob), where rob is the radius of the obstacle. Given the position of the centre of obstacle rQ/O, and pose (position and orientation) of the platform, the position of Q relative to P is rQ/P = [ rQ / O x x, rQ / O y y ]T. Using the orientation of rQ/P, i.e., nc = tan 1(rQ/Py / rQ/Px) , the orientation of the normal contact force Fnc is nc = + (nc + ). Then,J nc is substituted into equation (4) to construct the solution of wire tensions as given by equation (5).

Multi-Objective Trajectory Planning in Wire-Actuated Manipulators

429

4. Trajectory Planning and Impact Reduction When collision occurs, the given trajectory of the mobile platform has to be modified while keeping positive tension in each wire and satisfying additional desirable criteria, i.e., minimizing the norm of tension in the wires for a given normal contact force Fnc and minimizing the trajectory deviation from the desired trajectory of the mobile platform. In the work presented, before collision occurs, redundancy will be resolved at the torque level while tracing the given trajectory of the mobile platform. Therefore, the optimization problem before collision is formulated as follows: minimize T C 12 C n2 (7) subject to p N C I min For each pose of the mobile platform, the value of HC is calculated such that minimum wire tensions are maintained subject to positive tension in each wire. When there is no external force/moment, [ Fextx , Fext y , M extz ] = [0, 0, 0]. When collision occurs, the minimum positive tension in the wires is determined for each time instant such that the normal contact force Fnc does not exceed a certain value. For the subsequent instants, the trajectory of the mobile platform is modified while collision is avoided and the modified trajectory traces the given mobile platform trajectory as close as possible. Then, using the redundancy of the manipulator, the minimum tension in the wires is calculated such that the tension of each wire is positive. If the positive tension in each wire is not achieved, the trajectory is modified again and the tension minimization procedure is repeated. For this purpose, the trajectory of the mobile platform is x(t ) Fx

Mx o (t ) t N tc y GT L Kx o (t ) x var (t )

tc O t N toc

(8)

where tc is the instant at which the minimum distance d becomes zero and collision occurs, toc is the time instant to clear the obstacle. The given trajectory of the mobile platform xo(t) is chosen to be a fifth order polynomial, and xvar(t) = [xvar(t), yvar(t), var(t)]T is referred to as the variable portion of the mobile platform trajectory and is calculated at each time instant after collision. Once the mobile platform clears the obstacle, xvar(t) = 0. The fifth order trajectories for xo, yo and o are chosen to satisfy eighteen initial and final boundary conditions of the trajectory, x, and its derivatives. In order to have continuous xvar and yvar for t > tc, when collision occurs, the original trajectory of the mobile platform is changed in a way that xvar and yvar are modified dependently considering the slope of the line PQ , as shown in Figure 1(a). The trajectory is modified at t = tc + t using yvar = xvar tan ( + nc), where t is the time step. For t > (tc + t), similar to the collision instant, the modified trajectory is along the line that is perpendicular to PQ and tangential to

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both circular objects, and is such that PQ I rBi/P rob P th , where th is the minimum distance threshold. Using xvar, the constraint function represented in equation (7) will become a function of xvar, yvar and var. The goals after collision are to trace the given trajectory as close as possible and minimize the tension in 2 2 2 y var var the wires, so the objective functions are to minimize x Tvar x var x var T 2 2 and C 1 C n , one at a time, subject to C(t, xvar, yvar, var) I Cmin. In the simulation, the wires are allowed to cross over the obstacle which could occur when the obstacle and the wires are on different planes. After calculating xvar, yvar and var, at each time instant (t > tc), the minimum H is calculated that maintains positive tension in each wire. The velocity and acceleration of the platform are then approximated using backward difference method as x (ti) = (x(ti) x(ti1))/(ti1 ti) and x (ti) = ( x (ti) x (ti1))/(ti1 ti), where i corresponds to the iteration number for the time step. The results are then substituted into equation (5) for tension optimization purposes.

5. Simulation For the simulation, the anchor positions are {a1, a2, a3, a4} = {( 1, 0.75), (1, 0.75), (1, 0.75), ( 1, 0.75)} (units in meters), and the connection points of wires on the mobile platform are {1, 2, 3, 4} = {225, 315, 45, 135} (units in degree). The mass m, moment of inertia Iz, and radius rBi/P of the mobile platform are respectively 2 kg, 0.0144 kg.m2, and 0.125 m. The radius and position of the obstacle are rob = 0.05 m and (rQ/Ox, rQ/Oy) = (0.25 m, 0.1 m). The initial and final boundary conditions ( x, x , x, y, y , y, , , ) are (0, 0, 0, 0, 0, 0, 0, 0, 0)0 and (0.25 m, 0, 0, 0.25 m, 0, 0, 2 deg, 0, 0)f, respectively, with tQ[0, 10] s and time step of t = 0.01 s. The minimum allowable tension of each wire is Cmin = 2 N. The upper limit on the actuator torques is calculated based on the maximum allowable wire tension of 525 N (McColl and Notash 2009). With four wires, the constraint function in equation (7) is reduced to four linear inequalities in terms of HC, where HC is reduced to a scalar. At the instant of collision, the maximum allowable normal contact force Fnc is 5 N. The termination tolerances placed on constraint violations are chosen as 0.001 N, on the objective function as 0.001 N2, and on the estimated parameter values (i.e., HC) as 0.0001 N. The simulation results are obtained using the fmincon function in MATLAB. Figure 2 shows the trajectories x and xo, and xvar = x – xo. Figure 2(b) shows how close x is to xo. Collision has occurred at instant tc = 3.92 s. For t > 3.92 s, the minimization of xvar has resulted in the x component of the trajectory with lower magnitude compared to xo. Therefore, due to location of the obstacle, the collision has delayed tracing the x component of the trajectory. The displacement of the platform in Y direction is not affected by the location of the obstacle noticeably.

x xo

0.2 0 0 0.4

t

y [m] xvar [m] Rvar [deg] var

0.4

0 -0.05 -0.1 0 0.04 0.02 0 0 1 0 -1 0

c

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o

y, y [m] x, x [m] o

Multi-Objective Trajectory Planning in Wire-Actuated Manipulators

0 0 4

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time [sec]

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(a) (b) Figure 2. (a) Original and modified poses, (b) variable portion of the trajectory.

0.4

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[m/s ]

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[m/s ]

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(a) (b) Figure 3. (a) Change in the configuration, (b) original and modified accelerations. 50

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(a) (b) Figure 4. (a) Minimum H, (b) wire tensions.

Figure 3(a) shows how the mobile platform moves around the obstacle until it clears the obstacle and then continues the original trajectory. Figure 3(b) depicts the actual acceleration of the platform. Collision results in jumps in the velocity and acceleration of the platform at tc = 3.92 s. For t > 3.92 s, the trajectory of the platform is modified such that the magnitude of the acceleration should not exceed 3g m/s2. The magnitude of the maximum linear acceleration is less than 0.5g m/s2. Due to the redundancy of the manipulator, the orientation of the platform is remained unchanged and so are its derivatives. Figure 4(a) shows the minimum H

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that guarantees positive tension in each wire. Figure 4(b) illustrates the tension histories. It can be seen that the second wire is not critical (except for the collision instant) for maintaining the configurations shown in Figure 3(a). The peaks in the acceleration components of Figure 3(b) have resulted in discontinuous wire tensions at corresponding time instants.

6. Discussion and Conclusions An approach to resolve the actuation redundancy of planar wire-actuated parallel manipulators was investigated to minimize the wire tensions and deviation in the trajectory while clearing an obstacle with positive tension in each wire. Before collision, a prescribed trajectory of the mobile platform was followed and the norm of wire tensions was minimized. The infinite solutions for wire tensions were utilized to identify a vector of wire tensions such that the contact force was kept below an assigned value while the wire tensions were minimized. After collision, the desired trajectory was modified at each time instant such that the platform traced its predefined trajectory as close as possible. Then, using the redundancy of the manipulator, the minimum norm tension in the wires was determined. The simulation of a 3-DOF planar wire-actuated parallel manipulator was developed satisfying objective functions and constraints. Based on the optimization criteria, the tolerances used in the optimization routine, the optimization scheme, and the desired continuous trajectories, continuous wire tensions were produced before and after collision.

References Agahi, M. & Notash, L. (2009). Redundancy Resolution of Wire-Actuated Parallel Manipulators. special edition of the Transactions of the Canadian Society of Mechanical Engineers. 33(4):561-575. Doty, K.I., Melchiorri, C. & Bonivento, C. (1995). A Theory of Generalized Inverses Applied to Robotics. International Journal of Robotics Research. 12(1):1-19. Kawamura, S., Hitoshi, K. & Choe, W. (2000). High-speed manipulation by using parallel wire-driven robots. Robotica. 18:13-21. McColl, D. & Notash, L. (2009). Extension of Antipodal Theorem to Workspace Analysis of Planar Wire-Actuated Manipulators. Proc. of 5th IFToMM Int. Workshop on Computational Kinematics. 9-16. Notash, L. & Kamalzadeh, A. (2007). Inverse dynamics of wire-actuated parallel manipulators with a constraining linkage. Mechanism and Machine Theory. 42(9):1103-1118. Oh, S. R. & Agrawal, S.K. (2005). Cable Suspended Planar Robots with Redundant Cables: Controllers with Positive Tensions. IEEE Transactions on Robotics. 22(3):457-465. Williams II, R.L. & Gallina, P. (2003). Translational Planar Cable-Direct-Driven Robots. Journal of Intelligent and Robotic Systems. 37(1):69-96.

