Structural Synthesis of Parallel Robots Methodology [part 1] 0-7923-7008-2, 0-7923-7038-4, 0-7923-7164-X, 0-7923-7165-8

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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

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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

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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 4.5.1 Cross-coupling indices .......................................................... 256 4.5.2 Indices of input-output propensity......................................... 262 4.5.3 Kinetostatic indices defined in connection with manipulability ellipsoids and polytops ........................................... 266 4.6 Design Jacobian and motion decoupling ...................................... 274 4.6.1. Parallel robots with coupled motions ................................... 276 4.6.2. Parallel robots with decoupled motions................................ 285 4.6.3. Parallel robots with uncoupled motions ............................... 293 4.6.4. Maximally regular parallel robots ........................................ 296 5 Structural synthesis ............................................................................ 299 5.1 Structural synthesis: a systematic approach in mechanism design299 5.2. Morphological and evolutionary approaches............................... 304 5.2.1 Morphological approaches .................................................... 305 5.2.2 Evolutionary algorithms ........................................................ 307 5.3 Evolutionary morphology............................................................. 310 5.3.1 Design objectives................................................................... 310 5.3.2 Constituent elements.............................................................. 311 5.3.3 Morphological operators........................................................ 313 5.3.4 Set of solutions ...................................................................... 314 5.3.5 General structure of the evolutionary morphology................ 314 5.4 General approach to structural synthesis of parallel robots .......... 317 5.4.1 General conditions for structural synthesis of parallel robots via theory of linear transformations..................................... 317 5.4.2 General approach to structural synthesis of parallel robots via evolutionary morphology ............................................................... 319 6 Limbs with two degrees of connectivity ............................................ 329 6.1 Limbs with two translational motions .......................................... 329 6.2 Limbs with two rotational motions............................................... 333 6.3 Limbs with one translational and one rotational motion .............. 334

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6.4 Other limbs with two degrees of connectivity .............................. 335 6.5 Redundant limbs with two degrees of connectivity...................... 338 7 Limbs with three degrees of connectivity ......................................... 341 7.1 Limbs with three translational motions ........................................ 341 7.2 Planar limbs with one rotational and two translational motions... 352 7.3 Non planar limbs with one rotational and two translational motions ............................................................................................... 359 7.4 Limbs with one translational and two rotational motions............. 364 7.5 Limbs with three rotational motions............................................. 364 7.6 Other limbs with three degrees of connectivity ............................ 365 7.7 Redundant limbs with three degrees of connectivity.................... 375 8 Limbs with four degrees of connectivity........................................... 377 8.1 Limbs with Schönflies motion...................................................... 377 8.2 Limbs with two translational and two rotational motions ............ 425 8.3 Limbs with one translational and three rotational motions........... 435 8.4 Other limbs with four degrees of connectivity ............................. 436 8.5 Redundant limbs with four degrees of connectivity ..................... 442 9 Limbs with five degrees of connectivity ............................................ 445 9.1 Limbs with two rotational and three translational motions .......... 445 9.2 Limbs with two translational and three rotational motions .......... 637 9.3 Other limbs with five degrees of connectivity.............................. 644 9.4 Redundant limbs with five degrees of connectivity...................... 650 10 Limbs with six degrees of connectivity ........................................... 653 10.1 Limbs with three translational and three rotational motions ...... 653 10.2 Redundant limbs with six degrees of connectivity ..................... 661 References............................................................................................... 665 Index ....................................................................................................... 693

Preface

“…if the new theory is to lay claim to general interest, it must be capable of producing something new; it must make problems solvable which before could not be solved in a systematic way." Reuleaux, F., Theoretische Kinematik, Braunschweig: Vieweg, 1875 Reuleaux, F., The Kinematics of Machinery, London: Macmil lan, 1876 and New York: Dover, 1963 (translated by B.W. Kennedy)

Parallel robotic manipulators can be considered a well-established option for many different applications of manipulation, machining, guiding, testing, control, tracking, haptic force feed-back, etc. A typical parallel robotic manipulator (PM) consists of a mobile platform connected to the base (fixed platform) by at least two kinematic chains called limbs. The mobile platform can achieve between one and three independent translations (T) and one to three independent rotations (R). Parallel manipulators have been the subject of study of much robotic research during the last two decades. Early research on parallel manipulators has concentrated primarily on six degrees of freedom (DoFs) GoughStewart-type PMs introduced by Gough for a tire-testing device, and by Stewart for flight simulators. In the last decade, PMs with fewer than 6DoFs attracted researchers’ attention. Lower mobility PMs are suitable for many tasks requiring less than six DoFs. The motion freedoms of the end-effector are usually coupled together due to the multi-loop kinematic structure of the parallel manipulator. Hence, motion planning and control of the end-effector for PMs usually become very complicated. With respect to serial manipulators, such mechanisms can offer advantages in terms of stiffness, accuracy, load-toweight ratio, dynamic performances. Their disadvantages include a smaller workspace, complex command and lower dexterity due to a high motion coupling, and multiplicity of singularities inside their workspace. Uncoupled, fully-isotropic and maximally regular PMs can overcome these disadvantages.

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Isotropy of a robotic manipulator is related to the condition number of its Jacobian matrix, which can be calculated as the ratio of the largest and the smallest singular values. A robotic manipulator is fully-isotropic if its Jacobian matrix is isotropic throughout the entire workspace, i.e., the condition number of the Jacobian matrix is equal to one. We know that the Jacobian matrix of a robotic manipulator is the matrix mapping (i) the actuated joint velocity space on the end-effector velocity space, and (ii) the static load on the end-effector and the actuated joint forces or torques. The isotropic design aims at ideal kinematic and dynamic performance of the manipulator. We distinguish five types of PMs (i) maximally regular PMs, if the Jacobian J is an identity matrix throughout the entire workspace, (ii) fullyisotropic PMs, if the Jacobian J is a diagonal matrix with identical diagonal elements throughout the entire workspace, (iii) PMs with uncoupled motions if J is a diagonal matrix with different diagonal elements, (iv) PMs with decoupled motions, if J is a triangular matrix and (v) PMs with coupled motions if J is neither a triangular nor a diagonal matrix. Maximally regular and fully-isotropic PMs give a one-to-one mapping between the actuated joint velocity space and the external velocity space. The first solution for a fully-isotropic T3-type translational parallel robot was developed at the same time and independently by Carricato and Parenti-Castelli at University of Genoa, Kim and Tsai at University of California, Kong and Gosselin at University of Laval, and the author of this work at the French Institute of Advanced Mechanics. In 2002, the four groups published the first results of their works. The general methods used for structural synthesis of parallel mechanisms can be divided into three approaches: the method based on displacement group theory, the methods based on screw algebra, and the method based on the theory of linear transformations. The method proposed in this book is based on the theory of linear transformations and the evolutionary morphology and allows us to obtain the structural solutions of decoupled, uncoupled, fully-isotropic and maximally regular PMs with two to six DoFs in a systematic way. The new formulae for mobility, connectivity (spatiality), redundancy and overconstraint of PMs proposed recently by the author are integrated into the synthesis approach developed in this book. Various solutions of TaRb-type PMs are known today. In this notation, a=1,2,3 indicates the number of independent translations and b=1,2,3 the number of independent rotations of the moving platform. The parallel robots actually proposed by the robot industry have coupled and decoupled motions and just some isotropic positions in their workspace. As far as we

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are aware, this is the first book on robotics presenting solutions of uncoupled, fully-isotropic and maximally regular PMs. Non-redundant/redundant, overconstrained/isostatic solutions of uncoupled and fully-isotropic/maximally regular PMs with elementary/complex limbs actuated by linear/rotary actuators with/without idle mobilities and two to six DoFs are present in a systematic approach of structural synthesis. These solutions are derived from families of PMs with decoupled motions. A serial kinematic chain is associated with each elementary limb and at least one closed loop is integrated in each complex limb. The synthesis methodology and the solutions of PMs presented in this book represent the outcome of some recent research developed by the author in the last four years in the framework of the projects ROBEA-MAX (2002-2003) and ROBEA-MP2 (2004-2005) supported by the National Center for Scientific Research (CNRS). These results have been partially published by the author in the last two years. In these works the author has proposed the following for the first time in the literature: a) new formulae for calculating the degree of mobility, the degree of connectivity(spatiality), the degree of redundancy and the number of overconstraints of parallel robotic manipulators that overcome the drawbacks of the classical Chebychev-Grübler-Kutzbach formulae, b) a new approach to systematic innovation in engineering design called evolutionary morphology, c) solutions to follow fully-isotropic parallel robots: parallel wrists with two and three degrees of mobility, planar parallel robots, parallel robots of types T2R1 and T3R2, parallel robots with Schönflies motions, hexapods with six degrees of freedom. The various solutions proposed by the author belong to a modular family called IsoglideN-TxRy with a+b=n with 2 ≤ n ≤ 6, a=1,2,3 and b=1,2,3. The mobile platform of these robots can have any combination of n independent translations (T) and rotations (R). The Isogliden-TaRb modular family was developed by the author and his research team at the French Institute of Advanced Mechanics (IFMA) in Clermont-Ferrand. This work is organized in two parts published in two distinct books. Part 1 presents the methodology proposed for structural synthesis and Part 2 the various topologies of parallel robots systematically generated by the structural synthesis approach. The originality of this work resides in combining the new formulae for mobility connectivity, redundancy and overconstraints, and the evolutionary morphology in a unified approach of structural synthesis giving interesting innovative solutions for parallel robots. Part 1 is organized in ten chapters. The first chapter is intended to introduce the main concepts, definitions and components of the mechanical robotic system. Chapter 2 reviews the contributions in mobility calculation

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systematized in the so called Chebychev-Grübler-Kutzbach mobility formulae. The drawbacks and the limitations of these formulae are discussed, and the new formulae for mobility, connectivity, redundancy and overconstraint are demonstrated via an original approach based on the theory of linear transformations. These formulae are applied in chapter 3 for the structural analysis of parallel robots with simple and complex limbs. The new formulae are also applied to calculate the mobility and other structural parameters of single and multi-loop mechanisms that do not obey the classical Chebychev-Grübler-Kutzbach formulae, such as the mechanisms proposed by De Roberval, Sarrus, Bennett, Bricard and other so called “paradoxical mechanisms”. We have shown that these mechanisms completely obey the definitions, the theorems and the formulae proposed in the previous chapter. There is no reason to continue to consider them as “paradoxical”. Chapter 4 presents the main models and performance indices used in parallel robots. We emphasize the Jacobian matrix, which is the main issue in defining robot kinematics, singularities and performance indices. New kinetostatic performance indices are introduced in this section to define the motion decoupling and input-output propensity in parallel robots. Structural parameters introduced in the second chapter are integrated in the structural synthesis approach founded on the evolutionary morphology (EM) presented in chapter 5. The main paradigms of EM are presented in a closed relation with the biological background of morphological approaches and the synthetic theory of evolution. The main difference between the evolutionary algorithms and the EM are also discussed. The evolutionary algorithms are methods for solving optimization-oriented problems, and are not suited to solving conceptual design-oriented problems. They always start from a given initial population of solutions and do not solve the problem of creating these solutions. The first stage in structural synthesis of parallel robots is the generation of the kinematic chains called limbs used to give some constrained or unconstrained motion to the moving platform. The constrained motion of the mobile platform is obtained by using limbs with less than six degrees of connectivity. The various solutions of simple and complex limbs with two to six degrees of connectivity are systematically generated by the structural synthesis approach and presented in chapters 6-10. We focus on the solutions with a unique basis of the operational velocity space that are useful for generating various topologies of decoupled, uncoupled, fully-isotropic and maximally regular parallel robots presented in Part 2. Limbs with multiple bases of the operational velocity space and redundant limbs are also presented in these chapters. These limb solutions are systematized with respect to various combinations of independent motions of the distal link. They are defined by symbolic notations and illustrated in about 250 figures

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containing more than 1500 structural diagrams. Special attention was paid to graphic quality of structural diagrams to ensure a clear correspondence between the symbolic and graphic notation of joints and the relative position of their axes. The graphic illustration of these solutions is associated with the author’s conviction that a good structural diagram really “is worth a thousand words”, especially when you are trying to disseminate the result of the structural synthesis of kinematic chains. The kinematic chains presented in chapters 6-10 are useful as innovative solutions of limbs in parallel, serial and hybrid robots. In fact, serial and hybrid robots may be considered as a particular case of parallel robots with only one limb which can be a simple, complex or hybrid kinematic chain. Many serial robots actually combine closed loops in their kinematic structure. The various types of kinematic chains generated in chapters 6-10 are combined in Part 2 to set up innovative solutions of parallel robots with two to six degrees of mobility and various sets of independent motions of the moving platform. Part 2 is organized in 16 chapters presenting the following: translational PMs T2-type (chapter 1), T1R1-type PMs with screw motion (chapter 2), other T1R1-type PMs (chapter 3), R2-type spherical parallel wrists (chapter 4), translational PMs T3-type (chapter 5), T2R1-type PMs with planar motion (chapter 6), other T2R1-type PMs (chapter 7), T1R2-type PMs (chapter 8), R3-type spherical parallel wrists (chapter 9), T3R1-type PMs with Schönflies motion (chapter 10), other T3R1-type PMs (chapter 11), T2R2-type PMs (chapter 12), T1R3-type PMs (chapter 13), T3R2-type PMs (chapter 14), T2R3-type PMs (chapter 15) and PMs with six degrees of freedom (chapter 16). Parallel robots with coupled, decoupled and uncoupled motions, along with fully-isotropic and maximally regular solutions, are systematically presented in each chapter. Innovative solutions of overconstrained or nonoverconstrained, non-redundant or redundantly actuated parallel robots with simple or complex limbs actuated by linear or rotating motors are set up in Part 2 by applying the structural synthesis methodology presented in this first part. The writing of part 2 is still in progress and will soon be finalized. Many solutions for parallel robots obtained through this systematic approach of structural synthesis are presented here for the first time in the literature. The author had to make a difficult and challenging choice between protecting these solutions through patents, and releasing them directly into the public domain. The second option was adopted by publishing them in various recent scientific publications and mainly in this book. In this way, the author hopes to contribute to a rapid and widespread implementation of these solutions in future industrial products.

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Acknowledgements The scientific environment of the projects ROBEA-MAX (2002-2003) and ROBEA-MP2 (2004-2005) supported by the CNRS was the main source of encouragement and motivation to pursue the research on the structural synthesis of parallel robots and to write this book. Deep gratitude is expressed here to Dr. François Pierrot, Deputy Director of LIRMM and coordinator of both ROBEA projects, and also to all colleagues involved in these projects from the research laboratories LIRMM, INRIA, IRCCyN LASMEA and LaMI for the valuable scientific exchanges during the joint work in these projects. Moreover, financial support from the CNRS, FR TIMS and IFMA for developing the innovative Isoglide-family of parallel robots are duly acknowledged. Furthermore, Mrs. Nathalie Jacobs, Springer Dordrecht Engineering Editor is gratefully acknowledged for her availability and encouragement in pursuing this book project. Ms. Sarah Davey and Prof. Graham M.L. Gladwell are also gratefully acknowledged for linguistic reviewing of the manuscript. Prof. Gladwell not only took great care of the book language but also made numerous suggestions for improving and clarifying some fine points of its content. May I also acknowledge the excellent facilities and research environment provided by LaMI and IFMA which contributed actively to the completion of this project. To conclude, I cannot forget my wife Iléana and my son Christian for their love, affection and encouragement, providing the fertile ambience for this sustained work very often prolonged late into the evening and mostly during week-ends and holidays.

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List of abbreviations and notations ai – axial offset of the ith-link A (1A-2A-…-nA) and A (1-2-…-n) – simple open kinematic chain and its links A (0≡1A-2A-3A-4A-5A) simple open kinematic chain with link 1A fixed to base 0 A (1A≡0-2A-…nA≡1A) – simple closed kinematic chain with link 1A fixed to base 0 A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) simple open kinematic chain obtained by serial concatenation of two simple chains A1 and A2 AI – artificial intelligence bi – length of the i-link B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) single-loop kinematic chain obtained by connected the distal links (1A1≡1A2 and nA1≡nA2) of two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2) c – condition number C ← A1-A2-…Ak parallel robot with k simple limbs Ai (i=1, 2,…,k) C – cylindrical pair CNRS - Centre National de la Recherche Scientifique (National Center for Scientific Research) CGK - Chebychev-Grübler-Kutzbach Ci - instantaneous degree of motion coupling C - global degree of motion coupling d - linear dispalcement D ← A1-A2-…Ak1-E1-E2-…-Ek2 parallel robot with k1 simple limbs Ai (i=1, 2,…,k1) and k2 complex limbs Ei (i=1, 2,…,k2) DoF - degree-of-freedom DPs – design parameters eA and eA1 - link of A1-limb (e=1,2,3,…,n) eB and eA2 - link of A2-limb (e=1,2,3,…,n) eC and eA3 - link of A3-limb (e=1,2,3,…,n) eD and eA4 - link of A4-limb (e=1,2,3,…,n) E = (ε1 ,..., ε n ) - set of constituent elements EA – evolutionary algorithm EM – evolutionary morphology ES - Evolution strategies F - linear transformation f - degree of mobility of a joint in the general motion space with six dimensions

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F ← G1-G2-…Gk general notation for the kinematic chain associated to a serial, parallel or hybrid robot with k simple and/or complex limbs Gi (i=1, 2,…,k) FRs – functional requirements FR TIMS - Fédération de Recherche Technologies de l’Information et de la Mobilité, de la Sûreté GA – genetic algorithm Gi (1Gi-2Gi-…nGi) the kinematic chain associated to the ith limb (i=1,2,…,k). H – characteristic point of the distal link/end-effector H- helical pair IFMA - Institut Français de Mécanique Avancée (French Institute of Advanced Mechanics) INRIA - Institut National de Recherche en Informatique et en Automatique (The French National Institute for Research in Computer Science and Control) IRCCyN - Institut de Recherche en Communications et Cybernétique de Nantes I n×n - n×n identity matrix J - Jacobian matrix JT – transpose of the Jacobian matrix J-1 – inverse of the Jacobian matrix JS - direct kinematics matrix/Jacobian JP - inverse kinematics matrix/Jacobian k - total number of limbs of the parallel manipulator k1 - number of simple limbs of a parallel manipulator k2 - number of complex limbs of a parallel manipulator KP - kinematic pair KC - kinematic chain Q - generic notation for a kinematic chain or a mechanism LaMI - Laboratoire de Mécanique et Ingénieries (Mechanical Engineering Research Group) LASMEA - Laboratoire des Sciences et Matériaux pour l'Electronique, et d'Automatique (Laboratory of Sciences and Materials for Electronic, and of Automatic) LIRMM - Laboratoire d'Informatique, de Robotique et de Microélectronique de Montpellier (Montpellier Laboratory of Computer Science, Robotics, and Microelectronics) Lc - characteristic length m - total number of links (mechanism elements) including the fixed base mQ - number of joints in kinematic chain Q

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M - mobility MQ - mobility of kinematic chain/mechanism Q i M - instantaneous mobility i MQ - instantaneous mobility of kinematic chain/mechanism Q N – number of overconstraints n – number of kinematic elements (links) excepting the fixed link Ο t = ( ο1 ,...,ο n ) - set of morphological operators applicable at each generation t O0x0y0z0 - reference frame O0On - position vector of point On in fixed reference frame p - number of joints pp - number of passive joints p - operational velocity vector P - prismatic joint/pair P - actuated prismatic joint Pa - parallelogram loop Pat - telescopic parallelogram loop Pn2 - planar close loop with two degrees of mobility Pn4 - planar close loop with three degrees of mobility PM - parallel manipulator q - number of independent closed loops q - joint velocity vector qi - finite displacement in the ith actuated joint r - number of joint parameters that lost their independence rQ - number of joint parameters that lost their independence in kinematic chain/mechanism B rl - number of joint parameters that lost their independence in the closed loops integrated in the complex limbs of a parallel robot Gi rl number of joint parameters that lost their independence in the closed loops which may exist in ith limb Gi of a parallel robot R – revolute joint/pair R – actuated revolute joint Rb – rhombus loop RF - the range of linear transformation F RQ - vector space of operational velocities of the kinematic chain/mechanism Q RaQ/ b - vector space of relative velocities between links a and b in kinematic chain/mechanism Q Q ( Ra / b ) - base of the vector space RaQ/ b R - reangularity

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S aQ/ b - connectivity between links a and b in a kinematic ckain/mechanism Q SQ - connectivity of kinematic chain/mechanism Q i SQ - instantaneous connectivity of kinematic chain/mechanism Q S . - semangularity T - structural redundancy Τ t = ( τ 1 ,...,τ n ) - set of evolution criteria from generation t to generation t+1 U - vector space of joint velocities UQ - vector space of joint velocities of kinematic chain/mechanism Q v - translational velocity vector w - general velocity vector of the moving platform w1, w2, …, w6 - general coordinates W - vector space of external velocities WQ - vector space of external velocities of kinematic chain/mechanism Q xP, yP, zP - coordinates of point P in the reference frame α , β ,δ ,ϕ - rotation angles

α , β ,δ ,ϕ - time derivatives of the rotation angles ∆q - vector of the infinitesimal motions of joints ∆ w - vector of infinitesimal motions of the end-effector ι : ιi → {true, false} - termination criterion for evolutionary morphology υ1 - conditioning index υ2 - minimum conditioning index υ3 and υ4 - manipulability indexes ξi - square root of the eigenvalues of (JJT)-1 λ - scalar / proportionality factor

µ - vector of the operational forces τ - vector of the joint forces ψ - velocity transmission factor ω - angular velocity Σt = (σ 1 ,...,σ n ) - set of solutions at each generation t Φ = ( φ1 ,...,φn ) - set of design objectives 0 - fixed base of a kinematic chain/mechanism 1-2-…-n - links (elements) of a kinematic chain/mechanism 1Q-2Q-…-nQ - links (elements) of kinematic chain/mechanism

1 Introduction

This section introduces the terminology and presents the main objectives of the book. The family of parallel robots is placed in the general field of robots and the structural synthesis in the general context of robotics.

1.1 Robot The origin of the term robot can be found in a science fiction play written by Karel Čapek (1920). This play titled R.U.R. was published in Czech in 1920 as an abbreviation of Rozum's Universal Robots, a collective drama in a comic prologue and three acts first performed in National Theatre in Prague on January 25, 1921. We note that in Czech “rozum” means wisdom and “robota” means servitude/forced labour/drudgery. Since the word robot was apparently coined by the author's brother, cubist painter and writer Joseph Čapek, the term is used to refer to a wide range of man-made things from devices to programs that are used to do automatically and autonomously various tasks by replacing humans. Human-made analogs of living systems could be found long ago in Antiquity despite the recent useage of the word robot. The pigeon conceived by the Greek mathematician Archytas of Tarentum in 425 BC and chronicled by Favorinus and by Aulus Gellius in his Attic Nights could be considered the first “flying robot” and the south-seeking chariot, possibly used by the Chinesse Yellow Emperor Hunag Ti, the first “field robot”. Other legendary solutions such as the “walking machines” called wooden ox and gliding horse of Chu-ko Liang were invented to transport army food supplies across rough terrain during the period of China’s Three Kingdoms in 209 AC (Yan 1999). The automatic machines (Al-Jazary 1973) constructed by Al-Jazari (1136-1206), the designs of mechanical knight (Rosheim 2006) and flying machines (Da Vinci) of Leonardo da Vinci (1452-1519), the mechanical duck and the flute player – the first automatons (Balpe and Drye 1994) constructed by Jacques de Vaucanson (1709-1782) and the androids (the draughtsman, the musicians and the writer) constructed in 1774 by Pierre Jacquet-Droz (1771-1790) and his

2

1 Introduction

son Henri-Louis (1753-1791) could also be considered as precursors of the current robots from the pre-digital era. Hundreds of robot definitions are known today and hundreds of millions of web sites could be referenced today by a simple search on the word robot. According to the American Heritage Dictionary a robot is a mechanical device that is capable of performing a variety of often complex human tasks on command or by being programmed in advance. Other definitions of the word robot proposed by on-line dictionaries are systematized in Table 1.1. Robots can be found today in the manufacturing industry, agricultural, military and domestic applications, space exploration, medicine, education, information and communication technologies, entertainment, etc. We can see that the word robot is mainly used to refer to a wide range of mechanical devices or mechanisms, the common feature of which is that they are all capable of movement and can be used to perform physical tasks. Robots take on many different forms, ranging from humanoid, which mimic the human form and mode of movement, to industrial, whose appearance is dictated by the function they are to perform. Robots can be categorized as robotic manipulators, wheeled robots, legged robots swimming robots, flying robots, androids and self reconfigurable robots which can conform themselves to a given task. This book focuses on the parallel robotic manipulators which are the counterparts of the serial robots. We also note that the term robot has recently morphed to also refer to “bots”, which are automated programs used in several online functions in computer science and artificial intelligence. The term has also entered into the informal language to define certain forms of human behaviour or even a traffic light in South Africa.

1.1 Robot

3

Table 1.1. Various definitions of the term robot Dictionary Encarta® World English Dictionary, North American Edition1

Definition 1. Programmable machine for performing tasks: a mechanical device that can be programmed to carry out instructions and perform complicated tasks usually done by people. 2. Imaginary machine: a machine that resembles a human in appearance and can function like a human, especially in science fiction. 3. Person like a machine: somebody who works or behaves mechanically and emotionlessly. 4. South Africa traffic light: a set of automatic traffic lights (informal).

Compact Oxford English Dictionary2

A machine capable of carrying out a complex series of actions automatically, especially one programmable by a computer.

Merriam-Webster's Online Dictionary, 10th Edition3

1a. A machine that looks like a human being and performs various complex acts (as walking or talking) of a human being; also: a similar but fictional machine whose lack of capacity for human emotions is often emphasized. 1b. An efficient insensitive person who functions automatically. 2. A device that automatically performs complicated often repetitive tasks. 3. A mechanism guided by automatic controls.

Cambridge International Dictionary of English4

1. A machine used to perform jobs automatically, which is controlled by a computer. 2. Disapproving someone who does things in a very quick and effective way but never shows their emotions. 3. A traffic light in South Africa.

Cambridge Dictionary A mechanical device that works automatically or by comof American English5 puter control. Columbia Encyclope- Mechanical device designed to perform the work generally dia, Sixth Edition6 done by a human being. 1

http://encarta.msn.com/encnet/features/dictionary/dictionaryhome.aspx http://www.AskOxford.com 3 http://www.m-w.com/ 4 http://dictionary.cambridge.org/ 5,6 http://dictionary.cambridge.org/ 2

4

1 Introduction

Table 1.1. (cont.) The American Heritage® Dictionary of the English Language7

1. A mechanical device that sometimes resembles a human and is capable of performing a variety of often complex human tasks on command or by being programmed in advance. 2. A machine or device that operates automatically or by remote control. 3. A person who works mechanically without original thought, especially one who responds automatically to the commands of others.

The Wordsmyth English DictionaryThesaurus8

1. A mechanical device, esp. one controlled electronically, that can perform physical tasks of a human being, in industrial assembly or the like. 2. A person who moves or acts mechanically, without thought or feeling.

Infoplease Dictionary9

1. A machine that resembles a human and does mechanical, routine tasks on command. 2. A person who acts and responds in a mechanical, routine manner, usually subject to another's will; automaton. 3. Any machine or mechanical device that operates automatically with humanlike skill.

Dictionary.com10

1. A mechanical device that sometimes resembles a human and is capable of performing a variety of often complex human tasks on command or by being programmed in advance. 2. A machine or device that operates automatically or by remote control. 3. A person who works mechanically without original thought, especially one who responds automatically to the commands of others.

Wiktionary11

1. A machine that resembles humans in shape or scope of function. 2. A machine that operates automatically. 3. A machine controlled by a fundamentally ingrained computer. 7 http://www.bartleby.com/61/s0.html 8 http://www.wordsmyth.net/ 9 http://www.infoplease.com/dictionary.html 10 http://www.dictionary.com/ 11 http://www.wiktionary.org

1.1 Robot

5

Table 1.1. (cont.) A mechanism that can move automatically. Rhymezone12 LookWAYup Translating Dictioary /Thesaurus13 WordNet 1.7 Vocabulary Helper14 Wikipedia, the Free Encyclopedia15

1. A robot is a type of device. 2. Robot can also refer to Internet bot, an automated software program used on the Internet.

AllWords.com Multi-Lingual Dictionary16

1. Especially in science-fiction: a machine that vaguely resembles a human being and which can be programmed to carry out tasks. Compare android. 2. An automatic machine that can be programmed to perform specific tasks. 3. Someone who works efficiently but who lacks human warmth or sensitivity. 4. An automatic traffic signal.

WebmasterWorld Webmaster and Search Engine Glossary17

Program that automatically does "some action" without user intervention. In the context of search engines, it usually refers to a program that mimics a browser to download web pages automatically. A spider is a type of robot. Some times referred to as Webbots.

Netlingo18

Traditionally, it's a device that can move and react to sensory input… The term "robot" has morphed to also refer to bots, which are automated programs used in several online functions.

Webopedia19

1. A device that responds to sensory input. 2. A program that runs automatically without human intervention. 12 http://www.rhymezone.com/ 13 http://lookwayup.com/free 14 http://poets.notredame.ac.jp/cgi-bin/wn 15 http://www.wikipedia.org 16 http://www.allwords.com/ 17 http://www.webmasterworld.com/glossary/ 18 http://www.netlingo.com/index.cfm 19 http://www.pcwebopaedia.com/

6

1 Introduction

Table 1.1. (cont.) CCI Computer20

1. A mechanical device that performs a task that would otherwise be done by a human. Robots can be useful for jobs that are boring or dangerous for humans to perform. The simplest robots are capable only of repeating a programmed motion; the most sophisticated models can use sensors and artificial intelligence to distinguish between objects, understand natural language, and make decisions. Robots can be programmed or operated by remote control. 2. A computer program that performs intelligent tasks such as retrieving World Wide Web documents and indexing references. 3. A program that performs a programmed communication function such as automated email answering, responding to newsgroup message, or regulatory functions in IRC, graphical chat, and other online environments

Computer Telephony & Electronics Dictionary and Glossary21

1. A device that responds to sensory input. They are often just called "Bots". 2. A program that runs automatically without human intervention. Typically, a robot is endowed with some artificial intelligence so that it can react to different situations it may encounter. Two common types of robots are agents and spiders.

Internet Terms22

A robot is a program that is designed to automatically go out and explore the Internet for a specific purpose. Robots that record and index all of the contents of the network to create searchable databases are sometimes called spiders or Worms. WebCrawler and Lycos are popular examples of robots.

TECHNICAL23

Robot - Besides being a mechanical device used to mimic human form, usually to accomplish some repetitive task, this term refers to a computer program that scans Web pages and links. Like a similar spider program, robots are used to scan Web pages and index them.

20

http://www.currents.net/resources/dictionary/index.html http://www.csgnetwork.com/glossary.html 22 http://www.learnthenet.com/english/glossary/glossary.htm 23 http://www.ugeek.com/glossary/glossary_search.htm 21

1.2 Robotics

7

1.2 Robotics The science of robotics has undergone an explosive growth since the word robotics was coined by the writer Isaac Asimov in 1950. In his science fiction book “I, Robot” he presented the famous three laws of robotics: (i) a robot may not injure a human being, or, through inaction, allow a human being to come to harm; (ii) a robot must obey the orders given it by human beings except where such orders would conflict with the first law and (iii) a robot must protect its own existence as long as such protection does not conflict with the first or second law. According to the American Heritage Dictionary, robotics is the science or study of the technology associated with the design, fabrication, theory, and application of robots. This technology combines mechanical devices, actuators, sensors, controllers, software and computers. Today robots get closer to humans and new issues rise in matters related to human robot interaction, in advances and experiences of robots and automation at home, at work, for education, as well as in other emerging areas. The next generation of robots will be capable of interacting with humans on a sophisticated level, making decisions, helping, caring and giving companionship. Recent research by the Japan Robotics Association (JPA), United Nations Economic Commission (UNEC) and the International Federation of Robotics (IFR) indicates that the nascent personal and service robotics market will exhibit exceptional near term growth. The service and personal robotics marketplaces together will equal the size of the industrial robotics market (the combination of manufacturing and bio-industrial) by 2005, and will be twice the size of the industrial robotics market by 2010, and almost 4X its size by 2025. Many of the technologies required to build functional personal and service robots already exist, and markets for these products are already in place. Some of the most salient enabling technologies include advances in microprocessor technology, wireless technology, image processing, speech recognition, motion sensor technology, and embedded systems development tools. Hundreds of universities worldwide have research programs in robotics and many are awarding degrees in robotics. Various definitions of the word robotics are systematized in Table 1.2. We can see that they converge towards the integration of the design and the end use in the studies related to robotics. This book focuses on the conceptual design of parallel robots.

8

1 Introduction

Table 1.2. Various definitions of the term robotics Dictionary Encarta® World English Dictionary, North American Edition1

Definition Design and use of robots: the science and technology relating to computer-controlled mechanical devices such as the automated tools commonly found on automobile assembly lines.

Compact Oxford English Dictionary2

The branch of technology concerned with the design, construction, and application of robots.

Merriam-Webster's Online Technology dealing with the design, construction, and Dictionary, 10th Edition3 operation of robots in automation. Cambridge International Dictionary of English4

The science of making and using robots.

Cambridge Dictionary of American English5

The science of designing and operating robots.

Columbia Encyclopedia, Sixth Edition6

Science and technology of general purpose, programmable machine systems.

The American Heritage® Dictionary of the English Language7,10

The science or study of the technology associated with the design, fabrication, theory, and application of robots.

The Wordsmyth English Dictionary-Thesaurus8

The technology of designing and using robots for various, usually industrial tasks.

Infoplease Dictionary9

The use of computer-controlled robots to perform manual tasks, especially on an assembly line.

Wiktionary11,15

The science and technology of robots, their design, manufacture, and application.

1

http://encarta.msn.com/encnet/features/dictionary/dictionaryhome.aspx http://www.AskOxford.com 3 http://www.m-w.com/ 4 http://dictionary.cambridge.org/ 5,6 http://dictionary.cambridge.org/ 7 http://www.bartleby.com/61/s0.html 8 http://www.wordsmyth.net/ 9 http://www.infoplease.com/dictionary.html 10 http://www.dictionary.com/ 11 http://www.wiktionary.org 2

1.2 Robotics

9

Table 1.2. (cont.) The area of AI concerned with the practical use of roRhymezone12 LookWAYup Translating bots Dictioary/ Thesaurus13 WordNet 1.7 Vocabulary Helper14 AllWords.com Multi-Lingual Dictionary16

1. The branch of engineering concerned with the design, construction, operation and use of industrial robots, and which incorporates many of the concepts used in artificial intelligence. 2. A form of dancing in which dancers imitate the stiff jerky movements of robots.

The On-line Medical Dictionary17

The application of electronic, computerised control systems to mechanical devices designed to perform human functions. Formerly restricted to industry, but nowadays applied to artificial organs controlled by bionic (bioelectronic) devices, like automated insulin pumps and other prostheses.

Netlingo18,19

The field of computer science and engineering concerned with creating robots. Robotics is a branch of artificial intelligence.

CCI Computer20

The design, manufacture, and use of robots.

Computer Telephony & The field of computer science and engineering conElectronics Dictionary and cerned with creating robots or bots, devices that can Glossary21 move and react to sensory input. Robotics is one branch of artificial intelligence. 12 http://www.rhymezone.com/ 13 http://lookwayup.com/free 14 http://poets.notredame.ac.jp/cgi-bin/wn 15 http://www.wikipedia.org 16 http://www.allwords.com/ 17 http://cancerweb.ncl.ac.uk/omd/ 18 http://www.netlingo.com/index.cfm 19 http://www.pcwebopaedia.com/ 20 http://www.currents.net/resources/dictionary/index.html 21 http://www.csgnetwork.com/glossary.html

10

1 Introduction

1.3 Parallel Robot Although the appearance and capabilities of robots vary vastly, all robots share the features of a mechanical, movable structure under some form of control. The structure of a robot is usually mostly mechanical and takes the form of a mechanism having as constituent elements the links connected by kinematic joints. Serial or parallel kinematic chains are concatenated in the robot mechanism. The serial kinematic chain is formed by links connected sequentially by joints. Links are connected in series as well as in parallel making one or more closed-loops in a parallel mechanism. The members of a mechanism may take various forms: bars, plates, platforms, cams, gears, sliders, sliding blocks, etc. The mechanical architecture of parallel robots is based on parallel mechanisms in which a member called a mobile platform is connected to a reference element by at least two limbs that can be elementary (simple) or complex. The robot actuators are integrated in the limbs (also called legs) usually closed to the fixed member, also called the base or the fixed platform. The mobile platform positions the robot end-effector in space and could have between two and six degrees of freedom. Usually the number of actuators coincides with the degrees of freedom of the mobile platform, exceeding them only in the case of redundantly-actuated parallel robots. The paradigm of parallel robots is the hexapod-type robot, which has six degrees of freedom, but recently the machine industry has discovered the potential applications of lower-mobility parallel robots with just 2, 3, 4 or 5 degrees of freedom. Indeed, the study of this type of parallel manipulators is very important. They exhibit interesting features if compared to hexapods, such as simpler architecture, simpler control system, high-speed performance, low manufacturing and operating costs. Furthermore, for several parallel manipulators with decoupled or uncoupled motions the kinematic model can be easily solved to obtain algebraic expressions, which are well suited for an implementation in optimum design problems. Parallel mechanisms can be considered a well-established solution for many different applications of manipulation, machining, guiding, testing, control, etc.

1.4 Terminology The terminology used in this book and defined in this section is mainly established in accordance with the terminology adopted by the International Federation for the Promotion of Mechanism and Machine Science

1.4 Terminology

11

(IFToMM) and published in (Ionescu 2003). The main terms used in this book concerning kinematic pairs (joints), kinematic chains and robot kinematics are defined in Tables 1.3-1.5. They are completed by some complementary remarks, notations and symbols used in this book. In the standard terminology, a kinematic chain is an assembly of links (mechanism elements) and joints, and a mechanism is a kinematic chain in which one of its links is taken as a “frame”. In this definition the “frame” is a mechanism element deemed to be fixed. In this book, we use the notion of reference element to define the “frame” element. The reference element can be fixed or may merely be deemed to be fixed with respect to other mobile elements. The fixed base is denoted in this book by 0. A mobile element in a kinematic chain Q is denoted by nQ (n=1, 2, …). Two or more links connected together in the same link such that no relative motion can occur between them are considered as one link. The identity symbol “≡” is used between the links to indicate that they are welded together in the same link. For example, the notation 1Q≡0 is used to indicate that the first link 1Q is connected to the fixed base by a common link. A kinematic chain Q is denoted by the sequence of its links. The notation Q (1Q≡0-2Q…nQ) indicates a kinematic chain in which the first member is fixed and the notation Q (1Q-2Q-…nQ) a kinematic chain with no fixed member. We will use the notion of mechanism to qualify the whole mechanical system, and the notion of kinematic chain to qualify the sub-systems of a mechanism. So, in this book, the same assembly of links and joints Q1 is considered to be a kinematic chain when it is integrated as a sub-system in another assembly of links and joints Q2, and it is considered a mechanism when Q1 represents the whole system. The systematization, the definitions and the formulae presented in this book are also valuable for mechanisms and kinematic chains. We use the term mechanism element or link to name a component (member) of a mechanism with any form (bar, gear, cam, etc.). In this text, unless otherwise stated, we consider all links to be rigid. We distinguish the following types of links: a) monary link - a mechanism element connected in the kinematic chain by only one joint (a link which carries only one kinematic pairing element), b) binary link - a mechanism element connected in the kinematic chain by two joints (a link connected to two other links), c) polinary link - a mechanism element connected in the kinematic chain by more than two joints (ternary link - if the link is connected by three joints, quaternary link if the link is connected by four joints,…).

12

1 Introduction

Table 1.3. IFToMM terminology with respect to the kinematic pairs Term Pairing element

Definition Assembly of surfaces, lines or points of a solid body through which it may contact with another solid body.

