Structural Synthesis of Parallel Robots: Part 5: Basic Overconstrained Topologies with Schönflies Motions [1 ed.] 978-94-007-7400-1, 978-94-007-7401-8

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Solid Mechanics and Its Applications

Grigore Gogu

Structural Synthesis of Parallel Robots Part 5: Basic Overconstrained Topologies with Schönflies Motions

Solid Mechanics and Its Applications Volume 206

Series Editor G. M. L. Gladwell, Department of Civil Engineering, University of Waterloo, Waterloo, Canada

For further volumes: http://www.springer.com/series/6557

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.

Grigore Gogu

Structural Synthesis of Parallel Robots Part 5: Basic Overconstrained Topologies with Schönflies Motions

123

Grigore Gogu Institut Pascal-UMR 6602 Institut Français de Mécanique Avancée Aubiere Cedex France

ISSN 0925-0042 ISBN 978-94-007-7400-1 DOI 10.1007/978-94-007-7401-8

ISBN 978-94-007-7401-8

(eBook)

Springer Dordrecht Heidelberg New York London Library of Congress Control Number: 2013945770  Springer Science+Business Media Dordrecht 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

The ideas by which alone the nature of a simple mechanism can be arrived at may be very complex, or in certain circumstances may be quite the reverse. But equally whether these ideas be simple or complex, if it be desired to examine and understand the simple apparatus scientifically, it is necessary to work through the whole succession of them, passing from each one to the next higher from which it was developed, until really general principles are arrived at. Reuleaux, F., Theoretische Kinematik, Braunschweig: Vieweg, 1875 Reuleaux, F., The Kinematics of Machinery, London: Macmillan, 1876 and New York: Dover, 1963 (translated by A. B. W. Kennedy)

This book represents the fifth part of a larger work dedicated to the Structural Synthesis of Parallel Robots. Part 1 [1] presented the methodology of structural synthesis and the systematisation of structural solutions of simple and complex limbs with two to six degrees of connectivity systematically generated by the structural synthesis approach. Part 2 [2] presented structural solutions of translational parallel robotic manipulators with two and three degrees of mobility. Part 3 [3] focussed on structural solutions of parallel robotic manipulators with planar motion of the moving platform. Part 4 [4] presented structural solutions of other topologies of parallel robotic manipulators with two and three degrees of freedom of the moving platform. This book offers basic structural solutions of overconstrained parallel robotic manipulators with Schönflies motions of the moving platform systematically generated by using the structural synthesis approach proposed in Part 1. The originality of this work resides in the fact that it combines 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 mechanisms. Parallel robotic manipulators can be considered a well-established option for many different applications of manipulation, machining, guiding, testing, control, tracking, haptic force feedback, etc. A typical parallel robotic manipulator 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

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concentrated primarily on six degrees of freedom (DoFs) Gough-Stewart-type PMs introduced by Gough for a tire-testing device, and by Stewart for flight simulators. In the last decade, PMs with fewer than six DoFs 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-to-weight ratio, dynamic performances. Their disadvantages include a smaller workspace, complex command, and lower dexterity due to high motion coupling, and multiplicity of singularities inside their workspace. Uncoupled, fully isotropic, and maximally regular PMs can overcome these disadvantages. 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 endeffector 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) fully isotropic 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 parallel robot was a T3-type translational parallel mechanism. It 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 independently 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 work is based on the theory of linear transformations and the evolutionary morphology and allows us to obtain the structural solutions of coupled, 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

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proposed recently by the Author are integrated into the synthesis approach developed in this work. Various solutions of TaRb-type PMs are known today. In this notation, a = 1–3 indicates the number of independent translations and b = 1–3 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 work on robotics presenting solutions of uncoupled, fully isotropic, and maximally regular PMs along with coupled and decoupled solutions obtained by a systematic approach of structural synthesis. Non-redundant/redundant, overconstrained/isostatic solutions of coupled, decoupled, 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. A serial kinematic chain is associated with each elementary limb and at least one closed loop is integrated in each complex limb. The various solutions of maximally regular PMs proposed by the author belong to a modular family called Isoglide n - TaRb with a ? b = n with 2 B n B 6, a = 1–3 and b = 1–3. The mobile platform of these robots can have any combination of n independent translations (T) and rotations (R). The Isoglide n - TaRb modular family was developed by the Author and his research team of the Mechanical Engineering Research Group (LaMI/Pascal Institute), Blaise Pascal University and French Institute of Advanced Mechanics (IFMA) in Clermont-Ferrand. The synthesis methodology and the solutions of PMs presented in this work represent the outcome of some recent research initiated by the Author in the framework of the projects ROBEA-MAX and ROBEA-MP2 supported by the National Center for Scientific Research (CNRS). These results have been partially published by the Author in the last 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) innovative solutions of TaRb-type fully isotropic and maximally regular PMs for any combination of 0 B a B 3 independent translations and 0 B b B 3 independent rotations of the moving platform. Part 1 of this work [1] is organized into 10 chapters. Chapter 1 introduced the main concepts, definitions, and components of the mechanical robotic system. Chapter 2 reviewed the contributions in mobility calculation 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,

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connectivity, redundancy, and overconstraint are demonstrated via an original approach based on the theory of linear transformations. These formulae are applied in Chap. 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, and thus there is no reason to continue to consider them as ‘‘paradoxical.’’ Chapter 4 presented the main models and performance indices used in parallel robots. We put particular emphasis on 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 Chap. 2 are integrated in the structural synthesis approach founded on the evolutionary morphology (EM) presented in Chap. 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 Chaps. 6–10. We focused on the solutions with a unique combination of translational and rotational velocities in the 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 Parts 2–4. Limbs with multiple combinations of translational and rotational velocities in the basis 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 containing more than 1,500 structural diagrams. The kinematic chains presented in Chaps. 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 Chaps. 6–10 of Parts 2–4 are combined in Parts 2–5 to set up innovative solutions of parallel robots with two to

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four degrees of mobility and various sets of independent motions of the moving platform. Part 2 of this work [2] is organized into 7 chapters. Chapter 1 recalled the main concepts, the new formulae used to calculate the main structural parameters of PMs, and the original approach of structural synthesis. Chapter 2 focused on the structural synthesis of T2-type translational parallel manipulators (TPMs) with two degrees of freedom used in pick-and-place operations. Overconstrained/isostatic solutions of coupled, decoupled, uncoupled, and fully isotropic/maximally regular PMs with elementary/complex limbs actuated by linear/rotary actuators with/ without idle mobilities are presented. Chapter 3 presented the structural synthesis of overconstrained T3-type translational parallel manipulators with three degrees of freedom and coupled motions. Basic and derived solutions with linear or rotating actuators are presented. The basic solutions do not combine idle mobilities and/or idle pairs. Idle mobilities/pairs are used to reduce the degree of overconstraint in the derived solutions. The structural synthesis of non-overconstrained T3-type TPMs with decoupled motions is presented in Chap. 4. Basic and derived solutions with linear or rotating actuators are on hand. Chapters 5 and 6 presented the structural synthesis of overconstrained and non-overconstrained T3-type TPMs with uncoupled motions. Basic and derived solutions with rotating actuators and identical limbs are presented. Chapter 7 focused on the structural synthesis of overconstrained and non-overconstrained maximally regular T3-type TPMs. Basic and derived solutions with linear actuators and identical limbs are on hand. About 1,000 solutions of TPMs are illustrated in 550 figures. The structural parameters of these solutions are systematized in 134 tables. Part 3 of this work [3] is organized into 8 chapters. Chapter 1 recalls, the main concepts, the new formulae used to calculate the main structural parameters of PMs, and the original approach of structural synthesis applied to parallel robots with planar motion of the moving platform. In such a robot, the moving platform can undergo two independent translational motions T2 and one rotational motion R1 around an axis perpendicular to the plane of translations. This motion can be obtained by using planar or spatial parallel mechanisms. Chapters 2 and 3 presented the structural synthesis of overconstrained and non-overconstrained planar parallel robots with coupled motions. Basic and derived fully parallel and nonfully parallel solutions are on hand. The structural synthesis of overconstrained and non-overconstrained planar parallel robots with uncoupled motions is presented in Chap. 4. Chapter 5 focused on the structural synthesis of overconstrained and non-overconstrained maximally regular planar parallel robots. Chapters 6 and 7 presented the structural synthesis of basic and derived solutions of overconstrained and non-overconstrained spatial parallel robots with coupled and uncoupled planar motions of the moving platform. Chapter 8 focused on the structural synthesis of overconstrained and non-overconstrained maximally regular spatial parallel robots with planar motion of the moving platform. About 750 solutions are illustrated in 400 figures. The structural parameters of these solutions are systematized in 150 tables.

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Part 4 of this work [4] is organized into 9 chapters. Chapter 1 recalls the main concepts, the new formulae used to calculate the main structural parameters of PMs, and the original approach of structural synthesis applied to other parallel robotic mechanisms with two and three degrees of freedom of the moving platform. Chapter 2 presented the structural synthesis of overconstrained and non overconstrained parallel mechanisms with cylindrical motion of the moving platform. They make possible one independent translation (T1) and one independent rotation (R1) of the moving platform. The direction of the translation coincides with the rotation axis. These solutions are useful in applications that require positioning and orienting a body on an axis. Chapter 3 presented the structural synthesis of other T1R1-type parallel mechanisms that are useful in applications that require translating a body on a line and rotating it around an axis that is perpendicular to the direction of the translation. Chapters 4 and 8 presented the structural synthesis of parallel wrists (PWs) useful for orienting a body in space by two (R2) or three (R3) independent rotations of the mobile platform about a fixed point. Basic and derived fully parallel and non-fully parallel solutions of overconstrained and not overconstrained spatial parallel manipulators T2R1-type were on hand in Chaps. 5 and 6. These solutions are the parallel counterparts of the 3-DOF PPR serial robots, in which the moving platform can rotate about an axis undergoing a planar translation. They are used in various applications that require two independent planar translations (T2) and one independent rotation (R1) of the mobile platform around an axis lying in the plane of translation. Chapter 7 presented the structural synthesis of overconstrained and non-overconstrained T1R2type spatial parallel manipulators giving a unidirectional translation (T1) and the orientation of a body in space with two independent rotations (R2) of the mobile platform about a fixed point. In the solutions presented in Chaps. 2–8 of Part 4, the limbs constrain the characteristic point of the moving platform to carry out just independent motions. Chapter 9 focused on the structural synthesis of parallel mechanisms with some dependent motions combined with two or three independent motions of the moving platform. More than 700 solutions are illustrated in 350 figures. The structural parameters of these solutions are systematized in 170 tables. This book representing Part 5 is organised into 6 chapters. Chapter 1 recalls the main concepts, the new formulae used to calculate the main structural parameters of PMs, and the original approach of structural synthesis applied to basic overconstarined solutions of parallel robotic mechanisms with Schönflies motions of the moving platform. The parallel robotic mechanisms with Schönflies motions make possible three spatial independent translations (T3) and one independent rotation (R1) of the moving platform around an axis of fixed direction. These solutions are useful in applications that require positioning a body in space and orienting it on an axis. The structural synthesis of fully-parallel T3R1-type parallel mechanisms with coupled Schönflies motions are presented in Chap. 3. Chapter 4 presents the structural synthesis of overactuated topologies with coupled Schönflies motions. Three overactuated joints exist in these solutions. The overactuation of up to three joints contributes to a better repartition of the payload on the limbs.

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If overactuation is missing, the topologies presented in this section are reduced to a fully parallel solution, similar to those illustrated in Chap. 2 presenting one uncoupled motion. The structural synthesis of T3R1-type parallel mechanisms with decoupled Schönflies motions are presented in Chap. 4. The Jacobian matrix of these solutions is a triangular matrix. Fully-parallel solutions along with redundantly actuated solutions with uncoupled Schönflies motions are presented in Chap. 5. The Jacobian matrix of these solutions is a diagonal matrix. Usually, redundancy in parallel manipulators is used to eliminate some singular configurations, to minimize the joint rates, to optimize the joint torques/forces, to increase dexterity workspace, stiffness, eigenfrequencies, kinematic, and dynamic accuracy, to improve both kinematic and dynamic control algorithms. In this chapter, redundancy is used for decoupling motion as proposed for the first time in [5]. Chapter 6 focuses on the structural synthesis of maximally regular parallel mechanisms with Schönflies motions. More than 640 solutions are illustrated in 470 figures. The structural parameters of these solutions are systematized in 60 tables. 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 the various solutions is associated with the Author’s conviction that a good structural diagram really ‘‘is worth a thousand words,’’ especially when we are trying to disseminate the result of the structural synthesis of kinematic chains. Many solutions for parallel robots obtained through this systematic approach of structural synthesis are presented, in this work, 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 work. In this way, the author hopes to contribute to a rapid and widespread implementation of these solutions in future industrial products.

References 1. Gogu G (2008) Structural synthesis of parallel robots: part 1-methodology. Springer, Dordrecht 2. Gogu G (2009) Structural synthesis of parallel robots: part 2-translational topologies with two and three degrees of freedom. Springer, Dordrecht 3. Gogu G (2010) Structural synthesis of parallel robots: part 3-topologies with planar motion of the moving platform. Springer, Dordrecht Heidelberg London New York 4. Gogu G (2012) Structural synthesis of parallel robots: part 4-other topologies with two and three degrees of freedom. Springer, Dordrecht Heidelberg London New York 5. Gogu G (2006) Fully-isotropic redundantly-actuated parallel manipulators with five degrees of freedom. In: Husty M, Schröcker HP (eds) Proceeding of the first european conference on mechanism science, Obergurgl

Acknowledgments

The scientific environment of the projects ROBEA-MAX and ROBEA-MP2 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 finalize this work. Deep gratitude is expressed here to Dr. François Pierrot, 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 on these projects. Moreover, financial support from the CNRS, FR TIMS and IFMA for developing the innovative Isoglide-family of parallel robots is duly acknowledged. Furthermore, Prof. Graham M. L. Gladwell, the Series Editor of Solids Mechanics and Its Applications, and Mrs. Nathalie Jacobs, Springer Dordrecht Engineering Editor are gratefully acknowledged for their availability and encouragement in pursuing this publishing project. Ms. Sarah Davey is also gratefully acknowledged for the linguistic reviewing of this manuscript. May I also acknowledge the excellent facilities and research environment provided by LaMI/Pascal Institute 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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Contents

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Links, Joints and Kinematic Chains. . . . . . . . . . . . . . . 1.1.2 Serial, Parallel and Hybrid Robots . . . . . . . . . . . . . . . 1.2 Methodology of Structural Synthesis . . . . . . . . . . . . . . . . . . . 1.2.1 New Formulae for Mobility, Connectivity, Redundancy and Overconstraint of Parallel Robots . . . . . . . . . . . . . 1.2.2 Evolutionary Morphology Approach . . . . . . . . . . . . . . 1.2.3 Types of Parallel Robots with Respect to Motion Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Parallel Robots with Schönflies Motions of the Moving Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Fully-Parallel Topologies with Coupled Schönflies Motions 2.1 Topologies with Simple Limbs . . . . . . . . . . . . . . . . . . 2.2 Topologies with Complex Limbs. . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Overactuated Topologies with Coupled Schönflies Motions 3.1 Topologies with Simple Limbs . . . . . . . . . . . . . . . . . . 3.2 Topologies with Complex Limbs. . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Fully-Parallel Topologies with Decoupled 4.1 Topologies with Simple Limbs . . . . . 4.2 Topologies with Complex Limbs. . . . References . . . . . . . . . . . . . . . . . . . . . . .

Schönflies Motions. ............... ............... ...............

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35 35 61 182

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183 184 247 365

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367 367 379 430

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Topologies with Uncoupled Schönflies Motions 5.1 Fully-Parallel Topologies . . . . . . . . . . . . . 5.2 Redundantly Actuated Topologies . . . . . . . 5.2.1 Topologies with Simple Limbs . . . . 5.2.2 Topologies with Complex Limbs . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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431 431 543 544 554 578

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Maximally Regular Topologies with Schönflies Motions. . . . . 6.1 Fully-Parallel Topologies with Simple Limbs . . . . . . . . . . 6.2 Fully-Parallel Topologies with Simple and Complex Limbs 6.3 Fully-Parallel Topologies with Complex Limbs. . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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579 579 591 621 646

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abbreviations and Notations

C C* C C CNRS DoF eA and eG1 eB and eG2 eC and eG3 eD and eG4 EM fi F / G1-G2-…-Gk

FR TIMS Gi (1Gi-2Gi-…n H IFMA IFToMM INRIA

IRCCyN Inn J

Gi)

Cylindrical joint Cylindrical joint with one or two idle mobilities Actuated cylindrical joint with just one actuated translational of rotational motion Actuated cylindrical joint with both translational and rotational motions actuated Centre National de la Recherche Scientifique (National Center for Scientific Research) Degree of freedom Link of G1-limb (e = 1, 2, 3,…, n) Link of G2-limb (e = 1, 2, 3,…, n) Link of G3-limb (e = 1, 2, 3,…, n) Link of G4-limb (e = 1, 2, 3,…, n) Evolutionary morphology Degree of mobility of the ith joint General notation for the kinematic chain associated to a parallel mechanism with k simple and/or complex limbs Gi (i = 1, 2,…, k) Fédération de Recherche Technologies de l’Information, de la Mobilité et de la Sûreté Kinematic chain associated to the ith limb Characteristic point of the distal link/end-effector Institut Français de Mécanique Avancée (French Institute for Advanced Mechanics) International Federation for the Promotion of Mechanism and Machine Science Institut National de Recherche en Informatique et en Automatique (The French National Institute for Research in Computer Science and Control) Institut de Recherche en Communications et Cybernétique de Nantes n 9 n identity matrix Jacobian matrix

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k k1 k2 LaMI LASMEA

LIRMM

m MF MGi NF n : nGi O0x0y0z0 p pGi P P P* Pa Pa Pa* or Pacs Pac Pasu Pas Pass Pat Patcs Pau Pauu

Abbreviations and Notations

Total number of limbs in the parallel manipulator Number of simple limbs in the parallel manipulator Number of complex limbs in the parallel manipulator Laboratoire de Mécanique et Ingénieries (Mechanical Engineering Research Group) Laboratoire des Sciences et Matériaux pour l’Electronique, et d’Automatique (Laboratory of Sciences and Materials for Electronic, and of Automatic) Laboratoire d’Informatique, de Robotique et de Microélectronique de Montpellier (Montpellier Laboratory of Computer Science, Robotics, and Microelectronics) Total number of links including the fixed base Mobility of parallel mechanism F Mobility of the kinematic chain associated with limb Gi Number of overconstraints in the parallel mechanism F Moving platform in the parallel mechanism F / G1-G2…-Gk, Reference frame Total number of joints in the parallel mechanism Number of joints in Gi-limb Prismatic joint Actuated prismatic joint Idle prismatic joint R||R||R||R-type planar parallelogram loop R||R||R||R-type parallelogram loop with one actuated revolute joint R||R||CS-type parallelogram loop with three idle mobilities combined in a cylindrical and a spherical joint R||R||R||C-type parallelogram loop with one idle mobility combined in a cylindrical joint Parallelogram loop with three idle mobilities combined in a spherical and a revolute joint R||R||RS-type parallelogram loop with two idle mobilities combined in a spherical joint R||RSS-type parallelogram loop with idle mobilities combined in two spherical joints adjacent to the same link R\P\||R||R\P\||R-type telescopic planar parallelogram loop Telescopic parallelogram loop with three idle mobilities combined in a cylindrical and a spherical joint Parallelogram loop with one idle mobility combined in a universal joint Parallelogram loop with two idle mobilities combined in two universal joints

Abbreviations and Notations

PM Pn Pn* or Pncs Pnss

Pn2 Pn2* or Pn2cs

Pn3 Pn3* or Pn3cs

PPM q q_ qi rF

rl

rGi R R R* Rb Rb* or Rbcs RF

(RF) RGi

xix

Parallel manipulator One degree of mobility planar close loop Close loop with one degree of mobility and three idle mobilities combined in a cylindrical and a spherical joint Closed loop with one degree of mobility and three idle mobilities combined two spherical joints adjacent to the same link Planar closed loop with two degrees of mobility Close loop with two degrees of mobility and three idle mobilities combined in a cylindrical and a spherical joint Planar closed loop with three degrees of mobility Closed loop with three degrees of mobility and three idle mobilities combined in a cylindrical and a spherical joint Planar parallel manipulator Number of independent closed loops in the parallel mechanism Joint velocity vector Finite displacement in the ith actuated joint Total number of joint parameters that lose their independence in the closed loops combined in parallel mechanism F Total number of joint parameters that lose their independence in the closed loops combined in the k limbs Number of joint parameters that lost their independence in the closed loops combined in Gi-limb Revolute joint Actuated revolute joint Idle revolute joint Rhombus loop Planar rhombus loop with three idle mobilities combined in a cylindrical and a spherical joint The vector space of relative velocities between the mobile and the reference platforms in the parallel mechanism F / G1-G2-…-Gk, The basis of vector space RF The vector space of relative velocities between the mobile and the reference platforms in the kinematic chain Gi disconnected from the parallel mechanism F / G1-G2-…-Gk,

xx

(RGi) S S* SF SGi

SPM TF TPM U U* v, v1 , v2 , v3 x, y, z x_ ; y_ ; z_ a; b; d _ d_ _ b; a; x, xa , xb , xd 0 1 : 1Gi 1Gi-2Gi-…-nGi 1A-2A-…-nA 1B-2B-…-nB 1C-2C-…-nC 1D-2D-…-nD 1 and 2 in the notation 2PRR-1RPaPa ||

\

\|| in the notation R\P\||C

Abbreviations and Notations

The basis of vector space RGi Spherical joint Spherical joint with idle mobilities The connectivity between the mobile and the reference platforms in the parallel mechanism F / G1-G2-…-Gk The connectivity between the mobile and the reference platforms in the kinematic chain Gi disconnected from the parallel mechanism F / G1-G2-…-Gk Spatial parallel manipulator Degree of structural redundancy of parallel mechanism F Translational parallel manipulator Universal joint Universal joint with an idle mobility Translational velocity vectors Coordinates of characteristic point H Time derivatives of coordinates Rotation angles Time derivatives of the rotation angles Angular velocity vectors Fixed base of a kinematic chain/mechanism Fixed platform in the parallel mechanism F/ G1-G2…-Gk, Links of limb Gi Links of limb G1 Links of limb G2 Links of limb G3 Links of limb G4 The parallel mechanism has two limbs of type PRR and one limb of type RPaPa Parallel position of joint axes/directions, for example the notation Pa||Pass indicates the fact that the axes of the revolute joints of the parallelogram loops Pa and Pass are parallel Perpendicular position of joint axes/directions, for example the notation P\Pa indicates the fact that the axes of revolute joints in the parallelogram loop are perpendicular to the direction of the prismatic joint The axis of the cylindrical joint is perpendicular to the direction of the actuated prismatic joint and parallel to the direction of the revolute joint

Abbreviations and Notations

\|| in the notation R\Pa\||Pa \\ in the notation R\Pa\\Pa \\ in the notation Pass\R||R\\Pa

xxi

The revolute axes of the second parallelogram loop are perpendicular to the revolute axes of the first parallelogram loop and parallel to the axis of the actuated revolute joint The revolute axes of the second parallelogram loop are perpendicular to the revolute axes of the first parallelogram loop and also to the axis of the actuated prismatic joint The revolute axes of parallelogram loop Pa are perpendicular to the axes of the parallel revolute joints R||R and also to the axes of the revolute joints of parallelogram loop Pass

Chapter 1

Introduction

This book represents Part 5 of a larger work on the structural synthesis of parallel robots. The originality of this work resides in combining new formulae for the structural parameters and the evolutionary morphology in a unified approach of structural synthesis giving interesting innovative solutions for parallel robots. Part 1 [1] presented the methodology of structural synthesis and the systematisation of structural solutions of simple and complex limbs with two to six degrees of connectivity systematically generated by the structural synthesis approach. Part 2 [2] presented structural solutions of translational parallel robotic manipulators with two and three degrees of mobility. Part 3 [3] presented structural solutions of parallel robotic manipulators with planar motion of the moving platform. Part 4 [4] presented structural solutions of other parallel robotic manipulators with two and three degrees of freedom of the moving platform. Part 5 of this work focuses on the basic structural solutions of overconstrained parallel robotic manipulators with Schönflies motions of the moving platform. This section recalls the terminology, the new formulae for the main structural parameters of parallel robots (mobility, connectivity, redundancy and overconstraint) and the main features of the methodology of structural synthesis based on the evolutionary morphology presented in Part 1.

1.1 Terminology 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 have presented in Part 1 various definitions of the word robot and we have seen that it 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 G. Gogu, Structural Synthesis of Parallel Robots, Solid Mechanics and Its Applications 206, DOI: 10.1007/978-94-007-7401-8_1,  Springer Science+Business Media Dordrecht 2014

1

2

1 Introduction

perform. Robots can be categorized as robotic manipulators, wheeled robots, legged robots, swimming robots, flying robots, androids and self reconfigurable robots which can apply themselves to a given task. This book focuses on parallel robotic manipulators which are the counterparts to the serial robots. The various definitions of robotics converge towards the integration of the design and the end use in the studies related to robots. This book focuses on the conceptual design of parallel robots. Although the appearance and capabilities of robots vary greatly, 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 joints.

1.1.1 Links, Joints and Kinematic Chains 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 mechanical architecture of parallel robots is based on parallel mechanisms in which a member called a moving platform is connected to a reference member by at least two limbs that can be 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 moving platform positions the robot end-effector in space and may have anything 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 redundantlyactuated 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 manipulator is very important. They exhibit interesting features when compared to hexapods, such as a simpler architecture, a simpler control system, high-speed performance, low manufacturing and operating costs. Furthermore, for several parallel manipulators with decoupled/uncoupled motions and maximally regular as well, the kinematic model can be easily solved to obtain algebraic expressions, which are well suited for 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. The terminology used in this book is mainly established in accordance with the terminology adopted by the International Federation for the Promotion of Mechanism and Machine Science (IFToMM) and published in [5]. The main terms used in this book concerning kinematic pairs (joints), kinematic chains and robot kinematics are defined in Tables 1.1, 1.2 and 1.3 in Part 1 of this work. They are

1.1 Terminology

3

Table 1.1 Parallelogram loops with idle mobilities and their corresponding number of overconstraints N No. Parallelogram N Idle mobilities loop 1. 2. 3. 4. 5. 6. 7. 8. 9.

Pa (Fig. 1.2a) Pac (Fig. 1.2b) Pau (Fig. 1.2c) Pas (Fig. 1.2d) Pauu (Fig. 1.2e) Pacu (Fig. 1.2f)

3 2 2 1 1

No idle mobilities One translational idle mobility combined in a cylindrical joint One rotational idle mobility combined in a universal joint Two rotational idle mobilities combined in a spherical joint Two rotational idle mobilities combined in two universal joints

1 One translational idle mobility combined in a cylindrical joint and one rotational idle mobilities combined in a universal joint 0 One translational idle mobility combined in a cylindrical joint and two Pacs, Pa* (Fig. 1.2g) rotational idle mobilities combined in a spherical joint 0 Three rotational idle mobilities combined in one revolute joint and one Pasu (Fig. 1.2h) spherical joint Pass (Fig. 1.2i) 0 Three idle mobilities combined in two spherical joints adjacent to the same link with a complementary internal rotational mobility of the link adjacent to the two spherical joints

completed by some complementary remarks, notations and symbols used in this book. IFToMM terminology [5] defines a link as a mechanism element (component) carrying kinematic pairing elements and a joint is a physical realization of a kinematic pair. The pairing element represents the assembly of surfaces, lines or points of a solid body through which it may contact with another solid body. The kinematic pair is the mechanical model of the connection of two pairing elements having relative motion of a certain type and degree of freedom. 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 G is denoted by nG (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 1G : 0 is used to indicate that the first link 1G of the kinematic chain G is the fixed base. A kinematic chain G is denoted by the sequence of its links. The notation G (1G : 0-2G -  - nG) indicates a kinematic chain in which the first link is fixed and the notation G (1G - 2G -  nG) a kinematic chain with no fixed link. 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,

4

1 Introduction

in this book, the same assembly of links and joints G will be considered to be a kinematic chain when integrated as a sub-system in another assembly of links and joints and will be considered a mechanism when G represents the whole system. The systematization, the definitions and the formulae presented in this book are valuable for mechanisms and kinematic chains. We use the term mechanism element or link to name a component (member) of a mechanism. In this book, 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). 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 is connected in the kinematic chain by just one joint. In a simple open kinematic chain (open-loop mechanism) only monary and binary links are connected. In a complex kinematic chain at least one polynary link 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 distal links are the moving and the reference platform. The two platforms are connected by at least two simple or complex kinematic chains called limbs. Each limb contains at least one joint. A simple 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. IFToMM terminology [5] uses the term kinematic pair to define the mechanical model of the connection of links 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 1.1—Part 1. The graphical representations used in this book for the lower pairs are presented in Fig. 1.1a–f. Universal joints and homokinetic joints are also currently used in the mechanical structure of the parallel

1.1 Terminology

5

robots to transmit the rotational 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 (HJ) 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. [6–9]. The graphical representations used in this book for the universal homokinetic joints are presented in Fig. 1.1g–h. Joints with idle mobilities are commonly used to reduce the number of overconstraints in a mechanism. The idle mobility is a potential mobility of a joint that is not used by the mechanism and does not influence mechanism’s mobility in the hypothesis of perfect manufacturing and assembling precision. In theoretical conditions, when no errors exist with respect to parallel, perpendicular or intersecting positions of joint axes, motion amplitude associated with 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 mechanism. For example, the idle mobilities which can be combined in the parallelogram loop in Fig. 1.2 are systematized in Table 1.1 along with the number r of parameters that lose their independence in the closed loop and the number of overconstraints N of the corresponding linkage. A joint can combine idle and non idle (effective) mobilities. A joint combining only idle mobilities is called idle joint. A parallel mechanism is a single or multi-loop linkage in which a moving link called characteristic link or moving platform is connected to a reference link (fixed base) by at least two non interconnected kinematic chains called limbs. The number of limbs, k, of a parallel mechanism is established by taking into account the following relation between the total number of joints, p, of the parallel mechanism and the total number of joints pGi of ith limb (i = 1,2, … ,k) p¼

k X

pGi :

ð1:1Þ

i¼1

Equation (1.1) indicates that the limbs of the parallel mechanism F G1-G2-Gk must be defined in such a way that a joint must belong to just one limb; that is the same joint cannot be combined in two or more limbs. This is the meaning of the fact that, in a parallel mechanism, the end-effector is linked to the base by ‘‘several independent kinematic chains’’ as coined by Merlet [10] in the definition of a generalized parallel manipulator. The ‘‘independence’’ of the kinematic chains associated with the limbs of a parallel mechanism must be limited to the structural point of view. This ‘‘independence’’ is not valid for the kinematic and static point of view.

6

1 Introduction

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, h homokinetic joint, i two superposed revolute joints (1, 2) and (2, 3) with the same axis, j superposed cylindrical (1, 2) and revolute (2, 3) joints with the same axis, k superposed revolute (1, 2) and cylindrical (2, 3) joints with the same axis, and l two superposed cylindrical joints (1, 2) and (2, 3) with the same axis

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 a CAD model, structural diagram and structural graph. Figure 1.3 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. 1.3a is a photograph of the parallel robot built by Deltalab (http://www.deltalab.fr/). In a CAD model (Fig. 1.3b) the links and the joints are represented as being as close as possible to the physical implementation (Fig. 1.3a). In a structural diagram (Fig. 1.3c) 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. 1.3d) 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. 1.3c 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 parallel mechanism 6-SPS-type associated with the Gough-Stewart type parallel robot is a complex mechanism with a multi-loop associated graph (Fig. 1.3d). It has six simple limbs of type SPS. The actuated pair is underlined. The simple open kinematic chain associated with A-limb is denoted by A (1A : 0–2A–3A– 4A : 4)—Fig. 1.3e and its associated graph is tree-type (Fig. 1.3f).

1.1 Terminology

7

Fig. 1.2 Parallelogram loops of types Pa (a), Pac (b), Pau (c), Pas (d), Pauu (e), Pacu (f), Pa* (g), Pasu (h), Pass (i) and the number of r parameters that lost their independence in the closed loop

8

1 Introduction

1.1.2 Serial, Parallel and Hybrid Robots We consider the general case of a robot in which the end-effector is connected to the reference link by k C 1 kinematic chains. The end-effector is a binary or polynary link called a mobile platform in the case of parallel 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. A serial robot F / G1 is a robot in which the end-effector n : nG1 is connected to the reference link 1 : 1G1 by just one simple open kinematic chain Gi (1G1-2G1-nG1) called a serial kinematic chain. A parallel robot F / G1-G2--Gk is a robot in which the end-effector n : nGi is connected in parallel to the reference link 1 : 1Gi by k C 2 kinematic chains Gi (1Gi-2Gi-…-nGi) called limbs or legs. 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 G1 (1G1-2G1- nG1) called complex limb or complex leg. A fully-parallel robot F / G1-G2- Gk is a parallel robot in which the number of limbs is equal to the robot mobility (k = M C 2), and just one actuator exist in each limb. A non fully-parallel robot F / G1-G2- Gk is a parallel robot with fewer number of limbs than the robot mobility (k \ M), and at least one limb has more than one actuator.

1.2 Methodology of 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

1.2 Methodology of Structural Synthesis

9

Fig. 1.3 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)

10

1 Introduction

activities in the invention and the 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 entire set of mechanical architecture to meet the required structural parameters. The mechanical architecture or topology is defined by number, type and relative position of joint axes in the parallel robot. The structural parameters are mobility, connectivity, redundancy and the number of overconstraints. They define the number of actuators, the degrees of freedom and the motion-type of the moving platform. A systematic approach of structural synthesis founded on the theory of linear transformations and an evolutionary morphology has been proposed in Part 1 [1]. The approach integrates the new formulae for mobility, connectivity, redundancy and overconstraint of parallel manipulators [11, 12] and a new method of systematic innovation [13].

