Parallel Robots Second Edition SOLID MECHANICS AND ITS APPLICATIONS Volume 128 Series Editor: G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L3GI Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, andHow much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. For a list of related mechanics titles, see final pages. Parallel Robots (Second Edition) by J.-P. MERLET INRIA, Sophia-Antipolis, France AC.I.P. Catalogue record for this book is available from the Library of Congress. ISBN-10 1-4020-4132-2 (HB) ISBN-13 978-1-4020-4132-7 (HB) ISBN-10 1-4020-4133-0 (e-book) ISBN-13 978-1-4020-4133-4 (e-book) Published by Springer, P.O. Box 17, 3300 AADordrecht, 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. Table of Contents Preface xv Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii 1 Introduction 1 1.1 Characteristics of classical robots . . . . . . . . . . . . . . . 1 1.2 Other types of architecture . . . . . . . . . . . . . . . . . . 4 1.3 Needs for robotics . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Parallel robots: definition . . . . . . . . . . . . . . . . . . . 12 1.4.1 Generalized parallel manipulators: definition . . . . 12 1.4.2 Parallel manipulators . . . . . . . . . . . . . . . . . 12 1.4.3 Fully parallel manipulators . . . . . . . . . . . . . . 13 1.4.4 Fully parallel manipulators: analysis . . . . . . . . . 13 1.4.4.1 Planar robots. . . . . . . . . . . . . . . . . 14 1.4.4.2 General case . . . . . . . . . . . . . . . . . 15 1.5 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2 Structural synthesis and architectures 19 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Structural synthesis methods . . . . . . . . . . . . . . . . . 20 2.2.1 Graph theory . . . . . . . . . . . . . . . . . . . . . . 20 2.2.2 Group theory approach . . . . . . . . . . . . . . . . 21 2.2.2.1 The Lie group and subgroups of displacement 21 2.2.2.2 Subgroup motion generators . . . . . . . . 22 2.2.2.3 Type synthesis based on group theory . . . 23 2.2.3 The screw approach . . . . . . . . . . . . . . . . . . 23 2.2.3.1 Basics of screw theory . . . . . . . . . . . . 24 2.2.3.2 Type synthesis based on screw theory . . . 24 2.2.4 Structural synthesis and other kinematic performances 25 2.2.5 Structural synthesis and uncertainties . . . . . . . . 25 2.2.6 Notation for parallel robots . . . . . . . . . . . . . . 26 2.3 Planar robots . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.1 3 d.o.f. manipulators . . . . . . . . . . . . . . . . . . 27 2.4 Spatial motion robots . . . . . . . . . . . . . . . . . . . . . 29 v vi TABLE OF CONTENTS 2.4.1 Joints and actuators . . . . . . . . . . . . . . . . . . 29 2.4.2 Classification of parallel robots. . . . . . . . . . . . . 30 2.4.3 3 d.o.f. manipulators . . . . . . . . . . . . . . . . . . 31 2.4.3.1 Translation manipulators . . . . . . . . . . 31 2.4.3.2 Orientation manipulators . . . . . . . . . . 35 2.4.3.3 Mixed degrees of freedom manipulators . . 39 2.4.4 4 d.o.f. manipulators . . . . . . . . . . . . . . . . . . 43 2.4.5 5 d.o.f. manipulators . . . . . . . . . . . . . . . . . . 44 2.4.6 6 d.o.f. manipulators . . . . . . . . . . . . . . . . . . 47 2.4.6.1 UPS chain robot . . . . . . . . . . . . . . 48 2.4.6.2 PUS chain robots . . . . . . . . . . . . . . 51 2.4.6.3 RUS chain robots . . . . . . . . . . . . . . 52 2.4.6.4 Robots with miscellaneous chains . . . . . 54 2.4.6.5 Three-legged robots . . . . . . . . . . . . . 56 2.4.6.6 Decoupled robots . . . . . . . . . . . . . . 59 2.5 Redundant robots . . . . . . . . . . . . . . . . . . . . . . . 62 2.6 Articulated truss and binary actuation . . . . . . . . . . . . 62 2.7 MEMS and micro-positioning robots . . . . . . . . . . . . . 66 2.8 Wire robots . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.9 Examples of applications . . . . . . . . . . . . . . . . . . . 