Mechanism element

Solid body or fluid component of a mechanism.

Frame

Mechanism element deemed to be fixed.

Base

Body (link) of a robot that carries the first joint(s) of the kinematic chain of a manipulator or pedipulator.

Rigid body

Theoretical model of a solid body in which the distances between particles are considered to be constant, regardless of any forces acting upon the body.

Link

1. Mechanism element (component) carrying kinematic pairing elements. 2. Element of a linkage.

Limit position of a link

Position of a link for which a coordinate which describes its position relative to an adjacent link is a maximum or a minimum.

Input (driving) Link whereby motion and force are imparted to a mechanism. link Output (driven) link Link from which required forces and motions are obtained. (follower) Bar Link that carries only revolute joints. Crank

Link able to rotate completely about a fixed axis.

Rocker

Link that oscillates within a limited angle of rotation about a fixed axis.

Coupler (floating link) Slider

Link that is not connected directly to the frame. Link that forms a prismatic pair with one link and a revolute pair with another link.

1.4 Terminology

13

Table 1.3. (cont.) Sliding block

Compact element of a prismatic pair which slides along a guiding element (e.g. in a slot).

Guide

Element of a prismatic pair that is fixed to the frame and constrains the motion of a sliding block.

Crosshead

Component between a piston and a connecting rod which, by forming a prismatic joint with the frame, provides a reaction to the component of force in the connecting rod normal to the line of stroke of the piston.

Connecting rod

Coupler between a piston or a crosshead and a crank shaft.

Cam

Component with a curved profile or surface whereby it imparts a displacement either by point or line contact with a cam follower.

Kinematic pair

Mechanical model of the connection of two pairing elements having relative motion of a certain type and degree of freedom. Note: see synonym at joint.

Joint

1. Physical realization of a kinematic pair, including connection via intermediate mechanism elements. 2. Kinematic pair.

Degree of freedom (connectivity) of a kinematic pair Closure of a kinematic pair

The number of independent coordinates needed to describe the relative positions of pairing elements. Process of constraining two rigid bodies to form a kinematic pair by force (force closure), geometric shape (form closure), or flexible materials (material closure).

Force-closed (open) pair

Kinematic pair the elements of which are held in contact by means of external forces.

Form-closed pair

Kinematic pair the elements of which are constrained to contact each other by means of particular geometric shapes.

Lower pair

Kinematic pair that is formed by surface contact between its elements.

14

1 Introduction

Table 1.3. (cont.) Higher pair

Kinematic pair that is formed by point or line contact between its elements.

Revolute pair (hinge) Prismatic pair

Pair that allows only a rotary motion between two links. Pair that allows only a rectilinear translation between two links.

Helical (screw) pair

Pair that allows only a screw motion between two links.

Cylindrical pair

Pair for which the degree of freedom is two and that allows a rotation about a particular axis together with an independent translation in the direction of this axis.

Spherical pair

Pair for which the degree of freedom is three and that allows independent relative rotations about three mutually orthogonal axes.

Planar contact (sandwich) pair

Pair for which the degree of freedom is three and that allows relative motion in parallel planes.

Universal (Cardan, Kinematic joint connecting two shafts with intersecting axes. Hooke’s) joint (universal coupling) Pin joint Joint using a pin as the connecting component between two rigid bodies. Joint variable

Quantity that describes the relative motion between two consecutive links of a manipulator.

Joint space

Space defined by a vector whose components are joint variables.

1.4 Terminology

15

Table 1.4. IFToMM terminology with respect to the kinematic chains Term Kinematic chain

Definition Assembly of links and joints.

Closed kinematic chain

Kinematic chain each link of which is connected with at least two other links.

Open kinematic chain

Kinematic chain in which there is at least one link which carries only one kinematic pairing element.

Loop

Subset of links that forms a closed circuit.

Tree (mobile)

Kinematic chain that contains no loops.

Linkage

Kinematic chain whose joints are equivalent to lower pairs only.

Four-bar linkage

Linkage with four binary links.

Kinematic joint

Kinematic chain whose kinematical properties are equivalent in some respects to those of a kinematic pair.

Mechanism

1. System of bodies designed to convert motions of, and forces on, one or several bodies into constrained motions of, and forces on, other bodies. 2. Kinematic chain with one of its components (links) taken as a frame.

Kinematic inversion

Transformation of one mechanism into another by choosing a different member to be the frame.

Limit position of a mechanism

Configuration of a mechanism in which one of its links is in a limit position

Planar mechanism Mechanism in which all points of its links describe paths located in parallel planes. Spherical mechanism

Mechanism in which all points of its links describe paths located on concentric spheres.

16

1 Introduction

Table 1.4. (cont.) Spatial mechanism

Mechanism in which some points of some of its links describe non-planar paths, or paths located in non-parallel planes.

Guidance mechanism

Mechanism that guides a link through a prescribed sequence of positions.

Four-bar mechanism

Mechanism with four binary links.

Structure (of a mechanism)

Number and kinds of elements in a mechanism (members and joints) and the sequence of their contact.

Isomorphism

Equality of structures in respect of the numbers of members and joints, and the sequence of their interconnections.

Equivalent mechanism

Mechanism whose kinematical properties are equivalent in some respects to those of another mechanism with a different structure.

Degree of freedom Number of independent coordinates needed to define the con(mobility) of a figuration of a kinematic chain or mechanism. kinematic chain or mechanism Constraint

Any condition that reduces the degree of freedom of a system.

Drive

System of mutually connected devices for setting in motion one or several parts of a machine or a mechanism.

Actuator

Device that creates direct physical action on the process, other mechanical devices or the surrounding environment to perform some useful function.

1.4 Terminology

17

Table 1.5. IFToMM terminology with respect to the robot kinematics Term Robot

Definition Mechanical system under automatic control that performs operations such as handling and locomotion.

Manipulator

Device for gripping and the controlled movement of objects.

Anthropomorphic robot

Robot with a manipulator containing rotary joints similar to those of a human arm.

End-effector

Device attached to the distal end of a robot arm by which objects can be grasped or acted upon.

Gripper

End-effector which grasps, holds and releases objects.

Module

Self-contained assembly of robot elements which can be connected in various ways with other, not necessarily identical, modules to form a robot.

Parallel manipulator Manipulator that controls the motion of its end-effector by means of at least two kinematic chains joining the endeffector to the frame. Walking machine

Robot that performs locomotion functions similar to human beings or animals.

Pedipulator

Articulated leg of a walking machine.

Proximal

Close to the body (away from the end-effector) of a robot arm.

Distal

Away from the base (towards the end-effector) of a robot.

Motion

Changing position of a body relative to a frame of reference.

Absolute motion

Motion with respect to a fixed frame of reference.

Relative motion

Motion with respect to a moving frame of reference.

Inverse motion

Motion of a frame of reference relative to a moving body.

Frame motion (transportation)

Motion of a moving frame of reference.

18

1 Introduction

Table 1.5. (cont.) Displacement

Change of position of a body with respect to a fixed frame of reference.

Path (trajectory)

Line that a moving point describes in a given system of reference.

Axis of rotation

Straight line in a rotating rigid body whose points have zero displacement either in a finite time interval, or in an infinitesimal time interval, with respect to a frame of reference.

Screw axis

Straight line in rigid body whose points are displaced relative to a frame of reference, either in a finite or an infinitesimal time interval, coaxial with the line itself.

Translation

Motion (or component of the motion) of a rigid body in which each straight line rigidly connected with the body remains parallel to its initial direction.

Rectilinear transla- Translation in which the paths of points of a rigid body are tion straight lines. Rotation

Motion (or a component of the motion of) a rigid body in which all its points move on circular arcs centred on the same axis.

Angular displacement Angle of rotation

Displacement of a rigid body in rotation. Angle turned through by any line rigidly connected to a rotating body and perpendicular to the axis of rotation.

Precession

Rotation of a rigid body about an axis fixed in space when combined with rotation about an axis fixed in the body, the two axes being concurrent.

Nutation

Motion of a rigid body occurring simultaneously with a precession, when the angle between the axes of rotation and precession varies in time.

Orientation

Movement or manipulation of a rigid body into a predetermined attitude.

Pose

Combination of position and attitude.

1.4 Terminology

19

Table 1.5. (cont.) Reference point

Point chosen for position reference in defining a pose.

Planar (plane) motion

Motion of a rigid body in which its points describe curves located in parallel planes.

Spatial motion

Motion of a body in which at least one of its points describes a spatial curve.

Screw motion

Motion consisting of a rotation combined with a translation that is parallel to the rotation.

Spherical motion

Spatial motion of a body in which all points of the body move on concentric spheres.

Limit position

Configuration of a mechanism in which the position of a particular link, such as the output, is in some significant respect a maximum or a minimum.

Dead point

Configuration of a mechanism in which the input cannot be moved without assistance to the motion of another link.

Velocity

Rate of displacement with respect to time.

Absolute velocity

Velocity with respect to a fixed frame of reference.

Relative velocity

Velocity with respect to a moving frame of reference.

Frame (transporta- Absolute velocity of a particular point of a moving frame of tion) velocity reference. Angular velocity

Rate of angular displacement with respect to time.

Instantaneous screw Locus of the points in a rigid body in general spatial motion, axis whose linear velocity is parallel to the angular velocity vector of the body at the given instant. Working range

Range of any variable for normal operation of a robot.

Working space (volume)

Totality of points that can be reached by the reference point of a robot arm.

20

1 Introduction

Table 1.5. (cont.) Singular configura- Special position of the robot links which implies a reduction tion of mobility of the end-effector. Direct task

Computation of the pose, motion and forces at the endeffector of a robot arm from given actuator forces, displacements, velocities and accelerations.

Inverse task

Computation of actuator forces, displacements, velocities and accelerations from given forces, pose and motion of the endeffector of a robot.

Redundant mobility Amount by which the degree of freedom of a robot exceeds (of a robot) the number of independent variables that are necessary to define the task to be performed. Redundancy

Existence of more than one means to accomplish a given function.

Manoeuvrability

Ability of a robot with redundant mobility to solve a task by using various combinations of movements of its links.

Kinematics

Branch of theoretical mechanics dealing with the geometry of motion, irrespective of the causes that produce the motion.

Kinematic analysis Analysis of the kinematic aspects of mechanisms.

The IFToMM terminology defines open/closed kinematic chains and mechanisms, but it does not introduce the notions of simple (elementary) and complex kinematic chains and mechanisms. A closed kinematic chain is a kinematic chain in which each link is connected with at least two other links, and an open kinematic chain is a kinematic chain in which there is at least one link which carries only one kinematic pairing element (see Tables 1.3 and 1.4). In a simple open kinematic chain (open-loop mechanism) only monary and binary links are connected. In a complex kinematic chain at least one ternary element exists. We designate in each mechanism two extreme elements called reference element and final element. They are also called distal links. In an open kinematic chain, these elements are situated at the extremities of the chain. In a single-loop kinematic chain, the final element can be any element of the chain except the reference element. In a parallel mechanism, the two extreme elements are the mobile and the reference platform. The two platforms are connected by at least two elemen-

1.4 Terminology

21

tary or complex kinematic chains called limbs. Each limb contains at least one joint. An elementary limb is composed of a simple open kinematic chain in which the final element is the mobile platform. A complex limb is composed of a complex kinematic chain in which the final element is also the mobile platform. As we can see in Table 1.3, IFToMM terminology (Ionescu 2003) use the term kinematic pair to define the mechanical model of the connection of two pairing elements having relative motion of a certain type and degree of freedom. The word joint is used as a synonym for the kinematic pair and also to define the physical realization of a kinematic pair, including connection via intermediate mechanism elements. Both synonymous terms are used in this text. Usually, in parallel robots, lower pairs are used: revolute R, prismatic P, helical H, cylindrical C, spherical S and planar pair E. The definitions of these kinematic pairs are presented in Table 1.3. The graphical representations used in this book for the lower pairs are presented in Fig. 1.1(a)-(f). Universal joints and homokinetic joints are also currently used in the mechanical structure of the parallel robots to transmit the rotation motion between two shafts with intersecting axes. If the instantaneous velocities of the two shafts are always the same, the kinematic joint is homokinetic (from the Greek “homos” and “kinesis” meaning “same” and “movement”). We know that the universal joint (Cardan joint or Hooke’s joint) are heterokinetic joints. Various types of homokinetic joints are known today: Tracta, Weiss, Bendix, Dunlop, Rzeppa, Birfield, Glaenzer, Thompson, Triplan, Tripode, UF (undercut-free) ball joint, AC (angular contact) ball joint, VL plunge ball joint, DO (double offset) plunge ball joint, AAR (angular adjusted roller), helical flexure U-joints, etc. (Dudiţǎ et al. 2001a,

Fig. 1.1. Symbols used to represent the lower kinematic pairs and the kinematic joints: (a) revolute pair, (b) prismatic pair, (c) helical pair, (d) cylindrical pair, (e) spherical pair, (f) planar contact pair, (g) universal joint and (h) homokinetic joint

22

1 Introduction

Fig. 1.2. Various representations of a Gough-Stewart type parallel robot: physical implementation (a), CAD model (b), structural diagram (c) and its associated graph (d), A-limb (e) and its associated graph (f)

1.5 Structural synthesis

23

2001b). The graphical representations used in this book for the universal homokinetic joints are presented in Fig. 1.1(g)-(h). A parallel robot can be illustrated by a physical implementation or by an abstract representation. The physical implementation is usually illustrated by robot photography and the abstract representation by CAD model, structural diagram and structural graph. Figure 1.2 gives an example of the various representations of a Gough-Stewart type parallel robot largely used today in industrial applications. The physical implementation in Fig. 1.2a is a photograph of the parallel robot built by Deltalab (http://www.deltalab.fr/). In a CAD model (Fig. 1.2b) the links and the joints are represented as being as close as possible to the physical implementation (Fig. 1.2a). In a structural diagram (Fig. 1.2c) they are represented by simplified symbols, such as those introduced in Fig. 1.1, respecting the geometric relations defined by the relative positions of joint axes. A structural graph (Fig. 1.2d) is a network of vertices or nodes connected by edges or arcs with no geometric relations. The links are noted in the nodes and the joints on the edges. We can see that the Gough-Stewart type parallel robot has six identical limbs denoted in Fig. 1.2c by A, B, C, D , E and F. The final link is the mobile platform 4≡4A≡4B≡4C≡4D≡4E≡4F and the reference member is the fixed platform 1A≡1B≡1C≡1D≡1E≡1F≡0. Each limb is connected to both platforms by spherical pairs. A prismatic pair is actuated in each limb. The spherical pairs are not actuated and are called passive pairs. The two platforms are polinary links, the other two links of each limb are binary links. The mechanism associated with the GoughStewart type parallel robot is a complex mechanism with a multi-loop associated graph (Fig. 1.2d). The simple open kinematic chain associated with A-limb is denoted by A (1A≡0-2A-3A-4A≡4) – Fig. 1.2e and its associated graph is tree-type (Fig. 1.2f).

1.5 Structural synthesis Recent advances in research on parallel robots have contributed mainly to expanding their potential use to both terrestrial and space applications including areas such as high speed manipulation, material handling, motion platforms, machine tools, medical applications, planetary and underwater exploration. Therefore, the need for methodologies devoted to the systematic design of highly performing parallel robots is continually increasing. Structural synthesis is directly related to the conceptual phase of robot design, and represents one of the highly challenging subjects in recent robotics research. One of the most important activities in the invention and the

24

1 Introduction

design of parallel robots is to propose the most suitable solutions to increase the performance characteristics. The challenging and difficult objective of structural synthesis is to find a method to set up the mechanical architecture achieving the required structural parameters. The mechanical architecture is defined by number, type and relative position of joint axes in the parallel robot. The structural parameters are mobility, connectivity, redundancy and degree of overconstraint. They define the number of actuators, the degrees of freedom and the motion-type of the moving platform. In general, parallel manipulators’ performances are highly dependent on their mechanical architecture, so that structural synthesis becomes the central problem in the conceptual design phase, but only a few works can be found in the literature on this topic. We note that this is the first book focusing on the structural synthesis of the mechanical architecture of parallel robots. The topic of this book addresses the problem of structural, also called topological, synthesis of parallel robotic manipulators in a systematic way. This is an “urgent need” put forward by robot manufacturers and scientists. Torgny Brogårdh (2002) from ABB Automation Technology Products / Robotics, in his conference paper entitled “PKM Research - Important Issues, presented as seen from a Product Development Perspective at ABB Robotics” delivered in the Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, said: “This paper will address some of the most important PKM research issues as seen from a robot manufacturer’s point of view. In section 2 the importance of linking research activities to the application requirements is discussed and in section 3, which is the most important part of this paper, the urgent need for a systematic topology synthesis is put forward. The rest of the paper is structured according to a proposed development process for industrial parallel kinematic robots.” In his keynote speech entitled “Still a long way to go on the road for parallel mechanisms” presented in ASME 2002 DETC Conference, JeanPierre Merlet from INRIA Sophia Antipolis, France, the author of the first books on parallel robots (Merlet 1990, 1997, 2006) has stated (Merlet 2002): “Synthesis of parallel robot is an open field (there is a very limited number of papers addressing this issue) and, in my opinion, one of the main issue for the development of parallel robots in practice”. The importance of design synthesis defined by structural and dimensional synthesis is also reiterated by J-P. Merlet (2005): “ Synthesis of robots may be decomposed into two processes: structural synthesis (determine the general arrangement of the mechanical struc-

1.6 The objectives and originality of this book

25

ture such as the type and number of joints and the way they will be connected) and dimensional synthesis (determine the length of the links, the axis and location of the joints, the necessary maximal joint forces/torques). The performances that may be obtained for a robot are drastically dependent on both synthesis.” In this paper J-P. Merlet has put the emphasis on dimensional synthesis. In this book, we look mainly at structural synthesis which we consider as an essential step in robot design and optimization.

1.6 The objectives and originality of this book Early research on parallel manipulators (PMs) concentrated on the six degrees of freedom Gough-Stewart-type mechanism introduced by Gough for a tire-testing device (Gough and Whitehall, 1962) and by Stewart for flight simulators (Stewart, 1965). In the last decade, parallel manipulators with fewer than six degrees of freedom have attracted researchers’ attention. Lower mobility solutions are suitable for many tasks requiring less than six independent motions of the end-effector. Parallel mechanisms have been the subject of study of much robotic research during the last two decades. Some important studies are now available to support the design of parallel manipulators, such as the pioneering works of Hunt (1973, 1978, 1983) that showed various kinematic architectures of parallel robots, the comprehensive enumerations presented by Merlet (1990, 1997, 2000, 2006) and Tsai (1999, 2001), and the various methods for systematic synthesis of PMs presented in the latest editions of Advances in Robot Kinematics (Lenarčič J, Parenti-Castelli V 1996; Lenarčič and Husty 1998; Lenarčič J, Stanišić MM 2000; Lenarčič and Thomas 2002; Lenarčič and Galletti, 2004; Lenarčič and Roth, 2006). The motion freedoms of the end-effector are usually coupled together due to the multi-loop kinematic structure of the parallel manipulator(PM). Hence, motion planning and control of the end-effector for PMs usually become very complicated. With respect to serial manipulators, such mechanisms can offer advantages in terms of stiffness, accuracy, load-toweight ratio, dynamic performances. Their disadvantages include a smaller workspace, complex command and lower dexterity due to a high motion coupling, and multiplicity of singularities inside their workspace. Uncoupled, fully-isotropic and maximally regular PMs can overcome these disadvantages. Redundancy in parallel manipulators is used to eliminate some singular configurations (Wang and Gosselin 2004; Kurtz and Hayward 1992; Mer-

26

1 Introduction

let 1997; Firmani and Podhorodeski 2004, Alberich et al. 2006), to minimize the joint rates, to optimize the joint torques/forces (Dasgupta and Mruthyunjaya 1998b; Bruckman et al 2006; Nokleby et al. 2005) to increase dexterity workspace (Marquet et al 2001a,b), stiffness (Chakarov 2004), eigenfrequencies, kinematic and dynamic accuracy (Valasek et al. 2004), to improve velocity performances (Krut et al. 2004) and both kinematic and dynamic control algorithms (Liu et al. 2001; Cheng at al. 2003), to develop large forces in micro electro-mechanical systems (Mukherjee and Murlidhar 2001), decoupling the orientations and the translations (Jin et al. 2004), to obtain reconfigurable platforms (Mohamed and Gosselin 2005) and limbs (Fischer et al. 2004) or combined advantages (Nahon and Angeles 1991; Zanganach and Angeles 1994a,b; Kim 1997; Kock and Scumacher 1998, 2000; Mohamed 2003a,b). For the first time in the literature, the author recently presented new formulae for calculating the degree of redundancy of parallel robots, along with the use of redundancy for motion decoupling of parallel robots (Gogu 2006a). Redundancy is achieved by introducing additional actuated joints within the limbs beyond the minimum necessary for actuating the manipulator. Isotropy of a robotic manipulator is related to the condition number of its Jacobian matrix, which can be calculated as the ratio of the largest and the smallest singular values. A robot manipulator is said to be isotropic if the condition number of its Jacobian matrix is equal to one. A robot manipulator is said to be fully-isotropic if it is isotropic throughout the entire workspace. We know that the Jacobian matrix of a robotic manipulator is the matrix mapping (i) the actuated joint velocity space on the end-effector velocity space, and (ii) the static load on the end-effector and the actuated joint forces or torques. Thus, the condition number of the Jacobian matrix is a useful performance indicator characterizing the distortion of a unit hypersphere under this linear mapping. The condition number of the Jacobian matrix was first used by Salisbury and Craig (1982) to design mechanical fingers and developed by Angeles (1997) as a kinetostatic performance index of robotic mechanical systems. The isotropic design aims at ideal kinematic and dynamic performance of the manipulator (Zanganeh and Angeles 1997; Takeda and Funabashi 1998; Baron and Barnier 2001; Baron et al. 2002; Mohammadi Daniali and Zsombor-Murray 1999; Badescu et al. 2002; Fattah and Ghasemi 2002; Krut et al. 2002; Caro et al. 2003; Fassi et al. 2003, 2005; Tsai and Hunag 2003; Carricato and ParentiCastelli 2002; Carricato 2005; Liu et al. 2005; Pashkevich et al. 2005). The author has proposed a systematization of PMs based on the form of the Jacobian matrix J. We distinguish five types of PMs (Gogu, 2004a; Gogu, 2007a): (i) maximally regular PMs, if the Jacobian is an identity

1.6 The objectives and originality of this book

27

matrix throughout the entire workspace, (ii) fully-isotropic PMs, if the Jacobian is a diagonal matrix with identical diagonal elements throughout the entire workspace, (iii) PMs with uncoupled motions if J is a diagonal matrix with different diagonal elements, (iv) PMs with decoupled motions, if J is a triangular matrix and (v) PMs with coupled motions if J is neither a triangular nor a diagonal matrix. Maximally regular and fully-isotropic PMs give a one-to-one mapping between the actuated joint velocity space and the external velocity space. The first solution for a fully-isotropic T3-type translational parallel robot was developed at the same time and independently by Carricato and Parenti-Castelli at University of Genoa, Kim and Tsai at University of California, Kong and Gosselin at University of Laval, and the author of this work at the French Institute of Advanced Mechanics. In 2002, the four groups published the first results of their works (Carricato and ParentiCastelli 2002; Kim and Tsai 2002; Kong and Gosselin 2002b; Gogu 2002b). Each of the last three groups has built a prototype of this robot in their research laboratories and called this robot CPM (Kim and Tsai 2002), Orthogonal Tripteron (Gosselin et al. 2004) or Isoglide3-T3 (Gogu 2004b). The first practical implementation of this robot was the CPM developed at the University of California. The systematic synthesis of spatial mechanisms used in parallel robots is mainly founded on four approaches: group theory, screw theory, velocityloop equations and theory of linear transformations (Table 1.6). The method proposed in this book is founded on the theory of linear transformations and evolutionary morphology (Gogu 2005a) and allows us to obtain the structural solutions of decoupled, uncoupled, fully-isotropic and maximally regular PMs with two to six DoFs in a systematic way. The new formulae for mobility, connectivity (spatiality), redundancy and overconstraint of PMs proposed recently by the author, Gogu (2005a, 2005b, 2005c, 2005d) are integrated into the synthesis approach developed in this book. Various solutions of TaRb-type PMs are known today. In this notation, a=1,2,3 indicates the number of independent translations and b=1,2,3 the number of independent rotations of the moving platform. The parallel robots actually proposed by the robot industry have coupled and decoupled motions and just some isotropic positions in their workspace. As far as we are aware, this is the first book of robotics presenting solutions of uncoupled, fully-isotropic and maximally regular PMs.

28

1 Introduction

Table 1.6. Approaches used in the systematic synthesis of spatial mechanisms and parallel robots Approach Group theory

References Hervé 1978, 1995, 1999, 2004; Hervé and Sparacino 1991; Karouia and Hervé 2000, 2002, 2004; Angeles 2004; Li et al. 2004; Huynh and Hervé 2005; Lee and Hervé 2006; Rico et al. 2006.

Screw theory

Tsai 1999; Fang and Tsai 2002; Frisoli et al. 2000; Kong and Gosselin 2001, 2004a, 2004b, 2004c, 2004d, 2006; Huang and Li 2002, 2003; Carricato 2005

Velocity-loop equations Di Gregorio and Parenti-Castelli 1998; Di Gregorio 2002, Carricato and Parenti-Castelli 2002, 2003 Theory of linear transformations

Gogu 2002, 2003, 2004a, 2004b, 2004c, 2004d, 2005e, 2005f, 2005g, 2005h

Non-redundant/redundant, overconstrained/isostatic solutions of uncoupled and fully-isotropic PMs with elementary/complex limbs actuated by linear/rotary actuators with/without idle mobilities and two to six DoFs are present in a systematic approach of structural synthesis. These solutions are derived from families of PMs with decoupled motions. A serial kinematic chain is associated with each elementary limb, and at least one closed loop is integrated in each complex limb. The synthesis methodology and the solutions of PMs presented in this book represent the outcome of some recent research developed by the author in the last four years in the framework of the projects ROBEA-MAX (2002-2003) and ROBEA-MP2 (2004-2005) supported by the French National Council of Scientific Research (CNRS). These results have been partially published by the author in the last two years. In these works the author has proposed the following for the first time in the literature: a) new formulae for calculating the degree of mobility, the degree of connectivity(spatiality), the degree of redundancy and the number of over-

1.6 The objectives and originality of this book

29

constraints of parallel robotic manipulators that overcome the drawbacks of the formulae founded on the Chebychev-Grübler-Kutzbach (CGK) mobility formula (Gogu 2005b-e), b) a new approach to systematic innovation in engineering design called evolutionary morphology (Gogu 2005a), c) solutions to follow fully-isotropic parallel robots: parallel wrists with two (Gogu 2005f) and three (Gogu 2007c) degrees of mobility, planar parallel robots (Gogu 2004c), parallel robots of T1R2-type (Gogu 2005i), parallel robots with Schönflies motions (Gogu 2004a, 2005g, 2006c, 2007a), parallel robots of T3R2 type (Gogu 2006a,b,d), hexapods with six degrees of freedom (Gogu 2006e). The various solutions proposed by the author belong to a modular family called IsoglideN-TxRy with a+b=n with 2 ≤ n ≤ 6, a=1,2,3 and b=1,2,3. The mobile platform of these robots (Gogu 2007b) can have any combination of n independent translations (T) and rotations (R).The IsoglidenTaRb modular family was developed by the author and his research team at the French Institute of Advanced Mechanics (IFMA) in ClermontFerrand. The two parts of this work gather the author’s contributions, and develop them in a general and organic treatment of structural synthesis of parallel robotic manipulators, including new solutions presented for the first time in the literature. The main objectives of this book are as follows: • to present an original method of defining and calculating the structural parameters of parallel robots (mobility, connectivity, redundancy, degree of overconstraint) founded on the theory of linear transformations, • to present an original method for structural synthesis of parallel robots founded on the theory of linear transformations and evolutionary morphology, • to propose innovative solutions of parallel manipulators with 2 to 6 degrees of freedom and various degrees of motion coupling, with the emphasis on decoupled, uncoupled, fully-isotropic and maximally regular solutions. The innovative solutions presented in this book are useful in the design of new parallel robots for many applications: haptic devices, machine-tools called parallel-kinematics-machines (PKM), robot-assisted surgery, surveillance, telescope and pointing devices, motion simulation, testing machines, etc. The book will be useful for students, engineers and researchers working in associated fields of engineering. Many solutions of parallel robots obtained through this systematic approach of structural synthesis are presented here for the first time in the lit-

30

1 Introduction

erature. The author had to make a difficult and challenging choice between protecting these solutions through patents, and releasing them directly into the public domain. The second option was adopted by publishing them in various recent scientific publications and mainly in this book. In this way, the author hopes to contribute to a rapid and widespread implementation of these solutions in future industrial products.

2 Structural parameters

This section introduces the main structural parameters used in the structural synthesis of parallel robots: connectivity, mobility, redundancy, degree of overconstraint. We recall that IFToMM definitions of mobility, connectivity, redundancy and constraint are presented in Tables 1.3-1.5. In this chapter, we present an original approach based on the theory of linear transformations to define and to demonstrate new formulae for the calculation of these parameters along with the state of the art before the first publication of this new approach by the author of this work (Gogu 2005be). We will show that the classical Chebychev-Grübler-Kutzbach formula for global mobility calculation of multi-loop mechanisms does not fit many parallel manipulators. We know that a mobility formula is an explicit relationship between mobility and the structural parameters of the mechanism. Usually, these structural parameters are easily determined by inspection without the need to develop the set of constraint equations used in instantaneous mobility calculation. We introduce a mathematical framework for defining and calculating the global mobility of parallel robots. With this formal structure, combined with ideas from the theory of linear transformations, we define a new mobility formula for parallel manipulators with simple or complex limbs. Limb connectivity and mobile platform connectivity are the key parameters of the new mobility formula. These parameters are used also to calculate the degree of redundancy and the degree of overconstraint of a parallel manipulator. They are also useful to determine the necessary conditions to obtain a non-overconstrained solution. As far as we are aware, this is the first book to present the general relations between mobility, redundancy and overconstraint of parallel manipulators and the connectivity of the mobile platform. These relations are demonstrated via the theory of linear transformations and are used mainly in the structural analysis and synthesis presented in the following chapters. Examples of practical applications of the new formulae in structural analysis of parallel robots and other single and multi-loop mechanisms are presented in chapter 3.

32

2 Structural parameters

2.1 Critical review of mobility calculation Mobility is the main structural parameter of a mechanism and also one of the most fundamental concepts in the kinematic and the dynamic modelling of mechanisms. IFToMM terminology defines the mobility or the degree of freedom as the number of independent coordinates required to define the configuration of a kinematic chain or mechanism (Ionescu 2003). Mobility M is used to verify the existence of a mechanism (M>0), to indicate the number of independent parameters in robot modelling and to determine the number of inputs needed to drive the mechanism. Earlier works on the mobility of mechanisms go back to the second half of the nineteenth century. During the twentieth century, sustained efforts were made to find general methods for the determination of the mobility of any rigid body mechanism. Various formulae and approaches were derived and presented in the literature. Contributions have continued to emerge in the last years. Mobility calculation still remains a central subject in the theory of mechanisms. In the calculation of a mechanism’s mobility, the following key controlling parameters are used: the number of joints, p; the number of mechanism elements n (n=m-1; m denotes the total number of elements, including the fixed base); the mobility f; the degree of constraint, c ,of the joint; the motion parameter,b; the constraint parameter d=6-b of the mechanism; and the number of independent closed loops of the mechanism, q. The various methods proposed in the literature for mobility calculation of the closed loop mechanisms fall into two basic categories: a) approaches for mobility calculation based on setting up the kinematic constraint equations and calculating their rank for a given position of the mechanism with specific joint locations, b) formulae for a quick calculation of mobility without the need to develop the set of constraint equations. The approaches used for mobility calculation based on setting up the kinematic constraint equations and their rank calculation are valid without exception. The major drawback of these approaches is that the mobility cannot be determined quickly without setting up the kinematic model of the mechanism. Usually this model is expressed by the closure equations that must be analyzed for dependency. There is no way to derive information about mechanism mobility without performing position, velocity or static analysis by using analytical tools (screw system theory, linear algebra, affine geometry, Lie algebra, etc). For this reason, the real and practical value of these approaches is very limited in spite of their valuable theoretical foundations. The rank of the constraint equations is calculated in a

2.1 Critical review of mobility calculation

33

given position of the mechanism with specific joint locations. The mobility calculated in relation to a given configuration of the mechanism is an instantaneous mobility which can be different from the general mobility (global mobility or gross mobility). Global mobility has a single value for a given mechanism; it is a global parameter characterizing the mechanism in all its configurations except its singular ones. Instantaneous mobility is a local parameter characterizing the mechanism in a given configuration including singular ones. In a singular configuration the instantaneous mobility could be different from the global mobility. The formula for a quick calculation of mobility is an explicit relationship between structural parameters of the mechanism: the number of links and joints, the motion/constraint parameters of joints and of the mechanism. Usually, these structural parameters are easily determined by inspection without any need to develop the set of kinematic constraint equations. We note that the classical formulae for a quick calculation of mobility do not fit many classical mechanisms such as the mechanisms proposed by De Roberval 1670; Sarrus 1853; Delassus 1900, 1902, 1922; Bennett 1902; Bricard 1927; Myard 1931a,b; Goldberg 1943; Altman 1952; Baker 1978; Baker et al. 1982, Waldron 1979; Wohlhart 1987, 1991, 1995; etc., or for many recent parallel robots. Special geometric conditions play a significant role in the determination of the mobility of these mechanisms. When these mechanisms were limited to special examples, considered as “curiosities”, they were called by various names: paradoxical mechanisms (Bricard, 1927), paradoxical chains (Hervé 1999, 1978a,b; Norton 1999), exceptions (Myszka 2001), special cases (Mabie 1987); linkage with a paradox between “practical degrees of freedom” and “computed degrees of freedom” (Eckhardt 1998), overconstrained yet mobile linkage (Phillips 1984, 1990; Waldron and Kinzel 1999), linkages with anomalous mobility (Waldron and Kinzel 1999). The formulae known in the literature for the determination of mobility are not applicable to many types of recent parallel robots, see for example the robots Delta (Clavel 1988, 1990), Star (Hervé and Sparacino 1992, 1993), H4 (Pierrot and Company 1999), Orthoglide (Wenger and Chablat 2000), CPM (Kim and Tsai 2002), Tripteron (Gosselin et al. 2004) and other parallel manipulators presented in the literature (Kong and Gosselin 2002, Carricato and Parenti-Castelli 2002) or recently proposed by the author belonging to Isoglide family (Gogu 2002b, 2004a-d, 2005f-j, 2006a-e, 2007a-c). We cannot continue to consider the mechanisms that do not fit the various methods or equations for determining mobility as having structural flaws. We have to review the formulae and consider the flaw to be that of the formula rather than of the mechanism. In this chapter we will

34

2 Structural parameters

establish limits of these formulae, and propose a new formula for quick calculation of the mobility of parallel robots. Many formulae/approaches proposed in the last 150 years for the calculation of the mechanism mobility can be reduced to the same original formula. Previous works on mobility calculation (Wittenbauer 1923; Federhofer 1932; Artobolevskii 1953; Boden 1962; Bogolyubov 1964; Mruthyunjaya 2003) have mentioned various formulae but they have not analysed their genesis and similarities. In this section, we focus on a brief presentation and a critical analysis of these formulae/approaches to clearly identify the different contributions in this very important subject of the theory of mechanisms. Thirty five contributions are reviewed. To ensure unity of presentation, the original notations of some of the authors were not maintained; this also made it easier to distinguish between the different formulae. These contributions had been published before the first publication, in 2005, of the new approach proposed by the author (Gogu 2005b-e). We apply each formula to determine the degree of mobility of a recent parallel manipulator with simple limbs, the parallel Cartesian robotic manipulator CPM (Kim and Tsai 2002; Kong and Gosselin 2002, Carricato and Parenti-Castelli 2002, Gogu 2002b) presented in Fig. 2.1, which only contains revolute (R) and prismatic (P) joints. A serial kinematic chain is associated with each simple limb linking the mobile platform to the fixed base. The mechanism has n=10 kinematic elements (links), p0=3 prismatic joints adjacent to the fixed element, pn=9 revolute joints non-adjacent to the fixed element and q=2 independent closed loops. Each loop contains the same number and the same type of joints. The three limbs are identical

Fig. 2.1. Parallel Cartesian robotic manipulator: (a) kinematic chain; (b) associated graph

2.1 Critical review of mobility calculation

35

from a structural point of view. In each limb the direction of the prismatic joint and the axes of the revolute joints are parallel (P||R||R||R). Kim and Tsai (2002) have mentioned that this mechanism has 3 degrees of freedom, in spite of the fact that the general degrees-of-freedom equation presented by Tsai (2000) predicts that the mechanism has zero degrees of freedom. We analyse the results obtained by using the different formulae to determine the degree of mobility of the CPM robotic manipulator, and we draw conclusions on the limits to the various formulae. 2.1.1 Chebychev’s contribution In the second half of the nineteenth century, the mathematician P. A. Chebychev published an article (concerning a 4-bar linkage) in two parts in the proceedings of the Russian Royal Academy of Science of SaintPetersburg. Chebychev was the first scientist who proposed a mathematical formalisation for the calculation of mechanism mobility. The first part of the article (Chebychev 1854) was published under the impulse of Watt’s invention, known at the end of the eighteenth century. In this article, Chebychev applied his method of polynomial function approximation to find the lengths of the elements of the 4-bar mechanism so that it could describe a given couple curve with the least error. In the second part of the article, Chebychev (1869) proposed the first formula for the calculation of the number of independent variables in a mechanism. This contribution was quickly integrated into the first treatises on the theory of mechanisms published at the end of the nineteenth century (Dwelshauvres-Dery 1876, Laboulaye 1878). Chebychev expressed his formula in the form: 3n -2(p0+pn)=1

(2.1)

where 3n represents the number of variables required to describe the position and the orientation of the n kinematic (mobile) bars in the plane and 2(p0+pn) is the number of constraint equations imposed by the p=(p0+pn) revolute joints of the mechanism that can be adjacent (p0) or non-adjacent (pn) to the fixed base. It is known that each revolute joint introduces two constraint equations in a planar mechanism. Chebychev applied this formula to simple planar mechanisms with p0=2 and to complex planar mechanisms (p0=3) having only revolute joints and one degree of mobility. He did not extend the use of Eq. (2.1) to other types of mechanisms. This formula could be extended in the form M= 3n -2(p0+pn)

(2.2)

36

2 Structural parameters

to any planar mechanism with one degree of freedom joints (helical, prismatic and revolute joints). The well-known λ 4-bar mechanisms were obtained by Chebychev to describe couple curves containing a straight line segment by using Eq. (2.1). Supposing that we try to extrapolate Eq. (2.2), we obtain M=6 for the mechanism presented in Fig. 2.1 (with n=10, p0=3 and pn=9). This result is obviously erroneous, Chebychev’s formula (2.2) being limited to the planar mechanisms (mechanisms in which the axes of all the revolute joints are parallel and perpendicular to the plane containing the directions of the prismatic pairs). 2.1.2 Sylvester’s contribution J. J. Sylvester (1874) presented a modified form of Eq. (2.1) as a structural condition for one degree of freedom pin-connected planar mechanisms: 3m-2p-4=0.

(2.3)

In Eq. (2.3), m is the total number of links of a mechanism including the fixed element. Sylvester stated Eq. (2.3) in the following words: “In order for a combination of links to fulfil this so to say fatalistic condition of constrained motion and to entitle it to the name of a linkage in the speaker’s sense, a numerical relation must be satisfied between the number of links and the number of joints, viz., three times the number of links must be four greater than twice the number of joints. In applying this rule it must be understood that, if three links are jointed together, the junction counts for two joints; if four are jointed together, for three joints; and so on.” We can see that Eq. (2.3) can be obtained by simply replacing n by m-1 in the Chebychev’s formula and, in the general case, joints connecting a links count as a-1 joints.