1.2.1 New Formulae for Mobility, Connectivity, Redundancy and Overconstraint of Parallel Robots 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. We note that the mobility of a mechanism can be defined by the number of independent finite and/or infinitesimal displacements in the joints needed to define the configuration of the mechanism [1]. Mobility of a mechanism represents the sum of internal and external mobilities. The internal mobilities are localized to the level of a link or a group of links. They can be associated with finite or infinitesimal motions. The external mobilities are associated with the independent finite motions transmitted by the mechanism between the actuators and the end-effector. 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 few years. Mobility calculation still remains a central subject in the theory of mechanisms. In Part 1 [1] we have shown that the various methods proposed in the literature for mobility calculation of the closed loop mechanisms fall into two basic categories:

1.2 Methodology of Structural Synthesis

11

(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. The information about mechanism mobility is derived by performing position, velocity or static analysis by using analytical tools (screw 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. Moreover, the rank of the constraint equations is calculated in a 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, full-cycle mobility). The general mobility represents the minimum value of the instantaneous mobility in a free-of-singularity workspace. For a given mechanism, general mobility has a unique value for a free-of-singularity workspace. It is a global parameter characterizing the mechanism in all its configurations of the workspace 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 general mobility. In this book, unless otherwise stated, general mobility is simply called mobility. Note 1. In a kinematotropic mechanism with branching singularities, full-cycle mobility is associated with each branch [14–17, 18]. In this case, the full-cycle mobility (global mobility) is replaced by the branch mobility which represents the minimum value of the instantaneous mobility inside the same free-of-singularity branch. As each branch has its own mobility, a single value for global mobility cannot be associated with the kinematotropic mechanisms [14, 15]. The term kinematotropic was coined by Wohlhart [19] to define the linkages that permanently change their full-cycle mobility when passing by an instantaneous singularity from one branch to another. Various single and multi-loop kinematotropic mechanisms have been presented in the literature [19–22]. A formula for quick calculation of mobility is an explicit relationship between the following structural parameters: 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 constraint equations. In Part 1, we have shown that several dozen approaches proposed in the last 150 years for the calculation of mechanism mobility can be reduced to the same original formula that we have called the Chebychev-Grübler-Kutzbach (CGK) formula in its original or extended forms. These formulae have been critically

12

1 Introduction

reviewed [23] and a criterion governing mechanisms to which this formula can be applied has been set up in [24]. We have explained why this well-known formula does not work for some multi-loop mechanisms. New formulae for quick calculation of mobility have been proposed in [11] and demonstrated via the theory of linear transformations. More details and a development of these contributions have been presented in Part 1. The connectivity between two links of a mechanism represents the number of independent finite and/or infinitesimal displacements allowed by the mechanism between the two links. The number of overconstraints of a mechanism is given by the difference between the maximum number of joint kinematic parameters that could lose their independence in the closed loops, and the number of joint kinematic parameters that actually lose their independence in the closed loops. The structural redundancy of a kinematic chain represents the difference between the mobility of the kinematic chain and connectivity between its distal links. Let us consider the case of the parallel mechanism F / G1-G2--Gk in which the mobile platform n : nGi is connected to the reference platform 1 : 1Gi by k simple and/or complex kinematic chains Gi (1Gi -2Gi --nGi) called limbs. In Part 1, the following parameters have been associated with the parallel mechanism F G1-G2--Gk: RGi–the vector space of relative velocities between the mobile and the reference platforms, nGi and 1Gi, in the kinematic chain Gi disconnected from the parallel mechanism F, RF–the vector space of relative velocities between the mobile and the reference platforms, n : nGi and 1 : 1Gi, in the parallel mechanism F G1-G2--Gk, whose basis is ðRF Þ ¼ ðRG1 \ RG2 \    \ RGk Þ;

ð1:2Þ

SGi–the connectivity between the mobile and the reference platforms, nGi and 1Gi, in the kinematic chain Gi disconnected from the parallel mechanism F, SF–the connectivity between the mobile and the reference platforms, n : nGi and 1 : 1Gi, in the parallel mechanism F G1-G2- Gk. We recall that the connectivity is defined by the number of independent motions between the mobile and the reference platforms. The notation 1 :1Gi: 0 is used when the reference platform is the fixed base. The vector spaces of relative velocities between the mobile and the reference platforms are also called operational velocity spaces. The following formulae demonstrated in Chap. 2-Part 1 [1] for mobility MF, connectivity SF, number of overconstraints NF and redundancy TF of the parallel mechanism F G1-G2--Gk are used in structural synthesis of parallel robotic manipulators:

1.2 Methodology of Structural Synthesis

MF ¼

13 p X

fi rF ;

ð1:3Þ

i¼1

NF ¼ 6q  rF ;

ð1:4Þ

T F ¼ M F  SF ;

ð1:5Þ

SGi ¼ dimðRGi Þ;

ð1:6Þ

SF ¼ dimðRF Þ ¼ dimðRG1 \ RG2 \ . . . \ RGk Þ;

ð1:7Þ

where

rF ¼

k X

SGi  SF þ rl ;

ð1:8Þ

i¼1

q ¼ p  m þ 1;

ð1:9Þ

and rl ¼

k X

rGi

ð1:10Þ

i¼1

We note that pGi represents the number of joints of Gi-limb, p the total number of joints of parallel mechanism F, m the total number of links in mechanism F including the moving and reference platforms, q the total number of independent closed loops in the sense of graph theory, fi the mobility of the ith joint, rF the total number of joint parameters that lose their independence in mechanism F, rGi 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 k limbs of mechanism F. In Eqs. (1.6) and (1.7), dim denotes the dimension of the vector spaces. We denote by k1 the number of simple limbs and by k2 the number of complex limbs (k = k1 ? k2). In Chap. 5-Part 1 the following structural conditions have been established: (a) for the non redundant parallel robots (TF = 0) SF ¼ MF  MGi ði ¼ 1; . . .; kÞ;

ð1:11Þ

MGi ¼ SGi  6ði ¼ 1; . . .; kÞ;

ð1:12Þ

(b) for the redundant parallel robots with TF [ 0

14

1 Introduction

SF \ MF  MGi ði ¼ 1; . . .; kÞ;

ð1:13Þ

MGi [ SGi  6ði ¼ 1; . . .; kÞ;

ð1:14Þ

(c) for the non overconstrained parallel robots (NF = 0) MF ¼

p X

fi  6q;

ð1:15Þ

i¼1

(d) for the overconstrained parallel robots with NF [ 0 MF [

p X

fi  6q:

ð1:16Þ

i¼1

We recall that MGi ¼

pGi X

fi rGi :

ð1:17Þ

i¼1

We note that the intersection in Eqs. (1.2) and (1.7) is consistent if the vector spaces RGi are defined by the velocities of the same point situated on the moving platform with respect to the same reference frame. This point is called the characteristic point, and denoted by H. It is the point with the most restrictive motions of the moving platform. The connectivity SF of the moving platform n : nGi in the mechanism F / G1-G2--Gk is less than or equal to the mobility MF of mechanism F. The basis of the vector space RF of relative velocities between the moving and reference platforms in the mechanism F / G1-G2--Gk must be valid for any point of the moving platform n : nGi. Note 2. The bases of vector spaces RGj and RF may contain up to 6 independent velocity vectors vx, vy, vz, xa, xb and xd. We denote by vx, vy and vz the independent linear velocities of the characteristic point H of the moving platform and by xa, xb and xd the independent angular velocities of the moving platform. For example the basis of vector space RGj of a planar limb with three revolute joints is always (RGj) = (vx, vy, xd) if the three joint axes are parallel to z0-axis. For the same dimension SGj, the basis of vector space RGj of certain kinematic chains may be defined by different combinations of velocity vectors vx, vy, vz, xa, xb and xd. For example, in a spatial limb with three revolute joints with orthogonal axes and non zero distance between the joint axes adjacent to the same link, vector space RGj has always three dimensions, but the basis can be defined by various combination of three out of six vectors vx, vy, vz, xa, xb and xd. In these cases, the bases of RGj in Eqs. (1.2) and (1.7) are selected such as the minimum value of SF is obtained by Eq. (1.7). By this choice, the result of Eq. (1.3) fits in with general mobility definition as the minimum value of the instantaneous mobility in a free-

1.2 Methodology of Structural Synthesis

15

of-singularity branch. In the same way, in certain parallel robots, for the same dimension SF, the basis of vector space RF may be defined by different combinations of velocity vectors vx, vy, vz, xa, xb and xd. These solutions are called parallel robots with various combinations of rotational and translational velocities of the moving platform. In this case, the moving platform can make more than SF translational and/or rotational motions but just SF of them are independent motions. The three independent translational velocities are also denoted in this book by v1, v2, and v3. The parameters used in the new formulae (1.1)–(1.17) can be easily obtained by inspection with no need to calculate the rank of the homogeneous linear set of constraint equations associated with loop closure or with the rank of the complete screw system associated to the joints of the mechanism. An analytical method to compute these parameters has also been developed in Part 1 just for verification and for a better understanding of the meaning of these parameters. The following steps can be used for the calculation of structural parameters of a parallel mechanism based on formulae (1.1)–(1.10). Step 1: Identify the total number of links m (including the fixed base and the moving platform) and the total number of joints p in the parallel mechanism. Step 2: Calculate the number of independent closed loops q in the parallel mechanism, q = p - m ? 1. Step 3: Determine the number of limbs k connecting the moving platform to the fixed base such that no joint belongs to more than one limb, and check Eq. (1.1). Step 4: Identify the basis of RGj (j = 1,2,…,k) by observing the independent motions between distal link nGj and 1Gj in the kinematic chain associated with Gj– limb disconnected from the parallel mechanism. Step 5: Calculate the connectivity between distal links nGj and 1Gj in the kinematic chain Gj disconnected from the parallel mechanism, SGj = dim(RGj). If necessary, calculate the rank of the forward velocity Jacobian JGj of Gj-limb disconnected from the parallel mechanism, SGj = rank(JGj). Step 6: Calculate the connectivity between the distal links n : nGj and 1 : 1Gj in the parallel mechanism given by Eq. (1.7). Step 7: Determine the number of joint parameters that lose their independence in the closed loops that may exist in each limb. Step 8: Calculate the total number of joint parameters that lose their independence in the closed loops that may exist in the limbs of the parallel mechanism given by Eq. (1.10). Step 9: Calculate the total number of joint parameters that lose their independence in the parallel mechanism given by Eq. (1.8). Step 10: Calculate mobility MF, number of overconstraints NF and redundancy TF of the parallel mechanism given by Eqs. (1.3)–(1.5). Note 3. In the case of serial concatenated loops (loops with no common joints) the number of joint parameters that lose their independence in the closed loops that may exist in a limb is obtained by adding the number of joint parameters that lose their independence in each loop. Equation (1.8) can be used to calculate the number of joint parameters that lose their independence in each serial

16

1 Introduction

concatenated loop by considering the kinematic chain associated with the closed loop as a parallel kinematic chain with two limbs connecting a distal link to a reference link of the loop. Any two distinct loop members can be chosen as distal and reference links including two adjacent links. In this case, one limb is formed by just one joint (the joint connecting the two adjacent links) and the second limb combines the remaining joints of the loop (see Sect. 3.2—Part 1). Note 4. In the case of parallel concatenated loops (loops with some common joints) Eq. (1.8) can also be used to calculate the number of joint parameters that lose their independence in these closed loops, by considering the kinematic chain associated with these closed loops as a parallel kinematic chain connecting a polynary link to a reference link (see Sects. 3.3 and 3.4—Part 1). The formulae (1.1)–(1.10) have been successfully applied in Part 1 [1] to structural analysis of various mechanisms including so called ‘‘paradoxical’’ mechanisms. These formulae are also useful for the structural synthesis of various types of parallel mechanisms with 2 B MF B 6 and various combinations of independent motions of the moving platform. These solutions are obtained in a systematic approach of structural synthesis by using the limbs generated by the method of evolutionary morphology presented in Part 1.

1.2.2 Evolutionary Morphology Approach Evolutionary morphology (EM) is a new method of systematic innovation in engineering design proposed by the author in [13]. EM is formalized by a 6-tuple of design objectives, protoelements (initial components), morphological operators, evolution criteria, morphologies and a termination criterion. The design objectives are the structural solutions, also called topologies, defined by the required values of mobility, connectivity, overconstrains and redundancy and the level of motion coupling. The protoelements are the revolute and prismatic joints. The morphological operators are: (re)combination, mutation, migration and selection. These operators are deterministic and are applied at each generation of EM. At least MF = SF generations are necessary to evolve by successive combinations from the first generation of protoelements to a first solution satisfying the set of design objectives. Morphological migration could introduce new constituent elements formed by new joints or combinations of joints into the evolutionary process. Evolutionary morphology is a complementary method with respect to evolutionary algorithms that starts from a given initial population to obtain an optimum solution with respect to a fitness function. EM creates this initial population to enhance the chance of obtaining a ‘‘more global optimum’’. Evolutionary algorithms are optimization oriented methods; EM is a conceptual design oriented method. A detailed presentation of the evolutionary morphology can be found in Chap. 5—Part 1.

1.2 Methodology of Structural Synthesis

17

1.2.3 Types of Parallel Robots with Respect to Motion Coupling Various levels of motion coupling have been introduced in Chap. 4—Part 1 in relation with the Jacobian matrix of the robotic manipulator which is the matrix mapping (1) the actuated joint velocity space and the end-effector velocity space, and (2) the static load on the end-effector and the actuated joint forces or torques. Five types of parallel robotic manipulators (PMs) are introduced in Part 1: (1) maximally regular PMs, if the Jacobian J is an identity matrix throughout the entire workspace, (2) fully-isotropic PMs, if J is a diagonal matrix with identical diagonal elements throughout the entire workspace, (3) PMs with uncoupled motions if J is a diagonal matrix with different diagonal elements, (4) PMs with decoupled motions, if J is a triangular matrix and (5) PMs with coupled motions if J is neither a triangular nor a diagonal matrix. The term maximally regular parallel robot was recently coined by Merlet [10, 25] to define isotropic robots. We use this term to define just the particular case of fully-isotropic PMs, when the Jacobian matrix is an identity matrix throughout the entire workspace. 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 one. Thus, the condition number of the Jacobian matrix is an interesting performance index characterizing the distortion of a unit sphere under this linear mapping. The condition number of the Jacobian matrix was first used by Salsbury and Craig [26] to design mechanical fingers and developed by Angeles [27, 28] as a kinetostatic performance index of the robotic mechanical systems. The isotropic design aims at ideal kinematic and dynamic performance of the manipulator Fattah and Ghasemi [29]. In an isotropic configuration, the sensitivity of a manipulator is minimal with regard to both velocity and force errors and the manipulator can be controlled equally well in all directions. The concept of kinematic isotropy has been used as a criterion in the design of various parallel manipulators [30, 31]. Fully-isotropic PMs give a one-to-one mapping between the actuated joint velocity space and the operational velocity space. The condition number and the determinant of the Jacobian matrix being equal to one, the manipulator performs very well with regard to force and motion transmission. The various kinetostatic performance indices introduced in Sect. 4.5-Part 1 have optimal values for fullyisotropic PMs [1, 32, 33]. The first solutions of maximally regular and implicitly fully-isotropic parallel robot were developed at the same time and independently by Carricato and Parenti-Castelli at University of Genoa, Kim and Tsai at University of California, Gosselin and Kong at University of Laval, and the author at the French Institute for Advanced Mechanics (IFMA). In 2002, the four groups published the first results of their works [34–38]. Each of the last three groups has built a prototype of this

18

1 Introduction

T3-type translational parallel robot in their research laboratories and has called this robot CPM [35], Orthogonal Tripteron [39] or Isoglide3-T3 [40]. The first physical implementation of this robot was the CPM developed at University of California by Kim and Tsai [35]. An innovative solution of fully-isotropic T3-type translational parallel robot called Pantopteron was later proposed by Briot and Bonev [41]. In this solution based on pantograph linkages, the moving platform moves several times faster than its linear actuators. Various other types of maximally regular and implicitly fully-isotropic parallel robotic manipulators have been proposed in the last years (see Table 1.2). These solutions can be applied in machining applications [42] or haptic devices [43]. Overconstrained and non overconstrained solutions of parallel manipulators with coupled, decoupled and uncoupled motions of the moving platform along with maximally regular solutions are presented in the following sections of this work. These solutions are actuated by linear and/or rotating motors situated on the fixed base or on a moving link. Basic and derived topologies are presented in this work. Basic topologies do not combine idle mobilities and the number of limbs is equal to the number of independent motions of moving platform k = SF. To reduce the number of overconstraints in the parallel robot, derived solutions are used. They are obtained from the basic topologies by combining various idle mobilities in the kinematic

Table 1.2 Literature dedicated to maximally-regular and implicitly fully-isotropic parallel robotic manipulators No. Type of parallel robotic References manipulator 1.

T3-type

2. 3. 4.

R2-type parallel wrist R3-type parallel wrist R3-type redundantly-actuated parallel wrists Planar T2R1-type Spatial T2R1-type Spatial T2R1-type with planar motion of the moving platform T1R2-type T3R1-type with Schönflies motions T2R2-type T1R3-type T1R3-type redundantly-actuated T2R3-type redundantly-actuated T3R2-type T3R2-type redundantly-actuated T3R3-type hexapod

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Carricato and Parenti-Castelli [34], Gogu [38, 40, 74], Gosselin and Kong [36], Kim and Tsai [35], Kong and Gosselin [37, 75, 76], Rizk et al. [77], Stan et al. [78], Wu et al. [79] Gogu [80] Gogu [81] Gogu [82] [83, 84] Gogu [16, 85, 86], Zhang et al. [87] Gogu [84, 88] Gogu [89] Carricato [70], Gogu [68, 90, 90, 91, Gogu 42] Gogu [92] Gogu [93, 94] Gogu [95] Gogu [96] Gogu [91, 97, 98] Gogu [99] Gogu [97]

1.2 Methodology of Structural Synthesis

19

pairs. Derived solutions can also be obtained by reducing the number of limbs and integrating more than one actuator in some limbs. The basic topologies presented in this work can be fully parallel, overactuated or redundantly actuated solutions. The overactuated solutions have MF = SF and TF = 0. Overactuation is used in some additional motors to obtain a better payload repartition on the actuated joints of the parallel robot. The motion of the additional motors is strictly correlated with the motion of the MF basic motors. In the absence of this correlation, additional constraints are introduced in the links of the parallel robot. The redundantly actuated solutions are characterized by MF [ SF and TF [ 0. In these solutions TF actuators are used to control some internal mobilities in the limbs. Usually, redundancy in parallel manipulators is used to eliminate some singular configurations, to minimize the joint rates, to optimize the joint torques/forces, to increase dexterity workspace, stiffness, eigenfrequencies, kinematic and dynamic accuracy, to improve both kinematic and dynamic control algorithms, see [44–62]. In this work, a new use of redundancy is presented for motion decoupling, as presented for the first time in Gogu [63]. The following structural parameters associated with each solution presented in this book are systematized in the various tables of each chapter: m number of links including the fixed base, pGi number of joints in the Gi-limb, p total number of joints in the parallel mechanism given by Eq. (1.1), q number of independent closed loops in the parallel mechanism given by Eq. (1.9), k1 number of simple limbs, number of complex limbs, k2 k total number of limbs k = k1 ? k2, basis of the vector space of relative velocities between the moving and (RGi) reference platforms in Gi-limb disconnected from the parallel mechanism, SGi connectivity between the moving and reference platforms in Gi-limb disconnected from the parallel mechanism, given by Eq. (1.6), number of joint parameters that lost their independence in the closed rGi loops combined in Gi-limb, MGi mobility of Gi-limb, given by Eq. (1.17), (RF) basis of the vector space of relative velocities between the moving and reference platforms in the parallel mechanism given by Eq. (1.2), connectivity between the mobile and reference platforms in the parallel SF mechanism given by Eq. (1.7), rl total number of joint parameters that lose their independence in the closed loops combined in the k limbs given by Eq. (1.10), total number of joint parameters that lose their independence in the rF closed loops combined in the parallel mechanism given by Eq. (1.8), MF mobility of the parallel mechanism given by Eq. (1.3), number of overconstraints in the parallel mechanism given by Eq. (1.4), NF TF degree of structural redundancy of the parallel mechanism given by Eq. (1.5),

20

1 Introduction

Ppi

j¼1 fj

Pp

j¼1 fj

total number of degrees of mobility in the pi joints of limb i, where fj is the mobility of joint j, total number of degrees of mobility in the p joints of the parallel mechanism

1.3 Parallel Robots with Schönflies Motions of the Moving Platform This book focuses on the structural synthesis of basic overconstrained parallel robotic manipulators (PMs) with Schönflies motions of the moving platform. Parallel manipulators with Schönflies motions are the parallel counterparts of the well-known SCARA robots. The end-effector of these robots has four degrees of freedom, which are three independent translations (T3) and one rotation (R1) around an axis of fixed direction. This motion T3R1-type was study by the German mathematician Arthur Moritz Schönflies (Schoenflies) and is usually called Schönflies motion [64]. The referenced book of Schönflies [64] is the translation from German into French of ‘‘Geometrie der Bewegung in synthetischer Darstellung’’, Teubner, Leipzig, 1886 where author’s name is also spelt Schoenflies. This is why, probably for the first time, Bottema and Roth [65] named a special motion type Schoenflies motion. In Schönflies’ book as well as in Bottema and Roth book, a so-called Schoenflies motion is described as being a rigid-body motion having axodes (surfaces generated by the instantaneous screw axes in any time-dependent motion) that are cylinders (or prismatic surfaces) rather than being a four-degree-of-freedom (4-dof) motion. Schönflies is more known for his important contribution in the theory of 230 discrete (non-continuous) subgroups of displacements as a basic tool in crystallography developed independently by Schönflies and Fedorov. Today, continuous (non-discrete) groups of transformations are classically called Lie groups. The relation between Schönflies subgroup and the 4-dof motion SCARA robot (T3R1-type) appears in Hervé [66]. Moreover, the comprehensive enumeration of 4-dof serial kinematic chains generating Schönflies motions made of one dof kinematic pairs and plane-hinged parallelogram is presented by Hervé and Sparacino [67]. The first solutions of fully-isotropic parallel manipulators with Schönflies motions have been recently proposed [11, 68, 69], Caricato [70]. The parallel manipulators with Schönflies motions of the moving platform give three translational velocities v1, v2 and v3 along with one rotational velocity xi, (i = a, b or d) in the basis of the operational velocity vector space (RF) = (v1, v2, v3,xi). The connectivity between the moving and fixed platforms in the parallel manipulators with Schönflies motions is SF = 4. This kind of parallel robots are useful in pick-and-place operations when the end-effector only needs to undergo spatial translational motion and one rotation.

1.3 Parallel Robots with Schönflies Motions of the Moving Platform

21

Pick and place parallel robots are also typically used in light industries such as the electronics and packaging industries. They have to repeat accurately a simple transfer operation many times over at a relatively high speed in three degrees of freedom spatial motion and orienting the moving platform about a fixed direction. The direct kinematic model of the T3R1-type parallel robots becomes 2 3 2 3 v1 q_ 1 6 v2 7 6 q_ 2 7 6 7 ¼ ½J  6 7 ð1:18Þ 44 4 q 4 v3 5 _3 5 xi q_ 4 where:v1 = vx= x_ , v2 = vy= y_ and v3 = vz= z_ are the translational velocities of the characteristic point H of the moving platform, xi, (i = a, b or d) is the rotational motion of the moving platform, q_ 1 , q_ 2 , q_ 3 and q_ 4 are the velocities of the actuated joints, J44 is the Jacobian matrix. The various architectures of PMs with Schönflies motions presented in the literature use the following types of kinematic pairs: revolute R, prismatic P, helical H, cylindrical C, spherical S, planar contact E, universal joint U as well as the parallelogram loop Pa which can be considered as a complex pair of circular translation [71–73]. The fully-parallel overconstrained solutions of parallel robotic manipulator with Pp Schönflies motions and q independent loops meet the condition 1 fi \4 þ 6q. Table 1.3 presents examples of implemented PMs with Schönflies motions and Table 1.4 focuses on the literature dedicated to the study of this kind of parallel robots. The various methods used in their structural synthesis are systematized in Table 1.5.

Table 1.3 Examples of implemented parallel robotic manipulators with Schönflies motion of the moving platform No. Robot name References 1. 2. 3. 4.

HELIA4 Isoglide4-T3R1 I4 H4

Kanuk Manta McGill Schönfliesmotion generator 8. Quadriglide 9. Quadrupteron 10. PAMINSA 11. Par4 12. Pantopteron-4 5. 6. 7.

Krut et al. [100] Gogu [1, 42, 68, 69, 90, 101], Rizk et al. [102] Krut et al. [103] Andreff et al. [104], Choi et al. [105], Pierrot and Company [106], Pierrot et al. [107, 108] Rolland [109] Rolland [109] Alizadeh et al. [110], Morozov and Angeles [165], Caro et al. [112], Gauthier et al. [113], Morozoc and Angeles (2007) Ancuta et al. [114] Gosselin [115] Arakelian et al. [116], Brito et al. 117, 118, 119] Nablat et al. [120] Briot and Bonev [121]

22

1 Introduction

Table 1.4 Literature dedicated to the study of parallel manipulators with Schönflies motion of the moving platform No. Type of study References 1.

Accuracy

2. 3. 4.

Calibration Control Dynamic modelling

5. 6.

Identification Kinematic analysis

7.

Kinetostatics

8.

Optimal design

9.

Singularity analysis

10. Stiffness 11. Structural analysis 12. Structural synthesis

13. Workspace analysis and optimization

Briot and Bonev [122], Guo et al. [123], Andreff et al. [104], Rizk et al. [102] Andreff et al. [104] Pierrot et al. [107, 108], Choi et al. [105] Altuzarra et al. [124], Arakelian et al. [116], Cammarata et al. [125], Corral et al. [126], Company et al. [127], Gosselin [115], Gao et al. [214], Kong and Gosselin [128], Nablat et al. [120, 129] Nablat et al. [120], Renaud et al. [130] Briot and Bonev [121], Company et al. [131], Forgo [132], Gosselin et al. [133], Kim et al. [134, 135], Kong and Gosselin [136], Li et al. [137], Richard et al. [138], Salgado et al. [139, 140], Tanev [141], Wang and Gosselin [142], Yuan and Zhang [143], Zhao and Huang [144], Zhao et al. [145] Alizadeh et al. [110], Arakelian et al. [116], Gauthier et al. [113], Lu et al. [146] Altuzzara et al. [147, 148], Al-Widyan and Angeles [149], Ancuta et al. [114], Angeles [150], Angeles et al. [1, 151, 152] Angeles and Morozov [111], Briot et al. [117], Briot et al. [118], Briot and Bonev [41], Caro et al. [112], Choi et al. [105], Clavel [119], Company and Pierrot [153], Company et al. [107, 154–156], Hernández et al. [157], Hesselbach et al. [158], Kim et al. [135], Kokikabushiki et al. [159], Kong and Gosselin [128, 160, 161], Krut et al. [100, 103, 162], Lenarcˇicˇ et al. [163], Liu and Wang [73], Liu et al. [164], Morozov and Angeles [165], Morozov et al. [166], Nablat et al. [167–170], Pierrot and Company [106], Pierrot et al. [107, 108, 171– 174], Richard et al. [138], Rolland [109], Salgado et al. [139], Tyves et al. [175], Wang et al. [176], Yi et al. [177], Zlatanov and Gosselin [178] Amine et al. [179, 180], Briot and Bonev [121], Company et al. [131], Kim et al. [181], Lee and Lee [182], Lu et al. [183], Tyves et al. [175], Wang and Gosselin [142], Zhou et al. [184] Company et al. [185], Pinto et al. [186], Rizk et al. [77] Angeles [150], Chen [187], Chen et al. [188], Liu and Shi [189] Alizade and Bayram [190], Altuzarra et al. [191], Angeles [150], Campos et al. [192], Caro et al. [112], Carricato [70], Chen [193], Chen and Li [194], Fang and Tsai [195], Gao et al. [196], Glazunov [197], Glazunov et al. [198], Gogu [79, 88, 91, 93, 99], Guo et al. [199], Hervé [66, 200], Huang and Li [72, 201], Jin and Yang [202] Altuzarra et al. [203], Fang et al. [204, 205], Lee and Lee [182]

References

23

Table 1.5 Approaches used in the structural synthesis of parallel robotic manipulators with Schönflies motion of the moving platform No Approach References 1. 2. 3. 4. 5.

Assur groups CAD functionalities Constraint method Evolutionary morphology Group theory

6. 7.

Pattern analysis Screw theory

8.

Special Plücker coordinates 9. Structural groups 10. Structural parameters 11. Theory of linear transformations

Campos et al. [192] Lu [206] Huang and Li [72] Gogu [79, 88, 91, 93, 99] Altuzzara et al. [191], Angeles [150, 207], Caro et al. [112], Hervé [66, 200], Lee and Hervé [208–210], Lee et al. [211], Salgado et al. [140, 212] Chen [193] Carricato [70], Chen and Li [194], Fang and Tsai [195], Glazunov [197], Guo et al. [199], Huang and Li [72], Kong and, Gosselin [128, 213] Gao et al. [196] Alizade and Bayram [190] Gogu [42, 68, 69, 90, 101] Gogu [42, 68, 69, 90, 101]

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186. Pinto C, Corral J, Altuzarra O, Hernandez A (2010) A methodology for static stiffness mapping in lower mobility parallel manipulators with decoupled motions. Robotica 28(5):719–735 187. Chen C (2010) Mobility analysis of parallel manipulators and pattern of transform matrix. ASME J Mech Rob 2(4):041003 188. Chen Q, Li Q, Wu C, Hu X, Huang Z (2009) Mobility analysis of 4-RPRPR and 4-RRRPR parallel mechanisms with bifurcation of Schoenflies motion by screw theory. In: Dai JS, Zoppi M, Kong X (eds) Reconfigurable Mech Rob. KC Edizioni, Genova, pp 285–290 189. Liu PA, Shi XH (2011) Topology analyses and position solution based on asymmetric and fewer DOF parallel robots. In: J Gao (Ed) Advanced design technology, Trans Tech Publications, pp 1252–1257 190. Alizade R, Bayram C (2004) Structural synthesis of parallel manipulators. Mech Mach Theor 39:857–870 191. Altuzarra O, Loizaga M, Pinto C, Petuya V (2010) Synthesis of partially decoupled multilevel manipulators with lower mobility. Mech Mach Theor 46:106–118 192. Campos A, Budde C, Hesselbach J (2008) A type synthesis method for hybrid robot structures. Mech Mach Theor 43(8):984–995 193. Chen C (2010) A new Schöenfiles motion parallel manipulator. In: IEEE/ASME international conference on mechatronics and embedded systems and applications (MESA), pp 103–108 194. Chen Y, Li B (2009) Topology synthesis and classification of a novel family of parallel manipulator with 3-dimension translation and 1-dimension rotation, international conference on information and automation, Zhuhai, Macau 195. Fang Y, Tsai LW (2002) Structure synthesis of a class of 4-dof and 5-dof parallel manipulators with identical limb structures. Int J Rob Res 21(9):799–810 196. Gao F, Li W, Zhao X, Jin Z, Zhao H (2002) New kinematic structures for 2-, 3-, 4-, and 5DOF parallel manipulator designs. Mech Mach Theor 37(11):1395–1411 197. Glazunov V (2010) Design of decoupled parallel manipulators by means of the theory of screws. Mech Mach Theor 45(2):239–250 198. Glazunov V, Levin S, Shalyukhin K, Hakkyoglu M, Tung VD (2010) Development of mechanisms of parallel structure with four degrees of freedom and partial decoupling. J Mach Manuf Reliab 39(5):407–411 199. Guo S, Fang Y, Qu H (2012) Type synthesis of 4-DOF nonoverconstrained parallel mechanisms based on screw theory. Robotica 30(1):31–37 200. Hervé JM (1995) Design of parallel manipulators via the displacement group. In: Proceedings of the 9th world congress on the theory of machines and mechanisms. Milan, pp 2079–2082 201. Huang Z, Li QC (2002) General methodology for type synthesis of symmetrical lowermobility parallel manipulators and several novel manipulators. Int J Rob Res 21(2):131–145 202. Jin Q, Yang TL (2002) Structure synthesis of parallel manipulators with three-dimensional translation and one-dimensional rotation. In: Proceedings on ASME design engineering technical conference, Paper MECH-34307, Montreal 203. Altuzarra O, Pinto C, Sandru B, Hernandez A (2011) Optimal dimensioning for parallel manipulators: workspace, dexterity, and energy. Trans.ASME J Mech Design 133, 041007 204. Fang H, Chen J, Qu H (2009) Research on a novel four-degree-of-freedom parallel manipulator with scaling unit. In: Proceedings of IEEE international conference on robotics and biomimetics (ROBIO), pp 1762–1767, Guilin 205. Fang H, Chen J, Wang C (2009) Mechanism analysis of a novel four-degree-of-freedom parallel manipulator based on larger workspace. In: IEEE international conference on automation and logistics, Shenyang 206. Lu Y (2004) Using CAD functionalities for the kinematics analysis of spatial parallel manipulators with 3-, 4-, 5-, 6-linearly driven limbs. Mech Mach Theor 39:41–60

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207. Angeles J (2005) The degree of freedom of parallel robots: a group-theoretic approach. In: Proceedings of IEEE international conference robotics and automation, Barcelona, pp 1005–1012 208. Lee CC, Hervé JM (2008) Uncoupled actuation of overconstrained 3T-1R hybrid parallel manipulators. Robotica 27(1):103–117 209. Lee CC, Hervé JM (2009) On some applications of primitive Schönflies-motion generators. Mech Mach Theor 44(12):2153–2163 210. Lee CC, Hervé JM (2011). Isoconstrained parallel generators of Schoenflies motion. J. Mech. Rob 3(2), paper 021006-1 211. Lee PC, Lee JJ, Lee CC (2010) Four novel pick-and-place isoconstrained manipulators and their inverse kinematics. In: Proceedings on ASME Design Eng Tech Conf 212. Salgado O, Altuzarra O, Petuya V, Hernandez A (2007) Type synthesis of a family of 3T1R fully-parallel manipulators using a group-theoretic approach. In: Proceedings of the 12th world congress in mechanism and machine science, Besançon 213. Kong X, Gosselin CM (2007) Type synthesis of parallel mechanisms. Springer, Berlin 214. Guo X, Liu S, Wang Z (2006) Analysis for dynamics performance indices of 4-RR(RR)R parallel mechanism. Int J Innovative Comput, Inf Control 2(4):875–884

Chapter 2

Fully-Parallel Topologies with Coupled Schönflies Motions

In the general case, in a parallel robotic manipulator with coupled Schönflies motions each operational velocity depends on four actuated joint velocities vi ¼ vi ðq_ 1 ; q_ 2 ; q_ 3 ; q_ 3 ; q_ 4 Þ, i = 1, 2, 3 and xd ¼ xd ðq_ 1 ; q_ 2 ; q_ 3 ; q_ 4 Þ. In some specific solutions, one or two operational velocities depend on just one or two actuator velocities. We note that, in this particular case, if the Jacobian matrix in Eq. (1.18) is not triangular the parallel robot always has coupled motions. They could have just a few partially decoupled motions. The fully-parallel overconstrained topologies of parallel robotic manipulators with coupled Schönflies motions may have identical limbs or limbs with different topologies and could be actuated by linear or rotating motors. The limbs can be simple or complex kinematic chains and can also combine idle mobilities. The actuators can be mounted on the fixed base or on a moving link. The first solution has the advantage of reducing the moving masses and large workspace. The second solution would be more compact. There are no idle mobilities in the fully-parallel basic topologies presented in this section. The solutions presented in this section are obtained by using the methodology of structural synthesis proposed in Part 1 [1] and also used in Parts 2–4 of this work [2–4]. This original methodology combines new formulae for mobility connectivity, redundancy and over constraints, and the evolutionary morphology in a unified approach of structural synthesis of parallel robotic manipulators.

2.1 Topologies with Simple Limbs In the fully-parallel topologies of PMs with coupled Schönflies motions F / G1G2-G3-G4 presented in this section, the moving platform n : nGi (i = 1, 2, 3, 4) is connected to the reference platform 1 : 1Gi : 0 by four spatial simple limbs with five degrees of connectivity. The various types of simple limbs with five degrees of connectivity used in the fullyparallel basic topologies illustrated in this section are presented in Fig. 2.1. The simple limbs combine only revolute, prismatic and cylindrical joints. One actuator is combined G. Gogu, Structural Synthesis of Parallel Robots, Solid Mechanics and Its Applications 206, DOI: 10.1007/978-94-007-7401-8_2,  Springer Science+Business Media Dordrecht 2014

35

36

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.1 Simple limbs for fully-parallel PMs with coupled Schönflies motions defined by MG ¼ SG ¼ 5; ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa ; xd Þ

in each limb. The actuated joint is underlined in the structural graph. In the cylindrical joint denoted by C just one motion is actuated. This can be the translational or the rotational motion and it is indicated in the structrural diagram by a linear or circular arrow.

2.1 Topologies with Simple Limbs

37

Fig. 2.1 (continued)

Various topologies of PMs with coupled Schönflies motions and no idle mobilities can be obtained by using four limbs with identical or different topology presented in Fig. 2.1. Only topologies with four identical limb types are illustrated

38

Fig. 2.1 (continued)

2

Fully-Parallel Topologies with Coupled Schönflies Motions

2.1 Topologies with Simple Limbs

Fig. 2.1 (continued)

39

40

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.2 Fully-parallel PMs with coupled Schönflies motions of types 4RRRRR (a) and 4RRRRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R\R||R||R\k R (a) and R||R\R||R\R (b)

2.1 Topologies with Simple Limbs

41

Fig. 2.3 4RRRRR-type fully-parallel PMs with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R||R||R\R||R

42

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.4 Fully-parallel PMs with coupled Schönflies motions of types 4RRRRR (a) and 4PRRRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R||R\R||R||R (a) and P\R\R||R||R (b)

2.1 Topologies with Simple Limbs

43

Fig. 2.5 Fully-parallel PMs with coupled Schönflies motions of types 4RRRPR (a) and 4RRPRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R||R\R\P\||R (a) and R||R||P\R||R (b)

44

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.6 4PRRRR-type fully-parallel PMs with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology P\R\R||R\R (a) and P||R\R||R\R (b)

2.1 Topologies with Simple Limbs

45

Fig. 2.7 Fully-parallel PMs with coupled Schönflies motions of types 4RRPRR (a) and 4PRRRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R\R\P\kR\R (a) and P\R||R\R||R (b)

46

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.8 Fully-parallel PMs with coupled Schönflies motions of types 4PRRRR (a) and 4RPRRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology P||R||R||R||R\R (a) and R||P||R||R\R (b)

2.1 Topologies with Simple Limbs

47

Fig. 2.9 Fully-parallel PMs with coupled Schönflies motions of types 4RRPRR (a) and 4RRRPR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R||R||P||R\R (a) and R||R||R||P\R (b)

48

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.10 Fully-parallel PMs with coupled Schönflies motions of types 4RPRRR (a) and 4RRPRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R||P||R||R\R (a) and R||R||P||R\R (b)

2.1 Topologies with Simple Limbs

49

Fig. 2.11 Fully-parallel PMs with coupled Schönflies motions of types 4RRRRP (a) and 4RPRRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R||R||R\R\P (a) and R\P\kR\R||R (b)

50

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.12 Fully-parallel PMs with coupled Schönflies motions of types 4RRRRP (a) and 4RRRPR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R\R||R||R||P (a) and R\R||R||P||R (b)

2.1 Topologies with Simple Limbs

51

Fig. 2.13 Fully-parallel PMs with coupled Schönflies motions of types 4RRPRR (a) and 4RPRRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R\R||P||R||R (a) and R\P||R||R||R (b)

52

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.14 Fully-parallel PMs with coupled Schönflies motions of types 4RRRPR (a) and 4RRPRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R\R||R||P||R (a) and R\R||P||R||R||R (b)

2.1 Topologies with Simple Limbs

53

Fig. 2.15 Fully-parallel PMs with coupled Schönflies motions of types 4RPRRP (a) and 4PRRRP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R\P\kR\R\P (a) and P\R||R\R\P (b)

54

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.16 4PRRPR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology P\R\R\P\kR

in Figs. 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, 2.11, 2.12, 2.13, 2.14, 2.15, 2.16, 2.17, 2.18, 2.19, 2.20. The limb topology and connecting conditions of the solutions in Figs. 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, 2.11, 2.12, 2.13, 2.14, 2.15, 2.16, 2.17, 2.18, 2.19, 2.20 are systematized in Table 2.1, as are their structural parameters in Table 2.2.

2.1 Topologies with Simple Limbs

55

Fig. 2.17 4CRRR-type fully-parallel PMs with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology C\R||R\R (a) and C||R||R\R (a)

56

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.18 Fully-parallel PMs with coupled Schönflies motions of types 4RRCR (a) and 4RCRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R||R||C\R (a) and R||C||R\R (b)

2.1 Topologies with Simple Limbs

57

Fig. 2.19 Fully-parallel PMs with coupled Schönflies motions of types 4RRRC (a) and 4RCRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R\R||R||C (a) and R\C||R||R (b)

58

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.20 Fully-parallel PMs with coupled Schönflies motions of types 4RRCR (a) and 4RCRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R\R||C||R (a) and R||C\R||R (b)

2.1 Topologies with Simple Limbs

59

Table 2.1 Limb topology and connecting conditions of the fully-parallel solutions with no idle mobilities presented in Figs. 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, 2.11, 2.12, 2.13, 2.14, 2.15, 2.16, 2.17, 2.18, 2.19, 2.20 No. PM type Limb topology Connecting conditions 1.

4RRRRR (Fig. 2.2a)

R\R||R||R\k R (Fig. 2.1a)

2.

4RRRRR (Fig. 2.2b)

R||R\R||R\R (Fig. 2.1b)

3.

4RRRRR (Fig. 2.3a)

R||R||R\R||R (Fig. 2.1c)

4.

4RRRRR (Fig. 2.3b)

R||R||R\R||R (Fig. 2.1c)

5.

4RRRRR (Fig. 2.4a)

R||R\R||R||R (Fig. 2.1d)

6.

4PRRRR (Fig. 2.4b)

P\R\R||R||R (Fig. 2.1e)

7. 8. 9.

4RRRPR (Fig. 2.5a) 4RRPRR (Fig. 2.5b) 4PRRRR (Fig. 2.6a)

R||R\R\P\kR (Fig. 2.1f) R||R||P\R||R (Fig. 2.1g) P\R\R||R\R (Fig. 2.1h)

10. 11. 12. 13.