69 2.9.1 Spatial applications . . . . . . . . . . . . . . . . . . 70 2.9.2 Vibration . . . . . . . . . . . . . . . . . . . . . . . . 73 2.9.3 Medical applications . . . . . . . . . . . . . . . . . . 75 2.9.4 Simulators . . . . . . . . . . . . . . . . . . . . . . . . 77 2.9.5 Industrial applications . . . . . . . . . . . . . . . . . 79 2.9.5.1 Machine-tool . . . . . . . . . . . . . . . . . 80 2.9.5.2 Positioning devices . . . . . . . . . . . . . . 86 2.9.5.3 Other industrial applications . . . . . . . . 88 2.9.6 Miscellaneous applications . . . . . . . . . . . . . . . 91 2.10 Robots studied in this book . . . . . . . . . . . . . . . . . . 93 2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3 Inverse kinematics 95 3.1 Inverse kinematics . . . . . . . . . . . . . . . . . . . . . . . 95 3.1.1 General methods . . . . . . . . . . . . . . . . . . . . 95 3.1.1.1 Analytic method . . . . . . . . . . . . . . . 95 3.1.1.2 Geometrical method . . . . . . . . . . . . . 96 3.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . 97 3.1.2.1 Planar manipulators . . . . . . . . . . . . . 97 3.1.2.2 3-UPU manipulator . . . . . . . . . . . . 98 3.1.2.3 6-UPS manipulator . . . . . . . . . . . . . 99 3.1.2.4 6-PUS manipulator . . . . . . . . . . . . . 100 TABLE OF CONTENTS vii 3.1.2.5 6-RUS manipulator . . . . . . . . . . . . . 101 3.1.2.6 General conclusion . . . . . . . . . . . . . . 102 3.1.3 Extrema of the joint coordinates . . . . . . . . . . . 102 3.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4 Direct kinematics 105 4.1 Planar robots . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.1.1 The 4-bar mechanism . . . . . . . . . . . . . . . . . 106 4.1.2 Coupler curve and circularity . . . . . . . . . . . . . 106 4.1.3 Direct kinematics of the 3-RPR robot . . . . . . . . 107 4.1.3.1 Assembly modes . . . . . . . . . . . . . . . 108 4.1.3.2 Polynomial direct kinematics . . . . . . . . 108 4.1.3.3 Particular cases . . . . . . . . . . . . . . . 111 4.1.4 Other planar robots . . . . . . . . . . . . . . . . . . 111 4.2 Robots with 3 translational d.o.f. . . . . . . . . . . . . . . 112 4.3 Robots with 6 d.o.f. . . . . . . . . . . . . . . . . . . . . . . 113 4.3.1 Example of analysis: the TSSM . . . . . . . . . . . 113 4.3.1.1 Upper bound on the number of assembly modes . . . . . . . . . . . . . . . . . . . . . 113 4.3.1.2 Polynomial formulation . . . . . . . . . . . 114 4.3.1.3 Example of TSSM with 16 assembly modes 117 4.3.2 Analysis of other space mechanisms . . . . . . . . . 119 4.3.2.1 3 degrees of freedom wrist . . . . . . . . . 120 4.3.2.2 MSSM . . . . . . . . . . . . . . . . . . . . 121 4.3.2.3 6−PUS robot and Stewart platform . . . 121 4.3.2.4 Manipulators PPP-3S,PRR-3S,PPR-3S 122 4.3.3 Special cases of the 6−UPS robot . . . . . . . . . . 123 4.3.3.1 6-5 manipulators . . . . . . . . . . . . . . 123 4.3.3.2 6-4 manipulators . . . . . . . . . . . . . . 123 4.3.3.3 6-3 manipulators . . . . . . . . . . . . . . 124 4.3.3.4 5-5 manipulators . . . . . . . . . . . . . . 124 4.3.3.5 5-4 manipulators . . . . . . . . . . . . . . 124 4.3.3.6 4-4 manipulators . . . . . . . . . . . . . . 126 4.3.3.7 Manipulators with 5 aligned points . . . . 127 4.3.4 The SSM . . . . . . . . . . . . . . . . . . . . . . . . 127 4.3.5 General case of the 6−UPS robot . . . . . . . . . . 128 4.3.5.1 Maximum number of assembly modes . . . 128 4.3.5.2 Determination of the solutions . . . . . . . 128 4.3.5.3 Example with 40 real solutions . . . . . . . 129 4.3.6 Summary of results . . . . . . . . . . . . . . . . . . . 130 4.4 Systematic method for UPS robots . . . . . . . . . . . . . 130 4.4.1 Manipulators with 9 legs . . . . . . . . . . . . . . . 130 viii TABLE OF CONTENTS 4.4.2 Manipulators with 7 and 8 legs . . . . . . . . . . . . 