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Evaluation of the KOBIAN and HABIAN Emotion Expression Humanoid Robots with European Elderly People1 Massimiliano Zecca1,2,3,4, Giovanna Macrì5, Yu Mizoguchi6, Vito Monaco5, Nobutsuna Endo4,6, Kazuko Itoh3,7, Paolo Dario3,5, Atsuo Takanishi1,2,3,4,7,8 1

Institute for Biomedical Engineering, ASMeW, Waseda University, Tokyo, Japan 2 Humanoid Robotics Institute (HRI), Waseda University, Tokyo, Japan 3 Italy-Japan Joint Laboratory on Humanoid and Personal Robotics, Tokyo, Japan 4 Global Robot Academia, Waseda University, Tokyo, Japan 5 ARTS Lab, Scuola Superiore Sant’Anna, Pisa, Italy 6 Graduate School of Science and Engineering, Waseda University, Tokyo, Japan 7 Advanced Research Institute for Science and Engineering, Waseda Univ., Tokyo, Japan 8 Department of Modern Mechanical Engineering, Waseda University, Tokyo, Japan

1.

Introduction

Our world is getting older and older (OECD 2009), and therefore there is considerable expectation for more widespread home, medical, and nursing care services to assist not self-sufficient elderly people, both from the physical and psychological points of view. In this context, robots are expected to perform human tasks such as operating equipment designed for humans in dangerous environments, providing personal assistance, social care, cognitive therapy, entertainment, and education (Kozima, Nakagawa et al. 2007). These robots should be capable of a smooth and natural adaptation and interaction with their human partners as well as with the environment. They should also be able to communicate naturally with humans, especially in home and personal assistance applications for elderly and/or handicapped persons. Moreover, these devices should be safe and, more in general, should never have a negative effect on their human partners, neither physical nor emotional. —————————— 1

Research commissioned by The New Energy and Industrial Technology Development Organization (NEDO) and conducted at Humanoid Robotics Institute, Waseda University. Part of the research supported by a Grant-in-Aid for the WABOT-HOUSE Project by Gifu Prefecture. Partial support by: ASMeW Priority Research C Grant #11; JSPS Grant-in-aid for Scientific Research #19700389; Waseda University Grant for Special Research Projects (No. 266740). KOBIAN and HABIAN designed by 3D CAD software “SolidWorks”. Special thanks to: SolidWorks Japan K.K. KURARAY Co., Ltd.; the Italian Ministry of Foreign Affairs; Okino Industries LTD; STMicroelectronics; Japan ROBOTECH LTD. A special thank also to Mr. Kimura of GADPA and Mr. Okubo for their help in the realization of the emotional patterns. Eventually, the authors would also like the Università della Terza Età of Pontedera, PI, Italy and its members who kindly accepted to be part of this experiment.

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In recent years, several robots have been developed to investigate the socio-emotional aspects of human-robot interactions, in particular in Asian countries. Among them, we can find: animaloid robots such as the therapeutic robot PARO or SONY AIBO to interact with human beings and make them feel emotional attachment; humanoid robots like Honda ASIMO or Kawada HRP-2 to cooperate effectively with humans; humanoid robots like SONY QRIO, Waseda WE series and KOBIAN, Albert HUBO, android robots like Kokoro Actroid, SAYA, and Ishiguro’s Geminoid, and even other kinds of robot like NEC’s PaPeRo or MIT’s Kismet to explore socio-emotive face-to-face interactions with people; and so on. Our group has been investigating the fundamental technologies of RT service system that shares the living environment with elderly people with the aim to make them more comfortable during the normal activities of daily living (Itoh, Miwa et al. 2006; Zecca, Roccella et al. 2006; Endo, Momoki et al. 2008). To achieve this important goal, the robot appearance should be as human like as possible. We propose that humanoids should be designed to balance human-ness, to facilitate social interaction, and robot-ness, to avoid false expectations about the robots’ abilities (Mori 1970; DiSalvo, Gemperle et al. 2002). For this purpose, in the last years we have developed the whole body emotion expression humanoid robot KOBIAN (Fig. 1, left). The preliminary evaluation of its emotional expressions showed that the whole-body posture clearly improves the emotion recognition (Zecca, Endo et al. 2008). However, since the obtained results were not entirely satisfactory, we asked a professional photographer and a professional cartoonist to help us in the creation of new emotional poses. Specific objective of this paper was the evaluation of the effectiveness of the emotional expressions and the acceptability of the two emotional robots with European Elderly people. Unlike Japan and Asia, in fact, in Europe humanoid robots are not usually well accepted, and European people do not like human-like artificial beings (Dario, Guglielmelli et al. 1999; Nisbett 2003). This paper is organized as follows. Section 2 presents the development of an emotional wheeled robot, named HABIAN, with the main aim to compare its acceptability with respect to that of a biped emotional robot (2.A). The new emotional poses for KOBIAN and HABIAN were also realized in collaboration with one photographer and one cartoonist (2.B). A professional Japanese actor has also been asked to replicate the 14 emotional patterns (2.C). Section 3 presents the experimental evaluation, in particular, the recognition ratio of each emotional pattern (3.A) and the impression elicited by the two robots (3.B). Eventually, Section 4 presents the discussion and the conclusions.

2.

Materials and Methods

A. Wheeled-type emotion expression humanoid robot HABIAN The wheeled-type emotional humanoid robot HABIAN (Fig. 1, right) has been

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developed based on the whole-body humanoid robot KOBIAN with a Blackship mobile base (SGI Japan 2008) instead of the two legs. A summary of the DOFs of KOBIAN and HABIAN is presented in Table 1. Table 1: Main characteristics of KOBIAN and HABIAN.

B. Creation of the new KOBIAN HABIAN emotional patterns Height 1400 mm 1400 mm To improve the Width 520 mm 520 mm recognition ratio of the Weight 62 kg 59 kg emotional patterns, we DOFs 48: 37: Head 7 Head 7 asked one professional Neck 4 Neck 4 cartoonist and one Arm 7x2 Arm 7x2 professional photographer Hand 4x2 Hand 4x2 to design their original Trunk 1 Trunk 1 emotional poses for the 7 Waist 2 Waist 1 selected emotions (Anger, Leg 6x2 Leg 2 Disgust, Fear, Happiness, Perplexity, Sadness, and Surprise). Each artist spent one day each playing with the robot in order to find the best possible posture for each given emotion. The results are presented in Fig. 2. As can be clearly seen from the pictures, some of the poses realized by the two artists were very similar, while others were completely different. In any case most of the poses were completely different from the ones prepared by the students in our lab (Zecca, Endo et al. 2008).

Fig. 1: KOBIAN (left), Japanese professional actor (center) and HABIAN (right) showing the same emotional pattern.

C. Human actor A professional Japanese actor (Fig. 1, center) was asked to replicate the emotional expressions of the two robots. The 14 different expressions (7 for the Cartoonist, 7 for the Photographer) were repeated several times. The best matching videoclips were then selected and used for the evaluation phase.

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Fig. 2: Emotional postures realized by the cartoonist (left) and by the photographer (right).

3.

Experimental evaluation

A group of 19 people (male:8; female: 11; age 70.6±8.9), all members of the “Università della Terza Età” of Pontedera, Pisa, Italy, with absolutely no previous experience with humanoid robots and/or emotional robots agreed to participate to the evaluation after giving their informed consent to the experiment. 42 short videoclips of 7 emotions (Anger, Disgust, Fear, Happiness, Perplexity, Sadness, and Surprise) x 3 performers (human actor, HABIAN, KOBIAN) x 2 types (Cartoonist, Photographer) were shown to the subjects. The order of these videos was randomly selected before the experiment but was not disclosed to the participants. Each videoclip was consecutively shown 3 to 6 times, depending on the reaction time of the subjects, before moving to the next one. In particular the first videos were shown for a longer time, to make the users better understand the experimental protocol. The subjects were asked to choose the emotion that they thought the video robot was expressing among a predetermined list. A. Evaluation of the Emotion Recognition Ratio The recognition ratio for each video was calculated as

r (vi , e j ) Ni , j / N

where vi = video #i (i =1...42), ej = emotion #j ( j = 1...7 ) (6 emotions + perplexity); Ni,j : number of right responses (vi, ej); N: total number of people. The collected data were transcripted in Excel (Microsoft) and saved for further processing, which was done in Matlab® (The Mathworks, Inc., MA, USA). The results of the experiment are presented in Fig. 3, Fig. 4, and Table 2. In them, the following abbreviations are used: ANG: Anger; DIS: Disgust; FEA: Fear; HAP: Happiness; PER: Perplexity; SAD: Sadness; SUR: Surprise. Overall, the recognition ratio of emotional patterns was quite low. In total, only 57.1% of the videos were correctly labelled; however, the average recognition ratio for these videos was only 50.2%. On average 14.7% of the responses were

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“other” (i.e. 6 videos out of 42). 17 videos (40%) received very low recognition ratio, less than 22%: 4 for the actor, 7 for HABIAN, and 6 for KOBIAN.