2.1.3 Grübler’s contribution M. Grübler (1883, 1885) presented a structural condition for one degree of freedom planar mechanisms identical to Eq. (2.3). Later, Grübler proposed a structural condition for one degree of freedom spatial complex mechanisms with helical joints (Grübler 1916, 1917): 5h-6m+7=0 were h is the total number of helical joints.

(2.4)

2.1 Critical review of mobility calculation

37

2.1.4 Somov’s contribution P. O. Somov (1887) proposed the following structural condition for one degree of freedom mechanisms: m-q(b-1)=2.

(2.5)

Somov introduced for the first time the motion parameter (b), called mobility number, for planar (b=3) and for spatial mechanisms (b=6). 2.1.5 Hochman’s contribution Hochman made an important contribution to the structural analysis of mechanisms in his book on the kinematics of machinery (Hochman 1890). A detailed presentation of Hochman’s contribution can be found in (Bogolyubov 1964; Dudiţă and Diaconescu 1987). He proposed a relation between the total number of links m, the total number of joints p and the number of independent closed loops q in a complex mechanism (m+q-p=1) .This is Euler’s formula in the theory of graphs. He also proposed various structural conditions for the existence of one degree of freedom mechanisms:

with

b(m-1)-C=1

(2.6)

F- qb=1

(2.7)

b (p-q)-C=1

(2.8)

C = ∑ iCib

(2.9)

F = ∑ iFi b

(2.10)

F+C=pb

(2.11)

b −1 i =1

b −1 i =1

where C is the total number of constraints imposed by the joints and F – the total number of degrees of freedom of the joints. Cib represents the

number of joints having the degree of constraint cib = i , and Fi b is the

number of pairs having the degree of connectivity fi b = i . Hochman de-

38

2 Structural parameters

fined the motion parameter b as the mechanism category given by the number of simple existing motions between any two links (b0 if each of its loops have a mobility Mi≥M and a partial mobility if it has at least one loop of mobility Mir.

(2.110)

Eq. (2.110) follows directly from Eq. (2.68) by taking into consideration that a jointed system is a mechanism if M>0. In this case, the jointed system is mobile. If the jointed system is immobile (M=0) it is called a jointed structure or simply a structure (see Fig. 2.8a). In this section we will define M and r by using the theory of linear transformations. As we have seen in section 2.2.3, the kinematic equation of an

2.3 Mobility and connectivity of parallel robots

81

simple open chain can be described by a matrix transformation from the finite dimensional vector space U (called joint velocity space) into the finite dimensional vector space W (called external velocity space) of basis (W)=( v x ,v y ,v z ,ω x ,ω y ,ω z ): F :U → W . q F( q ) The linear transformation F is the transformation multiplication by the p matrix [ J ]6 × N , where N = ∑ i =1 f i . That is to each vector q , in the joint vector space U, the linear transformation F assigns a uniquely determined vector F ( q ) in the vector space W

F( q )= [ J ]6 × N [ q ]N ×1 .

(2.111)

Matrix [ J ]6 × N is usually called the Jacobian matrix. Vector q represents

the joint velocity vector and it is composed of the relative velocities in the joints of the mechanism. Vector F ( q ) in W is the image of vector q in U under transformation F. The image of U in W represents the range RF of F and we call it the operational space. For a mechanism Q, its operational space will be denoted by RFQ or simply by RQ or RQ. As F is a matrix transformation, with matrix [ J ]6 × N , then we can write dim(U ) = N = ∑ fi p

(2.112)

i =1

dim(RF)=rank(F)=rank( [ J ]6 × N ).

(2.113)

For closed chains with N = ∑ i =1 f i freedoms in joints and q independp

ent loops, the closure equations of the q loops involve F ( q ) =0 and the Jacobian becomes a 6q×N matrix. For a closed simple (q=1) or complex (q>1) mechanism, the number of joint parameters that lose their independence after loop closures is r= dim(RF)=rank(F)=rank( [ J ]6 q× N ).

(2.114)

Let us consider a general mechanism Q (that can be simple or complex, open or closed).

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2 Structural parameters

Definition 2.2A (Geometric definition of connectivity between two links of a mechanism). Connectivity S aQ/ b between two links a and b in a mechanism Q represents the number of independent finite displacements allowed by the mechanism between the two links. Definition 2.2B (Kinematic definition of connectivity between two links of a mechanism). Connectivity S aQ/ b between two links a and b in a mechanism Q represents the number of independent infinitesimal displacements allowed by the mechanism between the two links. Connectivity proposition 1. Connectivity S aQ/ b between two links (a and b) in a mechanism Q is given by the dimension of the vector space RaQ/ b of the relative velocities between links a and b in mechanism Q

S aQ/ b = dim( RaQ/ b ) .

(2.115)

The proof of Eq. (2.115) results directly from Definition 2.2 and the definition of the vector space RaQ/ b . We notice that RaQ/ b is a sub-space of R Q and implicitly a sub-space of W. Any mechanism gives a linear transformation F and consequently Eq. (2.68) is valid without exception. It is applicable to any type of mechanism (simple or complex, open or closed). In the following section we will present and demonstrate formulae for calculating the dimension of the range RF for simple open mechanisms without calculating the rank of the screw system associated with the joints of the mechanism. The new formulae, demonstrated by using linear transformations, are based on connectivity of a simple open kinematic chain. This important structural parameter is easily determined by inspection. An analytical method will also be presented just for verification and for a better understanding of the meaning of this parameter.

2.3.2 Mobility and connectivity of simple open kinematic chains

A simple open kinematic chain A (1-2-…-n) is a serial kinematic chain with n links and p joints (Fig. 2.9). We denote the reference link by 1 and the final link by n. The reference link can be fixed (1≡0) or mobile. All intermediate links (2, …,n-1) are binary links. We denote by M=MA the mobility of a simple open kinematic chain given by Eq. (2.68) with r=0. For a given simple open kinematic chain, MA has a constant positive value. By analogy with the definition of mobility, we can define connectivity from both a geometrical and a kinematical point of view. Connectivity SA of a simple open kinematic chain A (1-2-…-n) is given by the connectivity

2.3 Mobility and connectivity of parallel robots

83

Fig. 2.9. General case of a simple open kinematic chain

between the final and reference links n and 1. Links n and 1 are the extreme links of the simple open kinematic chain. Definition 2.3A. (Geometric definition of connectivity of a simple open kinematic chain). Connectivity of a simple open kinematic chain A (1-2-…-n) represents the number of independent finite displacements between the extreme links n and 1 of A. Definition 2.3B. (Kinematic definition of connectivity of a simple open kinematic chain). Connectivity of a simple open kinematic chain A (1-2-…-n) represents the number of independent infinitesimal displacements between the extreme links n and 1 of A S A = SnA/1 .

(2.116)

By analogy with mobility, we distinguish instantaneous connectivity iSA and general connectivity SA. The instantaneous connectivity characterizes the kinematic chain in a given position i. The general connectivity represents the maximum value of the instantaneous connectivity SA=max(iSA).

(2.117)

A given kinematic chain has an infinite number of positions, and with each position we can associate an instantaneous connectivity. In these conditions, general connectivity calculation is theoretically impossible by using Eq. (2.117). In this section we will present methods for calculating the general connectivity without calculating the instantaneous connectivity. General connectivity will be called simply connectivity. For the simple open kinematic chain, SA≥ iSA. For a simple open kinematic chain A (1- 2- …-n), the linear transformation FA and Eq. (1.111) becomes (Gogu 2002a):

v  ω =  

[ J ] [q ]

(2.118)

where: v and ω are the translation and angular velocity vectors of the distal link n, q is the joint velocity vector composed of the relative veloci-

ties in the joints of the mechanism and [ J ] = [ J ]6 × N is the Jacobian matrix.

84

2 Structural parameters

Vectors v and ω belong to the velocity vector space RnA/ 1 . This space defines the operational space of the simple open kinematic chain and dim( RnA/ 1 )=dim(RA)

(2.119)

Mobility proposition 2. The mobility of the simple open kinematic chain A with p joints is given by the dimension of its joint space

MA=dim (UA) = ∑ fi . p

(2.120)

i =1

Eq. (2.120) results directly from Eqs. (2.68). and (2.112). Connectivity proposition 2A. The connectivity of the simple open kinematic chain A (1- 2- …-n) is given by the dimension of its operational space SA=dim (RA).

(2.121)

Connectivity proposition 2B. The connectivity of the simple open kinematic chain A (1- 2- …-n) is given by the rank of the Jacobian matrix of the linear transformation FA associated with A

SA=rank (JA).

(2.122)

Eqs. (2.121) and (2.122) result directly from Eqs. (2.113) and (2.116). Connectivity indicates the number of independent velocities of the final link related to the reference link. For a given simple open kinematic chain, SA depends on the geometric configuration of the chain, defined by the finite displacements in the joints, and on the geometric parameters of the links. We recall that the rank of the Jacobian matrix can be calculated numerically or symbolically. The numerical calculation gives us the instantaneous connectivity in a given position of the kinematic chain defined by numerical values of joint variables and geometric parameters. The symbolic calculation gives us the general connectivity in an arbitrary position of the kinematic chain without indicating numerical values of joint variables and geometric parameters. Connectivity proposition 3. The connectivity of the simple open kinematic chain A is less than or equal to its mobility: SA≤MA .

This is a direct result of Eqs. (2.118), (2.120) and (2.122). The joint velocity space UA is a product space

(2.123)

2.3 Mobility and connectivity of parallel robots

U A = ∏U i = U 1 × U 2 × ⋅⋅⋅× U p p

85

(2.124)

i =1

where Ui is the velocity space of the ith joint (the joint between links i and i-1). Let us consider that the simple open kinematic chain A (1-2-…-i-…-n) is obtained by serial concatenation of two simple open kinematic chains A1 (1-2-…-i) and A2.(i-…-n). In this case, A is denoted by A≡A1-A2. We consider the sub-spaces UA1 and UA2 of the joint velocity space UA and their corresponding operational sub-spaces RA1 and RA2 of the operational space RA≡A1-A2 related to the linear transformation FA≡FA≡A1-A2. The sub-spaces UA1 and UA2 have the property U A1 ∩ U A2 = {0U } , that is the intersection of the two sub-spaces UA1 and UA2 is the zero vector of the linear space UA. The operational space RA≡A1-A2 is an addition space RA≡A1-A2=RA1+RA2 .

(2.125)

The vector spaces UA, Ui, RA, RA≡A1-A2, RA1 and RA2 have finite dimensions. To determine the dimensions of the joint space UA and of the operational spaces RA and RA≡A1-A2, the following properties of the vector spaces are useful: Property 3 (Dimension of the joint space). The dimension of the joint space is given by the dimension of the product space UA dim(U A ) = ∑ dim(U Ai ) . p

(2.126)

i =1

From Eqs. (2.68) and (1.126) we can see, as a particular case, that the dimension of the velocity space of a joint is given by its degree of mobility (dim(Ui)= fi). Property 4 (Dimension of the operational space). The dimension of the operational space RA≡A1-A2 is given by the dimension of the addition space: dim( RA= A1− A2 ) = dim( RA1 + RA2 ) =

dim( RA1 ) + dim( RA2 ) − dim( RA1 ∩ RA2 )

(2.127)

where RA1 ∩ RA2 is the common range of the sub-spaces RA1 and RA2 of the operational space RA=A1-A2 associated with the simple open mechanism A≡A1-A2. We call RA1 ∩ RA2 the common operational space of A1 and A2. Connectivity proposition 4A. The connectivity of a simple open kinematic chain A≡A1-A2, obtained by serial concatenation of two open simple

86

2 Structural parameters

kinematic chains A1 and A2, is given by the dimension of its addition operational space S A≡ A1− A2 = dim( RA≡ A1− A2 ) = dim( RA1 + RA2 ) =

dim( RA1 ) + dim( RA2 ) − dim( RA1 ∩ RA2 )

.

(2.128)

Connectivity proposition 4B. The connectivity of a simple open kinematic chain A≡A1-A2, obtained by serial concatenation of two open simple kinematic chains A1 and A2, is given by the rank of the Jacobian matrix of the linear transformation FA≡A1-A2 associated to A≡A1-A2

S A≡ A1− A2 = rank( J )A≡ A1− A2 .

(2.129)

Eqs. (2.128) and (2.129) result directly from Eqs. (2.121), (2.122) and (2.127). Singular configurations and redundancy of a simple open kinematic chain can be defined by using its mobility and connectivity. Definition 2.4 (Structural singularity of a simple open kinematic chain). When the general connectivity of a simple open kinematic chain A is smaller than its general mobility (SA rB .

(2.167)

Eq. (2.167) is a particular case of (2.110) when q=1. Mechanism existence condition 3A. A single-loop mechanism B (1-2…-n-(n+1)≡1) exists if and only if the mobility of the simple open chain A (1-2-…-n-(n+1)) associated with B is higher than the connectivity of A MA>SA .

(2.168)

Eq. (2.168) results directly from Eqs. (2.120), (2.140) and (2.167). Mechanism existence condition 3B. A single-loop mechanism B ← A1A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) exists if and only if the mobility of the simple open chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with B ← A1-A2 is higher than the connectivity of A≡A1-A2 MA>SA≡A1-A2 .

(2.169

Eq. (2.169) results directly from Eqs. (2.120), (2.144) and (2.167). Mechanism existence condition 4A. A single-loop mechanism B (1-2…-n-(n+1)≡1) exists if and only if the simple open chain A (1-2-…-n(n+1)) associated with B is structurally redundant

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2 Structural parameters

TA>0 .

(2.170)

Eq. (2.170) results directly from (2.154). Mechanism existence condition 4B. A single-loop mechanism B ← A1A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) exists if and only if the simple open chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with B ← A1-A2 is structurally redundant TA≡A1-A2 >0 .

(2.171)

Eq. (2.171) results directly from (2.159). Note 5. Formulae (2.140)-(2.142), (2.148)-(2.154), (2.160)-(2.162), (2.168) and (2.170) apply to the single-loop kinematic chain B (1-2-…-n(n+1)≡1) obtained by connecting in the same link the extreme links 1 and (n+1) of the simple open kinematic chain A (1-2-…-n-(n+1)) associated with B. Note 6. Formulae (2.144)-(2.146), (2.155)-(2.159), (2.163)-(2.166), (2.169) and (2.171) apply to the single-loop kinematic chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) obtained by connecting the extreme elements (1A1≡1A2 and nA1≡nA2) of two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2). In this case the three chains A1, A2 and A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) are called simple open kinematic chains associated with B ← A1-A2. Note 7. Formulae (2.142), (2.146), (2.150)-(2.153), (2.157), (2.158), (2.162), (2.166) involve the calculation of the Jacobian matrices. Numerical calculation of the rank and nullity of the Jacobian matrix allows us to calculate only instantaneous values of the structural parameters. Symbolic calculation of the rank and nullity could overcome this drawback but both cases are very time-consuming. For this reason, these formulae also have limited practical value for the quick calculation of mobility. Note 8. Formulae (2.140), (2.141), (2.144), (2.145), (2.148), (2.149), (2.154)-(2.156), (2.159)-(2.161), (2.163)-(2.165) and (2.168)-(2.171) are based on the connectivity of the simple open chain associated with the closed mechanism. This parameter can be easily determined through inspection by observing the independent motions given by the associated open chain between its extreme links. 2.3.4 Connectivity between two elements of a single-loop kinematic chain

Let us consider a single-loop kinematic chain B ← A1-A2 (1A1-2A1-…nA1≡nA2-…-2A2-1A2≡1A1) obtained by connecting the extreme elements

2.3 Mobility and connectivity of parallel robots

97

(1≡1A1≡1A2 and n≡nA1≡nA2) of two simple open kinematic chains A1 (1A12A1-…-nA1) and A2 (1A2-2A2-…-nA2) to the same link - Fig. 2.10b. We consider UA1 and UA2 the joint velocity sub-spaces of the open simple kinematic chains A1 and A2 and their corresponding operational spaces RA1 and RA2. The sub-spaces UA1 and UA2 have the property U A1 ∩ U A2 = {0U } . The

common space of RA1 and RA2 is the vector space RnB/ 1 = RA1 ∩ RA2 of relative velocities between links n≡nA1≡nA2 and 1≡1A1≡1A2 in the single-loop kinematic chain B ← A1-A2. Connectivity proposition 7A. The connectivity S nB/ 1 between links n≡nA1≡nA2 and 1≡1A1≡1A2 in the single-loop kinematic chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) obtained by connecting the extreme elements 1≡1A1≡1A2 and n≡nA1≡nA2 of two open simple kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2), to the same link, is given by the dimension of the common vector space of A1 and A2 S nB/ 1 = dim( RA1 ∩ RA2 ) .

(2.172)

Proof. In the open simple kinematic chain A1, the number of relative independent infinitesimal displacements or velocities between extreme links nA1 and 1A1 is given by SA1=dim (RA1). The number of relative independent velocities between extreme links nA2 and 1A2 in the simple open kinematic chain A2 is given by SA2=dim (RA2). As we have considered 1A1≡1A2 and nA1≡nA2, the number of relative independent velocities between two elements 1≡1A1≡1A2 and n≡nA1≡nA2 in the single-loop mechanism B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) is given by the dimension of the intersection of operational vector spaces RA1 and RA2. Connectivity proposition 7B. The connectivity S nB/ 1 between links n≡nA1≡nA2 and 1≡1A1≡1A2 in the single-loop kinematic chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1), obtained by connecting the extreme elements (1≡1A1≡1A2 and n≡nA1≡nA2) of two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2), to the same link is given by the difference between the sum of the connectivities of A1 and A2 and the connectivity of the simple open chain A=A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with the closed chain B ← A1-A2 S nB/ 1 = S A1 + S A2 − S A≡ A1− A2 .

(2.173)

Proof. According to Eq. (2.172), Eq. (2.128) can be written as S A≡ A1− A2 = S A1 + S A2 − S nB/ 1 .

(2.174)

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2 Structural parameters

Eq. (2.174) expresses the fact that connectivity of the simple open chain A=A1-A2 associated with the closed chain B ← A1-A2 is given by the difference between the sum of connectivities of A1 and A2 and the connectivity between links n≡nA1≡nA2 and 1≡1A1≡1A2 in single-loop kinematic chain B ← A1-A2. Eq. (2.173) follows directly from (2.174). A similar equation to (2.173) was proposed by Voinea and Atanasiu (1960) to define the connectivity of a link in a single-loop mechanism by using screw theory. Connectivity proposition 8A. Two links n≡nA1≡nA2 and 1≡1A1≡1A2 in the single-loop kinematic chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A21A2≡1A1), obtained by connecting the extreme elements 1≡1A1≡1A2 and n≡nA1≡nA2 of two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2) to the same link, have relative movements if the dimension of the common operational space of A1 and A2 is larger than zero

If

S nB/ 1 >0 if dim( RA1 ∩ RA2 ) > 0 .

(2.175)

dim( RA1 ∩ RA2 ) = 0

(2.176)

two links n≡nA1≡nA2 and 1≡1A1≡1A2 are relatively immobile ( S nB/ 1 =0). Eqs. (2.175) and (2.176) result directly from Eq. (2.172). Connectivity proposition 8B. Two links n≡nA1≡nA2 and 1≡1A1≡1A2 in the single-loop kinematic chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A21A2≡1A1), obtained by connecting the extreme elements (1≡1A1≡1A2 and n≡nA1≡nA2) of two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2) to the same link, have relative movements if the sum of connectivities of A1 and A2 is greater than the connectivity of the simple open chain A=A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with the closed chain B ← A1-A2

If

S nB/ 1 >0 if S A1 + S A2 > S A≡ A1− A2 .

(2.177)

S A1 + S A2 = S A≡ A1− A2

(2.178)

two links n≡nA1≡nA2 and 1≡1A1≡1A2 are relatively immobile ( S nB/ 1 =0). Eqs. (2.177) and (2.178) result directly from Eq. (2.173). Connectivity proposition 9. Connectivity S nB/ 1 between two links n≡nA1≡nA2 and 1≡1A1≡1A2 in the single-loop kinematic chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1), obtained by connecting the extreme

2.3 Mobility and connectivity of parallel robots

99

elements (1≡1A1≡1A2 and n≡nA1≡nA2) of two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2) to the same link, is less than or equal to the mobility MB of the single-loop mechanism B ← A1-A2 S nB/ 1 ≤ M B .

(2.179)

The proof of Eq. (2.179) is founded on definitions 1 and 2 of mobility and connectivity. If S nB/ 1 > M B , MB independent parameters are not sufficient to describe the relative motion between two elements 1≡1A1≡1A2 and n≡nA1≡nA2 of the single-loop kinematic chain B ← A1-A2 ;this contradicts the definition of mobility. Connectivity proposition 10A. The basis of the vector space RnB/ 1 of relative velocities between the links n≡nA1≡nA2 and 1≡1A1≡1A2 in the single-loop kinematic chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) does not vary with with the position of the characteristic point on link n≡nA1≡nA2. Proof. We recall that the characteristic point is the point where link n≡nA1≡nA2 is split up to get associated open chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2). The independent velocities of this point define the basis ( RnB/ 1 ). Link n≡nA1≡nA2 has an infinite number of points and the velocity of each point can be expressed by a linear combination of the independent velocities of the same basis. The basis ( RnB/ 1 ) is valid for any point of the mobile platform including the points with the most restrictive motion. Connectivity proposition 10B. The connectivity S nB/ 1 between the links n≡nA1≡nA2 and 1≡1A1≡1A2 in the single-loop kinematic chain B ← A1A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) obtained by connecting the extreme elements (1≡1A1≡1A2 and n≡nA1≡nA2) of two simple open kinematic chains A1 (1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2) to the same link, does not vary with the position of the characteristic point on link n≡nA1≡nA2. Connectivity proposition 10B is a direct consequence of proposition 10A. Connectivity proposition 11A. The basis of operational space RA≡A1-A2 of simple open kinematic chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated to single-loop chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A21A2≡1A1) does not vary with the choice of split up links 1 and n. Proof. We have seen that the simple open kinematic chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with the single-loop kinematic chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) could be obtained by splitting up the link 1A1≡1A2 into 1A1 and 1A2. Two simple open chains A1

100

2 Structural parameters

(1A1-2A1-…-nA1) and A2 (1A2-2A2-…-nA2) can also be associated with A≡A1A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) by splitting up the link nA1≡nA2 into nA1 and nA2. The operational vector space RA≡A1-A2 is an addition space RA≡A1A2=R1A+RA2 independently of the choice of 1A1≡1A2 and nA1≡nA2 to be split up. Connectivity proposition 11B. The connectivity SA≡A1-A2 of the simple open kinematic chain A≡A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2) associated with the single-loop chain B ← A1-A2 (1A1-2A1-…-nA1≡nA2-…-2A2-1A2≡1A1) is does not vary with the choice of split up links 1 and n. Connectivity proposition 11B is a direct consequence of proposition 11A. The following remark is very important for the choice of the bases of operational vector spaces RA1 and RA2 when there are various ways to choose them. Remark 1. (Choice of the bases of operational vector spaces RA1 and RA2). If there are different ways to choose the bases of the operational vector spaces RA1 and/or RA2, the choice must be made by respecting the following: a) connectivity propositions 7-9 stating that 0 < S nB/ 1 ≤ M B

(2.180)

b) connectivity propositions 10A and B and 11A and B, c) propositions 2 and 3 of the existence of the mobile single-loop mechanism and the corresponding conditions (2.167)-(2.171), d) definition of connectivity (in the sense of general connectivity) as the maximum value of the instantaneous connectivity - Eq. (2.117), e) permanent constraints in the relative motions of the links generated by the closed loop with special geometric conditions. The formalisation for mobility and connectivity of simple open kinematic chains and single-loop mechanisms will be used in the next sections to demonstrate new formulae for quick mobility calculation of parallel robots with simple and complex limbs. 2.3.5. Mobility and connectivity of parallel robots with simple limbs

In this section we consider parallel robots with simple limbs. They are complex closed mechanisms composed of two polinary links and many binary links. The two polinary links are the reference link and the final link. They represent the reference and mobile platforms.

2.3 Mobility and connectivity of parallel robots

101

Fig. 2.11. The parallel mechanism C=A1-A2-...-AK with k elementary limbs

Definition 2.7 (Parallel robot with simple limbs). A parallel robot with simple limbs is a complex mechanism C ← A1-A2-…Ak with two polinary elements 1C≡1Ai and nC≡nAi connected by k simple open chains Ai (1Ai-2Ai-…-nAi), i=1,2,…k (Fig. 2.11). The two polinary elements are the reference platform 1C and the mobile platform nC. The simple open kinematic chain Ai is called a simple limb. The complex mechanism C ← A1-A2-…Ak has p joints and k-1 structurally independent closed loops concatenated in-parallel. We denote by UAi and RAi the joint velocity space and the operational velocity space of the simple open kinematic chain Ai, and by UA1-Ai and RA1-Ai the joint velocity space and the operational velocity space of the simple open kinematic chain A1Ai (1A1-2A1-…-nA1≡nAi-(n-1)Ai-(n-2)Ai-…-2Ai-1Ai) associated with each closed loop Bi ← A1-Ai (1A1-2A1-…-nA1≡nAi-(n-1)Ai-(n-2)Ai-…-2Ai-1Ai≡1A1), i=2,3,… ,k. We denote by r=rC the number of joint parameters that lose their independence in the k-1 structurally independent closed loops concatenated inparallel in the complex mechanism C ← A1-A2-…Ak. We note that the simple limbs Ai are detached from the parallel mechanism C ← A1-A2-…Ak when defining the associated parameters UAi, RAi and SAi. Todisconnect them the mobile platform is cut up in the proximity of each limb. Definition 2.8. (Connectivity of the mobile platform in the parallel robot with simple limbs). The connectivity S nC/ 1 of the mobile platform n≡nAi in a parallel mechanism C ← A1-A2-…Ak with k simple limbs Ai (1Ai2Ai-…-nAi), i=1,2,…k is given by the number of independent velocities be-

102

2 Structural parameters

tween the mobile platform n≡nAi and the reference platform 1=1Ai (i=1,2,…k) in the parallel mechanism C. Connectivity proposition 12 (Connectivity of the mobile platform in the parallel robot with simple limbs). The connectivity S nC/ 1 of the mobile platform n≡nAi in a parallel mechanism C ← A1-A2-…Ak with k simple limbs Ai (1Ai-2Ai-…-nAi), i=1,2,…k is given by the dimension of the common operational spaces of the simple limbs Ai associated with the parallel mechanism C S nC/ 1 = dim( RnC/ 1 ) = dim( RA1 ∩ RA2 ∩ ... ∩ RAk )

(2.181)

where

S nC/ 1 = SnA1/ 1− A2 − ...− Ak denotes the connectivity between the mobile platform

n≡nAi and the reference platform 1≡1Ai in the parallel mechanism C ← A1A2-…Ak, RnC/ 1 = RnA1/ 1− A2 − ...− Ak - operational velocity vector space of parallel mechanism C ← A1-A2-…Ak defined by the relative velocities between the mobile platform n≡nAi and the reference platform 1≡1Ai in the parallel mechanism C ← A1-A2-…Ak. The proof of the Eq. (2.181) is obtained by recurrence S nA1/ 1− A2 = dim( RnA1/ 1− A2 ) = dim( RA1 ∩ RA2 )

(2.182)

S nA1/ 1− A2 − A3 = dim( RnA1/ 1− A2 − A3 ) = dim( RnA1/ 1− A2 ∩ RA3 ) =

(2.183)

S nA1/ 1− A2 − ...− Ai = dim( RnA1/ 1− A2 − ...− Ai ) = dim( RnA1/ 1− A2 − ...− A( i −1 ) ∩ RAi ) =

(2.184)

dim( RA1 ∩ RA2 ∩ RA3 )

dim( RA1 ∩ RA2 ∩ ... ∩ RA( i −1 ) ∩ RAi )

where S nA1/ 1− A2 is the connectivity and the between the mobile platform n≡nAi and the reference platform 1≡1Ai (i=1,2) in the parallel mechanism C ← A1A2 with only two limbs, S nA1/ 1− A2 − A3 is the connectivity between the mobile platform n≡nAi and the reference platform 1≡1Ai (i=1,2,3) in the parallel mechanism C ← A1-A2-A3 with only three limbs,

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103

S nA1/ 1− A2 − ...− Ai is the connectivity between the mobile platform n≡nAi and the reference platform 1≡1Ai (i=1,2,…k) in the parallel mechanism C ← A1A2-…Ai with i limbs, RnA1/ 1− A2 is the operational velocity space of the parallel mechanism C ← A1-A2 with only two limbs, RnA1/ 1− A2 − A3 - operational velocity space of the parallel mechanism C ← A1A2-A3 with only three limbs, RnA1/ 1− A2 − ...− Ai - operational velocity space of the parallel mechanism C ← A1-A2-…-Ai with i limbs. Conectivity proposition 13A. The number of joint parameters that lose their independence in a parallel mechanism C ← A1-A2-…Ak with k simple limbs Ai (1Ai-2Ai-…-nAi) is given by the difference between the sum of the dimensions of the operational spaces RAi of the simple open chains Ai (i=1,2,…,k) associated with the simple limbs and the connectivity of the mobile platform S nC/ 1 in the parallel mechanism C ← A1-A2-…Ak rC = rA1− A2 − ...− Ak = ∑ dim( RAi ) − dim( RA1 ∩ RA2 ∩ ... ∩ RAk ) . k

(2.185)

i =1

Proof. The proof of Eq. (2.185) is also obtained by recurrence. We initially consider the complex open mechanism associated with the parallel mechanism by considering that the reference links 1Ai of the simple limbs Ai are not integrated into the reference platform 1Ai≠1Aj (i,j=1,2,…k, i≠j). At this initial step the reference platform does not exist. It is divided into small parts corresponding to the links 1Ai of each limb. Only the links nAi of the simple limbs Ai are connected in the mobile platform n=nAi (i=1,2,…k) - see Fig. 2.12. The complex open kinematic chain has no closed loops. All mobility parameters of the joints are independent. In the first step, we connect only links 1A1 and 1A2 of limbs A1 and A2 to form the reference platform 1 (1≡1A1≡1A2). The complex kinematic chain in Fig. 2.13 has only one closed loop A1-A2. By taking into consideration connectivity proposition 6B, the number of joint parameters that lose their independence in the closed loop B ← A1-A2 (1A1-2A1-…-(n-1)A1-nA1≡nA2-(n1)A2 -…-2A2-1A2≡1A1) is given by the connectivity of the simple open chain A1-A2 (1A1-2A1-…-(n-1)A1-nA1≡nA2-(n-1)A2 -…-2A2-1A2) associated with the closed loop B ← A1-A2

rA1− A2 = dim( RA1− A2 ) = dim( RA1 + RA2 ) =

dim( RA1 ) + dim( RA2 ) − dim( RA1 ∩ RA2 )

.

(2.186)

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2 Structural parameters

In the second step, we connect link 1A3 of limb A3 with the reference platform 1 (1≡1A1≡1A2≡1A3) - see Fig. 2.14. The number of joint parameters that lose their independence in the closed loops A1-A2-A3 is

Fig. 2.12. Complex open kinematic chain with k branches

Fig. 2.13. Complex kinematic chain with one closed loop and k-2 open branches

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105

rA1− A2 − A3 = rA1− A2 + S( A1− A2 )− A3 = rA1− A2 + dim( RnA1/ 1− A2 + RA3 ) =

rA1− A2 + dim( RnA1/ 1− A2 ) + dim( RA3 ) − dim( RnA1/ 1− A2 ∩ RA3 ) =

rA1− A2 + dim( RA1 ∩ RA2 ) + dim( RA3 ) − dim( RA1 ∩ RA2 ∩ RA3 ) =

(2.187)

dim( RA1 ) + dim( RA2 ) + dim( RA3 ) − dim( RA1 ∩ RA2 ∩ RA3 ).

In Eq. (2.187), we denoted by S(A1-A2)-A3 the connectivity of the complex open kinematic chain (A1-A2)-A3 obtained by serial concatenation of the simple closed kinematic chain B ← A1-A2 and the simple open kinematic chain A3 in Fig. 2.13. S(A1-A2)-A3 is given by the connectivity between the link 1A3 and the reference link 1≡1A1≡1A2 in the complex kinematic chain (A1-A2)-A3 by taking into consideration the existence of the closed loop B ← A1-A2 S( A1− A2 )− A3 = dim( RnA1/ 1− A2 + RA3 ) =

dim( RnA1/ 1− A2 ) + dim( RA3 ) − dim( RnA1/ 1− A2 ∩ RA3 ) =

dim( RA1 ∩ RA2 ) + dim( RA3 ) − dim( RA1 ∩ RA2 ∩ RA3 ).

(2.188

In the third step, we connect link 1A4 of limb A4 with the reference platform 1≡1A1≡1A2≡1A3≡1A4 - see Fig. 2.15. The number of joint parameters that lose their independence in the closed loops A1-A2-A3-A4 is rA1− A2 − A3 − A4 = rA1− A2 − A3 + S( A1− A2 − A3 )− A4 = rA1− A2 − A3 + dim( RnA1/ 1− A2 − A3 + RA4 ) =

rA1− A2 − A3 + dim( RnA1/ 1− A2 − A3 ) + dim( RA4 ) − dim( RnA1/ 1− A2 − A3 ∩ RnA4/ 1 ) = rA1− A2 − A3 + dim( RA1 ∩ RA2 ∩ RA3 ) + dim( RA4 ) − dim( RA1 ∩ RA2 ∩ RA3 ∩ RA4 ) =

(2.189)

dim( RA1 ) + dim( RA2 ) + dim( RA3 ) + dim( RA4 ) − dim( RA1 ∩ RA2 ∩ RA3 ∩ RA4 ).

The connectivity of the complex open kinematic chain (A1-A2-A3)-A4 obtained by serial concatenation of the complex closed kinematic chain A1A2-A3 and the simple open kinematic chain A4 in Fig. 2.14 is denoted by S(A1-A2-A3)-A4 in Eq. (2.189). S(A1-A2-A3)-A4 is given by the connectivity between link 1A4 and the reference link 1≡1A1≡1A2≡1A3 in the complex open kinematic chain (A1-A2-A3)-A4 by taking into consideration the closed loops of the complex closed kinematic chain A1-A2-A3

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2 Structural parameters

S( A1− A2 − A3 )− A4 = dim( RnA1/ 1− A2 − A3 + RA4 ) =

dim( RnA1/ 1− A2 − A3 ) + dim( RA4 ) − dim( RnA1/ 1− A2 − A3 ∩ RA4 ) =

dim( RA1 ∩ RA2 ∩ RA3 ) + dim( RA4 ) − dim( RA1 ∩ RA2 ∩ RA3 ∩ RA4 )

. (2.190)

Fig. 2.14. Complex kinematic chain with two structurally independent closed loops and k-3 open branches

Fig. 2.15. Complex open kinematic chain with three structurally independent closed loops and k-4 open branches

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107

At the end of the recursive procedure we obtain Eq. (2.185). Connectivity proposition 13B. The number of joint parameters that lose their independence in the parallel mechanism C ← A1-A2-…Ak with k simple limbs Ai (1Ai-2Ai-…-nAi) is given by the difference between the sum of the connectivities SAi of the open simple chains Ai (i=1,2,…,k) associated with the simple limbs Ai ,and the platform connectivity S nC/ 1 in the parallel mechanism C ← A1-A2-…Ak rC = rA1− A2 − ...− Ak = ∑ S Ai − S nC/ 1

(2.191

M C = ∑ dim(U Ai ) − ∑ dim( RAi ) + dim( RA1 ∩ RA2 ∩ ... ∩ RAk )

(2.192)

M C = ∑ f i − ∑ S Ai + SnC/ 1 .

(2.193)

k

i =1

where SAi=dim(RAi), S nC/ 1 = d im( RA1 ∩ RA2 ∩ ... ∩ RAk ) , RAi is the operational vector space of the simple open kinematic chain Ai associated with the simple limb Ai. The proof of Eq. (2.191) is obtained directly from Eq. (2.185) and from the definition of the connectivity of a simple open chain as the relative connectivity between the extreme links nAi and 1 of the elementary open Ai chain Ai (SAi= S nAi / 1 ). We can see that Eq. (2.174) represents a particular case of Eq. (2.191) when k=2. In this case the parallel mechanism with two limbs becomes a single-loop mechanism. Mobility proposition 6. The mobility of a parallel mechanism C ← A1A2-…Ak with p joints and k simple limbs Ai (1Ai-2Ai-…-nAi), i=1,2,…k, is given by

or

k

k

i =1

i =1

p

k

i =1

i =1

where: UAi and RAi are the joint velocity space and the operational velocity space of the simple open kinematic chain Ai (1Ai-2Ai-…-nAi) associated with Ai- limb, fi is the mobility of the ith joint, SAi is the connectivity of simple open kinematic chain Ai associated with Ai-limb, S nC/ 1 is the connectivity of mobile platform in the parallel mechanism C. Eqs. (2.192) and (2.193) are obtained directly by integrating Eqs. (2.126), (2.185) and (2.191) in Eq. (2.68).

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2 Structural parameters

Connectivity proposition 14. Mobile and reference platforms n≡nAi and 1≡1Ai of a parallel mechanism C ← A1-A2-…Ak with k simple limbs Ai (1Ai-2Ai-…-nAi), i=1,2,…k have relative movements if the dimension of the common operational space of Ai is greater than zero

If

S nC/ 1 >0 if dim( RA1 ∩ RA2 ∩ ... ∩ RAk ) > 0 .

(2.194)

dim( RA1 ∩ RA2 ∩ ... ∩ RAk ) = 0

(2.195)

two platforms are relatively immobile ( S nC/ 1 =0). Eqs. (2.194) and (2.195) result directly from Eq. (2.181). Connectivity proposition 15.The connectivity S nC/ 1 between the mobile and reference platforms n≡nAi and 1≡1Ai of the parallel mechanism C ← A1A2-…Ak with k simple limbs Ai (1Ai-2Ai-…-nAi), i=1,2,…k is less than or equal to the mobility MC of the parallel mechanism C ← A1-A2-…Ak S nC/ 1 ≤ M C .

(2.196)

The proof of Eq. (2.196) is founded on definitions 1 and 2 of mobility and connectivity. If S nc / 1 > M c , Mc independent parameters are not sufficient to describe the relative motion between 1≡1Ai and n≡nAi of the parallel mechanism C ← A1-A2-…Ak which is contradictory to the definition of mobility. Connectivity proposition 16. The basis of the vector space RnC/ 1 of relative velocities between the mobile and reference platforms n≡nAi and 1≡1Ai in the parallel mechanism C ← A1-A2-…Ak with k simple limbs Ai (1Ai2Ai-…-nAi), i=1,2,…k does not vary with the position of the characteristic point on the mobile platform n≡nAi. Proof. The independent velocities of the characteristic point define the basis ( RnC/ 1 ). The mobile platform n≡nAi has an infinite number of points, and the velocity of each point can be expressed by a linear combination of the independent velocities of the same basis. The basis ( RnC/ 1 ) is valid for any point of the mobile platform including the points with the most restrictive motion. The following remark is very important for the choice of the bases of operational vector spaces RAi when multiple possibilities exist to define them. Remark 2. (Choice of the bases of operational vector spaces RAi). If there are different ways to choose the bases of the operational vector spaces RAi ,the choice must be made by respecting the following:

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109

a) connectivity propositions 14 and 15 stating that 0 < SnC/ 1 ≤ M C

(2.197)

b) connectivity proposition 16, c) definition of mobility (in the sense of general mobility) as the minimum value of the instantaneous mobility - Eq. (2.109), d) permanent constraints in the relative motions of the links generated by the closed loops with special geometric conditions. We note that condition c) implies that the bases in Eq. (2.181) are selected so that the minimum value of S nC/ 1 is obtained. By this choice, Eqs. (2.192) and (2.193) fit in with Eq. (2.109) setting up general mobility as the minimum value of instantaneous mobility. 2.3.6. Mobility and connectivity of parallel robots with complex limbs

In this section we consider parallel robots with complex limbs. They are complex closed mechanisms integrating more than two polynary links. Two polynary links are the mobile and the reference platforms. The remaining polinary links represents intermediary platforms integrated into the complex limbs. Definition 2.9 (Parallel robot with complex limbs). A parallel robot with complex limbs denoted by D ← A1-A2-…Ak1-E1-E2-…-Ek2 is a complex mechanism with p joints in which the mobile platform n≡nAi≡nEj is connected to the reference platform 1≡1Ai≡1Ej by k≥2 limbs, of which at least one limb is complex. A complex limb contains at least one closed loop. We denoted by Ai (i=1,2,…k1) the elementary limbs, and by Ej (j=1,2,…k2) the complex limbs (k1+k2=k). We can see that parallel mechanisms with simple limbs represents a particular case of D when k2=0. If k1=0 all limbs are complex. We denote by UAi, UEj and RAi, REj the joint velocity spaces and the operational velocity spaces associated with limbs Ai and Ej. We also denote by r=rD the number of joint parameters that lose their independence in the closed loops of the complex mechanism D ← A1-A2…Ak1-E1-E2-…-Ek2. The joint velocity space UEj and the operational velocity space REj associated with the complex limb Ej can be introduced by analogy with the corresponding parameters UAi and RAi defined for the simple open kinematic chains in section 2.3.2.