4PRRRR 4RRPRR 4PRRRR 4PRRRR

(Fig. (Fig. (Fig. (Fig.

2.6b) 2.7a) 2.7b) 2.8a)

P||R\R||R\R (Fig. 2.1i) R\R\P\kR\R (Fig. 2.1j) P\R||R\R||R (Fig. 2.1k) P||R||R||R||R\R (Fig. 2.1l)

14. 15. 16. 17. 18. 19.

4RPRRR 4RRPRR 4RRRPR 4RPRRR 4RRPRR 4RRRRP

(Fig. (Fig. (Fig. (Fig. (Fig. (Fig.

2.8b) 2.9a) 2.9b) 2.10a) 2.10b) 2.11a)

R||P||R||R\R (Fig. 2.1m) R||R||P||R\R (Fig. 2.1n) R||R||R||P\R (Fig. 2.1o) R||P||R||R\R (Fig. 2.1p) R||R||P||R\R (Fig. 2.1q) R||R||R\R\P (Fig. 2.1r)

20. 21.

4RPRRR (Fig. 2.11b) 4RRRRP (Fig. 2.12a)

R\P\kR\R||R (Fig. 2.1s) R\R||R||R||P (Fig. 2.1t)

22. 23. 24. 25. 26. 27. 28. 29. 30.

4RRRPR (Fig. 2.12b) 4RRPRR (Fig. 2.13a) 4RPRRR (Fig. 2.13b) 4RRRPR (Fig. 2.14a) 4RRPRR (Fig. 2.14b) 4RPRRP (Fig. 2.15a) 4PRRRP (Fig. 2.15b) 4PRRPR (Fig. 2.16) 4CRRR (Fig. 2.17a)

R\R||R||P||R Fig. 2.1u) R\R||P||R||R (Fig. 2.1v) R\P||R||R||R (Fig. 2.1w) R\R||R||P||R (Fig. 2.1x) R\R||P||R||R||R (Fig. 2.1y) R\P\kR\R\P (Fig. 2.1z) P\R||R\R\P (Fig. 2.1z1) P\R\R\P\kR (Fig. 2.1a0 ) C\R||R\R (Fig. 2.1d0 )

The first and the last revolute joints of the four limbs have parallel axes The first, the second and the last revolute joints of the four limbs have parallel axes The two last revolute joints of the four limbs have parallel axes The three first revolute joints of the four limbs have parallel axes The two first revolute joints of the four limbs have parallel axes. The second joints of the four limbs have parallel axes Idem No. 5 Idem No. 5 The second and the last joints of the four limbs have parallel axes Idem No. 9 Idem No. 1 Idem No. 3 The last revolute joints of the four limbs have parallel axes Idem No. 13 Idem No. 13 Idem No. 13 Idem No. 13 Idem No. 13 The before last revolute joints of the four limbs have parallel axes Idem No. 3 The first revolute joints of the four limbs have parallel axes Idem No. 21 Idem No. 21 Idem No. 21 Idem No. 21 Idem No. 21 Idem No. 19 Idem No. 19 Idem No. 5 Idem No. 1 (continued)

60

2

Table 2.1 (continued) No. PM type 31. 32. 33. 34. 35. 36. 37.

4CRRR 4RRCR 4RCRR 4RRRC 4RCRR 4RRCR 4RCRR

(Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig.

2.17b) 2.18a) 2.18b) 2.19a) 2.19b) 2.20a) 2.20b)

Fully-Parallel Topologies with Coupled Schönflies Motions

Limb topology C||R||R\R R||R||C\R R||C||R\R R\R||R||C R\C||R||R R\R||C||R R||C\R||R

(Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig.

Connecting conditions 2.1e0 ) 2.1f0 ) 2.1g0 ) 2.1h0 ) 2.1l0 ) 2.1b0 ) 2.1c0 )

Idem Idem Idem Idem Idem Idem Idem

No. No. No. No. No. No. No.

13 13 13 21 21 21 21

Table 2.2 Structural parameters of parallel mechanisms in Figs. 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, 2.11, 2.12, 2.13, 2.14, 2.15, 2.16, 2.17, 2.18, 2.19, 2.20 No. Structural parameter Solution

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

m pi (i = 1,…, 4) p q k1 k2 k (RG1) (v1 ; v2 ; v3 ; xb ; xd ) (RG2) (v1 ; v2 ; v3 ; xa ; xd ) (RG3) (v1 ; v2 ; v3 ; xb ; xd ) (RG4) (v1 ; v2 ; v2 ; xa ; xd ) SGi (i = 1,…,4) rGi (i = 1,…,4) MGi (i = 1,…,4) (RF) SF rl rF MF NF TF Pp1 fj Ppj¼1 2 j¼1 fj

Figures 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, 2.11, 2.12, 2.13, 2.14, 2.15, 2.16

Figures 2.17, 2.18, 2.19, 2.20

18 5 20 3 4 0 4 (v1 ; v2 ; v3 ; xb ; xd )

14 4 16 3 4 0 4

(v1 ; v2 ; v3 ; xa ; xd ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) 5 0 5 (v1 ; v2 ; v3 ; xd ) 4 0 16 4 2 0 5 5

5 0 5 (v1 ; v2 ; v3 ; xd ) 4 0 16 4 2 0 5 5 (continued)

2.2 Topologies with Complex Limbs Table 2.2 (continued) No. Structural parameter

61

Solution Figures 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, 2.11, 2.12, 2.13, 2.14, 2.15, 2.16

24. 25. 26.

Pp3

fj Ppj¼1 4 f j Ppj¼1 f j¼1 j

Figures 2.17, 2.18, 2.19, 2.20

5

5

5 20

5 20

m number of links including the fixed base, pGi number of joints in the Gi-limb, p total number of joints in the parallel mechanisma , q number of independent closed loops in the parallel mechanismb , k1 number of simple limbs, k2 number of complex limbs, k total number of limbsc , (RGi) basis of the vector space of relative velocities between the moving and reference platforms in Gi-limb disconnected from the parallel mechanism, SGi connectivity between the moving and reference platforms in Gi-limb disconnected from the parallel mechanismd , rGi number of joint parameters that lost their independence in the closed loops combined in Gi-limb, MGi mobility of Gi-limbe , (RF) basis of the vector space of relative velocities between the moving and reference platforms in the parallel mechanismf , SF connectivity between the mobile and reference platforms in the parallel mechanismg , rl total number of joint parameters that lose their independence in the closed loops combined in the k limbsh , rF total number of joint parameters that lose their independence in the closed loops combined in the parallel mechanismi , MF mobility of the parallel mechanismj , NF number of over constraints in the parallel mechanismk , TF degree of structural redundancy of the parallel mechanisml , fj mobility of jth joint. P a p ¼ ki¼1 pGi ; b q = p - m ? 1, c k = k1 ? k2, d SGi = dim(R P Gi Gi), i = 1, 2,…,k, e MGi ¼ pj¼1 fj  rGi , i = 1, 2,…,k, f (RF) = (RG1) \… \ (RGk), g SF = dim(RF), P h rl ¼ ki¼1 rGi ; P i rF ¼ ki¼1 SGi  SF þ rl ; P j MF ¼ pi¼1 fi  rF ; k NF ¼ 6q  rF ; l TF ¼ MF  SF :

2.2 Topologies with Complex Limbs In the fully-parallel topologies of PMs with coupled Schönflies motions F / G1G2-G3-G4 presented in this section, the moving platform n : nGi (i = 1, 2, 3, 4) is connected to the reference platform 1 : 1Gi : 0 by four spatial complex limbs with four or five degrees of connectivity. The complex limbs combine only revolute, prismatic and cylindrical joints. One actuator is combined in each limb. The actuated joint is underlined in the structural graph. In the cylindrical joint denoted by C just one motion is actuated. This can be the translational or the rotational motion and it is indicated in the structrural diagram by a linear or a circular arrow.

62

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.21 Complex limbs for fully-parallel PMs with coupled Schönflies motions, combining one parallelogram loop, defined by MG ¼ SG ¼ 4; ðRG Þ ¼ ðv1 ; v2 ; v3 ; xd Þ

2.2 Topologies with Complex Limbs

Fig. 2.21 (continued)

63

64

Fig. 2.21 (continued)

2

Fully-Parallel Topologies with Coupled Schönflies Motions

2.2 Topologies with Complex Limbs

65

Fig. 2.22 Complex limbs for fully-parallel PMs with coupled Schönflies motions, combining two parallelogram loops, defined by MG ¼ SG ¼ 4; ðRG Þ ¼ ðv1 ; v2 ; v3 ; xd Þ

66

Fig. 2.22 (continued)

2

Fully-Parallel Topologies with Coupled Schönflies Motions

2.2 Topologies with Complex Limbs

Fig. 2.22 (continued)

67

68

Fig. 2.22 (continued)

2

Fully-Parallel Topologies with Coupled Schönflies Motions

2.2 Topologies with Complex Limbs

Fig. 2.22 (continued)

69

70

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.22 (continued)

The complex limbs with four degrees of connectivity presented in Figs. 2.21, 2.22, 2.23 combine one, two or three planar parallelogram loops. One or two planar parallelogram loops are combined in the complex limbs with five degrees of connectivity presented in Figs. 2.24 and 2.25. Each planar parallelogram loop Pa introduces three degrees of over constraint. Various topologies of PMs with coupled Schönflies motions and no idle mobilities can be obtained by using four limbs with identical or different topology presented in Figs. 2.21, 2.22, 2.23, 2.24, 2.25. Only topologies with four identical limb types are illustrated in Figs. 2.26, 2.27, 2.28, 2.29, 2.30, 2.31, 2.32, 2.33, 2.34, 2.35, 2.36, 2.37, 2.38, 2.39, 2.40, 2.41, 2.42, 2.43, 2.44, 2.45, 2.46, 2.47, 2.48, 2.49, 2.50, 2.51, 2.52, 2.53, 2.54, 2.55, 2.56, 2.57, 2.58, 2.59, 2.60, 2.61, 2.62, 2.63, 2.64, 2.65, 2.66, 2.67, 2.68, 2.69, 2.70, 2.71, 2.72, 2.73, 2.74, 2.75, 2.76, 2.77, 2.78, 2.79, 2.80, 2.81, 2.82, 2.83, 2.84, 2.85, 2.86, 2.87, 2.88, 2.89, 2.90, 2.91, 2.92, 2.93, 2.94, 2.95, 2.96, 2.97, 2.98, 2.99, 2.100, 2.101, 2.102, 2.103, 2.104, 2.105, 2.106, 2.107, 2.108, 2.109, 2.110, 2.111, 2.112, 2.113, 2.114, 2.115, 2.116, 2.117, 2.118, 2.119. The limb topology and connecting conditions of the solutions in Figs. 2.26, 2.27, 2.28, 2.29, 2.30, 2.31, 2.32, 2.33, 2.34, 2.35, 2.36, 2.37, 2.38, 2.39, 2.40, 2.41, 2.42, 2.43, 2.44, 2.45, 2.46, 2.47, 2.48, 2.49, 2.50, 2.51, 2.52, 2.53, 2.54, 2.55, 2.56, 2.57, 2.58, 2.59, 2.60, 2.61, 2.62, 2.63, 2.64, 2.65, 2.66, 2.67, 2.68,

2.2 Topologies with Complex Limbs

71

Fig. 2.23 Complex limbs for fully-parallel PMs with coupled Schönflies motions, combining three parallelogram loops, defined by MG ¼ SG ¼ 4; ðRG Þ ¼ ðv1 ; v2 ; v3 ; xd Þ

72

Fig. 2.23 (continued)

2

Fully-Parallel Topologies with Coupled Schönflies Motions

2.2 Topologies with Complex Limbs

Fig. 2.23 (continued)

73

74

Fig. 2.23 (continued)

2

Fully-Parallel Topologies with Coupled Schönflies Motions

2.2 Topologies with Complex Limbs

Fig. 2.23 (continued)

75

76

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.24 Complex limbs for fully-parallel PMs with coupled Schönflies motions, combining one parallelogram loop, defined by MG ¼ SG ¼ 5; ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa ; xd Þ or ðRG Þ ¼ ðv1 ; v2 ; v3 ; xb ; xd Þ

2.2 Topologies with Complex Limbs

Fig. 2.24 (continued)

77

78

Fig. 2.24 (continued)

2

Fully-Parallel Topologies with Coupled Schönflies Motions

2.2 Topologies with Complex Limbs

79

Fig. 2.25 Complex limbs for fully-parallel PMs with coupled Schönflies motions, combining two parallelogram loops, defined by MG ¼ SG ¼ 5; ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa ; xd Þ or ðRG Þ ¼ ðv1 ; v2 ; v3 ; xb ; xd Þ

80

Fig. 2.25 (continued)

2

Fully-Parallel Topologies with Coupled Schönflies Motions

2.2 Topologies with Complex Limbs

81

Fig. 2.26 Fully-parallel PMs with coupled Schönflies motions of types 4RRPaR (a) and 4RRRPa (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology R||R\Pa\kR (a) and R||R||R\Pa (b)

82

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.27 Fully-parallel PMs with coupled Schönflies motions of types 4PRPaR (a) and 4PPPaR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology P\R\Pa\kR (a) and P\P\kPa\R (b)

2.2 Topologies with Complex Limbs

83

Fig. 2.28 Fully-parallel PMs with coupled Schönflies motions of types 4PRPPa (a) and 4PPRPa (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology P||R\P||Pa (a) and P\P||R\Pa (b)

84

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.29 Fully-parallel PMs with coupled Schönflies motions of types 4PRPPa (a) and 4PRPaP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology P\R||P\Pa (a) and P\R\Pa \kP (b)

2.2 Topologies with Complex Limbs

85

Fig. 2.30 4PPPaR-type fully-parallel PMs with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology P\P\kPa\kR (a) and P\P||Pa\\R (b)

86

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.31 Fully-parallel PMs with coupled Schönflies motions of types 4PPaRP (a) and 4PPaPR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology P||Pa\R||P (a) and P||Pa\P|| R (b)

2.2 Topologies with Complex Limbs

87

Fig. 2.32 Fully-parallel PMs with coupled Schönflies motions of types 4PPaPR (a) and 4PPaRP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology P\Pa||P\\R (a) and P\Pa\kR\kP (b)

88

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.33 Fully-parallel PMs with coupled Schönflies motions of types 4RPPaP (a) and 4RPPPa (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology R\P||Pa\kP (a) and R||P\P||Pa (b)

2.2 Topologies with Complex Limbs

89

Fig. 2.34 4PaPPR-type fully-parallel PMs with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology Pa||P\P||R (a) and Pa\P\kP\kR (b)

90

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.35 Fully-parallel PMs with coupled Schönflies motions of types 4PaRPP (a) and 4PaPRP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology Pa\R||P\P (a) and Pa||P\R||P (b)

2.2 Topologies with Complex Limbs

91

Fig. 2.36 Fully-parallel PMs with coupled Schönflies motions of types 4PaPRP (a) and 4PaRPP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology Pa\P||R\P (a) and Pa\R\P\kP (b)

92

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.37 Fully-parallel PMs with coupled Schönflies motions of types 4PCPa (a) and 4PPaC (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology P\C\Pa (a) and P||Pa\C (b)

2.2 Topologies with Complex Limbs

93

Fig. 2.38 Fully-parallel PMs with coupled Schönflies motions of types 4CPPa (a) and 4PaPC (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology C\P||Pa (a) and Pa||P\C (b)

94

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.39 4PaCP-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology Pa\C\P

2.2 Topologies with Complex Limbs

95

Fig. 2.40 4PPaPaR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology P\Pa\\Pa\||R

96

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.41 4PPaRPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology P\Pa||R\Pa

2.2 Topologies with Complex Limbs

97

Fig. 2.42 4PRPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology P\R||Pa\Pa

98

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.43 4RPPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology R\P\kPa\\Pa

2.2 Topologies with Complex Limbs

99

Fig. 2.44 4PaPPaR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa\P\\Pa\\R

100

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.45 4PaPRPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa\P\kR\Pa

2.2 Topologies with Complex Limbs

101

Fig. 2.46 4PaRPPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa||R\P\\Pa

102

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.47 4RPaPPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology R||Pa\P\\Pa

2.2 Topologies with Complex Limbs

103

Fig. 2.48 4PaPaPR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa\Pa\\P\\R

104

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.49 4RPaPaR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology R\Pa||Pa\kR

2.2 Topologies with Complex Limbs

105

Fig. 2.50 4RPaPaR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology R\Pa||Pa\kR

106

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.51 4RPaRPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology R\Pa\kR\Pa

2.2 Topologies with Complex Limbs

107

Fig. 2.52 4PaRPPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa\R\P||Pa

108

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.53 4PaPaRP-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa||Pa\R\P

2.2 Topologies with Complex Limbs

109

Fig. 2.54 4PaPaPR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa||Pa||P\R

110

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.55 4PRPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology P\R\Pa||Pa

2.2 Topologies with Complex Limbs

111

Fig. 2.56 4PPaRPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology P||Pa\R\Pa

112

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.57 4PPaPaR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology P||Pa||Pa\R

2.2 Topologies with Complex Limbs

113

Fig. 2.58 4PaPPaR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa||P||Pa\R

114

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.59 4RPPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology R\P||Pa||Pa

2.2 Topologies with Complex Limbs

115

Fig. 2.60 4PaPPaR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa\P\\Pa\kR

116

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.61 4PaRPaP-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa\R\Pa\kR

2.2 Topologies with Complex Limbs

117

Fig. 2.62 4PaRPPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa\R||P\Pa

118

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.63 4PaPRPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa\P||R\Pa

2.2 Topologies with Complex Limbs

119

Fig. 2.64 4PaPaPR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa\Pa\\P||R

120

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.65 4PaPaRP-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa\Pa\\R||P

2.2 Topologies with Complex Limbs

121

Fig. 2.66 4RPaPaP-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology R\Pa\\Pa\\P

122

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.67 4RPaPPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology R\Pa\kP\\Pa

2.2 Topologies with Complex Limbs

123

Fig. 2.68 4RPPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology R||P\Pa\\Pa

124

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.69 4PRPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology P||R\Pa\\Pa

2.2 Topologies with Complex Limbs

125

Fig. 2.70 4PaCPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa\C\Pa

126

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.71 4CPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology C\Pa\\Pa

2.69, 2.70, 2.71, 2.72, 2.73, 2.74, 2.75, 2.76, 2.77, 2.78, 2.79, 2.80, 2.81, 2.82, 2.83, 2.84, 2.85, 2.86, 2.87, 2.88, 2.89, 2.90, 2.91, 2.92, 2.93, 2.94, 2.95, 2.96, 2.97, 2.98, 2.99, 2.100, 2.101, 2.102, 2.103, 2.104, 2.105, 2.106, 2.107, 2.108, 2.109, 2.110, 2.111, 2.112, 2.113, 2.114, 2.115, 2.116, 2.117, 2.118, 2.119 are systematized in Tables 2.3 and 2.4 as are their structural parameters in Tables 2.5, 2.6, 2.7, 2.8.

2.2 Topologies with Complex Limbs

127

Fig. 2.72 4PaPaC-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa\Pa\\C

128

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.73 4PaPaPaR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa\Pa||Pa\kR

2.2 Topologies with Complex Limbs

129

Fig. 2.74 4PaPaRPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa\Pa\kR\Pa

130

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.75 4PaRPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa||R\Pa||Pa

2.2 Topologies with Complex Limbs

131

Fig. 2.76 4RPaPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology R||Pa\Pa||Pa

132

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.77 4PaPaPaR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa\Pa\kPa\\R

2.2 Topologies with Complex Limbs

133

Fig. 2.78 4PaPaRPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa\Pa\\R\Pa

134

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.79 4PaRPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa\R\Pa\\Pa

2.2 Topologies with Complex Limbs

135

Fig. 2.80 4RPaPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology R\Pa\\Pa\kPa

136

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.81 4PaPaPaR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa\Pa||Pa\\R

2.2 Topologies with Complex Limbs

137

Fig. 2.82 4PaRPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa\R\Pa||Pa

138

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.83 4RPaPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology R\Pa\\Pa||Pa

2.2 Topologies with Complex Limbs

139

Fig. 2.84 4PaPaPaR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa||Pa\Pa||R

140

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.85 4PaPaRPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa||Pa\R||Pa

2.2 Topologies with Complex Limbs

141

Fig. 2.86 4PaRPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa\R\Pa\kPa

142

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.87 4RPaPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology R\Pa||Pa\kPa

2.2 Topologies with Complex Limbs

143

Fig. 2.88 4PaPaPaR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa\Pa||Pa||R

144

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.89 4PaPaRPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa\Pa||R||Pa

2.2 Topologies with Complex Limbs

145

Fig. 2.90 4PaRPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa\R||Pa||Pa

146

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.91 4RPaPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology R\Pa\kPa||Pa

2.2 Topologies with Complex Limbs

147

Fig. 2.92 4PaPaPaR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa\Pa\kPa||R

148

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.93 4PaPaRPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa\Pa\kR||Pa

2.2 Topologies with Complex Limbs

149

Fig. 2.94 4PaRPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa||R\Pa\kPa

150

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.95 4RPaPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology R||Pa\Pa\kPa

2.2 Topologies with Complex Limbs

151

Fig. 2.96 4PaPaPaR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa||Pa\Pa\kR

152

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.97 4PaPaRPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa||Pa||R\Pa

2.2 Topologies with Complex Limbs

153

Fig. 2.98 4PaRPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology Pa||R||Pa\Pa

154

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.99 4RPaPaPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 42, limb topology R||Pa||Pa\Pa

2.2 Topologies with Complex Limbs

155

Fig. 2.100 4PaRRRR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology Pa\R\R||R\R (a) and Pa||R||R\R||R (b)

156

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.101 4PaRRRR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology Pa\R||R||R\R

2.2 Topologies with Complex Limbs

157

Fig. 2.102 4RPaRRR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R\Pa\kR||R\R

158

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.103 4RRPaRR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ), TF = 0, NF = 14, limb topology R||R\Pa\kR\R

2.2 Topologies with Complex Limbs

159

Fig. 2.104 4RRRRPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R||R||R\R||Pa

160

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.105 4RRRRPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R\R||R||R\kPa

2.2 Topologies with Complex Limbs

161

Fig. 2.106 Fully-parallel PMs with coupled Schönflies motions of types 4RRRPaR (a) and 4RRPaRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R\R||R\Pa\kR (a) and R\R\Pa\kR||R (b)

162

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.107 4PaRRRR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology Pa||R\R||R||R

2.2 Topologies with Complex Limbs

163

Fig. 2.108 Fully-parallel PMs with coupled Schönflies motions of types 4RPaRRR (a) and 4PaRRPR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R\Pa\kR\R||R (a) and Pa||R\R\P\kR (b)

164

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.109 4PRPaRR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology P||R\Pa\kR\R

2.2 Topologies with Complex Limbs

165

Fig. 2.110 Fully-parallel PMs with coupled Schönflies motions of types 4RPRRPa (a) and 4PRPaRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R\P\kR\R||Pa (a) and P\R\Pa\kR\R (b)

166

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.111 4PRRRPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology P\R||R\R||Pa

2.2 Topologies with Complex Limbs

167

Fig. 2.112 4CPaRR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology C\Pa\kR\R

168

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.113 4PaRPaRR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 26, limb topology Pa\R\Pa\kR\R

2.2 Topologies with Complex Limbs

169

Fig. 2.114 4PaRPaRR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 26, limb topology Pa||R\Pa\kR\R

170

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.115 4RPaRPaR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 26, limb topology R\Pa\kR||Pa\R

2.2 Topologies with Complex Limbs

171

Fig. 2.116 4PaRRRPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 26, limb topology Pa||R||R\R||Pa

172

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.117 4RRPaRPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 26, limb topology R||R||Pa\R||Pa

2.2 Topologies with Complex Limbs

173

Fig. 2.118 4RRPaRPa-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 26, limb topology R\R\Pa\kR||Pa

174

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Fig. 2.119 4PaRPaRR-type fully-parallel PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 26, limb topology Pa||R\Pa||R||R

2.2 Topologies with Complex Limbs

175

Table 2.3 Limb topology and connecting conditions of the fully-parallel solutions with no idle mobilities presented in Figs. 2.26, 2.27, 2.28, 2.29, 2.30, 2.31, 2.32, 2.33, 2.34, 2.35, 2.36, 2.37, 2.38, 2.39, 2.40, 2.41, 2.42, 2.43, 2.44, 2.45, 2.46, 2.47, 2.48, 2.49, 2.50, 2.51, 2.52, 2.53, 2.54, 2.55, 2.56, 2.57, 2.58, 2.59, 2.60, 2.61, 2.62, 2.63, 2.64, 2.65, 2.66, 2.67, 2.68, 2.69, 2.70, 2.71, 2.72 No. PM type Limb topology Connecting conditions 1.

4RRPaR (Fig. 2.26a) R||R\Pa\kR (Fig. 2.21a)

2.

4RRRPa (Fig. 2.26b) R||R||R\Pa (Fig. 2.21b)

3.

4PRPaR (Fig. 2.27a) P\R\Pa\kR (Fig. 2.21c)

4.

4PPPaR (Fig. 2.27b) P\P\kPa\R (Fig. 2.21d)

5.

4PRPPa (Fig. 2.28a) P||R\P||Pa (Fig. 2.21e)

6.

4PPRPa (Fig. 2.28b) P\P||R\Pa (Fig. 2.21f)

7. 8. 9. 10. 11.

4PRPPa 4PRPaP 4PPPaR 4PPPaR 4PPaRP

(Fig. (Fig. (Fig. (Fig. (Fig.

2.29a) 2.29b) 2.30a) 2.30b) 2.31a)

P\R||P\Pa (Fig. 2.21g) P\R\Pa \kP (Fig. 2.21h) P\P\kPa\kR (Fig. 2.21i) P\P||Pa\R (Fig. 2.21j) P||Pa\R||P (Fig. 2.21k)

12. 13. 14. 15.

4PPaPR 4PPaPR 4PPaRP 4RPPaP

(Fig. (Fig. (Fig. (Fig.

2.31b) 2.32a) 2.32b) 2.33a)

P||Pa\P|| R (Fig. 2.21l) P\Pa||P\\R (Fig. 2.21m) P\Pa\kR\kP (Fig. 2.21n) R\P||Pa\kP (Fig. 2.21o)

16. 17. 18. 19. 20. 21. 22. 23. 24.

4RPPPa (Fig. 2.33b) 4PaPPR (Fig. 2.34a) 4PaPPR (Fig. 2.34b) 4PaRPP (Fig. 2.35a) 4PaPRP (Fig. 2.35b) 4PaPRP (Fig. 2.36a) 4PaRPP (Fig. 2.36b) 4PCPa (Fig. 2.37a) 4PPaC (Fig. 2.37b)

R||P\P||Pa (Fig. 2.21p) Pa||P\P||R (Fig. 2.21q) Pa\P\kP\kR (Fig. 2.21r) Pa\R||P\P (Fig. 2.21s) Pa||P\R||P (Fig. 2.21t) Pa\P||R\P (Fig. 2.21u) Pa\R\P\kP (Fig. 2.21v) P\C\Pa (Fig. 2.21w) P||Pa\C (Fig. 2.21x)

25. 4CPPa (Fig. 2.38a) 26. 27. 28. 29.

4PaPC (Fig. 2.38b) 4PaCP (Fig. 2.39) 4PPaPaR (Fig. 2.40) 4PPaRPa (Fig. 2.41)

C\P||Pa (Fig. 2.21y) Pa||P\C (Fig. 2.21z) Pa\C\P (Fig. 2.21z1) P\Pa\\Pa\kR (Fig. 2.22a) P\Pa||R\Pa (Fig. 2.22b)

The first two and the last revolute joints of the four limbs have parallel axes The three first revolute joints of the four limbs have parallel axes The second and the last joints of the four limbs have parallel axes The last revolute joints of the four limbs have parallel axes The second joints of the four limbs have parallel axes The third joints of the four limbs have parallel axes Idem No. 5 Idem No. 5 Idem No. 4 Idem No. 4 The before last joints of the four limbs have parallel axes Idem No. 4 Idem No. 4 Idem No. 4 The first revolute joints of the four limbs have parallel axes Idem No. 15 Idem No. 4 Idem No. 4 Idem No. 11 Idem No. 5 Idem No. 11 Idem No. 5 Idem No. 5 The last joints of the four limbs have parallel axes The first joints of the four limbs have parallel axes Idem No. 24 Idem No. 11 Idem No. 4 The axes of the revolute joints connecting the two parallelogram loops of the four limbs are parallel (continued)

176

2

Table 2.3 (continued) No. PM type 30. 31. 32. 33.

4PRPaPa 4RPPaPa 4PaPPaR 4PaPRPa

(Fig. (Fig. (Fig. (Fig.

2.42) 2.43) 2.44) 2.45)

34. 4PaRPPa (Fig. 2.46)

35. 4RPaPPa (Fig. 2.47) 36. 4PaPaPR (Fig. 2.48) 37. 4RPaPaR (Fig. 2.49) 38. 39. 40. 41.

4RPaPaR 4RPaRPa 4PaRPPa 4PaPaRP

42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58.

4PaPaPR (Fig. 2.54) 4PRPaPa (Fig. 2.55) 4PPaRPa (Fig. 2.56) 4PPaPaR (Fig. 2.57) 4PaPPaR (Fig. 2.58) 4RPPaPa (Fig. 2.59) 4PaPPaR (Fig. 2.60) 4PaRPaP (Fig. 2.61) 4PaRPPa (Fig. 2.62) 4PaPRPa (Fig. 2.63) 4PaPaPR (Fig. 2.64) 4PaPaRP (Fig. 2.65) 4RPaPaP (Fig. 2.66) 4RPaPPa (Fig. 2.67) 4RPPaPa (Fig. 2.68) 4PRPaPa (Fig. 2.69) 4PaCPa (Fig. 2.70)

(Fig. (Fig. (Fig. (Fig.

2.50) 2.51) 2.52) 2.53)

59. 4CPaPa (Fig. 2.71) 60. 4PaPaC (Fig. 2.72)

Fully-Parallel Topologies with Coupled Schönflies Motions

Limb topology P\R||Pa\Pa (Fig. 2.22c) R\P\kPa\\Pa (Fig. 2.22d) Pa\P\\Pa\\R (Fig. 2.22e) Pa\P\kR\Pa (Fig. 2.22f)

Connecting conditions

Idem No. 5 Idem No. 15 Idem No. 4 The axes of the revolute joints connecting links 5 and 6 of the four limbs are parallel Pa||R\P\\Pa (Fig. 2.22g) The axes of the revolute joints connecting links 4 and 5 of the four limbs are parallel R||Pa\P\\Pa (Fig. 2.22h) Idem No. 15 Pa\Pa\\P\\R (Fig. 2.22i) Idem No. 4 R\Pa||Pa\kR (Fig. 2.22j) The first and the last revolute joints of the four limbs are parallel R\Pa||Pa\kR (Fig. 2.22k) Idem No. 37 R\Pa\kR\Pa (Fig. 2.22l) Idem No. 15 Pa\R\P||Pa (Fig. 2.22m) Idem No. 34 Pa||Pa\R\P (Fig. 2.22n) The axes of the revolute joints connecting links 7 and 8 of the four limbs are parallel Pa||Pa||P\R (Fig. 2.22o) Idem No. 4 P\R\Pa||Pa (Fig. 2.22p) Idem No. 5 P||Pa\R\Pa (Fig. 2.22q) Idem No. 33 P||Pa||Pa\R (Fig. 2.22r) Idem No. 4 Pa||P||Pa\R (Fig. 2.22s) Idem No. 4 Idem No. 15 R\P||Pa||Pa (Fig. 2.22t) Pa\P\\Pa\kR (Fig. 2.22u) Idem No. 4 Pa\R\Pa\kR (Fig. 2.22v) Idem No. 34 Pa\R||P\Pa (Fig. 2.22w) Idem No. 34 Pa\P||R\Pa (Fig. 2.22x) Idem No. 33 Pa\Pa\\P||R (Fig. 2.22y) Idem No. 4 Pa\Pa\\R||P (Fig. 2.22z) Idem No. 41 R\Pa\\Pa\\P (Fig. 2.22a0 ) Idem No. 15 R\Pa\kP\\Pa (Fig. 2.22b0 ) Idem No. 15 R||P\Pa\\Pa (Fig. 2.22c0 ) Idem No. 15 P||R\Pa\\Pa (Fig. 2.22d0 ) Idem No. 5 Pa\C\Pa (Fig. 2.22e0 ) The axes of the cylindrical joints of the four limbs are parallel C\Pa\\Pa (Fig. 2.22f 0 ) Idem No. 58 Pa\Pa\\C (Fig. 2.22g0 ) Idem No. 58

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

4PaPaPaR 4PaPaRPa 4PaRPaPa 4RPaPaPa 4PaPaPaR 4PaPaRPa 4PaRPaPa 4RPaPaPa 4PaPaPaR 4PaRPaPa 4RPaPaPa 4PaPaPaR 4PaPaRPa 4PaRPaPa 4RPaPaPa 4PaPaPaR 4PaPaRPa 4PaRPaPa 4RPaPaPa 4PaPaPaR 4PaPaRPa 4PaRPaPa 4RPaPaPa 4PaPaPaR 4PaPaRPa

(Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig.

2.73) 2.74) 2.75) 2.76) 2.77) 2.78) 2.79) 2.80) 2.81) 2.82) 2.83) 2.84) 2.85) 2.86) 2.87) 2.88) 2.89) 2.90) 2.91) 2.92) 2.93) 2.94) 2.95) 2.96) 2.97)

Pa\Pa||Pa\kR (Fig. 2.23a) Pa\Pa\kR\Pa (Fig. 2.23b) Pa||R\Pa||Pa (Fig. 2.23c) R||Pa\Pa||Pa (Fig. 2.23d) Pa\Pa\kPa\\R (Fig. 2.23e) Pa\Pa\\R\Pa (Fig. 2.23f) Pa\R\Pa\\Pa (Fig. 2.23g) R\Pa\\Pa\kPa (Fig. 2.23h) Pa\Pa||Pa\\R (Fig. 2.23i) Pa\R\Pa||Pa (Fig. 2.23j) R\Pa\\Pa||Pa (Fig. 2.23k) Pa||Pa\Pa||R (Fig. 2.23l) Pa||Pa\R||Pa (Fig. 2.23m) Pa\R\Pa\kPa (Fig. 2.23n) R\Pa||Pa\kPa (Fig. 2.23o) Pa\Pa||Pa||R (Fig. 2.23p) Pa\Pa||R||Pa (Fig. 2.23q) Pa\R||Pa||Pa (Fig. 2.23r) R\Pa\kPa||Pa (Fig. 2.21s) Pa\Pa\kPa||R (Fig. 2.23t) Pa\Pa\kR||Pa (Fig. 2.23u) Pa||R\Pa\kPa (Fig. 2.23v) R||Pa\Pa\kPa (Fig. 2.23w) Pa||Pa\Pa\kR (Fig. 2.23x) Pa||Pa||R\Pa (Fig. 2.23z)

(continued)

The last revolute joints of the four limbs have parallel axes The axes of the revolute joints connecting links 7 and 8 of the four limbs are parallel The axes of the revolute joints connecting links 4 and 5 of the four limbs are parallel The first revolute joints of the four limbs have parallel axes Idem No. 1 Idem No. 2 Idem No. 3 Idem No. 4 Idem No. 1 Idem No. 3 Idem No. 4 Idem No. 1 Idem No. 2 Idem No. 3 Idem No. 4 Idem No. 1 Idem No. 2 Idem No. 3 Idem No. 4 Idem No. 1 Idem No. 2 Idem No. 3 Idem No. 4 Idem No. 1 Idem No. 2

Table 2.4 Limb topology and connecting conditions of the fully-parallel solutions with no idle mobilities presented in Figs. 2.73, 2.74, 2.75, 2.76, 2.77, 2.78, 2.79, 2.80, 2.81, 2.82, 2.83, 2.84, 2.85, 2.86, 2.87, 2.88, 2.89, 2.90, 2.91, 2.92, 2.93, 2.94, 2.95, 2.96, 2.97, 2.98, 2.99, 2.100, 2.101, 2.102, 2.103, 2.104, 2.105, 2.106, 2.107, 2.108, 2.109, 2.110, 2.111, 2.112, 2.113, 2.114, 2.115, 2.116, 2.117, 2.118, 2.119 No. PM type Limb topology Connecting conditions

2.2 Topologies with Complex Limbs 177

26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51.

