STRUCTURAL SYNTHESIS OF PARALLEL ROBOTS SOLID MECHANICS AND ITS APPLICATIONS Volume 149 Series Editor: G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How 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. Structural Synthesis of Parallel Robots Part 1: Methodology By GRIGORE GOGU Mechanical Engineering Research Group, French Institute of Advanced Mechanics and Blaise Pascal University, Clermont-Ferrand, France A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-1-4020-5102-9 (HB) ISBN 978-1-4020-5710-6 (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 © 2008 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. Contents Preface ..................................................................................................... IX Acknowledgements ...........................................................................XIV List of abbreviations and notations..................................................... XV 1 Introduction.............................................................................................1 1.1 Robot ................................................................................................1 1.2 Robotics............................................................................................7 1.3 Parallel Robot .................................................................................10 1.4 Terminology ...................................................................................10 1.5 Structural synthesis.........................................................................23 1.6 The objectives and originality of this book ....................................25 2 Structural parameters ..........................................................................31 2.1 Critical review of mobility calculation ...........................................32 2.1.1 Chebychev’s contribution........................................................35 2.1.2 Sylvester’s contribution...........................................................36 2.1.3 Grübler’s contribution .............................................................36 2.1.4 Somov’s contribution ..............................................................37 2.1.5 Hochman’s contribution ..........................................................37 2.1.6 Somov-Malytsheff’s formula ..................................................38 2.1.7 Koenigs’ formula.....................................................................39 2.1.8 Kutzbach’s mobility equation..................................................39 2.1.9 Dobrovolski’s mobility equation.............................................40 2.1.10 Contribution of Y.F. Moroskine............................................40 2.1.11 Contribution of R. Voinea and M. Atanasiu..........................41 2.1.12 Kolchin’s mobility equation ..................................................42 2.1.13 Rössner’s contribution...........................................................42 2.1.14 Boden’s mobility equation.....................................................42 2.1.15 Manafu’s formula ..................................................................43 2.1.16 Ozol’s formula........................................................................ 43 2.1.17 Contribution of K. J. Waldron...............................................44 2.1.18 Contribution of N. Manolescu ...............................................44 2.1.19 Contribution of C. Bagci .......................................................45 VI Contents 2.1.20 Contribution of P. Antonescu ................................................46 2.1.21 Contribution of F. Freudenstein and R. Alizade....................47 2.1.22 Hunt’s contribution................................................................48 2.1.23 Hervé’s contribution..............................................................49 2.1.24 Gronowicz’s contribution......................................................50 2.1.25 Baker’s contribution ..............................................................50 2.1.26 Davies’s contribution.............................................................51 2.1.27 Contribution of V.P. Agrawal and J.S. Rao...........................52 2.1.28 Contribution of J. Angeles and C. Gosselin ..........................52 2.1.29 Contribution of F. Dudiţă and D. Diaconescu .......................53 2.1.30 Contribution of P. Fanghella and C. Galletti .........................55 2.1.31 Fayet’s contribution...............................................................55 2.1.32 Tsai’s formula........................................................................56 2.1.33 McCarthy’s formula ..............................................................56 2.1.34 Contribution of Z. Huang, L.F. Kong and Y.F. Fang............57 2.1.35 Contribution of J.M. Rico, J. Gallardo and B. Ravani...........57 2.2 Chebychev-Grübler-Kutzbach mobility formulae..........................58 2.2.1 The original Chebychev-Grübler-Kutzbach formula...............58 2.2.2 The extended Chebychev-Grübler-Kutzbach formula.............61 2.2.3 Limits of applicability of CGK formulae ................................62 2.3 Mobility and connectivity of parallel robots ..................................78 2.3.1 General definitions and formulae for mobility and connectivity of mechanisms .............................................................79 2.3.2 Mobility and connectivity of simple open kinematic chains ...82 2.3.3 Mobility and connectivity of single-loop kinematic chains.....88 2.3.4 Connectivity between two elements of a single-loop kinematic chain.................................................................................96 2.3.5. Mobility and connectivity of parallel robots with simple limbs ...............................................................................................100 2.3.6. Mobility and connectivity of parallel robots with complex limbs ...............................................................................................109 2.3.7. General formulae for robot mobility and connectivity .........114 2.4 Overconstraints in parallel robots.................................................120 2.5 Redundancy in parallel robots ......................................................125 2.6 General formulae for structural parameters..................................127 3 Structural analysis..............................................................................131 3.1 Simple open kinematic chains ......................................................131 3.2 Single-loop kinematic chains........................................................137 3.3 Parallel mechanisms with simple limbs........................................148 3.4 Parallel mechanisms with complex limbs.....................................168 3.5 Other multi-loop kinematic chains ...............................................228 Contents VII 4 Kinematic analysis ..............................................................................235 4.1. Decoupling in axiomatic design ..................................................236 4.2. Geometric modeling ....................................................................238 4.3 Kinematic modeling .................................................................241 4.3.1 Direct and inverse kinematics matrices used in Jacobian calculation.......................................................................................242 4.3.2 Design and conventional Jacobian matrices..........................243 4.4 Types of workspaces and singularities .........................................248 4.4.1 Types of workspaces .............................................................248 4.4.2 Types of singularities.............................................................249 4.5. Kinetostatic performance indices ................................................253