ADVANCES IN ROBOT KINEMATICS Advances in Robot Kinematics Edited by I Jo ef Stefan Institute Ljubljana, Slovenia and B. ROTH Stanford University California, U.S.A. Mechanisms and Motion ý ý ž JADRAN LENAR A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN-10 1-4020-4940-4 (HB) ISBN-13 978-1-4020-4940-8 (HB) ISBN-10 1-4020-4941-2 (e-book) ISBN-13 978-1-4020-4941-5 (e-book) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com Printed on acid-free paper All Rights Reserved © 2006 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands. Preface This is the tenth book in the series of Advances in Robot Kinematics. Two were produced as workshop proceedings, Springer published one book in 1991 and since 1994 Kluwer published a book every two years without interruptions. These books deal with the theory and practice of robot kinematics and treat the motion of robots, in particular robot manipulators, without regard to how this motion is produced or con- trolled. Each book of Advances in Robot Kinematics reports the most recent research projects and presents many new discoveries. The issues addressed in this book are fundamentally kinematic in nature, including synthesis, calibration, redundancy, force control, dex- terity, inverse and forward kinematics, kinematic singularities, as well as over-constrained systems. Methods used include line geometry, quater- nion algebra, screw algebra, and linear algebra. These methods are ap- plied to both parallel and serial multi-degree-of-freedom systems. The en application. All the contributions had been rigorously reviewed by independent reviewers and fifty three articles had been recommended for publica- tion. They were introduced in seven chapters. The authors discussed their results at the tenth international symposium on Advances in Robot Kinematics which was held in June 2006 in Ljubljana, Slovenia. The symposium was organized by Jozef Stefan Institute, Ljubljana, under the patronage of IFToMM - International Federation for the Promotion of Mechanism and Machine Science. We are grateful to the authors for their contributions and for their efficiency in preparing the manuscripts, and to the reviewers for their timely reviews and recommendations. We are also indebted to the per- sonnel at Springer for their excellent technical and editorial support. Jadran Lenarˇciˇc and Bernard Roth, editors results should interest researchers, teachers and students, in fields of gineering and mathematics related to robot theory, design, control and Contents Methods in Kinematics J. Andrade-Cetto, F. Thomas Wire-based tracking using mutual information 3 G. Nawratil The control number as index for Stewart Gough platforms 15 C. Innocenti, D. Paganelli Determining the 3×3 rotation matrices that satisfy three linear equations in the direction cosines 23 P.M. Larochelle A polar decomposition based displacement metric for a finite region of SE(n)33 J P. Merlet, P. Donelan On the regularity of the inverse Jacobian of parallel robots 41 P. Fanghella, C. Galletti, E. Giannotti Parallel robots that change their group of motion 49 A.P. Murray, B.M. Korte, J.P. Schmiedeler Approximating planar, morphing curves with rigid-body linkages 57 M. Zoppi, D. Zlatanov, R. Molfino On the velocity analysis of non-parallel closed chain mechanisms 65 Properties of Mechanisms H. Bamberger, M. Shoham, A. Wolf Kinematics of micro planar parallel robot comprising large joint clearances 75 H.K. Jung, C.D. Crane III, R.G. Roberts Stiffness mapping of planar compliant parallel mechanisms in a serial arrangement 85 Y. Wang, G.S. Chirikjian Large kinematic error propagation in revolute manipulators 95 A. Pott, M. Hiller A framework for the analysis, synthesis and optimization of par allel kinematic machines 103 Z. Luo, J.S. Dai Searching for undiscovered planar straight-line linkages 113 X. Kong, C.M. Gosselin Type synthesis of three-DOF up-equivalent parallel manipula tors using a virtual-chain approach 123 A. De Santis, P. Pierro, B. Siciliano The multiple virtual end-effectors approach for human-robot in teraction 133 Humanoids and Biomedicine J. Babiˇc, D. Omrˇcen, J. Lenarˇciˇc Balance and control of human inspired jumping robot 147 J. Park, F.C. Park A convex optimization algorithm for stabilizing whole-body mo tions of humanoid robots 157 R. Di Gregorio, V. Parenti-Castelli Parallel mechanisms for knee orthoses with selective recovery action 167 S. Ambike, J.P. Schmiedeler Modeling time invariance in human arm motion coordination 177 M. Veber, T. Bajd, M. Munih