4PaRPaPa (Fig. 2.98) 4RPaPaPa (Fig. 2.99) 4PaRRRR (Fig. 2.100a) 4PaRRRR (Fig. 2.100b) 4PaRRRR (Fig. 2.101) 4RPaRRR (Fig. 2.102) 4RRPaRR (Fig. 2.103) 4RRRRPa (Fig. 2.104) 4RRRRPa (Fig. 2.105) 4RRRPaR (Fig. 2.106a) 4RRPaRR (Fig. 2.106b) 4PaRRRR (Fig. 2.107) 4RPaRRR (Fig. 2.108a) 4RPaRRR (Fig. 2.108b) 4PRPaRR (Fig. 2.109) 4RPRRPa (Fig. 2.110a) 4PRPaRR (Fig. 2.110b) 4PRRRPa (Fig. 2.111) 4CPaRR (Fig. 2.112) 4PaRPaRR (Fig. 2.113) 4PaRPaRR (Fig. 2.114) 4RPaRPaR (Fig. 2.115) 4PaRRRPa (Fig. 2.116) 4RRPaRPa (Fig. 2.117) 4RRPaRPa (Fig. 2.118) 4PaRPaRR (Fig. 2.119)

Table 2.4 (continued) No. PM type

Limb topology

Pa||R||Pa\Pa (Fig. 2.23y) R||Pa||Pa\Pa (Fig. 2.23z1) Pa\R\R||R\R (Fig. 2.24a) Pa||R||R\R||R (Fig. 2.24b) Pa\R||R||R\R (Fig. 2.24c) R\Pa\kR||R\R (Fig. 2.24d) R||R\Pa\kR\R (Fig. 2.24e) R||R||R\R||Pa (Fig. 2.24f) R\R||R||R\kPa (Fig. 2.24g) R\R||R\Pa\kR (Fig. 2.24h) R\R\Pa\kR||R (Fig. 2.24i) Pa||R\R||R||R (Fig. 2.24j) R\Pa\kR\R||R (Fig. 2.24k) Pa||R\R\P\kR (Fig. 2.24l) P||R\Pa\kR\R (Fig. 2.24m) R\P\kR\R||Pa (Fig. 2.24o) P\R\Pa\kR\R (Fig. 2.24n) P\R||R\R||Pa (Fig. 2.24p) C\Pa\kR\R (Fig. 2.24q) Pa\R\Pa\kR\R (Fig. 2.25a) Pa||R\Pa\kR\R (Fig. 2.25b) R\Pa\kR||Pa\R (Fig. 2.25c) Pa||R||R\R||Pa (Fig. 2.25d) R||R||Pa\R||Pa (Fig. 2.25e) R\R\Pa\kR||Pa (Fig. 2.25f) Pa||R\Pa||R||R (Fig. 2.25g)

Idem No. 3 Idem No. 4 The revolute joints connecting links 4–5 and 7–8 of the four limbs have parallel axes The two last revolute joints of the four limbs have parallel axes Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 3 Idem No. 4 Idem No. 4 Idem No. 4 Idem No. 3 Idem No. 4 Idem No. 3 Idem No. 1 Idem No. 3 Idem No. 1 Idem No. 3 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 The axes of the revolute joints connecting links 6 and 7 of the four limbs are parallel Idem No. 48 Idem No. 4 Idem No. 3

Connecting conditions

178 2 Fully-Parallel Topologies with Coupled Schönflies Motions

2.2 Topologies with Complex Limbs

179

Table 2.5 Structural parametersa of parallel mechanisms in Figs. 2.26, 2.27, 2.28, 2.29, 2.30, 2.31, 2.32, 2.33, 2.34, 2.35, 2.36, 2.37, 2.38, 2.39 No. Structural parameter Solution

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. a

m pi (i = 1,…,4) p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1,…,4) rGi (i = 1,…,4) MGi (i = 1,…,4) (RF) SF rl rF MF NF TF Pp1 fj Ppj¼1 2 f j Ppj¼1 3 f j Ppj¼1 4 j¼1 fj Pp j¼1 fj

Figures 2.26, 2.27, 2.28, 2.29, 2.30, 2.31, 2.32, 2.33, 2.34, 2.35, 2.36

Figures 2.37, 2.38, 2.39

22 7 28 7 0 4 4 (v1 ; (v1 ; (v1 ; (v1 ; 4 3 4 (v1 ; 4 12 24 4 18 0 7 7

18 6 24 7 0 4 4 (v1 ; (v1 ; (v1 ; (v1 ; 4 3 4 (v1 ; 4 12 24 4 18 0 7 7

v2 ; v2 ; v2 ; v2 ;

v3 ; v3 ; v3 ; v3 ;

xd ) xd ) xd ) xd )

v2 ; v3 ; x d )

7 7

7 7

28

28

See footnote of Table 2.2 for the nomenclature of structural parameters

v2 ; v2 ; v2 ; v2 ;

v3 ; v3 ; v3 ; v3 ;

xd ) xd ) xd ) xd )

v2 ; v3 ; xd )

180

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Table 2.6 Structural parametersa of parallel mechanisms in Figs. 2.40, 2.41, 2.42, 2.43, 2.44, 2.45, 2.46, 2.47, 2.48, 2.49, 2.50, 2.51, 2.52, 2.53, 2.54, 2.55, 2.56, 2.57, 2.58, 2.59, 2.60, 2.61, 2.62, 2.63, 2.64, 2.65, 2.66, 2.67, 2.68, 2.69, 2.70, 2.71, 2.72 No. Structural parameter Solution

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. a

m pi (i = 1,…,4) p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1,…,4) rGi (i = 1,…,4) MGi (i = 1,…,4) (RF) SF rl rF MF NF TF P p1 fj Ppj¼1 2 fj Ppj¼1 3 j¼1 fj Pp4 fj Ppj¼1 f j¼1 j

Figures 2.40, 2.41, 2.42, 2.43, 2.44, 2.45, 2.46, 2.47, 2.48, 2.49, 2.50, 2.51, 2.52, 2.53, 2.54, 2.55, 2.56, 2.57, 2.58, 2.59, 2.60, 2.61, 2.62, 2.63, 2.64, 2.65, 2.66, 2.67, 2.68, 2.69

Figures 2.70, 2.71, 2.72

30 10 40 11 0 4 4 (v1 ; v2 ; v3 ; xd ) (v1 ; v2 ; v3 ; xd ) (v1 ; v2 ; v3 ; xd ) (v1 ; v2 ; v3 ; xd ) 4 6 4 (v1 ; v2 ; v3 ; xd ) 4 24 36 4 30 0 10

26 9 36 11 0 4 4 (v1 ; v2 ; v3 ; xd ) (v1 ; v2 ; v3 ; xd ) (v1 ; v2 ; v3 ; xd ) (v1 ; v2 ; v3 ; xd ) 4 6 4 (v1 ; v2 ; v3 ; xd ) 4 24 36 4 30 0 10

10 10

10 10

10 40

10 40

See footnote of Table 2.2 for the nomenclature of structural parameters

2.2 Topologies with Complex Limbs

181

Table 2.7 Structural parametersa of parallel mechanisms in Figs. 2.73, 2.74, 2.75, 2.76, 2.77, 2.78, 2.79, 2.80, 2.81, 2.82, 2.83, 2.84, 2.85, 2.86, 2.87, 2.88, 2.89, 2.90, 2.91, 2.92, 2.93, 2.94, 2.95, 2.96, 2.97, 2.98, 2.99, 2.100, 2.101, 2.102, 2.103, 2.104, 2.105, 2.106, 2.107, 2.108, 2.109, 2.110, 2.111 No. Structural parameter Solution

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. a

m pi (i = 1,…,4) p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1,…,4) rGi (i = 1,…,4) MGi (i = 1,…,4) (RF) SF rl rF MF NF TF P p1 fj Ppj¼1 2 f j Ppj¼1 3 f j Ppj¼1 4 j¼1 fj Pp j¼1 fj

Figures 2.73, 2.74, 2.75, 2.76, 2.77, 2.78, 2.79, 2.80, 2.81, 2.82, 2.83, 2.84, 2.85, 2.86, 2.87, 2.88, 2.89, 2.90, 2.91, 2.92, 2.93, 2.94, 2.95, 2.96, 2.97, 2.98, 2.99

Figures 2.100, 2.101, 2.102, 2.103, 2.104, 2.105, 2.106, 2.107, 2.108, 2.109, 2.110, 2.111

38 13 52 15 0 4 4 (v1 ; v2 ; v3 ; xd ) (v1 ; v2 ; v3 ; xd ) (v1 ; v2 ; v3 ; xd ) (v1 ; v2 ; v3 ; xd ) 4 9 4 (v1 ; v2 ; v3 ; xd ) 4 36 48 4 42 0 13

26 8 32 7 0 4 4 (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) 5 3 5 (v1 ; v2 ; v3 ; xd ) 4 12 28 4 14 0 8

13 13

8 8

13

8

52

32

See footnote of Table 2.2 for the nomenclature of structural parameters

182

2

Fully-Parallel Topologies with Coupled Schönflies Motions

Table 2.8 Structural parametersa of parallel mechanisms in Figs. 2.112, 2.113, 2.114, 2.115, 2.116, 2.117, 2.118, 2.119 No. Structural parameter Solution

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. a

m pi (i = 1,…,4) p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1,…,4) rGi (i = 1,…,4) MGi (i = 1,…,4) (RF) SF rl rF MF NF TF Pp1 fj Pj¼1 p2 j¼1 fj Pp3 fj Pj¼1 p4 fj Pj¼1 p j¼1 fj

Figure 2.112

Figures 2.113, 2.114, 2.115, 2.116, 2.117, 2.118, 2.119

22 7 28 7 0 4 4 (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) 5 3 5 (v1 ; v2 ; v3 ; xd ) 4 12 28 4 14 0 8 8

34 11 44 11 0 4 4 (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) 5 6 5 (v1 ; v2 ; v3 ; xd ) 4 24 40 4 26 0 11 11

8 8

11 11

32

44

See footnote of Table 2.2 for the nomenclature of structural parameters

References 1. Gogu G (2008) Structural synthesis of parallel robots: part 1-methodology. Springer, Dordrecht 2. Gogu G (2009) Structural synthesis of parallel robots: part 2-translational topologies with two and three degrees of freedom. Springer, Dordrecht 3. Gogu G (2010) Structural synthesis of parallel robots: part 3-topologies with planar motion of the moving platform. Springer, Dordrecht 4. Gogu G (2012) Structural synthesis of parallel robots: part 4-other topologies with two and three degrees of freedom. Springer, Dordrecht

Chapter 3

Overactuated Topologies with Coupled Schönflies Motions

In the overactuated parallel robotic manipulator with coupled Schönflies motions presented in this section each operational velocity depends on maximum three independent actuated joint velocities v1 ¼ v1 ðq_ 1 ; q_ 2 ; q_ 3 Þ; v2 ¼ v2 ðq_ 1 ; q_ 2 ; q_ 3 Þ; v3 ¼ v3 ðq_ 4 Þ and xd ¼ xd ðq_ 1 ; q_ 2 ; q_ 3 Þ. In these topologies the operational velocity v3 is uncoupled, but the Jacobian matrix in Eq. (1.18) is not triangular and the parallel robot always has coupled motions. Three overactuated joints exist in these topologies. The dependent actuated joint variables q5, q6 and q7 of these three overactuated joints are equal to the independent joint variable q4 and implicitely q_ 5 ¼ q_ 6 ¼ q_ 7 ¼ q_ 4 . The overactuation of up to three joints contributes to a better repartition of the payload on the four limbs. Overactution can be reduced to just one or two joints. If overactuation is missing, the topologies presented in this section become fully-parallel solution, similar to those illustrated in Chap. 2 presenting one uncoupled motion. The overactuated and overconstrained topologies of parallel robotic manipulator with coupled Schönflies motions may have identical limbs or limbs with different structures and could be actuated by linear or rotating motors. The limbs can be simple or complex kinematic chains and can also combine idle mobilities. The actuators can be mounted on the fixed base or on a moving link. The first solution has the advantage of reducing the moving masses and large workspace. The second solution would be more compact. There are no idle mobilities in the topologies presented in this section. The solutions presented in this section are obtained by using the methodology of structural synthesis proposed in Part 1 [1] and also used in Parts 2–4 of this work [2–4]. This original methodology combines new formulae for mobility connectivity, redundancy and overconstraints, and the evolutionary morphology in a unified approach of structural synthesis of parallel robotic manipulators.

G. Gogu, Structural Synthesis of Parallel Robots, Solid Mechanics and Its Applications 206, DOI: 10.1007/978-94-007-7401-8_3,  Springer Science+Business Media Dordrecht 2014

183

184

3 Overactuated Topologies with Coupled Schönflies Motions

3.1 Topologies with Simple Limbs In the overactuated topologies of PMs with coupled Schönflies motions F / G1-G2-G3-G4 presented in this section, the moving platform n : nGi (i = 1, 2, 3, 4) is connected to the reference platform 1 : 1Gi : 0 by four spatial simple limbs with four or five degrees of connectivity. Two actuators are combined in a revolute, prismatic or cylindrical pair of limbs G1, G2 and G3 and just one actuator in limb G4. Limbs G1, G2 and G3 are overactuated. The actuated joint is underlined in the structural graph. In the cylindrical joint denoted by C just one motion is actuated. This can be the translational or the rotational motion and is indicated in the structrural diagram by a linear or circular arrow. Both translational and rotational motions are acteuated in the cylindrical joint denoted by C. The various types of simple limbs with four and five degrees of connectivity used in the overactuated topologies illustrated in this section are presented in Figs. 3.1, 3.2, 3.3. The simple limbs combine only revolute, prismatic and cylindrical joints. Various topologies of overactuated PMs with coupled Schönflies motions and no idle mobilities can be obtained by using four limbs with identical or different topology presented in Figs. 3.1, 3.2, 3.3. Only topologies with four identical limb types are illustrated in Figs. 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 3.10, 3.11, 3.12, 3.13, 3.14, 3.15, 3.16, 3.17, 3.18, 3.19, 3.20, 3.21, 3.22, 3.23, 3.24, 3.25, 3.26, 3.27, 3.28, 3.29, 3.30, 3.31, 3.32, 3.33, 3.34, 3.35, 3.36, 3.37, 3.38, 3.39, 3.40, 3.41, 3.42, 3.43, 3.44, 3.45, 3.46, 3.47, 3.48, 3.49. The limb topology and connecting conditions of the solutions in Figs. 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 3.10, 3.11, 3.12, 3.13, 3.14, 3.15, 3.16, 3.17, 3.18, 3.19, 3.20, 3.21, 3.22, 3.23, 3.24, 3.25, 3.26, 3.27, 3.28, 3.29, 3.30, 3.31, 3.32, 3.33, 3.34, 3.35, 3.36, 3.37, 3.38, 3.39, 3.40, 3.41, 3.42, 3.43, 3.44, 3.45, 3.46, 3.47, 3.48, 3.49 are systematized in Tables 3.1 and 3.2 and their structural parameters in Tables 3.3 and 3.4.

3.1 Topologies with Simple Limbs

185

Fig. 3.1 Simple limbs for overactuated PMs with coupled Schönflies motions defined by MG ¼ SG ¼ 4; ðRG Þ ¼ ðv1 ; v2 ; v3 ; xd Þ

186

Fig. 3.1 (continued)

3 Overactuated Topologies with Coupled Schönflies Motions

3.1 Topologies with Simple Limbs

Fig. 3.1 (continued)

187

188

Fig. 3.1 (continued)

3 Overactuated Topologies with Coupled Schönflies Motions

3.1 Topologies with Simple Limbs

189

Fig. 3.2 Simple limbs combining a cylindrical joint for overactuated PMs with coupled Schönflies motions defined by MG ¼ SG ¼ 4; ðRG Þ ¼ ðv1 ; v2 ; v3 ; xd Þ

190

Fig. 3.2 (continued)

3 Overactuated Topologies with Coupled Schönflies Motions

3.1 Topologies with Simple Limbs

191

Fig. 3.3 Simple limbs for overactuated PMs with coupled Schönflies motions defined by MG ¼
SG ¼ 5; ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa ; xd Þ or ðRG Þ ¼ v1 ; v2 ; v3 ; xb ; xd

192

Fig. 3.3 (continued)

3 Overactuated Topologies with Coupled Schönflies Motions

3.1 Topologies with Simple Limbs

Fig. 3.3 (continued)

193

194

Fig. 3.3 (continued)

3 Overactuated Topologies with Coupled Schönflies Motions

3.1 Topologies with Simple Limbs

Fig. 3.3 (continued)

195

196

Fig. 3.3 (continued)

3 Overactuated Topologies with Coupled Schönflies Motions

3.1 Topologies with Simple Limbs

197

Fig. 3.4 Overactuated PMs with coupled Schönflies motions of types 4PRRR (a) and 4RPRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology P||R||R||R (a) and R||P||R||R (b)

198

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.5 Overactuated PMs with coupled Schönflies motions of types 4RPRR (a) and 4RRPR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology R||P||R||R (a) and R||R||P||R (b)

3.1 Topologies with Simple Limbs

199

Fig. 3.6 Overactuated PMs with coupled Schönflies motions of types 4RRPR (a) and 4PPRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology R||R||P||R (a) and P\P\kR||R (b)

200

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.7 Overactuated PMs with coupled Schönflies motions of types 4RRRP (a) and 4RPRP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology R||R||R||P (a) and R\P\kR||P (b)

3.1 Topologies with Simple Limbs

201

Fig. 3.8 Overactuated PMs with coupled Schönflies motions of types 4PPRR (a) and 4PRPR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology P\P||R||R (a) and P\R||P||R (b)

202

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.9 Overactuated PMs with coupled Schönflies motions of types 4RRPP (a) and 4RPRP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology R||R||P\P (a) and R||P||R\P (b)

3.1 Topologies with Simple Limbs

203

Fig. 3.10 4RPPR-type overactuated PMs with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology R||P\P\kR (a) and R\P\kP||R (b)

204

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.11 Overactuated PMs with coupled Schönflies motions of types 4PRRP (a) and 4PRPR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology P\R||R||P (a) and P\R||P||R (b)

3.1 Topologies with Simple Limbs

205

Fig. 3.12 Overactuated PMs with coupled Schönflies motions of types 4PRRP (a) and 4RPRP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology P||R||R\P (a) and R||P||R\P (b)

206

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.13 Overactuated PMs with coupled Schönflies motions of types 4PRPR (a) and 4RPPR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology P||R\P\kR (a) and R||P\P\kR (b)

3.1 Topologies with Simple Limbs

207

Fig. 3.14 Overactuated PMs with coupled Schönflies motions of types 4RPPR (a) and 4RPRP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology R\P\kP||R (a) and R\P\kR||P (b)

208

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.15 4PPPR-type overactuated PMs with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology P\P\\P\kR (a) and P\P\\P\R (b)

3.1 Topologies with Simple Limbs

209

Fig. 3.16 Overactuated PMs with coupled Schönflies motions of types 4PPPR (a) and 4PPRP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology P\P\\P||R (a) and P\P||R\P (b)

210

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.17 4PPRP-type overactuated PMs with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology P\P\\R||P (a) and P\P\kR\P (b)

3.1 Topologies with Simple Limbs

211

Fig. 3.18 4PRPP-type overactuated PMs with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology P\R||P\P (a) and P\R\P\kP (b)

212

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.19 Overactuated PMs with coupled Schönflies motions of types 4PRPP (a) and 4RPPP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology P||R\P\\P (a) and R\P\\P\\P (b)

3.1 Topologies with Simple Limbs

213

Fig. 3.20 4RPPP-type overactuated PMs with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology R\P\kP\\P (a) and R||P\P\\P (b)

214

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.21 Overactuated PMs with coupled Schönflies motions of types 4RRC (a) and 4RCR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology R||R||C (a) and R||C||R (b)

3.1 Topologies with Simple Limbs

215

Fig. 3.22 Overactuated PMs with coupled Schönflies motions of types 4PCP (a), 4CPP (b) and 4PPC (c) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology P\C\P (a), C\P\\P (b) and P\P\\C (c)

216

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.23 Overactuated PMs with coupled Schönflies motions of types 4PCR (a) and 4RCP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology P\C||R (a) and R||C\P (b)

3.1 Topologies with Simple Limbs

217

Fig. 3.24 Overactuated PMs with coupled Schönflies motions of types 4RPC (a) and 4PRC (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology R\P\kC (a) and P\R||C (b)

218

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.25 Overactuated PMs with coupled Schönflies motions of types 4CRP (a) and 4CPR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology C||R\P (a) and C\P\kR (b)

3.1 Topologies with Simple Limbs

219

Fig. 3.26 Overactuated PMs with coupled Schönflies motions of types 4RPC (a) and 4CRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 6, limb topology R\P\kC (a) and C||R||R (b)

220

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.27 4RRRRR-type overactuated PMs with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R\R||R\R||R (a) and R||R\R||R\R (b)

3.1 Topologies with Simple Limbs

221

Fig. 3.28 4RRRRR-type overactuated PMs with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R||R||R\R||R (a) and R||R\R||R||R (b)

222

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.29 Overactuated PMs with coupled Schönflies motions of types 4PRRRR (a) and 4RPRRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology P||R||R\R||R (a) and R||P||R\R||R (b)

3.1 Topologies with Simple Limbs

223

Fig. 3.30 Overactuated PMs with coupled Schönflies motions of types 4RPRRR (a) and 4RRPRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R||P||R\R||R (a) and R||R||P\R||R (b)

224

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.31 Overactuated PMs with coupled Schönflies motions of types 4RRRPR (a) and 4RRRRP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R||R\R\P\kR (a) and R||R\R||R\kP (b)

3.1 Topologies with Simple Limbs

225

Fig. 3.32 Overactuated PMs with coupled Schönflies motions of types 4PRRRR (a) and 4RPRRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology P\R\R||R\R (a) and R\P\R||R\R (b)

226

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.33 Overactuated PMs with coupled Schönflies motions of types 4RRPRR (a) and 4RRRPR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R\R\P\kR\R (a) and R\R||R\P\kR (b)

3.1 Topologies with Simple Limbs

227

Fig. 3.34 Overactuated PMs with coupled Schönflies motions of types 4RRRRP (a) and 4PRRRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R\R||R\R\P (a) and P\R||R\R||R (b)

228

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.35 Overactuated PMs with coupled Schönflies motions of types 4RPRRR (a) and 4RRRPR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R\P\kR\R||R (a) and R||R\R||P||R (b)

3.1 Topologies with Simple Limbs

229

Fig. 3.36 Overactuated PMs with coupled Schönflies motions of types 4RRRPR (a) and 4RRRRP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R||R\R||P||R (a) and R||R\R||R||P (b)

230

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.37 Overactuated PMs with coupled Schönflies motions of types 4RRRRP (a) and 4PRRRP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R||R\R||R\P (a) and P\R\R||R\P (b)

3.1 Topologies with Simple Limbs

231

Fig. 3.38 Overactuated PMs with coupled Schönflies motions of types 4RRPRP (a) and 4RPRRP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R\R||P||R\P (a) and R\P||R||R\P (b)

232

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.39 Overactuated PMs with coupled Schönflies motions of types 4RRPRP (a) and 4RRRPP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R\R||P||R\P (a) and R\R||R||P\P (b)

3.1 Topologies with Simple Limbs

233

Fig. 3.40 Overactuated PMs with coupled Schönflies motions of types 4PRRRP (a) and 4RPRRP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology P||R||R\R\P (a) and R||P||R||R\P (b)

234

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.41 Overactuated PMs with coupled Schönflies motions of types 4RPRRP (a) and 4RRPRP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R||P||R\R\P (a) and R||R||P\R\P (b)

3.1 Topologies with Simple Limbs

235

Fig. 3.42 Overactuated PMs with coupled Schönflies motions of types 4RRRPP (a) and 4PRRPR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R||R\R\P\\P (a) and P||R||R\P\\R (b)

236

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.43 Overactuated PMs with coupled Schönflies motions of types 4RPRPR (a) and 4RRPPR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R||P||R\P\\R (a) and R||R||P\P\\R (b)

3.1 Topologies with Simple Limbs

237

Fig. 3.44 Overactuated PMs with coupled Schönflies motions of types 4RPRPR (a), 4PPRRR (b) and 4PPRRR (c) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R||P||R\P\\R (a), P\P\kR||R\\R (b) and P\P||R||R\\R (c)

238

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.45 Overactuated PMs with coupled Schönflies motions of types 4CRRR (a) and 4RCRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology C||R\R||R (a) and R||C\R||R (b)

3.1 Topologies with Simple Limbs

239

Fig. 3.46 Overactuated PMs with coupled Schönflies motions of types 4RRCR (a) and 4RRRC (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R||R\C||R (a) and R||R\R||C (b)

240

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.47 Overactuated PMs with coupled Schönflies motions of types 4RCRP (a) and 4RRCP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology R\C||R\\P (a) and R\R||C\\P (b)

3.1 Topologies with Simple Limbs

241

Fig. 3.48 Overactuated PMs with coupled Schönflies motions of types 4CRRP (a) and 4RCRP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology C||R\R\P (a) and R||C\R\P (b)

242

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.49 Overactuated PMs with coupled Schönflies motions of types 4CRPR (a) and 4PCRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 2, limb topology C||R\P\\R (a) and P\C||R\\R (b)

3.1 Topologies with Simple Limbs

243

Table 3.1 Limb topology and connecting conditions of the overactuated solutions with no idle mobilities presented in Figs. 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 3.10, 3.11, 3.12, 3.13, 3.14, 3.15, 3.16, 3.17, 3.18, 3.19, 3.20 No. PM type Limb topology Connecting conditions 1.

4PRRR (Fig. 3.4a)

2.

4RPRR (Fig. 3.4b)

3. 4. 5. 6. 7. 8. 9.

4RPRR 4RRPR 4RRPR 4PPRR 4RRRP 4RPRP 4PPRR

(Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig.

3.5a) 3.5b) 3.6a) 3.6b) 3.7a) 3.7b) 3.8a)

10. 11. 12. 13. 14. 15.

4PRPR 4RRPP 4RPRP 4RPPR 4RPPR 4PRRP

(Fig. (Fig. (Fig. (Fig. (Fig. (Fig.

3.8b) 3.9a) 3.9b) 3.10a) 3.10b) 3.11a)

16. 4PRPR (Fig. 3.11b) 17. 4PRRP (Fig. 3.12a) 18. 19. 20. 21. 22. 23. 24. 25. 26.

4RPRP 4PRPR 4RPPR 4RPPR 4RPRP 4PPPR 4PPPR 4PPPR 4PPRP

(Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig.

3.12b) 3.13a) 3.13b) 3.14a) 3.14b) 3.15a) 3.15b) 3.16a) 3.16b)

27. 28. 29. 30. 31. 32. 33. 34.

4PPRP 4PPRP 4PRPP 4PRPP 4PRPP 4RPPP 4RPPP 4RPPP

(Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig.

3.17a) 3.17b) 3.18a) 3.18b) 3.19a) 3.19b) 3.20a) 3.20b)

P||R||R||R (Fig. 3.1a)

The prismatic joints of the four limbs have parallel directions R||P||R||R (Fig. 3.1b) The first revolute joints of the four limbs have parallel axes R||P||R||R (Fig. 3.1c) Idem No. 2 R||R||P||R (Fig. 3.1d) Idem No. 2 R||R||P||R (Fig. 3.1e) Idem No. 2 Idem No. 1 P\P\kR||R (Fig. 3.1i) R||R||R||P (Fig. 3.1f) Idem No. 2 R\P\kR||P (Fig. 3.1g) Idem No. 2 P\P||R||R (Fig. 3.1h) The last revolute joints of the four limbs have parallel axes P\R||P||R (Fig. 3.1p) Idem No. 9 R||R||P\P (Fig. 3.1j) Idem No. 2 R||P||R\P (Fig. 3.1k) Idem No. 2 Idem No. 2 R||P\P\kR (Fig. 3.1l) R\P\kP||R (Fig. 3.1m) Idem No. 2 P\R||R||P (Fig. 3.1n) The last prismatic joints of the four limbs have parallel directions P\R||P||R (Fig. 3.1o) Idem No. 9 P||R||R\P (Fig. 3.1v) The first prismatic joints of the four limbs have parallel directions R||P||R\P (Fig. 3.1q) Idem No. 2 Idem No. 1 P||R\P\kR (Fig. 3.1r) R||P\P\kR (Fig. 3.1s) Idem No. 2 R\P\kP||R (Fig. 3.1t) Idem No. 2 R\P\kR||P (Fig. 3.1u) Idem No. 2 P\P\\P\kR (Fig. 3.1h0 ) Idem No. 9 P\P\\P\\R (Fig. 3.1w) Idem No. 9 P\P\\P||R (Fig. 3.1x) Idem No. 9 P\P||R\P (Fig. 3.1y) The revolute joints of the four limbs have parallel axes P\P\\R||P (Fig. 3.1z) Idem No. 26 P\P\kR\P (Fig. 3.1a0 ) Idem No. 26 P\R||P\P (Fig. 3.1b0 ) Idem No. 26 P\R\P\kP (Fig. 3.1c0 ) Idem No. 26 P||R\P\\P (Fig. 3.1d0 ) Idem No. 26 R\P\\P\\P (Fig. 3.1e0 ) Idem No. 26 R\P\kP\\P (Fig. 3.1f0 ) Idem No. 26 R||P\P\\P (Fig. 3.1g0 ) Idem No. 26

244

3 Overactuated Topologies with Coupled Schönflies Motions

Table 3.2 Limb topology and connecting conditions of the overactuated solutions with no idle mobilities presented in Figs. 3.21, 3.22, 3.23, 3.24, 3.25, 3.26, 3.27, 3.28, 3.29, 3.30, 3.31, 3.32, 3.33, 3.34, 3.35, 3.36, 3.37, 3.38, 3.39, 3.40, 3.41, 3.42, 3.43, 3.44, 3.45, 3.46, 3.47, 3.48, 3.49 No. PM type Limb topology Connecting conditions 1.

4RRC (Fig. 3.21a)

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

4RCR (Fig. 3.21b) 4PCP (Fig. 3.22a) 4CPP (Fig. 3.22b) 4PPC (Fig. 3.22c) 4PCR (Fig. 3.23a) 4RCP (Fig. 3.23b) 4RPC (Fig. 3.24a) 4PRC (Fig. 3.24b) 4CRP (Fig. 3.25a) 4CPR (Fig. 3.25b) 4RPC (Fig. 3.26a) 4CRR (Fig. 3.26b) 4RRRRR (Fig. 3.27a)

15. 4RRRRR (Fig. 3.27b) 16. 4RRRRR (Fig. 3.28a) 17. 4RRRRR (Fig. 3.28b) 18. 19. 20. 21. 22. 23. 24.

4PRRRR 4RPRRR 4RPRRR 4RRPRR 4RRRPR 4RRRRP 4PRRRR

(Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig.

3.29a) 3.29b) 3.30a) 3.30b) 3.31a) 3.31b) 3.32a)

25. 26. 27. 28.

4RPRRR 4RRPRR 4RRRPR 4RRRRP

(Fig. (Fig. (Fig. (Fig.

3.32b) 3.33a) 3.33b) 3.34a)

29. 4PRRRR (Fig. 3.34b)

R||R||C (Fig. 3.2a)

The cylindrical joints of the four limbs have parallel axes R||C||R (Fig. 3.2b) Idem No. 1 P\C\P (Fig. 3.2c) Idem No. 1 C\P\\P (Fig. 3.2d) Idem No. 1 P\P\\C (Fig. 3.2e) Idem No. 1 P\C||R (Fig. 3.2f) Idem No. 1 R||C\P (Fig. 3.2g) Idem No. 1 R\P\kC (Fig. 3.2h) Idem No. 1 P\R||C (Fig. 3.2i) Idem No. 1 C||R\P (Fig. 3.2j) Idem No. 1 C\P\kR (Fig. 3.2k) Idem No. 1 R\P\kC (Fig. 3.2i) Idem No. 1 C||R||R (Fig. 3.2m) Idem No. 1 R\R||R\R||R (Fig. 3.3a) The first and the last revolute joints of the four limbs have parallel axes R||R\R||R\R (Fig. 3.3b) Idem No. 14 R||R||R\R||R (Fig. 3.3c) The first revolute joints of the four limbs have parallel axes R||R\R||R||R (Fig. 3.3d) The last revolute joints of the four limbs have parallel axes P||R||R\R||R (Fig. 3.3e) Idem No. 17 R||P||R\R||R (Fig. 3.3f) Idem No. 17 R||P||R\R||R (Fig. 3.3g) Idem No. 17 R||R||P\R||R (Fig. 3.3h) Idem No. 17 R||R\R\P\kR (Fig. 3.3i) Idem No. 17 R||R\R||R\kP (Fig. 3.3j) Idem No. 17 P\R\R||R\R (Fig. 3.3k) The second and the last joints of the four limbs have parallel axes R\P\R||R\R (Fig. 3.3l) Idem No. 14 R\R\P\kR\R (Fig. 3.3m) Idem No. 14 R\R||R\P\kR (Fig. 3.3n) Idem No. 14 R\R||R\R\P (Fig. 3.3o) The first and the fourth revolute joints of the four limbs have parallel axes P\R||R\R||R (Fig. 3.3p) The second joints of the four limbs have parallel axes (continued)

3.1 Topologies with Simple Limbs Table 3.2 (continued) No. PM type 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.

4RPRRR 4RRRPR 4RRRPR 4RRRRP 4RRRRP 4PRRRP 4RRPRP 4RPRRP 4RRPRP 4RRRPP 4PRRRP

(Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig.

3.35a) 3.35b) 3.36a) 3.36b) 3.37a) 3.37b) 3.38a) 3.38b) 3.39a) 3.39b) 3.40a)

41. 42. 43. 44.

4RPRRP 4RPRRP 4RRPRP 4RRRPP

(Fig. (Fig. (Fig. (Fig.

3.40b) 3.41a) 3.41b) 3.42a)

45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.

4PRRPR (Fig. 3.42b) 4RPRPR (Fig. 3.43a) 4RRPPR (Fig. 3.43b) 4RPRPR (Fig. 3.44a) 4RPRPR (Fig. 3.44b) 4PPRRR (Fig. 3.44c) 4CRRR (Fig. 3.45a) 4RCRR (Fig. 3.45b) 4RRCR (Fig. 3.46a) 4RRRC (Fig. 3.46b) 4RCRP (Fig. 3.47a) 4RRCP (Fig. 3.47b) 4CRRP (Fig. 3.48a) 4RCRP (Fig. 3.48b) 4CRPR (Fig. 3.49a) 4PCRR (Fig. 3.49b)

Limb topology R\P\kR\R||R (Fig. 3.3q) R||R\R||P||R (Fig. 3.3r) R||R\R||P||R (Fig. 3.3s) R||R\R||R||P (Fig. 3.3t) R||R\R||R\P (Fig. 3.3u) P\R\R||R\P (Fig. 3.3v) R\R||P||R\P (Fig. 3.3w) R\P||R||R\P (Fig. 3.3x) R\R||P||R\P (Fig. 3.3y) R\R||R||P\P (Fig. 3.3z) P||R||R\R\P (Fig. 3.3z1)

245

Connecting conditions

Idem No. 16 Idem No. 16 Idem No. 16 Idem No. 16 Idem No. 16 Idem No. 29 Idem No. 16 Idem No. 16 Idem No. 16 Idem No. 16 The fourth joints of the four limbs have parallel axes R||P||R||R\P (Fig. 3.3a) Idem No. 40 R||P||R\R\P (Fig. 3.3b0 ) Idem No. 40 Idem No. 40 R||R||P\R\P (Fig. 3.3c0 ) R||R\R\P\\P (Fig. 3.3d0 ) The third joints of the four limbs have parallel axes P||R||R\P\\R (Fig. 3.3e0 ) Idem No. 17 R||P||R\P\\R (Fig. 3.3f0 ) Idem No. 17 R||R||P\P\\R (Fig. 3.3g0 ) Idem No. 17 R||P||R\P\\R (Fig. 3.3h0 ) Idem No. 17 P\P\kR||R\\R (Fig. 3.3i0 ) Idem No. 17 P\P||R||R\\R (Fig. 3.3j0 ) Idem No. 17 C||R\R||R (Fig. 3.3k0 ) Idem No. 17 R||C\R||R (Fig. 3.3l0 ) Idem No. 17 R||R\C||R (Fig. 3.3m0 ) Idem No. 16 R||R\R||C (Fig. 3.3n0 ) Idem No. 16 R\C||R\\P (Fig. 3.3o0 ) Idem No. 16 R\R||C\\P (Fig. 3.3p0 ) Idem No. 16 C||R\R\P (Fig. 3.3q0 ) Idem No. 44 R||C\R\P (Fig. 3.3r0 ) Idem No. 44 C||R\P\\R (Fig. 3.3s0 ) Idem No. 17 P\C||R\\R (Fig. 3.3t0 ) Idem No. 17

246

3 Overactuated Topologies with Coupled Schönflies Motions

Table 3.3 Structural parametersa of parallel mechanisms in Figs. 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 3.10, 3.11, 3.12, 3.13, 3.14, 3.15, 3.16, 3.17, 3.18, 3.19, 3.20, 3.21, 3.22, 3.23, 3.24, 3.25, 3.26 No. Structural parameter Solution Figures 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 3.10, Figures 3.21, 3.11, 3.12, 3.13, 3.14, 3.15, 3.16, 3.17, 3.22, 3.23, 3.24, 3.18, 3.19, 3.20 3.25, 3.26 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. a

m pi (i = 1,…,4) p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1,…,4) rGi (i = 1,…,4) MGi (i = 1,…,4) (RF) SF rl rF MF NF TF Pp1 fj Ppj¼1 2 f j Ppj¼1 3 f j Ppj¼1 4 j¼1 fj Pp j¼1 fj

14 4 16 3 4 0 4 ðv1 ; v2 ; v3 ; xd Þ ðv1 ; v2 ; v3 ; xd Þ ðv1 ; v2 ; v3 ; xd Þ ðv1 ; v2 ; v3 ; xd Þ 4 0 4 ðv1 ; v2 ; v3 ; xd Þ 4 0 12 4 6 0 4 4

10 3 12 3 4 0 4 ðv1 ; v2 ; v3 ; xd Þ ðv1 ; v2 ; v3 ; xd Þ ðv1 ; v2 ; v3 ; xd Þ ðv1 ; v2 ; v3 ; xd Þ 4 0 4 ðv1 ; v2 ; v3 ; xd Þ 4 0 12 4 6 0 4 4

4 4

4 4

14

14

See footnote of Table 2.2 for the nomenclature of structural parameters

3.2 Topologies with Complex Limbs

247

Table 3.4 Structural parametersa of parallel mechanisms in Figs. 3.27, 3.28, 3.29, 3.30, 3.31, 3.32, 3.33, 3.34, 3.35, 3.36, 3.37, 3.38, 3.39, 3.40, 3.41, 3.42, 3.43, 3.44, 3.45, 3.46, 3.47, 3.48, 3.49 No. Structural parameter Solution Figures 3.27, 3.28, 3.29, 3.30, 3.31, Figures 3.44, 3.32, 3.33, 3.34, 3.35, 3.36, 3.37, 3.38, 3.45, 3.46, 3.47, 3.39, 3.40, 3.41, 3.42, 3.43, 3.44 3.48, 3.49 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. a

m pi (i = 1,…,4) p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1,…,4) rGi (i = 1,…,4) MGi (i = 1,…,4) (RF) SF rl rF MF NF TF Pp1 fj Ppj¼1 2 f j Ppj¼1 3 j¼1 fj Pp4 fj Ppj¼1 j¼1 fj

18 5 20 3 4 0 4 ðv1 ; v2 ; v3 ; xa ; xd Þ
v1 ; v2 ; v3 ; xb ; xd ðv1 ; v2 ; v3 ; xa ; xd Þ
v1 ; v2 ; v3 ; xb ; xd 5 0 5 ðv1 ; v2 ; v3 ; xd Þ 4 0 16 4 2 0 5

14 4 16 3 4 0 4 ðv1 ; v2 ; v3 ; xa ; xd Þ
v1 ; v2 ; v3 ; xb ; xd ðv1 ; v2 ; v3 ; xa ; xd Þ
v1 ; v2 ; v3 ; xb ; xd 5 0 5 ðv1 ; v2 ; v3 ; xd Þ 4 0 16 4 2 0 5

5 5

5 5

5

5

20

20

See footnote of Table 2.2 for the nomenclature of structural parameters

3.2 Topologies with Complex Limbs In the overactuated topologies of PMs with coupled Schönflies motions F / G1-G2-G3-G4 presented in this section, the moving platform n : nGi (i = 1, 2, 3, 4) is connected to the reference platform 1 : 1Gi : 0 by four spatial complex limbs with four or five degrees of connectivity. Two actuators are combined in a revolute, prismatic or cylindrical pair of limbs G1, G2 and G3 and just one actuator in limb G4. Limbs G1, G2 and G3 are overactuated. The actuated joint is underlined in the structural graph. In the cylindrical joint denoted by C just one motion is actuated. This can be the translational or the rotational motion and

248

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.50 Complex limbs for overactuated PMs with coupled Schönflies motions, combining a parallelogram loop, defined by MG ¼ SG ¼ 4; ðRG Þ ¼ ðv1 ; v2 ; v3 ; xd Þ

3.2 Topologies with Complex Limbs

Fig. 3.50 (continued)

249

250

Fig. 3.50 (continued)

3 Overactuated Topologies with Coupled Schönflies Motions

3.2 Topologies with Complex Limbs

Fig. 3.50 (continued)

251

252

Fig. 3.50 (continued)

3 Overactuated Topologies with Coupled Schönflies Motions

3.2 Topologies with Complex Limbs

Fig. 3.50 (continued)

253

254

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.50 (continued)

it is indicated in the structrural diagram by a linear or circular arrow. Both translational and rotational motions are acteuated in the cylindrical joint denoted by C. The complex limbs with four or five degrees of connectivity presented in Figs. 3.50, 3.51, 3.52, 3.53 combine one or two planar parallelogram loops. Each planar parallelogram loop Pa introduces three degrees of overconstraint. Various topologies of overactuated PMs with coupled Schönflies motions and no idle mobilities can be obtained by using four limbs with identical or different topology presented in Figs. 3.50, 3.51, 3.52, 3.53. Only topologies with four identical limb types are illustrated in Figs. 3.54, 3.55, 3.56, 3.57, 3.58, 3.59, 3.60, 3.61, 3.62, 3.63, 3.64, 3.65, 3.66, 3.67, 3.68, 3.69, 3.70, 3.71, 3.72, 3.73, 3.74, 3.75, 3.76, 3.77, 3.78, 3.79, 3.80, 3.81, 3.82, 3.83, 3.84, 3.85, 3.86, 3.87, 3.88, 3.89, 3.90, 3.91, 3.92, 3.93, 3.94, 3.95, 3.96, 3.97, 3.98, 3.99, 3.100, 3.101, 3.102, 3.103, 3.104, 3.105, 3.106, 3.107, 3.108, 3.109, 3.110, 3.111, 3.112, 3.113, 3.114, 3.115, 3.116, 3.117, 3.118, 3.119, 3.120, 3.121, 3.122, 3.123, 3.124, 3.125, 3.126, 3.127, 3.128, 3.129, 3.130, 3.131, 3.132, 3.133, 3.134, 3.135, 3.136, 3.137, 3.138,

3.2 Topologies with Complex Limbs

255

Fig. 3.51 Complex limbs for overactuated PMs with coupled Schönflies motions, combining two parallelogram loops, defined by MG ¼ SG ¼ 4; ðRG Þ ¼ ðv1 ; v2 ; v3 ; xd Þ

256

Fig. 3.51 (continued)

3 Overactuated Topologies with Coupled Schönflies Motions

3.2 Topologies with Complex Limbs

257

Fig. 3.51 (continued)

3.139, 3.140, 3.141, 3.142, 3.143, 3.144, 3.145, 3.146, 3.147. The limb topology and connecting conditions of the solutions in Figs. 3.54, 3.55, 3.56, 3.57, 3.58, 3.59, 3.60, 3.61, 3.62, 3.63, 3.64, 3.65, 3.66, 3.67, 3.68, 3.69, 3.70, 3.71, 3.72, 3.73, 3.74, 3.75, 3.76, 3.77, 3.78, 3.79, 3.80, 3.81, 3.82, 3.83, 3.84, 3.85, 3.86, 3.87, 3.88, 3.89, 3.90, 3.91, 3.92, 3.93, 3.94, 3.95, 3.96, 3.97, 3.98, 3.99, 3.100, 3.101, 3.102, 3.103, 3.104, 3.105, 3.106, 3.107, 3.108, 3.109, 3.110, 3.111, 3.112, 3.113, 3.114, 3.115, 3.116, 3.117, 3.118, 3.119, 3.120, 3.121, 3.122, 3.123, 3.124, 3.125, 3.126, 3.127, 3.128, 3.129, 3.130, 3.131, 3.132, 3.133, 3.134, 3.135, 3.136, 3.137, 3.138, 3.139, 3.140, 3.141, 3.142, 3.143, 3.144, 3.145, 3.146, 3.147 are systematized in Tables 3.5 and 3.6 as are their structural parameters in Tables 3.7, 3.8, 3.9.