Fig. 3: Recognition ratio for the patterns prepared by the cartoonist (left) and by the photographer (right).

The recognition ratio for KOBIAN, HABIAN and Actor was 31.0 %, 25.6%, and 31.7% respectively for the patterns prepared by the cartoonist; 22.2%, 38.8% and 55.0% for the patterns prepared by the photographer. The combined recognition ratio – i.e. obtained by combining the best emotional expressions – was 38.7%, 46.7%, and 57.4%, respectively. Among the different emotions, Anger and Perplexity were the most difficult to recognize. It is worth noticing, however, that the emotions expressed by the actors weren’t always recognized well either. This could possibly indicate a negative effect due to the different culture and different age of the subjects (Nisbett 2003). Further experiments in this sense are therefore necessary. B. Evaluation of the robot impression 28 semantic differential questions (Kanda, Ishiguro et al. 2001) were used to evaluate the impression of the two robots and of the human actor: (translated from Japanese) Gentle, Scary; Positive sensation, Negative sensation; Friendly, Hostile; Safe, Dangerous; Warm, Cold; Lovely, Hateful; Outgoing, Formal; Easy to understand, Difficult to understand; Easy to approach, Difficult to approach; Bright, Dark; Considerate, Selfishness; Human, Mechanical; Substantial, Empty; Interesting, Uninteresting; Pleasant, Unpleasant; Like, Dislike; Interesting, Boring; Good, Wrong; Complicated, Simple; Quick, Slow; Agile, Slow; Violent, Calm; Positive, Negative; Strong, Weak; Showy, Simple; Cheerful, Gloomy; Sensitive, Insensitive; Wise, Silly. The questionnaire was presented in Italian after a translation from the original Japanese. A 7-point Likert scale ranging from -3 to +3 was used for the evaluation, with 0 meaning the neutral response. A Kruscal-Wallis one-way analysis of variance showed that there are significant differences between the human actor and the two robots in several parameters (see Fig. 4), while the two robots are evaluated almost the same except for the pair Considerate/Selfish, in which HABIAN was perceived as more considerate than KOBIAN (p<0.05) and as considerate as the actor. As expected, the human actor is always perceived as more “positive” than the two robots.

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DISGUST

ANGER

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Sadness

A Ca H K A Ph H K A Ca H K A Ph H K A Ca H K A Ph H K

Surprise

Fear

Disgust

Anger

Emotion Artist Performer

Table 2: Summary of the results of the evaluation. In the “Artist” column, Ca: Cartoonist; Ph: Photographer. In the “Performer” column: A: Actor; H: HABIAN: K: KOBIAN.

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Fig. 4: Significative parameters. Significance level is indicated by *: p < 0.05; **: p<0.01.

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C. Free comments The users were also asked to provide some personal opinion on the robots. Most of the comments were definitely negative, such as “why do we need a biped robot in our house? There is no reason for it!”, “I’m scared1”, “I do not want a human being-like object in my house!”, “why do you need to give it emotions? It frightens me!”, and so on. These comments clearly show that the negative results are mostly due to the different cultural background, as in previous experiments with Japanese senior citizens we always got positive feedbacks (Zecca, Endo et al. 2008).

4.

Conclusions

In this elderly-dominated society, Personal Robots and Robot Technology (RT)-based assistive devices are expected to play a major role, both for joint activities with their human partners and for participation in community life. So far, several different personal robots have been developed. However, it is not yet clear what kind of ability is necessary for personal robot. We think that emotional expression of robot is effective for joint activities of humans and robots. The robot should express in particular happiness and perplexity, which we are considered to be fundamental emotional expressions for a smooth and natural interaction with humans. To this purpose we developed a whole body bipedal humanoid robot, named KOBIAN, which is also capable to express human-like emotions. This novel robot is based on the previously developed Biped Humanoid Robot WABIAN-2 for the lower body, and on the Emotion Expression Humanoid Robot WE-4R for the head. We also developed a new wheeled type emotion expressing humanoid robot named HABIAN, which is based on KOBIAN for the upper body, with a Blackship mobile base instead of the legs. We asked two professional artists (a cartoonist and a photographer) to help us in the realization of new emotional patterns; we also asked a professional actor to perform the same patterns. In this paper we presented the preliminary evaluation of the new emotional expressions with a group of European elderly people. The preliminary results showed that – as expected – the emotions are not always correctly recognized. The two robots, nonetheless, are capable of expressing these emotions, and most of their expressions were correctly labeled as “emotional” and not neutral. It is worth noticing that the emotions expressed by the actors weren’t always recognized well either. This could possibly indicate a negative effect due to the different culture and different age of the subjects, as Western people are not familiar with Japanese expressions (Dario, Guglielmelli et al. 1999; Nisbett 2003). We also received several negative comments; which clearly show that the negative results are mostly due to the different cultural background, as in previous experiments with Japanese senior citizens we always got positive feedbacks. For sure the expressiveness of the robots must be increased in their next

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versions. We would also like to evaluate the new emotional expressions with more people from different countries and of different ages. In addition, we will investigate the influence of outer covering and the effects of sound, movement, arms, and whole body. Furthermore, we will investigate what kind of robot is effective in human living environment by human-robot interaction experiment. References Dario, P., E. Guglielmelli, et al. (1999). "MOVAID: a personal robot in everyday life of disabled and elderly people " Technology and Disability 10(2): 77-93. DiSalvo, C., F. Gemperle, et al. (2002). "All robots are not created equal: the design and perception of humanoid robot heads." Proceedings of the conference on Designing interactive systems: processes, practices, methods, and techniques. Endo, N., S. Momoki, et al. (2008). Development of Whole-Body Emotion Expression Humanoid Robot. 2008 IEEE International Conference on Robotics and Automation (ICRA). Pasadena, CA, USA: 2140-2145. Itoh, K., H. Miwa, et al. (2006). Mechanisms and Functions for a Humanoid Robot to Express Human-like Emotions. IEEE International Conference on Robotics and Automation - ICRA 2006. Kanda, T., H. Ishiguro, et al. (2001). Psychological Evaluation on Interactions between People and Robot. 19th Annual Conference of the Robotics Society of Japan (RSJ2001), Tokyo, Japan. Kozima, H., C. Nakagawa, et al. (2007). "Children-robot interaction: a pilot study in autism therapy." Prog Brain Res 164: 385-400. Mori, M. (1970). "Bukimi no tani [The Uncanny Valley]." Energy 7(4): 33-35 (Originally in Japanese). Nisbett, R. (2003). The Geography of Thought : How Asians and Westerners Think Differently...and Why Nicholas Brealey Publishing. OECD (2009). Population statistics. http://www.oecd.org, Organisation for Economic Co-operation and Development. SGI Japan. (2008). "Blackship, http://www.sgi.co.jp/solutions/robot/blackship/." Zecca, M., K. Endo, et al. (2008). Design and Evaluation of The Soft Hand WSH-1 For The Emotion Expression Humanoid Robot KOBIAN. 6th International Conference of the International Society for Gerontechnology (Gerontechnology2008). Pisa, Italy. Zecca, M., N. Endo, et al. (2008). Design of the humanoid robot KOBIAN –preliminary analysis of facial and whole body emotion expression capabilities–. IEEE-RAS International Conference on Humanoid Robots HUMANOIDS2008. Daejeon, South Korea. 1: 487-492. Zecca, M., S. Roccella, et al. (2006). From the Human Hand to a Humanoid Hand: Biologically-Inspired Approach for the Development of RoboCasa Hand #1. ROMANSY 16 Robot Design, Dynamics and Control, CISM Lecture Note #487. T. Zielinska and C. Zielinski, Springer Wien: 287-294.

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Modular Design of Emotion Expression Humanoid Robot KOBIAN Nobutsuna Endo1, Keita Endo1, Massimiliano Zecca2,3, and Atsuo Takanishi2,3,4,5 1

Graduate School of Science and Engineering, Waseda University, Tokyo, Japan Institute for Biomedical Engineering, ASMeW, Waseda University, Tokyo, Japan 3 Italy-Japan joint laboratory on Humanoid and Personal Robotics “RoboCasa”, Tokyo, Japan 4 Humanoid Robotics Institute (HRI), Waseda University, Tokyo, Japan 5 Advanced Research Institute for Science and Engineering, Waseda University, Tokyo, Japan 2

Abstract. Personal robots and Robot Technology (RT)-based assistive devices are expected to play a substantial role in our society, largely populated by elders; they will play an active role in joint works and community life with humans. In particular, these robots are expected to play an important role for the assistance of elderly and disabled people during normal activities of daily living (ADLs). To achieve this result, personal robots should be also capable of human-like emotion expressions. In this perspective we developed a whole body bipedal humanoid robot, named KOBIAN, which is also capable to express human-like emotions. In this paper we present the mechanical and modular design of KOBAIN.