133 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.6 Fast numerical methods . . . . . . . . . . . . . . . . . . . . 135 4.6.1 Newton schemes . . . . . . . . . . . . . . . . . . . . 136 4.6.1.1 Principle . . . . . . . . . . . . . . . . . . . 136 4.6.1.2 Implementation for the direct kinematics . 137 4.6.1.3 DrawbacksoftheNewtonschemesandreal- time issues . . . . . . . . . . . . . . . . . . 139 4.6.1.4 Convergence of the Newton schemes . . . . 140 4.6.1.5 Extending the unicity domain: the inflation 142 4.6.2 Interval analysis scheme . . . . . . . . . . . . . . . . 142 4.6.3 Methods efficiency and computation time . . . . . . 143 4.6.4 Path tracking . . . . . . . . . . . . . . . . . . . . . . 144 4.7 Direct kinematics with extra sensors . . . . . . . . . . . . . 145 4.7.1 Type and location of the extra sensors . . . . . . . . 146 4.7.2 Maximal number of sensors . . . . . . . . . . . . . . 146 4.7.2.1 Addition of angular sensors . . . . . . . . . 146 4.7.2.2 Addition of linear sensors . . . . . . . . . . 147 4.7.2.3 Combination of angular and linear sensors 148 4.7.3 Relationship between sensors accuracy and pose ac- curacy . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5 Velocity, accuracy and acceleration analysis 153 5.1 Kinematics relations . . . . . . . . . . . . . . . . . . . . . . 153 5.2 Inverse jacobian matrix . . . . . . . . . . . . . . . . . . . . 153 5.2.1 Euler angles inverse jacobian . . . . . . . . . . . . . 155 5.2.1.1 Example: 6−UPS manipulator . . . . . . . 155 5.2.2 Inverse kinematic jacobian . . . . . . . . . . . . . . . 156 5.2.2.1 Example: planar 3-RPR manipulator . . . 157 5.2.2.2 Example: 3−UPU manipulator . . . . . . 158 5.2.2.3 Example: 3−PUS rotational wrist . . . . 159 5.2.2.4 Example: 6−UPS manipulator . . . . . . . 160 5.2.2.5 Example: 6−PUS manipulator . . . . . . . 161 5.2.3 Inverse jacobian and Plu¨cker line coordinates . . . . 161 5.3 Jacobian matrix . . . . . . . . . . . . . . . . . . . . . . . . 162 5.4 Kinetostatic performance indices . . . . . . . . . . . . . . . 163 5.4.1 Manipulability and the kinematics polyhedron . . . 163 5.4.2 Condition number and other indices . . . . . . . . . 165 5.4.2.1 Manipulability index and condition number 165 5.4.2.2 Validity of the condition number . . . . . 167 5.4.2.3 Isotropy . . . . . . . . . . . . . . . . . . . . 169 TABLE OF CONTENTS ix 5.4.2.4 Global conditioning indices . . . . . . . . 169 5.4.2.5 Other accuracy indices . . . . . . . . . . . 170 5.5 Determination of the joint velocities and twist . . . . . . . 171 5.5.1 Determination of the joint velocities . . . . . . . . . 171 5.5.2 Determination of the twist. . . . . . . . . . . . . . . 171 5.6 Extrema of the velocities in a workspace . . . . . . . . . . . 172 5.6.1 Extrema of the twist . . . . . . . . . . . . . . . . . 172 5.6.2 Extrema of the joint velocities . . . . . . . . . . . . 173 5.7 Accelerations analysis . . . . . . . . . . . . . . . . . . . . . 173 5.7.1 6−UPS robot. . . . . . . . . . . . . . . . . . . . . . 173 5.7.2 6−PUS robot. . . . . . . . . . . . . . . . . . . . . . 174 5.8 Accuracy analysis . . . . . . . . . . . . . . . . . . . . . . . 175 5.8.1 Geometrical errors . . . . . . . . . . . . . . . . . . . 175 5.8.2 Thermal errors . . . . . . . . . . . . . . . . . . . . . 176 5.8.3 Gravity induced errors . . . . . . . . . . . . . . . . . 176 5.8.4 Dynamics errors . . . . . . . . . . . . . . . . . . . . 176 5.8.5 Worst poses for accuracy . . . . . . . . . . . . . . . 176 5.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 6 Singular configurations 179 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.2 Singularity influence and classification . . . . . . . . . . . . 179 6.2.1 Singularities and velocities . . . . . . . . . . . . . . . 179 6.2.2 Singularities and statics . . . . . . . . . . . . . . . . 