258

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.52 Complex limbs for overactuated PMs with coupled Schönflies motions, combining one parallelogram loop, defined by MG ¼ SG ¼ 5; ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa ; xd Þ or ðRG Þ ¼ ðv1 ; v2 ; v3 ; xb ; xd Þ

3.2 Topologies with Complex Limbs

Fig. 3.52 (continued)

259

260

Fig. 3.52 (continued)

3 Overactuated Topologies with Coupled Schönflies Motions

3.2 Topologies with Complex Limbs

Fig. 3.52 (continued)

261

262

Fig. 3.52 (continued)

3 Overactuated Topologies with Coupled Schönflies Motions

3.2 Topologies with Complex Limbs

Fig. 3.52 (continued)

263

264

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.53 Complex limbs for overactuated PMs with coupled Schönflies motions, combining one parallelogram loop, defined by MG ¼ SG ¼ 5; ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa ; xd Þ or ðRG Þ ¼ ðv1 ; v2 ; v3 ; xb ; xd Þ

3.2 Topologies with Complex Limbs

265

Fig. 3.54 Overactuated PMs with coupled Schönflies motions of types 4PRPPa (a) and 4PRPaP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology P||R\P\kPa (a) and P||R||Pa\P (b)

266

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.55 Overactuated PMs with coupled Schönflies motions of types 4PPaRP (a) and 4PRRPa (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology P||Pa||R\P (a) and P||R||R||Pa (b)

3.2 Topologies with Complex Limbs

267

Fig. 3.56 Overactuated PMs with coupled Schönflies motions of types 4PPRPa (a) and 4PPRPa (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology P\P||R||Pa (a) and P\P\kR||Pa (b)

268

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.57 Overactuated PMs with coupled Schönflies motions of types 4PRPPa (a) and 4PRPaP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology P\R||P||Pa (a) and P\R||Pa||P (b)

3.2 Topologies with Complex Limbs

269

Fig. 3.58 Overactuated PMs with coupled Schönflies motions of types 4RPRPa (a) and 4RRPPa (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology R||P||R||Pa (a) and R||R||P||Pa (b)

270

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.59 RRPaP overactuated PM with coupled Schönflies motions defined MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology R||R||Pa||P

by

3.2 Topologies with Complex Limbs

271

Fig. 3.60 Overactuated PMs with coupled Schönflies motions of types 4RPaPP (a) and 4RPPPa (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology R||Pa\P\kP (a) and R||P\P\kPa (b)

272

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.61 Overactuated PMs with coupled Schönflies motions of types 4PaRPP (a) and 4RPaPP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology Pa||R||P\P (a) and R||Pa||P\P (b)

3.2 Topologies with Complex Limbs

273

Fig. 3.62 Overactuated PMs with coupled Schönflies motions of types 4RPPaP (a) and 4PPPaR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology R\P\kPa||P (a) and P\P\kPa||R (b)

274

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.63 Overactuated PMs with coupled Schönflies motions of types 4PPPaR (a) and 4PPaRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology P\P||Pa||R (a) and P||Pa||R||R (b)

3.2 Topologies with Complex Limbs

275

Fig. 3.64 4PPaRP-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology P\Pa||R||P

276

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.65 4PPaPR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology P\Pa||P||R

3.2 Topologies with Complex Limbs

277

Fig. 3.66 Overactuated PMs with coupled Schönflies motions of types 4PPaPR (a) and 4PPaRP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology P||Pa\P\kR (a) and P||Pa\R\P (b)

278

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.67 4RPPaP-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology R||P||Pa\P

3.2 Topologies with Complex Limbs

279

Fig. 3.68 4RPRPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology R||P||R||Pa

280

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.69 Overactuated PMs with coupled Schönflies motions of types 4PaPPR (a) and 4PaPPR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology Pa||P\P\kR (a) and Pa\P\kP||R (b)

3.2 Topologies with Complex Limbs

281

Fig. 3.70 4PaPRP-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology Pa\P\kR||P

282

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.71 4PaRRP-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology Pa||R||R||P

3.2 Topologies with Complex Limbs

283

Fig. 3.72 4PaRPR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology Pa||R||P||R

284

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.73 Overactuated PMs with coupled Schönflies motions of types 4PaPRP (a) and 4PaRPP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology Pa||P||R\P (a) and Pa||R\P\kP (b)

3.2 Topologies with Complex Limbs

285

Fig. 3.74 4PRPPa-type overactuated PMs with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology P\R\P||Pa (a) and P\R\P\Pa (b)

286

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.75 Overactuated PMs with coupled Schönflies motions of types 4PRPaP (a) and 4PPRPa (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology P\R\Pa\\P (a) and P\P\\R\Pa (b)

3.2 Topologies with Complex Limbs

287

Fig. 3.76 4PPPaR-type overactuated PMs with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology P\P||Pa\\R (a) and P\P\kPa\\R (b)

288

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.77 4PaPPR-type overactuated PMs with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology Pa||P\P\\R (a) and Pa\P\kP\\R (b)

3.2 Topologies with Complex Limbs

289

Fig. 3.78 4PaRPR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology Pa\R\P\kR

290

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.79 4PaPRR-type overactuated PMs with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology Pa||P\R||R (a) and Pa\P\\R||R (b)

3.2 Topologies with Complex Limbs

291

Fig. 3.80 Overactuated PMs with coupled Schönflies motions of types 4PaRRP (a) and 4RPaRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology Pa\R||R\P (a) and R\Pa\kR||R (b)

292

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.81 4PaRRR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology Pa\R||R||R

3.2 Topologies with Complex Limbs

293

Fig. 3.82 4PaPRP-type overactuated PMs with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology Pa||P\R\P (a) and Pa\P\\R\P (b)

294

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.83 Overactuated PMs with coupled Schönflies motions of types 4CPaP (a) and 4PaCP (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology C||Pa\P (a) and Pa||C\P (b)

3.2 Topologies with Complex Limbs

295

Fig. 3.84 Overactuated PMs with coupled Schönflies motions of types 4CRPa (a) and 4PCPa (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology C||R||Pa (a) and P\C||Pa (b)

296

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.85 Overactuated PMs with coupled Schönflies motions of types 4PPaC (a) and 4RCPa (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology P\Pa||C (a) and R||C||Pa (b)

3.2 Topologies with Complex Limbs

297

Fig. 3.86 Overactuated PMs with coupled Schönflies motions of types 4CPPa (a) and 4PaPC (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology C\P\kPa (a) and Pa\P\kC (b)

298

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.87 Overactuated PMs with coupled Schönflies motions of types 4PaRC (a) and 4PaCR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology Pa||R||C (a) and Pa\C\P (b)

3.2 Topologies with Complex Limbs

299

Fig. 3.88 4PaCR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 18, limb topology Pa||C||R

300

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.89 4PPaPaR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology P||Pa||Pa||R

3.2 Topologies with Complex Limbs

301

Fig. 3.90 4PPaRPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology P||Pa||R||Pa

302

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.91 4PRPaPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology P||R||Pa||Pa

3.2 Topologies with Complex Limbs

303

Fig. 3.92 4RPPaPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology R||P||Pa||Pa

304

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.93 4PaPPaR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa||P||Pa||R

3.2 Topologies with Complex Limbs

305

Fig. 3.94 4PaPRPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa||P||R||Pa

306

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.95 4RPaPPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology R||Pa||P||Pa

3.2 Topologies with Complex Limbs

307

Fig. 3.96 4PaPaPR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa||Pa||P||R

308

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.97 4PaPaRP-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa||Pa||R||P

3.2 Topologies with Complex Limbs

309

Fig. 3.98 4PaRPaP-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa||R||Pa||P

310

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.99 4PaRPPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa||R||P||Pa

3.2 Topologies with Complex Limbs

311

Fig. 3.100 4RPaPaP-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology R||Pa\Pa\\P

312

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.101 4PaRPaR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa||R\Pa\kR

3.2 Topologies with Complex Limbs

313

Fig. 3.102 4PaPaRR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa\Pa||R||R

314

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.103 4PaRRPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa\R||R||Pa

3.2 Topologies with Complex Limbs

315

Fig. 3.104 4PaPPaR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa\P\\Pa||R

316

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.105 4PaRPaP-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa||R\Pa\\P

3.2 Topologies with Complex Limbs

317

Fig. 3.106 4CPaPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology C||Pa||Pa

318

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.107 4PaPaC-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa||Pa||C

3.2 Topologies with Complex Limbs

319

Fig. 3.108 4PaCPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 30, limb topology Pa||C||Pa

320

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.109 4RPaRRR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R\Pa\kR\R||R

3.2 Topologies with Complex Limbs

321

Fig. 3.110 4RRRRPa-type overactuated PMs with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R||R\R||R||Pa (a) and R\R||R\R||Pa (b)

322

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.111 4PaRRRR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology Pa||R\R||R\kR

3.2 Topologies with Complex Limbs

323

Fig. 3.112 4PaRRRR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology Pa||R||R\R||R

324

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.113 4RRRPaR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R||R\R\Pa\kR

3.2 Topologies with Complex Limbs

325

Fig. 3.114 4RRRRPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R||R\R||R\kPa

326

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.115 4PRRRPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology P\R\R||R\kPa

3.2 Topologies with Complex Limbs

327

Fig. 3.116 4RRPRPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R\R||P||R\kPa

328

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.117 4RPRRPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R\P||R||R\kPa

3.2 Topologies with Complex Limbs

329

Fig. 3.118 4RRPRPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R\R||P||R\kPa

330

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.119 4RRRPPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R\R||R||P\kPa

3.2 Topologies with Complex Limbs

331

Fig. 3.120 4PRRRPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology P||R||R\R||Pa

332

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.121 4RPRRPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R||P||R\R||Pa

3.2 Topologies with Complex Limbs

333

Fig. 3.122 4RPRRPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R||P||R\R||Pa

334

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.123 4RRPRPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R||R||P\R||Pa

3.2 Topologies with Complex Limbs

335

Fig. 3.124 4RRRPPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R||R\R\P\kPa

336

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.125 4PRRPaR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology P||R||R\Pa||R

3.2 Topologies with Complex Limbs

337

Fig. 3.126 4RPRPaR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R||P||R\Pa||R

338

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.127 4RPRPaR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R||P||R\Pa||R

3.2 Topologies with Complex Limbs

339

Fig. 3.128 4RRPPaR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R||R||P\Pa||R

340

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.129 Overactuated PMs with coupled Schönflies motions of types 4PPaRRR (a) and 4PaPRRR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology P\Pa\kR||R||\kR (a) and Pa\P\\R||R\\R (b)

3.2 Topologies with Complex Limbs

341

Fig. 3.130 4PPaRRR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology P\Pa\\R||R\kR

342

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.131 4PaPRRR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology Pa\P||R||R\kR

3.2 Topologies with Complex Limbs

343

Fig. 3.132 4PaRPRR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology Pa\R||P||R\kR

344

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.133 4PaRPRR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology Pa\R||P||R\kR

3.2 Topologies with Complex Limbs

345

Fig. 3.134 4PaRRPR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology Pa\R||R||P\kR

346

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.135 4RCRPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R\C||R\kPa

3.2 Topologies with Complex Limbs

347

Fig. 3.136 4RRCPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R\R||C\kPa

348

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.137 4CRRPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology C||R\R||Pa

3.2 Topologies with Complex Limbs

349

Fig. 3.138 4RCRPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R||C\R||Pa

350

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.139 4CRPaR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology C||R\Pa||R

3.2 Topologies with Complex Limbs

351

Fig. 3.140 4RCPaR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology R||C\Pa||R

352

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.141 Overactuated PMs with coupled Schönflies motions of types 4PaCRR (a) and 4PaRCR (b) defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 14, limb topology Pa\C||R\kR (a) and Pa\R||C\kR (b)

3.2 Topologies with Complex Limbs

353

Fig. 3.142 4PaRRRPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 26, limb topology Pa\R||R\kR||Pa

354

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.143 4RPaRRPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 26, limb topology R\Pa\kR\kR||Pa

3.2 Topologies with Complex Limbs

355

Fig. 3.144 4RRRPaPa-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 26, limb topology R||R\R||Pa||Pa

356

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.145 4RPaRPaR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 26, limb topology R\Pa\kR\kPa||R

3.2 Topologies with Complex Limbs

357

Fig. 3.146 4PaPaRRR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 26, limb topology Pa||Pa\R||R\kR

358

3 Overactuated Topologies with Coupled Schönflies Motions

Fig. 3.147 4PaPaRRR-type overactuated PM with coupled Schönflies motions defined by MF ¼ SF ¼ 4; ðRF Þ ¼ ðv1 ; v2 ; v3 ; xd Þ, TF = 0, NF = 26, limb topology Pa||Pa\R||R\kR

3.2 Topologies with Complex Limbs

359

Table 3.5 Limb topology and connecting conditions of the overactuated solutions with no idle mobilities presented in Figs. 3.54, 3.55, 3.56, 3.57, 3.58, 3.59, 3.60, 3.61, 3.62, 3.63, 3.64, 3.65, 3.66, 3.67, 3.68, 3.69, 3.70, 3.71, 3.72, 3.73, 3.74, 3.75, 3.76, 3.77, 3.78, 3.79, 3.80, 3.81, 3.82, 3.83, 3.84, 3.85, 3.86, 3.87, 3.88 No. PM type Limb topology Connecting conditions 1.

4PRPPa (Fig. 3.54a)

P||R\P\kPa (Fig. 3.50a)

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

4PRPaP (Fig. 3.54b) 4PPaRP (Fig. 3.55a) 4PRRPa (Fig. 3.55b) 4PPRPa (Fig. 3.56a) 4PPRPa (Fig. 3.56b) 4PRPPa (Fig. 3.57a) 4PRPaP (Fig. 3.57b) 4RPRPa (Fig. 3.58a) 4RRPPa (Fig. 3.58b) RRPaP (Fig. 3.59) 4RPaPP (Fig. 3.60a) 4RPPPa (Fig. 3.60b) 4PaRPP (Fig. 3.61a) 4RPaPP (Fig. 3.61b) 4RPPaP (Fig. 3.62a) 4PPPaR (Fig. 3.62b) 4PPPaR (Fig. 3.63a) 4PPaRR (Fig. 3.63b) 4PPaRP (Fig. 3.64) 4PPaPR (Fig. 3.65) 4PPaPR (Fig. 3.66a) 4PPaRP (Fig. 3.66b)

P||R||Pa\P (Fig. 3.50b) P||Pa||R\P (Fig. 3.50c) P||R||R||Pa (Fig. 3.50d) P\P||R||Pa (Fig. 3.50e) P\P\kR||Pa (Fig. 3.50f) P\R||P||Pa (Fig. 3.50g) P\R||Pa||P (Fig. 3.50h) R||P||R||Pa (Fig. 3.50i) R||R||P||Pa (Fig. 3.50j) R||R||Pa||P (Fig. 3.50k) R||Pa\P\kP (Fig. 3.50l) R||P\P\kPa (Fig. 3.50m) Pa||R||P\P (Fig. 3.50n) R||Pa||P\P (Fig. 3.50o) R\P\kPa||P (Fig. 3.50p) P\P\kPa||R (Fig. 3.50q) P\P||Pa||R (Fig. 3.50r) P||Pa||R||R (Fig. 3.50s) P\Pa||R||P (Fig. 3.50t) P\Pa||P||R (Fig. 3.50u) P||Pa\P\kR (Fig. 3.50v) P||Pa\R\P (Fig. 3.50i0 )

24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

4RPPaP 4RPRPa 4PaPPR 4PaPPR 4PaPRP 4PaRRP 4PaRPR 4PaPRP 4PaRPP 4PRPPa

R||P||Pa\P (Fig. 3.50w) R||P||R||Pa (Fig. 3.50x) Pa||P\P\kR (Fig. 3.50y) Pa\P\kP||R (Fig. 3.50z) Pa\P\kR||P (Fig. 3.50z1) Pa||R||R||P (Fig. 3.50a0 ) Pa||R||P||R (Fig. 3.50b0 ) Pa||P||R\P (Fig. 3.50c0 ) Pa||R\P\kP (Fig. 3.50d0 ) P\R\P||Pa (Fig. 3.50e0 )

(Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig.

3.67) 3.68) 3.69a) 3.69b) 3.70) 3.71) 3.72) 3.73a) 3.73b) 3.74a)

34. 4PRPPa (Fig. 3.74b) 35. 4PRPaP (Fig. 3.75a) 36. 4PPRPa (Fig. 3.75b)

P\R\P\Pa (Fig. 3.50f0 ) P\R\Pa\\P (Fig. 3.50g0 ) P\P\\R\Pa (Fig. 3.50h0 )

37. 4PPPaR (Fig. 3.76a)

P\P||Pa\\R (Fig. 3.50j0 )

The revolute joints of the four limbs have parallel axes Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 The before last joints of the four limbs have parallel axes Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 The second joints of the four limbs have parallel axes Idem No. 33 Idem No. 33 The third joints of the four limbs have parallel axes The last joints of the four limbs have parallel axes (continued)

360

3 Overactuated Topologies with Coupled Schönflies Motions

Table 3.5 (continued) No. PM type

Limb topology

Connecting conditions

38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49.

4PPPaR (Fig. 3.76b) 4PaPPR (Fig. 3.77a) 4PaPPR (Fig. 3.77b) 4PaRPR (Fig. 3.78) 4PaPRR (Fig. 3.79a) 4PaPRR (Fig. 3.79b) 4PaRRP (Fig. 3.80a) 4RPaRR (Fig. 3.80b) 4PaRRR (Fig. 3.81) 4PaPRP (Fig. 3.82a) 4PaPRP (Fig. 3.82b) 4CPaP (Fig. 3.83a)

P\P\kPa\\R (Fig. 3.50k0 ) Pa||P\P\\R (Fig. 3.50l0 ) Pa\P\kP\\R (Fig. 3.50m0 ) Pa\R\P\kR (Fig. 3.50n0 ) Pa||P\R||R (Fig. 3.50o0 ) Pa\P\\R||R (Fig. 3.50p0 ) Pa\R||R\P (Fig. 3.50q0 ) R\Pa\kR||R (Fig. 3.50s0 ) Pa\R||R||R (Fig. 3.50r0 ) Pa||P\R\P (Fig. 3.50t0 ) Pa\P\\R\P (Fig. 3.50u0 ) C||Pa\P (Fig. 3.50v0 )

50. 51. 52. 53. 54. 55. 56. 57. 58. 59.

4PaCP 4CRPa 4PCPa 4PPaC 4RCPa 4CPPa 4PaPC 4PaRC 4PaCR 4PaCR

Pa||C\P (Fig. 3.50w0 ) C||R||Pa (Fig. 3.50x0 ) P\C||Pa (Fig. 3.50y0 ) P\Pa||C (Fig. 3.50z0 ) R||C||Pa (Fig. 3.50z0 1) C\P\kPa (Fig. 3.50a00 ) Pa\P\kC (Fig. 3.50b00 ) Pa||R||C (Fig. 3.50c00 ) Pa\C\P (Fig. 3.50e00 ) Pa||C||R (Fig. 3.50d00 )

Idem No. 37 Idem No. 37 Idem No. 37 Idem No. 37 Idem No. 37 Idem No. 37 Idem No. 23 Idem No. 37 Idem No. 37 Idem No. 23 Idem No. 23 The cylindrical joints of the four limbs have parallel directions Idem No. 49 Idem No. 49 Idem No. 49 Idem No. 49 Idem No. 49 Idem No. 49 Idem No. 49 Idem No. 49 Idem No. 49 Idem No. 49

(Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig.

3.83b) 3.84a) 3.84b) 3.85a) 3.85b) 3.86a) 3.86b) 3.87a) 3.87b) 3.88)

Table 3.6 Limb topology and connecting conditions of the overactuated solutions with no idle mobilities presented in Figs. 3.89, 3.90, 3.91, 3.92, 3.93, 3.94, 3.95, 3.96, 3.97, 3.98, 3.99, 3.100, 3.101, 3.102, 3.103, 3.104, 3.105, 3.106, 3.107, 3.108, 3.109, 3.110, 3.111, 3.112, 3.113, 3.114, 3.115, 3.116, 3.117, 3.118, 3.119, 3.120, 3.121, 3.122, 3.123, 3.124, 3.125, 3.126, 3.127, 3.128, 3.129, 3.130, 3.131, 3.132, 3.133, 3.134, 3.135, 3.136, 3.137, 3.138, 3.139, 3.140, 3.141, 3.142, 3.143, 3.144, 3.145, 3.146, 3.147 No. PM type Limb topology Connecting conditions 1.

4PPaPaR (Fig. 3.89)

P||Pa||Pa||R (Fig. 3.51a)

2. 3. 4. 5. 6. 7. 8.

4PPaRPa 4PRPaPa 4RPPaPa 4PaPPaR 4PaPRPa 4RPaPPa 4PaPaPR

P||Pa||R||Pa P||R||Pa||Pa R||P||Pa||Pa Pa||P||Pa||R Pa||P||R||Pa R||Pa||P||Pa Pa||Pa||P||R

(Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig.

3.90) 3.91) 3.92) 3.93) 3.94) 3.95) 3.96)

(Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig.

3.51b) 3.51c) 3.51d) 3.51e) 3.51f) 3.51g) 3.51h)

The revolute joints of the four limbs have parallel axes Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 (continued)

3.2 Topologies with Complex Limbs Table 3.6 (continued) No. PM type

361

Limb topology

Connecting conditions

3.97) 3.98) 3.99) 3.100)

Pa||Pa||R||P (Fig. 3.51i) Pa||R||Pa||P (Fig. 3.51j) Pa||R||P||Pa (Fig. 3.51k) R||Pa\Pa\\P (Fig. 3.51l)

13. 4PaRPaR (Fig. 3.101)

Pa||R\Pa\kR (Fig. 3.51m)

14. 4PaPaRR (Fig. 3.102) 15. 4PaRRPa (Fig. 3.103)

Pa\Pa||R||R (Fig. 3.51n) Pa\R||R||Pa (Fig. 3.51o)

16. 4PaPPaR (Fig. 3.104) 17. 4PaRPaP (Fig. 3.105)

Pa\P\\Pa||R (Fig. 3.51p) Pa||R\Pa\\P (Fig. 3.51q)

18. 4CPaPa (Fig. 3.106)

C||Pa||Pa (Fig. 3.51r)

Idem No. 1 Idem No. 1 Idem No. 1 The first revolute joints of the four limbs have parallel axes The last revolute joints of the four limbs have parallel axes Idem No. 13 The revolute joints of the parallelogram loops connecting the four limbs to the moving plateform have parallel axes Idem No. 13 The revolute joints of the parallelogram loops connecting the four limbs to the fixed base have parallel axes The cylindrical joints of the four limbs have parallel axes Idem No. 18 Idem No. 18 Idem No. 13 Idem No. 15 Idem No. 15 Idem No. 13 Idem No. 17 Idem No. 12 Idem No. 12 The second joints of the four limbs have parallel axes Idem No. 12 Idem No. 12 Idem No. 12 Idem No. 12 Idem No. 15 Idem No. 15 Idem No. 15 Idem No. 15 Idem No. 15

9. 10. 11. 12.

4PaPaRP 4PaRPaP 4PaRPPa 4RPaPaP

(Fig. (Fig. (Fig. (Fig.

19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

4PaPaC (Fig. 3.107) 4PaCPa (Fig. 3.108) 4RPaRRR (Fig. 3.109) 4RRRRPa (Fig. 3.110a) 4RRRRPa (Fig. 3.110b) 4PaRRRR (Fig. 3.111) 4PaRRRR (Fig. 3.112) 4RRRPaR (Fig. 3.113) 4RRRRPa (Fig. 3.114) 4PRRRPa (Fig. 3.115)

Pa||Pa||C (Fig. 3.51s) Pa||C||Pa (Fig. 3.51t) R\Pa\kR\R||R (Fig. 3.52a) R||R\R||R||Pa (Fig. 3.52b) R\R||R\R||Pa (Fig. 3.52d) Pa||R\R||R\kR (Fig. 3.52c) Pa||R||R\R||R (Fig. 3.52e) R||R\R\Pa\kR (Fig. 3.52f) R||R\R||R\kPa (Fig. 3.52g) P\R\R||R\kPa (Fig. 3.52h)

29. 30. 31. 32. 33. 34. 35. 36. 37.

4RRPRPa 4RPRRPa 4RRPRPa 4RRRPPa 4PRRRPa 4RPRRPa 4RPRRPa 4RRPRPa 4RRRPPa

R\R||P||R\kPa (Fig. 3.52i) R\P||R||R\kPa (Fig. 3.52j) R\R||P||R\kPa (Fig. 3.52k) R\R||R||P\kPa (Fig. 3.52l) P||R||R\R||Pa (Fig. 3.52m) R||P||R\R||Pa (Fig. 3.52n) R||P||R\R||Pa (Fig. 3.52o) R||R||P\R||Pa (Fig. 3.52p) R||R\R\P\kPa (Fig. 3.52q)

(Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig. (Fig.

3.116) 3.117) 3.118) 3.119) 3.120) 3.121) 3.122) 3.123) 3.124)

(continued)

362

3 Overactuated Topologies with Coupled Schönflies Motions

Table 3.6 (continued) No. PM type 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62.

4PRRPaR (Fig. 3.125) 4RPRPaR (Fig. 3.126) 4RPRPaR (Fig. 3.127) 4RRPPaR (Fig. 3.128) 4PPaRRR (Fig. 3.129a) 4PaPRRR (Fig. 3.129b) 4PPaRRR (Fig. 3.130) 4PaPRRR (Fig. 3.131) 4PaRPRR (Fig. 3.132) 4PaRPRR (Fig. 3.133) 4PaRRPR (Fig. 3.134) 4RCRPa (Fig. 3.135) 4RRCPa (Fig. 3.136) 4CRRPa (Fig. 3.137) 4RCRPa (Fig. 3.138) 4CRPaR (Fig. 3.139) 4RCPaR (Fig. 3.140) 4PaCRR (Fig. 3.141a) 4PaRCR (Fig. 3.141b) 4PaRRRPa (Fig. 3.142) 4RPaRRPa (Fig. 3.143) 4RRRPaPa (Fig. 3.144) 4RPaRPaR (Fig. 3.145) 4PaPaRRR (Fig. 3.146) 4PaPaRRR (Fig. 3.147)

Limb topology

Connecting conditions

P||R||R\Pa||R (Fig. 3.52r) R||P||R\Pa||R (Fig. 3.52s) R||P||R\Pa||R (Fig. 3.52t) R||R||P\Pa||R (Fig. 3.52u) P\Pa\kR||R||\kR (Fig. 3.52v) Pa\P\\R||R\\R (Fig. 3.52w) P\Pa\\R||R\kR (Fig. 3.52x) Pa\P||R||R\kR (Fig. 3.52y) Pa\R||P||R\kR (Fig. 3.52z) Pa\R||P||R\kR (Fig. 3.52a0 ) Pa\R||R||P\kR (Fig. 3.52b0 ) R\C||R\kPa (Fig. 3.52c0 ) R\R||C\kPa (Fig. 3.52d0 ) C||R\R||Pa (Fig. 3.52e0 ) R||C\R||Pa (Fig. 3.52f0 ) C||R\Pa||R (Fig. 3.52g0 ) R||C\Pa||R (Fig. 3.52h0 ) Pa\C||R\kR (Fig. 3.52i0 ) Pa\R||C\kR (Fig. 3.52j0 ) Pa\R||R\kR||Pa (Fig. 3.53a) R\Pa\kR\kR||Pa (Fig. 3.53b) R||R\R||Pa||Pa (Fig. 3.53c) R\Pa\kR\kPa||R(Fig. 3.53d) Pa||Pa\R||R\kR (Fig. 3.53e) Pa||Pa\R||R\kR (Fig. 3.53f)

Idem Idem Idem Idem Idem Idem Idem Idem Idem Idem Idem Idem Idem Idem Idem Idem Idem Idem Idem Idem Idem Idem Idem Idem Idem

No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No.