110

2 Structural parameters

Definition 2.10. (Joint velocity space associated with a complex limb). Joint velocity space UEj associated with complex limb Ej is the joint space of the complex open kinematic chain associated with the limb by opening all closed loops integrated in the limb. Definition 2.11. (Operational velocity space associated with a complex limb). The operational velocity space REj associated with the complex limb Ej is defined by the relative velocities between the mobile platform n≡nEj and the reference platform 1≡1Ej in the complex cinematic chain associated with limb Ej. We note that the constraints generated by the closed loops integrated in the complex limb Ej are taken into account in the operational velocity space REj ,and are ignored in the joint velocity space UEj. Definition 2.12. (Connectivity of a complex limb) The connectivity SEj of a complex limb is given by the number of independent motions between the mobile platform n≡nEj and the reference platform 1≡1Ej in the complex cinematic chain associated with limb Ej. We note that simple and complex limbs Ai and Ej are detached from the parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2 when defining the associated parameters UAi, RAi, SAi and UEj, REj, SEj. To disconnect them, the mobile platform is cut up near each limb. Definition 2.13. (Connectivity of the mobile platform in a parallel robot with complex limbs). The connectivity S nD/ 1 of the mobile platform n≡nAi≡nEj in a parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2 with k1 simple limbs Ai (1Ai-2Ai-…-nAi), i=1,2,…k1 and and k2 complex limbs Ej (j=1,2,…k2) is defined as the number of independent motions between the mobile platform n≡nAi≡nEj and the reference platform 1≡1Ai≡1Ej in the parallel mechanism D. Connectivity proposition 17 (Connectivity of the mobile platform in a parallel mechanism with complex limbs). The connectivity of the mobile platform n≡nAi≡nEj in the parallel mechanism D ← A1-A2-…Ak1-E1-E2…-Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) is defined as the dimension of the common operational spaces of the simple and complex limbs associated with the parallel mechanism D

S nD/ 1 = dim( RnD/ 1 ) = dim( RA1 ∩ ... ∩ RAk 1 ∩ RE1 ∩ ... ∩ REk 2 ) .

(2.198)

where

S nD/ 1 = SnA1/ 1− A2 − ...− Ak 1− E1− E 2 − ...− Ek 2 denotes the connectivity between the mo-

bile platform n≡nAi≡nEj and the reference platform 1≡1Ai≡1Ej in the parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2,

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111

RnD/ 1 = RnA1/ 1− A2 − ...− Ak 1− E1− E 2 − ...− Ek 2 is the operational velocity vector space of

the parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2 defined by the relative velocities between the mobile platform n≡nAi≡nEj and the reference platform 1≡1Ai≡1Ej in the parallel mechanism D ← A1-A2-…Ak1-E1-E2-…Ek2, Eq. (2.198) is obtained by recurrence, as in Eq. (2.181). Connectivity proposition 18A. The number of joint parameters that lose their independence in the parallel mechanism D ← A1-A2-…Ak1-E1-E2…-Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) is given by rD = rA1− ...− Ak 1− E1− ...− Ek 2 = ∑ dim( RAi ) + ∑ dim( REj ) − dim( RnD/ 1 ) + rl k1

k2

i =1

j =1

(2.199)

rl= ∑ j =1 rl Ej additional joint parameters lose their independence in the

Eq. (2.199) follows from Eq. (2.185) by taking into account the fact that k2

closed loops integrated in the complex limbs. By rl Ej we have denoted the joint parameters that lose their independence in the closed loops integrated in complex limb Ej. We note that in complex limb Ej, the closed loops could be concatenated serially or in parallel. Two closed loops are serially concatenated if they do not have any common joint. One or more common joints could be shared by closed loops concatenated in parallel. The number of joint parameters that lose their independence in a single closed loop or in a closed loop serially concatenated in Ej-limb is given by connectivity propositions 5A, B, C and 6A, B, C – see Eqs. (2.140)-(2.142) and (2.144)(2.146). The number of joint pareameters that lose their independence in closed loops concatenated in parallel is given by connectivity propositions 13A and B – see Eqs. (2.185) and (2.191). Connectivity proposition 18B. The number of joint parameters that lose their independence in the parallel mechanism D ← A1-A2-…Ak1-E1-E2…-Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) is given by rD = rA1− ...− Ak 1− E1− ...− Ek 2 = ∑ S Ai + ∑ S Ej − S nD/ 1 + rl k1

k2

i =1

j =1

(2.200)

where: SAi and SEj are the connectivities of limbs Ai and Ej. Eq. (2.200) is obtained directly from Eq. (2.199) and from the definition of the connectivity of simple and complex chains associated with the limbs, as the number of independent motions between the extreme links (nAi and 1Ai of limb Ai and nEj and 1Ej of limb Ej). We can see that Eqs. (2.181), (2.185) and

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2 Structural parameters

(2.191) represents particular cases of Eqs. (2.198), (2.199) and (2.200) when k2=0. In this case the parallel mechanism with complex limbs D ← A1-A2-…Ak1-E1-E2-…-Ek2 becomes a parallel mechanism with simple limbs C ← A1-A2-…Ak1. Serially and parallel concatenated closed loops can be integrated in a limb. A closed loop is serially concatenated in a limb if the joints of the loop do not belong to other closed loops of the same limb. A group of closed loops are parallel concatenated in a limb if they have at least one common joint. The additional number of joint parameters that lose their independence in a closed loop serially concatenated in a complex limb is easily obtained by inspection by using the connectivity propositions 5A-C and 6A-C and their associated equations (2.140)-(2.142) and (2.144)(2.146). The additional number of joint parameters losing their independence in a group of closed loops that are parallel concatenated in a complex limb is also easily obtained by inspection by using the connectivity propositions 13A-B and their associated equations (2.185) and (2.191). We note that the closed loops parallel concatenated form a parallel kinematic chain integrated in the complex limb. Mobility proposition 7. The mobility of a parallel mechanism D ← A1A2-…Ak1-E1-E2-…-Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) is given by M D = ∑ dim(U Ai ) + ∑ dim(U Ej ) − ∑ dim( RAi ) − ∑ dim( REj ) + k1

i =1

D n/1

dim( R

or

) − rl

k2

k1

k2

j =1

i =1

j =1

M D = ∑ fi −∑ S Ai − ∑ S Ej + S nD/ 1 − rl p

k1

k2

i =1

i =1

j =1

(2.201)

(2.202)

where: UAi and RAi are the joint velocity space and the operational velocity space of the simple kinematic chain Ai associated with Ai-limb, UEj and REj are the joint velocity space and the operational velocity space of complex kinematic chain Ej associated with Ej-limb, fi is the mobility of the ith joint and p the total number of joints of the parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2 SAi is the connectivity of simple kinematic chain Ai associated with Ailimb, SEj is the connectivity of complex kinematic chain Ej associated with Ejlimb,

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113

S nD/ 1 is the connectivity of the mobile platform in the parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2. Eqs. (2.201) and (2.202) are obtained directly by integrating Eqs. (2.199) and (2.200) in Eq. (2.68). By analogy with connectivity propositions 14, 15 and 16 applicable to paralle robots with simple limbs, the following propositions can be formulated for parallel robots with complex limbs. Connectivity proposition 19. The mobile and reference platforms n≡nAi≡nEj and 1≡1Ai≡1Ej of a parallel mechanism D ← A1-A2-…Ak1-E1-E2…-Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) have relative motions if the dimension of the common operational space of Ai and Ej is greater than zero S nD/ 1 >0 ⇔

If

dim( RA1 ∩ ... ∩ RAk 1 ∩ RE1 ∩ ... ∩ REk 2 ) > 0 .

dim( RA1 ∩ ... ∩ RAk 1 ∩ RE1 ∩ ... ∩ REk 2 ) = 0

(2.203)

(2.204)

two platforms are relatively immobile ( SnD/ 1 =0). Eqs. (2.203) and (2.204) result directly from Eq. (2.198). Connectivity proposition 20. The connectivity S nD/ 1 between the mobile and reference platforms n≡nAi≡nEj and 1≡1Ai≡1Ej of a parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) is less than or equal to the mobility MD of the parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2 S nD/ 1 ≤ M D .

(2.205)

The proof of Eq. (2.205) is founded on the definitions 1 and 2 of mobility and connectivity. If Eq. (2.205) is not satisfied, MD independent parameters are not enough to describe the relative motion between n≡nAi≡nEj and 1≡1Ai≡1Ej of parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2 which contradicts the definition of mobility. Connectivity proposition 21. The basis of vector space RnD/ 1 of relative velocities between the mobile and reference platforms n≡nAi≡nEj and 1≡1Ai≡1Ej in the parallel mechanism D ← A1-A2-…Ak1-E1-E2-…-Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) does not vary with the position of the characteristic point on the mobile platform n≡nAi≡nEj. Proof. The independent velocities of the characteristic point define the basis ( RnD/ 1 ). The mobile platform n≡nAi has an infinite number of points,

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2 Structural parameters

and the velocity of each point can be expressed by a linear combination of the independent velocities of the same basis. The basis ( RnD/ 1 ) is valid for any point of the mobile platform including the points with the most restrictive motion. The following remark is also very important for the choice of the bases of operational vector spaces RAi and REj when when there are various ways to choose them. Remark 3. (Choice of the bases of operational vector spaces RAi and REj). If there are different ways to choose the bases of the operational vector spaces RAi and REj , the choice must be made by respecting the following: a) connectivity propositions 19 and 20 stating that 0 < S nD/ 1 ≤ M D

(2.206)

b) connectivity proposition 21, c) definition of mobility (in the sense of general mobility) as the minimum value of the instantaneous mobility - Eq. (2.109), d) permanent constraints in the relative motions of the links generated by the closed loops with special geometric conditions. We note that condition c) implies that the bases in Eq. (2.198) are selected so that the minimum value of SnD/ 1 is obtained. By making this choice, Eqs. (2.201) and (2.202) fit in with Eq. (2.109) setting up the general mobility as the minimum value of the instantaneous mobility. 2.3.7. General formulae for robot mobility and connectivity

The formulae for mobility and connectivity demonstrated in the previous section for parallel robots with complex limbs can be easily applied to parallel robots with simple limbs (k2=0), to single loop mechanisms (k2=0 and k1=2) or to simple open kinematic chains (k2=0 and k1=1). In this way, the corresponding formulae for mobility and connectivity introduced in sections 2.3.2-2.3.5 can be obtained as particular cases of general formulae demonstrated in section 2.3.6. To facilitate the correspondences and to bring together the different notations, in this section we present the formulae demonstrated in section 2.3.6 for parallel robots with complex limbs in a unified approach applicable to any kind of serial, parallel or hybrid serial-parallel robot. We consider the general case of a robot in which the end-effector is connected to the reference link by k≥1 kinematic chains. We recall that the end-effector is a polynary link called a mobile platform in the case of par-

2.3 Mobility and connectivity of parallel robots

115

allel robots, and a monary link for serial robots. The reference link may either be the fixed base or may be deemed to be fixed. The kinematic chains connecting the end-effector to the reference link can be simple or complex. They are called limbs or legs in the case of parallel robots. A serial robot can be considered to be a parallel robot with just one simple limb, and a hybrid robot a parallel robot with just one complex limb. We denote by F ← G1-G2-…Gk the kinematic chain associated with a general serial, parallel or hybrid robot, and by Gi (1Gi-2Gi-…nGi) the kinematic chain associated with the ith limb (i=1,2,…,k). The end effector is n≡nGi and the reference link 1≡1Gi. If the reference link is the fixed base, it is denoted by 1≡1Gi≡0. The total number of robot joints is denoted by p. Definition 2.14 (Serial robot). A serial robot F ← G1 is a robot in which end-effector n≡nG1 is connected to reference link 1≡1G1 by just one simple open kinematic chain Gi (1Gi-2Gi-…nGi) called a serial kinematic chain. Definition 2.15 (Parallel robot). A parallel robot F ← G1-G2-…Gk is a robot in which end-effector n≡nGi is connected in parallel to reference link 1≡1Gi by k≥2 kinematic chains Gi (1Gi-2Gi-…nGi) called limbs or legs. Definition 2.16 (Hybrid serial-parallel robot). A hybrid serial-parallel robot F ← G1 is a robot in which end-effector n≡nG1 is connected to reference link 1≡1G1 by just one complex kinematic chain Gi (1Gi-2Gi-…nGi) called complex limb or complex legs. Definition 2.17 (Fully-parallel robot). A fully-parallel robot F ← G1G2-…Gk is a parallel robot in which the number of limbs is equal to the robot mobility (k=M≥2), and each limb integrates just one actuator. Definition 2.18. (Joint velocity space associated with a kinematic chain). The joint velocity space UGi associated with a kinematic chain Gi is the joint velocity space of the open kinematic chain associated with Gi by opening all closed loops which may exist in Gi. Definition 2.19. (Operational velocity space associated with a kinematic chain). Operational velocity space RGi associated with a kinematic chain Gi (1Gi-2Gi-…nGi) is the vector space of relative velocities between extreme links nGi and 1Gi. We note that the constraints generated by the closed loops which may exist in Gi are taken into account in the operational velocity space RGi and are ignored in the joint velocity space UGi. Definition 2.20. (Operational velocity space associated with a robot). The operational velocity space RF associated with a robot is defined by the relative velocities between the end-effector and the reference link in the mechanism F ← G1-G2-…Gk in which the end-effector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi).

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2 Structural parameters

Definition 2.21. (Connectivity of a kinematic chain) Connectivity SGi of a kinematic chain Gi (1Gi-2Gi-…nGi) is given by the number of independent motions between extreme links nGi and 1Gi. We note that the kinematic chain Gi is detached from the mechanism F ← G1-G2-…Gk when defining the associated parameters UGi, RGi and SGi. To disconnect them the mobile platform is cut up near each limb. For serial robots Gi and F represent the same kinematic chain. Definition 2.22. (Connectivity of robot end-effector). The connectivity SF of the robot end-effector in the mechanism F ← G1-G2-…Gk in which the end-effector n≡nGi is connected to reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi), is defined as the number of independent motions between the end-effector and the reference link in the mechanism F. The connectivity SF of the robot-end effector is also called the robot connectivity, and represents the connectivity of the kinematic chain F ← G1-G2-…Gk. Connectivity proposition 22. (Connectivity of robot end-effector). The connectivity SF of the robot end-effector in the mechanism F ← G1-G2…Gk in which the end-effector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi), is given by the dimension of the common operational spaces of the kinematic chains Gi

S F = dim( RF ) = dim( RG1 ∩ RG 2 ∩ ... ∩ RGk ) .

(2.207)

where RGi is the operational velocity space of the kinematic chain Gi ,and RF is the operational velocity space of the kinematic chain F. Eq. (2.207) indicates that the constraints introduced by all k≥1 kinematic chains Gi (1Gi-2Gi-…nGi) are taken into account when defining the connectivity of robot end-effector. Eq. (2.207) represents a generalization of Eqs. (2.121), (2.172), (2.181) and (2.198) demonstrated in the previous sections. Connectivity proposition 23A. The number of joint parameters that lose their independence in themechanism F ← G1-G2-…Gk in which the endeffector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi) is given by r = ∑ dim( RGi ) − dim( RF ) + rl k

(2.208)

i =1

where RGi and RF are the operational velocity spaces of the kinematic chains Gi and F ,and rl represents the additional joint parameters that lose their independence in the closed loops which may exist in the kinematic chains Gi.

2.3 Mobility and connectivity of parallel robots

117

We note that rl = ∑ rlGi , where rlGi represents the additional joint pak

i =1

rameters that lose their independence in the closed loops which may exist in ith kinematic chain Gi. Eq. (2.208) represents a generalization of of Eqs. (2.145), (2.185) and (2.199) demonstrated in the previous sections. Connectivity proposition 23B. The number of joint parameters that lose their independence in the mechanism F ← G1-G2-…Gk in which the end-effector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi) is given by r = ∑ SGi − S F + rl k

(2.209)

i =1

where: SGi and SF are the connectivities of the kinematic chains Gi ,and F and rl represents the additional joint parameters that lose their independence in the closed loops which may exist in the kinematic chains Gi. Eq. (2.209) represents a generalization of Eqs. (2.140), (2.191) and (2.200) demonstrated in the previous sections. We recall that serially and parallel concatenated closed loops can be integrated in the kinematic chain Gi. A closed loop is serially concatenated in Gi if the joints of the loop do not belong to other closed loops of Gi. A group of closed loops are concatenated in parallel in Gi if they have at least one common joint. The additional number of joint parameters losing their independence in a closed loop serially concatenated in Gi is easily obtained by inspection by using the connectivity propositions 5A-C and 6A-C and their associated equations (2.140)-(2.142) and (2.144)-(2.146). The additional number of joint parameters losing their independence in a group of closed loops that are concatenated in parallel in a complex limb is also easily obtained by inspection by using the connectivity propositions 13A-B and their associated equations (2.185) and (2.191). We note that the closed loops concatenated in parallel form a parallel kinematic chain integrated in Gi. Mobility proposition 8A. The mobility of a mechanism F ← G1-G2-…Gk in which the end-effector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi) is given by M = ∑ dim(U Gi ) − ∑ dim( RGi ) + dim( RF ) − rl . k

k

i =1

i =1

(2.210)

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2 Structural parameters

Mobility proposition 8A. The mobility of a mechanism F ← G1-G2-…Gk in which the end-effector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi) is given by M = ∑ f i −∑ SGi + S F − rl p

k

i =1

i =1

(2.211)

where: UGi and RGi are the joint velocity space and operational velocity space of the kinematic chain Gi, RF is the operational velocity space of the kinematic chain F ← G1-G2…Gk, fi is the mobility of the ith joint, and p the total number of joints of the mechanism F, SGi is the connectivity of the kinematic chain Gi, SF is the connectivity of the kinematic chain F ← G1-G2-…Gk, rl represents the additional joint parameters that lose their independence in the closed loops which may exist in the kinematic chains Gi. Eqs. (2.210) and (2.211) represent the generalization of of Eqs. (2.120), (2.160), (2.161), (2.192), (2.193), (2.201) and (2.202) demonstrated in the previous sections. Connectivity proposition 24. A mechanism F ← G1-G2-…Gk in which the end-effector is connected to the reference link by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi) permits relative motions between end-effector n≡nGi and reference link 1≡1Gi if the dimension of the common operational space of Gi (i=1,2,…,k) is greater than zero SF>0 ⇔

If

dim( RG1 ∩ RG 2 ∩ ... ∩ RGk ) > 0 .

dim( RG1 ∩ RG 2 ∩ ... ∩ RGk ) = 0

(2.212)

(2.213)

the end-effector and reference link are relatively immobile (SF=0). Eqs. (2.212) and (2.213) represent the generalization of Eqs. (2.175), (2.176), (2.194), (2.195), (2.203) and (2.204) demonstrated in the previous sections. Connectivity proposition 25. The connectivity SF of a mechanism F ← G1-G2-…Gk in which the end-effector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi) is less than or equal to the mobility M of the mechanism F ← G1-G2-…Gk SF ≤ M .

(2.214)

2.3 Mobility and connectivity of parallel robots

119

Eq. (2.214) represents the generalisation of Eqs. (2.179), (2.196) and (2.205) demonstrated in the previous sections. Connectivity proposition 26. The basis of the vector space RF of relative velocities between the end-effector and the reference link in the mechanism F ← G1-G2-…Gk in which the end-effector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi), does not vary with the position of the characteristic point on the end-effector. This connectivity proposition represents a generalization of connectivity propositions 10A, 16 and 21 demonstrated in sections 2.3.4-2.3.6. The following remark is also very important for the choice of the bases of operational vector spaces RGi when there are various ways to choose them. Remark 4. (Choice of the bases of operational vector spaces RGi). If there are various ways to choose the bases of the operational vector spaces RGi ,the choice must be made by respecting the following: a) connectivity propositions 24 and 25 stating that 0 < SF ≤ M

(2.215)

b) connectivity proposition 26, c) definition of mobility (in the sense of general mobility) as the minimum value of the instantaneous mobility - Eq. (2.109), d) permanent constraints in the relative motions of the links generated by the closed loops with special geometric conditions. We note that condition c) implies that the bases in Eq. (2.207) are selected so that the minimum value of S F is obtained. By this choice, Eqs. (2.210) and (2.211) fit in with Eq. (2.109) setting up the general mobility as the minimum value of the instantaneous mobility. Remark 4 represents a generalization of previous remarks presented in sections 2.3.4-2.3.6. We note that the new formulae presented in this section are applicable to the following: (i) Serial robots, when k=1, rl=0 and G1 is a simple open kinematic chain. In this case F and G1 represent the same kinematic chain F ← G1 ,and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi). (ii) Parallel robots with simple limbs, when k≥2, rl≠0 and all Gi (i=1,2,…,k) are simple open kinematic chains. (iii) Parallel robots with complex limbs, when k≥2, rl≠0 and at least one of Gi (i=1,2,…,k) is a complex kinematic chain. (iv) Serial robots with closed loops and parallel hybrid serialparallel robots, when k=1, rl≠0 and G1 is a complex open kine-

120

2 Structural parameters

matic chain. In this case F and G1 also represent the same kinematic chain F ← G1 and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi). The application of the new formulae proposed in this section will be illustrated in chapter 3 for the various types of robots. In the following chapters these formulae will be used for structural synthesis of parallel robots with simple and complex limbs. Along with the morphological approach, they represent the cornerstone of the structural synthesis method.

2.4 Overconstraints in parallel robots Usually overconstrained mechanisms are defined as mechanisms that, without satisfying the Chebychev-Grübler-Kutzbach mobility criterion, have full cycle mobility. Overconstrained mechanisms have mobility over a finite range of motions even though they should be structures (M≤0) according to the CGK formula given in Eq. (2.64). This definition was introduced to designate single and multi-loop mechanisms that do not obey the well-known mobility formulae presented in section 2.2 (see for examples the mechanisms presented in Figs. 2.1, 2.6, 2.7 and 2.8b). The first overconstrained multi-loop mechanism which appeared in the literature was proposed by the French mathematician Gilles Personne (or Personier) De Roberval and dates from 1669. The original drawings presented by De Roberval in “Mémoires de l'Académie royale des sciences: depuis 1666, jusqu'à 1699” are presented in Fig. 2.16 (De Roberval 1669). This mechanism is known as the De Roberval scale, and is used in the standard laboratory double-pan balance. The horizontal arms together with both vertical pan supports make a double parallelogram mechanism. In this way the pans will stay horizontal in the full range of movement. More importantly, it prevents the pans from rotating as they move up and down, and this is the principle that ensures that the position of weights on the pans has no effect on the balance. De Roberval’s scale (Fig. 2.8b) could be considered both as the first overconstrained mechanism and as the first “parallel mechanism with one degree of mobility” and three simple limbs. We will analyse this mechanism in the next chapter. Since than, other overconstrained mechanisms have been proposed (Sarrus 1853; Delassus 1900, 1902, 1922; Bennett 1902, 1914; Bricard 1927; Myard 1931a,b; Goldberg1943; Altman 1952; Baker 1978; Waldron 1968,1969 1979; Wohlhart 1987, 1991, 1995; Dietmaier 1995; Fang and Tsai 2004a; Mavroidis and Roth 1995) et al . References can be found to

2.4 Overconstraints in parallel robots

121

Fig. 2.16. De Roberval scale published in 1669: front cover of «Mémoires de l'Académie royale des sciences: depuis 1666, jusqu'à 1699» (a), original drawings (b) and (c)

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2 Structural parameters

almost all known overconstrained mechanisms in (Baker 1980a, 1984; Waldron 1973a,b; Phillips 1990). In this section we revisit the definition of overconstrained mechanisms independently of the CGK formulae. The author presented the new general formulae for the calculation of the number of overconstrains in parallel robots for the first time in (Gogu 2005e). We consider a mechanism with p joints, n mobile links (reference link not included) and q structurally independent loops. Overconstraint proposition 1A. The number of overconstraints N of a mechanism with mobility M is given by the difference between the number of kinematic constraints introduced by the joints of the mechanism before and after closing the loops N = ∑ ci − (6n − M ) p

(2.216)

i =1

Proof. The number of kinematic constraints introduced by the joints of the mechanism before closing the loops is given by the sum of the degrees

of joint constraints

∑

p

i =1 i

c . The total number of constraints existing in the

mechanism after closing the loops is given by the difference between the number of independent motions of the n mobile members considered free in the general kinematic space with six dimensions, and the number of independent motion parameters remaining after connecting the members by joints and closing the q independent loops of the mechanism. Eq. (2.216) could also be interpreted from the static point of view. In p this case, ∑ i =1 ci represents the number of independent forces acting in the

joints of the mechanism, 6n indicates the number of static equations for the n mobile members, and M indicates the number of forces in the actuated joints. Overconstraint proposition 1B. The number of overconstraints N of a mechanism with mobility M is given by the difference between the maximum number of joint kinematic parameters that could lose their independence in the q closed loops, and the number of joint kinematic parameters that actually lose their independence in the closed loops:

N=6q-r

(2.217)

Proof. Eq. (2.217) is obtained from Eq. (2.216) by taking into consideration Eq. (2.68), the definition of q given by the theory of graphs (q=pn) and the relation between the degree of mobility and the degree of constraint of a joint (f+c=6).

2.4 Overconstraints in parallel robots

123

If the number of overconstraints is zero (N=0) the mechanism is called non-overconstrained or isostatic. Eqs. (2.216) and (2.217) are applicable to single and multi-loop mechanisms. For a single-loop mechanism (q=1) we obtain

N=6-r.

(2.218)

We note that a simple open kinematic chain cannot be overconstrained. Let us consider the case of the mechanism F ← G1-G2-…Gk in which endeffector n≡nGi is connected to reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi). The operational vector spaces of F and Gi are RGi and RF with dim(RF)=SF and dim(RGi)=SGi. We recall that RGi is the vector space of relative velocities between extreme links nGi and 1Gi in kinematic chain Gi, and RF the vector space of relative velocities between extreme links n≡nGi and 1≡1Gi in the mechanism F ← G1-G2-…Gk. The dimensions of these spaces give connectivities SF and SGi of F and Gi (see definitions 19-22 in section 2.3.7 for more details). Overconstraint proposition 2A. The number of overconstraints of a mechanism F ← G1-G2-…Gk in which end-effector n≡nGi is connected to reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi) is given by

N = 6q − ∑ dim( RGi ) + dim( RF ) − rl k

(2.219)

i =1

where RGi and RF are the operational velocity spaces of kinematic chains Gi and F and rl represent the additional joint parameters that lose their independence in the closed loops which may exist in kinematic chains Gi. Eq. (2.219) results directly from Eqs. (2.217) and (2.185) by taking into consideration the fact that the number of structurally independent loops formed between k kinematic chains in-parallel concatenated is q=k-1. Additional loops may exist in kinematic chains Gi. Overconstraint proposition 2A. The number of overconstraints of a mechanism F ← G1-G2-…Gk in which end-effector n≡nGi is connected to reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi) is given by

N = 6q − ∑ SGi + S F − rl k

(2.220)

i =1

where: SGi and SF are the connectivities of kinematic chains Gi and F, and rl represents the additional joint parameters that lose their independence in the closed loops which may exist in kinematic chains Gi.

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2 Structural parameters

Eq. (2.220) follows directly from Eq. (2.219) by taking into account the definition of connectivities SGi and SF as the dimensions of velocity vector spaces SGi and SF. The number of overconstraints gives the degree of overconstraint of the mechanism. To obtain a non-overconstrained mechanism, idle mobilities are introduced in the joints of the mechanism. Definition 2.23. (Idle mobility). Idle mobility is a joint mobility that does not affect a mechanism’s mobility. Idle mobilities could change just SGi, SF, and consequently the number of overconstraints N. In theoretical conditions, when no errors exist with respect to parallel and perpendicular positions of joint axes, motion amplitude in an idle mobility is zero. Real life manufacturing and assembling processes introduce errors in the relative positions of the joint axes and, in this case, the idle mobilities become effective mobilities usually with small amplitudes, depending on the precision of the parallel robot. We note that formulae (2.219) and (2.220) are applicable to the following: (i) Serial robots, when k=1, rl=0 and G1 is a simple open kinematic chain. In this case F and G1 represent the same kinematic chain F ← G1 and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi). (ii) Parallel robots with simple limbs, when k≥2, rl≠0 and all Gi (i=1,2,…,k) are simple open kinematic chains. (iii) Parallel robots with complex limbs when k≥2, rl≠0 and at least one of Gi (i=1,2,…,k) is a complex kinematic chain. (v) Serial robots with closed loops and parallel hybrid serialparallel robots, when k=1, rl≠0 and G1 is a complex open kinematic chain. In this case F and G1 also represent the same kinematic chain F ← G1 and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi). The application of these new formulae along with the selection of idle mobilities to obtain non-overconstrained mechanisms will be illustrated in chapter 3 for the various types of robots. In the following chapters these formulae will be used for structural synthesis of overconstrained and nonoverconstrained parallel robots with simple and complex limbs.

2.5 Redundancy in parallel robots

125

2.5 Redundancy in parallel robots Redundant robots are mechanismas with more degrees of freedom than required for doing the prescribed task in the task space. Redundant actuation in parallel manipulators can be achieved by: (i) actuating some of the passive joints within the existing limbs, (ii) introducing additional actuated limbs beyond the minimum necessary to actuate the manipulator and (iii) introducing some additional actuated joints within the limbs beyond the minimum necessary to actuate the manipulator. We note that actuator redundancy does not affect the connectivity of the end-effector and only increases the number of actuators. In all cases, only M actuators have independent motion. The other actuators have dependent motions. In extreme cases, dependency consists in actuator locking. A selective choice of actuators with independent motion could have many advantages. Redundancy in parallel manipulators is used to eliminate some singular configurations (Wang and Gosselin 2004; Kurtz and Hayward 1992; Merlet 1997; Firmani and Podhorodeski 2004, Alberich et al. 2006), to minimize the joint rates, to optimize the joint torques/forces (Dasgupta and Mruthyunjaya 1998b; Bruckman et al 2006; Nokleby et al. 2005) to increase dexterity workspace (Marquet et al 2001a,b), stiffness (Chakarov 2004), eigenfrequencies, kinematic and dynamic accuracy (Valasek et al. 2004), to improve velocity performances (Krut et al. 2004) and both kinematic and dynamic control algorithms (Liu et al. 2001; Cheng at al. 2003), to develop large forces in micro electro-mechanical systems (Mukherjee et al. 2001), decoupling the orientations and the translations (Jin et al. 2004), to obtain reconfigurable platforms (Mohamed and Gosselin 2005) and limbs (Fischer et al. 2004) or combined advantages (Nahon and Angeles 1991; Zanganach and Angeles 1994a,b; Kim 1997; Kock and Scumacher 1998, 2000; Mohamed 2003a,b). The author was the first to present new formulae for calculating the degree of redundancy of parallel robots (Gogu 2006a). Redundancy is achieved by introducing additional actuated joints within the limbs, beyond the minimum necessary to actuate the manipulator. In this way we reduce motion coupling in parallel manipulators. Let us consider again the case of the mechanism F ← G1-G2-…Gk in which end-effector n≡nGi is connected to reference link 1≡1Gi by k simple or complex kinematic chains Gi (1Gi-2Gi-…nGi). The mechanism F ← G1-G2…Gk is characterized by: (i) UGi - the vector space of joint velocities in kinematic chain Gi, (ii) RGi - the vector space of relative velocities between extreme links nGi and 1Gi in kinematic chain Gi,

126

2 Structural parameters

(iii)

RF - the vector space of relative velocities between extreme links n≡nGi and 1≡1Gi in parallel mechanism F ← G1-G2-…Gk, (iv) SGi=dim(RGi) - the connectivity of Gi, (v) SF=dim(RF) - the connectivity of F, (see definitions 18-22 in section 2.3.7 for more details). Definition 2.24. (Structural redundancy of a kinematic chain). The structural redundancy of the kinematic chain represents the difference between its mobility and connectivity. Structural redundancy proposition 3A. The structural redundancy of a mechanism F ← G1-G2-…Gk in which end-effector n≡nGi is connected to reference link 1≡1Gi by k kinematic chains Gi (1Gi-2Gi-…nGi) is given by T = ∑ dim(U Gi ) − ∑ dim( RGi ) − rl k

k

i =1

i =1

(2.221)

Structural redundancy proposition 3B. The structural redundancy of a mechanism F ← G1-G2-…Gk in which end-effector n≡nGi is connected to reference link 1≡1Gi by k kinematic chains Gi (1Gi-2Gi-…nGi) is given by

T = ∑ f i −∑ SGi − rl p

k

i =1

i =1

(2.222)

where: UGi and RGi are the joint velocity space and operational velocity space of kinematic chain Gi, fi is the mobility of the ith joint, and p the total number of joints of the mechanism F, SGi is the connectivity of the kinematic chain Gi, rl represents the additional joint parameters losing their independence in the closed loops which may exist in kinematic chains Gi. Eqs. (2.221) and (2.222) results directly from structural redundancy definition and Eqs. (2.210) and (2.211). In a parallel robot, the structural redundancy gives the internal mobilities in the limbs. We note that formulae (2.221) and (2.211) are applicable to the following: (iv) Serial robots, when k=1, rl=0 and G1 is a simple open kinematic chain. In this case F and G1 represent the same kinematic chain F ← G1 and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi). (v) Parallel robots with simple limbs, when k≥2, rl≠0 and all Gi (i=1,2,…,k) are simple open kinematic chains.

2.6 General formulae for structural parameters

127

Parallel robots with complex limbs, when k≥2, rl≠0 and at least one of Gi (i=1,2,…,k) is a complex kinematic chain. (vi) Serial robots with closed loops and parallel hybrid serialparallel robots, when k=1, rl≠0 and G1 is a complex open kinematic chain. In this case F and G1 also represent the same kinematic chain F ← G1 and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi). The application of these new formulae will be illustrated in chapter 3 for the various types of robots. In the following chapters these formulae will be used for structural synthesis of redundant and non redundant parallel robots with simple and complex limbs. (vi)

2.6 General formulae for structural parameters The propositions and formulae demonstrated in the previous sections for mobility, connectivity, overconstraint and redundancy of various types of robots can be extrapolated to single and multi-loop mechanisms. In this case the end-effector n≡nGi is replaced by a link called a characteristic link, and the reference link is the fixed base 1≡1Gi≡0. Theoretically any link except the reference one can be considered as a characteristic link. In this way, a mechanism open or closed, single or multi-loop can be defined as a kinematic chain F ← G1-G2-…Gk with p≥2 joints. Definition 2.25. (Mechanism). A mechanism F ← G1-G2-…Gk is a kinematic chain in which a characteristic link denoted by n≡nGi (i=1,2,…k) is connected to reference link 1≡1Gi≡0 by k≥1 sub-kinematic chains Gi (1Gi-2Gi-…nGi). To simplfy the terminology, sub-kinematic chains Gi that are parts of the mechanism F ← G1-G2-…Gk are simply called kinematic chains. Definition 2.26. (Simple mechanism). A simple mechanism is a mechanism F ← G1-G2-…Gk with just monary and binary links. Definition 2.27. (Complex mechanism). A complex mechanism is a mechanism F ← G1-G2-…-Gk with at least one ternary link. We recall the following:(i) a monary link is a link which carries only one kinematic pairing element, (ii) a binary link is a link connected in the kinematic chain by two joints and (iii) a polynary link is a link connected in the kinematic chain by more than two joints –see section 1.4. Definition 2.28. (Serial mechanism). A serial mechanism F ← G1 is a kinematic chain in which characteristic link n≡nG1 is connected to reference link 1≡1G1≡0 by just one simple open kinematic chain G1 (1G1-2G1…nG1).

128

2 Structural parameters

Definition 2.29. (Single-loop mechanism). A single-loop mechanism F ← G1-G2 is a kinematic chain in which characteristic link n≡nGi (i=1,2) is connected to reference link 1≡1Gi≡0 by k=2 serial kinematic chains Gi (1Gi-2Gi-…nGi). Definition 2.30. (Multi-loop mechanism). A multi-loop mechanism F ← G1-G2-…Gk is a kinematic chain, containing at least one ternary link, in which characteristic link n≡nGi (i=1,2,…k) is connected to reference link 1≡1Gi≡0 by k≥2 kinematic chains Gi (1Gi-2Gi-…nGi). The ternary link can be the characteristic link or any other link of the kinematic chain. We can see that serial and single-loop mechanisms are simple, and multi-loop mechanisms are complex mechanisms. We note that joint velocity space UGi, ,operational velocity space RGi , and connectivity SGi of the kinematic chain Gi (1Gi-2Gi-…nGi) are defined in the previous section (see definitions 18, 19 and 21). The kinematic chain Gi is detached from the mechanism F ← G1-G2-…Gk when defining associated parameters UGi, RGi and SGi. To disconnect it the characteristic link is cut after the last joint of Gi. Definition 2.31. (Connectivity of the characteristic link). Connectivity SF of characteristic link n≡nGi in the mechanism F ← G1-G2-…Gk is defined by the number of independent motions between n≡nGi and reference link 1≡1Gi≡0 allowed by the mechanism F. A mechanism F ← G1-G2-…Gk permits relative motions between endeffector n≡nGi and reference link 1≡1Gi≡0 when n≡nGi and 1≡1Gi≡0 are connected by k≥1 kinematic chains Gi (1Gi-2Gi-…nGi) if and only if the dimension of the common operational space of Gi (i=1,2,…,k) is higher than zero – see Eq. (2.212). The general formule demonstrated in this chapter for mobility M, degree of overconstraint N and redundancy T of parallel robots are also applicable to the mechanism F ← G1-G2-…Gk. To conclude this chapter we recall these formulae and the meaning of the parameters used to define them. M = ∑ f i −r

(2.223)

N=6q-r

(2.224)

T=M-SF

(2.225)

p

i =1

where

2.6 General formulae for structural parameters

129

r = ∑ SGi − S F + rl

(2.226)

S F = dim( RF ) = dim( RG1 ∩ RG 2 ∩ ... ∩ RGk )

(2.227)

rl = ∑ rlGi .