Introduction In our society getting older and older, there is considerable expectation for more widespread home, medical and nursing care services to assist not self-sufficient elderly people, both from the physical and psychological points of view. In this context, robots are expected to perform human tasks such as: 1) operating equipment designed for humans in dangerous environments, 2) providing personal assistance, social care, cognitive therapy, entertainment, and education. These robots should be capable of a smooth and natural adaptation and interaction with their human partners as well as with the environment. They should also be able to communicate naturally with humans, especially in home and personal assistance applications for elderly and/or handicapped persons. Moreover, these devices should be safe and, more in general, should never have a negative effect on their human partners, neither physical nor emotional. Our group is investigating the fundamental technologies of RT service system that shares the living environment with elderly people with the aim to make them more comfortable during the normal activities of daily living. To achieve this important goal, the robot appearance should be as human like as possible. We propose that humanoids should be designed to balance human-ness, to facilitate

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social interaction, and robot-ness, to avoid false expectations about the robots’ abilities (DiSalvo et al., 2002, Mori, 1970). For this purpose, in the last years we have developed the whole body emotion expression humanoid robot KOBIAN (Endo et al., 2008) based on the bipedal humanoid robot WABIAN-2 (Ogura et al., 2006) and emotion expression humanoid robot WE-4RII. KOBIAN is 1400 mm tall and 520 mm wide. Its weight is 62 kg. The preliminary evaluation of the whole body emotion expressions showed that the body of the robot clearly improves the emotion recognition. KOBIAN’s head was designed based on WE-4RII’s one. WE-4RII’s head has 22 degrees of freedom, but it is too bulky and heavy to mount on KOBIAN’s body which was developed based on WABIAN-2’s one, so we reduced the DOFs of KOBIAN’s head by 7-DOFs, downsized, and trim weight. In order to evaluate the effect of DOF reduction, we designed the appearance of KOBIAN’s face about the same as WE-4RII’s. However, we can not predicate that the current appearance of KOBIAN’s face is best suited to KOBIAN because people either like this appearance design or don’t like it. For this reason, KOBIAN’s head is designed as driving mechanism of facial expression and outer covering are separated, and it is easy to adapt to various outer covering. And, the driving parts are modularized and it is easy to change trajectories of each facial parts. In this paper, we describe the modular designs of KOBIAN.

Design Design Concept. As mentioned above, it is fundamental for the robot which is active in human living environment to be capable of biped walking. In addition, this kind of robot needs also the capability of moving in natural ways, as they share the living environment with human. Therefore, we developed the whole body emotion expression humanoid robot KOBIAN, based on the biped humanoid robot WABIAN-2 and WE-4RII (see Figure 1). Concerning the interaction at emotional level, a mechanism to tell the user whether the robot understand human words or not is important for a smooth communication and interaction. In addition, it is necessary to transmit emotion for the emotional communication. Therefore the robot should be capable of facial expression for promotion of interaction, communication of meaning of the sensitivity, and indication of internal states. In particular, smile that indicates the robot feel pleasant and perplexity that indicates the robot cannot understand human words are essential. As mentioned above, there are existing various types of humanoid robots and their appearance designs are varied. This is caused by variety of designers and developers, and in future, much more variety of robot’s appearance will be required when humanoid robots come to be more imminent for a human being.

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With most of facial expression robots, the driving mechanism of facial parts and the appearance are tied closely because the facial expressions are made by each facial part such as eyebrows, eyelid, and mouth driven mechanically. In this way, however, developers have to design the driving mechanism from the beginning in order to adapt to various types of appearance, and we think of it as inefficient. Likewise in WE-4RII, driving wires of eyebrow pass through the forehead and it is difficult to change the appearance and it is also difficult to maintain. So, with KOBIAN’s head, we decided to separate the driving mechanism of facial parts and outer covering and also decided to conduct modular design in order to make the one head easier to adapt to various types of outer covering. Mechanical configuration. Human’s eyebrow grows on the surface of skin and move with skin. But, we focused on the change of the eyebrow shape, and decide to move KOBIAN’s eyebrows independently from forehead. The eyebrow is molded with thermoplastic resin SEPTON (KURARAY Co. Ltd.). The both ends of the eyebrow are free turns, and an arbitrary point between the ends is assumed a control point. A neodymium magnet is mounted on the back of control point. The outer covering for forehead is molded with hard resin, and a neodymium magnet and a mechanism which drive the magnet are mounted on the corresponding position of control point of eyebrow. The magnet of mechanism side is vertically driven by wires, and the control point of eyebrow is driven in the same direction. The driving mechanism is shown in Figure 2, and the exploded view is shown in Figure 3. The basement of eyebrow driving mechanism can be fixed with moved in longitudinal direction and pitch direction, and easily adapt to change of the position of the control point of the eyebrow and the change of the inclination of the forehead. So, we can easily change to various types of the shapes of eyebrow and outer covering.

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WE-4RII has 5-DOFs at the mouth (open and close of the jaw, extending, up and down of the angles of mouth). The lip is made by using springs of spindle type and actuated by wires, and has high expression performance. However, assembly and maintenance are hard. KOBIAN has 2 -DOFs at the mouth (open and close of the jaw, up and down of the angles of mouth). The lip is molded with SEPTON. The mechanism of the mouth is shown in Figure 4. This mechanism is much easier to assembly and maintain than WE-4RII’s one. In addition, motor drivers (Tokushu Denso Co., Ltd) for the head’s 7-DOFs can be stored into the back of the head. The lines for rotation speed command to motor driver and the lines for encoder pulses are collected at a junction circuit on the back of the head, and connect to a control PC on KOBIAN’s back through one harness. All of electric interface are power line for the motor drivers (48V), power line for the encoders (5V), and the harness to junction circuit. By modularizing the entire head, it is easy to mount the head on other robot systems. The head is mounted on its body via neck. The neck is developed based on WE-4RII’s neck (Miwa, et al., 2004) and has 4-DOFs. Figure 5 shows a CAD design of the head and Figure 6 shows an exploded view of the head and neck. Figure 7 shows examples of KOBIAN’s facial expressions. KOBIAN expresses them by using 4-DOFs at eyebrows, upper eyelids, and mouth.

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Figure 7. Examples of the facial expressions of the head We have also developed a new anthropomorphic robotic hand WSH-1 (Waseda Soft Hand-No.1) for KOBIAN’s hand (Zecca et al., 2008). WSH-1’s palm consists of a framework structure and soft material (Hitohada gel, Exseal Co., Ltd.). WSH-1’s finger is molded by silicone (ELASTOSIL M8520, Wacker Asahikasei Silicone Co., Ltd.) , wears a skin molded by Hitohada gel, and performs bending and stretching by a wire. Figure 8 and Figure 9 show the structure and picture.

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System configuration. The system configuration of KOBIAN is shown in Figure 10. The control PC (Pentium M 1.6GHz, RAM 2GB) is mounted on KOBIAN’s back.

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Storage is a Compact Flash memory 2GB, and an operating system (QNX Neutrino 6.3.0) is installed in it. The CPU communicates to HRP Interface Board (Zuco, Co., Ltd.) via PCI Back Plain Board. The control program give speed commands to motor drivers via D/A convertor on HRP Interface Board and get encoders’ pulses via pulse counter on HRP Interface Board. Nitta 6-axis Force/Torque Sensors mounted on both feet send data via PCI Receiver Board. The control PC is operated by using remote desktop via Ethernet or Wireless LAN IEEE 802.11g. Figure 11 shows examples of the whole-body emotion expressions. External PC

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Figure 11. Examples of the whole-body emotion expression

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Discussion A modular design is a guidance to reduce costs for various product line developments. Naturally, module granularity depends on a desired product lineup. The modular design guidance of KOBIAN’s head is to adapt to various outer covering and easy to maintenance. At present, unfortunately, it has not been confirmed yet about the adaptation. We think this will be confirmed by collaborating with many appearance designers and making various types of outer covering. About the maintainability, it becomes easier rather than WE-4RII. In order to evaluate quantitatively, it is necessary to measure maintenance time. KOBIAN’s head is a module that provides a electric interface consist of motor speed commands and encoder pulses of 7-DOFs. However, we cannot assume it as the best modular design because it requires PID control of each motor to PC. A module that connection interface is completely free does not exist. A module is a thing comprising some kind of limited connection interface. In this sense, the performance of the modularization depends on availability and compatibility of the connection interface. On KOBIAN’s head, it can be assumed as more-modularized by using communication interface used well such as RS-232C, RS-485, Ethernet and USB instead of the interface that requires many lines such as motor speed commands and encoder pulses. Furthermore, availability will be improved by using not only speed commands but also commands for angles and angular accelerations in the protocol. So, we think it is better way to mount a processor which executes PID control in each module.