13 13 13 13 13 13 13 13 13 13 13 12 12 15 15 13 13 13 13 15 15 15 13 13 13

Table 3.7 Structural parametersa of parallel mechanisms in Figs. 3.54, 3.55, 3.56, 3.57, 3.58, 3.59, 3.60, 3.61, 3.62, 3.63, 3.64, 3.65, 3.66, 3.67, 3.68, 3.69, 3.70, 3.71, 3.72, 3.73, 3.74, 3.75, 3.76, 3.77, 3.78, 3.79, 3.80, 3.81, 3.82, 3.83, 3.84, 3.85, 3.86, 3.87, 3.88 No. Structural parameter Solution Figures 3.54, 3.55, 3.56, 3.57, 3.58, 3.59, Figures 3.83, 3.60, 3.61, 3.62, 3.63, 3.64, 3.65, 3.66, 3.84, 3.85, 3.86, 3.67, 3.68, 3.69, 3.70, 3.71, 3.72, 3.73, 3.87, 3.88 3.74, 3.75, 3.76, 3.77, 3.78, 3.79, 3.80, 3.81, 3.82 1. 2. 3. 4. 5. 6. 7. 8.

m pi (i = 1,…,4) p q k1 k2 k (RG1)

22 7 28 7 0 4 4 ðv1 ; v2 ; v3 ; xd Þ

18 6 24 7 0 4 4 ðv1 ; v2 ; v3 ; xd Þ (continued)

3.2 Topologies with Complex Limbs Table 3.7 (continued) No. Structural parameter

363

Solution Figures 3.54, 3.55, 3.56, 3.57, 3.58, 3.59, Figures 3.83, 3.60, 3.61, 3.62, 3.63, 3.64, 3.65, 3.66, 3.84, 3.85, 3.86, 3.67, 3.68, 3.69, 3.70, 3.71, 3.72, 3.73, 3.87, 3.88 3.74, 3.75, 3.76, 3.77, 3.78, 3.79, 3.80, 3.81, 3.82

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. a

(RG2) (RG3) (RG4) SGi (i = 1,…,4) rGi (i = 1,…,4) MGi (i = 1,…,4) (RF) SF rl rF MF NF TF P p1 fj Ppj¼1 2 f j Ppj¼1 3 f j Ppj¼1 4 j¼1 fj Pp j¼1 fj

ðv1 ; v2 ; v3 ; xd Þ ðv1 ; v2 ; v3 ; xd Þ ðv1 ; v2 ; v3 ; xd Þ 4 3 4 ðv1 ; v2 ; v3 ; xd Þ 4 12 24 4 18 0 7 7

ðv1 ; v2 ; v3 ; xd Þ ðv1 ; v2 ; v3 ; xd Þ ðv1 ; v2 ; v3 ; xd Þ 4 3 4 ðv1 ; v2 ; v3 ; xd Þ 4 12 24 4 18 0 7 7

7 7

7 7

28

28

See footnote of Table 2.2 for the nomenclature of structural parameters

Table 3.8 Structural parametersa of parallel mechanisms in Figs. 3.89, 3.90, 3.91, 3.92, 3.93, 3.94, 3.95, 3.96, 3.97, 3.98, 3.99, 3.100, 3.101, 3.102, 3.103, 3.104, 3.105, 3.106, 3.107, 3.108 No. Structural parameter Solution Figures 3.89, 3.90, 3.91, 3.92, 3.93, 3.94, Figures 3.106, 3.95, 3.96, 3.97, 3.98, 3.99, 3.100, 3.101, 3.107, 3.108 3.102, 3.103, 3.104, 3.105 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

m pi (i = 1,…,4) p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1,…,4) rGi (i = 1,…,4)

30 10 40 11 0 4 4 ðv1 ; v2 ; v3 ; xd Þ ðv1 ; v2 ; v3 ; xd Þ ðv1 ; v2 ; v3 ; xd Þ ðv1 ; v2 ; v3 ; xd Þ 4 6

26 9 36 11 0 4 4 ðv1 ; v2 ; v3 ; xd Þ ðv1 ; v2 ; v3 ; xd Þ ðv1 ; v2 ; v3 ; xd Þ ðv1 ; v2 ; v3 ; xd Þ 4 6 (continued)

364

3 Overactuated Topologies with Coupled Schönflies Motions

Table 3.8 (continued) No. Structural parameter

Solution Figures 3.89, 3.90, 3.91, 3.92, 3.93, 3.94, Figures 3.106, 3.95, 3.96, 3.97, 3.98, 3.99, 3.100, 3.101, 3.107, 3.108 3.102, 3.103, 3.104, 3.105

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. a

MGi (i = 1,…,4) (RF) SF rl rF MF NF TF P p1 fj Ppj¼1 2 f j Ppj¼1 3 j¼1 fj Pp4 fj Ppj¼1 f j¼1 j

4 ðv1 ; v2 ; v3 ; xd Þ 4 24 36 4 30 0 10 10

4 ðv1 ; v2 ; v3 ; xd Þ 4 24 36 4 30 0 10 10

10 10

10 10

40

40

See footnote of Table 2.2 for the nomenclature of structural parameters

Table 3.9 Structural parametersa of parallel mechanisms in Figs. 3.109, 3.110, 3.111, 3.112, 3.113, 3.114, 3.115, 3.116, 3.117, 3.118, 3.119, 3.120, 3.121, 3.122, 3.123, 3.124, 3.125, 3.126, 3.127, 3.128, 3.129, 3.130, 3.131, 3.132, 3.133, 3.134, 3.135, 3.136, 3.137, 3.138, 3.139, 3.140, 3.141, 3.142, 3.147 No. Structural parameter Solution

1. 2. 3. 4. 5. 6. 7. 8. 9.

m pi (i = 1,…,4) p q k1 k2 k (RG1) (RG2)

Figures 3.109, 3.110, 3.111, 3.112, 3.113, 3.114, 3.115, 3.116, 3.117, 3.118, 3.119, 3.120, 3.121, 3.122, 3.123, 3.124, 3.125, 3.126, 3.127, 3.128, 3.129, 3.130, 3.131, 3.132, 3.133, 3.134

Figures 3.135, 3.136, 3.137, 3.138, 3.139, 3.140, 3.141

Figures 3.142, 3.143, 3.144, 3.145, 3.146, 3.147

26 8 32 7 0 4 4 ðv1 ; v2 ; v3 ; xa ; xd Þ
v1 ; v2 ; v3 ; xb ; xd

22 7 28 7 0 4 4 ðv1 ; v2 ; v3 ; xa ; xd Þ
v1 ; v2 ; v3 ; xb ; xd

34 11 44 11 0 4 4 ðv1 ; v2 ; v3 ; xa ; xd Þ
v1 ; v2 ; v3 ; xb ; xd (continued)

References

365

Table 3.9 (continued) No. Structural parameter Solution

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. a

(RG3) (RG4) SGi (i = 1,…,4) rGi (i = 1,…,4) MGi (i = 1,…,4) (RF) SF rl rF MF NF TF P p1 fj Ppj¼1 2 j¼1 fj Pp3 fj Ppj¼1 4 f j Ppj¼1 f j¼1 j

Figures 3.109, 3.110, 3.111, 3.112, 3.113, 3.114, 3.115, 3.116, 3.117, 3.118, 3.119, 3.120, 3.121, 3.122, 3.123, 3.124, 3.125, 3.126, 3.127, 3.128, 3.129, 3.130, 3.131, 3.132, 3.133, 3.134

Figures 3.135, 3.136, 3.137, 3.138, 3.139, 3.140, 3.141

Figures 3.142, 3.143, 3.144, 3.145, 3.146, 3.147

ðv1 ; v2 ; v3 ; xa ; xd Þ
v1 ; v2 ; v3 ; xb ; xd 5 3 5 ðv1 ; v2 ; v3 ; xd Þ 4 12 28 4 14 0 8

ðv1 ; v2 ; v3 ; xa ; xd Þ
v1 ; v2 ; v3 ; xb ; xd 5 3 5 ðv1 ; v2 ; v3 ; xd Þ 4 12 28 4 14 0 8

ðv1 ; v2 ; v3 ; xa ; xd Þ
v1 ; v2 ; v3 ; xb ; xd 5 6 5 ðv1 ; v2 ; v3 ; xd Þ 4 24 40 4 26 0 11

8 8

8 8

11 11

8

8

11

32

32

44

See footnote of Table 2.2 for the nomenclature of structural parameters

References 1. Gogu G (2008) Structural synthesis of parallel robots: Part 1-methodology. Springer, Dordrecht 2. Gogu G (2009) Structural synthesis of parallel robots: part 2-translational topologies with two and three degrees of freedom. Springer, Dordrecht 3. Gogu G (2010) Structural synthesis of parallel robots: part 3-topologies with planar motion of the moving platform. Springer, Dordrecht 4. Gogu G (2012) Structural synthesis of parallel robots: part 4-other topologies with two and three degrees of freedom. Springer, Dordrecht

Chapter 4

Fully-Parallel Topologies with Decoupled Schönflies Motions

In the parallel robotic manipulators with decoupled Schönflies motions presented in this chapter each translational velocity of the moving platform depends on one actuated joint velocity vi ¼ vi ðq_ i Þ, i = 1, 2, 3 and the rotational velocity on two actuated joint velocities xd ¼ xd ðq_ 3 ; q_ 4 Þ. The Jacobian matrix in Eq. (1.18) is triangular and the parallel robot has decoupled motions. The fully-parallel overconstrained topologies of parallel robotic manipulator with decoupled Schönflies motions have four limbs actuated by linear or rotating motors. The limbs can be simple or complex kinematic chains and can also combine idle mobilities. The actuators can be mounted on the fixed base or on a moving link. The first solution has the advantage of reducing the moving masses and large workspace. The second solution would be more compact. There are no idle mobilities in the fully-parallel basic topologies presented in this chapter. The solutions presented in this section are obtained by using the methodology of structural synthesis proposed in Part 1 [1] and also used in Parts 2–4 of this work [2–4]. This original methodology combines new formulae for mobility connectivity, redundancy and overconstraints, and the evolutionary morphology in a unified approach of structural synthesis of parallel robotic manipulators.

4.1 Topologies with Simple Limbs In the fully-parallel topologies of PMs with decoupled Schönflies motions F G1 G2 G3 G4 presented in this section, the moving platform n : nGi (i = 1, 2, 3, 4) is connected to the reference platform 1 : 1Gi : 0 by four spatial simple limbs with four or five degrees of connectivity. The various types of simple limbs with four and five degrees of connectivity used in the fully-parallel basic topologies illustrated in this section are presented in Figs. 4.1 and 4.2. The simple limbs combine only revolute, prismatic and cylindrical joints. One actuator is combined in a prismatic or cylindrical pair of each limb. In the structural graph, the actuated joint is underlined. In this section, just the translational motion is actuated in the cylindrical joint, denoted by C. G. Gogu, Structural Synthesis of Parallel Robots, Solid Mechanics and Its Applications 206, DOI: 10.1007/978-94-007-7401-8_4,  Springer Science+Business Media Dordrecht 2014

367

368

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.1 Simple limbs for fully-parallel PMs with decoupled Schönflies motions defined by MG = SG = 4, ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa Þ

Various topologies of PMs with decoupled Schönflies motions and no idle mobilities can be obtained by combining four limbs presented in Figs. 4.1 and 4.2.

4.1 Topologies with Simple Limbs

369

Fig. 4.1 (continued)

Only topologies with three identical limbs and one slightly different limb are illustrated in Figs. 4.3, 4.4, 4.5, 4.6, 4.7.

370

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

The limb topology and connecting conditions of the solutions in Figs. 4.3, 4.4, 4.5, 4.6, 4.7 are systematized in Table 4.1, and their structural parameters in Tables 4.2 and 4.3.

Fig. 4.2 Simple limbs for fully-parallel
PMs with decoupled Schönflies motions defined by MG = SG = 5, ðRG Þ ¼ v1 ; v2 ; v3 ; xa ; xb

4.1 Topologies with Simple Limbs

371

Fig. 4.3 4PPPR-type fully-parallel PMs with decoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 0, NF = 6, limb topology P ? P ?? P ?? R and P ?P ?? P ?||R (a), P ? P ?? P||R (b)

372

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.4 Fully-parallel PMs with decoupled Schönflies motions of types 1PPRR-3PPRRR (a) and 1PRPR-3PRPRR (b) defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 0, NF = 3, limb topology P ? P ?||R||R and P ? P ?||R||R ? R (a), P||R ? P ?||R and P||R ? P ?||R ? R (b)

4.1 Topologies with Simple Limbs

373

Fig. 4.5 Fully-parallel PMs with decoupled Schönflies motions of types 1PRRP-3PRRPR (a) and 1PRRR-3PRRRR (b) defined by MF = SF = 4, (ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ), TF = 0, NF = 3, limb topology P||R||R ? P and P||R||R ? P ?? R (a), P||R||R||R and P||R||R||R ? R (b)

374

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.6 Fully-parallel PMs with decoupled Schönflies motions of types 1PPPR-3PPC (a) and 1CPR-3CPRR (b) defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 0, NF = 6 (a), NF = 3 (b), limb topology P ? P ?? P ?? R and P ? P ?? C (a), C ? P ?||R and C ? P ?||R ? R (b)

4.1 Topologies with Simple Limbs

375

Fig. 4.7 Fully-parallel PMs with decoupled Schönflies motions of types 1CRP-3CRPR (a) and 1CRR-3CRRR (b) defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 0, NF = 3, limb topology C||R ? P and C||R ? P ?? R (a), C||R||R and C||R||R ? R (b)

376

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Table 4.1 Limb topology and connecting conditions of the fully-parallel solutions with no idle mobilities presented in Figs. 4.3, 4.4, 4.5, 4.6, 4.7 No. PM type Limb topology Connecting conditions 1.

4PPPR (Fig. 4.3a)

2.

4PPPR (Fig. 4.3b)

3.

1PPRR-3PPRRR (Fig. 4.4a)

4.

1PRPR-3PRPRR (Fig. 4.4b)

5.

1PRRP-3PRRPR (Fig. 4.5a)

6.

1PRRR-3PRRRR (Fig. 4.5b) 1PPPR-3PPC (Fig. 4.6a)

7.

1CPR-3CPRR (Fig. 4.6b) 9. 1CRP-3CRPR (Fig. 4.7a) 10. 1CRR-3CRRR (Fig. 4.7b) 8.

The last revolute joints of the four P ?P ?? P ?? R limbs have parallel axes. The (Fig. 4.1b) actuated prismatic joints of limbs P ?P ?? P ?|R (Fig. 4.1a) G1, G2 and G3 hvave orthogonal directions. The actuated prismatic joints of limbs G3 and G4 have parallel directions P ?P ?? P ?? R (Fig. 2.1b) Idem No. 1 P ?P ?? P||R (Fig. 4.1c) Idem No. 1 P ?P ?|R||R (Fig. 4.1d) P ?P ?|R||R ?R (Fig. 4.2a) Idem No. 1 P||R ?P ?|R (Fig. 4.1e) P||R ?P ?|R ?R (Fig. 4.2b) P||R||R ?P (Fig. 4.1f) Idem No. 1 P||R||R ?P ?? R (Fig. 4.2c) P||R||R||R (Fig. 4.1g) Idem No. 1 P||R||R||R ?R (Fig. 4.2d) The last joints of the four limbs have P ?P ?? P ?? R parallel axes. The actuated (Fig. 4.1b) prismatic joints of limbs G1, G2 and P ?P ?? C (Fig. 4.1h) G3 hvave orthogonal directions. The actuated prismatic joints of limbs G3 and G4 have parallel directions Idem No. 1 C ?P ?|R (Fig. 4.1i) C ?P ?|R ?R (Fig. 4.2e) Idem No. 1 C||R ?P (Fig. 4.1j) C||R ?P ?? R (Fig. 4.2f) Idem No. 1 C||R||R (Fig. 4.1k) C||R||R ?R (Fig. 4.2g)

4.1 Topologies with Simple Limbs

377

Table 4.2 Structural parametersa of parallel mechanisms in Figs. 4.3, 4.4, 4.5 No. Structural parameter Solution Figure 4.3 Figures 4.4 and 4.5 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. a

m p1 pi (i = 2, 3, 4) p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SG1 SGi (i = 2, 3, 4) rGi (i = 1,…,4) MG1 MGi (i = 2, 3, 4) (RF) SF rl rF MF NF TF Pp1 fj Pj¼1 p2 j¼1 fj Pp3 fj Pj¼1 p4 fj Pj¼1 p j¼1 fj

14 4 4 16 3 4 0 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) 4 4 0 4 4 (v1 ; v2 ; v3 ; xa ) 4 0 12 4 6 0 4 4

17 4 5 19 3 4 0 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xa ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) 4 5 0 4 5 (v1 ; v2 ; v3 ; xa ) 4 0 15 4 3 0 4 5

4

5

4 16

5 19

See footnote of Table 2.2 for the nomenclature of structural parameters

378

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Table 4.3 Structural parametersa of parallel mechanisms in Figs. 4.6 and 4.7 No. Structural parameter Solution Figure 4.6a Figures 4.6b and 4.7 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. a

m p1 pi (i = 2, 3, 4) p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SG1 SGi (i = 2, 3, 4) rGi (i = 1,…,4) MG1 MGi (i = 2,3,4) (RF) SF rl rF MF NF TF Pp1 fj Ppj¼1 2 j¼1 fj Pp3 fj Ppj¼1 4 f j Ppj¼1 f j¼1 j

11 4 3 13 3 4 0 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) 4 4 0 4 4 (v1 ; v2 ; v3 ; xa ) 4 0 12 4 6 0 4 4

13 3 4 15 3 4 0 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xa ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) 4 5 0 4 5 (v1 ; v2 ; v3 ; xa ) 4 0 15 4 3 0 4 5

4

5

4 16

5 19

See footnote of Table 2.2 for the nomenclature of structural parameters

4.2 Topologies with Complex Limbs

379

4.2 Topologies with Complex Limbs In the fully-parallel topologies of PMs with decoupled Schönflies motions F G1 G2 G3 G4 presented in this section, the moving platform n : nGi (i = 1, 2, 3, 4) is connected to the reference platform 1 : 1Gi : 0 by four spatial complex limbs with four or five degrees of connectivity. One actuator is combined in each limb. In the structural graph, the actuated joint is underlined. In the topologies presented in this section, just the translational motion is actuated in the cylindrical joint denoted by C. The complex limbs with four or five degrees of connectivity presented in Figs. 4.8, 4.9, 4.10, 4.11, 4.12, 4.13 combine one two or three planar loops. The Pa-type planar loops have one degree of mobility and the Pn2 and Pn3-types planar loops have two and, respectively, three degrees of mobility. Each planar loop introduces three degrees of overconstraint. Various topologies of PMs with decoupled Schönflies motions and no idle mobilities can be obtained by combining four limbs presented in Figs. 4.8, 4.9, 4.10, 4.11, 4.12, 4.13. Only topologies with three identical limbs and one slightly different limb are illustrated in Figs. 4.14, 4.15, 4.16, 4.17, 4.18, 4.19, 4.20, 4.21, 4.22, 4.23, 4.24, 4.25, 4.26, 4.27, 4.28, 4.29, 4.30, 4.31, 4.32, 4.33, 4.34, 4.35, 4.36, 4.37, 4.38, 4.39, 4.40, 4.41, 4.42. The limb topology and connecting conditions of the solutions in Figs. 4.14, 4.15, 4.16, 4.17, 4.18, 4.19, 4.20, 4.21, 4.22, 4.23, 4.24, 4.25, 4.26, 4.27, 4.28, 4.29, 4.30, 4.31, 4.32, 4.33, 4.34, 4.35, 4.36, 4.37, 4.38, 4.39, 4.40, 4.41, 4.42 are systematized in Table 4.4, as are their structural parameters in Tables 4.5, 4.6, 4.7.

380

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.8 Complex limbs for fully-parallel PMs with decoupled Schönflies motions, combining one closed planar loop, defined by MG = SG = 4, ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa Þ

4.2 Topologies with Complex Limbs

Fig. 4.8 (continued)

381

382

Fig. 4.8 (continued)

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

4.2 Topologies with Complex Limbs

Fig. 4.8 (continued)

383

384

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.9 Complex limbs for fully-parallel PMs with decoupled Schönflies motions, combining
one planar closed loop, defined by MG = SG = 5, ðRF Þ ¼ v1 ; v2 ; v3 ; xa ; xb

4.2 Topologies with Complex Limbs

Fig. 4.9 (continued)

385

386

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.10 Complex limbs for fully-parallel PMs with decoupled Schönflies motions, combining two closed loops, defined by MG = SG = 4, ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa Þ

4.2 Topologies with Complex Limbs

Fig. 4.10 (continued)

387

388

Fig. 4.10 (continued)

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

4.2 Topologies with Complex Limbs

Fig. 4.10 (continued)

389

390

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.11 Complex limbs for fully-parallel PMs with decoupled Schönflies motions, combining
two closed loops, defined by MG = SG = 5, ðRG Þ ¼ v1 ; v2 ; v3 ; xa ; xb

4.2 Topologies with Complex Limbs

Fig. 4.11 (continued)

391

392

Fig. 4.11 (continued)

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

4.2 Topologies with Complex Limbs

Fig. 4.11 (continued)

393

394

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.12 Complex limbs for fully-parallel PMs with decoupled Schönflies motions, combining three closed loops, defined by MG = SG = 4, ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa Þ

4.2 Topologies with Complex Limbs

395

Fig. 4.12 (continued)

Fig. 4.13 Complex limbs for fully-parallel PMs with decoupled Schönflies
motions, combining three closed loops, defined by MG = SG = 5, ðRG Þ ¼ v1 ; v2 ; v3 ; xa ; xb

396

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.14 4PPPaR-type fully-parallel PMs with decoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 0, NF = 18, limb topology P ? P ?|Pa||R and P ? P ?||Pa ?||R (a), P ? P ?||Pa ?? R (b)

4.2 Topologies with Complex Limbs

397

Fig. 4.15 4PPaPR-type fully-parallel PMs with decoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 0, NF = 18, limb topology P||Pa ? P ?||R and P||Pa ? P ?? R (a), P||Pa ? P||R (b)

398

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.16 1PPaRR-3PPaRRR-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 0, NF = 15, limb topology P||Pa||R||R and P||Pa||R||R ? R

4.2 Topologies with Complex Limbs

399

Fig. 4.17 1PaRPR-3PaRPRR-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 0, NF = 15, limb topology Pa ? R ? P ?||R and Pa ? R ? P ?||R ? R

400

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.18 1PRRbR-3PRRbRR-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 0, NF = 15, limb topology P||R||Rb||R and P||R||Rb||R ? R

4.2 Topologies with Complex Limbs

401

Fig. 4.19 1PPn2R-3PPn2RR-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 0, NF = 15, limb topology P||Pn2||R and P||Pn2||R ? R

402

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.20 1PPn2R-3PPn2RR-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 0, NF = 15, limb topology P||Pn2||R and P||Pn2||R ? R

4.2 Topologies with Complex Limbs

403

Fig. 4.21 1PPn3-3PPn3R-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 0, NF = 15, limb topology P||Pn3 and P||Pn3 ? R

404

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.22 1PPn3-3PPn3R-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 0, NF = 15, limb topology P||Pn3 and P||Pn3 ? R

4.2 Topologies with Complex Limbs

405

Fig. 4.23 1CRbR-3CRbRR-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 0, NF = 15, limb topology C||Rb||R and C||Rb||R ? R

406

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.24 4PPaPaR-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 0, NF = 30, limb topology P||Pa||Pa||R and P||Pa||Pa ? R

4.2 Topologies with Complex Limbs

407

Fig. 4.25 1PaPaRR-3PaPaRRR type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 0, NF = 27, limb topology Pa ? Pa||R||R and Pa ? Pa||R||R ?? R

408

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.26 1PaPaRR-3PaPaRRR type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xa ), TF = 0, NF = 27, limb topology Pa ? Pa||R||R and Pa ? Pa||R||R ?|| R

4.2 Topologies with Complex Limbs

409

Fig. 4.27 1PRRbRbR-3PRRbRbRR-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xa ), TF = 0, NF = 27, limb topology P||R||Rb||Rb||R and P||R||Rb||Rb||R ? R

410

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.28 1PaRRbR-3PaRRbRR-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xa ), TF = 0, NF = 27, limb topology Pa ? R||Rb||R and Pa ? R||Rb||R ?? R

4.2 Topologies with Complex Limbs

411

Fig. 4.29 1PaRRbR-3PaRRbRR-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xa ), TF = 0, NF = 27, limb topology Pa ? R||Rb||R and Pa ? R||Rb||R ?|| R

412

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.30 1PaPn2R-3PaPn2RR-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xa ), TF = 0, NF = 27, limb topology Pa ? Pn2||R and Pa ? Pn2||R ?? R

4.2 Topologies with Complex Limbs

413

Fig. 4.31 1PaPn2R-3PaPn2RR-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xa ), TF = 0, NF = 27, limb topology Pa ? Pn2||R and Pa ? Pn2||R ?||R

414

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.32 1PaPn2R-3PaPn2RR-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xa ), TF = 0, NF = 27, limb topology Pa ? Pn2||R and Pa ? Pn2||R ?|| R

4.2 Topologies with Complex Limbs

415

Fig. 4.33 1PaPn2R-3PaPn2RR-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xa ), TF = 0, NF = 27, limb topology Pa ? Pn2||R and Pa ? Pn2||R ?? R

416

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.34 1PaPn3-3PaPn3R-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xa ), TF = 0, NF = 27, limb topology Pa ? Pn3||R and Pa ? Pn3 ?? R

4.2 Topologies with Complex Limbs

417

Fig. 4.35 1PaPn3-3PaPn3R-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xa ), TF = 0, NF = 27, limb topology Pa ? Pn3||R and Pa ? Pn3 ?|| R

418

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.36 1PaPn3-3PaPn3R-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xa ), TF = 0, NF = 27, limb topology Pa ? Pn3||R and Pa ? Pn3 ?? R

4.2 Topologies with Complex Limbs

419

Fig. 4.37 1PaPn3-3PaPn3R-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xa ), TF = 0, NF = 27, limb topology Pa ? Pn3||R and Pa ? Pn3 ?|| R

420

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.38 1CRbRbR-3CRbRbRR-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xa ), TF = 0, NF = 27, limb topology C||Rb||Rb||R and C||Rb||Rb||R ? R

4.2 Topologies with Complex Limbs

421

Fig. 4.39 4PaPaPaR-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xa ), TF = 0, NF = 42, limb topology Pa ? Pa||Pa||R and Pa ? Pa||Pa ?? R

422

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.40 4PaPaPaR-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xa ), TF = 0, NF = 42, limb topology Pa ? Pa||Pa||R and Pa ? Pa||Pa ?||R

4.2 Topologies with Complex Limbs

423

Fig. 4.41 1PaRRbRbR-3PaRRbRbRR-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xa ), TF = 0, NF = 39, limb topology Pa ? R||Rb||Rb||R and Pa ? R||Rb||Rb||R ?? R

424

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Fig. 4.42 1PaRRbRbR-3PaRRbRbRR-type fully-parallel PM with decoupled Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xa ), TF = 0, NF = 39, limb topology Pa ? R||Rb||Rb||R and Pa ? R||Rb||Rb||R ?|| R

4.2 Topologies with Complex Limbs

425

Table 4.4 Limb topology and connecting conditions of the fully-parallel solutions with no idle mobilities presented in Figs. 4.14, 4.15, 4.16, 4.17, 4.18, 4.19, 4.20, 4.21, 4.22, 4.23, 4.24, 4.25, 4.26, 4.27, 4.28, 4.29, 4.30, 4.31, 4.32, 4.33, 4.34, 4.35, 4.36, 4.37, 4.38, 4.39, 4.40, 4.41, 4.42 No. PM type Limb topology Connecting conditions 1.

4PPPaR (Fig. 4.14a)

P ?? P ? Pa||R (Fig. 4.8c) P ? P ?|Pa ?? R (Fig. 4.8b)

2.

4PPPaR (Fig. 4.14b)

3.

4PPaPR (Fig. 4.15a)

4.

4PPaPR (Fig. 4.15b)

5.

1PPaRR-3PPaRRR (Fig. 4.16)

6.

1PaRPR–3PaRPRR (Fig. 4.17)

P ? P ?? Pa||R (Fig. 4.8c) P ? P ?? Pa ?? R (Fig. 4.8a) P||Pa ? P ?? R (Fig. 4.8e) P||Pa ?P ?? R (Fig. 4.8d) P||Pa ? P ?? R (Fig. 4.8e) P||Pa ? P||R (Fig. 4.8f) P||Pa||R||R (Fig. 4.8g) P||Pa||R||R ? R (Fig. 4.9a) Pa ? R ? P ?? R (Fig. 4.8h) Pa ? R ? P ?? R ? R (Fig. 4.9b)

7.

1PRRbR–3PRRbRR (Fig. 4.18)

8.

1PPn2R–3PPn2RR (Fig. 4.19)

9.

1PPn2R–3PPn2RR (Fig. 4.20)

10. 1PPn3–3PPn3R (Fig. 4.21)

11. 1PPn3–3PPn3R (Fig. 4.22)

P||R||Rb||R (Fig. 4.8i) P||R||Rb||R ? R (Fig. 4.9c) P||Pn2||R (Fig. 4.8j) P||Pn2||R ? R (Fig. 4.9d) P||Pn2||R (Fig. 4.8k) P||Pn2||R ?R (Fig. 4.9e) P||Pn3 (Fig. 4.8l) P||Pn3 ? R (Fig. 4.9f) P||Pn3 (Fig. 4.8m) P||Pn3 ? R (Fig. 4.9g)

The last revolute joints of the four limbs have parallel axes. The actuated prismatic joints of limbs G1, G2 and G3 have orthogonal directions. The actuated prismatic joints of limbs G3 and G4 have parallel directions Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

The last revolute joints of the four limbs have parallel axes. The revolute joints between links 4 and 5 of limbs G1, G2 and G3 hvave orthogonal axes. The revolute joints between links 4 and 5 of limbs G3 and G4 have parallel axes Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

(continued)

426 Table 4.4 (continued) No. PM type 12. 1CRbR–3CRbRR (Fig. 4.23)

13. 4PPaPaR (Fig. 4.24)

14. 1PaPaRR–3PaPaRRR (Fig. 4.25)

15. 1PaPaRR–3PaPaRRR (Fig. 4.26)

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Limb topology

Connecting conditions

C||Rb||R (Fig. 4.8n) C||Rb||R ? R (Fig. 4.9h) P||Pa||Pa||R (Fig. 4.10b) P||Pa||Pa ? R (Fig. 4.10a) Pa ? Pa||R||R (Fig. 4.10c) Pa ? Pa||R||R ?? R (Fig. 4.11a)

Idem No. 1

Pa ? Pa||R||R (Fig. 4.10d) Pa ? Pa||R||R ?? R (Fig. 4.11b) 16. 1PRRbRbR–3PRRbRbRR P||R||Rb||Rb||R (Fig. 4.27) (Fig. 4.10e) P||R||Rb||Rb||R ? R (Fig. 4.11c) 17. 1PaRRbR-3PaRRbRR Pa ? R||Rb||R (Fig. 4.28) (Fig. 4.10f) Pa ? R||Rb||R ?? R (Fig. 4.11d) 18. 1PaRRbR-3PaRRbRR Pa ? R||Rb||R (Fig. 4.29) (Fig. 4.10g) Pa ? R||Rb||R ?? R (Fig. 4.11e) 19. 1PaPn2R–3PaPn2RR Pa ? Pn2||R (Fig. 4.30) (Fig. 4.10h) Pa ? Pn2||R ?? R (Fig. 4.11f) Pa ? Pn2||R 20. 1PaPn2R–3PaPn2RR (Fig. 4.31) (Fig. 4.10i) Pa ? Pn2||R ?? R (Fig. 4.11g) Pa ? Pn2||R 21. 1PaPn2R–3PaPn2RR (Fig. 4.32) (Fig. 4.10j) Pa ? Pn2||R ?? R (Fig. 4.11h) Pa ? Pn2||R 22. 1PaPn2R–3PaPn2RR (Fig. 4.33) (Fig. 4.10k) Pa ? Pn2||R ?? R (Fig. 4.11i)

Idem No. 1

The last revolute joints of the four limbs have parallel axes. The revolute joints between links 7 and 8 of limbs G1, G2 and G3 hvave orthogonal axes. The revolute joints between links 7 and 8 of limbs G3 and G4 have parallel axes Idem No. 14

Idem No. 1

Idem No. 6

Idem No. 1

Idem No. 14

Idem No. 14

Idem No. 14

Idem No. 14

(continued)

4.2 Topologies with Complex Limbs Table 4.4 (continued) No. PM type 23. Pa ?Pn2||R ?? R (Fig. 4.34)

24. 1PaPn3–3PaPn3R (Fig. 4.35)

25. 1PaPn3–3PaPn3R (Fig. 4.36)

26. 1PaPn3–3PaPn3R (Fig. 4.37)

27. 1CRbRbR–3CRbRbRR (Fig. 4.38)

28. 4PaPaPaR (Fig. 4.39)

29. 4PaPaPaR (Fig. 4.40)

30. 1PaRRbRbR– 3PaRRbRbRR (Fig. 4.41) 31. 1PaRRbRbR– 3PaRRbRbRR (Fig. 4.42)

427

Limb topology

Connecting conditions

Pa ? Pn3||R (Fig. 4.10l) Pa ? Pn3 ?? R (Fig. 4.11j) Pa ? Pn3||R (Fig. 4.10m) Pa ? Pn3 ?? |R (Fig. 4.11k) Pa ? Pn3||R (Fig. 4.10n) Pa ?Pn3 ?? R (Fig. 4.11l) Pa ? Pn3||R (Fig. 4.10o) Pa ? Pn3 ?? R (Fig. 4.11m) C||Rb||Rb||R (Fig. 4.10p) C||Rb||Rb||R ? R (Fig. 4.11n) Pa ? P||Pa||R (Fig. 4.12a) Pa ? Pa||Pa ?? R (Fig. 4.12c) Pa ? Pa||Pa||R (Fig. 4.12a) Pa ? Pa||Pa ?? R (Fig. 4.12b) Pa ? R||Rb||Rb||R (Fig. 4.12d) Pa ? R||Rb||Rb||R ?? R (Fig. 4.13a) Pa ? R||Rb||Rb||R (Fig. 4.12e) Pa ? R||Rb||Rb||R ?? R (Fig. 4.13b)

Idem No. 6

Idem No. 6

Idem No. 6

Idem No. 6

Idem No. 6

Idem No. 6

Idem No. 6

Idem No. 6

Idem No. 6

428

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Table 4.5 Structural parametersa of parallel mechanisms in Figures 4.14, 4.15, 4.16, 4.17, 4.18, 4.19, 4.20, 4.21, 4.22, 4.23 No. Structural Solution parameter Figures 4.14 Figures 4.16, 4.17, 4.18, 4.19, 4.20, Figure 4.23 and 4.15 4.21, 4.22 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. a

m p1 pi (i = 2, 3, 4) p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SG1 SGi (i = 2, 3, 4) rGi (i = 1,…,4) MG1 MGi (i = 2,3,4) (RF) SF rl rF MF NF TF Pp1 fj Pj¼1 p2 fj Pj¼1 p3 fj Pj¼1 p4 fj Pj¼1 p j¼1 fj

22 7 7 28 7 0 4 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) 4 4 3 4 4 (v1 ; v2 ; v3 ; xa ) 4 12 24 4 18 0 7

25 7 8 31 7 0 4 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xa ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) 4 5 3 4 5 (v1 ; v2 ; v3 ; xa ) 4 12 27 4 15 0 7

21 6 7 27 7 0 4 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xa ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) 4 5 0 4 5 (v1 ; v2 ; v3 ; xa ) 4 12 27 4 15 0 7

7 7

8 8

8 8

7 28

8 31

8 31

See footnote of Table 2.2 for the nomenclature of structural parameters

4.2 Topologies with Complex Limbs

429

Table 4.6 Structural parametersa of parallel mechanisms in Figs. 4.24, 4.25, 4.26, 4.27, 4.28, 4.29, 4.30, 4.31, 4.32, 4.33, 4.34, 4.35, 4.36, 4.37, 4.38 No. Structural parameter Solution

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. a

m p1 pi (i = 2,3,4) p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SG1 SGi (i = 2, 3, 4) rGi (i = 1,…,4) MG1 MGi (i = 2, 3, 4) (RF) SF rl rF MF NF TF Pp1 fj Pj¼1 p2 fj Pj¼1 p3 fj Pj¼1 p4 j¼1 fj Pp j¼1 fj

Figure 4.24

Figures 4.25, 4.26, 4.27, 4.28, 4.29, 4.30, 4.31, 4.32, 4.33, 4.34, 4.35, 4.36, 4.37

Figure 4.38

30 10 10 40 11 0 4 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) 4 4 6 4 4 (v1 ; v2 ; v3 ; xa ) 4 24 36 4 30 0 10 10

33 10 11 43 11 0 4 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xa ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) 4 5 6 4 5 (v1 ; v2 ; v3 ; xa ) 4 24 39 4 27 0 10 11

29 9 10 39 11 0 4 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xa ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) 4 5 0 4 5 (v1 ; v2 ; v3 ; xa ) 4 24 39 4 27 0 10 11

10 10

11 11

11 11

40

43

43

See footnote of Table 2.2 for the nomenclature of structural parameters

430

4 Fully-Parallel Topologies with Decoupled Schönflies Motions

Table 4.7 Structural parametersa of parallel mechanisms in Figs. 4.39, 4.40, 4.41, 4.42 No. Structural parameter Solution 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. a

m p1 pi (i = 2,3,4) p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SG1 SGi (i = 2, 3, 4) rGi (i = 1,…,4) MG1 MGi (i = 2, 3, 4) (RF) SF rl rF MF NF TF Pp1 fj Ppj¼1 2 j¼1 fj Pp3 fj Ppj¼1 4 f j Ppj¼1 j¼1 fj

Figures 4.39 and 4.40

Figures 4.41 and 4.42

38 13 13 52 15 0 4 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) 4 4 9 4 4 (v1 ; v2 ; v3 ; xa ) 4 36 48 4 42 0 13

41 13 14 55 15 0 4 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xa ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) 4 5 0 4 5 (v1 ; v2 ; v3 ; xa ) 4 36 51 4 39 0 13

13 13

14 14

13

14

52

55

See footnote of Table 2.2 for the nomenclature of structural parameters

References 1. Gogu G (2008) Structural synthesis of parallel robots: part 1-methodology. Springer, Dordrecht 2. Gogu G (2009) Structural Synthesis of Parallel Robots: Part 2-Translational Topologies with Two and Three Degrees of Freedom. Springer, Dordrecht 3. Gogu G (2010) Structural synthesis of parallel robots: part 3-topologies with planar motion of the moving platform. Springer, Dordrecht Heidelberg London New York 4. Gogu G (2012) Structural synthesis of parallel robots: part 4-other topologies with two and three degrees of freedom. Springer, Dordrecht Heidelberg London New York

Chapter 5

Topologies with Uncoupled Schönflies Motions

In the parallel robotic manipulators with uncoupled Schönflies motions presented in this chapter each independent velocity of the moving platform depends on one actuated joint velocity vi ¼ vi ðq_ i Þ, i = 1, 2, 3 and xd ¼ xd ðq_ 4 Þ. The Jacobian matrix in Eq. (1.18) is diagonal and the parallel robot has uncoupled motions. Fully-parallel solutions along with redundantly actuated solutions are presented in this chapter. The limbs can be simple or complex kinematic chains and can also combine idle mobilities. The actuators can be mounted on the fixed base or on a moving link. The first solution has the advantage of reducing the moving masses and large workspace. The second solution would be more compact. The solutions presented in this section are obtained by using the methodology of structural synthesis proposed in Part 1 [1] and also used in Parts 2–4 of this work [2–4]. This original methodology combines new formulae for mobility connectivity, redundancy and overconstraints, and the evolutionary morphology in a unified approach of structural synthesis of parallel robotic manipulators.

5.1 Fully-Parallel Topologies In the fully-parallel topologies of PMs with uncoupled Schönflies motions F / G1G2-G3-G4 presented in this section, the moving platform n : nGi (i = 1, 2, 3, 4) is connected to the reference platform 1 : 1Gi : 0 by four simple or complex limbs with four, five or six degrees of connectivity. One actuator is combined in a revolute or cylindrical pair of each limb. The actuated joint is underlined in the structural graph. In the cylindrical joint denoted by C just the rotational motion is actuated. The limbs G1, G2 and G3 are used for positioning the moving platform and limb G4 for orienting it. There are no idle mobilities in these basic solutions.

G. Gogu, Structural Synthesis of Parallel Robots, Solid Mechanics and Its Applications 206, DOI: 10.1007/978-94-007-7401-8_5,  Springer Science+Business Media Dordrecht 2014

431

432

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.1 Simple limbs for fully-parallel PMs with uncoupled Schönflies motions defined by
rG = 0, ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, MG = SG = 4 (a) and rG = 0, ðRG Þ ¼ v1 ; v2 ; v3 ; xa ; xb ; xd , MG = SG = 6 (b)

The various types of simple and complex limbs used in the fully-parallel basic solutions illustrated in this section are presented in Figs. 5.1, 5.2, 5.3, 5.4, 5.5, 5.6. The simple limbs combine only revolute and prismatic joints (Fig. 5.1). A cylindrical joint is also used in the complex limbs to replace two consecutive revolute and prismatic joints. The simple limb in Fig. 5.1b combines a homokinetic double universal joint with telescopic intermediary shaft. One (Figs. 5.2 and 5.3), two (Figs. 5.4 and 5.5) or three (Fig. 5.6) planar parallelogram loops are combined in the complex limbs. A planar telescopic parallelogram loop is combined in the limbs in Fig. 5.4l, n and p. Three joint parameters loose their independence in each parallelogram loop. Various topologies of PMs with uncoupled Schönflies motions of the moving platform and no idle mobilities can be obtained by using different topologies presented in Figs. 5.1, 5.2, 5.3, 5.4, 5.5, 5.6. Fully-parallel topologies with at least two identical limbs are illustrated in Figs. 5.7, 5.8, 5.9, 5.10, 5.11, 5.12, 5.13, 5.14, 5.155.16, 5.17, 5.18, 5.19, 5.20, 5.21, 5.22, 5.23, 5.24, 5.25, 5.26, 5.27, 5.28, 5.29, 5.30, 5.31, 5.32, 5.33, 5.34, 5.35, 5.36, 5.37, 5.38, 5.39, 5.40, 5.41, 5.42, 5.43, 5.44, 5.45, 5.46, 5.47, 5.48, 5.49, 5.50, 5.51, 5.52, 5.53, 5.54, 5.55, 5.56, 5.57, 5.58, 5.59, 5.60, 5.61, 5.62, 5.63, 5.64, 5.65, 5.66, 5.67, 5.68, 5.69, 5.70, 5.71, 5.72, 5.73, 5.74, 5.75, 5.76, 5.77, 5.78, 5.79, 5.80, 5.81, 5.82, 5.83, 5.84, 5.85, 5.86, 5.87, 5.88, 5.89, 5.90, 5.91, 5.92. These topologies are actuated by rotating motors mounted on the fixed base.