(2.228)

k

i =1

and

k

i =1

We recall that p represents the total number of joints, k the total number of independent closed loops in the sense of graph theory, fi the mobility of the ith joint, r the number of joint parameters that lose their independence in the mechanism F, rlGi the number of joint parameters that lose their independence in the closed loops of limb Gi, rl the total number of joint parameters that lose their independence in the closed loops that may exist in the limbs of mechanism F, SF connectivity of characteristic link n≡nGi in the mechanism F, SGi the connectivity of characteristic link n≡nGi in limb Gi disconnected from the mechanism F, RG operational velocity space of the mechanism F ← G1-G2-…Gk and RGi the operational velocity space of limb Gi disconnected from the mechanism F. We note that the intersection in Eq. (2.227) is consistent if the operational velocity spaces RGi are defined by the velocities of the same point situated on the characteristic link. This point is called the characteristic point, and denoted in this work by H. It is a point with the most restrictive motions of the characteristic link. The connectivity SF of the characteristic link n≡nGi in the mechanism F ← G1-G2-…Gk is less than or equal to the mobility M of the mechanism F. The basis of the vector space RF of relative velocities between the characteristic link n≡nGi and the reference link 1≡1Gi≡0 in the mechanism F ← G1-G2-…Gk dos not vary with the position of the characteristic point on n≡nGi. When there are various ways to choose the basis of the operational spaces, the bases of RGi in Eq. (2.227) are selected such as the minimum value of S F is obtained. By this choice, the result of Eq. (2.223) fits in with Eq. (2.109) setting up the general mobility as the minimum value of the instantaneous mobility. Formulae (2.223)-(2.228) are applicable to any kind of mechanism F ← G1-G2-…Gk:

130

(i)

2 Structural parameters

Serial mechanisms, when k=1, q=0, rl=0 and G1 is a simple open kinematic chain. In this case F and G1 represent the same kinematic chain F ← G1 and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi). (ii) Single-loop mechanisms, when k=2, q=1, rl≠0 and Gi (i=1,2) are simple open kinematic chains. (iii) Multi-loop mechanisms, when k≥2, q≥2, rl≠0 and mechanism F ← G1-G2-…Gk contains at least one ternary link. Kinematic inversion and change of the distal link may be applied to associate a parallel mechanism with any multi-loop mechanism. Although relative motions between components remain the same, structural parameters (mobility, degree of overconstraint and degree of structural redundancy) of mechanisms are invariant with respect to kinematic inversion or the change of the distal link. Kinematic inversion changes the reference link by repositionning the “observer”. The following two properties may be formulated: Property 5 (Invariance of structural parameters with the change of the reference link). Mobility, overconstraints and redundancy of the mechanism are invariants under the change of the reference link. Property 6 (Invariance of structural parameters with the change of the distal link). Mobility, overconstraints and redundancy of the mechanism are invariants under the change of the distal link. The application of these new formulae will be illustrated in chapter 3 for the various types of mechanism to overcome the drawbacks of the CGK mobility formulae widely used for mobility calculation of multi-loop mechanisms. These new formulae do not involve setting up the instantaneous constraint systems and are successfully applied in chapter 3 to various mechanisms including so called “paradoxical” mechanisms.

3 Structural analysis

Structural analysis of parallel robotic manipulators aims at determining the main structural parameters: connectivity, mobility, redundancy, degree of overconstraint. The following examples will illustrate the definitions, the theorems and the properties presented in the previous chapter. We emphasise the difference between mobility M and connectivity S; these are often confused in the literature under the name of degree of freedom (DoF). As we have seen, mobility and connectivity are the key structural parameters also used to calculate the degree of redundancy and the degree of overconstraint. We begin with simple examples of open and closed-loop kinematic chains associated with the limbs of parallel robots and we finish with structural analysis of various types of parallel robots with simple or complex limbs. We illustrate the fact that the formulae demonstrated in the previous chapter for parallel robots are also applicable to single or multiloop mechanisms that do not obey the classical Chebychev-GrüblerKutzbach (CGK) formulae.

3.1 Simple open kinematic chains We recall that an open kinematic chain is a chain in which there is at least one link which carries only one kinematic pairing element, called a monary link. In a simple open kinematic chain (open-loop mechanism) a link is connected by one, or a maximum of two, joints, which means only monary and binary links exist. Simple open kinematic chains are associated with simple limbs of parallel robotic manipulators. Example 1. The simple open kinematic chain A (1-2-…-n+1) presented in Fig. 3.1a has n+1 links, and p=n cylindrical joints with the same axis. The mobility of each joint is fi=2 (i=1,2,…,p). This kinematic chain has MA=2p=2n, SA=2 and TA=MA-SA=2p-2. The mobility is given by Eq.(2.120). The independent finite displacements in the joints necessary to define the configuration of the simple open kinematic chain are ϕi +1,i and di + 1,i (i=1,2,…,n,n+1) - see definition 1 in chapter 2. The independent velocities in the joints required to define the velocities of the distal link (n+1)

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3 Structural analysis

are ω i + 1,i = ϕ i + 1 ,i and vi + 1 ,i = d i + 1,i (i=1,2,…,n,n+1) - see definition 2. They

form the basis of the joint space (UA)=( ϕ 21 , d 21 , …, ϕ i + 1,i , d i + 1,i ,…, ϕ n + 1,n , mobility of the mechanism M A = dim(U A ) = ∑ i =1 f = 2m . The connectivity

d n+1,n ) having the cardinal 2m. The dimension of this joint space gives the m

is given by Eq.(2.121). The dimension of the operational space can easily be determined by inspection. We can see that the independent finite displacements between the distal links 1 and n+1 are x and ϕ x (see definition 3). The independent velocities between the distal links are v x and ω x (see definition 4). They form the basis of the operational space (RA)=( v x ,ω x ) having the cardinal 2. The dimension of this space gives the connectivity SA=dim(RA)=2. The kinematic chain achieves the addition of linear and angular velocities of the joints ( v x = vn1 = v21 + v32 + ... + vn +1 ,n and ω x = ω n1 = ω 21 + ω 32 + ... + ω n +1 ,n ). Structural redundancy TA=2(m-1) is given by Eq. (2.130). Figure 3.1b presents the particular case where the p cylindrical joints are replaced by p revolute joints with the same axis. The mobility of each joint

Fig. 3.1. Simple open kinematic chains with n+1 links and p=n joints: (a) MA=2p=2n, SA=2, TA=2(n-1); (b and c) MA=p=n, SA=1, TA=n-1

3.1 Simple open kinematic chains

133

is fi=1 (i=1,2,…,p). The mechanism has MA=p=n, SA=1 and TA=MA-SA= n-1. The only independent velocity between the distal links is ω x , and (RA)=( ω x ), SA=dim(RA)=1. To control this velocity we have to know the MA=p independent velocities in the revolute joints of the mechanism. The mechanism achieves the addition of joint angular velocities ( ω x = ω n1 = ω 21 + ω 32 + ... + ω n +1 ,n ). The kinematic chain presented in Fig. 3.1c represents another particular case in which the p cylindrical joints are replaced by p prismatic joints having the same direction. The mobility of each joint is fi=1 (i=1,2,…,p). The mechanism has MA=p=n, S=1 and TA=MA-SA=n-1. The only relative independent velocity between the distal and proximal links is v x . We obtain (RA)=( v x ) and SA=dim(RA)=1. To control this velocity we have to know the MA=p independent velocities in the prismatic joints of the mechanism. The mechanism achieves the addition of joint linear velocities ( v x = vn1 = v21 + v32 + ... + vn +1 ,n ). Example 2. The simple open kinematic chain in Fig. 3.2 has n+1 links and p=n (n≥3) revolute joints R with parallel axes. The mobility of each joint is fi=1 (i=1,2,…,n). The mechanism has MA=p=n, SA=3 and TA=MASA=n-3. The relative independent velocities between the distal links 1 and n+1 are v x ,v y ,ω z , than (RA)=( v x ,v y ,ω z ) and SA=dim(RA)=3. To control these velocities we have to know MA=p=n independent velocities in the revolute joints of the mechanism.

Fig. 3.2. Simple open kinematic chain with n+1 links and p=n revolute joints with parallel axes: MA=p=n, SA=3, TA=n-3

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3 Structural analysis

Example 3. Each simple open kinematic chain associated with the three limbs of the parallel Cartesian robotic manipulator presented in Fig. 2.1 has MA1=MA2=MA3=4, SA1=SA2=SA3=4 and TA1=TA2=TA3=0. The independent finite displacements in the joints needed to define the configuration of the simple open kinematic chain associated with limb A1 (0A≡1A-2A-3A-4A5A) are d 2 A , ϕ 3 A , ϕ 4 A and ϕ 5 A - Fig. 2.2a. The independent velocities in the joints needed to define the velocities of any link of A1-limb are d 2 A ,ϕ 3 A ,ϕ 4 A and ϕ 5 A . They form the basis of the joint velocity space

(UA1)=( d 2 A ,ϕ 3 A ,ϕ 4 A ,ϕ 5 A ) having the cardinal 4. The dimension of this space gives the mobility of the simple open mechanism MA1=dim (UA1)=4. The relative independent velocities between the distal links (0A≡1A and 5) of the simple open kinematic chain associated with A1-limb are v x ,v y ,v z

and ω x . They form the basis of the operational velocity space (RA1)=( v x ,v y ,v z ,ω x ) having the cardinal 4. The dimension of this opera-

tional space gives the connectivity of the simple open kinematic chain associated with A1-limb, SA1=dim (RA1)=4. Similar results are obtained for the A2-limb (0B≡1B-2B-3B-4B-5B) - Fig. 2.2b - and for A3-limb (0C≡1C-2C3C-4C-5C) - Fig. 2.2c. We have (RA2)=( v x ,v y ,v z ,ω y ) and (RA3)=( v x ,v y ,v z ,ω z ) and SA1=SA2=SA3=4. These results can easily be verified by calculating the rank of the Jacobian matrix associated with each limb. If we express the translation velocity of point H and the angular velocity of link 5A in the reference coordinate system O0x0y0z0 (Fig. 2.2a) we obtain the following kinematic equation for the chain associated with A1-limb

 vx

vy

vz

ω x ω y ω z  = [ J ] A1  d 2 A ϕ 3 A ϕ 4 A ϕ 5 A  T

T

(3.1)

with

[ J ]A1

0 1 0 -b S -b S 3A 3A 4A 43A  0 b3AC3A +b4AC43A = 1 0 0 0  0 0

0 -b4A S43A b4AC43A 1 0 0

0 0  0 . 1 0  0 

(3.2)

3.1 Simple open kinematic chains

135

Similar equations can be written for the simple open kinematic chains associated with A2-limb (Fig. 2.2a) and with A3-limb (Fig. 2.2b):

 vx

vy

vz

ω x ω y ω z  = [ J ] A2  d 2 B ϕ 3 B ϕ 4 B ϕ 5 B 

T

(3.3)

vy

vz

ω x ω y ω z  = [ J ] A3  d 2C ϕ 3C ϕ 4C ϕ 5C 

T

(3.4)

and

 vx

T

T

with the following Jacobian matrices:

[ J ]A2

0 b3B C3B +b4B C43B 1 0  0 -b3B S3B -b4B S43B = 0 0 0 1  0 0

b4B C43B 0 -b4B S43B 0 1 0

0 0  0  0 1  0 

(3.5)

[ J ]A3

0 -b3C S3C -b4C S43C 0 b C +b C 3C 3C 4C 43C  1 0 = 0 0 0 0  1 0

-b4C S43C b4C C43C 0 0 0 1

0 0  0  0 0  1

(3.6)

and

The connectivity of the simple open mechanism associated with each limb of the parallel Cartesian robotic manipulator in Fig. 2.2 can also be obtained by using the rank of the Jacobian matrices: SA1=rank ( [ J ] A1 )=4,

SA2=rank ( [ J ]A2 )=4, SA3=rank ( [ J ]A3 )=4. By analyzing the Jacobian matrices (3.2), (3.5) and (3.6) we can see that each simple kinematic chain in Fig. 2.2 has an operational space with a unique basis. After eliminating the two lines containing only zero terms, we can calculate the determinants of

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3 Structural analysis

Fig. 3.3. Singular configurations of simple open kinematic chain A1: structural singularities (a), (b) and (c) and instantaneous singularity (d)

Jacobian matrices [ J ] Ai( 4×4 ) (i=1,2,3). For example, the determinant of the

Jacobian matrix [ J ]A1( 4×4 ) is d=b3Ab4AS4A. This simple open kinematic chain

has structural singularities when b3A=0 and/or b4A=0 (see Fig. 3.3a-c) and an instantaneous singularity when S4A=0, that is ϕ 4 A = 0 (see Fig. 3.3d). These singularities do not affect the mobility that remains MA1=4 for all cases in Fig. 3.3. The singularities affect only the connectivity of this simple open kinematic chain that become SA1=3 (Fig. 3.3a, b and d) and SA1=2 - Fig. 3.3c. Example 4. The simple open kinematic chains presented in Fig. 2.3 are obtained by serial concatenation of two limbs of the parallel Cartesian robotic manipulator from Fig. 2.1. The independent velocities of distal link 1B with respect to reference link 1A≡0A in the simple open chain A≡A1-A2 (0A≡1A-2A-…-5A≡5A-…2B-1B) are: v x ,v y ,v z ,ω x , and ω y . They form the basis of the operational space (RA≡A1-A2)=( v x ,v y ,v z ,ω x ,ω y ) having the cardinal 5. The dimension of this operational space gives the connectivity of the open

3.2 Single-loop kinematic chains

137

simple chain A≡A1-A2: SA1-A2=dim (RA1-A2)=5. This result can also be obtained by using Eq. (2.127) : SA≡A1-A2=dim(RA≡A1-A2)= dim(RA1)+dim(RA2)-dim(RA1 ∩ RA2)=4+4-3=5. This kinematic chain has the following structural parameters: MA=8, SA=5, TA=3. Similar results may be obtained for the simple open kinematic chain in Fig. 2.4. The independent velocities of the distal link 1C relative to the reference link 1A≡0A in the simple open chain A≡A1-A3 (0A≡1A-2A-…-5A≡5C…2C-1C) are: v x ,v y ,v z ,ω x , and ω z . They form the basis of the operational space (RA≡A1-A3)=( v x ,v y ,v z ,ω x ,ω z ) also having the cardinal 5. Implicitly, the connectivity of the open chain A≡A1-A3 is SA≡A1-A3= dim(RA≡A1-A3)=5. This result can also be obtained as follows: SA≡A1-A3=dim(RA≡A1-A3)= dim(RA1)+dim(RA3)-dim(RA1 ∩ RA3)=4+4-3=5. We can see that, in general, the dimensions of the velocity joint space and the operational velocity space can easily be determined by inspection for the simple open kinematic chains. The calculation of the rank of the Jacobian matrix is useful just for verification, if necessary.

3.2 Single-loop kinematic chains We recall that, in a single-loop kinematic chain each link is connected with two other links, which means only binary links exist in a single-loop mechanism. Example 5. The single-loop kinematic chain in Fig. 3.4 has p=8 onedegree-of-freedom joints and m=8 links, including the fixed reference link (1A≡1B≡0). The single-loop kinematic chain B1 (1A≡0-2A-…-5A≡5B-…2B1B≡0) can be obtained by fixing together the distal links (1A≡0 and 1B) of the simple open kinematic chain A≡A1-A2 (1A≡0-2A-…-5A≡5B-…2B-1B) from Fig. 2.3. For the single-loop kinematic chain B1, Eq. (2.140) gives rB1=SA≡A1-A2=dim(RA≡A1-A2)=5 (see example 4) and the mobility given by Eq. (2.160) is MB1=8-5=3. We may obtain the same result by considering that the single-loop kinematic chain B1 ← A1-A2 (1A≡0-2A-…-5A≡5B-…2B1B≡0) is obtained by connecting together the distal links 1A≡0 with 1B≡0 and 5A with 5B of simple open kinematic chains A1 (0A≡1A-2A-3A-4A-5A) and A2 (0B≡1B-2B-3B-4B-5B) in Fig. 2.2a-b and by applying Eqs. (2.144) and (2.164). The unique basis of the operational spaces of simple open kinematic chains A1 and A2 are (RA1)=( v x ,v y ,v z ,ω x ) and (RA2)=( v x ,v y ,v z ,ω y ). We recall that (RA1) and (RA2) represent the bases of the vector spaces of the ve-

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3 Structural analysis

locities of point H in open kinematic chains A1 and A2 in Fig. 2.2a-b. The connectivity between links 5≡5A≡5B and 1≡1A≡1B≡0 in the single-loop mechanism B1 ← A1-A2 given by Eq. (2.172) is S5B1/ 1 = dim( RA1 ∩ RA2 ) = 3 . The relative independent velocities allowed by the single-loop mechanism B1 ← A1-A2 between links 5≡5A≡5B and 1≡1A≡1B≡0 are vx, vy and vz. Similar results are obtained for the single-loop kinematic chain B2 (0A≡1A-2A-…-5A≡5C-…2C-1C≡0) in Fig. 3.5: rB2=SA≡A1-A3=5 and MB2=85=3. This kinematic chain may be obtained by fixing together the distal links (1A≡0 and 1C) of the simple open kinematic chain A≡A1-A3 (0A≡1A2A-…-5A≡5C-…2C-1C) from Fig. 2.4 or by connecting together the distal

Fig. 3.4 Single-loop kinematic chain B1 ← A1-A2 associated with limbs A and B of the parallel Cartesian robotic manipulator: (a) structural diagram, (b) associated graph

Fig. 3.5. Single-loop kinematic chain B1 ← A1-A3 associated with limbs A and C of the parallel Cartesian robotic manipulator: (a) structural diagram, (b) associated graph

3.2 Single-loop kinematic chains

139

links 1A≡0 with 1C≡0 and 5A with 5C of simple open kinematic chains A1 (0A≡1A-2A-3A-4A-5A) and A2 (0B≡1B-2B-3B-4B-5B) in Fig. 2.2a and c (see example 4 for the calculation of SA≡A1-A3 and dim(RA≡A1-A3)). The basis of the operational space of open simple kinematic chain A3 is (RA3)=( v x ,v y ,v z ,ω z ) and S5B/21 = dim( RA1 ∩ RA3 ) = 3 . The relative independent velocities allowed by single-loop mechanism B2 ← A1-A3 between the links 5 and 1≡0 are also the translational velocities vx, vy and vz. The degree of overconstraint of these single-loop mechanisms given by Eq. (2.218) is N=5. Five joint parameters lose their independence in each closed loop (r=rB1=rB2=5). Example 6. The single-loop mechanism B (1A≡0-2A-…-4A≡4B-…2B1B≡0) in Fig. 3.6a has p=6 one-degree-of-freedom revolute joints and m=6 links including the fixed reference link (1A≡1B≡0). It represents an orthogonal version of the Sarrus mechanism (Sarrus 1853), and may be obtained by eliminating the prismatic joints from the kinematic chain in Fig. 3.4. The simple open kinematic chain A≡A1-A2 (1A≡0-2A-…-4A≡4B-…2B1B) associated with the closed loop mechanism B has rB=SA≡A1A2=dim(RA≡A1-A2)=5 (Fig. 3.6b). The same five relative independent motions ( v x , v y , v z , ω x , ω y ) exist between the distal links 1B and 1A≡0 in the simple open kinematic chains presented in Figs. 2.3 and 3.6b. The mobility of the single-loop mechanism B given by Eq. (2.160) is MB=6-5=1. We could obtain the same result by considering that the single-loop kinematic chain B ← A1-A2 (1A≡0-2A-…-4A≡4B-…2B-1B≡0) is obtained by connecting together the distal links 1A≡0 with 1B≡0 and 4A with 4B of the simple open kinematic chains A1 (0A≡1A-2A-3A-4A) and A2 (0B≡1B-2B-3B-4B) in Fig. 3.6c-d. Three relative independent motions ( v y , v z , ω x ) exist between

distal links 4A and 1A≡0 in the simple open kinematic chain A1 (Fig. 3.6c) and ( v x , v z , ω y ) between distal links 4B and 1B≡0 in the simple open kinematic chain A2 (Fig. 3.6d). Eq. (2.144) gives rB=SA≡A1-A2= dim(RA≡A1A2)=dim(RA1)+dim(RA2)-dim(RA1 ∩ RA2)=3+3-1=5. We can see that, in general, the mobility of a simple closed-loop mechanism can easily be calculated if we know the dimension of the velocity operational space of the simple open kinematic chain associated with the closed-loop mechanism. The unique bases of the operational spaces of simple open kinematic chains A1 and A2 are (RA1)=( v y ,v z ,ω x ) and (RA2)=( v x ,v z ,ω y ). The connectivity between links 4≡4A≡4B and 1≡1A≡1B≡0 in single-loop mechanism B ← A1-A2 given by Eq. (2.172) is S4B/ 1 = dim( RA1 ∩ RA2 ) = 1 . The relative

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3 Structural analysis

Fig. 3.6. Orthogonal version of Sarrus mechanism: structural diagram and graph of the single loop-mechanism (a), associated simple open chain A≡A1-A2 (b), associated simple open chain A1 (c) and associated simple open chain A1 (d)

3.2 Single-loop kinematic chains

141

independent velocity allowed by the single-loop mechanism B ← A1-A2 between links 4≡4A≡4B and 1≡1A≡1B≡0 is vz. The degree of overconstraint of this single-loop mechanism given by Eq. (2.218) is N=5. Five joint parameters lose their independence in the closed loop (r=rB=5). Note. The following examples illustrate the application of the formulae presented in this section by emphasizing the choice of the bases of operational vector spaces RA1 and RA2 when there are various ways to choose them. Example 7. Four-bar planar mechanism B (1≡0-2-3-4-5≡1) presented in Fig. 3.7a, has p=4 revolute joints with parallel axes and m=4 links

Fig. 3.7. Four-bar mechanism B (1≡0-2-3-4-5≡1) (a) and its associated simple open kinematic chain A (1≡0-2-3-4-5) (b)

Fig. 3.8. Four-bar planar mechanism B (1≡0-2-3-4-5≡1) (a) and its associated simple open chains A1 and A2 obtained by splitting up link 4 (b), link 3 (c) and link 2 (d)

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3 Structural analysis

Fig. 3.9. Four-bar planar parallelogram mechanism B (1≡0-2-3-4-5≡1) (a) and its associated simple open chains A1 (1A≡0-2A-3A) and A2 (1B≡0-2B-3B) (b)

Table 3.1. Valid bases of operational vector spaces RA1 and RA2 for the four-bar planar mechanism B ← A1-A2 No. A1 and A2 1 Fig. 3.8b

(RA1) ( v x ,v y ,ω z )

2

Fig. 3.8c

( v x ,v y )

3

Fig. 3.8c

( v x ,v y )

4

Fig. 3.8d

(ω z )

(RA2) (ω z )

(RA1 ∩ RA2) (ω z )

( v x ,ω z )

( vy )

( v y ,ω z ) ( v x ,v y ,ω z )

( vx )

(ω z )

including the fixed reference link (1≡5≡0). It can be obtained by connecting in the same link the distal links of the simple open kinematic chain A (1≡0-2-3-4-5) in Fig. 3.7b. We can simply observe by inspection that rA=SA=3 and (RA)=( v x ,v y ,ω z ). The mobility given by Eq. (2.160) is MB=4-3=1. The same result can be obtained by considering that the singleloop four-bar mechanism B (1≡0-2-3-4-5≡1) can be obtained by connecting in the same links the distal links of the two simple open kinematic chains A1 and A2 (Fig. 3.8). The simple chains A1 and A2 can be obtained by splitting up links 2, 3 or 4. The bases of vector spaces (RA1) and (RA2) used in Eq. (2.174) to calculate SA1, SA2 and S nB/ 1 are systematized in Table 3.1. These bases respect Eq. (2.180) and the conditions mentioned in Remark 1. We note that these bases are also valid for a four-bar planar parallelogram mechanism (Fig. 3.9). Special geometric conditions concerning the lengths of opposite links do not allow link 3 to rotate with respect to reference link 1≡0. In the parallelogram planar mechanism, link 3 makes a translational circular motion around the reference link with a constant orientation. Example 8. The four-bar spatial mechanism B (1≡0-2-3-4-5≡1) presented in Fig. 3.10a has p=4 spherical joints and m=4 links including the fixed reference link (1≡5≡0). It can be obtained by connecting in the same

3.2 Single-loop kinematic chains

143

link the distal links of the simple open kinematic chain A (1≡0-2-3-4-5) in Fig. 3.10b. We can simply observe by inspection that six independent motions exist between distal links 5 and 1 of the open kinematic chain A (Fig. 3.10b). We thus obtain (RA)=( v x ,v y ,v z ,ω x ,ω y ,ω z ) and rB=SA=6. The mobility given by Eq. (2.160) is MB=12-6=6. We may obtain the same result by considering that the single-loop fourbar spatial mechanism B (1≡0-2-3-4-1) is obtained by connecting in the same link the distal links of two simple open kinematic chains A1 (1A≡02A-3A) and A2 (1B≡0-2B-3B). In Fig. 3.10c, the simple chains A1 and A2 are obtained by splitting up link 3. The bases of the vector spaces (RA1) and (RA2) used in Eq. (2.174) to calculate SA1, SA2 and S3B/ 1 are: (RA2)=( v x ,v z ,ω x ,ω y ,ω z ) or (RA2)= (RA1)=( v x ,v y ,v z ,ω x ,ω y ) and ( v y ,v z ,ω x ,ω y ,ω z ). These bases respect Eq. (2.180) and the conditions mentioned in Remark 1. We recall that (RA1) and (RA2) represent the bases of the vector spaces of the velocities of point H in the open kinematic chains A1 and A2. We thus obtain S3B/ 1 =dim(RA1 ∩ RA2)=4. Eq. (2.144) gives rB=SA≡A1-A2=dim(RA≡A1-A2)= dim(RA1)+dim(RA2)-dim(RA1 ∩ RA2)=5+5-4=6.

Fig. 3.10. Four-bar spatial mechanism B (1≡0-2-3-4-5≡1) (a), its associated simple open chains (b) and (c) and the particular case of four-bar spatial parallelogram (d)

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3 Structural analysis

Link 3 may have four independent motions in a four-bar spatial mechanism: two translations and two rotations: v x ,v z ,ω x ,ω y or v y ,v z ,ω x ,ω y . If we consider link 3 as the characteristic link we obtain SF= S3B/ 1 =4, and Eq. (2.225) indicates two degrees of redundancy for this mechanism. They are given by the rotations of links 2 and 4 about the axes passing through the centres of the spherical joints adjacent to each link. We may consider that this mechanism has two internal and four external mobilities. We note that these structural analyses are also valid for the four-bar spatial mechanism with equal lengths of the opposite links (Fig. 10c). This mechanism is also called a four-bar spatial parallelogram. Example 9. The slider-crank spatial mechanism B (1≡0-2-3-4-5≡1) presented in Fig. 3.11a has p=4 joints RSSP-type and m=4 links including the fixed reference link (1≡5≡0). It can be obtained by connecting in the same link the distal links of the simple open kinematic chain A (1≡0-2-3-4-5) in Fig. 2.11b. We can simply observe by inspection that six independent motions exist between distal links 5 and 1 of the open kinematic chain A (Fig. 3.11b) and (RA)=( v x ,v y ,v z ,ω x ,ω y ,ω z ) and rB=SA=6.

Fig. 3.11. The slider-crank spatial mechanism B (1≡0-2-3-4-5≡1) of type RSSP (a) and its associated simple open chains (b) and (c)

3.2 Single-loop kinematic chains

145

The mobility given by Eq. (2.160) is MB=8-6=2. We can obtain the same result by considering that the single-loop slider-crank spatial mechanism B (1≡0-2-3-4-1) is obtained by connecting in the same link the distal links of two simple open kinematic chains A1 (1A≡0-2A-3A) and A2 (1B≡02B-3B). In Fig. 3.11c, the simple chains A1 and A2 are obtained by splitting up link 3. The vector space RA1 may have one of the following basis: (RA1)=( v x ,v y ,v z ,ω x ), (RA1)=( v x ,v y ,v z ,ω y ) or (RA1)=( v x ,v y ,v z ,ω z ). The vector space RA2 has the unique basis (RA1)=( v y ,ω x ,ω y ,ω z ). These bases respect Eq. (2.180) and the conditions mentioned in Remark 1. We obtain S3B/ 1 =dim(RA1 ∩ RA2)=2. Equation (2.144) gives rB=SA≡A1-A2=dim(RA≡A1A2)= dim(RA1)+dim(RA2)-dim(RA1 ∩ RA2)=4+4-2=6. Link 3 may have two independent motions in this spatial mechanism: one translation and one rotation. The translation is v y and the rotation may be any component ( ω x , ω y or ω z ) of the angular velocity ω of link 3 about the axis passing through the centre of the two spherical joints. If we consider link 3 as the characteristic link, we obtain SF= S3B/ 1 =2 and Eq. (2.225) indicates zero degrees of redundancy. Two independent parameters have to be known to control the motion of all links. One of these parameters is the rotation of link 3 about the axis passing through the centre of the two spherical joints. The second parameter may be the translation of slider 4, or the rotation of crank 2. We can consider that this mechanism has one internal and one external mobility. Example 10. The Bennett mechanism B (1≡0-2-3-4-5≡1) presented in Fig. 3.12a has p=4 joints and m=4 links including the fixed reference link (1≡5≡0). It can be obtained by connecting distal links 2A and 4B of the simple open kinematic chains A1 (1≡0-2A) and A2 (1B≡0-2B-3B-4B) in Fig. 3.12b. We can simply observe by inspection that SA1=1 and SA2=3. In accordance with connectivity theorem 8a and Eq. (2.175), link 2 of the Bennett four-link mechanism B can have relative motions with respect to the reference link only if the bases of the operational spaces RA1 and RA2 have at least one common velocity, that is S 2B/ 1 = 1 . We may consider that (RA1)=( ω z ) and (RA2) is any basis of cardinal 3 obtained by combining ω z with two other rotational and translational velocities. Eq. (2.174) gives SA=A1-A2 =1+3-1=3. The mobility of the Bennett mechanism given by Eq. (2.163) is MB=4-3=1. One degree of freedom motion of the Bennett mechanism is due to its special geometric configuration (see Fig. 3.12a): (i) the opposite links (i.e., links that are not concurrent) have the same length denoted by a and b, (ii) the twist angles α and β are equal on opposite sides but with different signs and (iii) the link length and twist angles

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Fig. 3.12. Bennett mechanism B (1≡0-2-3-4-5≡1) (a) and its associated simple open chains A1 (1A≡0-2A) and A2 (1B≡0-2B-3B-4B) (b)

3.2 Single-loop kinematic chains

147

Fig. 3.13. Bricard mechanism B (1≡0-2-3-4-5-6-7≡1) (a) and its associated simple open chains A1 (1A≡0-2A) and A2 (1B≡0-2B-3B-4B-5B-6B) (b)

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must satisfy the relation: asinβ =bsinα. We can see that the Bennett mechanism completely obeys the definitions, the theorems and the formulae presented in chapter 2 for quick mobility calculation of single-loop mechanisms. There is no reason to continue to consider this mechanism as “paradoxical” more than one hundred years after its invention (Bennett 1902). Example 11. The general trihedral 6R single-loop mechanism of Bricard B (1≡0-2-3-4-5-6-7≡1) in Fig. 3.13a has p=6 joints and m=6 links including the fixed reference link (1≡7≡0). It can be obtained by connecting distal links 2A and 6B of the open simple kinematic chains A1 (1≡0-2A) and A2 (1B≡0-2B-3B-4B-5B-6B) in Fig. 3.13b. We can simply observe by inspection that SA1=1 and SA2=5. In accordance with connectivity theorem 8a and Eq. (2.175), link 2 of the Bricard mechanism B can have relative motions related to the reference link only if the bases of the operational spaces RA1 and RA2 have at least one common velocity, that is S 2B/ 1 = 1 . We may consider that (RA1)=( ω z ), and (RA2) is any basis of cardinal 5 obtained by combining three rotational and two translational velocities. Eq. (2.174) gives SA=A1-A2 =1+5-1=5. The mobility of the Bricard mechanism given by Eq. (2.163) is MB=6-5=1. One degree of freedom motion of the Bricard mechanism is due to its special geometric configuration defined by the existence of two pairs of three joint axes intersecting in two different points. In this way, a common line intersects all joint axes. We can see that the Bricard mechanism completely obeys the definitions, the theorems and the formulae presented in this section for quick mobility calculation of singleloop mechanisms. As with the Bennett mechanism, there is no reason to continue to consider the Bricard mechanism as “paradoxical” for so long (Bricard 1927).

3.3 Parallel mechanisms with simple limbs We recall that a parallel mechanism with simple limbs is a complex mechanism C ← A1-A2-…Ak with two polinary links 1≡1Ai and n≡nAi connected by k simple chains Ai (1Ai-2Ai-…-nAi), i=1,2,…k. The two polinary links are the reference platform 1 and the mobile platform n. The simple open kinematic chain Ai is called a simple limb. When the reference platform is fixed to the base, we use the notation 1≡1Ai≡0. We also illustrate the calculation of the structural parameters of various complex closed mechanisms that can be considered as parallel mechanisms with simple limbs. We emphasize examples of robotic manipulators and

3.3 Parallel mechanisms with simple limbs

149

complex closed mechanisms that do not obey the classical criteria for quick calculation of mobility presented in sections 2.1 and 2.2. Example 12. The parallel mechanism C ← A1-A2-A3 in Fig. 3.14a is obtained by parallel concatenation of 3 simple limbs A1 (1A≡0-2A-3A-4A-5A), A2 (1B≡0-2B-3B-4B-5B) and A3 (1C≡0-2C-3C-4C). The mobile platform is 5≡5A≡5B≡4C and the reference platform is fixed to the base (1≡1A≡1B≡1C≡0). The mechanism has m=10 links including the fixed base, p=11 joints, two independent loops (q=11-10+1=2), k1=3 and k2=0. The limbs are composed of revolute (R) and prismatic (P) joints. Limbs A1 and A2 are of type P||R||R||R and limb A3 of type R||R||R. The axes of the revolute pairs are parallel to the O0x0-axis in limb A1, O0y0-axis in limb A2 and O0z0-axis in limb A3.

Fig. 3.14. Translational parallel mechanism Isoglide2-T2 of type 2PRRR-RRR: structural diagram (a), associated graph (b) and the simple open kinematic chains associated with its simple limbs A1 (c), A2 (d) and A3(e)

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Fig. 3.15. CAD model of Isoglide2-T2 of type 2PRRR-RRR

This robot belongs to a family of innovative and modular parallel robotic manipulators with 2-6 degrees of mobility and decoupled motions, proposed by the author (Gogu 2002b), and developed under the name of Isogliden-TaRb at the French Institute of Advanced Mechanics (IFMA). In this notation, a represents the number of independent translational motions and b the number of independent rotational motions of the end-effector. The integer numbers a, b and n satisfy the relation n=a+b with 0≤a≤3, 0≤b≤3 and 2≤n≤6. When a or b equal zero, the corresponding notation T or R is missing in Isogliden-TaRb. This is the case for Isoglide2-T2 with a= 2 and b=0. Figure 3.15 presents a CAD model of this parallel robotic manipulator. Other solutions of parallel robots of this family will be analyzed in the following examples. We simply observe by inspection that the simple limbs (Fig. 3.14b, c, d) have the following unique bases of their operational velocity spaces: (RA1)=( v x ,v y ,v z ,ω x ), (RA2)=( v x ,v y ,v z ,ω y ) and (RA3)=( v x ,v y ,ω z ). The dimensions of these three operational spaces give the connectivity of the simple open chains associated with each limb Ai of the parallel mechanism C ← A1-A2-A3 (SA1= SA2=4 and SA3=3). The connectivity of the mobile platform in parallel mechanism C ← A1-A2-A3, given by Eq. (2.181), is S5C/ 1 = dim( RA1 ∩ RA2 ∩ RA3 ) = 2 . Only two relative independent velocities ( v x and v y ) exist between mobile platform 5 and reference base 1≡0 in parallel mechanism C ← A1-A2-A3. The dimension of the joint vector space U is p dim(U)= ∑ i =1 fi =11. The number of joint parameters that lose their inde-

pendence in the parallel mechanism C ← A1-A2-A3 given by Eq. (2.185) is rC=11-2=9. The mobility of the parallel mechanism C ← A1-A2-A3 given by Eq. (2.193) is MC=11-11+2=2. Two independent parameters are necessary

3.3 Parallel mechanisms with simple limbs

151

to define the position of any link in the mechanism. In the parallel mechanism C ← A1-A2-A3 in Fig. 3.14a, these parameters are the two displacements q1 and q2 in the linear actuators mounted on the fixed base. We have seen in section 2.2 that this mechanism does not obey the classical CGK formulae but it perfectly complies with the formulae presented in this section for mobility calculation of parallel mechanisms with simple limbs. The degree of overconstraint of this parallel robot given by Eq. (2.217) is NC=12-9=3 and the degree of structural redundancy given by Eq. (2.222) is TC=0. We note that in a parallel robot with simple limbs, rl=0. Example 13. Planar parallel mechanism C ← A1-A2-A3 in Fig. 3.16a is obtained by parallel concatenation of 3 simple limbs A1 (1A≡0-2A-3A-4A), A2

Fig. 3.16. Translational parallel mechanism 3PPR: structural diagram (a), associated graph (b) and the simple open kinematic chains associated with its simple limbs A1 (c), A2 (d) and A3(e)

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(1B≡0-2B-3B-4B) and A3 (1C≡0-2C-3C-4C). The mobile platform is 4≡4A≡4B≡4C and the reference platform is fixed to the base (1≡1A≡1B≡1C≡0). The mechanism has m=8 links including the fixed base, p=9 joints, two independent loops (q=9-8+1=2), k1=3 and k2=0. The mobile platform is 4≡4A≡4B≡4C and the reference platform is fixed to the base (1≡1A≡1B≡1C≡0). The limbs are composed of revolute R and prismatic P joints. The three limbs are planar P ⊥ P ⊥ ||R-type. The axes of the revolute pairs are parallel to O0z0. We simply observe by inspection that the simple open chains associated with each limb Ai of the parallel mechanism C ← A1-A2-A3 (Fig. 3.16c-e) have the same basis for their operational velocity spaces (RA1)=(RA2)=(RA3)= ( v x ,v y ,ω z ) and the same connectivity SA1=SA2= SA3=3. The connectivity of the mobile platform in parallel mechanism C ← A1-A2-A3, given by Eq. (2.181), is S4C/ 1 = dim( RA1 ∩ RA2 ∩ RA3 ) = 3 . Three relative independent velocities v x ,v y and ω z exist between mobile platform 4 and the reference base 1≡0 in the parallel mechanism C ← A1-A2-A3. The dimension of the joint vector space U is p dim(U)= ∑ i =1 fi =9. The number of joint parameters that lose their inde-

pendence in the parallel mechanism C ← A1-A2-A3, given by Eq. (2.191) is rC=9-3=6. The mobility of the parallel mechanism C ← A1-A2-A3, given by Eq. (2.193) is MC=9-9+3=3. Three independent parameters are necessary to define the position of any link in the mechanism. In the parallel mechanism C ← A1-A2-A3 in Fig. 3.16a, these parameters may be the three displacements q1, q2 and q3 in the linear actuators mounted on the fixed base. This parallel mechanism obeys theorem 3 and Eq. (2.94). For this reason, the mobility of this parallel mechanism can also be obtained by using the classical CGK formulae presented in section 2.2. The degree of overconstraint of this parallel robot given by Eq. (2.217) is NC=12-6=6 and the degree of structural redundancy given by Eq. (2.222) is TC=9-9=0. We note that rl=0. Example 14. Results similar to those in example 13 are also obtained for the planar parallel mechanism C ← A1-A2-A3 in Fig. 3.17. This mechanism has only revolute pairs with parallel axes. It is used in planar parallel robots actuated by three rotating motors usually mounted on the fixed base. Example 15. The parallel Cartesian robotic manipulator in Fig. 2.1 is a parallel mechanism C ← A1-A2-A3 obtained by parallel concatenation of 3 simple limbs A1 (1A≡0-2A-3A-4A-5A), A2 (1B≡0-2B-3B-4B-5B) and A3 (1C≡0-

3.3 Parallel mechanisms with simple limbs

153

Fig. 3.17. Planar parallel mechanism with rotation actuators C ← A1-A2-A3 (a) and the simple open kinematic chains associated with its simple limbs A1 (b), A2 (c) and A3(d)

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Fig. 3.18. CAD model of Isoglide3-T3

2C-3C-4C-5C). As we have seen in example 3 – section 2.2, we simply observe by inspection (Fig. 2.2) that (RA1)=( v x ,v y ,v z ,ω x ), (RA2)= ( v x ,v y ,v z ,ω y ), (RA3)=( v x ,v y ,v z ,ω z ) and SAi=4 (i=1,2,3). The connectivity of the mobile platform 5≡5A≡5B≡5B in the parallel mechanism C ← A1-A2A3, given by Eq. (2.181), is S5C/ 1 = dim( RA1 ∩ RA2 ∩ RA3 ) = 3 . Three relative independent translational velocities ( v x , v y and v z ) exist between the mobile and fixed platforms. The dimension of the joint vector space U is p dim(U)= ∑ i =1 fi =12. The number of joint parameters that lose their inde-

pendence in the parallel mechanism C ← A1-A2-A3 given by Eq. (2.191) is rC=12-3=9. The mobility of parallel mechanism C ← A1-A2-A3 given by Eq. (2.193) is MC=12-12+3=3. Three independent parameters are necessary to define the position of any link in the mechanism, for example the displacements q1, q2 and q3 in the linear actuators mounted on the fixed base. We have seen in section 2.2 that this mechanism does not obey the classical CGK formulae but it completely obeys to new formulae for mobility calculation of parallel mechanisms with simple limbs. The degree of overconstraint of this parallel robot given by Eq. (2.217) is NC=12-9=3 and the degree of structural redundancy given by Eq. (2.222) is TC=12-12=0 with rl=0. The parallel robotic manipulator in Fig. 2.1 also belongs to the Isoglide family. It represents Isoglide3-T3 of type T3 (Fig. 3.18). Example 16. Four versions of the parallel robotic manipulators with simple limbs Isoglide4-T3R1 are analysed in this example. They are obtained from the solution in Fig. 3.19 by changing the type of joints

3.3 Parallel mechanisms with simple limbs

155

(a)

(b)

Fig. 3.19. The parallel manipulator Isoglide4-T3R1 developed at the French Institute of Advanced Mechanics in a modular design approach: partial view (a) and general view (b)

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3 Structural analysis

Fig. 3.20. Parallel mechanism with simple limbs Isoglide4-T3R1 of type 3-PRRRR+1PRRR: structural diagram (a), associated graph (b), simple open chains associated with limbs A1 (c), A2 (d), A3 (e) and A4 (f)

3.3 Parallel mechanisms with simple limbs

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Fig. 3.21. CAD model of 3-PRRRR+1PRRR-type Isoglide4-T3R1 parallel robot

connecting the four limbs to the moving platform. We can see that the solutions in Figs. 3.15 and 3.18 are particular cases derived from this solution. In the four solutions of Isoglide4-T3R1, the mobile platform has Schönflies motions that are three independent translations T and one rotation R about a fixed direction axis. Isoglide4-T3R1 (3-PRRRR+1PRRR-type) is a parallel mechanism C ← A1-A2-A3-A4 obtained by parallel concatenation of 4 simple limbs A1 (1A≡0-2A-3A-4A-5A-6A), A2 (1B≡0-2B-3B-4B-5B), A3 (1C≡0-2C-3C-4C-5C-6C) and A4 (1D≡0-2D-3D-4D-5D-6D) - see Figs. 3.20 and 3.21. The mechanism has three limbs of type P||R||R||R ⊥ R and one limb of type P||R||R||R. In each limb, the axes of the first three revolute joints are parallel to the direction of the prismatic joint. The last revolute joints of the four limbs have parallel axes. The actuated prismatic joints of limbs A1, A2 and A3 have orthogonal directions. The actuated prismatic joints of limbs A3 and A4 have parallel directions. For ϕ y ∈ (−π / 2, +π / 2) the simple open kinematic chains associated

with the four limbs have (Fig. 3.20c-f): (RA1)=( v x ,v y ,v z ,ω x ,ω y ), (RA3)=(RA4)= ( v x ,v y ,v z ,ω y ,ω z ), (RA2)= ( v x ,v y ,v z ,ω y ), and SA1=SA3=SA4

=5, SA2=4. The connectivity of the platform 6≡6A≡5B≡6C≡6D in parallel mechanism C ← A1-A2-A3-A4 is S6C/ 1 = dim( RA1 ∩ RA2 ∩ RA3 ∩ RA4 ) = 4 . Four relative independent velocities v x ,v y , v z and ω y exist between the

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(a)

3 Structural analysis

(b)

(c) Fig. 3.22. Parallel mechanism with simple limbs Isoglide4-T3R1 of type 4-PRRRR (a), associated graph (b), CAD model (c)

3.3 Parallel mechanisms with simple limbs

159

mobile and the fixed platforms. p The dimension of the joint vector space U is dim(U)= ∑ i =1 fi =19. The number of joint parameters that lose their independence in the parallel robot given by Eq. (2.191) is rC=19-4=15, and the mobility given by Eq. (2.193) is MC=19-15=4. Four independent joint parameters are necessary to completely describe the motion of the mechanism. In Isoglide4-T3R1 presented in Figs. 3.19-3.24 these parameters are the displacements of four linear actuators qi (i=1,2,…,4) mounted on the fixed base. The degree of overconstraint of this parallel robot given by Eq. (2.217) is NC=18-15=3 and the degree of structural redundancy given by Eq. (2.222) is TC=19-19=0 with rl=0. Figures 3.22 and 3.23 present solutions of Isoglide4-T3R1 with two and, respectively, one degree of overconstraint. The solution presented in Fig. 3.22 is not overconstrained (NC=0). Isoglide4-T3R1 of type 4-PRRRR -see Fig. 3.22- has four limbs of type P||R||R||R ⊥ R and the following structural parameters: (RA1)=(RA2)= ( v x ,v y ,v z ,ω x ,ω y ), (RA3)= (RA4)=( v x ,v y ,v z ,ω y ,ω z ), SAi=5 (i=1,2,3,4), S6C/ 1 = dim( RA1 ∩ RA2 ∩ RA3 ∩ RA4 ) = 4 , r=5+5+5+5-4=16, MC=20-6=4, NC=18-16=2. The last revolute joint of limb A2 has an idle mobility denoted by R¤. This limb is of type PRRRR¤.