Assessment of finger joint angles and calibration of instrumental glove 185 R. Konietschke, G. Hirzinger, Y. Yan All singularities of the 9-DOF DLR medical robot setup for min imally invasive applications 193 G. Liu, R.J. Milgram, A. Dhanik, J.C. Latombe On the inverse kinematics of a fragment of protein backbone 201 V. De Sapio, J. Warren, O. Khatib Predicting reaching postures using a kinematically constrained shoulder model 209 viii Contents Analysis of Mechanisms D. Chablat, P. Wenger, I.A. Bonev Self motions of special 3-RPR planar parallel robot 221 A. Degani, A. Wolf Graphical singularity analysis of 3-DOF planar parallel manip ulators 229 C. Bier, A. Campos, J. Hesselbach Direct singularity closeness indexes for the hexa parallel robot 239 A. Karger Stewart-Gough platforms with simple singularity surface 247 A. Kecskem´ethy, M. T¨andl A robust model for 3D tracking in object-oriented multibody systems based on singularity-free Frenet framing 255 P. Ben-Horin, M. Shoham Singularity of a class of Gough-Stewart platforms with three con current joints 265 T.K. Tanev Singularity analysis of a 4-DOF parallel manipulator using geo metric algebra 275 R. Daniel, R. Dunlop A geometrical interpretation of 3-3 mechanism singularities 285 Workspace and Performance J.A. Carretero, G.T. Pond Quantitative dexterous workspace comparisons 297 E. Ottaviano, M. Husty, M. Ceccarelli Level-set method for workspace analysis of serial manipulators 307 M. Gouttefarde, J P. Merlet, D. Daney Determination of the wrench-closure workspace of 6-DOF paral lel cable-driven mechanisms 315 G. Gogu Fully-isotropic hexapods 323 P. Last, J. Hesselbach A new calibration stategy for a class of parallel mechanisms 331 M. Krefft, J. Hesselbach The dynamic optimization of PKM 339 . Contents ix - J.A. Snyman On non-assembly in the optimal synthesis of serial manipulators performing prescribed tasks 349 Design of Mechanisms W.A. Khan, S. Caro, D. Pasini, J. Angeles Complexity analysis for the conceptual design of robotic archi tecture 359 D.V. Lee, S.A. Velinsky Robust three-dimensional non-contacting angular motion sensor 369 K. Brunnthaler, H P. Schr¨ocker, M. Husty Synthesis of spherical four-bar mechanisms using spherical kine matic mapping 377 R. Vertechy, V. Parenti-Castelli Synthesis of 2-DOF spherical fully parallel mechanisms 385 G.S. Soh, J.M. McCarthy Constraint synthesis for planar n-R robots 395 T. Bruckmann, A. Pott, M. Hiller Calculating force distributions for redundantly actuated tendon 403 P. Boning, S. Dubowsky A study of minimal sensor topologies for space robots 413 M. Callegari, M C. Palpacelli Kinematics and optimization of the translating 3-CCR/3-RCC parallel mechanisms 423 Motion Synthesis and Mobility C C. Lee, J.M. Herv´e Pseudo-planar motion generators 435 S. Krut, F. Pierrot, O. Company On PKM with articulated travelling-plate and large tilting angles 445 C.R. Diez-Mart´ınez, J.M. Rico, J.J. Cervantes-S´anchez, J. Gallardo Mobility and connectivity in multiloop linkages 455 K. Tcho´n, J. Jakubiak Jacobian inverse kinematics algorithms with variable steplength for mobile manipulators 465 x -based Stewart platforms Contents J. Zamora-Esquivel, E. Bayro-Corrochano Kinematics and grasping using conformal geometric algebra 473 R. Subramanian, K. Kazerounian Application of kinematics tools in the study of internal mobility of protein molecules 481 O. Altuzarra, C. Pinto, V. Petuya, A. Hernandez Motion pattern singularity in lower mobility parallel manipula tors 489 Author Index 497 Contents xi Methods in Kinematics J. Andrade-Cetto, F. Thomas Wire-based tracking using mutual information G. Nawratil C. Innocenti, D. Paganelli Determining the 3×3 rotation matrices that satisfy three linear equations in the direction cosines P.M. Larochelle A polar decomposition based displacement metric for a finite region of SE(n) J P. Merlet, P. Donelan On the regularity of the inverse Jacobian of parallel robots P. Fanghella, C. Galletti, E. Giannotti Parallel robots that change their group of motion A.P. Murray, B.M. Korte, J.P. Schmiedeler Approximating planar, morphing curves with rigid-body linkages M. Zoppi, D. Zlatanov, R. Molfino On the velocity analysis of non-parallel closed chain mechanisms 3 15 23 33 41 49 57 65 The control number as index for Stewart Gough platforms [...]