5.1 Fully-Parallel Topologies

433

Fig. 5.2 Complex limbs combining one parallelogram loop for fully-parallel PMs with uncoupled Schönflies motions defined by ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, MG = SG = 4, rG = 3

434

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.3 Complex limbs combining one parallelogram loop for fully-parallel
PMs with uncoupled Schönflies motions defined by MG = SG = 5, ðRG Þ ¼ v1 ; v2 ; v3 ; xa ; xb , rG = 3

5.1 Fully-Parallel Topologies

435

Fig. 5.4 Complex limbs combining two parallelogram loops for fully-parallel PMs with uncoupled Schönflies motions defined by MG = SG = 4, ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, rG = 6

436

Fig. 5.4 (continued)

5

Topologies with Uncoupled Schönflies Motions

5.1 Fully-Parallel Topologies

Fig. 5.4 (continued)

437

438

Fig. 5.4 (continued)

5

Topologies with Uncoupled Schönflies Motions

5.1 Fully-Parallel Topologies

439

Fig. 5.5 Complex limbs combining two parallelogram loops for fully-parallel PMs with uncoupled Schönflies motions defined by MG = SG = 5, ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa ; xd Þ, rG = 6

The topologies combining G4-limb presented in Figs. 5.1a and 5.4k, m, o give an unlimited rotational motion of the moving platform. The limb topology and connecting conditions of the solutions in Figs. 5.7, 5.8, 5.9, 5.10, 5.11, 5.12, 5.13, 5.14, 5.15, 5.16, 5.17, 5.18, 5.19, 5.20, 5.21, 5.22, 5.23, 5.24, 5.25, 5.26, 5.27, 5.28, 5.29, 5.30, 5.31, 5.32, 5.33, 5.34, 5.35, 5.36, 5.37, 5.38, 5.39, 5.40, 5.41, 5.42, 5.43, 5.44, 5.45, 5.46, 5.47, 5.48, 5.49, 5.50, 5.51, 5.52, 5.53, 5.54, 5.55, 5.56, 5.57, 5.58, 5.59, 5.60, 5.61, 5.62, 5.63, 5.64, 5.65, 5.66, 5.67, 5.68, 5.69, 5.70, 5.71, 5.72, 5.73, 5.74, 5.75, 5.76, 5.77, 5.78, 5.79, 5.80, 5.81, 5.82, 5.83, 5.84, 5.85, 5.86, 5.87, 5.88, 5.89, 5.90, 5.91, 5.92 are systematized in Tables 5.1, 5.2, 5.3, 5.4, 5.5, as are their structural parameters in Tables 5.6, 5.7, 5.8, 5.9, 5.10, 5.11, 5.12.

440

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.6 Complex limbs combining three parallelogram loops for fully-parallel PMs with uncoupled Schönflies motions defined by MG = SG = 4, ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, rG = 9

5.1 Fully-Parallel Topologies

441

Fig. 5.7 3PaPPR-1RPPP-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ), TF = 0, NF = 15, limb topology R\P\\P\\P and Pa||P\P\kR, Pa||P\P\\R (a), Pa\P\kP\kR, Pa\P\kP\\R (b)

442

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.8 2PaPRRR-1PaPRR-1RPPP-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 13, limb topology R\P\\P\\P and Pa\P\\R||R\\R, Pa\P\\R||R (a), Pa||P\R||R\\R, Pa||P\R||R (b)

5.1 Fully-Parallel Topologies

443

Fig. 5.9 2PaRPRR-1PaRPR-1RPPP-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 13, limb topology Pa\R\P\kR\R, Pa\R\P\kR and R\P\\P\\P

444

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.10 2PaRRRR-1PaRRR-1RPPP-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 13, limb topology Pa\R||R||R\kR, Pa\R||R||R and R\P\\P\\P

5.1 Fully-Parallel Topologies

445

Fig. 5.11 3PaPPR-1RUPU-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 13, limb topology R\R\R\P\kR\R and Pa||P\P\kR, Pa||P\P\\R (a), Pa\P\kP\kR, Pa\P\kP\\R (b)

446

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.12 2PaPRRR-1PaPRR-1RUPU-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 11, limb topology R\R\R\P\kR\R and Pa\P\\R||R\\R, Pa\P\\R||R (a), Pa||P\R||R\\R, Pa||P\R||R (b)

5.1 Fully-Parallel Topologies

447

Fig. 5.13 2PaRPRR-1PaRPR-1RUPU-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 11, limb topology Pa\R\P\kR\R, Pa\R\P\kR and R\R\R\P\kR\R

448

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.14 2PaRRRR-1PaRRR-1RUPU-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 11, limb topology Pa\R||R||R\kR, Pa\R||R||R and R\R\R\P\kR\R

5.1 Fully-Parallel Topologies

449

Fig. 5.15 3PaPPR-1RPaPaP-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, (RF) = ( :v1 ; v2 ; v3 ; xb :), TF = 0, NF = 21, limb topology R||Pa||Pa||P and Pa||P\P\kR, Pa||P\P\\R (a), Pa\P\kP\kR, Pa\P\kP\\R (b)

450

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.16 3PaPPR-1RPPaPa-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 21, limb topology R||P||Pa||Pa and Pa||P\P\kR, Pa||P\P\\R (a), Pa\P\kP\kR, Pa\P\kP\\R (b)

5.1 Fully-Parallel Topologies

451

Fig. 5.17 3PaPPR-1RPaPatP-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 21, limb topology R||Pa||Pat||P and Pa||P\P\kR, Pa||P\P\\R (a), Pa\P\kP\kR, Pa\P\kP\\R (b)

452

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.18 3PaPPR-1RPPaPat-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 21, limb topology R||P||Pa||Pat and Pa||P\P\kR, Pa||P\P\\R (a), Pa\P\kP\kR, Pa\P\kP\\R (b)

5.1 Fully-Parallel Topologies

453

Fig. 5.19 2PaPRRR-1PaPRR-1RPaPaP-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 19, limb topology R||Pa||Pa||P and Pa\P\\R||R\\R, Pa\P\\R||R (a), Pa||P\R||R\\R, Pa||P\R||R (b)

454

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.20 2PaPRRR-1PaPRR-1RPPaPa-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 19, limb topology R||P||Pa||Pa and Pa\P\\R||R\\R, Pa\P\\R||R (a), Pa||P\R||R\\R, Pa||P\R||R (b)

5.1 Fully-Parallel Topologies

455

Fig. 5.21 2PaPRRR-1PaPRR-1RPaPatP-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ), TF = 0, NF = 19, limb topology R||Pa||Pat||P and Pa\P\\R||R\\R, Pa\\P\\R||R (a), Pa||P\R||R\\R, Pa||P\R||R (b)

456

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.22 2PaPRRR-1PaPRR-1RPPaPat-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 19, limb topology R||P||Pa||Pat and Pa\P\\R||R\\R, Pa\P\\R||R (a), Pa||P\R||R\\R, Pa||P\R||R (b)

5.1 Fully-Parallel Topologies

457

Fig. 5.23 2PaRPRR-1PaRPR-1RPaPaP-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 19, limb topology Pa\R\P\kR\R, Pa\R\P\kR and R||Pa||Pa||P

458

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.24 2PaRPRR-1PaRPR-1RPPaPa-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ), TF = 0, NF = 19, limb topology Pa\R\P\kR\R, Pa\R\P\kR and R||P||Pa||Pa

5.1 Fully-Parallel Topologies

459

Fig. 5.25 2PaRPRR-1PaRPR-1RPaPatP-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 19, limb topology Pa\R\P\kR\R, Pa\R\P\kR and R||Pa||Pat||P

460

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.26 2PaRPRR-1PaRPR-1RPPaPat-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 19, limb topology Pa\R\P\kR\R, Pa\R\P\kR and R||P||Pa||Pat

5.1 Fully-Parallel Topologies

461

Fig. 5.27 2PaRRRR-1PaRRR-1RPaPaP-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 19, limb topology Pa\R||R||R\kR, Pa\R||R||R and R||Pa||Pa||P

462

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.28 2PaRRRR-1PaRRR-1RPPaPa-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 19, limb topology Pa\R||R||R\kR, Pa\R||R||R and R||P||Pa||Pa

5.1 Fully-Parallel Topologies

463

Fig. 5.29 2PaRRRR-1PaRRR-1RPaPatP-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 19, limb topology Pa\R||R||R\kR, Pa\R||R||R and R||Pa||Pat||P

464

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.30 2PaRRRR-1PaRRR-1RPPaPat-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 19, limb topology Pa\R||R||R\kR, Pa\R||R||R and R||P||Pa||Pat

5.1 Fully-Parallel Topologies

465

Fig. 5.31 3PaPPaR-1RPPP-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 24, limb topology R\P\\P\\P and Pa||P\Pa\kR, Pa||P\Pa||R (a), Pa\P\\Pa\kR, Pa\P\\Pa||R (b)

466

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.32 3PaPaPR-1RPPP-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 24, limb topology R\P\\P\\P and Pa\Pa\kP\\R, Pa\Pa\kP\kR (a), Pa\Pa\\P\\R, Pa\Pa\\P\kR (b)

5.1 Fully-Parallel Topologies

467

Fig. 5.33 2PaPaRRR-1PaPaRR-1RPPP-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 22, limb topology Pa\Pa||R||R\\R, Pa\Pa||R||R and R\P\\P\\P

468

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.34 2PaPaRRR-1PaPaRR-1RPPP-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 22, limb topology Pa\Pa||R||R\kR, Pa\Pa||R||R and R\P\\P\\P

5.1 Fully-Parallel Topologies

469

Fig. 5.35 3PaPPaR-1RUPU-type fully-parallel PMs with uncoupled Schönflies motions defined ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 22 limb topology by MF = SF = 4, R\R\R\P\kR\R and Pa||P\Pa\kR, Pa||P\Pa||R (a), Pa\P\\Pa\kR, Pa\P\\Pa||R (b)

470

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.36 3PaPaPR-1RUPU-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 22 limb topology R\R\R\P\kR\R and Pa\Pa\kP\\R, Pa\Pa\kP\kR (a), Pa\Pa\\P\\R, Pa\Pa\\P\kR (b)

5.1 Fully-Parallel Topologies

471

Fig. 5.37 2PaPaRRR-1PaPaRR-1RUPU-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 20, limb topology Pa\Pa||R||R\\R, Pa\Pa||R||R and R\R\R\P\kR\R

472

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.38 2PaPaRRR-1PaPaRR-1RUPU-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 20, limb topology Pa\Pa||R||R\kR, Pa\Pa||R||R and R\R\R\P\kR\R

5.1 Fully-Parallel Topologies

473

Fig. 5.39 3PaPPaR-1RPaPaP-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 30 limb topology R||Pa||Pa||P and Pa||P\Pa\kR, Pa||P\Pa||R (a), Pa\P\\Pa\kR, Pa\P\\Pa||R (b)

474

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.40 3PaPPaR-1RPPaPa-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 30 limb topology R||P||Pa||Pa and Pa||P\Pa\kR, Pa||P\Pa||R (a), Pa\P\\Pa\kR, Pa\P\\Pa||R (b)

5.1 Fully-Parallel Topologies

475

Fig. 5.41 3PaPPaR-1RPaPatP-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 30 limb topology R||Pa||Pat||P and Pa||P\Pa\kR, Pa||P\Pa||R (a), Pa\P\\Pa\kR, Pa\P\\Pa||R (b)

476

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.42 3PaPPaR-1RPPaPat-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 30 limb topology R||P||Pa||Pat and Pa||P\Pa\kR, Pa||P\Pa||R (a), Pa\P\\Pa\kR, Pa\P\\Pa||R (b)

5.1 Fully-Parallel Topologies

477

Fig. 5.43 3PaPaPR-1RPaPaP-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 30 limb topology Pa\Pa\kP\\R, Pa\Pa\kP\kR and R||Pa||Pa||P

478

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.44 3PaPaPR-1RPaPaP-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 30 limb topology Pa\Pa\\P\\R, Pa\Pa\\P\kR and R||Pa||Pa||P

5.1 Fully-Parallel Topologies

479

Fig. 5.45 3PaPaPR-1RPaPatP-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 30 limb topology Pa\Pa\kP\\R, Pa\Pa\kP\kR and R||Pa||Pat||P

480

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.46 3PaPaPR-1RPaPatP-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 30 limb topology Pa\Pa\\P\\R, Pa\Pa\\P\kR and R||Pa||Pat||P

5.1 Fully-Parallel Topologies

481

Fig. 5.47 3PaPaPR-1RPPaPa-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 30 limb topology Pa\Pa\kP\\R, Pa\Pa\kP\kR and R||P||Pa||Pa

482

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.48 3PaPaPR-1RPPaPa-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 30 limb topology Pa\Pa\\P\\R, Pa\Pa\\P\kR and R||P||Pa||Pa

5.1 Fully-Parallel Topologies

483

Fig. 5.49 3PaPaPR-1RPPaPat-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 30 limb topology Pa\Pa\kP\\R, Pa\Pa\kP\kR and R||P||Pa||Pat

484

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.50 3PaPaPR-1RPPaPat-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 30 limb topology Pa\Pa\\P\\R, Pa\Pa\\P\kR and R||P||Pa||Pat

5.1 Fully-Parallel Topologies

485

Fig. 5.51 2PaPaRRR-1PaPaRR-1RPaPaP-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 28, limb topology Pa\Pa||R||R\\R, Pa\Pa||R||R and R||Pa||Pa||P

486

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.52 2PaPaRRR-1PaPaRR-1RPaPaP-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 28, limb topology Pa\Pa||R||R\kR, Pa\Pa||R||R and R||Pa||Pa||P

5.1 Fully-Parallel Topologies

487

Fig. 5.53 2PaPaRRR-1PaPaRR-1RPaPatP-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 28, limb topology Pa\Pa||R||R\\R, Pa\Pa||R||R and R||Pa||Pat||P

488

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.54 2PaPaRRR-1PaPaRR-1RPaPatP-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 28, limb topology Pa\Pa||R||R\kR, Pa\Pa||R||R and R||Pa||Pat||P

5.1 Fully-Parallel Topologies

489

Fig. 5.55 2PaPaRRR-1PaPaRR-1RPPaPa-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 28, limb topology Pa\Pa||R||R\\R, Pa\Pa||R||R and R||P||Pa||Pa

490

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.56 2PaPaRRR-1PaPaRR-1RPPaPa-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 28, limb topology Pa\Pa||R||R\kR, Pa\Pa||R||R and R||P||Pa||Pa

5.1 Fully-Parallel Topologies

491

Fig. 5.57 2PaPaRRR-1PaPaRR-1RPPaPat-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 28, limb topology Pa\Pa||R||R\\R, Pa\Pa||R||R and R||P||Pa||Pat

492

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.58 2PaPaRRR-1PaPaRR-1RPPaPat-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 28, limb topology Pa\Pa||R||R\kR, Pa\Pa||R||R and R||P||Pa||Pat

5.1 Fully-Parallel Topologies

493

Fig. 5.59 3PaPaPaR-1RPPP-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 33, limb topology Pa\Pa||Pa\kR, Pa\Pa||Pa||R and R\P\\P\\P

494

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.60 3PaPaPaR-1RPPP-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 33, limb topology Pa\Pa||Pa\\R, Pa\Pa||Pa||R and R\P\\P\\P

5.1 Fully-Parallel Topologies

495

Fig. 5.61 3PaPaPaR-1RUPU-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 31, limb topology Pa\Pa||Pa\kR, Pa\Pa||Pa||R and R\R\R\P\kR\R

496

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.62 3PaPaPaR-1RUPU-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 31, limb topology Pa\Pa||Pa\\R, Pa\Pa||Pa||R and R\R\R\P\kR\R

5.1 Fully-Parallel Topologies

497

Fig. 5.63 3PaPaPaR-1RPaPaP-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 39, limb topology Pa\Pa||Pa\kR, Pa\Pa||Pa||R and R||Pa||Pa||P

498

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.64 3PaPaPaR-1RPaPaP-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 39, limb topology Pa\Pa||Pa\\R, Pa\Pa||Pa||R and R||Pa||Pa||P

5.1 Fully-Parallel Topologies

499

Fig. 5.65 3PaPaPaR-1RPaPatP-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 39, limb topology Pa\Pa||Pa\kR, Pa\Pa||Pa||R and R||Pa||Pat||P

500

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.66 3PaPaPaR-1RPaPatP-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 39, limb topology Pa\Pa||Pa\\R, Pa\Pa||Pa||R and R||Pa||Pat||P

5.1 Fully-Parallel Topologies

501

Fig. 5.67 3PaPaPaR-1RPPaPa-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 39, limb topology Pa\Pa||Pa\kR, Pa\Pa||Pa||R and R||P||Pa||Pa

502

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.68 3PaPaPaR-1RPPaPa-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 39, limb topology Pa\Pa||Pa\\R, Pa\Pa||Pa||R and R||P||Pa||Pa

5.1 Fully-Parallel Topologies

503

Fig. 5.69 3PaPaPaR-1RPPaPat-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 39, limb topology Pa\Pa||Pa\kR, Pa\Pa||Pa||R and R||P||Pa||Pat

504

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.70 3PaPaPaR-1RPPaPat-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 39, limb topology Pa\Pa||Pa\\R, Pa\Pa||Pa||R and R||P||Pa||Pat

5.1 Fully-Parallel Topologies

505

Fig. 5.71 3PaPPR-1CPaPa-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 21, limb topology C||Pa||Pa and Pa||P\P\kR, Pa||P\P\\R (a), Pa\P\kP\kR, Pa\P\kP\\R (b)

506

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.72 3PaPPR-1CPaPat-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 21, limb topology C||Pa||Pat and Pa||P\P\kR, Pa||P\P\\R (a), Pa\P\kP\kR, Pa\P\kP\\R (b)

5.1 Fully-Parallel Topologies

507

Fig. 5.73 2PaPRRR-1PaPRR-1CPaPa-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 19, limb topology C||Pa||Pa and Pa\P\\R||R\\R, Pa\P\\R||R (a), Pa||P\R||R\\R, Pa||P\R||R (b)

508

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.74 2PaPRRR-1PaPRR-1CPaPat-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 19, limb topology C||Pa||Pat and Pa\P\\R||R\\R, Pa\P\\R||R (a), Pa||P\R||R\\R, Pa||P\R||R (b)

5.1 Fully-Parallel Topologies

509

Fig. 5.75 2PaRPRR-1PaRPR-1CPaPa-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 19, limb topology Pa\R\P\kR\R, Pa\R\P\kR and C||Pa||Pa

510

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.76 2PaRPRR-1PaRPR-1CPaPat-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 19, limb topology Pa\R\P\kR\R, Pa\R\P\kR and C||Pa||Pat

5.1 Fully-Parallel Topologies

511

Fig. 5.77 2PaRRRR-1PaRRR-1CPaPa-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 19, limb topology Pa\R||R||R\kR, Pa\R||R||R and C||Pa||Pa

512

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.78 2PaRRRR-1PaRRR-1CPaPat-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 19, limb topology Pa\R||R||R\kR, Pa\R||R||R and C||Pa||Pat

5.1 Fully-Parallel Topologies

513

Fig. 5.79 3PaPPaR-1CPaPa-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 30 limb topology C||Pa||Pa and Pa||P\Pa\kR, Pa||P\Pa||R (a), Pa\P\\Pa\kR, Pa\P\\Pa||R (b)

514

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.80 3PaPPaR-1CPaPat-type fully-parallel PMs with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 30 limb topology C||Pa||Pat and Pa||P\Pa\kR, Pa||P\Pa||R (a), Pa\P\\Pa\kR, Pa\P\\Pa||R (b)

5.1 Fully-Parallel Topologies

515

Fig. 5.81 3PaPaPR-1CPaPa-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 30 limb topology Pa\Pa\kP\\R, Pa\Pa\kP\kR and C||Pa||Pa

516

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.82 3PaPaPR-1CPaPa-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 30 limb topology Pa\Pa\\P\\R, Pa\Pa\\P\kR and C||Pa||Pa

5.1 Fully-Parallel Topologies

517

Fig. 5.83 3PaPaPR-1CPaPat-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 30 limb topology Pa\Pa\kP\\R, Pa\Pa\kP\kR and C||Pa||Pat

518

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.84 3PaPaPR-1CPaPat-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 30 limb topology Pa\Pa\\P\\R, Pa\Pa\\P\kR and C||Pa||Pat

5.1 Fully-Parallel Topologies

519

Fig. 5.85 2PaPaRRR-1PaPaRR-1CPaPa-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 28, limb topology Pa\Pa||R||R\\R, Pa\Pa||R||R and C||Pa||Pa

520

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.86 2PaPaRRR-1PaPaRR-1CPaPa-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 28, limb topology Pa\Pa||R||R\kR, Pa\Pa||R||R and C||Pa||Pa

5.1 Fully-Parallel Topologies

521

Fig. 5.87 2PaPaRRR-1PaPaRR-1CPaPat-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 28, limb topology Pa\Pa||R||R\\R, Pa\Pa||R||R and C||Pa||Pat

522

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.88 2PaPaRRR-1PaPaRR-1CPaPat-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 28, limb topology Pa\Pa||R||R\kR, Pa\Pa||R||R and C||Pa||Pat

5.1 Fully-Parallel Topologies

523

Fig. 5.89 3PaPaPaR-1CPaPa-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 39, limb topology Pa\Pa||Pa\kR, Pa\Pa||Pa||R and C||Pa||Pa

524

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.90 3PaPaPaR-1CPaPa-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 39, limb topology Pa\Pa||Pa\\R, Pa\Pa||Pa||R and C||Pa||Pa

5.1 Fully-Parallel Topologies

525

Fig. 5.91 3PaPaPaR-1CPaPat-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 39, limb topology Pa\Pa||Pa\kR, Pa\Pa||Pa||R and C||Pa||Pat

526

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.92 3PaPaPaR-1CPaPat-type fully-parallel PM with uncoupled Schönflies motions defined by MF = SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, TF = 0, NF = 39, limb topology Pa\Pa||Pa\\R, Pa\Pa||Pa||R and C||Pa||Pat

5.1 Fully-Parallel Topologies

527

Table 5.1 Limb topology and connecting conditions of the fully-parallel solutions with no idle mobilities presented in Figs. 5.7, 5.8, 5.9, 5.10, 5.11, 5.12, 5.13, 5.14 No. PM type 1.

3PaPPR-1RPPP (Fig. 5.7a)

2.

3PaPPR-1RPPP (Fig. 5.7b)

3.

2PaPRRR1PaPRR1RPPP (Fig. 5.8a) 2PaPRRR1PaPRR1RPPP (Fig. 5.8b) 2PaRPRR1PaRPR1RPPP (Fig. 5.9) 2PaRRRR1PaRRR1RPPP (Fig. 5.10) 3PaPPR-1RUPU (Fig. 5.11a)

4.

5.

6.

7.

8.

3PaPPR-1RUPU (Fig. 5.11b)

9.

2PaPRRR1PaPRR1RUPU (Fig. 5.12a)

10. 2PaPRRR1PaPRR1RUPU (Fig. 5.12b) 11. 2PaRPRR1PaRPR1RUPU (Fig. 5.13)

12. 2PaRRRR1PaRRR1RUPU (Fig. 5.14)

Limb topology

Connecting conditions

Pa||P\P\kR (Fig. 5.2a) Pa||P\P\kR (Fig. 5.2d) R\P\\P\\P (Fig. 5.1a) Pa\P\kP\kR (Fig. 5.2b) Pa\P\kP\\R (Fig. 5.2c) R\P\\P\\P (Fig. 5.1a) Pa\P\\R||R\\R (Fig. 5.3a) Pa\P\\R||R (Fig. 5.2e) R\P\\P\\P (Fig. 5.1a)

The last joints of the four limbs have superposed axes/directions

Pa||P\R||R\\R (Fig. 5.3b) Pa||P\R||R (Fig. 5.2f) R\P\\P\\P (Fig. 5.1a) Pa\R\P\kR\R (Fig. 5.3c) Pa\R\P\kR (Fig. 5.2g) R\P\P\\P (Fig. 5.1a) Pa\R||R||R\kR (Fig. 5.3d) Pa\R||R||R (Fig. 5.2h) R\P\\P\\P (Fig. 5.1a)

Idem No. 1

The last joints of the four limbs have superposed axes/directions. The revolute joints between links 5 and 6 of limbs G1, G2 and G3 have orthogonal axes Idem No. 3

The last joints of the four limbs have superposed axes/directions. The revolute joints between links 4 and 5 of limbs G1, G2 and G3 have orthogonal axes Idem No. 5

Pa||P\P\kR (Fig. 5.2a) The first revolute joint of G4 limb and the last revolute joints of limbs G1, G2 and G3 Pa||P\P\\R (Fig. 5.2d) have parallel axes. The last joints of limbs R\R\R\P\kR\R (Fig. 5.1b) G1, G2 and G3 have superposed axes Pa\P\kP\kR (Fig. 5.2b) Idem No. 7 Pa\P\kP\\R (Fig. 5.2c) R\R\R\P\kR\R (Fig. 5.1b) Pa\P\\R||R\\R (Fig. 5.3a) The first revolute joint of G4 limb and the last revolute joints of limbs G1, G2 and G3 Pa\P\\R||R (Fig. 5.2e) have parallel axes. The last joints of of R\R\R\P\kR\R (Fig. 5.1b) limbs G1, G2 and G3 have superposed axes. The revolute joints between links 5 and 6 of of limbs G1, G2 and G3 have orthogonal axes Pa||P\R||R\\R (Fig. 5.3b) Idem No. 9 Pa||P\R||R (Fig. 5.2f) R\R\R\P\kR\R (Fig. 5.1b) Pa\R\P\kR\R (Fig. 5.3c) The first revolute joint of G4 limb and the last revolute joints of limbs G1, G2 and G3 Pa\R\P\kR (Fig. 5.2g) have parallel axes. The last joints of of R\R\R\P\kR\R (Fig. 5.1b) limbs G1, G2 and G3 have superposed axes, and their revolute joints between links 4 and 5 have orthogonal axes Pa\R||R||R\kR (Fig. 5.3d) Idem No. 11 Pa\R||R||R (Fig. 5.2h) R\R\R\P\kR\R (Fig. 5.1b)

528

5

Topologies with Uncoupled Schönflies Motions

Table 5.2 Limb topology and connecting conditions of the fully-parallel solutions with no idle mobilities presented in Figs. 5.15, 5.16, 5.17, 5.18, 5.19, 5.20, 5.21, 5.22, 5.23, 5.24, 5.25, 5.26, 5.27, 5.28, 5.29, 5.30 No. PM type Limb topology Connecting conditions 1.

3PaPPR-1RPaPaP (Fig. 5.15a)

2.

3PaPPR-1RPaPaP (Fig. 5.15b)

3.

3PaPPR-1RPPaPa (Fig. 5.16a)

4.

3PaPPR-1RPPaPa (Fig. 5.16b)

5.

3PaPPR1RPaPatP (Fig. 5.17a)

6.

3PaPPR1RPaPatP (Fig. 5.17b)

7.

3PaPPR-1RPPaPat (Fig. 5.18a)

8.

3PaPPR-1RPPaPat (Fig. 5.18b)

9.

2PaPRRR-1PaPRR1RPaPaP (Fig. 5.19a)

10. 2PaPRRR-1PaPRR1RPaPaP (Fig. 5.19b)

Pa||P\P\kR (Fig. 5.2a) Pa||P\P\\R (Fig. 5.2d) R||Pa||Pa||P (Fig. 5.4k) Pa\P\kP\kR (Fig. 5.2b) Pa\P\kP\\R (Fig. 5.2c) R||Pa||Pa||P (Fig. 5.4k) Pa||P\P\kR (Fig. 5.2a) Pa||P\P\\R (Fig. 5.2d) R||P||Pa||Pa (Fig. 5.4m)

Pa\P\kP\kR (Fig. 5.2b) Pa\P\kP\\R (Fig. 5.2c) R||P||Pa||Pa (Fig. 5.4m) Pa||P\P\kR (Fig. 5.2a) Pa||P\P\\R (Fig. 5.2d) R||Pa||Pat||P (Fig. 5.4l) Pa\P\kP\kR (Fig. 5.2b) Pa\P\kP\\R (Fig. 5.2c) R||Pa||Pat||P (Fig. 5.4l) Pa||P\P\kR (Fig. 5.2a) Pa||P\P\\R (Fig. 5.2d) R||P||Pa||Pat (Fig. 5.4n) Pa\P\kP\kR (Fig. 5.2b) Pa\P\kP\\R (Fig. 5.2c) R||P||Pa||Pat (Fig. 5.4n) Pa\P\R||R\\R (Fig. 5.3a) Pa\P\\R||R (Fig. 5.2e) R||Pa||Pa||P (Fig. 5.4k)

Pa||P\R||R\\R (Fig. 5.3b) Pa||P\R||R (Fig. 5.2f) R||Pa||Pa||P (Fig. 5.4k)

The last joints of the four limbs have superposed axes/directions Idem No. 1

The revolute joint between links 7D and 9 of G4 limb and the last revolute joints of limbs G1, G2 and G3 have superposed axes Idem No. 3

Idem No. 1

Idem No. 1

Idem No. 3

Idem No. 3

The last joints of the four limbs have superposed axes/directions. The revolute joints between links 5 and 6 of limbs G1, G2 and G3 have orthogonal axes Idem No. 9

(continued)

5.1 Fully-Parallel Topologies Table 5.2 (continued) No. PM type 11. 2PaPRRR-1PaPRR1RPPaPa (Fig. 5.20a) 12. 2PaPRRR-1PaPRR1RPPaPa (Fig. 5.20b) 13. 2PaPRRR-1PaPRR1RPaPatP (Fig. 5.21a) 14. 2PaPRRR-1PaPRR1RPaPatP (Fig. 5.21b) 15. 2PaPRRR-1PaPRR1RPPaPat (Fig. 5.22a) 16. 2PaPRRR-1PaPRR1RPPaPat (Fig. 5.22b) 17. 2PaRPRR-1PaRPR1RPaPaP (Fig. 5.23)

18. 2PaRPRR-1PaRPR1RPPaPa (Fig. 5.24) 19. 2PaRPRR-1PaRPR1RPaPatP (Fig. 5.25) 20. 2PaRPRR-1PaRPR1RPPaPat (Fig. 5.26)

529

Limb topology

Connecting conditions

Pa\P\\R||R\\R (Fig. 5.3a) Pa\P\\R||R (Fig. 5.2e) R||P||Pa||Pa (Fig. 5.4m) Pa||P\R||R\\R (Fig. 5.3b) Pa||P\R||R (Fig. 5.2f) R||P||Pa||Pa (Fig. 5.4m) Pa\P\\R||R\\R (Fig. 5.3a) Pa\P\\R||R (Fig. 5.2e) R||Pa||Pat||P (Fig. 5.4l) Pa||P\R||R\\R (Fig. 5.3b) Pa||P\R||R (Fig. 5.2f) R||Pa||Pat||P (Fig. 5.4l) Pa\P\\R||R\\R (Fig. 5.3a) Pa\P\\R||R (Fig. 5.2e) R||P||Pa||Pat (Fig. 5.4n) Pa||P\R||R\\R (Fig. 5.3b) Pa||P\R||R (Fig. 5.2f) R||P||Pa||Pat (Fig. 5.4n) Pa\R\P\kR\R (Fig. 5.3c) Pa\R\P\kR (Fig. 5.2g) R||Pa||Pa||P (Fig. 5.4k)

Idem No. 9

Pa\R\P\kR\R (Fig. 5.3c) Pa\R\P\kR (Fig. 5.2g) R||P||Pa||Pa (Fig. 5.4m) Pa\R\P\kR\R (Fig. 5.3c) Pa\R\P\kR (Fig. 5.2g) R||Pa||Pat||P (Fig. 5.4l) Pa\R\P\kR\R (Fig. 5.3c) Pa\R\P\kR (Fig. 5.2g) R||P||Pa||Pat (Fig. 5.4n)

Idem No. 9

Idem No. 9

Idem No. 9

Idem No. 9

Idem No. 9

The last joints of the four limbs have superposed axes/directions. The revolute joints between links 4 and 5 of limbs G1, G2 and G3 have orthogonal axes Idem No. 17

Idem No. 17

The revolute joint between links 7D and 9 of G4 limb and the last revolute joints of limbs G1, G2 and G3 have superposed axes. The revolute joints between links 4 and 5 of limbs G1, G2 and G3 have orthogonal axes (continued)

530

5

Table 5.2 (continued) No. PM type 21. 2PaRRRR-1PaRRR1RPaPaP (Fig. 5.27) 22. 2PaRRRR-1PaRRR1RPPaPa (Fig. 5.28) 23. 2PaRRRR-1PaRRR1RPaPatP (Fig. 5.29) 24. 2PaRRRR-1PaRRR1RPPaPat (Fig. 5.30)

Topologies with Uncoupled Schönflies Motions

Limb topology

Connecting conditions

Pa\R||R||R\kR (Fig. 5.3d) Pa\R||R||R (Fig. 5.2h) R||Pa||Pa||P (Fig. 5.4k) Pa\R||R||R\kR (Fig. 5.3d) Pa\R||R||R (Fig. 5.2h) R||P||Pa||Pa (Fig. 5.4m) Pa\R||R||R\kR (Fig. 5.3d) Pa\R||R||R (Fig. 5.2h) R||Pa||Pat||P (Fig. 5.4l) Pa\R||R||R\kR (Fig. 5.3d) Pa\R||R||R (Fig. 5.2h) R||P||Pa||Pat (Fig. 5.4n)

Idem No. 17

Idem No. 20

Idem No. 17

Idem No. 20

Table 5.3 Limb topology and connecting conditions of the fully-parallel solutions with no idle mobilities presented in Figs. 5.31, 5.32, 5.33, 5.34, 5.35, 5.36, 5.37, 5.38 No. PM type Limb topology Connecting conditions 1.

3PaPPaR1RPPP (Fig. 5.31a)

2.

3PaPPaR1RPPP (Fig. 5.31b)

3.

3PaPaPR1RPPP (Fig. 5.32a)

4.

3PaPaPR1RPPP (Fig. 5.32b)

5.

2PaPaRRR1PaPaRR1RPPP (Fig. 5.33)

6.

2PaPaRRR1PaPaRR1RPPP (Fig. 5.34) 3PaPPaR1RUPU (Fig. 5.35a)

7.

Pa||P\Pa\kR (Fig. 5.4a) Pa||P\Pa||R (Fig. 5.4d) R\P\\P\\P (Fig. 5.1a) Pa\P\\Pa\kR (Fig. 5.4b) Pa\P\\Pa||R (Fig. 5.4c) R\P\\P\\P (Fig. 5.1a) Pa\Pa\kP\\R (Fig. 5.4f) Pa\Pa\kP\kR (Fig. 5.4h) R\P\\P\\P (Fig. 5.1a) Pa\Pa\\P\\R (Fig. 5.4g) Pa\Pa\\P\kR (Fig. 5.4e) R\P\\P\\P (Fig. 5.1a) Pa\Pa||R||R\\R (Fig. 5.5a) Pa\Pa||R||R (Fig. 5.4i) R\P\\P\\P (Fig. 5.1a) Pa\Pa||R||R\kR (Fig. 5.5b) Pa\Pa||R||R (Fig. 5.4j) R\P\\P\\P (Fig. 5.1a)

The last joints of the four limbs have superposed axes/directions Idem No. 1

Idem No. 1

Idem No. 1

The last joints of the four limbs have superposed axes/directions. The revolute joints between links 7 and 8 of limbs G1, G2 and G3 have orthogonal axes Idem No. 5

Pa||P\Pa\kR (Fig. 5.4a) The first revolute joint of G4 limb and the last revolute joints of limbs G1, Pa||P\Pa||R (Fig. 5.4d) G2 and G3 have parallel axes. The R\R\R\P\kR\R (Fig. 5.1b) last joints of of limbs G1, G2 and G3 have superposed axes (continued)

5.1 Fully-Parallel Topologies

531

Table 5.3 (continued) No. PM type Limb topology 8.

3PaPPaR1RUPU (Fig. 5.35b)

9.

3PaPaPR1RUPU (Fig. 5.36a)

10. 3PaPaPR1RUPU (Fig. 5.36b) 11. 2PaPaRRR1PaPaRR1RUPU (Fig. 5.37) 12. 2PaPaRRR1PaPaRR1RUPU (Fig. 5.38)

Pa\P\\Pa\kR (Fig. 5.4b) Pa\P\\Pa||R (Fig. 5.4c) R\R\R\P\kR\R (Fig. 5.1b) Pa\Pa\kP\\R (Fig. 5.4f) Pa\Pa\kP\kR (Fig. 5.4h) R\R\R\P\kR\R (Fig. 5.1b) Pa\Pa\\P\\R (Fig. 5.4g) Pa\Pa\\P\kR (Fig. 5.4e) R\R\R\P\kR\R (Fig. 5.1b) Pa\Pa||R||R\\R (Fig. 5.5a) Pa\Pa||R||R (Fig. 5.4i) R\R\R\P\kR\R (Fig. 5.1b) Pa\Pa||R||R\kR (Fig. 5.5b) Pa\Pa||R||R (Fig. 5.4j) R\R\R\P\kR\R (Fig. 5.1b)

Connecting conditions Idem No. 7

Idem No. 7

Idem No. 7

The last joints of of limbs G1, G2 and G3 have superposed axes. The revolute joints between links 7 and 8 of limbs G1, G2 and G3 have orthogonal axes Idem No. 11

Table 5.4 Limb topology and connecting conditions of the fully-parallel solutions with no idle mobilities presented in Figs. 5.39, 5.40, 5.41, 5.42, 5.43, 5.44, 5.45, 5.46, 5.47, 5.48, 5.49, 5.50, 5.51, 5.52, 5.53, 5.54, 5.55, 5.56, 5.57, 5.58, 5.59, 5.60, 5.61, 5.62, 5.63, 5.64, 5.65, 5.66, 5.67, 5.68 No. PM type Limb topology Connecting conditions 1.

3PaPPaR-1RPaPaP (Fig. 5.39a)

2.

3PaPPaR-1RPaPaP (Fig. 5.39b)

3.