Fig. 3.23. Parallel mechanism with simple limbs Isoglide4-T3R1 of type 2-RRS+1-PRRRR+1-PRRR (a) and associated graph (b)

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(b)

(c) Fig. 3.24. Parallel mechanism with simple limbs Isoglide4-T3R1 of type 3-PRRS+1-PRRR (a), associated graph (b), CAD model (c)

3.3 Parallel mechanisms with simple limbs

161

Isoglide4-T3R1 of type 2-PRRS+1-PRRRR+1-PRRR -see Fig. 3.23- has two idle mobilities (one in each spherical joint S=RRR¤) and the following structural parameters: (RA1)=( v x ,v y ,v z ,ω x ,ω y ), (RA2)=( v x ,v y ,v z ,ω y ),

(RA3)=( v x ,v y ,v z ,ω x ,ω y ,ω z ),

(RA4)=( v x ,v y ,v z ,ω x ,ω y ,ω z ),

SA1=5,

SA2=4, SAi=6 (i=3,4), S6C/ 1 = dim( RA1 ∩ RA2 ∩ RA3 ∩ RA4 ) = 4 , rC=214=17, MC=21-17=4, NC=18-17=1. Isoglide4-T3R1 of type 3-PRRS+1-PRRR - see Fig. 3.24- has three idle mobilities (one in each spherical joint S=RRR¤) and the following structural parameters: (RA1)=( v x ,v y ,v z ,ω x ,ω y ,ω z ), (RA2)=( v x ,v y ,v z ,ω y ),

(RA3)=( v x ,v y ,v z ,ω x ,ω y ,ω z ), (RA4)=( v x ,v y ,v z ,ω x ,ω y ,ω z ), SA2=4, SAi=6

(i=1,3,4), S5C/ 1 = dim( RA1 ∩ RA2 ∩ RA3 ∩ RA4 ) = 4 , rC=22-4=18 MC=2218=4, NC=18-18=0. Example 17. The parallel robotic manipulator Isoglide5-T3R2 (Fig. 3.25) is another solution in the Isoglide family.

Fig. 3.25. Parallel mechanism with simple limbs Isoglide5-T3R2 (a) and its associated graph (b)

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Fig. 3.26. Simple open chains associated with limbs A1 (a), A2 (b), A3 (c), A4 (d) and A5 (e) of Isoglide5-T3R2

Parallel mechanism C ← A1-A2-A3-A4-A5 is obtained by parallel concatenation of 5 simple limbs A1 (1A≡0-2A-3A-4A- 5A-6A-7A), A2 (1B≡0-2B-3B-4B5B-6B), A3 (1C≡0-2C-3C-4C-5C-6C-7C), A4 (1D≡0-2D-3D-4D-5D-6D) and A5 (1E≡0-2E-3E-4E-5E-6E-7E).

3.3 Parallel mechanisms with simple limbs

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For α ∈ (0, π / 2) and β ∈ (−π / 2, +π / 2) the simple open kinematic chains associated with the five limbs have (Fig. 3.26c-e): (RA1)=( v x ,v y ,v z ,ω x ,ω α ,ω β ), (RA2)=(RA4)=( v x ,v y ,v z ,ω α ,ω β ), (RA3)= (RA5)=( v x ,v y ,v z ,ω z ,ω α ,ω β ) and SA1=SA3=SA5 =6, SA2= SA4=5.

The connectivity of the platform 7≡7A≡6B≡7C≡6D≡7E is S = dim( RA1 ∩ RA2 ∩ RA3 ∩ RA4 ∩ RA4 ) = 5 . Five relative independent velocities v x ,v y ,v z ,ω α and ω β exist between the mobile and the fixed C 7/1

platforms. The dimension of the joint vector space U is dim(U)= ∑ i =1 fi p

=28. The number of joint parameters that lose their independence in the parallel robot is rC=28-5=23 and the mobility MC=28-23=5. Five independent joint parameters are necessary to completely describe the motion of the mechanism. In the Isoglide5-T3R2 these parameters are the displacements of the five linear actuators qi (i=1,2,…,5) mounted on the fixed base. The degree of overconstraint of this parallel robot given by Eq. (2.217) is NC=24-23=1 and the degree of structural redundancy given by Eq. (2.222) is TC=28-28=0 with rl=0. Example 18. The Gough-Stewart parallel robot 6-UPS-type in Fig. 3.27a is a parallel mechanism C ← A1-A2-A3-A4-A5-A6 obtained by parallel concatenation of 6 simple identical limbs A1 (1A≡0-2A-3A-4A), A2 (1B≡0-2B3B-4B), A3 (1C≡0-2C-3C-4C), A4 (1D≡0-2D-3D-4D), A5 (1E≡0-2E-3E-4E) and A6 (1F≡0-2F-3F-4F). Each telescopic limb is connected to the fixed platform by a universal joint U and to the mobile platform by the spherical S joint (Fig. 3.27c). We have (RAi)=( v x ,v y ,v z ,ω x ,ω y ,ω z ) and SAi=6

(i=1,2,…,6). The connectivity of the mobile platform 4≡4A≡4B≡4C≡4D≡4E≡4F in the parallel mechanism C ← A1-A2-A3-A4-A5-A6 is S4C/ 1 = 6 . Six relative independent velocities v x ,v y ,v z ,ω x ,ω y and ω z exist between the mobile and the fixed platforms. The dimension of the p joint vector space U is dim(U)= ∑ i =1 fi =36. The number of joint parame-

ters that lose their independence in the parallel robot is rC=36-6=30 and the mobility M=36-30=6. Six independent joint parameters are necessary to completely describe the motion of the mechanism. In Gough-Stewart type parallel robots these parameters are the displacements of the six linear actuators qi (i=1,2,…,6) integrated in the prismatic joints. We can see that for this type of parallel robot the actuators are not mounted on the fixed base. This solution is not overconstrained (NA=30-30=0) and non redundant (TA=36-36=0).

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Fig. 3.27. Gough-Stewart parallel mechanism of type 6-UPS (a), associated graph (b), simple open chain associated with limbs A1 (c)

The Gough-Stewart parallel robot 6-SPS-type in Fig. 1.2 is obtained by parallel concatenation of 6 simple identical limbs SPS-type. Each telescopic limb is connected to the fixed and moving platforms by spherical joints. In this case, both the limb connectivity and the connectivity of the mobile platform 4≡4A≡4B≡4C≡4D≡4E≡4F in the parallel mechanism C ← A1-A2-A3-A4-A5-A6 are unchanged (RAi)=( v x ,v y ,v z ,ω x ,ω y ,ω z ), SAi=6 (i=1,2,…,6) and S4C/ 1 = 6 . The dimension of the joint vector space U is

dim(U)= ∑ i =1 fi =42. The number of joint parameters that lose their indep

pendence in the parallel robot is rC=30 and the mobility MC=42-30=12. Twelve independent joint parameters are necessary to completely describe the motion of all links in this mechanism. We note that, among these 12 joint parameters, 6 represent internal limb mobilities. Each limb has an internal mobility that is a rotation around the axis directed to the centers of its spherical joints. These internal mobilities do not influence the motion of the mobile platform. For this reason, just six joint parameters are in fact used to actuate the robot. This non overconstarined solution (NC=0) has six degrees of redundancy (TC=42-36=6). Example 19. The parallel mechanism C ← A1-A2-A3 in Fig. 3.28a is obtained by parallel concatenation of 3 simple limbs A1 (1A≡0-2A-3A-4A), A2 (1B≡0-2B- 3B-4B) and A3 (1C≡0-2C-3C-4C). The mobile platform is 4≡4A≡4B≡4C and the reference platform is fixed to the base 1≡1A≡1B≡1C≡0.

3.3 Parallel mechanisms with simple limbs

165

Fig. 3.28. General case of 3-RPS mechanism (a), associated graph (b), simple open chains associated with limbs A1 (c), A2 (d) and A3 (e)

166

3 Structural analysis

Each telescopic limb is connected to the base by a revolute pair and to the mobile platform by a spherical pair. This parallel mechanism called 3RPS was proposed by Hunt (1983) in his pioneering article on structural synthesis of parallel mechanisms. We can observe by inspection that RA1 has a unique basis (RA1)=( v y ,v z ,ω x ,ω y ,ω z ), and RA2 and RA3 have multiple bases. The basis of RA2 is formed by any combination of five independent velocities of the point H in the open chain associated with limb A2. We may consider one of the following sets of bases for RA2: ( v y ,v z ,ω x ,ω y ,ω z ), ( v x ,v z ,ω x ,ω y ,ω z ), ( v x ,v y ,ω x ,ω y ,ω z ),

( v x ,v y ,v z ,ω x ,ω z ), ( v x ,v y ,v z ,ω y ,ω z ) or ( v x ,v y ,v z ,ω x ,ω y ). The same five possibilities exist for the basis of RA3. In accordance with Remark 2 and Eq. (2.197), the bases of RA2 and RA3 are selected such that the minimum value of S4C/ 1 is obtained. We have

S4C/ 1 = dim( R4C/ 1 ) = dim( RA1 ∩ RA2 ∩ RA3 ) =3, SA1=SA2=SA3=5, rC=3

and MC=3. The complete set of possible bases for R4C/ 1 is: ( ω x ,ω y ,ω z ), ( v z ,ω x ,ω y ),

( v z ,ω x ,ω z ),

( v z ,ω y ,ω z ),

( v y ,ω x ,ω y ),

( v y ,ω x ,ω z ),

( v y ,ω y ,ω z ), ( v y ,v z ,ω x ), ( v y ,v z ,ω y ), ( v y ,v z ,ω z ). As we can see, the proposed method gives us, in a very simple way, a complete and correct result concerning the mobility and connectivity of a 3-RPS parallel manipulator without studying the instantaneous constraint system. As previously mentioned, a given parallel mechanism has an infinite number of positions and for each position we can associate an instantaneous constraint system. In these conditions, an instantaneous constraint study could offer just some partial results on the instantaneous connectivity of the mobile platform depending on the configuration of the mechanism (Huang and Fang 1996; Huang et al. 1996; Fang and Huang 1997; Huynh 2004). This parallel manipulator is not overconstrained (N=0) and is non redundant (T=0). Example 20. The parallel mechanism C ← A1-A2-A3 in Fig. 3.29a is obtained by parallel concatenation of 3 simple limbs A1 (1A≡0-2A-3A-4A-5A6A), A2 (1B≡0-2B-3B-4B-5B-6B) and A3 (1C≡0-2C-3C-4C-5C-6C). The mobile platform is 6≡6A≡6B≡6C and the reference platform is fixed to the base 1≡1A≡1B≡1C≡0. Each telescopic limb is connected to the base and to the mobile platform by two intersecting revolute pairs (Cardan joints). This parallel mechanism called 3-RRPRR or 3-UPU was proposed by (Tsai 1996; Tsai and Joshi 2000) as a translational parallel manipulator. In order to keep the moving platform from changing its orientation, Tsai pointed out the following two geometric conditions for the four revolute joint axes

3.3 Parallel mechanisms with simple limbs

167

Fig. 3.29. 3-UPU mechanism (a), associated graph (b), simple open chains associated with limbs A1 (c), A2 (d) and A3 (e)

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3 Structural analysis

of each limb: (i) the first revolute joint axis is parallel to the last revolute joint axis and (ii) the two intermediate revolute joint axes are parallel to one another. These geometric conditions do not imply the intersection of the first two revolute joint axes specific to the universal joint. Hence, any 3-RRPRR parallel manipulator whose revolute joint axes satisfy the above conditions will be a translational manipulator (Tsai and Joshi 2000). These geometric conditions were generalized later by Di Gregorio and Parenti-Castelli (1998, 2002). This parallel manipulator has multiple bases for RA1, RA2 and RA3. We can observe by inspection (Fig. 3.22c-e) that RA1 could have two different bases, RA2 and RA3 three different bases each. The two possible bases of RA1 are: ( v x ,v y ,v z ,ω x ,ω z ) or ( v x ,v y ,v z ,ω x ,ω y ). The three bases of RA2 and RA3 are: ( v x ,v y ,v z ,ω x ,ω z ), ( v x ,v y ,v z ,ω x ,ω y ), ( v x ,v y ,v z ,ω y ,ω z ). In accordance with remark 2 and

Eq. (2.197), the bases of RA1, RA2 and RA3 are selected so that the minimum value of S6C/ 1 is obtained. We thus obtain SA1=SA2=SA3=5, ( R6C/ 1 ) =

( RA1 ∩ RA2 ∩ RA3 ) =( v x ,v y ,v z ), S4C/ 1 = dim( R4C/ 1 ) =3, rC=12, MC=3,

NC=0, TC=0. The three independent translations of the mobile platform are obtained by actuating the prismatic pairs integrated in the limbs.

3.4 Parallel mechanisms with complex limbs We recall that a parallel robotic manipulator with complex limbs denoted by D ← A1-A2-…Ak1-E1-E2-…-Ek2 is a complex mechanism with p joints in which the mobile platform n≡nAi≡nEj is connected to the reference platform 1≡1Ai≡1Ej by k≥2 limbs, of which at least one limb is complex. A complex limb contains at least one closed loop. The elementary limbs are denoted by Ai (i=1,2,…k1) and the complex limbs by Ej (j=1,2,…k2) with k1+k2=k. When the reference platform is fixed to the base, we use the notation 1≡1Ai≡1Ej≡0. We have denoted by k1 the total number of simple limbs, and by k2 the total number of complex limbs. We illustrate the calculation of the structural parameters of parallel robotic manipulators with complex limbs which contain various combinations of planar or spatial closed loops. We emphasize examples which do not comply with the classical CGK formulae. Example 21. The parallel mechanism 2PPa-type in Fig 3.30a represents a maximally-regular and fully-isotropic translational robotic manipulator used in pick and place operations (Gogu 2002). The parallel mechanism D ← E1-E2 in Fig. 3.30 is obtained by parallel concatenation of two complex

3.4 Parallel mechanisms with complex limbs

169

Fig. 3.30. Fully-isotropic overconstrained translational parallel mechanism 2PPatype: kinematic structure (a), associated graph (b), complex chains associated with limbs A1 (c) and A2 (d)

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3 Structural analysis

Fig. 3.31. Fully-isotropic non overconstrained translational parallel mechanism PPass-PCPass-type: kinematic structure (a), associated graph (b), complex chains associated with limbs A1 (c) and A2 (d)

3.4 Parallel mechanisms with complex limbs

171

limbs E1 (1A≡0-2A-3A-4A-5A) and E2 (1B≡0-2B-3B-4B-5B). A planar parallelogram closed loop Pa is included in each complex limb. The mechanism has the following structural characteristics: m=8 links including the fixed base, p=10 joints, three independent loops (q=10-8+1=3), k1=0 and k2=2. The mobile platform is 5≡5A≡5B and the reference platform is fixed to the base (1≡1A≡1B≡0). The two limbs of type P ⊥ Pa are identical. Each limb is composed of a prismatic pair P and a planar parallelogram loop Pa. The axes of the revolute joints in the parallelogram loops are parallel to the O0z-axis, and the directions of the two prismatic pairs are parallel to the axes O0x and O0y. We simply observe by inspection (Fig. 3.30c,d) that the bases of the operational velocity spaces of two limbs are (RE1)=(RE2)=( v x ,v y ). The dimensions of these operational spaces give the connectivity SE1= SE2=2 of the kinematic chains associated with each limb of parallel mechanism D ← E1-E2. The connectivity of the mobile platform in the parallel mechanism D ← E1-E2, given by Eq. (2.198), is S5D/ 1 = dim( RE1 ∩ RE 2 ) = 2 . Just two independent translational velocities v x and v y exist between mobile platform 5 and reference base1≡1A≡1B≡0 in the parallel mechanism D ← E1-E2. The dimension of the joint vector space U is p dim(U)= ∑ i =1 fi =10. Three joint parameters lose their independence in each planar parallelogram (rl=6). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=4-2+6=8. The mobility of the parallel mechanism D ← E1-E2 given by Eq. (2.202) is MD=10-4+2-6=2. Two independent parameters are necessary to define the position of any link in the mechanism. These parameters are the two displacements q1 and q2 in the prismatic actuators mounted on the fixed base (underlined joints). Equations (2.217) and (2.222) indicate that the parallel mechanism has ten degrees of overconstraint (ND=18-4+2-6=10) and zero degrees of redundancy (TD=10-4-6=0). We note that the classical CGK formulae give an erroneous mobility (MD=1) for this parallel mechanism. Figure 3.31 presents a non overconstrained solution of type PPassPCPass. Two spherical joints S are included in each parallelogram loop. A cylindrical joint C is also integrated in the E2-limb. We have denoted by Pass the parallelogram loop containing two spherical joints. Six joint parameters lose their independence in a Pass-type parallelogram loop. This solution has the following structural parameters: m=9, p=11, q=3, k1=0, k2=2, (RE1)=( v x ,v y ,ω y ), (RE2)=( v x ,v y ,v z ,ω x ,ω z ), SE1=3, SE2=5, S5D/ 1 = 2, dim(U)=20, rl=12, rD=18, MD=2, ND=0, TD=0.

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3 Structural analysis

Example 22. The parallel mechanism 2PaPa-type in Fig 3.32a also represents a maximally-regular and fully-isotropic translational robotic manipulator used in pick and place operations (Gogu 2002). The parallel mechanism D ← E1-E2 in Fig. 3.32a is obtained by parallel concatenation of two complex limbs E1 (1A≡0-2A-3A-4A-5A-6A-7A) and E2 (1B≡0-2B-3B-4B-5B6A-7A). Two planar parallelogram loops Pa are included in each complex limb. The mechanism has the following structural characteristics: m=12 links including the fixed base, p=16 joints, five independent loops (q=1612+1=5), k1=0 and k2=2. The mobile platform is 7≡7A≡7B and the reference platform is fixed to the base (1≡1A≡1B≡0). The two limbs of type Pa||Pa are identical. Each limb is composed of two planar parallelogram loops Pa. The axes of the revolute joints in the parallelogram loops are parallel to parallel to O0z.

Fig. 3.32. Fully-isotropic overconstrained translational parallel mechanism 2PaPa-type: kinematic structure (a), associated graph (b), complex chains associated with limbs A1 (c) and A2 (d)

3.4 Parallel mechanisms with complex limbs

173

We simply observe by inspection (Fig. 3.32c,d) that the bases of the operational velocity spaces of both limbs are (RE1)=(RE2)=( v x ,v y ). The dimensions of these operational spaces give the connectivity SE1= SE2=2 of the kinematic chains associated with each limb of the parallel mechanism D ← E1-E2. The connectivity of the mobile platform in the parallel mechanism D ← E1-E2, given by Eq. (2.198), is S5D/ 1 = dim( RE1 ∩ RE 2 ) = 2 . Just two independent translational velocities v x and v y exist between mobile platform 7 and reference base 1≡1A≡1B≡0 in parallel mechanism D ← E1p E2. The dimension of the joint vector space U is dim(U)= ∑ i =1 fi =16.

Fig. 3.33. Fully-isotropic non overconstrained translational parallel mechanism Pa*Pa*C-Pa*Pa*S-type: kinematic structure (a), associated graph (b), complex chains associated with limbs A1 (c) and A2 (d)

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3 Structural analysis

Three joint parameters lose their independence in each planar parallelogram (rl=12). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=4-2+12=14. The mobility of the parallel mechanism D ← E1-E2 given by Eq. (2.202) is MD=16-4+2-12=2. Two independent parameters are necessary to define the position of any link in the mechanism. These parameters are the two displacements q1 and q2 in the revolute actuators mounted on the fixed base (underlined joints). Equations (2.217) and (2.222) indicate that the parallel mechanism has 16 degrees of overconstraint (ND=30-4+2-12=16) and zero degrees of redundancy (TD=16-4-12=0). We note that the classical CGK formulae give an erroneous mobility (MD=1) for this parallel mechanism. Figure 3.33 presents a non overconstrained solution Pa*Pa*CPa*Pa*S-type. One cylindrical and one spherical joint are introduced in each parallelogram loop denoted by Pa*. A cylindrical joint C is also integrated in the E1-limb, and a spherical joint in the E2-limb. Six joint parameters lose their independence in a Pa*-type parallelogram loop. This solution has the following structural parameters: m=14, p=18, q=5, k1=0, k2=2, (RE1)=( v x ,v y ,v z ), (RE2)=( v x ,v y ,ω x ,ω y ,ω z ), SE1=3, SE2=5, S5D/ 1 = 2, dim(U)=32, rl=24, rD=30, MD=2, ND=0, TD=0. Example 23. The parallel mechanism presented in Fig 3.34a represents an innovative maximally-regular and fully-isotropic spherical parallel wrist proposed by the author in (Gogu 2005f). The parallel mechanism D ← A1-E1 in Fig. 3.34 is obtained by parallel concatenation of a simple limb A1 (1A≡0-2A-3A) and a complex limb E1 (1B≡0-2B-3B-4B-5B-6B-7B-8B). A planar parallelogram closed loop Pa is included in the complex limb E1. The mechanism has the following structural characteristics: m=9 links including the fixed base, p=10 joints, two independent loops (q=109+1=2), one simple (k1=1) and one complex (k2=1) limb. The mobile platform is 8≡3A≡8B and the reference platform is fixed to the base (1≡1A≡1B≡0). The simple limb A1 is composed of two revolute pairs with orthogonal and intersecting axes R ⊥ R-type. The complex limb E1 is of typePa ⊥ R||R||R ⊥ R. The axes of the last revolute joints of the two limbs are parallel. The axes of the revolute joints in the parallelogram loop are parallel to the x0y0-plane and the three revolute pairs have axes parallel to O0z. We simply observe by inspection that the limbs (Fig. 3.34c,d) have the following bases of their operational velocity spaces: (RA1)=( ω xy ,ω z )

and (RE1)=( v x ,v y ,v z ,ω xy ,ω z ). The rotation axis of velocity ω xy lies in

the x0y0-plane. The dimensions of these operational spaces SA1=2 and

3.4 Parallel mechanisms with complex limbs

175

Fig. 3.34. Fully-isotropic overconstrained parallel wrist RR-PaRRRR-type with two degrees of freedom: kinematic structure (a), associated graph (b), kinematic chains associated with limbs A1 (c) and A2 (d)

SE1=5 give the connectivity of the kinematic chains associated with each limb of parallel mechanism D ← A1-E1. The connectivity of mobile platform in the parallel mechanism D ← A1-E1, given by Eq. (2.198), is S8D/ 1 = dim( RE1 ∩ RE 2 ) = 2 . Just two independent angular velocities ω xy and ω z exist between the

mobile platform 8 and the reference base 1≡0 in the parallel mechanism D ← A1-E1. The dimension of the joint vector space U is dim(U)= ∑ i =1 fi =10. Three joint parameters lose their independence in p

the planar parallelogram loop incorporated in limb E1 (rl=3). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=7-2+3=8. The mobility of parallel mechanism D ← A1-E1 given by Eq. (2.202) is MD=10-7+2-3=2. Two independent parameters are necessary to define the position and orientation of any link in the mechanism. These parameters are the two displacements q1 and q2 in the revolute actuators mounted on the fixed base (underlined joints). Equations (2.217) and (2.222) indicate that the parallel mechanism has four degrees of overconstraint (ND=12-8=4) and zero degrees of redundancy (TD=10-7-3=0).

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3 Structural analysis

Fig. 3.35. Fully-isotropic non overconstrained parallel wrist RR-Pa*RRS-type with two degrees of freedom: kinematic structure (a), associated graph (b), kinematic chains associated with limbs A1 (c) and A2 (d)

We note that the classical CGK formulae give an erroneous mobility (MD=1) for this parallel mechanism. Figure 3.35 presents a non overconstrained solution RR-Pa*RRS-type. One cylindrical and one spherical joint are introduced in the parallelogram loop denoted by Pa*. The complex limb E2 of type Pa*RRS includes a Pa*-type parallelogram loop, two revolute joints with parallel axes and a spherical joint. This solution has the following structural parameters: m=8, p=9, q=2, k1=1, k2=1, (RE1)=( ω xy ,ω z ), (RE2)=( v x ,v y ,ω x ,ω y ,ω z ), SA1=2,

SE1=6, S5D/ 1 = 2, dim(U)=14, rl=6, rD=12, MD=2, ND=0, TD=0. Example 24. Orthoglide (Fig. 3.36) is a translational parallel robot (Wanger and Chablat 2000; Chablat and Wanger 2003) developed by IRCCyN (Institut de Recherche en Communications et Cybernétique de Nantes). The parallel mechanism D ← E1-E2-E3 in Figs. 3.36a and 3.37a is obtained by parallel concatenation of 3 complex limbs E1 (1A≡0-2A-3A-4A5A), E2 (1B≡0-2B-3B-4B-5B) and E3 (1C≡0-2C-3C-4C-5C). The mechanism has the following structural characteristics: m=17 links including the fixed base, p=21 revolute and prismatic joints, five independent loops (q=2117+1=5), three complex limbs (k1=0, k2=3). The mobile platform is

3.4 Parallel mechanisms with complex limbs

177

5≡5A≡5B≡5C and the reference platform is fixed to the base (1≡1A≡1B≡1C≡0). Each complex limb P ⊥ R ⊥ Pa ⊥ R-type is composed of one prismatic and two revolute pairs combined with a planar parallelogram loop. The axes of the revolute joints are perpendicular to the axes of the parallelogram loop. In each limb, the prismatic pair is actuated and its direction is perpendicular to the axis of the revolute pair. The directions of the three actuated prismatic pairs are also perpendicular to each other. We simply observe by inspection that the complex limbs (Fig. 3.37c-e) have the following unique bases of their operational velocity spaces: (RE1)=( v x ,v y ,v z ,ω y ), (RE2)=( v x ,v y ,v z ,ω z ) and (RE3)=( v x ,v y ,v z ,ω x ).

The dimensions of these three operational spaces SEi=4 (i=1,2,3) give the connectivity of the kinematic chains associated with each limb Ei of the parallel mechanism D ← E1-E2-E3. The connectivity of mobile platform in

Fig. 3.36. Translational parallel robot Orthoglide (courtesy of CNRS-IRCCyN)

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3 Structural analysis

Fig. 3.37. Parallel robot Orthoglide: structural diagram (a), associated graph (b), complex open chains associated with limbs A1 (c), A2 (d) and A3 (e)

3.4 Parallel mechanisms with complex limbs

179

the parallel mechanism D ← E1-E2-E3, given by Eq. (2.198), is S7D/ 1 = dim( RE1 ∩ RE 2 ∩ RE 3 ) = 3 . Just three translational velocities v x , v y and v z exist between the mobile platform 7 and the reference base 1≡0 in the parallel mechanism D ← E1-E2-E3. p The dimension of the joint vector space U is dim(U)= ∑ i =1 fi =21. Three

joint parameters lose their independence in each planar parallelogram loop (rl=9). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=12-3+9=18. The mobility of the parallel mechanism D ← E1-E2-E3 given by Eq. (2.202) is MD=2112+3-9=3. Three independent parameters are necessary to define the position of any link in the mechanism. In the Orthoglide robot, these parameters are the three displacements q1, q2 and q3 of the linear actuators mounted on the fixed base. Equation (2.217) indicates that the Orthoglide parallel mechanism has twelve degrees of overconstraint (ND=30-12+39=12) and Eq. (2.222) gives zero degrees of redundancy (TD=21-12-9=0). Example 25. The fully-isotropic translational parallel mechanism 3PPaPa-type (Gogu 2004b) in Fig. 3.38 is obtained by parallel concatenation of 3 complex limbs E1 (1A≡0-2A-3A-4A-5A-6A-7A-8A), E2 (1B≡0-2B-3B4B-5B-6B-7B-8B) and E3 (1C≡0-2C-3C-4C-5C-6C-7C-8C). The mechanism has the following structural characteristics: m=20 links including the fixed base, p=27 revolute and prismatic joints, eight independent loops (q=2720+1=8) and three complex limbs (k1=0, k2=3). The mobile platform is 8≡8A≡8B≡8C and the reference platform is fixed to the base (1≡1A≡1B≡1C≡0). Each complex limb P||Pa||R-type is composed of one prismatic joint and two planar parallelogram loops. The prismatic joints of the three limbs are actuated and their directions are mutually perpendicular. We simply observe by inspection that the complex limbs (Fig. 3.38c-e) have the following unique bases of their operational velocity spaces: (RE1)=( v x ,v y ,v z ), (RE2)=( v x ,v y ,v z ) and (RE3)=( v x ,v y ,v z ). The dimensions of these three operational spaces SEi=3 (i=1,2,3) give the connectivity of the kinematic chains associated with each limb Ei of parallel mechanism D ← E1-E2-E3. The connectivity of the mobile platform in the parallel mechanism D ← E1-E2-E3, given by Eq. (2.198), is S7D/ 1 = dim( RE1 ∩ RE 2 ∩ RE 3 ) = 3 . Just three translational velocities v x , v y and v z exist between the mobile platform 8 and the reference base 1≡0 in the parallel mechanism D ← E1E2-E3.

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3 Structural analysis

Fig. 3.38. Fully-isotropic overconstrained translational parallel mechanism 3PPaPa-type: kinematic structure (a), associated graph (b), complex chains associated with limbs A1 (c), A2 (d) and A3 (e)

3.4 Parallel mechanisms with complex limbs

181

Fig. 3.39. Fully-isotropic non overconstrained translational parallel mechanism 3PRPa*Pa*-type: kinematic structure (a), associated graph (b)

The dimension of the joint vector space U is dim(U)= ∑ i =1 fi =27. Three p

joint parameters lose their independence in each planar parallelogram loop (rl=18). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=9-3+18=24. The mobility of the parallel mechanism D ← E1-E2-E3 given by Eq. (2.202) is MD=27-9+3-18=3. Three independent parameters are necessary to define the position of any link in the mechanism. These parameters are the three displacements q1, q2 and q3 of the linear actuators mounted on the fixed base. Equation (2.217) indicates that the parallel mechanism in Fig. 3.38 has 24 degrees of overconstraint (ND=48-9+3-18=24) and Eq. (2.222) gives zero degrees of redundancy (TD=27-9-18=0).

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3 Structural analysis

Fig. 3.40. Complex chains associated with limbs A1 (a), A2 (b) and A3 (c) of the fully-isotropic non overconstrained translational parallel mechanism 3PRPa*Pa*-type

3.4 Parallel mechanisms with complex limbs

183

Figure 3.39 presents a non overconstrained solution 3-PRPa*Pa*Rtype. One cylindrical and one spherical joint are introduced in the parallelogram loop denoted by Pa*. Two revolute joints with perpendicular axes are introduced in each complex limb outside the parallelogram loops. The first revolute joints of the three limbs are mutually orthogonal. The last revolute joints of the three limbs are also mutually orthogonal. We simply observe by inspection that the complex limbs isolated from the parallel mechanism (Fig. 3.40a-c) have the following bases of their operational velocity spaces: (RE1)=( v x ,v y ,v z ,ω x ,ω y ), (RE2)=( v x ,v y ,v z ,ω y ,ω z ) and (RE3)= ( v x ,v y ,v z ,ω x ,ω z ). Six joint parameters lose their independence

in each parallelogram loop. This solution has the following structural parameters: m=26, p=33, q=8, k1=0, k2=3, SE1= SE2=SE3=5, S10D / 1 = 3, dim(U)=51, rl=36, rD=48, MD=3, ND=0, TD=0. Example 26. The Delta parallel mechanism 3-RPa4s-type invented by Clavel (1990) is widely used today in translational parallel robots. The solution presented in Fig. 3.41a (Clavel 1990) represent a hybrid robot in which a translational mechanism Delta-type with rotating actuators (Fig. 3.41a) is serially concatenated with a revolute joint. The translational parallel mechanism (Fig. 3.42a) is obtained by parallel concatenation of 3 complex limbs E1 (1A≡0-2A-3A-4A-5A), E2 (1B≡0-2B-3B-4B-5B) and E3 (1C≡02C-3C-4C-5C). The mechanism has the following structural characteristics: m=11 links including the fixed base, p=15 joints, five independent loops (q=15-11+1=5) and three complex limbs (k1=0, k2=3). The mobile platform is 5≡5A≡5B≡5C and the reference platform is fixed to the base

Fig. 3.41. Hybrid mechanism Delta-type invented by R. Clavel and presented in US Patent 4976582 (Clavel 1990) (a) and its associated translational parallel mechanism Delta-type with rotating actuators (b)

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3 Structural analysis

Fig. 3.42. Translational Delta mechanism 3-RPa4s-type: kinematic structure (a), associated graph (b), complex chains associated with limbs A1 (c), A2 (d) and A3 (e)

(1≡1A≡1B≡1C≡0). Each complex limb RPa4s-type is composed of one revolute and a spatial parallelogram loop containing 4 spherical joints. The revolute joints of the three limbs are actuated and the directions of their rotation axes lie in a plane. Moving platform 5 in each complex limb isolated from the Delta parallel mechanism (Fig. 3.42c-e) may realize three translation velocities v x ,v y ,v z and two angular velocities ω i and ω i+1 (i=1 for Alimb, i=3 for B-limb and i=5 for C-limb): (RE1)= ( v x ,v y ,v z ,ω 1 ,ω 2 )

(RE2)=( v x ,v y ,v z ,ω 3 ,ω 4 ) and (RE3)= ( v x ,v y ,v z ,ω 5 ,ω 6 ),

3.4 Parallel mechanisms with complex limbs

185

The dimensions of these three operational spaces SEi=5 (i=1,2,3) give the connectivity of the kinematic chains associated with each limb Ei of parallel mechanism D ← E1-E2-E3. The connectivity of the mobile platform in the parallel mechanism D ← E1-E2-E3, given by Eq. (2.198), is S5D/ 1 = dim( RE1 ∩ RE 2 ∩ RE 3 ) = 3 . Just three translational velocities v x , v y and v z exist between the mobile platform 5 and the reference base 1≡0 in the parallel mechanism D ← E1-E2-E3. p The dimension of the joint vector space U is dim(U)= ∑ i =1 fi =39. Six

joint parameters lose their independence in each parallelogram loop Pa4Stype (rl=18). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=15-3+18=30. The mobility of parallel mechanism D ← E1-E2-E3 given by Eq. (2.202) is MD=39-15+3-18=9. Nine independent parameters have to be known to define the position and the orientation of each link of the parallel mechanism. Equation (2.222) gives six degrees of redundancy (TD=39-15-18=6) for the Delta parallel mechanism. Each parallelogram loop Pa4S-type contains two internal mobilities introduced by the rotation of links 3 and 4 of each limb along a rotation axis passing through the centre of the two spherical joints adjacent to links 3 and 4. These six degrees of redundancy represent internal mobilities in the limbs that cannot be transmitted from the input to the output of the parallel mechanism. The remaining three degrees of mobility are external mobilities that are transmitted from the input to the output of the parallel mechanism. They represent the independent parameters

Fig. 3.43. Parallel mechanism linear Delta-type invented by R. Clavel and presented in US Patent 4976582 (Clavel 1990) (a) and its associated translational parallel mechanism Delta-type with linear actuators (b)

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3 Structural analysis

Fig. 3.44. Translational Delta mechanism 3-PPa4s-type: kinematic structure (a), associated graph (b), complex chains associated with limbs A1 (c), A2 (d) and A3 (e)

that are necessary to define the position of the moving platform. These parameters are the three angular displacements q1, q2 and q3 of the rotating actuators mounted on the fixed base. Equation (2.217) indicates that the parallel mechanism in Fig. 3.42 is not overconstraint (ND=30-15+3-18=0). The same structural analysis and parameters are applicable to the translational parallel Delta-type mechanism with linear actuators in Figs. 3.43b and 3.44a. The only difference is that the three degrees of external mobilities are associated with three linear displacements in the linear actuators mounted on the fixed base instead of rotating actuators (Fig. 3.44).