... example, in Nanua and Waldron, 19 90 and Hunt and Primrose, 19 93 Its direct kinematics can be solved in closed-form by using three consecutive trilateration operations yielding 8 solutions, as 5 Wire-based Tracking Using Mutual Information z p yA platform a1 1 a ¯ base (a) (b) (c) a3 θA xA x a2 (d) Figure 1 The main two configurations used for wire-based tracking devices: (a) the “ 3-2 -1 , (b) the “ 2-2 -2 ”, and. .. space: applications in robot kinematics and surface reconstruction, Advances in Robot Kinematics: Analysis & Control (J Lenarcic, M Husty, eds.), pp 403– 412 , Kluwer Salisbury, J.K and Craig, J.J (19 82), Articulated Hands: Force Control and Kinematic Issues, Int J of Robotics Research, vol 1, no 1, pp 4 17 Wolf, A., and Shoham, M (2003), Investigations of Parallel Manipulators Using Linear Complex Approximation,... possible 14 J Andrade-Cetto and F Thomas Acknowledgements J Andrade-Cetto completed this work as a Juan de la Cierva Postdoctoral Fellow of the Spanish Ministry of Education and Science under project TIC200 3-0 92 91 and was also supported in part by projects DPI 200 4-0 5 414 , and the EU PACO-PLUS project FP 6-2 004-IST- 4-2 7657 F Thomas was partially supported by the Spanish Ministry of Education and Science,... the proposed tracking device, with (d) the rotating base in Thomas et al., 2005 The 2-2 -2 configuration was first proposed in Jeong et al., 19 99 for a wire-based tracking device The kinematics of this configuration was studied, for example, in Griffis and Duffy, 19 89, Nanua et al., 19 90, and Parenti-Castelli and Innocenti, 19 90 where it was shown that its forward kinematics has 16 solutions In other words,... = 1, , 6 15 J Lenar i and B Roth (eds.), Advances in Robot Kinematics, 15 –22 © 2006 Springer Printed in the Netherlands 16 G Nawratil Kinematic meaning of the Jacobian The velocity vector v(Pi ) of Pi due to the instantaneous screw (= twist) q := (q, q) of the platform Σ against Σ0 can be decomposed in a component vL (Pi ) along the ith leg Li and in a component v⊥ (Pi ) orthogonal to it (see Fig 1) ,... mean and 1 mm standard deviation Moreover, readings of the base orientation are also 12 J Andrade-Cetto and F Thomas Table 1 frames Coordinates of the attaching points (in meters) in their local coordinate x y z a1 0.3000, 0.0000, 0.0000 a2 – 0 .15 00, 0.2598, a3 – 0 .15 00, – 0.2598, y z b1 0 .10 00, 0.0000, 0.0000 0.0000 b2 – 0.0500, 0.0866, 0.0000 0.0000 b3 – 0.0500, – 0.0866, 0.0000 −3 6 l1 l 2 1. 05 x 10 ... Position error (m) 3 1. 04 1. 03 1. 02 2 0 −2 −4 3 2 Time (sec) 1 −6 5 4 (a) Wire Lengths −0 .1 0 5 4 3 1 3 2 Time (sec) 5 4 (c) Orientation Error 2 x 10 2 Base angle error (rad) 1. 5 1 0.5 0 −0.5 1 0 1 −2 1. 5 1 0.5 −3 1 1 3 2 Time (sec) (d) Base Angle Figure 4 platform −0.05 −3 2 0 0 (b) Position Error 2.5 1. 5 3 2 Time (sec) 1 0 roll pitch yaw 0.05 Mutual information 1. 01 0 Base angle (rad) 0 .1 x y z 4 l Orientation... Robotic Systems, 6(6):703–720 Hunt, K and Primrose, E (19 93) Assembly configurations of some in- parallel-actuated manipulators Mechanism and Machine Theory, 28 (1) : 31 42 Jeong, J., Kim, S., and Kwak, Y (19 99) Kinematics and workspace analysis of a parallel wire mechanism for measuring a robot pose Mechanism and Machine Theory, 34(6):825–8 41 MacKay, D.J C (19 92) Information based objective functions for... J P (2006) Parallel Robots, volume 12 8 of Solid Mechanics and its Applications Springer, New York, 2nd edition Nanua, P and Waldron, K (19 90) Direct kinematics solution of a special parallel robot structure In Proceedings of the 8th CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators, pages 13 4 14 2, Warsaw Nanua, P., Waldron, K., and Murthy, V (19 90) Direct kinematic solution of... vol 17 , no 10 , pp 10 81 10 90 Merlet, J.-P (19 92), Singular Configurations of Parallel Manipulators and Grassmann Geometry, Int J of Robotics Research, vol 8, no 5, pp 45–56 Nawratil, G (2006), New Performance Indices for 6R Robot Postures, CD-Proc of the 1st EuCoMeS (M Husty, H.P Schr¨cker, eds.), 12 pp., Obergurgl, Austria o Pottmann, H., Peternell, M., and Ravani, B (19 98), Approximation in line space: . Motion ý ý ž JADRAN LENAR A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN -1 0 1- 4 02 0-4 94 0-4 (HB) ISBN -1 3 97 8 -1 -4 02 0-4 94 0-8 (HB) ISBN -1 0 1- 4 02 0-4 94 1- 2 (e-book) ISBN -1 3 . Lenarþiþ and B. Roth (eds.), Advances in Robot Kinematics, 3 14 . © 2006 Springer. Printed in the Netherlands. Wire-based tracking devices are an affordable alternative to costly track- Tracking devices,. students, in fields of gineering and mathematics related to robot theory, design, control and Contents Methods in Kinematics J. Andrade-Cetto, F. Thomas Wire-based tracking using mutual information