Conclusion and Future Works In this elderly-dominated society, Personal Robots and Robot Technology (RT)-based assistive devices are expected to play a major role, both for joint activities with their human partners and for participation in community life. So far, several different personal robots have been developed. However, it is not yet clear what kind of ability is necessary for personal robot. We think that emotional expression of robot is effective for joint activities of humans and robots. To this purpose we developed a whole body bipedal humanoid robot, named KOBIAN, which is also capable to express human-like emotions. This novel robot is based on the previously developed WABIAN-2 for the lower body, and on WE-4RII for the head. We consider an appearance of humanoid robot, particularly face, is that people either like it or don’t like it. It is inefficient to design a driving mechanism of robot head from the beginning in order to adapt to various preferences. Therefore, we conduct modular design of driving part and outer covering. And, modular design

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contributes to maintainability. In addition, we described the system configurations of KOBIAN, and discussed about modularizing and interface. In the future, we would like to collaborate with appearance designers, develop varies types of outer covering, compare the difference of the impressions on humans, and approach to clarification of designerÿs sense. In addition, we will improve the modular design of the humanoid robots in order to make them easier to adapt to various requirements and designs. Acknowledgement This research was commissioned by the New Energy and Industrial Technology Development Organization (NEDO) and conducted at Humanoid Robotics Institute, Waseda University. Special thanks to SolidWorks Japan K.K. for the software contribution, and KURARAY Co., Ltd. for the SEPTON contribution. This work was supported in part by Global COE Program "Global Robot Academia" from the Ministry of Education, Culture, Sports, Science and Technology of Japan. References Miwa, H., Okuchi, T., Takanobu, H. and Takanishi, A. (2002). Development of a new human-like head robot WE-4. Proceedinds of the IEEE/RSJ International Conference on Intelligent Robots and System, 2002, 2443-2448. Miwa, H., Itoh, K., Matsumoto, M., Zecca, M., Takanobu, H., Roccella, S., Carrozza, M. C., Dario, P., Takanishi, A. (2004). Effective Emotional Expressions with Emotion Expression Humanoid Robot WE-4RII. Proceedings of 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems, September 28 - October 2, 2004, Sendai, Japan, 2203-2208. Endo, N., Momoki, S., Zecca, M., Saito, M., Mizoguchi, Y., Itoh, K., and Takanishi, A. (2008). Development of Whole-Body Emotion Expression Humanoid Robot. Proceedings of the 2008 IEEE International Conference on Robotics and Automation, Pasadena, CA, USA, 2008, 2140-2145. DiSalvo, C., Gemperle, F., Forlizzi, J., and Kiesler, S. (2002). All robots are not created equal: the design and perception of humanoid robot heads. Proceedings of the conference on Designing interactive systems: processes, practices, methods, and techniques, 2002. Mori, M. (1970). Bukimi no tani [The Uncanny Valley]. Energy, vol. 7 (Originally in Japanese), 1970, 33-35. Zecca, M., Endo, K., Endo, N.,Mizoguchi, Y., Kusano, T., Itoh, K., and Takanishi, A. (2008). Design and Evaluation of The Soft Hand WSH-1 For The Emotion Expression Humanoid Robot KOBIAN. Proceedings of the 6th International Conference of the International Society for Gerontechnology, Pisa, Italy, June, 2008.

A Random-Profile Approach for Trajectory Planning of Wheeled Mobile Robots With non-Slipping Constraint M. Haddad and H.E. Lehtihet. Laboratory of Structure Mechanics EMP, BP 17 Bordj El-Bahri 16111, Algiers, Algeria. [email protected] [email protected]

1. Introduction Due to increased needs to improve the productivity in various industries, several methods have been proposed to solve the minimum-time trajectory-planning problem for a nonholonomic multi-body Wheeled Mobile Robot (WMR), subjected to move in a structured workspace under kinematics constraints (see, e.g., Jacobs et al., 1991, Reister and Pin, 1994, Renaud and Fourquet, 1997, Aydin and Temeltas, 2002, Balkcom and Mason, 2002). However, dynamics is also important, particularly if velocities are high or if it is needed to compute correctly inputs of the system control. For instance, most trajectory planners for WMR suppose that the friction coefficient (wheels /ground) is sufficiently high. In other words, it is assumed that the tangential contact force always remains inside the friction cone. However, typically in the case of minimum-time trajectories undertaken on a slippery floor, dynamic states may exist where such a hypothesis would not hold. When that happens, the wheel concerned by the violation will slip and the robot will subsequently deviate from its reference trajectory. Of course, such an undesirable behavior cannot be avoided adequately just by using kinematics. This can be properly handled only by incorporating an additional dynamics constraint in the trajectory-planning problem. Methods that handle dynamics without using any simplified model are quite few. Of interest here, is the Random-Profile Approach (RPA) that considers full dynamics and that is able to account for constraints on: – Geometry (obstacle avoidance, bounded path curvature / joint positions); – Kinematics (nonholonomic constraints, bounded velocities / accelerations); – Dynamics (bounded torques, stability issue). RPA is a flexible off-line planner that involves inverse dynamics only. Initially proposed for fixed-base manipulators (Chettibi and Lehtihet, 2002), it was later extended to WMRs (Haddad et al., 2005, 2007a) and to wheeled mobile manipulators (Haddad et al., 2006, 2007b, 2009).

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Our goal is to show that RPA is versatile enough to handle non-slipping restriction. Thus, we will be concerned mainly by this newly introduced constraint, as the main development of this method is already available in the above-cited 2007a reference.

2. Description of the problem Let ℜ = (O, x, y, z) be the fixed frame of the world coordinates. The robot considered here is a wheeled platform constituted of a rigid chassis supported by non deformable wheels. The platform type may belong to any of the nonholonomic WMRs discussed in Campion et al. (1996). We assume that the motion is confined to the (O, x, y) plan. We note ℜ′ = (O′, x′, y′, z′) the moving frame associated to the platform, such as z′ // z. The platform state in ℜ can be described by a vector X = [x, y, θ]T, where x, y are the coordinates of O′ in ℜ while θ is the orientation of the platform in ℜ. The generalized coordinates of the system are given by the n-vector q = [XT, qwT]T, where qw is the (n – 3)-vector of independent coordinates related to the rotation/orientation of the wheels in ℜ′. The WMR is to move from XSTART = [xS, yS, θS]T to XGOAL =[xG, yG, θG]T. The general trajectory problem consist of finding the trajectory q(t), the time history of vector Γ(t) of actuator efforts and the travel time T so that boundary conditions are matched, all constraints are respected and a given cost function J is minimized (here, we consider only the minimum-time problem, so that: J ≡ T). The set of feasible trajectories may be restricted by numerous constraints that must be satisfied during the travel from X START to XGOAL. Boundary conditions (1a) – Position/Orientation X(t = 0) = X&START and X(t = T) = &XGOAL q (t = 0) = 0 and q (t = T) = 0 (1b) – Velocity Physical limitations The following constraints on active-joint torques must be verified for t ∈ [0, T]: Γi (t ) ≤ Γimax

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All of the above restrictions (1), including nonholonomic constraints and collision issues, are already handled by the basic RPA scheme (see Haddad et al., 2007a). Non-slipping constraint Such a constraint is defined according to Coulomb’s law as follows: Fti ≤ η i = 1, ... nw with Fti = Fxi2 + Fyi2 (2) Fzi Here Fxi, Fyi and Fzi are the components of the contact force of the ith wheel with the ground. These efforts can be easily calculated by Newton-Euler formalism. nw is the number of wheels. η is the friction coefficient of the wheels/ground contact.

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3. Proposed method 3.1. Overview of the basic RPA scheme First, RPA treats the problem without integrating the set of differential equations governing the behavior of the robot. Indeed, RPA considers that T and q(t) are the only unknowns of the problem. Once these unknowns are found, the vector Γ(t) of generalized efforts is easily deduced via inverse dynamics. Second, RPA builds time-scaled trajectory profiles to map real trajectories over the unit interval. Indeed, any given trajectory q(t) can be uniquely characterized by its travel time T and its time-evolution shape Q: q(t) = Q(ξ(t)) = Q(ξ) o ξ(t)

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The trajectory profile Q(ξ) gives the shape of the time history of q from the start of the motion (ξ = 0) to its end (ξ = 1). Hereafter, the prime symbol will be reserved to indicate derivatives with respect to ξ. We note that any given profile Q uniquely defines a whole class of trajectories. All member q of a given class Q will share the same time-evolution shape but will have distinctive travel times T. Third, all constraints of the problem can be grouped into two categories: class-related and member-related constraints. The former depend solely on the profile Q whereas the later further depend on the travel time T. For example, obstacle-avoidance constraints and other geometric constraints never depend on the travel time. They belong therefore to the first category. It can be shown that, for a given Q, (in other words, within a given class), all member-related constraints, such as those on kinematics and dynamics, can be converted to bounds on admissible values of T. Fourth, given a class Q, we can easily extract the specific travel time TQ that will distinguish the best member q within this selected class while taking into account all constraints. Hence, the problem of finding the overall best trajectory is converted to finding the class to which belongs this desired trajectory. In other words, RPA is a method that probes the solution space directly at the class level. This problem is solved via a nested master/optimization whose final output is the best member of the best class, among those classes that would have been tested. The slave routine is a simple deterministic optimizer that actually plays the role of a workhorse fitness function evaluating any input class and extracting its best member. The master routine is a stochastic optimizer that plays the role of a class selector. It is based on a simulated-annealing strategy (Hajek, 1985) that randomly probes the class space to propose candidate profile for evaluation. Its build-in Metropolis algorithm gradually targets those promising candidate that would most likely hold the optimal trajectory.