3.4 Parallel mechanisms with complex limbs

187

The hybrid mechanism in Fig. 3.41a may be considered a complex open kinematic chain F with the following structural characteristics: m=12 links including the fixed base, p=16 joints, five independent loops (q=1511+1=5) and just one complex limb (k1=0, k2=1, k=1). In this case, SG1=SF and Eqs. (2.226) and (2.228) give r=rl=rG1. The number of joint parameters that lose their independence in the five closed loops of this mechanism is r=rD=30, where rD was calculated above for the mechanism in Fig 3.42a. Equations (2.223)-(2.225) give the following structural parameters for the mechanism in Fig. 3.44: M=10, N=0, T=6. The six degrees of redundancy represent internal mobilities in the limbs and the remaining four degrees of mobility are external mobilities given by the angular displacements of the four revolute actuated joints. Example 27. The Delta parallel mechanism in Fig. 3.45 represents another solution invented by R. Clavel [Clavel US patent 4976582]. This robot may be considered as a parallel mechanism D ← E1-E2 obtained by parallel concatenation of 2 complex limbs E1 and E2 (Fig. 3.46a). Limb E1 (Fig. 3.46b) is the hybrid Delta mechanism analysed in Example 26, and limb E2 of type RUPU (Fig. 3.46c) is composed of a homokinetic double universal joint U with a telescopic intermediary shaft (1D≡0-2D-3D-4D-5D6D-7D). As we have seen in Example 26, limb E1 of type 3RPa4s-R is composed by serial concatenation of a translational parallel mechanism 3RPa4s and a revolute joint. Five independent closed loops exist in limb E1, and

Fig. 3.45. Parallel mechanism Delta-type with rotating actuators invented by R. Clavel and presented in US Patent 4976582 (Clavel 1990)

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3 Structural analysis

Fig. 3.46. Parallel mechanism Delta-type with rotating actuators: kinematic structure (a), associated graph (b), complex chains associated with limbs E1 (c) and E2 (d)

four in limb E2. We have seen in the previous example that 30 joint parameters lose their independence in the closed loops of limb E1 ( rl E1 =30). The connectivity theorem 5A and Eq. (2.140) in chapter 2, and example 1 in this chapter also indicate that one joint parameter loses its independence

3.4 Parallel mechanisms with complex limbs

189

Fig. 3.47. Parallel mechanism Delta-type with linear and rotating actuators: kinematic structure (a), associated graph (b), complex chains associated with limbs E1 (c) and E2 (d)

in each closed loop composed of two revolute joints of same axis of the universal joints ( rl E 2 =4). The parallel mechanism in Fig. 3.46a has the following structural characteristics: m=17 links including the fixed base, p=26 joints, ten independent loops (q=26-17+1=10) and two complex limbs (k1=0, k2=2). The end-effector is 7≡6A≡6B≡6C≡7D, and the reference platform is fixed to the base (1≡1A≡1B≡1C≡1D≡0). The bases and the dimensions of the operational velocities spaces associated with the two com-

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3 Structural analysis

plex limbs are: (RE1)=( v x ,v y ,v z ,ω z ), (RE2)=( v x ,v y ,v z ,ω x ,ω y ,ω z ), SE1=4 and SE2=6. The connectivity of the end-effector in parallel mechanism D ← E1-E2, given by Eq. (2.198), is S7D/ 1 = dim( RE1 ∩ RE 2 ) = 4 . Three translational velocities v x , v y and v z and one angular velocity ω z exist between the end-effector 7 and the reference base 1≡0 in the parallel mechanism D ← E1-E2. p The dimension of the joint vector space U is dim(U)= ∑ i =1 fi =50. The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=10-4+34=40. The mobility of parallel mechanism D ← E1-E2 in Fig. 3.46a given by Eq. (2.202) is MD=5010+4-34=10. Ten joint parameters have to be known to define the position and the orientation of each link of the parallel mechanism. Equation (2.222) gives six degrees of redundancy (TD=50-10-34=6) for Delta parallel mechanism in Fig. 3.46a. Each parallelogram loop Pa4S-type contains two internal mobilities introduced by the rotation of the links 3 and 4 of each limb along a rotation axis passing through the centre of the two spherical joints adjacent to links 3 and 4. These six degrees of redundancy represent internal mobilities in the limbs that cannot be transmitted from the input to the output of the parallel mechanism. The remaining four degrees of mobility are external mobilities that are transmitted from the input to the output of the parallel mechanism. They represent the independent parameters that are necessary to define the position of the moving platform. These parameters are the three angular displacements qi (i=1,…,4) in the rotating actuators mounted on the fixed base. Equation (2.217) indicates that the parallel mechanism in Fig. 3.46a has ND=60-10+4-34=20. Five overconstraints are introduced in each closed loop with two revolute joints of same axis of the universal joints. Usually these overconstraints in the universal joints are eliminated by a precise alignment of rotation axes and the technical solutions used for the revolute joints. To avoid these overconstraints in static and dynamic analysis of the mechanisms with universal joints, just one revolute joint kinematically equivalent to the two revolute joints of same axis is taken into account. In this case, the mechanism in Fig. 46a may be considered as a parallel mechanism D ← E1-A1 with one complex limb E1 as in the previous analysis, and a simple limb A1 (1D≡0-2D-3D-4D-5D-6D-7D). The following structural parameters are associated with mechanism D ← E1-A1: m=17, p=22, q=6, k1=1, k2=1, (RE1)=( v x ,v y ,v z ,ω z ), (RE2)=( v x ,v y ,v z ,ω x ,ω y ,ω z ), SE1=4, SE2=6, S7D/ 1 = 4 dim(U)= ∑ i =1 fi =46, rl=30, rD=36, MD=10, TD=6 p

3.4 Parallel mechanisms with complex limbs

191

and ND=0. With this hypothesis, mechanism D ← E1-A1 in Fig. 46a may be considered non overconstrained. The same structural analysis and parameters are applicable to the parallel Delta-type mechanism in Figs. 3.43a and 3.47a. The only difference is that the four degrees of external mobilities are associated with three linear displacements in the linear actuators and one angular displacement in the rotating motor mounted on the fixed base (Fig. 3.47). Example 28. The mechanism in Fig. 3.48 represents a translational Delta-type parallel robot with universal joints also invented by Clavel (1990). This translational parallel mechanism D ← E1-E2-E3 is obtained by parallel concatenation of 3 complex limbs E1 (1A≡0-2A-3A-4A-5A-6A), E2 (1B≡0-2B-3B-4B-5B-6B) and E3 (1C≡0-2C-3C-4C-5C-6C) – Fig. 3.49a. The mechanism has the following structural characteristics: m=14 links including the fixed base, p=27 revolute joints, 14 independent closed loops (q=27-14+1=14) and three complex limbs (k1=0, k2=3). The mobile platform is 6≡6A≡6B≡6C and the reference platform is

Fig. 3.48. Translational Delta parallel mechanism 3RUU-type invented by R. Clavel and presented in US Patent 4976582 (Clavel 1990)

Table 3.2. Valid bases of operational vector spaces RE1, RE2 and RE3 of complex limbs of the Delta parallel mechanism No. (RE1) 1 2 3

(RE2)

(RE3)

(

∩

3

i =1

REi )

( v x ,v y ,v z ,ω x ,ω y ) ( v x ,v y ,v z ,ω y ,ω z ) ( v x ,v y ,v z ,ω z ,ω x ) ( v x ,v y ,v z ) ( v x ,v y ,v z ,ω y ,ω z ) ( v x ,v y ,v z ,ω z ,ω x ) ( v x ,v y ,v z ,ω x ,ω y ) ( v x ,v y ,v z ) ( v x ,v y ,v z ,ω z ,ω x ) ( v x ,v y ,v z ,ω x ,ω y ) ( v x ,v y ,v z ,ω y ,ω z ) ( v x ,v y ,v z )

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3 Structural analysis

Fig. 3.49. Delta Parallel mechanism 3RUU-type: kinematic structure (a), associated graph (b), complex chain associated with a limb (c)

fixed to the base (1≡1A≡1B≡1C≡0). Each complex limb RUU-type is composed of nine revolute joints. Two closed loops with two revolute joints with the same axis exist in each universal joint. The moving platform 6 in each complex limb isolated from the Delta parallel mechanism (Fig. 3.49c) may realize three translation velocities v x ,v y ,v z and two angular velocities ω i and ω i+1 (i=1 for A-limb, i=3 for B-limb and i=5 for C-limb). In the general case, each angular velocity has

3.4 Parallel mechanisms with complex limbs

193

three components ω x ,ω y ,ω z but just one can be independent. Various bases of the operational velocity space of each complex limb are systematized in Table 3.2. They are set up by taking into account Eq. (2.206) and Remark 3 in chapter 2. The dimensions of these three operational spaces SEi=5 (i=1,2,3) give the connectivity of the kinematic chains associated with each limb Ei of parallel mechanism D ← E1-E2-E3. The connectivity of the mobile platform in the parallel mechanism D ← E1-E2-E3, given by Eq. (2.198), is S6D/ 1 = dim( RE1 ∩ RE 2 ∩ RE 3 ) = 3 . Just three translational velocities v x , v y and v z exist between the mobile platform 6 and the reference base 1≡0 in the parallel mechanism D ← E1-E2-E3. p The dimension of the joint vector space U is dim(U)= ∑ i =1 fi =27.One joint parameter loses its independence in each closed loop of the universal joints (rl=12). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=15-3+12=24. The mobility of the parallel mechanism D ← E1-E2-E3 given by Eq. (2.202) is MD=27-15+3-12=3. Three independent parameters have to be known to define the position and the orientation of each link of the parallel mechanism. These parameters are the three angular displacements q1, q2 and q3 in the rotating actuators mounted on the fixed base. Equation (2.222) gives zero degrees of redundancy (TD=27-15-12=0). Equation (2.217) gives ND=84-15+3-12=60. Each closed loop in the universal joint introduces five overconstraints. Usually these overconstraints in the universal joints are eliminated by a precise alignment of rotation axes and the technical solutions used for the revolute joints. As we have shown in the previous example, to avoid these overconstraints in static and dynamic analysis of the mechanisms with universal joints, just one revolute joint kinematically equivalent to the two revolute joints of same axis is taken into account. With this hypothesis, the mechanism in Fig. 3.49a may be considered as a parallel mechanism C ← A1-A2-A3 with three simple limbs A1 (1A≡0-2A-3A-4A-5A), A2 (1B≡0-2B-3B-4B-5B) and A3 (1C≡0-2C-3C-4C-5C). The following structural parameters are associated with mechanism C ← A1-A2-A3: m=14, p=15, q=2, k1=3, k2=0, p SA1=SA2=SA3=5, S6C/ 1 = 3, dim(U)= ∑ i =1 fi =15, rl=0, rC=12, MC=3, TC=0

and NC=0. In this case, mechanism C ← A1-A2-A3 in Fig. 49a may be considered non overconstrained. Example 29. The mechanism in Fig. 3.50 represents a translational Tricept-type parallel robot invented by Neumann (1998). This translational parallel mechanism D ← E1-E2-E3-E4 is obtained by parallel concatenation of 4 complex limbs E1 (1A≡0-2A-3A-4A-5A),

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3 Structural analysis

Fig. 3.50. Tricept parallel mechanism invented by K-E. Neumann and presented in US Patent 4732525 (Neumann 1988)

Fig. 3.51. Tricept parallel mechanism: CAD model (a), kinematic structure (b), associated graph (c) – courtesy of Vincent Robin for the CAD model (a)

3.4 Parallel mechanisms with complex limbs

195

E2 (1B≡0-2B-3B-4B-5B), E3 (1C≡0-2C-3C-4C-5C) and E4 (1D≡0-2D-3D-4D) – Fig. 3.51a. The mechanism has the following structural characteristics: m=13 links including the fixed base, p=20 joints, 8 independent closedloops (q=20-13+1=8) and four complex limbs (k1=0, k2=4). The mobile platform is 5≡5A≡5B≡5C≡4D, and the reference platform is fixed to the base (1≡1A≡1B≡1C≡1D≡0). Five closed loops with two revolute joints with the same axis exist in the mechanism, one in each limb Ai (i=1,2,3) and two in the universal joint of limb A4. The moving platform 5 in each limb A1, A2 and A3 isolated from the Tricept parallel mechanism may realize six independent velocities, v x ,v y ,v z ,ω x ,ω y and ω z . In limb A4 isolated from the Tricept parallel mechanism, the moving platform 5 may realize just three independent motions which may be three translations, or combinations of rotations and translations. Various bases may be set up for the operational velocity space of limb A4 isolated from the Tricept parallel mechanism: ( v x ,v y ,v z ), ( v x ,v y ,ω x ), ( v x ,v y ,ω y ), ( v y ,v z ,ω z ), ( v x ,ω x ,ω z ), ( v y ,ω x ,ω z ), ( v z ,ω x ,ω z ), ( v x ,ω y ,ω z ), ( v y ,ω y ,ω z ) or ( v z ,ω y ,ω z ). The dimensions of the four operational spaces give the connectivity of the kinematic chains associated with each limb Ei of parallel mechanism D ← E1-E2-E3-E4: SEi=6 (i=1,2,3) and SE4=3. The connectivity of the mobile platform in the parallel mechanism D ← E1-E2-E3-E4, given by Eq. (2.198), is S5D/ 1 = dim( RE1 ∩ RE 2 ∩ RE 3 ∩ RE 4 ) = 3 . Just three independent velocities exist between the mobile platform 5 and the reference base 1≡0 in parallel mechanism D ← E1-E2-E3-E4. The basis of the operational velocity space of the moving platform 5 in the Tricept parallel mechanism coincides with the basis of the operational velocity space of limb A4 isolated from the Tricept parallel mechanism above mentioned. p The dimension of the joint vector space U is dim(U)= ∑ i =1 fi =25. One

joint parameter loses its independence in each closed loop, with just two revolute joints of same axis (rl=5). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=21-3+5=23. The mobility of the parallel mechanism D ← E1-E2-E3-E4 given by Eq. (2.202) is MD=26-21+3-5=3. Three independent parameters have to be known to define the position and the orientation of each link of the parallel mechanism. These parameters are the three displacements q1, q2 and q3 of the linear actuators mounted in the telescopic limbs A1, A2 and A3. These actuators are not adjacent to the fixed base. Equation (2.222) gives zero degrees of redundancy (TD=26-21-5=0). Equation (2.217) gives ND=48-21+3-5=25. These overconstraints are introduced in the five

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3 Structural analysis

Fig. 3.52. Kinematic structure of Tricept robot considered as a parallel mechanism with four simple limbs

closed loops with two revolute joints of same axis. Each closed loop introduces 5 overconstraints. Usually these overconstraints are eliminated by a precise alignment of rotation axes, and by the technical solutions used for the revolute joints. As we have shown in the previous example, to avoid these overconstraints in static and dynamic analysis of such a mechanisms, just one revolute joint kinematically equivalent to the two revolute joints of same axis is taken into account. With this hypothesis, the Tricept parallel mechanism (Fig. 3.52) may be considered as a parallel mechanism C ← A1-A2-A3A4 with four simple limbs A1 (1A≡0-2A-3A-4A-5A), A2 (1B≡0-2B-3B-4B-5B), A3 (1C≡0-2C-3C-4C-5C) and E4 (1D≡0-2D-3D-4D). The following structural parameters are associated with mechanism C ← A1-A2-A3-A4: m=13, p=15, p q=3, k1=0, k2=1, k=1, S5C/ 1 = 3, dim(U)= ∑ i =1 fi =21, rl=0, rC=18, MC=3,

TC=0 and NC=0. The parallel mechanism C ← A1-A2-A3 in Fig. 3.52 is non overconstrained. Example 30. The mechanism in Fig. 3.53 represents the hybrid robot IRB 940 fabricated by ABB by concatenating a Tricept parallel mechanism and a serial wrist RRR-type. This hybrid mechanism may be considered to be a complex open kinematic chain F with the following structural characteristics: m=16 links including the fixed base, p=23 joints, eight independent loops (q=23-16+1=6) and just one complex limb (k1=0, k2=1, k=1). In this case, SG1=SF and Eqs. (2.226) and (2.228) give r=rl=rG1. The number of joint parameters that lose their independence in the eight closed

3.4 Parallel mechanisms with complex limbs

197

Fig. 3.53. Hybrid mechanism obtained by concatenating a Tricept parallel mechanism and a serial wrist: CAD model (a), kinematic structure (b), associated graph (c) – courtesy of Vincent Robin for the CAD model (a)

loops of this mechanism is r=rD=23, where rD was calculated above for the mechanism in Fig 3.51 in the previous example. Equations (2.223)(2.225) give the following structural parameters for the mechanism in Fig. 3.53: M=6, N=25, T=0. As in the previous example, the 25 overconstraints are introduced by the five closed loops with just two revolute joints with the same axis. If we take into account just one revolute joint kinematically equivalent to the two revolute joints of same axis (Fig. 3.54) the following structural parameters are associated: m=13, p=15, q=3, k1=0, k2=1, k=1, p SG1=SF=3, dim(U)= ∑ i =1 fi =21. The number of joint parameters that lose their independence in the eight closed loops is r=rC=18,

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3 Structural analysis

Fig. 3.54. A hybrid mechanism obtained by concatenating a Tricept parallel mechanism with simple limbs and a serial wrist: kinematic structure (a), associated graph (b)

where rC was calculated above for the mechanism in Fig 3.52 in the previous example. Equations (2.223)-(2.225) give the following structural parameters for the mechanism in Fig. 3.54: M=3, T=0 and N=0. The mechanism is non overconstrained. Example 31. The parallel robot H4 (Fig. 3.55) was developed by Montpellier Laboratory of Computer Science, Robotics, and Microelectronics – LIRMM (Pierrot and Company 1999; Company and Pierrot 1999; Pierrot et al. 2003, Company et al. 2003, 2006). This mechanism F ← G1-G2 is obtained by parallel concatenation of 2 hybrid limbs G1 and G2 with the moving platform 6 (Fig. 3.56a). Each limb Gi ← Di-Ri (i=1,2) is obtained by concatenating a complex kinematic chain Di with a revolute joint Ri. Complex chains D1 ← E1-E2 and D2 ← E3-E4 represent parallel mechanisms with two complex limbs E1-E2 (Fig. 3.56b) and E3-E4 (Fig. 3.56c) connecting the fixed base to two intermediary moving platforms 5≡5A≡5B and 5’≡5C≡5D. Each complex limb Ei combines a revolute joint and a spatial parallelogram loop with four spherical joints Pa4s: E1 (1A≡0-2A-3A-4A-5A), A2 (1B≡0-2B-3B-4B-5B), A3 (1C≡0-2C-3C-4C-5C) and E4 (1D≡0-2D-3D-4D-5D). The mechanism has the following structural characteristics: m=16 links including the fixed base, p=22 joints, 7 independent closed loops (q=2216+1=7) and two complex limbs (k1=0, k2=2, k=2).

3.4 Parallel mechanisms with complex limbs

199

Fig. 3.55. H4 parallel robot developed by Montpellier Laboratory of Computer Science, Robotics, and Microelectronics (a) and CAD models (b) and (c) (courtesy LIRMM)

The four complex limbs Ei isolated from the parallel robot (Figs. 3.57 and 3.58) have the following bases and dimensions for their operational velocities spaces: (RE1)=( v x ,v y ,v z ,ω 1 ,ω 2 ),(RE2)=( v x ,v y ,v z ,ω 3 ,ω 4 ), (RE1)=( v x ,v y ,v z ,ω 5 ,ω 6 ),(RE2)=( v x ,v y ,v z ,ω 7 ,ω 8 ) and SEi=5 (i=1,...,4). The connectivity of the intermediary platforms 5 in the parallel mechanisms D1 ← E1-E2, given by Eq. (2.198), is S5D1/ 1 = 3.

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3 Structural analysis

Fig. 3.56. H4 parallel robot, kinematic structure (a), associated graph (b), limbs G1 (c) and G2 (d) isolated from the parallel robot

Just three independent translational velocities v x , v y and v z exist between 5 and reference base 1A≡1B ≡0 in parallel mechanism D1 ← E1-E2 isolated from the parallel robot. Equation (2.209) indicates that 19 joint parameters lose their independence in the closed loops of parallel mechanism D1 ← E1E2 (rD1=10-3+12=19). We have taken into account that 6 joint parameters

3.4 Parallel mechanisms with complex limbs

201

Fig. 3.57. Parallel mechanism D1 ← E1-E2 (a) and its complex limbs E1 (b) and E2 (c) isolated from the parallel mechanism D1

lose their independence in each parallelogram closed loop integrated in complex limbs E1 and E2. The same results are obtained for limb D2: S5'/D 21 = 3 and rD2=19. In each hybrid limb G1 and G2 isolated from the H4 parallel robot (Fig. 3.56b and c), the moving platform 6 may realize four independent velocities, v x ,v y ,v z and ω z . We have: (RG1)=(RG2)=( v x ,v y ,v z ,ω z ) and SG1=SG2=4.

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3 Structural analysis

Fig. 3.58. Parallel mechanism D2 ← E3-E4 (a) and its complex limbs E3 (b) and E4 (c) isolated from the parallel mechanism D2

The connectivity of mobile platform 6 in the parallel robot H4 (Fig. 3.56a) is S6F/ 1 = dim( RG1 ∩ RG 2 ) = 4 . Four independent velocities exist between mobile platform 6 and reference base 1≡0 in parallel mechanism F ← G1-G2 associated with H4 robot. The basis of the operational velocity space of the moving platform 6 in H4 parallel robot coincides with the ba-

3.4 Parallel mechanisms with complex limbs

203

sis of the operational velocity space of limbs G1 and G1 isolated from the parallel mechanism F ← G1-G2 . p The dimension of the joint vector space U is dim(U)= ∑ i =1 fi =54. The number of joint parameters that lose their independence in the parallel mechanism F ← G1-G2 given by Eq. (2.226) is r=8-4+38=42. We have taken into account in Eq. (2.228) that 19 joint parameters lose their independence in each hybrid limb Gi (i=1,2) as we have calculated above. The mobility of the parallel mechanism F ← G1-G2 given by Eq. (2.223) is M=54-42=12. Twelve joint parameters have to be known to define the position and the orientation of each link of the parallel mechanism F ← G1-G2. Equation (2.225) gives eight degrees of redundancy (T=12-4=8) for H4 parallel robot. Each parallelogram loop Pa4S-type contains two internal mobilities introduced by the rotation of the links 3 and 4 of each limb along a rotation axis passing through the centre of the two spherical joints adjacent to links 3 and 4. These eight degrees of redundancy represent internal mobilities in the limbs that cannot be transmitted from the input to the output of the parallel mechanism. The remaining four degrees of mobility are external mobilities that are transmitted from the input to the output of the parallel mechanism. They represent the independent parameters that are necessary to define the position and the orientation of the moving platform 6. These parameters are the angular displacements in the four rotating actuators mounted on the fixed base. Equation (2.224) indicates that H4 parallel robot is not overconstrained N=42-42=0. Example 32. Isoglide4-T3R1 in Fig. 3.59 is the translational parallel robot with decoupled motions proposed in (Gogu 2002; 2005e). This mechanism F ← G1-G2 is obtained by parallel concatenation of a hybrid limb G1 and a simple limb G2 to the moving platform 6 (Fig. 3.59a,b). The hybrid limb G1 ← C-R (Fig. 3.59c) is obtained by serial concatenation of a complex kinematic chain C with a revolute joint R. The complex chain C ← A1-A2-A3 represents a translational parallel mechanisms with three simple limbs Isoglide3-T3 of type 3-P||R||R||R analysed in example 15. The intermediary moving platform 5 of Isoglide3-T3 is connected to the moving platform 6 by a revolute joint. The simple limb G2 (1D≡0-2D-3D-4D-5D6) is of type P||R||R||R ⊥ R (Fig. 3.49d). The revolute joints connecting the limbs G1 and G2 to the moving platform have parallel axes. The mechanism has the following structural characteristics: m=16 links including the fixed base, p=18 joints, 3 independent closed loops (q=1816+1=3), one simple limb and one complex limb (k1=1, k2=1, k=2).

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3 Structural analysis

Fig. 3.59. Parallel robot Isoglide4-T3R1 with one simple and one complex limb (a), associated graph (b), limbs G1 (c) and G2 (d) isolated from the parallel mechanism

3.4 Parallel mechanisms with complex limbs

205

Fig. 3.60. CAD model of parallel robot Isoglide4-T3R1 with one simple and one complex limb

The two limbs isolated from the parallel robot (Figs. 3.59c and d) have the following bases and dimensions for their operational velocities spaces: (RG1)=( v x ,v y ,v z ,ω y ), (RG2)=( v x ,v y ,v z ,ω y ,ω z ), and SG1=4, SG2=5. As we have seen in Example 15, nine joint parameters lose their independence in the closed loops of the translational parallel mechanism C ← A1-A2-A3 included in the hybrid limb G1 (rl= rlG1 =rC=9). The connectivity of the mobile platform 6 in Isoglide4-T3R1 (Fig. 3.59a) is S6F/ 1 = dim( RG1 ∩ RG 2 ) = 4 . Four independent velocities exist between the mobile platform 6 and the reference base 1≡1A≡1B≡1C≡1D≡0 in the parallel mechanism F ← G1-G2 associated with this parallel robot of the Isoglide family. p The dimension of the joint vector space U is dim(U)= ∑ i =1 fi =18. The number of joint parameters that lose their independence in the parallel mechanism F ← G1-G2 given by Eq. (2.226) is r=9-4+9=14. The mobility of the parallel mechanism F ← G1-G2 given by Eq. (2.223) is M=18-14=4. Four joint parameters have to be known to define the position and the orientation of each link of the parallel mechanism F ← G1-G2. These parameters are the linear displacements of the four linear actuators mounted on the fixed base. Equation (2.225) gives zero degrees of redundancy (T=4-4=0) and Eq. (2.224) indicates that parallel robot Isoglide4-T3R1 in Fig. 3.59a has four degrees of overconstraints (N=18-14=4).

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3 Structural analysis

Figure 3.60 presents a CAD model of the parallel robot Isoglide4-T3R1 with one hybrid and one simple limb. Example 33. Isoglide3-T2R1 in Fig. 3.61 is a fully-isotropic planar parallel robot proposed in (Gogu 2004c). This mechanism F ← G1-G2-G3 is obtained by parallel concatenation of two simple limbs G1 and G2 and a hybrid limb G3. The moving platform is denoted by 8, and the fixed base by 1≡1A≡1B≡1C≡0 (Fig. 3.61a,b). Simple limbs G1 (1A≡0-2A-3A-4A-5A-8), and G2 (1B≡0-2B-3B-4B-5B-8) are of type P||R||R||R ⊥ R (Figs. 3.61c and d). Hybrid limb G3 (Fig. 3.61e) is obtained by concatenating a revolute pair with two interconnected planar parallelogram loops (2C-3C-4C-6C) and (7C6C-4C-5C -8). The revolute pairs connecting the three limbs to the moving platform have the same axis. The mechanism has the following structural characteristics: m=16 links including the fixed base, p=19 joints, 4 independent closed loops (q=1916+1=4), two simple limbs and one complex limb (k1=2, k2=1, k=3). The three limbs isolated from the parallel robot (Figs. 3.61c,d,e) have the following bases and dimensions for their operational velocities spaces: (RG1)=( v x ,v y ,v z ,ω x ,ω z ), (RG2)=( v x ,v y ,v z ,ω y ,ω z ), (RG3)=( v x ,v y ,ω z ), and SG1= SG2=5, SG3=3. To calculate the number of joint parameters that lose their independence in the two closed loops integrated in hybrid limb G3 ← R-C, we consider the

Fig. 3.61. Fully-isotropic planar parallel robot Isoglide3-T2R1 with one hybrid and two simple limbs (a), associated graph (b)

3.4 Parallel mechanisms with complex limbs

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Fig. 3.62. Limbs G1 (c), G2 (d) and G3 (e) isolated from the parallel mechanism of Isoglide3-T2R1 in Fig. 3.61

kinematic chain formed by these two closed loops as a parallel kinematic chain C ← A1-A2-A3 with three simple limbs connecting the ternary links 6C and 4C (Fig. 3. 63). The three simple limbs are: A1 (6C-4C) of type R||R (Fig. 3.63b), A2 (6C-2C-3C-4C) of type R||R||R (Fig. 3.63c) and A3 (6C-7C-85C-4C) of type R||R||R||R (Fig. 3.63d).

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Fig. 3.63. Parallel kinematic chain C ← A1-A2-A3 with distal links 4C and 6C and associated with the closed loops in limb G3 ← R-C (a), limb A1 (b), limb A2 (c) and limb A3 (d)

The three limbs isolated from the parallel kinematic chain C ← A1-A2-A3 (Figs. 3.63b,c,d) have the following bases and dimensions for their operational velocities spaces: (RA1)=( ω z ), (RA2)= ( v x ,v y ,ω z ), (RA3)=( v x ,v y ,ω z ) and SA1=1, SA2= SA3=3. The connectivity of link 4C with respect to link 6C C in parallel kinematic chain C ← A1-A2-A3 (Fig. 3.62a) is S4C / 6C = dim( RA1 ∩ RA2 ∩ RA3 ) = 1 . Just one rotation about the z-axis is possible between the two links. The number of joint parameters that lose their independence in the parallel kinematic chain C ← A1-A2-A3 is rC=7-1=6. This value is given by Eq. (2. 191) by taking by taking into account that in the

3.4 Parallel mechanisms with complex limbs

209

parallel kinematic chain C ← A1-A2-A3 we have considered 6C as the reference link, and 4C as the distal link. We thus obtain rl = rlG 3 = rC = 6 . The connectivity of the mobile platform 8 in Isoglide3-T2R1 with two simple and one hybrid limb in Fig. 3.61a is S8F/ 1 = dim( RG1 ∩ RG 2 ∩ RG3 ) =3. Three independent velocities ( v x ,v y and ω z ) exist between mobile platform 8 and reference base 1≡1A≡1B≡1C≡0 in parallel mechanism F ← G1-G2-G3 associated with this parallel robot of Isoglide family. p The dimension of the joint vector space U is dim(U)= ∑ i =1 fi =19. The number of joint parameters that lose their independence in the parallel mechanism F ← G1-G2-G3 given by Eq. (2.226) is r=13-3+6=16. The mobility of the parallel mechanism F ← G1-G2-G3 given by Eq. (2.223) is M=19-16=3. Three joint parameters have to be known to define the position and the orientation of each link of the parallel mechanism F ← G1-G2G3. These parameters are the displacements (qi, i=1,2,3) in the two linear and one rotating actuators mounted on the fixed base. Equation (2.225) gives zero degrees of redundancy (T=3-3=0) and Eq. (2.224) indicates that parallel robot Isoglide3-T2R1 in Fig. 3.61a has eight degrees of overconstraint (N=24-16=8). Six of these overconstraints are in the two planar parallelogram loops. Example 34. Isoglide3-T2R1 in Fig. 3.64 is a slightly modified version of the fully-isotropic planar parallel robot presented in Fig. 3.61 (Gogu, 2004c). This mechanism F ← G1-G2 is obtained by parallel concatenation of two hybrid limbs G1 and G2. The moving platform is denoted by 8 and the fixed base by 1≡1A≡1B≡1C≡0 (Fig. 3.64a,b). This solution is derived from the solution in Fig. 3.61 by superposing two revolute joints in just one joint. Limb G1 ← B-R is obtained by serially concatenating a closed loop B ← A1-A2 with a revolute joint. Limbs G2 in Fig. 3.64a and G3 in Fig. 3.61a are identical. The revolute pairs connecting the two limbs to the moving platform have the same axis. The mechanism has the following structural characteristics: m=15 links including the fixed base, p=18 joints, 4 independent closed loops (q=18-15+1=4) and two hybrid limbs (k=2). The two limbs isolated from the parallel robot (Figs. 3.64c,d) have the following bases and dimensions for their operational velocities spaces: (RG1)=( v x ,v y ,v z ,ω z ), (RG2)=( v x ,v y ,ω z ) and SG1=4, SG2=3. Five joint parameters lose their independence in the closed loop of limb G1 (see Example 5), rlG1 = rB = 5 , and six in the closed loops of limb G2 (see Example 33), rlG 2 = 6 .

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Fig. 3.64. Fully-isotropic planar parallel robot Isoglide3-T2R1 with two hybrid limbs (a), associated graph (b), limb G1 (c) and limb G2 (d)

The connectivity of mobile platform 8 in Isoglide3-T2R1 with two hybrid limbs in (Fig. 3.64a) is S8F/ 1 = dim( RG1 ∩ RG 2 ∩ RG3 ) = 3 . Three independent velocities ( v x ,v y and ω z ) exist between the mobile platform 8 and the reference base 1≡1A≡1B≡1C≡0 in the parallel mechanism F ← G1-G2 associated with this parallel robot of the Isoglide family.

3.4 Parallel mechanisms with complex limbs

211

The dimension of the joint vector space U is dim(U)= ∑ i =1 fi =18. The p

number of joint parameters that lose their independence in the parallel mechanism F ← G1-G2 given by Eq. (2.226) is r=7-3+11=15. We have taken into account that Eq. (2.228) gives rl=11. The mobility of the parallel mechanism F ← G1-G2 given by Eq. (2.223) is M=18-15=3. Three joint parameters have to be known to define the position and the orientation of each link of the parallel mechanism F ← G1-G2. These parameters are the displacements (qi, i=1,2,3) in the two linear and one rotating actuators mounted on the fixed base. Equation (2.225) gives zero degrees of redundancy (T=3-3=0) and Eq. (2.224) indicates that the parallel robot Isoglide4-T3R1 in Fig. 3.63a has nine degrees of overconstraint (N=24-15=9). Six of these overconstraints are in the two planar parallelogram loops. Example 35. Isoglide3-T2R1 in Fig. 3.65 is a slightly modified version of the fully-isotropic planar parallel robot presented in Fig. 3.64 (Gogu, 2004c). The three limbs of this mechanism are interconnected in one complex kinematic chain F. The moving platform is denoted by 8 and the fixed base by 1≡1A≡1B≡1C≡0 (Fig. 3.65a,b). The mechanism has the following structural characteristics: m=15 links including the fixed base, p=18 joints, 4 independent closed loops (q=18-15+1=4) and one complex limb (k=1). The connectivity of mobile platform 8 in Isoglide3-T2R1 with one complex limb in (Fig. 3.65a) is S8F/ 1 = SG1 = 3. Three independent velocities ( v x ,v y and ω z ) exist between mobile platform 8 and reference base

Fig. 3.65. Fully-isotropic planar parallel robot Isoglide3-T2R1 with three interconnected limbs (a), associated graph (b)

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Fig. 3.66. Parallel kinematic chain F ← G1-G2 with distal link 7 associated with fully-isotropic parallel robot Isoglide3-T2R1 with three interconnected limbs (a), limb G1 (b) and limb G2 (c)

3.4 Parallel mechanisms with complex limbs

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Fig. 3.67. CAD model of Isoglide3-T2R1 with interconnected limbs

1≡1A≡1B≡1C≡0 in the mechanism F associated with this parallel robot of the Isoglide family. To calculate the number of joint parameters that lose their independence in the four closed loops, we may consider F ← G1-G2 as a parallel mechanism with two hybrid limbs G1 and G2 connecting reference link 1 and distal link 7 (Fig. 3.66a). As we have seen in chapter 2, the number of joint parameters that lose their independence in the closed loops of a mechanism does not depend of the choice of the distal links. The two hybrid limbs G1 and G2 in Figs. 3.66c,d have the same structural parameters with limbs G1 and G2 in Figs. 3.64c,d analysed in Example 34: rlG1 = 5 and rlG 2 = 6 . Eq. (2.228) gives rl=11 as in the previous example. The number of joint parameters that lose their independence in the parallel mechanism F ← G1-G2 given by Eq. (2.226) is r=7-3+11=15. p The dimension of the joint vector space U is dim(U)= ∑ i =1 fi =18 and

the mobility of the mechanism F in Fig. 3.65a, given by Eq. (2.223) is M=18-15=3. Three joint parameters have to be known to define the position and the orientation of each link of the mechanism F. These parameters are the displacements (qi, i=1,2,3) in the two linear and one rotating actuators mounted on the fixed base. Equation (2.225) gives zero degrees of redundancy (T=3-3=0), and Eq. (2.224) indicates that the parallel robot Isoglide3-T2R1 with interconnected limbs in Fig. 3.65a has nine degrees of overconstraint (N=2415=9). Six of these overconstraints are in the two planar parallelogram loops. The two versions of Isoglide4-T3R1 in Figs. 3.64a and 3.65a have the same structural parameters. A CAD model of Isoglide3-T2R1 with interconnected limbs is presented in Fig. 3.67.

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Example 36. Isoglide4-T3R1 in Fig. 3.68 is a fully-isotropic parallel robot with Schönflies motions (Gogu 2004a, 2007a). The three limbs of this mechanism are interconnected in one complex kinematic chain F. The moving platform is denoted by 9, and the fixed base by 1≡1A≡1B≡1C≡0 (Fig. 3.68a,b). The mechanism has the following structural characteristics: m=16 links including the fixed base, p=19 joints, 4 independent closed loops (q=19-16+1=4) and one complex limb (k=1). The connectivity of mobile platform 8 in the fully-isotropic Isoglide4-T3R1 in Fig. 3.68a is S9F/ 1 = SG1 = 4. Four independent velocities ( v x ,v y ,v z and ω z ) exist between the mobile platform 9 and the reference base in the mechanism F associated with this parallel robot of the Isoglide family. To calculate the number of joint parameters that lose their independence in the four closed loops, we may consider F ← G1-G2 as a parallel mechanism with two hybrid limbs G1 and G2 connecting the reference link 1 with the distal link 8 (Fig. 3.69a). As we have seen in chapter 2 and in the previous example, the number of joint parameters that lose their independence in the closed loops of a mechanism does not depend of the choice of the distal link. The hybrid limb G1 (Figs. 3.69b) has the same structural parameters as limb G1 in Figs. 3.64c and 3.66b analysed in Examples 34 and 35: (RG1)=( v x ,v y ,v z ,ω z ), SG1=4, and rlG1 = 5 . The hybrid limb G2 (Fig. 3.69c) is obtained by concatenating a prismatic joint with two planar parallelogram loops. We simply observe by inspection that (RG2)=( v x ,v y ,v z ,ω z ) and SG2=4.