181 6.2.3 Singularities and kinematics . . . . . . . . . . . . . . 182 6.2.4 Serial singularity . . . . . . . . . . . . . . . . . . . . 182 6.3 Parallel singularities . . . . . . . . . . . . . . . . . . . . . . 183 6.3.1 Motivations for the study of singularity . . . . . . . 183 6.3.2 Singularity analysis . . . . . . . . . . . . . . . . . . . 184 6.4 Grassmann geometry . . . . . . . . . . . . . . . . . . . . . . 185 6.4.1 Variety and geometry . . . . . . . . . . . . . . . . . 186 6.4.2 Examples of geometrical analysis . . . . . . . . . . . 189 6.4.2.1 Planar 3−RPR manipulator . . . . . . . . 189 6.4.2.2 3−UPU manipulator . . . . . . . . . . . . 190 6.4.2.3 MSSM . . . . . . . . . . . . . . . . . . . . 191 6.5 Motion associated with singularities . . . . . . . . . . . . . 200 6.5.1 Determination of the singularity motion . . . . . . . 201 6.5.2 Determination of the instantaneous rotation axis . . 201 6.5.3 Example: the MSSM . . . . . . . . . . . . . . . . . . 202 6.5.3.1 Type 3d configuration . . . . . . . . . . . 202 6.5.3.2 Type 5a and 5b configuration . . . . . . . 203 6.6 Singularity indices . . . . . . . . . . . . . . . . . . . . . . . 204 x TABLE OF CONTENTS 6.7 Singularity test . . . . . . . . . . . . . . . . . . . . . . . . . 206 6.8 Mechanisms in permanent singularity . . . . . . . . . . . . 208 6.9 Singularity-free path-planning and workspace enlargement 209 6.10 Singularity and design . . . . . . . . . . . . . . . . . . . . . 210 6.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7 Workspace 213 7.1 Workspace limits, representation and type . . . . . . . . . . 213 7.1.1 The different types of workspaces . . . . . . . . . . . 213 7.1.2 Orientation representation . . . . . . . . . . . . . . . 214 7.2 Workspace calculation methods . . . . . . . . . . . . . . . . 215 7.2.1 Geometrical approach . . . . . . . . . . . . . . . . . 215 7.2.2 Discretisation method . . . . . . . . . . . . . . . . . 216 7.2.3 Numerical methods . . . . . . . . . . . . . . . . . . . 217 7.3 Planar manipulators . . . . . . . . . . . . . . . . . . . . . . 219 7.3.1 Constant orientation workspace . . . . . . . . . . . 219 7.3.1.1 Joint coordinates limits . . . . . . . . . . . 219 7.3.1.2 Mechanical limits on the passive joints . . 220 7.3.1.3 Leg interference . . . . . . . . . . . . . . . 220 7.3.2 Orientation workspace . . . . . . . . . . . . . . . . . 221 7.3.3 Dextrous workspace . . . . . . . . . . . . . . . . . . 222 7.3.4 Maximal workspace . . . . . . . . . . . . . . . . . . 223 7.3.5 Inclusive orientation workspace . . . . . . . . . . . . 225 7.3.6 Total orientation workspace . . . . . . . . . . . . . . 227 7.4 3−UPU manipulator . . . . . . . . . . . . . . . . . . . . . . 228 7.5 6−UPS manipulator . . . . . . . . . . . . . . . . . . . . . . 228 7.5.1 Cross-sections of the constant orientation workspace 229 7.5.2 3D constant orientation workspace . . . . . . . . . . 230 7.5.2.1 Workspace area and volume . . . . . . . . 231 7.5.2.2 Mechanical limits on the joints . . . . . . . 233 7.5.2.3 Interference between links . . . . . . . . . . 237 7.5.3 Orientation workspace . . . . . . . . . . . . . . . . . 239 7.5.4 Dextrous workspace . . . . . . . . . . . . . . . . . . 240 7.5.5 Maximal workspace . . . . . . . . . . . . . . . . . . 240 7.5.6 Workspace for machine-tool . . . . . . . . . . . . . . 242 7.5.7 Comparison between architectures . . . . . . . . . . 244 7.6 Workspace performance indices . . . . . . . . . . . . . . . . 245 7.7 Trajectory verification . . . . . . . . . . . . . . . . . . . . . 246 7.7.1 Line segment verification . . . . . . . . . . . . . . . 246 7.7.1.1 Constraints on the link lengths . . . . . . . 246 7.7.1.2 Mechanical limits on the joints . . . . . . . 248 7.7.1.3 Example . . . . . . . . . . . . . . . . . . . 248