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Naturally, this difficult functional optimization is converted to a more tractable parametric problem via a discrete representation of trajectory profiles. Namely, each random profile is built from a finite set of free control nodes; continuity to a sufficient order is achieved by fitting a smooth model that accounts for boundary conditions and nonholonomic constraints. These free control nodes being the only unknowns, the whole problem boils down to finding their optimal setting. Since a trajectory may be written in terms of a geometric path and a motion on this path, it is then convenient to express a trajectory profile as follows: Q(ξ) = P(λ(ξ)) = P(λ) o λ(ξ)

(4)

P(λ) is a time-independent vector with same dimension as that of Q. It describes the geometric path pursued by the robot in the generalized-coordinate space when λ varies continuously from 0 (start) to 1 (end). The other form, (), is the motion profile. It is a monotonically increasing scalar function that describes how the robot actually moves on its path P. This partition path/motion of a trajectory profile Q greatly increases the flexibility of RPA. A random Q now can be built using two distinct sets of control nodes on two tailored fitting models: a cubic-spline model for () and an 4th-order B-spline model for P(λ) (to insure afterward the continuity of torques). Moreover, geometric constraints and boundary conditions now can be distributed conveniently. For instance, () will be concerned by boundary conditions on velocity; whereas P(λ) will be concerned by boundary conditions position/orientation, obstacle-avoidance and nonholonomic constraints. These constraints are handled either, a posteriori, by rejection or, a priori, by inclusion (Haddad et al., 2007a). The rejection is used exclusively for obstacle avoidance whereas the inclusion is used to handle nonholonomic constraints and boundary conditions.

3.2. Treatment of the non-slipping constraint In the RPA framework, given a profile Q, all kinodynamic constraints such as (1c) simply translate to bounds on admissible values of T (Haddad et al., 2007a). T ∈ [TL, TR]

(5)

The non-slipping constraint (2) is converted also into a set of subintervals of admissible values for T. Indeed, components Fxi, Fyi and Fzi can be written as follows: ⎧ Fxi = D xi (Q, Q′, Q′′) T −2 + H xi (Q) ⎪ −2 ⎨ Fy i = D yi (Q, Q′, Q′′) T + H yi (Q) ⎪ F = D (Q, Q′, Q′′) T − 2 + H (Q) zi zi ⎩ zi

(6)

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In the above expressions, the first terms (D) represent the effect of inertial, Coriolis and centrifugal forces while the second (H) regroups the effect of gravitational forces. Using (6), the constraint (2) may be rewritten under the following form: A0i + A2i T −2 + A4i T −4 ≤ 0

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)

A0i = H xi2 + H yi2 −η 2 H zi2 , A2i = 2 Dxi H xi + Dyi H yi −ηDzi H zi , A4i = Dxi2 + Dyi2 −η 2 Dzi2

Hence, each inequality in (7) may lead to, at most, two distinct intervals of admissible values of T: (8) T ∈ [TL1i, TR1i] ∪ [TL2i, TR2i] However, intersecting (5) and (8) for ξ in [0, 1] may yield to a set of k distinct windows: ∏ = [T1, T2] ∪ [T3, T4]∪…[T2k-1, T2k]. Consequently, for any trial Q, the optimal travel TQ must minimize J within ∏. Let Tm be the travel time minimizing the cost function JQ(T), for a given Q(ξ), without considering kinodynamic constraints. The value of Tm can be calculated easily via the minimization of a one-dimensional function (Haddad et al., 2007a). Therefore, the optimal TQ minimizing JQ(T) under kinodynamic constraints is deduced according to the position of Tm in the admissible interval ∏ : ⎧T TQ = ⎨ m ⎩T j /

if Tm ∈ ∏ J Q (T j ) = min (J Q (Ti ) for i = 1 2k ) if Tm ∉ ∏

(9)

4. Numerical example We consider the WMR described in Table 1 and in Figure 1. This robot has two independent driving wheels and two free caster wheels. Contact forces of the driving wheels with the ground are given by the following: Izr 1 ⎧ ⎪ Fx1 = 2T 2 (mr v′(ξ) + L θ′′(ξ)) w ⎪ mr v(ξ)θ′(ξ) ⎪ ⎨ Fy1 = 2T 2 ⎪ ⎪F = − rw (m + 2Iyw )v(ξ)θ′(ξ) + mr g ⎪ z1 2LwT 2 r 4 rw2 ⎩

Izr 1 ⎧ ⎪ Fx 2 = 2T 2 (mr v′(ξ) − L θ′′(ξ)) w ⎪ mr v(ξ)θ′(ξ) ⎪ , ⎨ Fy 2 = 2T 2 ⎪ ⎪F = rw (m + 2Iyw )v(ξ)θ′(ξ) + mr g ⎪ z 2 2LwT 2 r 4 rw2 ⎩

v(ξ) = x′(ξ) cos(θ(ξ)) + y′(ξ) sin(θ(ξ)) ,

with mr = m p + 2mw ,

Izr = Iz p + 2( Iz w + m L ) , v′(ξ) = x′′(ξ) cos(θ(ξ)) + y ′′(ξ) sin( θ(ξ)) . 2 w w

The workspace is a (24m×21m) flat floor with three obstacles (Figure 2a). Boundary conditions are as follows: XSTART = (3, 3, 0) and XGOAL = (22, 12, 0). The following results have been obtained using our RPA code running on a 1.6 GHz

478

M. Haddad and H.E. Lehtihet

Intel Centrino Duo PC. First, we solve the minimum-time trajectory problem under maximum-torque constraints but without considering non-slipping constraints. The solution is shown in Figure 2 with a score T = 8.79 s and a runtime of 63 s. The final path is given in Figure 2a. The time history of velocities, active-joint torques and ratio of contact forces is given, respectively, in Figure 2b, 2c and 2d. This result clearly indicates that such a solution is actually not feasible as the ratio of contact forces well exceeds the threshold value η = 0.7. Table 1. Parameters of the WMR.

Lw = 0.75 m

mp = 50.0 kg

Izp = 26.04 kg.m²

Γ1max = 5 Nm

Lv = 2.00 m

mw = 1.0 kg

Izw = 0.0025 kg.m²

Γ2max = 5 Nm

Iyw = 0.005 kg.m²

η = 0.7

rw = 0.10 m

The platform is a (2Lw×Lv) rectangle. The center of gravity is in the middle of the wheel axis. The inertia moment of the platform is considered only around the z’-axis. rw is the wheel radius. mp and mw are, respectively, the mass of the platform and that of the wheels. Izp is the inertia moment of the platform about the z′–axis. Izw and Iyw are the inertia moment of the wheels about the z′–axis and the y′–axis (passing through the wheel center). The friction coefficient η is the same for the two driving wheels. Inertia parameters of the free caster wheels are negligible.

We solve again the same problem including now the non-slipping constraint. The new solution is given in Figure 3 with a score T = 9.36 s and a runtime of 88 s. As shown in Figures 3c and 3d, both types of constraints are fully respected during the motion. As expected, the robot is compelled to decrease its speed significantly around the corner. However, RPA is a sufficiently flexible optimization technique that is able to compensate for this drawback by finding a new shorter path, now accessible to the robot due to its lower speed x'

rw z'

y'

Lv Lw

free caster wheel

Figure 1. Wheeled mobile robot

T = 9.36 s

G

G

q1 q 2

t[s] Γ[Nm]

Γ1 Γ2 t[s]

Ft /Fz

Ft1/Fz1 Ft2/Fz2

t[s]

Figure 2. Results obtained without considering non-slipping constraints

(b) Active-joint torques

q [rd/s]

(b) Active-joint velocities

S

(c) Ratio of contact forces

(c) Active-joint torques

(b) Active-joint velocities

S

(c) Ratio of contact forces

479

T = 8.79 s

(a) path

(a) path

A Random-Profile Approach… with non-Slipping Constraint

q [rd/s]

q1 q 2

t[s] Γ[Nm]

Γ1 Γ2 t[s]

Ft /Fz

Ft1/Fz1 Ft2/Fz2

t[s]

Figure 3. Results obtained while considering non-slipping constraints

480

M. Haddad and H.E. Lehtihet

5. Conclusion We have proposed an extension of the random-profile approach to handle a trajectory-planning problem for a wheeled mobile robot while considering a non-slipping constraint. This constraint complicates the problem due to the non-linearity of the forces at the wheels/ground contact. Our approach is based on the simultaneous search for the robot path and for the time history on this path. In this approach, the issue of non-slipping has been treated by transforming this constraint to a set of admissible subintervals for the travel time.