Fig. 3.68. Fully-isotropic parallel robot Isoglide4-T3R1 with three interconnected limbs (a), associated graph (b)

3.4 Parallel mechanisms with complex limbs

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Fig. 3.69. The parallel kinematic chain F ← G1-G2 with distal link 8 associated with the fully-isotropic parallel robot Isoglide4-T3R1 with three interconnected limbs (a), limb G1 (b) and limb G2 (c)

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3 Structural analysis

Fig. 3.70. The parallel kinematic chain C ← A1-A2-A3 with reference link 7C and distal link 5C associated with the closed loops in limb G2 ← P-C (a), limb A1 (b), limb A2 (c) and limb A3 (d)

To calculate the number of joint parameters that lose their independence in the two closed loops integrated in the hybrid limb G3 ← P-C, we consider the kinematic chain formed by these two closed loops as a parallel kinematic chain C ← A1-A2-A3 with three simple limbs connecting the ternary links 5C and 7C (Fig. 3. 70a). The three simple limbs are: A1 (7C-5C) of type R||R (Fig. 3.70b), A2 (7C-8-9-6C-5C) of type R||R||R||R (Fig. 3.70c) and A3 (7C-2C-3C-4C-5C) of type R||R||R||R (Fig. 3.70d). The three limbs isolated from the parallel kinematic chain C ← A1-A2-A3 (Figs. 3.70b,c,d) have the following bases and dimensions for their operational velocities spaces: (RA1)=( ω z ), (RA2)= ( v x ,v y ,ω z ), (RA3)=( v x ,v y ,ω z ) and SA1=1, SA2= SA3=3.

3.4 Parallel mechanisms with complex limbs

217

Fig. 3.71. CAD model of Isoglide4-T3R1 with interconnected limbs

The connectivity of link 5C with respect to link 7C in the parallel kineC matic chain C ← A1-A2-A3 is S5C (Fig. / 7 C = dim( RA1 ∩ RA2 ∩ RA3 ) = 1 3.62a). Just one rotation about the z-axis is possible between the two links. The number of joint parameters that lose their independence in the parallel kinematic chain C ← A1-A2-A3 is rC=7-1=6. This value is given by Eq. (2. 191) by taking into account that in the parallel kinematic chain C ← A1-A2A3 we have considered 7C as the reference link, and 5C as the distal link. We thus obtain rlG 2 = rC = 6 . The number of joint parameters that lose their independence in the parallel mechanism F ← G1-G2 given by Eq. (2.226) is r=8-4+11=15. We have taken into account that the number of joint parameters that lose their independence in the two limbs of the parallel mechanism F ← G1-G2, given by Eq. (2.228) is rl = rlG1 + rlG 2 = 11 . The dimension of the joint vector space U is dim(U)= ∑ i =1 fi =19, and p

the mobility of the mechanism F in Fig. 3.68a, given by Eq. (2.223) is M=19-15=4. Four joint parameters have to be known to define the position and the orientation of each link of the mechanism F. These parameters are the displacements (qi, i=1,2,3) of the three linear actuators mounted on the fixed base, and one rotating actuator mounted on link 2C in the proximity of the fixed base. Equation (2.225) gives zero degrees of redundancy (T=4-4=0) and Eq. (2.224) indicates that the parallel robot Isoglide4-T3R1 with three interconnected limbs in Fig. 3.68a has nine degrees of overconstraint (N=2415=9). Six of these overconstraints are in the two planar parallelogram loops. Figure 3.71 presents the CAD model of this solution.

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3 Structural analysis

Example 37. Isoglide4-T3R1 in Fig. 3.72 represents the first solution of a fully-isotropic parallel robot with Schönflies motions presented in the literature (Gogu 2004a). The limbs of this mechanism are interconnected in one complex kinematic chain F. The moving platform is denoted by 10, and the fixed base by 1≡1A≡1B≡1C≡0 (Fig. 3.72a,b). The mechanism has the following structural characteristics: m=17 links including the fixed base, p=21 joints, 5 independent closed loops (q=21-17+1=5) and one complex limb (k=1). The connectivity of the mobile platform 10 in the fully-isotropic Isoglide4-T3R1 in Fig. 3.72a is S10F / 1 = SG1 = 4. Four independent velocities ( v x ,v y ,v z and ω z ) exist between mobile platform 10 and the reference base in the mechanism F associated with this parallel robot of the Isoglide family. To calculate the number of joint parameters that lose their independence in the five closed loops, we may consider F ← G1-G2 as a parallel mechanism with two complex limbs G1 and G2 connecting reference link 1 with distal link 9 (Fig. 3.73a). As we have seen in chapter 2 and in the previous examples, the number of joint parameters that lose their independence in the closed loops of a mechanism does not depend of the choice of the distal link. The hybrid limb G1 (Figs. 3.73b) has the same structural parameters as the limb G1 in Figs. 3.64c, 3.66b and 3.69b analysed in Examples 34-36: (RG1)=( v x ,v y ,v z ,ω z ), SG1=4, and rlG1 = 5 . Three closed loops are combined in complex limb G2 (Fig. 3.73c). We simply observe by inspection that (RG2)=( v x ,v y ,v z ,ω z ) and SG2=4.

Fig. 3.72. The fully-isotropic parallel robot Isoglide4-T3R1 with three interconnected limbs and the actuators mounted on the fixed base (a), associated graph (b)

3.4 Parallel mechanisms with complex limbs

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Fig. 3.73. The parallel kinematic chain F ← G1-G2 with distal link 8 associated with the fully-isotropic parallel robot Isoglide4-T3R1 with three interconnected limbs and actuators on the fixed base (a), limb G1 (b) and limb G2 (c)

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3 Structural analysis

Fig. 3.74. Parallel kinematic chain D ← A1-A2-A3-E1 with reference link 8C and distal link 6C associated with the closed loops in limb G2 (a), limb A1 (b), limb A2 (c) and limb E1 (d)

3.4 Parallel mechanisms with complex limbs

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To calculate the number of joint parameters that lose their independence in the three closed loops integrated in the complex limb G3, we consider the kinematic chain formed by these three closed loops as a parallel kinematic chain D ← A1-A2-E1 with two simple limbs (A1 and A2) and one complex limb E1 connecting the ternary links 6C and 8C (Fig. 3. 74a). The two simple limbs are: A1 (8C-6C) of type R||R (Fig. 3.74b) and A2 (8C-9-10-7C6C) of type R||R||R||R (Fig. 3.74c). Link 6C is connected to link 8C by the complex limb E1 (Fig. 3.74d) including a closed loop B (1-2C-3C-4C-1). The three limbs isolated from the parallel kinematic chain D ← A1-A2-E1 (Figs. 3.74b,c,d) have the following bases and dimensions for their operational velocities spaces: (RA1)=( ω z ), (RA2)= ( v x ,v y ,ω z ), (RE1)=( v x ,v y ,ω z ) and SA1=1, SA2= SE1=3. We observe that the operational velocity space of relative velocities between links 6C and 8C in the complex limb E1 is not influenced by closed loop B. The three independent velocities in (RE1) are in fact given by the three revolute pairs with parallel axes outside the closed loop B. This closed loop B ← A1B − A2B (Fig. 3.75a) is obtained by putting together the distal links of two simple chains A1B and A2B of type P||R and R||P respec-

tively (Fig. 3.75b,c). Chains A1B and A2B have (RA1B)=(RA1B)=( v z ,ω z ) and SAIB=SA2B= 2. Two joint parameters lose their independence in the closed loop B (rB=2+2-2=2). The connectivity of link 6C with respect to link 8C in the parallel kinematic chain D ← A1-A2-E1 is S6DC / 8C = dim( RA1 ∩ RA2 ∩ RE1 ) = 1 (Fig. 3.73a). Just one rotation about the z-axis is possible between the two links. The number of joint parameters that lose their independence in the parallel kinematic chain D ← A1-A2-E1 is rD=7-1+2=8. This value is given by Eq. (2. 191) by taking by taking into account that in the parallel kinematic chain D ← A1-A2-E1 we have considered 8C as the reference link, and 6C as the distal link. We thus obtain rlG 2 = rD = 8 . The number of joint parameters that lose their independence in the parallel mechanism F ← G1-G2 given by Eq. (2.226) is r=8-4+13=17. We have taken into account that the number of joint parameters that lose their indpendence in the two limbs of parallel mechanism F ← G1-G2, given by Eq. (2.228) is rl = rlG1 + rlG 2 = 13 . The dimension of the joint vector space U is dim(U)= ∑ i =1 fi =21 and p

the mobility of the mechanism F in Fig. 3.68a, given by Eq. (2.223) is M=21-17=4. Four joint parameters have to be known to define the position and the orientation of each link of the mechanism F. These parameters

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3 Structural analysis

Fig. 3.75. Closed loop B ← A1B − A2B integrated in limb G2 (a) and its associated simple open kinematic chains A1B (b) and A2B (c)

are the displacements (qi, i=1,2,3,4) in the three linear and one rotating actuators mounted on the fixed base. Equation (2.225) gives zero degrees of redundancy (T=4-4=0) and Eq. (2.224) indicates that the parallel robot Isoglide4-T3R1 with three interconnected limbs and actuators on the fixed base in Fig. 3.72a has nine degrees of overconstraint (N=30-17=13). Six of these overconstraints are in the two planar parallelogram loops, and four in the closed loop B integrated in limb G2. Example 38. I4R in Fig. 3.76 is an innovative solution for a parallel robot with Schönflies motions for high-speed pick-and-place operations, developed by LIRMM - Montpellier Laboratory of Computer Science, Robotics, and Microelectronics (Krut et al. 2004b, Company et al. 2006). The limbs of this mechanism are interconnected in one complex kinematic chain F ← G1. The robot uses a two-part moving platform 5 and 5’ connected by a prismatic joint (Fig. 3.77).

3.4 Parallel mechanisms with complex limbs

(a)

223

(b)

(c)

Fig. 3.76. The I4R parallel robot developed by Montpellier Laboratory of Computer Science, Robotics, and Microelectronics (a) and CAD models (b) and (c) (courtesy LIRMM)

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Fig. 3.77. I4R parallel robot: kinematic structure (a) and associated graph (b)

3.4 Parallel mechanisms with complex limbs

225

Fig. 3.78. Pulley-cable system used in I4R parallel robot as technological solution for rolling without sliding joint (courtesy LIRMM)

The relative translation between the two parts of the moving platform is transformed in rotational motion of an end-effector 6 by a rolling-withoutsliding joint denoted by Rw. (Fig. 3.77). This solution gives a high tilting angle of the end-effector connected to haf-platform 5 by a revolute joint and to half-platform 5’ by the rolling-without-sliding joint. A pulley-cable system is used as a technological solution for the rolling without sliding joint (Fig. 3.78) Each half-platform is connected to two complex limbs RPa4S-type (Fig. 3.77). The complex mechanism associated with the I4R robot may be considered to be a parallel mechanism D ← A1-A2-E1 connecting a reference link 5 and a distal link 5’ (Fig. 3.79). This mechanism is obtained by kinematic inversion with respect to link 5. This inversion conserves the structural parameters of the mechanism. It is equivalent to positioning the “observer” on link 5 instead on the fixed base 1≡1A≡1B≡1C≡1D≡0. The parallel mechanism D ← A1-A2-E1 has one complex and two simple limbs. The two simple limbs are: A1 (5-6-5’) of type R||Rw (Fig. 3.80a) and A2 (55’) reduced to a prismatic joint (Fig. 3.80b). The complex limb E1 combines 6 closed loops (Fig. 3.80c). In Example 31, we have demonstrated that 38 joint parameters lose their independence in the closed loops combined in limbs G1 and G2 of the parallel robot H4. These closed loops are structurally identical with those of the complex limb E1 of I4R. Subsequently, rl= rl E1 =38 for the I4R parallel robot. The three limbs isolated from the parallel mechanism D ← A1-A2-E1 (Figs. 3.80) have the following bases and dimensions for their operational velocities spaces: (RA1)=( v x ,ω z ), (RA2)= ( v x ), (RE1)=( v x ,v y ,v z ) and SA1=2, SA2= 1, SE3=3. The connectivity of link 5’ with respect to link 5 in the parallel mechanism D ← A1-A2-E1 (Fig. 3.79) is S 5'/D 5 = dim( RA1 ∩ RA2 ∩ RE1 ) = 1 . This

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3 Structural analysis

Fig. 3.79. Parallel mechanism D ← A1-A2-E1 with reference link 5 and distal link 5’ associated with parallel robot I4R by kinematic inversion with respect to link 5 (a) and its associated graph (b)

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Fig. 3.80. Limbs isolated from parallel mechanism D ← A1-A2-E1 associated with parallel robot I4R by kinematic inversion with respect to link 5: simple limbs A1 (a) and A2 (b) and complex limb E1(c)

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3 Structural analysis

result holds because just one translation about the x-axis is possible between the two half-platforms 5 and 5’. The number of joint parameters that lose their independence in the parallel kinematic chain D ← A1-A2-E1 given by Eq. (2. 200) is r=rD=6-1+38=43. p The dimension of the joint vector space U is dim(U)= ∑ i =1 fi =55 and the mobility of the mechanism associated with I4R parallel robot given by Eq. (2.223) is M=55-43=12. Twelve joint parameters have to be known to define the position and the orientation of each link of the parallel mechanism I4R. Equation (2.225) gives eight degrees of redundancy (T=124=8). As for the H4 robot analysed in Example 31, each parallelogram loop Pa4S-type contains two internal mobilities introduced by the rotation of the links 3 and 4 of each limb about a rotation axis passing through the centre of the two spherical joints adjacent to links 3 and 4. These eight degrees of redundancy represent internal mobilities in the limbs that cannot be transmitted from the input to the output of the parallel mechanism. The remaining four degrees of mobility are external mobilities that are transmitted from the input to the output of the parallel mechanism. They represent the independent parameters that are necessary to define the position and the orientation of the end-effector 6. These parameters are the angular displacements of the four rotating actuators mounted on the fixed base. Equation (2.224) indicates that the I4R parallel robot has five degrees of overconstraint (N=48-43=5).

3.5 Other multi-loop kinematic chains The structural formulae demonstrated in chapter 2 for parallel robots may also be applied to various multi-loop mechanisms that do not obey to the classical CGK formulae. The three following examples illustrate the application of the new formulae presented in chapter 2 to complex mechanisms that may be considered to be parallel mechanisms with simple limbs. Example 39. The mechanism proposed by De Roberval (1670) may be considered as a parallel mechanism F ← G1-G2-G3 (Fig. 3.81) obtained by the concatenation of two planar parallelogram loops. The six revolute joints have parallel axes. The CGK formulae give zero degrees of mobility for this mechanism, which is clearly wrong.

3.5 Other multi-loop kinematic chains

229

Fig. 3.81. The De Roberval mechanism (a), associated graph (b), simple open chains associated with limbs G1 (c), G2 (d) and G3 (e) isolated from the mechanism

The ternary links 1 and 3 may be considered as the fixed and moving platforms. They are connected by three simple limbs R||R-type: G1(1A-2A3A), G2(1B-2B-3B) and G3(1C-2C-3C). The mechanism has the following structural characteristics: m=5 links including the fixed base, p=6 revolute joints, 2 structurally independent closed loops (q=6-5+1=2) and three simple limbs (k=3). We may observe by inspection that the three simple limbs have the following bases for their operational spaces: (RG1)=( v x ,v y ) (RG2)=( v x ,ω x ) or (RG2)=( v y ,ω x ) and (RG3)=( v x ,v y ) – see Fig. 3.81c-e. The dimensions of these operational vector spaces give the connectivity of the kinematic chains associated with each limb Gi of the parallel mechanism: SGi=2, i=1,2,3. We have considered the characteristic point H belonging to the moving platform, point O0 as the reference point, and the velocities of H are expressed in the coordinate system O0x0y0z0. In accordance with remark 2 and Eq. (2.197), the point H and the bases of RGi (i=1,2,3) are selected such that the minimum value of S3F/ 1 is obtained. By this choice, the connectivity of the mobile platform 3 in the parallel mechanism F, given by Eq. (2.227), is SF= S3F/ 1 = dim( RG1 ∩ RG 2 ∩ RG3 ) =1. Only one relative independent velocity v x or v y exists between the mobile platform 3 and the reference link 1≡0. The dimension of the joint

230

3 Structural analysis

Fig. 3.82. Parallel mechanism equivalent to De Roberval mechanism by kinematic inversion with respect to link 1 (a), associated graph (b)

vector space U is dim(U)=

∑

p

i =1

fi =6. The number of joint parameters that

lose their independence in De Roberval’s mechanism, given by Eq.(2.226) is r=6-1=5. The mobility of the mechanism, given by Eq. (2.223), is M=6-5=1. Equation (2.224) gives N=7 and T=0. De Roberval mechanism in Fig. 3.81 is non redundant with seven degrees of overconstraint. It is not necessary to treat this mechanism as a special mechanism with passive freedoms or mechanisms with over closed loops. No special equations are necessary to calculate the mobility of this mechanism; it obeys the general formulae proposed in Chapter 2 for calculating the degree of mobility of parallel mechanisms with simple limbs. The same results are obtained for the De Roberval mechanism in Fig. 3.82. This may be transformed into the mechanism in Fig. 3.81 by kinematic inversion with respect to link 1. As we have seen in chapter 2 kinematic inversion does not alter the mobility. Example 40. The mechanism F ← G1-G2 in Fig. 3.83 is obtained by interconnecting two De Roberval mechanisms. The eight revolute joints have parallel axes. The CGK formula gives a negative mobility for this mechanism, which is wrong. The ternary links 1 and 3 may be considered as the fixed and moving platforms. They are connected by one simple limb G1(1A-2A-3A) of type R||R, and a complex limb G2 analysed in Example 38. The mechanism has the following structural characteristics: m=6 links including the fixed base, p=8 revolute joints, 3 structurally independent closed loops (q=8-6+1=3) and two limbs (k=2). We may simply observe by inspection that the two limbs have the following bases and dimensions of their operational spaces: (RG1)=( v x ,v y ), (RG2)=( v x ) or (RG2)=( v y ) and SG1=2, SG2=1– see Fig. 3.83c, d. We have considered the characteristic point H belonging to the moving platform, point O0 as the reference point, and the velocities of H are

3.5 Other multi-loop kinematic chains

231

Fig. 3.83. Complex mechanism with one simple and one complex limb F ← G1-G2 (a), associated graph (b), kinematic chains associated with limbs G1 (c) and G2 (d) isolated from the mechanism

expressed in the coordinate system O0x0y0z0. In accordance with remark 2 and Eq. (2.197), point H and the bases of RGi (i=1,2,3) are selected so that the minimum value of S3F/ 1 is obtained. By this choice, the connectivity of the mobile platform 3 in the parallel mechanism F, given by Eq. (2.227), is SF= S3F/ 1 = dim( RG1 ∩ RG 2 ) =1. Only one relative independent velocity v x or v y exists between the mobile platform 3 and the reference link 1≡0. As we have seen in Example 38, five joint parameters lose their independence in the two closed loops in limb G2 (rl=5). The dimension of the joint vector p space U is dim(U)= ∑ i =1 fi =8. The number of joint parameters that lose their independence in the mechanism in Fig. 3.83 given by Eq.(2.226) is r=3-1+5=7. The mobility of the mechanism given by Eq. (2.223), is M=8-7=1. Equation (2.224) gives N=11 and T=0. The mechanism in Fig. 3.83 is non redundant with 11 degrees of overconstraint. Example 41. The complex mechanism presented in Fig. 3.84a can also be considered a parallel mechanism F ← G1-G2-G3 with three simple limbs,

232

3 Structural analysis

in which ternary links 1 and 4 are the reference and the “distal” links. They are connected by three simple limbs (Fig. 3.84a,b): G1(1A-2A-3A-4A) of type R||R||R , G2(1B-2B) reduced to a single revolute joint and G3(1C-2C-3C-4C) of type C||C||R. All revolute (R) and cylindrical (C) joints have parallel axes. The mechanism has the following structural characteristics: m=6 links including the fixed base, p=7 joints, 2 structurally independent closed loops (q=7-6+1=2) and three simple limbs (k=3). The CGK formulae give different results (M=1 or M=2) depending on the choice of the two structurally independent loops. If the closed loops (1-2A-3A-4-3C-2C-1) and (1-43C-2C-1) are selected, these formulae give the erroneous result M=1; the other two combination of two closed loops give M=2. Theoretically, any two loops of the graph presented in Fig. 3.84d can be considered as structurally independent. In fact, the two loops can be independent from a structural point of view, in the sense of graph theory, but they are not always independent from a kinematic point of view, as we have shown in chapter 2. This ambiguity can be avoided by using the new formulae. By inspection, we can see that the operational spaces RGi (i=1,2,3) of the simple open kinematic chains associated with the three limbs isolated from the mechanism (Fig. 3.84c-e) have the following bases and dimensions: (RG1)=( v x ,v y ,ω z ), (RG2)=( ω z ), (RG3)=( v x ,v y ,v z ,ω z ) and SG1=3, SG2=1,

Fig. 3.84. The complex mechanism F ← G1-G2-G3 with three simple limbs (a), associated graph (b), simple open chains associated with limbs G1 (c), G2 (d) and G3 (e) isolated from the mechanism

3.5 Other multi-loop kinematic chains

233

SG3=4. We have considered the characteristic point H situated on link 4 on the axis of the revolute joint connecting this link to the fixed base. The connectivity of link 4 in the parallel mechanism F ← G1-G2-G3 is S4F/ 1 = dim( RA1 ∩ RA2 ∩ RA3 ) = 1 . Only one relative independent velocity ( ω z ) exists between links 4 and 1. The dimension of the joint vector space

U is dim(U)= ∑ i =1 fi =9. The number of joint parameters that lose their inp

dependence in this complex mechanism is r=8-1=7 and the correct value of the mobility is given by Eq. (2.223) is M=9-7=2. The mechanism has one degree of redundancy (T=2-1=1) and five degrees of overconstraint (N=12-7=5). We can see that this mechanism also obeys Eq. (2.223), and the mobility is determined without any ambiguity. Example 42. The complex mechanism presented in Fig. 3.85a was proposed by Baker (1980b) and analysed by Davies (1981) and Fayet (1995a). It may be considered a parallel mechanism F ← G1-G2-G3 with three simple limbs, in which ternary links 1 and 4 are the reference and the distal links respectively. They are connected by three simple limbs (Fig. 3.85a): G1(12-4) of type C-S, G2(1-4) reduced to a planar joint Pn and G3(1-3-4) of

Fig. 3.85. Complex mechanism F ← G1-G2-G3 with three simple proposed by Baker (1980b) and analysed by Davies (1981) and Fayet (1995a) - (a), simple open chains associated with limbs G1 (b), G2 (c) and G3 (d) isolated from the mechanism

234

3 Structural analysis

type C-S. The mechanism has the following structural characteristics: m=4 links including the reference link, p=5 joints, 2 structurally independent closed loops (q=5-4+1=2) and three simple limbs (k=3). By inspection, we can see that the operational spaces RGi (i=1,2,3) of the simple open kinematic chains associated with the three limbs isolated from the mechanism (Fig. 3.85b-d) have the following bases and dimensions: (RG1)=( v y ,v z ,ω x ,ω y ,ω z ), (RG2)=( v x ,v y ,ω z ), (RG3)=( v y ,ω x ,ω y ,ω z ) and SG1=5, SG2=3, SG3=4. In accordance with remark 2 and Eq. (2.197), point H and the bases of RGi (i=1,2,3) have been selected such that the minimum value of S4F/ 1 to be obtained. By this choice, the connectivity of the mobile platform 3 in the parallel mechanism F, given by Eq. (2.227), is SF= S4F/ 1 = dim( RG1 ∩ RG 2 ∩ RG3 ) =2. Two independent velocity v y and ω z exist between the mobile platform 4 and the reference link 1. The dimenp sion of the joint vector space U is dim(U)= ∑ i =1 fi =13. The number of joint parameters that lose their independence in the mechanism in Fig. 3.85 given by Eq.(2.226) is r=12-2=10. The mobility of the mechanism given by Eq. (2.223), is M=13-10=3. Equation (2.224) gives N=2 and T=1. The mechanism in Fig. 3.85 has two degrees of overconstraint and one degree of redundancy. One internal mobility exists in limb G3 – the rotation of the link 3 that cannot be transmitted to link 4. The remaining two degrees of mobility are external mobilities that have to be actuated to obtain the translation and the rotation of distal link 4. Translation about the y-axis may be actuated in one of the three joints connected to the fixed base. Rotation about the z-axis may be actuated by a circular translation in the planar joint.

4 Kinematic analysis

Kinematics of parallel robots is very important for structural synthesis. We have seen in chapter 2 that the structural parameters of parallel robots are mainly kinematically defined. The duality between velocity and force allows us to give alternative definitions of structural parameters based on statics. The author has preferred the kinematic definition, considering that the kinematic nature of the structural parameters facilitates the understanding of their meaning. Apparently, we are more skilled in imagining the motion than imaging the static or dynamic forces. In this chapter, we will introduce the main models and performance indices used in parallel robots. In general, to study robot motions, geometric and kinematic models are used. We will present the general approach used to set up these models and we emphasize the Jacobian matrix which is the main issue in defining robot kinematics, singularities and performance indices. We saw in chapter 2 that the Jacobian is the matrix of the linear transformation between the velocity joint space and the operational space. This matrix contains very important information concerning robot behavior from a kinematic and static (kinetostatic) point of view. The Jacobian will be used to define and to characterize motion decoupling in parallel robots. In modern design theories, decoupling the functional requirements has become a basic design principle (Suh 1990). In robotics, this principle leads to significant advantages in design, modeling, motion planning and control (Vukobratovic and Kircanski 1983; Konstantinov et al. 1987; Patarinski et al. 1991; Patarinski and Uichiyama 1993; Youcef-Toumi 1992; Galabov et al. 1999). In many robotic applications the possibility of decoupling end-effector motion becomes a cornerstone criterion in the choice of the robot mechanism. The problem is that in the early stage of parallel robot design the Jacobian matrix is not yet known. For structural synthesis of parallel robots we use an alternative method to define motion decoupling and singularities, which does not need prior knowledge of the Jacobian matrix. This method is based on the main structural parameters introduced in chapter 2 via the theory of linear transformation.

236

4 Kinematic analysis

4.1. Decoupling in axiomatic design Axiomatic design is a new design theory developed by Suh (1990) to establish a scientific foundation for the design field, and provide a fundamental basis for the creation of products, processes, systems, software and organizations. This is a significant departure from the conventional design processes largely dominated by empiricism and intuition. Axiomatic design considers that the design objective is always stated in the functional domain, whereas the physical solution is always generated in the physical domain. These two domains are in permanent interconnection at every hierarchical level of the design process. To satisfy the design objectives defined in terms of functional requirements (FRs), a physical embodiment defined in terms of design parameters (DPs) must be created. In this way, the design is formally defined in axiomatic design as the creation of synthesized solutions in the form of products, processes or systems that satisfy the perceived needs through the mapping between the functional requirements in the functional domain and the design parameters of the physical domain, through the proper selection of DPs that satisfy FRs. This mapping process in not unique, and the actual outcome depends on the designer’s individual creative process. Therefore, there can be a large number of plausible design solutions and mapping techniques. The design axioms, theorems and corollaries formulated in axiomatic design provide the principles that the mapping techniques must satisfy to produce a good design, and offer a basis for comparing and selecting design solutions. Axiomatic design is based on two axioms that govern good design: the independence axiom and the information axiom. The independence axiom states that “in an acceptable design, the DPs and the FRs are related in such a way that a specific DP can be adjusted to satisfy its corresponding FR without affecting other functional requirements”. The information axiom states that “the best design is a functionally uncoupled design that has minimum information content” – see page 48 of “Principles of Design” by Suh (1990). We can see that decoupling is the keyword of both axioms. The independence axiom is mathematically represented by the design equation {FR}=[D]{DP}

(4.1)

where {FR} is the set of functional requirements, {DP} is the set of design parameters and [D] is a m × n matrix called the design matrix. The elements of this matrix are denoted by aij (i=1,2,…n and j=1,2,…,m). The left-hand side of the design equation represents “what we want in terms of design goals” and the right-hand side represents “how we hope to satisfy

4.1. Decoupling in axiomatic design

237

these goals in terms of a design solution”. We mention that in (Suh 1990), {FR} and {DP} are called “vectors”. In this work we call them “sets” because they may not always satisfy the mathematical definition of a vector. A design that can be represented by Eq. (4.1) with [D] being a diagonal matrix, i.e. aij=0 unless i=j, satisfies the first axiom since the independence of FRs is assured when each DP is changed. That is, FRi can be satisfied independently by simply changing the corresponding DPi without affecting any other FR. Such a design is called an uncoupled design. If the design matrix is triangular the design can be decoupled and is called a decoupled or quasi-coupled design. A truly coupled design is one in which the design matrix is neither diagonal nor triangular. Two general quantitative measures are proposed in (Suh 1990) for functional independence: reangularity R and semangularity S . They are the following functions of the design matrix elements aij: 2  2  n  a a   ∑ ki kj   k =1     R = ∏ 1 − n n    i =1,n −1 aki2  ∑ akj2   j =1+ i ,n  ∑    k =1  k =1   

1

and

  n  a jj S = ∏ 1 j =1   n 2 2   ∑ akj    k =1 

   .   

(4.2)

(4.3)

Reangularity measures the degree of orthogonality between the DPs; . the maximum value of the reangularity occurs when the DPs are orthogonal. As the degree of coupling increases, so the value of R decreases. The sums in the denominator of Eq. (4.2) are the squares of the magnitudes of the columns in the design matrix. These terms are used to normalize the columns of the matrix. When the columns are already normalized, the sums are equal to one. Normalization is necessary in order to make the measure insensitive to changes in the scale of any of the DPs. The sum in the numerator is the dot product of the ith and jth columns of the design matrix. This sum, when properly normalized by the denominator, can be thought of as the cosine of the angle between the vectors in the ith and jth columns of the design matrix. Since this cosine term is squared and sub-

238

4 Kinematic analysis

tracted from unity, the argument of the product operator is the square of the sine of the same angle. Semangularity characterizes the alignment of DPs with their corresponding FRs. The denominator of Eq. (4.3) is a normalizing factor that is equal to unity if the columns of the design matrix are normalized. When the design matrix is normalized, semangularity is simply the product of the absolute values of the diagonal elements of the design matrix. If the design matrix is properly normalized, then the diagonals are unity when all offdiagonal elements are zero. When the semangularity is unity, the DPs are aligned with the FRs. An uncoupled design has R =1 and S =1. By analogy with the three types of design, in the next section we will define parallel robots with coupled, decoupled and uncoupled motions by considering the Jacobian as the design matrix.

4.2. Geometric modeling In general in robot analysis, geometric modeling is set up in terms of finite motions; it is also called geometric analysis or pose analysis. Definition 4.1 (Geometric problem). The geometric problem in robot analysis involves setting up the implicit relation between the finite motions of the actuated joints and the moving platform. Usually, the geometric model is expressed in terms of linear and/or angular finite displacements of the actuated joints; moving platform pose is defined by its spatial position and orientation with respect to the reference frame. This model can be direct or inverse. In the direct geometric model the inputs are the finite displacement of the actuated joints, and the output is the pose of the moving platform. In the inverse geometric model the input is the pose of the moving platform and the outputs are the finite displacements of the actuated joints. Let us consider the general case of a parallel robot F ← G1-G2-…Gk in which the moving platform n≡nGi is connected to the reference link 1≡1Gi by k limbs Gi (1Gi-2Gi-…nGi), i=1,2,…k. The robot has mobility M, and the connectivity of the moving platform is SF. The kinematic chains associated with limbs Gi can be simple or complex, and each limb Gi could integrate gi linear and/or rotation actuators. We have seen in chapter 2 that the total number of actuators is equal to the mobility

∑g k

i =1

i

=M .

(4.4)

4.2. Geometric modeling

239

The geometric constraints imposed by the limbs on the motion of the moving platform give, for the ith limb, the following set of gi general relations fij ( qij ,w1 ,w2 ,...,w6 ) = 0 , i=1,2,…,k ;

j=1,2,…,gi.

(4.5)

where qij (i=1,2,…,k ; j=1,2,…,gi) represent the linear or angular displacements in the actuated joints and w1, w2, …, w6 represent the parameters used to describe the spatial position and orientation of the moving platform; these are called the general coordinates of the moving platform. We note that various parameters can be used to represent the spatial pose of the moving platform: quaternions, biquaternions, homogeneous operators, rotation matrices, complex matrices, etc. More details on these parameters are presented in the book (Gogu and Coiffet 1996). In general, relations (4.5) can be obtained by expressing the vector- loop equation of each limb in the fixed coordinate frame, and eliminating the variables of the passive joints. Setting up analytical closed-form relations (4.5) is possible when an analytical inverse geometric model is known for each limb. In the general case, setting up analytical closed-form relations (4.5) may not always be possible for all limb architectures. However, in parallel robots, chains Gi are usually very simple, and such relations can be easily obtained. A very common way to set up closed-form relations (4.5) is to start from the equalities

( xAi − xBi )

2

+ ( y Ai − yBi ) + ( z Ai − z Bi ) = hij ( qij ,w1 ,w2 ,...,w6 ) 2

2

(4.6)

written for each limb Gi (i=1,2,…,M). The distance constraints can be solved algebraically or geometrically (Thomas et al. 2004). The left-hand of Eq. (4.6) represents the square of the distance AiBi between the extreme points of each limb (Fig. 4.1). Points Ai are situated on the fixed base and have coordinates xAi, yAi and zAi. Points Bi are situated on the moving platform and have coordinates xBi, yBi and zBi. The coordinates of point Ai are the components of position vector O0Ai expressed in the fixed reference frame O0 x0 y0 z0 . The coordinates of point Bi are the components of vector O0Bi also expressed in the fixed reference frame by O0Bi=O0On+R0nOnBi

(4.7)

where OnBi is the position vector of point Bi in the mobile frame Onxnynzn attached to the moving platform, O0On is the position vector of point On in the fixed reference frame O0 x0 y0 z0 , and R0n is the 3 × 3 rotation matrix

240

4 Kinematic analysis

giving the relative orientation of the frame On xn yn zn with respect to O0 x0 y0 z0 . Vectors O0Ai and OnBi are constant and known for a given geometry of the fixed and moving platforms. In the general case, vector O0On depends on three generalized coordinates, these define the position of the characteristic point of a moving platform. The rotation matrix R0n depends on three generalized coordinates used to define the orientation of the moving platform. In this way, the coordinates of point Bi in the fixed reference frame depend on six general coordinates. The right-hand of Eq. (4.6) is a function depending on at least qij. Usually, in the parallel robots with telescopic limbs this function depends only on qij, that is hij=hij(qij). For example in Gough-type parallel robots gi=1 (i=1,2,…,6) and hij= qi12 . In other cases, this function could also depend on up to six general coordinates. Eqs. (4.5) could be used to solve the direct and inverse geometric problem of parallel manipulators. Definition 4.2 (Direct geometric problem). The direct geometric problem involves the calculation of the general coordinates w1, w2, …, w6 when the actuated joint parameters qij (i=1,2,…,k ; j=1,2,…,gi) are known in Eq. (4.5). Definition 4.3 (Inverse geometric problem). The inverse geometric problem involves the calculation of the actuated joint parameters qij (i=1,2,…,k ; j=1,2,…,gi) when the general coordinates w1, w2, …, w6 of the moving platform are known in Eq. (4.5).

Fig. 4.1. General case of a parallel robot with k limbs

4.3. Kinematic modeling

241

We note that contrary to the situation in serial robots, the direct geometric problem is generally more difficult than the inverse one. The inverse geometric problem can be easily solved analytically by using Eqs. (4.6). The direct geometric problem is usually solved numerically. Various methods can be used: elimination, homotopy, Gröbner basis, interval analysis or other ad-hoc methods (Merlet 2005). We will see in the examples presented at the end of this chapter that parallel robots with decoupled motions can have analytical solutions for both inverse and direct problems obtained in a very straightforward manner. This is an important advantage both for kinematic and dynamic modelling and path planning and control.

4.3 Kinematic modeling In general in robot analysis, kinematic modeling is set up in terms of infinitesimal motions; it is also called differential motion analysis. Definition 4.4 (Kinematic problem). The kinematic problem in robot analysis involves setting up the implicit relations between the infinitesimal motions of the actuated joints and the moving platform. Usually, the kinematic models are expressed in term of robot velocity and eventually robot acceleration. The kinematic model can also be direct and inverse. In the direct kinematic model the input is the velocity of the actuated joints and the output the velocity of the moving platform. In the inverse kinematic model the input is the velocity of the moving platform and the outputs are the velocities of the actuated joints. In robot kinematics, the linear mapping between the velocity of the actuated joints and moving platform is set up by using the Jacobian matrix. Various methods can be used to calculate the Jacobian of a parallel robot: velocity vectorloop equations (Mohamed et al. 1983, Khalil and Dombre 2002), theory of reciprocal screws (Mohamed and Duffy 1985), motor algebra (Sugimoto 1987), Grassmann geometry (Merlet 1989), Plücker line coordinates (Hunt et al. 1991), etc. Various types of Jacobian matrices are known: conventional Jacobian (Tsai 1999), screw-based Jacobian (Tsai 1998), and overall Jacobian (Joshi and Tsai 2002). The screw-based Jacobian is developed by using the concept of reciprocal screws introduced by Ball (1900). The screw-based Jacobian becomes the conventional Jacobian after interchanging the first three columns with the last three columns in the parallel Jacobian. The overall Jacobian or full Jacobian (Merlet 2005) was proposed by Joshi and Tsai (2002) to provide information about both architecture and constraint singularities of parallel robots with less than six degrees of freedom. In this section, we introduce the design and conventional Jacobian and we emphasize the difference between these two Jacobian matrices.

242

4 Kinematic analysis

The design Jacobian is useful for defining and understanding: (i) the mechanical nature of motion decoupling in parallel robots, (ii) the nature of conventional Jacobian matrix, and (iii) the formulation nature of certain singularities in parallel manipulators. 4.3.1 Direct and inverse kinematics matrices used in Jacobian calculation By differentiating Eq. (4.5) with respect to time, we obtain

∂f ij

∂qij or

qij +

∂f ij

∂qij

∂f ij

∂w1

qij = −

w1 +

∂f ij

∂w1

∂f ij

∂w2

w1 −

w2 + ... +

∂f ij

∂w2

∂f ij

∂w6

w2 − ... −

w6 = 0 ,

∂f ij

∂w6

(4.8)

(4.9)

w6 ,

j=1,2,…,gi.

i=1,2,…,k;

In matrix form, Eq. (4.9) can be written as follows where q =  q11

(4.10)

Js q =Jp w

ity vector of actuated joints, w = [ w1 of the moving platform,

... q1g1

q21

 ∂f11  ∂q  11   0 Js =     0  

... q2 g2

0 ∂f12 ∂q12

0

is the direct kinematic matrix and

w2

... qkgk  is the velocT

... w6 ] is the velocity vector

... qk 1

 0    0  ,   ∂f kgk  ∂qkgk 

T

(4.11)

4.3. Kinematic modeling

 ∂f11  ∂w  1  ∂f12  J p = −  ∂w1    ∂f kgk  ∂w  1

∂f11 ∂w2 ∂f12 ∂w2

∂f kgk ∂w2

∂f11  ∂w6   ∂f12  ∂w6    ∂f kgk  ∂w6 

243

(4.12)

is the parallel kinematic matrix. In the literature, Js and Jp are also named serial and parallel Jacobian matrices (Wanger and Chablat 2000) by analogy with the corresponding matrix of Eq. (4.8) written in matrix form. The latter is called Jacobian, although this is not correct from a strictly mathematical point of view. Matrices Js and Jp were used by Gosselin and Angeles (1990) for singularity analysis and were also called direct kinematics and inverse kinematics Jacobians in (Tsai, 1999). We use the names of direct and inverse kinematics matrices coined by Chablat and Wenger (1998). Eq. (4.10) represents the general kinematic model of parallel robots in which Js is a M × M matrix and Jp is a 6 × M matrix. If the mobility M is greater than connectivity SF, the parallel robot is structurally redundant. If SF