6. References Aydin, S., and Temeltas, H. (2002). A novel approach to smooth trajectory planning of a mobile robot. IEEE Int. Workshop on Advanced Motion Control, 472–477. Balkcom, D. J., and Mason, M. T. (2002). Extremal trajectories for bounded velocity mobile robots. In Proc. of IEEE Int. Conf. on Robotics & Automation. Campion, G., Bastin, G., and d’Andrea-Novel, B. (1996). Structural properties and classification of kinematic and dynamic Models of wheeled mobile robots. IEEE Transactions on Robotics and Automation, Vol. 12, No.1, 47–62. Chettibi, T., and Lehtihet, H.E. (2002). A new approach for point-to-point optimal motion planning problems of robotic manipulators. In Proc. 6th Biennial Conf. on Engineering Systems Design and Analysis, APM10. Haddad, M., Chettibi, T., Lehtihet, H. E., and Hanchi, S. (2005). A new approach for minimum-time motion planning problem of wheeled mobile robots. 16th IFAC World Congress, Prague. Haddad, M., Chettibi, T., Saidouni, T., Hanchi, S., and Lehtihet, H. E. (2006). Sub-optimal motion planner of mobile manipulators in generalized point-to-point task with stability constraint. In Romansy 16 Robot Design, Dynamics and Control, edited by Teresa Zielinska and Cezary Zielinski, Springer Wien New York, 171–178. Haddad, M., Chettibi, T., Hanchi, S., and Lehtihet, H. E. (2007a). A random-profile approach for trajectory planning of wheeled mobile robots. European J. of Mechanics A/Solids, Vol. 26, 519–540. Haddad, M., Hanchi, S., and Lehtihet, H. E. (2007b). Sub-optimal motion planner of a mobile manipulator with end-effector’s specified path. IFToMM World Congress. Haddad, M., Hanchi, S., and Lehtihet, H. E. (2009). Point-to-point trajectory planning of wheeled mobile manipulators with stability constraint. Extension of the random-profile approach. European J. of Mechanics, A/Solid, Vol. 28, 477–493. Hajek, B. (1985). A tutorial survey of theory and application of simulated annealing. In Proceeding of 24th Conference on Decision and Control, 755–760. Reister, D. B., and Pin, F. G. (1994). Time-optimal trajectories for mobile robots with two independently driven wheels. Int. J. of Robotic Research, Vol.13, No.1, 38–54. Renaud, M., and Fourquet, J. Y. (1997). Minimum-time motion of a mobile robot with two independent acceleration-driven wheels. IEEE Int. Conf. on Rob. Aut. NM, 2608–2613.

The Second Type of Singularity Analysis of the New Parallel Manipulator with 6 Degree-of-Freedom Zh. Baigunchekov, M. Izmambetov Kazakh-British Technical University, Almaty, Republic of Kazakhstan Abstract - The analytical method of the second type of singularity analysis of the new parallel manipulator with 6 degrees-of-freedom is worked out on the bases of Jacoby matrix connecting output and input generalized velocities.

Introduction As Merlet (2000) has noted, parallel manipulators (PM) in comparison with serial manipulators possess have big carrying capacity, high velocities and positioning accuracy. These advantages motivated the development of the new six-legged PM with six degree of freedom (DOF) by Baigunchekov et al. (2007, 2009). However, PM have many singularity configurations in which they cannot carry out their functional tasks. Gosselin and Angeles (1990), Tsai (1999) describe three types of singularity of PM. The Jacobian of a PM can be divided into two matrix: one matrix associated with the direct kinematics and the other matrix associated with the inverse kinematics. In the first type of singularity, when det B=0, a manipulator loses 1 or more DOF. In the second type of singularity, when det =0, a manipulator gains 1 or more DOF. In the third type of singularity the inverse and direct kinematic singularities are presented simultaneously. We (2010) obtained some results for identification of the first type of singularities, and also one of conditions of occurrence of the second type of singularities of the new PM with 6 DOF. In this paper we have presented results of more full and detailed analysis of the second type of singularities of this PM based on research of degeneration of Jacobian matrix A because of a linear dependence of its columns.

Geometry of the PM with 6 DOF The PM with 6 DOF contains a moving platform, connected with a fixed base by six legs of kind RSS, where R - a revolute active-joint, S - a spherical joint (Figure 1). This PM is intended for reproduction of movement of a mobile platform or the frame PxPyPzP attached to it, with respect to base frame OXYZ X P = X P (q(t )), YP = YP (q(t )), Z P = Z P (q(t ))⎫ ⎬

γ P = γ P (q(t )), α P = α P (q(t )), β P = β P (q(t )) ⎭

(1)

482

Zh. Baigunchekov and M. Izmambetov

Figure 1. Six-legged parallel manipulator with 6 DOF.

where q(t)=[θ1(t), θ2(t),…, θ6(t)]T - a vector of the input generalized coordinates; x(t)=[X, Y, Z, γP, αP, βP]T - a vector of the output coordinates; γP, αP βP - the components of relative orientation of coordinates systems PxPyPzP and OXYZ. Taking into structure symmetry of the PM with 6 DOF, all geometrical parameters are shown on the i-th leg OiAiBiP (Figure 2). These geometrical parameters are determined on basis of the Denavit-Hartenberg Homogeneous Transformation Matrix: lOi Ai = f i , l Ai Bi = g i - lengths of actuated and passive B

links, respectively; a0,i , b0,i , c0,i ,α 0,i , β 0,i , γ 0,i - parameters determining frame Oi xi yi zi attached to a motionless element of revolute active-joint i with respect to frame OXYZ; l PBi = hi , ϕi - polar coordinates of spherical joints Bi with respect to

local frame PxPyPzP. ψi, - a constant angle between link OiAi and an axis Oizi. Let the relative positioning of local frame PxPyPzP with respect to base frame OXYZ is defined by a set of parameters a0, P , b0, P , c0, P , α 0, P , β 0, P , γ 0, P , and the vector g i = Ai Bi is defined by the spherical coordinates gi , θ i' , θ i'' . Then for PM the following equation is fair A (x, q)x = B(x, q)q ,

(2)

where A and B - 6 × 6 Jacobian matrices depending only from a configuration of

[

PM; x = r PT , TP

]

T

[

- a vector of output velocities; q = θ1 , θ2 ,..., θ6

[

of the generalized velocities; rP = X P , YP , Z P

]

T

]T

- a vector

- a vector of velocity of the point ;

The Second Type of Singularity Analysis of the New Parallel…

483

Figure 2. Geometry of the i-th leg of the PM.

[

]

P = ω PX , ω PY , ω PZ T - a vector of angular velocities of a moving platform.

Elements aij and bij of matrices A and B are determined through constant geometrical and variable kinematic parameters of PM by the following equations

{

ai1 = g i ⋅ e X = g iX = g i cθ i"c(θ i + θ i' )(cγ 0i cβ 0i − sγ 0i cα 0i sβ 0i ) −

}

− cθ i" s (θ i + θ i' )(cγ 0i sβ 0i + sγ 0i cα 0i cβ 0i ) + sθ i" sγ 0i sα 0i , ai 2 = g i ⋅ eY = g iY =

{

g i cθ i"c(θ i

+ θ i' )(sγ 0i cβ 0i

+ cγ 0i cα 0i sβ 0i ) +

}

+ cθ i" s(θ i + θ i' )(cγ 0i cα 0i cβ 0i − sγ 0i sβ 0i ) − sθ i"cγ 0i sα 0i ,

{

ai3 = gi ⋅ e Z = giZ = gi cθ i"c(θ i + θ i' ) sα0i sβ 0i ) − − cθ i" s (θ i + θ i' ) sα 0i cβ 0i − sθ i" cα 0i

(3)

ai 4 = (h i × g i ) ⋅ e X = niX = hiY g iZ − hiZ g iY a i 5 = (h i × g i ) ⋅ e Y = niY = hiZ g iX − hiX g iZ ai 6 = (h i × g i ) ⋅ e Z = niZ = hiX g iY − hiY g iX

(4)

bii = e Ti ⋅ (f i × g i ) = f i ⋅ g i ⋅ sin ψ i cos θ i'' sin θ i' bij = 0, for i ≠ j , i = 1, 2, ... , 6 ; j = 1, 2, ... , 6 ,

(5)

},

484

Zh. Baigunchekov and M. Izmambetov

where e X , e Y , e Z - unit vectors indicating the direction of axes OX, OY and OZ, respectively; f i = Oi Ai , h i = PB i ; components of hi are determined by equations hiX = hi {(cγ 0 P cβ 0 P − sγ 0 P cα 0 P sβ 0 P ) cos ϕi −

− (cγ 0 P sα 0 P + sγ 0 P cα 0, P cβ 0 P ) sin ϕi },

hiY = hi {( sγ 0 P cβ 0 P + cγ 0 P cα 0 P sβ 0 P ) cos ϕi +

+ (cγ 0 P cα 0 P cβ 0 P − sγ 0 P sβ 0 P ) sin ϕi },

hiZ = hi {sα 0 P sβ 0 P cos ϕi + sα 0 P cβ 0 P sin ϕi } .

Singularity Analysis of the PM With 6 DOF The first type of singularity of PM is associated with degeneration of matrix B, and the second type of singularity is associated with degeneration of a matrix A, and it is located in inside of working space. There are nonzero output velocities x of the platform for a corresponding configuration of the PM, and they are transformed in zero vector by matrix A. These velocities of the platform are possible, even at immovable drives of input joints. We shall write matrix A in the following form ⎡ g1 X ⎢ g A = ⎢ 2X ⎢ ⎢ ⎣g6X

g1Y g 2Y

g1Z g 2Z

n1 X n2 X

n1Y n 2Y

g 6Y

g 6Z

n6 X

n6Y

n1Z ⎤ ⎥ n2 Z ⎥ , ⎥ ⎥ n6 Z ⎦

(6)

where elements of first three columns are components of vectors gi, and elements of last three columns are components of vectors n i = h i × g i . Degeneracy of a matrix A in consequence of linear dependence of arbitrary two lines with numbers j and k have considered Baigunchekov et al. (2010). One of conditions of occurrence of the second type of singularity configuration is established. According to this condition passive links of PM corresponding to two linearly dependent lines of a Jacobian A attain layout along a direct line which passes through attachment points of these links with a mobile platform. In the given paper we shall consider linear dependence of columns of a matrix A. As against line vectors, each column vector is defined by a set separate coordinates of vectors of one group consisting either from vectors g i , or n i = (h i × g i ) i = 1,2,...,6 . The following three cases of degeneration of matrix A are possible: Case 10. three columns of one vectors group are linearly dependent. Case 20. two columns of one vectors group are linearly dependent. Case 30. all vectors of one vectors group are zero vectors.

The Second Type of Singularity Analysis of the New Parallel…

485

Let's preliminary accept without the proof the following theorem. The theorem. That two vectors laying in various and mutually crossed planes, were parallel, it is necessary and enough that each of these vectors was parallel to a straight line being intersection of their planes. 10. Three columns of one vectors group are linearly dependent. This case means that all vectors of this group are parallel. Let's consider a case when all vectors ni will be parallel, i.e., last three columns of matrix A will be linearly dependent. Fairly following statement. The statement 1. Parallelism of all vectors ni i = 1,2,...,6 , is possible in that, and only in the event that each of these vectors is parallel to axis PzP of coordinates system PxPyPzP connected to a mobile platform. The proof of this statement is based on the above mentioned theorem. In turn given statement means that in this case, according to definition of vector multiplication, vectors gi, i = 1,2,...,6 should lay in a plane of a mobile platform. Thus, we find one of conditions of degeneracy of A: z-components of vectors gi with respect to local frame PxPyPzP should have zero values, i.e. g iz = 0 , i = 1,2,...,6

(7)

We can obtain a kind of the equations (2) by transformation of coordinates it is possible concerning the coordinates system PxPyPzP connected to a platform. Thus for a considered Case 10 the third, fourth and fifth columns of matrix A will be zero. Then the null-space of A is generated by a vector [0, 0, 1, 1, 1, 0]T. This null-space corresponds to set of local rotations of a platform about an arbitrary axis of plane PxPyP and to its local linear moving in a direction of axis PzP at motionless drives. I.e. the PM gains the three transitory additional DOF. Moreover, a force applied at the platform in a direction of axis PzP, and also a forces couple applied at the platform in a arbitrary plane of parallel axis PzP do not influence drives, i.e. the PM is not capable to withstand these loadings. 20. Two columns of one vectors group are linearly dependent. This case means that all of a vector of this group lay in parallel planes or are simultaneously parallel to one common plane. The statement 2. Vectors n i = (h i × g i ) i = 1,2,...,6 , can be simultaneously parallel to any plane. Let's accept such statement without the proof. Not losing a generality of received results, we shall consider a special case when these vectors are parallel to a plane of platform PxPyP. As vectors ni free, we admit, that ni ∈ PxPyP, i = 1,2,...,6 . Then, for the sixth column of matrix A composed concerning frame PxPyPzP niz = 0 , i = 1,2,...,6

(8)

486

Zh. Baigunchekov and M. Izmambetov

Let's define a plane Γi , as a plane containing points P, Ai and Bi (Figure 3). Then in view of that n i ∈ Px P y P , we shall obtain g i ∈ Γi ⇒ Γi ⊥Px P y P , ⇒ Pz P ∈ Γi i = 1,2,...,6

(9)

⎧1) (g i ) Pz P = Pi , Pz P ∈ Γi ⇒ ⎨ ⎩2) (g i ) | | Pz P ,

(10)

whence follows g i ∈ Γi ,

i = 1,2,...,6

where through (gi) the straight line containing a vector gi is denoted. Thus, at realization of a condition (8) links AiBi lay in the planes containing axis PzP. Moreover, cases when all lines of links AiBi either intersect axis PzP in corresponding points Pi or all these lines become parallel to axis PzP are possible. In the first case (Figure 3a), local rotation of a platform about an axis PzP takes place. This is so because according to (8) the last column of the A is equal to zero, and hence, the null-space of matrix A is spanned by [0, 0, 0, 0, 0, 1]T. This null-space corresponds to set of rotations of the platform about an axis PzP at motionless drives, i.e. PM gains the one additional DOF. This time the PM will not be able to withstand a torque applied at the platform in parallel to axis PzP. In the second case (Figure 3b), from (g i ) | | Pz P , i = 1,2,...,6 follows g ix P = 0, B

B

g iy P = 0 , i.e. except for (8) we have also the first and second columns of A with

zero elements. Then the null-space of A is spanned by [1, 1, 0, 0, 0, 1]T that corresponds here to set of rotations of a platform about an axis PzP and to local moving on plane PxPyP at motionless drives. Moreover, force and a forces couple applied at the platform in parallel to plane of a platform do not influence drives.

Figure 3. The second type of singularity of the PM.

The Second Type of Singularity Analysis of the New Parallel…

487

As position of pole of a platform is selected arbitrarily, and also vectors ni can be parallel to any plane in space the considered case of singularity takes place every time when all lines of links AiBi of the PM simultaneously intersect arbitrary, but the same straight line in space, or become parallel to this straight line. 30. All vectors of one group of vectors are zero. Obviously, any of vectors gi cannot be zero. And vectors ni, describing a relative positioning of a mobile platform and links AiBi, can be simultaneously zero, i.e. B

B

n i = (h i × g i ) = 0 , i = 1,2,...,6

(11)

Since hi ≠ 0, g i ≠ 0 , from (11) it is obtained that h i | | g i , i.e. lines of vectors gi converge in pole P (Figure 4). However, as the position of pole of a mobile platform is selected arbitrarily such singularity of a configuration occurs at any convergence of lines of passive links with lengths g i .

Figure 4. The second type of singularity of the PM.

488

Zh. Baigunchekov and M. Izmambetov

In a case when all vectors gi, i = 1,2,...,6 lay in a platform plane (Figure 4a), using an invariant kind of the equations (2) with respect to frame PxPyPzP, it is possible to show, that the third column of matrix A also will be zero. Then the null-space of matrix A is generated by a vector [0, 0, 1, 1, 1, 1]T. This null-space corresponds here to set of local rotations of a platform about its centroid and to local moving in a direction of axis PzP at motionless drives. Moreover, a force parallel to axis PzP, and also the any torque applied on a platform do not influence drives, i.e. the manipulator is not capable to counteract these loadings. If all lines of passive links converge outside of a platform plane local spherical movability takes place (Figure 4b). This is so because pole P of a platform can be moved to a point of a convergence of lines of links AiBi on axis PzP. Then in the corresponding Jacobian A constructed concerning a local frame with a new pole only the last three columns will be zero. The null-space of matrix A is generated by a [0, 0, 0, 1, 1, 1]T that corresponds to local spherical movability about a pole P. B

Conclusion Research of degeneracy of Jacoby matrix A of the PM with six DOF in consequence of linear dependence of its columns - vectors is carried out. Corresponding conditions of the second type singularity of configurations of the PM are obtained; their geometrical and physical interpretations are given. The obtained analytical conditions of singularities of a configuration can be expressed through constants and variable parameters of the manipulator that is important for the control of such configurations. The further publication of results of research of singularity configurations of the third type of the manipulator is planned.

References J.P. Merlet, 2000, Parallel Robots, Kluwer Academic Publishers, London. Zh. Baigunchekov, M. Ceccarelli, B. Nurakhmetov, N. Baigunchekov, Parallel Manipulator, Patent No. 20727, Republic of Kazakhstan, Bulletin No. 2, 2009. Zh. Baigunchekov, B. Nurakhmetov, B. Absadykov, K. Sartaev, M. Izmambetov, N. Baigunchekov "The New Parallel Manipulator with 6 Degree-of-Freedom", Proceedings of 12th IFToMM World Congress in Mechanism and Machine science, June 17-21, 2007, Besançon, France, Vol. 5, pp. 641-646. L.W. Tsai, 1999, Robot analysis: the mechanics of serial and parallel manipulators, John Wiley & Sons, Inc., New York. C. Gosselin and J. Angeles, 1990, "Singularity analysis of closed-loop kinematic chains," IEEE Trans. Robotics & Autom., Vol. 6, No. 3, pp. 281-290. Zh. Baigunchekov, I. Beisembetov, M. Izmambetov, 2010, "Singularity Analysis of the New Parallel Manipulator with 6 Degrees-of-Freedom ", ICME 2010, 30 June – 2 July, London, it is presented.

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