3PaPPaR-1RPPaPa (Fig. 5.40a)

Pa||P\Pa\kR (Fig. 5.4a) Pa||P\Pa||R (Fig. 5.4d) R||Pa||Pa||P (Fig. 5.4k) Pa\P\\Pa\kR (Fig. 5.4b) Pa\P\\Pa||R (Fig. 5.4c) R||Pa||Pa||P (Fig. 5.4k) Pa||P\Pa\kR (Fig. 5.4a) Pa||P\Pa||R (Fig. 5.4d) R||P||Pa||Pa (Fig. 5.4m)

The last joints of the four limbs have superposed axes/directions Idem No. 1

The revolute joint between links 7D and 9 of G4 limb and the last revolute joints of limbs G1, G2 and G3 have superposed axes (continued)

532

5

Table 5.4 (continued) No. PM type 4.

3PaPPaR-1RPPaPa (Fig. 5.40b)

5.

3PaPPaR1RPaPatP (Fig. 5.41a)

6.

3PaPPaR1RPaPatP (Fig. 5.41b)

7.

3PaPPaR-1RPPaPat (Fig. 5.42a)

8.

3PaPPaR-1RPPaPat (Fig. 5.42b)

9.

3PaPaPR-1RPaPaP (Fig. 5.43)

10. 3PaPaPR-1RPaPaP (Fig. 5.44) 11. 3PaPaPR1RPaPatP (Fig. 5.45) 12. 3PaPaPR1RPaPatP (Fig. 5.46) 13. 3PaPaPR-1RPPaPa (Fig. 5.47) 14. 3PaPaPR-1RPPaPa (Fig. 5.48) 15. 3PaPaPR-1RPPaPat (Fig. 5.49) 16. 3PaPaPR-1RPPaPat (Fig. 5.50) 17. 2PaPaRRR-1PaPaRR1RPaPaP (Fig. 5.51)

Topologies with Uncoupled Schönflies Motions

Limb topology

Connecting conditions

Pa\P\\Pa\kR (Fig. 5.4b) Pa\P\\Pa||R (Fig. 5.4c) R||P||Pa||Pa (Fig. 5.4m) Pa||P\Pa\kR (Fig. 5.4a) Pa||P\Pa||R (Fig. 5.4d) R||Pa||Pat||P (Fig. 5.4l) Pa\P\\Pa\kR (Fig. 5.4b) Pa\P\\Pa||R (Fig. 5.4c) R||Pa||Pat||P (Fig. 5.4l) Pa||P\Pa\kR (Fig. 5.4a) Pa||P\Pa||R (Fig. 5.4d) R||P||Pa||Pat (Fig. 5.4n) Pa\P\\Pa\kR (Fig. 5.4b) Pa\P\\Pa||R (Fig. 5.4c) R||P||Pa||Pat (Fig. 5.4n) Pa\Pa\kP\\R (Fig. 5.4f) Pa\Pa\kP\kR (Fig. 5.4h) R||Pa||Pa||P (Fig. 5.4k) Pa\Pa\\P\\R (Fig. 5.4g) Pa\Pa\\P\kR (Fig. 5.4e) R||Pa||Pa||P (Fig. 5.4k) Pa\Pa\kP\\R (Fig. 5.4f) Pa\Pa\kP\kR (Fig. 5.4h) R||Pa||Pat||P (Fig. 5.4l) Pa\Pa\\P\\R (Fig. 5.4g) Pa\Pa\\P\kR (Fig. 5.4e) R||Pa||Pat||P (Fig. 5.4l) Pa\Pa\kP\\R (Fig. 5.4f) Pa\Pa\kP\kR (Fig. 5.4h) R||P||Pa||Pa (Fig. 5.4m) Pa\Pa\\P\\R (Fig. 5.4g) Pa\Pa\\P\kR (Fig. 5.4e) R||P||Pa||Pa (Fig. 5.4m) Pa\PakP\\R (Fig. 5.4f) Pa\Pa\kP\kR (Fig. 5.4h) R||P||Pa||Pat (Fig. 5.4n) Pa\Pa\\P\\R (Fig. 5.4g) Pa\Pa\\P\kR (Fig. 5.4e) R||P||Pa||Pat (Fig. 5.4n) Pa\Pa||R||R\\R (Fig. 5.5a) Pa\Pa||R||R (Fig. 5.4i) R||Pa||Pa||P (Fig. 5.4k)

Idem No. 3

Idem No. 1

Idem No. 1

Idem No. 3

Idem No. 3

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 3

Idem No. 3

Idem No. 3

Idem No. 3

Idem No. 1

(continued)

5.1 Fully-Parallel Topologies Table 5.4 (continued) No. PM type 18. 2PaPaRRR-1PaPaRR1RPaPaP (Fig. 5.52) 19. 2PaPaRRR-1PaPaRR1RPaPatP (Fig. 5.53) 20. 2PaPaRRR-1PaPaRR1RPaPatP (Fig. 5.54) 21. 2PaPaRRR-1PaPaRR1RPPaPa (Fig. 5.55)

22. 2PaPaRRR-1PaPaRR1RPPaPa (Fig. 5.56) 23. 2PaPaRRR-1PaPaRR1RPPaPat (Fig. 5.57) 24. 2PaPaRRR-1PaPaRR1RPPaPat (Fig. 5.58) 25. 3PaPaPaR-1RPPP (Fig. 5.59) 26. 3PaPaPaR-1RPPP (Fig. 5.60) 27. 3PaPaPaR-1RUPU (Fig. 5.61)

28. 3PaPaPaR-1RUPU (Fig. 5.62) 29. 3PaPaPaR-1RPaPaP (Fig. 5.63)

533

Limb topology

Connecting conditions

Pa\Pa||R||R\kR (Fig. 5.5b) Pa\Pa||R||R (Fig. 5.4i) R||Pa||Pa||P (Fig. 5.4k) Pa\Pa||R||R\\R (Fig. 5.5a) Pa\Pa||R||R (Fig. 5.4i) R||Pa||Pat||P (Fig. 5.4l) Pa\Pa||R||R\kR (Fig. 5.5b) Pa\Pa||R||R (Fig. 5.4i) R||Pa||Pat||P (Fig. 5.4l) Pa\Pa||R||R\\R (Fig. 5.5a) Pa\Pa||R||R (Fig. 5.4i) R||P||Pa||Pa (Fig. 5.4m)

Idem No. 1

Pa\Pa||R||R\kR (Fig. 5.5b) Pa\Pa||R||R (Fig. 5.4i) R||P||Pa||Pa (Fig. 5.4m) Pa\Pa||R||R\\R (Fig. 5.5a) Pa\Pa||R||R (Fig. 5.4i) R||P||Pa||Pat (Fig. 5.4n) Pa\Pa||R||R\kR (Fig. 5.5b) Pa\Pa||R||R (Fig. 5.4i) R||P||Pa||Pat (Fig. 5.4n) Pa\Pa||Pa\kR (Fig. 5.6a) Pa\Pa||Pa||R (Fig. 5.6b) R\P\\P\\P (Fig. 5.1a) Pa\Pa||Pa\\R (Fig. 5.6c) Pa\Pa||Pa||R (Fig. 5.6b) R\P\\P\\P (Fig. 5.1a) Pa\Pa||Pa\kR (Fig. 5.6a) Pa\Pa||Pa||R (Fig. 5.6b) R\R\R\P\kR\R (Fig. 5.1b)

Pa\Pa||Pa\\R (Fig. 5.6c) Pa\Pa||Pa||R (Fig. 5.6b) R\R\R\P\kR\R (Fig. 5.1b) Pa\Pa||Pa\kR (Fig. 5.6a) Pa\Pa||Pa||R (Fig. 5.6b) R||Pa||Pa||P (Fig. 5.4k)

Idem No. 1

Idem No. 1

The revolute joint between links 7D and 10 of G4 limb and the last revolute joints of limbs G1, G2 and G3 have superposed axes Idem No. 21

Idem No. 21

Idem No. 21

Idem No. 1

Idem No. 1

The first revolute joint of G4 limb and the last revolute joints of limbs G1, G2 and G3 have parallel axes. The last joints of of limbs G1, G2 and G3 have superposed axes Idem No. 27

Idem No. 1

(continued)

534

5

Table 5.4 (continued) No. PM type 30. 3PaPaPaR-1RPaPaP (Fig. 5.64) 31. 3PaPaPaR1RPaPatP (Fig. 5.65) 32. 3PaPaPaR1RPaPatP (Fig. 5.66) 33. 3PaPaPaR-1RPPaPa (Fig. 5.67)

34. 3PaPaPaR-1RPPaPa (Fig. 5.68)

Topologies with Uncoupled Schönflies Motions

Limb topology

Connecting conditions

Pa\Pa||Pa\\R (Fig. 5.6c) Pa\Pa||Pa||R (Fig. 5.6b) R||Pa||Pa||P (Fig. 5.54k) Pa\Pa||Pa\kR (Fig. 5.6a) Pa\Pa||Pa||R (Fig. 5.6b) R||Pa||Pat||P (Fig. 5.4l) Pa\Pa||Pa\\R (Fig. 5.6c) Pa\Pa||Pa||R (Fig. 5.6b) R||Pa||Pat||P (Fig. 5.54l) Pa\Pa||Pa\kR (Fig. 5.6a) Pa\Pa||Pa||R (Fig. 5.6b) R||P||Pa||Pa (Fig. 5.4m)

Idem No. 1

Pa\Pa||Pa\\R (Fig. 5.6c) Pa\Pa||Pa||R (Fig. 5.6b) R||P||Pa||Pa (Fig. 5.54m)

Idem No. 1

Idem No. 1

The revolute joint between links 7D and 11 of G4 limb and the last revolute joints of limbs G1, G2 and G3 have superposed axes Idem No. 33

Table 5.5 Limb topology and connecting conditions of the fully-parallel solutions with no idle mobilities presented in Figs. 5.69, 5.70, 5.71, 5.72, 5.73, 5.74, 5.75, 5.76, 5.77, 5.78, 5.79, 5.80, 5.81, 5.82, 5.83, 5.84, 5.85, 5.86, 5.87, 5.88, 5.89, 5.90, 5.91, 5.92 No. PM type Limb topology Connecting conditions 1.

3PaPaPaR1RPPaPat (Fig. 5.69)

Pa\Pa||Pa\kR (Fig. 5.6a) Pa\Pa||Pa||R (Fig. 5.6b) R||P||Pa||Pat (Fig. 5.4n)

2.

3PaPaPaR1RPPaPat (Fig. 5.70)

3.

3PaPPR1CPaPa (Fig. 5.71a)

Pa\Pa||Pa\\R (Fig. 5.6c) Pa\Pa||Pa||R (Fig. 5.6b) R||P||Pa||Pat (Fig. 5.54n) Pa||P\P\kR (Fig. 5.2a) Pa||P\P\\R (Fig. 5.2d) C||Pa||Pa (Fig. 5.4o)

4.

3PaPPR1CPaPa (Fig. 5.71b)

5.

3PaPPR1CPaPat (Fig. 5.72a)

6.

3PaPPR1CPaPat (Fig. 5.72b)

Pa\P\kP\kR (Fig. 5.2b) Pa\P\kP\\R (Fig. 5.2c) C||Pa||Pa (Fig. 5.40) Pa||P\P\kR (Fig. 5.2a) Pa||P\P\\R (Fig. 5.2d) C||Pa||Pat (Fig. 5.4p) Pa\P\kP\kR (Fig. 5.2b) Pa\P\kP\\R (Fig. 5.2c) C||Pa||Pat (Fig. 5.4p)

The revolute joint between links 7D and 11 of G4 limb and the last revolute joints of limbs G1, G2 and G3 have superposed axes Idem No. 1

The revolute joint between links 6D and 8 of G4 limb and the last revolute joints of limbs G1, G2 and G3 have superposed axes Idem No. 3

Idem No. 3

Idem No. 3

(continued)

5.1 Fully-Parallel Topologies Table 5.5 (continued) No. PM type Limb topology 7.

8.

9.

10.

11.

12.

13.

14.

15.

2PaPRRR1PaPRR1CPaPa (Fig. 5.73a) 2PaPRRR1PaPRR1CPaPa (Fig. 5.73b) 2PaPRRR1PaPRR1CPaPat (Fig. 5.74a) 2PaPRRR1PaPRR1CPaPat (Fig. 5.74b) 2PaRPRR1PaRPR1CPaPa (Fig. 5.75) 2PaRPRR1PaRPR1CPaPat (Fig. 5.76) 2PaRRRR1PaRRR1CPaPa (Fig. 5.77) 2PaRRRR1PaRRR1CPaPat (Fig. 5.78) 3PaPPaR1CPaPa (Fig. 5.79a)

16. 3PaPPaR1CPaPa (Fig. 5.79b) 17. 3PaPPaR1CPaPat (Fig. 5.80a) 18. 3PaPPaR1CPaPat (Fig. 5.80b)

535

Connecting conditions

Pa\P\\R||R\\R (Fig. 5.3a) Idem No. 3 Pa\P\\R||R (Fig. 5.2e) C||Pa||Pa (Fig. 5.40) Pa||P\R||R\\R (Fig. 5.3b) Pa||P\R||R (Fig. 5.2f) C||Pa||Pa (Fig. 5.40)

Idem No. 3

Pa\P\\R||R\\R (Fig. 5.3a) Idem No. 3 Pa\P\\R||R (Fig. 5.2e) C||Pa||Pat(Fig. 5.4p) Pa||P\R||R\\R (Fig. 5.3b) Pa||P\R||R (Fig. 5.2f) C||Pa||Pat (Fig. 5.4p)

Idem No. 3

Pa\R\P\kR\R (Fig. 5.3c) Pa\R\P\kR (Fig. 5.2g) C||Pa||Pa (Fig. 5.40)

Idem No. 3

Pa\R\P\kR\R (Fig. 5.3c) Pa\R\P\kR (Fig. 5.2g) C||Pa||Pat (Fig. 5.4p)

Idem No. 3

Pa\R||R||R\kR (Fig. 5.3d) Pa\R||R||R (Fig. 5.2h) C||Pa||Pa (Fig. 5.40)

Idem No. 3

Pa\R||R||R\kR (Fig. 5.3d) Pa\R||R||R (Fig. 5.2h) C||Pa||Pat (Fig. 5.4p)

Idem No. 3

Pa||P\Pa\kR (Fig. 5.4a) Pa||P\Pa||R (Fig. 5.4d) C||Pa||Pa (Fig. 5.40)

The revolute joint between links 6D and 9 of G4 limb and the last revolute joints of limbs G1, G2 and G3 have superposed axes Idem No. 15

Pa\P\\Pa\kR (Fig. 5.4b) Pa\P\\Pa||R (Fig. 5.4c) C||Pa||Pa (Fig. 5.40) Pa||P\Pa\kR (Fig. 5.4a) Pa||P\Pa||R (Fig. 5.4d) C||Pa||Pat (Fig. 5.4p) Pa\P\\Pa\kR (Fig. 5.4b) Pa\P\\Pa||R (Fig. 5.4c) C||Pa||Pat (Fig. 5.4p)

Idem No. 15

Idem No. 15

(continued)

536

5

Topologies with Uncoupled Schönflies Motions

Table 5.5 (continued) No. PM type Limb topology 19. 3PaPaPR1CPaPa (Fig. 5.81) 20. 3PaPaPR1CPaPa (Fig. 5.82) 21. 3PaPaPR1CPaPat (Fig. 5.83) 22. 3PaPaPR1CPaPat (Fig. 5.84) 23. 2PaPaRRR1PaPaRR1CPaPa (Fig. 5.85) 24. 2PaPaRRR1PaPaRR1CPaPa (Fig. 5.86) 25. 2PaPaRRR1PaPaRR1CPaPat (Fig. 5.87) 26. 2PaPaRRR1PaPaRR1CPaPat (Fig. 5.88) 27. 3PaPaPaR1CPaPa (Fig. 5.89) 28. 3PaPaPaR1CPaPa (Fig. 5.90) 29. 3PaPaPaR1CPaPat (Fig. 5.91) 30. 3PaPaPaR1CPaPat (Fig. 5.92)

Pa\Pa\kP\\R (Fig. 5.4f) Pa\Pa\kP\kR (Fig. 5.4h) C||Pa||Pa (Fig. 5.4o) Pa\Pa\\P\\R (Fig. 5.4g) Pa\Pa\\P\kR (Fig. 5.4e) C||Pa||Pa (Fig. 5.40) Pa\Pa\kP\\R (Fig. 5.4f) Pa\Pa\kP\kR (Fig. 5.4h) C||Pa||Pat (Fig. 5.4p) Pa\Pa\\P\\R (Fig. 5.4g) Pa\Pa\\P\kR (Fig. 5.4e) C||Pa||Pat (Fig. 5.4p) Pa\Pa||R||R\\R (Fig. 5.5a) Pa\Pa||R||R (Fig. 5.4i) C||Pa||Pa (Fig. 5.40) Pa\Pa||R||R\kR (Fig. 5.5b) Pa\Pa||R||R (Fig. 5.4i) C||Pa||Pa (Fig. 5.40)

Connecting conditions Idem No. 15

Idem No. 15

Idem No. 15

Idem No. 15

The revolute joint between links 6D and 10 of G4 limb and the last revolute joints of limbs G1, G2 and G3 have superposed axes Idem No. 23

Pa\Pa||R||R\\R (Fig. 5.5a) Pa\Pa||R||R (Fig. 5.4i) C||Pa||Pat (Fig. 5.4p)

Idem No. 23

Pa\Pa||R||R\kR (Fig. 5.5b) Pa\Pa||R||R (Fig. 5.4i) C||Pa||Pat (Fig. 5.4p)

Idem No. 23

Pa\Pa||Pa\kR (Fig. 5.6a) Pa\Pa||Pa||R (Fig. 5.6b) C||Pa||Pa (Fig. 5.4o)

The revolute joint between links 6D and 11 of G4 limb and the last revolute joints of limbs G1, G2 and G3 have superposed axes Idem No. 27

Pa\Pa||Pa\\R (Fig. 5.6c) Pa\Pa||Pa||R (Fig. 5.6b) C||Pa||Pa (Fig. 5.54o) Pa\Pa||Pa\kR (Fig. 5.6a) Pa\Pa||Pa||R (Fig. 5.6b) C||Pa||Pat (Fig. 5.4p) Pa\Pa||Pa\\R (Fig. 5.6c) Pa\Pa||Pa||R (Fig. 5.6b) C||Pa||Pat (Fig. 5.54p)

Idem No. 27

Idem No. 27

5.1 Fully-Parallel Topologies

537

Table 5.6 Structural parametersa of parallel mechanisms in Figs. 5.7, 5.8, 5.9, 5.10, 5.11 No. Structural parameter Solution 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. a

m pi (i = 1, 3) p2 p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1, 3) SG2 SG4 rGi (i = 1, 2, 3) rG4 MGi (i = 1, 3) MG2 MG4 (RF) SF rl rF MF NF TF Pp1 fj Ppj¼1 2 f j Ppj¼1 3 f j Ppj¼1 4 j¼1 fj Pp j¼1 fj

Figure 5.7

Figures 5.8, 5.9, 5.10

Figures 5.11

20 7 7 4 25 6 1 3 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) 4 4 4 3 0 4 4 4 (v1 ; v2 ; v3 ; xb ) 4 9 21 4 15 0 7

22 8 7 4 27 6 1 3 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xb ) 5 4 4 3 0 5 4 4 (v1 ; v2 ; v3 ; xb ) 4 9 23 4 13 0 8

22 7 7 6 27 6 1 3 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xa ; xb ; xd ) 4 4 6 3 0 4 4 6 (v1 ; v2 ; v3 ; xb ) 4 9 23 4 13 0 7

7 7

7 8

7 7

4 25

4 27

6 27

See footnote of Table 2.2 for the nomenclature of structural parameters

538

5

Topologies with Uncoupled Schönflies Motions

Table 5.7 Structural parametersa of parallel mechanisms in Figs. 5.12, 5.13, 5.14, 5.15, 5.16, 5.17, 5.18, 5.19, 5.20, 5.21, 5.22, 5.23, 5.24, 5.25, 5.26, 5.27, 5.28, 5.29, 5.30 No. Structural parameter Solution

m pi (i = 1, 3) p2 p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1, 3) SG2 SG4 rGi (i = 1, 2, 3) rG4 MGi (i = 1, 3) MG2 MG4 (RF) SF rl rF MF NF TF Pp1 fj Pj¼1 p2 fj Pj¼1 p3 31. j¼1 fj Pp4 32. fj Ppj¼1 33. f j¼1 j

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

a

Figures 5.12, 5.13, 5.14

Figures 5.15, 5.16, 5.17, 5.18

Figures 5.19, 5.20, 5.21, 5.22, 5.23, 5.24, 5.25, 5.26, 5.27, 5.28, 5.29, 5.30

24 8 7 6 29 6 1 3 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xa ; xb ; xd ) 5 4 6 3 0 5 4 6 (v1 ; v2 ; v3 ; xb ) 4 9 25 4 11 0 8 7

24 7 7 10 31 8 0 4 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) 4 4 4 3 6 4 4 4 (v1 ; v2 ; v3 ; xb ) 4 15 27 4 21 0 7 7

26 8 7 10 33 8 0 4 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xb ) 5 4 4 3 6 5 4 4 (v1 ; v2 ; v3 ; xb ) 4 15 29 4 19 0 8 7

8 6

7 10

8 10

29

31

33

See footnote of Table 2.2 for the nomenclature of structural parameters

5.1 Fully-Parallel Topologies

539

Table 5.8 Structural parametersa of parallel mechanisms in Figs. 5.31, 5.32, 5.33, 5.34, 5.35, 5.36 No. Structural parameter Solution Figures 5.31 and 5.32 Figures 5.33 and 5.34 Figures 5.35 and 5.36 m pi (i = 1, 3) p2 p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1, 3) SG2 SG4 rGi (i = 1, 2, 3) rG4 MGi (i = 1, 3) MG2 MG4 (RF) SF rl rF MF NF TF Pp1 fj Pj¼1 p2 fj Pj¼1 p3 31. j¼1 fj Pp4 32. fj Pj¼1 p 33. j¼1 fj

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

a

26 10 10 4 34 9 1 3 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) 4 4 4 6 0 4 4 4 (v1 ; v2 ; v3 ; xb ) 4 18 30 4 24 0 10 10

28 11 10 4 36 9 1 3 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xb ) 5 4 4 6 0 5 4 4 (v1 ; v2 ; v3 ; xb ) 4 18 32 4 22 0 11 10

28 10 10 6 36 9 1 3 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xa ; xb ; xd ) 4 4 6 6 0 4 4 6 (v1 ; v2 ; v3 ; xb ) 4 18 32 4 22 0 10 10

10 4

11 4

10 6

34

36

36

See footnote of Table 2.2 for the nomenclature of structural parameters

540

5

Topologies with Uncoupled Schönflies Motions

Table 5.9 Structural parametersa of parallel mechanisms in Figs. 5.37, 5.38, 5.39, 5.40, 5.41, 5.42, 5.43, 5.44, 5.45, 5.46, 5.47, 5.48, 5.49, 5.50, 5.51, 5.52, 5.53, 5.54, 5.55, 5.56, 5.57, 5.58 No. Structural Solution parameter Figures 5.37 and 5.38 Figures 5.39, 5.40, 5.41, 5.42, Figures 5.51, 5.52, 5.43, 5.44, 5.45, 5.46, 5.47, 5.53, 5.54, 5.55, 5.56, 5.48, 5.49, 5.50 5.57, 5.58 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. a

m pi (i = 1, 3) p2 p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1, 3) SG2 SG4 rGi (i = 1, 2, 3) rG4 MGi (i = 1, 3) MG2 MG4 (RF) SF rl rF MF NF T PF p1 fj Pj¼1 p2 fj Pj¼1 p3 fj Pj¼1 p4 fj Ppj¼1 j¼1 fj

30 11

30 10

32 11

10 6 38 9 1 3 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xa ; xb ; xd ) 5

10 10 40 11 0 4 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) 4

10 10 42 11 0 4 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xb ) 5

4 6 6

4 4 6

4 4 6

0 5

6 4

6 5

4 6 (v1 ; v2 ; v3 ; xb ) 4 18 34 4 20 0 11 10

4 4 (v1 ; v2 ; v3 ; xb ) 4 24 36 4 30 0 10 10

4 4 (v1 ; v2 ; v3 ; xb ) 4 24 38 4 28 0 11 10

11 6

10 10

11 10

38

40

42

See footnote of Table 2.2 for the nomenclature of structural parameters

5.1 Fully-Parallel Topologies

541

Table 5.10 Structural parametersa of parallel mechanisms in Figs. 5.59, 5.60, 5.61, 5.62, 5.63, 5.64, 5.65, 5.66, 5.67, 5.68, 5.69, 5.70 No. Structural Solution parameter Figures 5.61 and 5.62 Figures 5.63, 5.64, 5.65, 5.66, 5.67, Figures 5.59 and 5.60 5.68, 5.69, 5.70 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. a

m pi (i = 1, 3) p2 p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1, 3) SG2 SG4 rGi (i = 1, 2, 3) rG4 MGi (i = 1, 3) MG2 MG4 (RF) SF rl rF MF NF TF Pp1 fj Pj¼1 p2 fj Pj¼1 p3 fj Pj¼1 p4 fj Ppj¼1 j¼1 fj

32 13 13 4 43 12 1 3 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) 4 4 4 9

34 13 13 6 45 12 1 3 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xa ; xb ; xd ) 4 4 6 9

36 13 13 10 49 14 0 4 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) 4 4 4 9

0 4

0 4

6 4

4 4 (v1 ; v2 ; v3 ; xb ) 4 27 39 4 33 0 13 13

4 6 (v1 ; v2 ; v3 ; xb ) 4 27 41 4 31 0 13 13

4 4 (v1 ; v2 ; v3 ; xb ) 4 33 45 4 39 0 13 13

13

13

13

4 43

6 45

10 49

See footnote of Table 2.2 for the nomenclature of structural parameters

542

5

Topologies with Uncoupled Schönflies Motions

Table 5.11 Structural parametersa of parallel mechanisms in Figs. 5.71, 5.72, 5.73, 5.74, 5.75, 5.76, 5.77, 5.78, 5.79, 5.80, 5.81, 5.82, 5.83, 5.84 No. Structural Solution parameter Figures 5.73, 5.74, 5.75, Figures 5.79, 5.80, 5.81, Figures 5.71 and 5.72 5.76, 5.77, 5.78 5.82, 5.83, 5.84 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. a

m pi (i = 1, 3) p2 p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1, 3) SG2 SG4 rGi (i = 1, 2, 3) rG4 MGi (i = 1, 3) MG2 MG4 (RF) SF rl rF MF NF TF Pp1 fj Pj¼1 p2 fj Pj¼1 p3 j¼1 fj Pp4 fj Ppj¼1 f j¼1 j

23 7 7 9 30 8 0 4 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) 4 4 4 3

25 8 7 9 32 8 0 4 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xb ) 5 4 4 3

29 10 10 9 39 11 0 4 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) 4 4 4 6

6 4 4 4 (v1 ; v2 ; v3 ; xb ) 4 15 27 4 21 0 7

6 5 4 4 (v1 ; v2 ; v3 ; xb ) 4 15 29 4 19 0 8

6 4 4 4 (v1 ; v2 ; v3 ; xb ) 4 24 36 4 30 0 10

7 7

7 8

10 10

10 31

10 33

10 40

See footnote of Table 2.2 for the nomenclature of structural parameters

5.2 Redundantly Actuated Topologies

543

Table 5.12 Structural parametersa of parallel mechanisms in Figs. 5.85, 5.86, 5.87, 5.88, 5.89, 5.90, 5.91, 5.92 No. Structural parameter Solution 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. a

m pi (i = 1, 3) p2 p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1, 3) SG2 SG4 rGi (i = 1, 2, 3) rG4 MGi (i = 1, 3) MG2 MG4 (RF) SF rl rF MF NF TF Pp1 fj Pj¼1 p2 fj Pj¼1 p3 j¼1 fj Pp4 fj Pj¼1 p j¼1 fj

Figures 5.85, 5.86, 5.87, 5.88

Figures 5.89, 5.90, 5.91, 5.92

31 11 10 9 41 11 0 4 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xb ) 5 4 4 6 6 5 4 4 (v1 ; v2 ; v3 ; xb ) 4 24 38 4 28 0 11 10

35 13 13 9 48 14 0 4 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) 4 4 4 9 6 4 4 4 (v1 ; v2 ; v3 ; xb ) 4 33 45 4 39 0 13 13

11 10

13 10

42

49

See footnote of Table 2.2 for the nomenclature of structural parameters

5.2 Redundantly Actuated Topologies The solutions presented in this section have five degrees of mobility and one degree of structural redundancy. One redundantly actuated prismatic joint exists in G4-limb of the solutions presented in this section. This limb has five degrees of mobility, MG4 = 5, four degrees of connectivity between the moving platform

544

5

Topologies with Uncoupled Schönflies Motions

isolated from the parallel mechanisms and the fixed base, SG4 = 4, and one degree of structural redundancy TG4 = MG4-SG4 = 1. This degree of structural redundancy is used for decoupling the translational motion on z0-axis and the rotational motion of the moving platform. To do this, the actuated joint variable q5 is equal to joint variable q4 and implicitely q_ 5 ¼ q_ 4 . Usually, redundancy in parallel manipulators is used to eliminate some singular configurations, to minimize the joint rates, to optimize the joint torques/forces, to increase dexterity workspace, stiffness, eigenfrequencies, kinematic and dynamic accuracy, to improve both kinematic and dynamic control algorithms. In this section, a new use of redundancy is presented for motion decoupling, as presented for the first time in Gogu (2006). Redundantly actuated topologies of parallel robotic manipulators with uncoupled Schönflies motions may have simple or complex limbs and could be actuated by linear or rotating motors. The actuators can be mounted on the fixed base or on a moving link. The first solution has the advantage of reducing the moving masses and large workspace. The second solution would be more compact. There are no idle mobilities in the topologies presented in this section.

5.2.1 Topologies with Simple Limbs In the redundantly actuated topologies of PMs with uncoupled Schönflies motions F / G1-G2-G3-G4 presented in this section, the moving platform n : nGi (i = 1, 2, 3, 4) is connected to the reference platform 1 : 1Gi : 0 by four spatial simple limbs with four or five degrees of connectivity. Two actuators are combined in the two first joints of G4-limb. Limbs G1, G2 and G3 are actuated by one linear motor mounted on the fixed base. The actuated joint is underlined in the structural graph. In the cylindrical joint denoted by C just one motion is actuated. This can be the translational or the rotational motion and is indicated in the structrural diagram by a linear or circular arrow. The various types of simple kinematic chains with four and five degrees of connectivity used in G4-limb of the Redundantly actuated topologies illustrated in this section are presented in Fig. 5.93. These simple limbs combine only revolute, prismatic and cylindrical joints. Various topologies of redundantly actuated PMs with uncoupled Schönflies motions and no idle mobilities can be obtained by using G1, G2 and G3 limbs with the topology, presented in Figs. 4.1 and 4.2, and G4-limb presented in Fig. 5.93. Only topologies with at least two identical limb types are illustrated in Figs. 5.94, 5.95, 5.96, 5.97, 5.98. The limb topology and connecting conditions of the solutions in Figs. 5.94, 5.95, 5.96, 5.97, 5.98 are systematized in Table 5.13 and their structural parameters in Tables 5.14 and 5.15.

5.2 Redundantly Actuated Topologies

545

Fig. 5.93 Simple limbs for G4-limb of redundantly actuated PMs with uncoupled Schönflies motions defined by ðRG Þ ¼ ðv1 ; v2 ; v3 ; xb Þ, MG = 5, SG = 4 (a, b, g) and ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa ; xd Þ), MG = 6, SG = 5 (c–f, h–j)

546

Fig. 5.93 (continued)

5

Topologies with Uncoupled Schönflies Motions

5.2 Redundantly Actuated Topologies

547

Fig. 5.94 3PPPR-1PPPPR type redundantly actuated PMs with uncoupled Schönflies motions defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 6, limb topology P\P\\P\\R, P\P\\P\kR, P||P\P\\P\kR (a) and P\P\\P\\R, P\P\\P||R, P||P\P\\P||R (b)

548

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.95 Redundantly actuated PMs with uncoupled Schönflies motions of types 1PPRR2PPRRR-1PPPRR (a) and 1PRPR-2PRPRR-1PPRPRR (b) defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 3, limb topology P\P\kR||R, P\P\kR||R\R, P||P\P\kR||R\R (a) and P||R\P\kR, P||R\P\kR\R, P||P||R\P\kR\R (b)

5.2 Redundantly Actuated Topologies

549

Fig. 5.96 Redundantly actuated PMs with uncoupled Schönflies motions of types 1PRRP2PRRPR-1PPRRPR (a) and 1PRRR-2PRRRR-1PPRRRR (b) defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 3, limb topology P||R||R\P, P||R||R\P\\R, P||P||R||R\P\\R (a) and P||R||R||R, P||R||R||R\R, P||P||R||R||R\R (b)

550

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.97 Redundantly actuated PMs with uncoupled Schönflies motions of types 1PPPR-2PPC1PPPC (a) and 1CPR-2CPRR-1PCPRR (b) defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 6 (a), NF = 3 (b), limb topology P\P\\P\\R, P\P\\C, P||P\P\\C (a) and C\P\kR, C\P\kR\R, P|| C\P\kR\R (b)

5.2 Redundantly Actuated Topologies

551

Fig. 5.98 Redundantly actuated PMs with uncoupled Schönflies motions of types 1CRP2CRPR-1PCRPR (a) and 1CRR-2CRRR-1PCRRR (b) defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 3, limb topology C||R\P, C||R\P\\R, P||C||R\P\\R (a) and C||R||R, C||R||R\R, P||C||R||R\R (b)

552

5

Topologies with Uncoupled Schönflies Motions

Table 5.13 Limb topology and connecting conditions of the fully-parallel solutions with no idle mobilities presented in Figs. 5.94, 5.95, 5.96, 5.97, 5.98 No. PM type Limb topology Connectingconditions 1.

3PPPR1PPPPR (Fig. 5.94a)

P\P\\P\\R (Fig. 4.1b) P\P\\P\kR (Fig. 4.1a) P||P\P\\P\kR (Fig. 5.93a)

2.

3PPPR1PPPPR (Fig. 5.94b)

3.

1PPRR2PPRRR1PPPRR (Fig. 5.95a) 1PRPR2PRPRR1PPRPRR (Fig. 5.95b) 1PRRP2PRRPR1PPRRPR (Fig. 5.96a) 1PRRR2PRRRR1PPRRRR (Fig. 5.96b) 1PPPR-2PPC1PPPC (Fig. 5.97a)

P\P\\P\\R (Fig. 4.1b) P\P\\P||R (Fig. 4.1c) P||P\P\\P||R (Fig. 5.93b) P\P\kR||R (Fig. 4.1d) Idem No. 1 P\P\kR||R\R (Fig. 4.2a) P||P\P\kR||R\R (Fig. 5.93d)

4.

5.

6.

7.

8.

1CPR-2CPRR1PCPRR (Fig. 5.97b)

9.

1CRP-2CRPR1PCRPR (Fig. 5.98a)

10. 1CRR-2CRRR1PCRRR (Fig. 5.98b)

The last revolute joints of the four limbs have parallel axes. The actuated prismatic joints of limbs G1, G2 and G3 hvave orthogonal directions. The actuated prismatic joints of limbs G3 and G4 have parallel directions Item No. 1

P||R\P\kR (Fig. 4.1e) Idem No. 1 P||R\P\kR\R (Fig. 4.2b) P||P||R\P\kR\R (Fig. 5.93e) P||R||R\P (Fig. 4.1f) Idem No. 1 P||R||R\P\\R (Fig. 4.2c) P||P||R||R\P\\R (Fig. 5.93c) P||R||R||R (Fig. 4.1 g) P||R||R||R\R (Fig. 4.2d) P||P||R||R||R\R (Fig. 5.93f)

Idem No. 1

P\P\\P\\R (Fig. 4.1b) P\P\\C (Fig. 4.1 h) P||P\P\\C (Fig. 5.93g) C\P\kR (Fig. 4.1i) C\P\kR\R (Fig. 4.2e) P|| C\P\kR\R (Fig. 5.93i) C||R\P (Fig. 4.1j) C||R\P\\R (Fig. 4.2f) P||C||R\P\\R (Fig. 5.93h) C||R||R (Fig. 4.1 k) C||R||R\R (Fig. 4.2 g) P||C||R||R\R (Fig. 5.93j)

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

5.2 Redundantly Actuated Topologies

553

Table 5.14 Structural parametersa of parallel mechanisms in Figs. 5.94, 5.95, 5.96 No. Structural parameter Solution 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. a

m p1 pi (i = 2, 3) p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SG1 SGi (i = 2, 3) SG4 rGi (i = 1,…,4) MG1 MGi (i = 2, 3) MG4 (RF) SF rl rF MF NF TF Pp1 fj Ppj¼1 2 f j Ppj¼1 3 f j Ppj¼1 4 j¼1 fj Pp j¼1 fj

Figure 5.94

Figures 5.95 and 5.96

15 4 4 5 17 3 4 0 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) 4 4 4 0 4 4 5 (v1 ; v2 ; v3 ; xa ) 4 0 12 5 6 1 4

18 4 5 6 20 3 4 0 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xa ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) 4 5 5 0 4 5 6 (v1 ; v2 ; v3 ; xa ) 4 0 15 5 3 1 4

4 4

5 5

5 17

6 20

See footnote of Table 2.2 for the nomenclature of structural parameters

554

5

Topologies with Uncoupled Schönflies Motions

Table 5.15 Structural parametersa of parallel mechanisms in Figs. 5.97 and 5.98 No. Structural parameter Solution 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. a

m p1 pi (i = 2, 3) p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SG1 SGi (i = 2, 3) SG4 rGi (i = 1,…,4) MG1 MGi (i = 2, 3) MG4 (RF) SF rl rF MF NF TF Pp1 fj Ppj¼1 2 f j Ppj¼1 3 f j Ppj¼1 4 j¼1 fj Pp j¼1 fj

Figure 5.97a

Figure 5.97b and 5.98

12 3 3 5 14 3 4 0 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) 4 4 4 0 4 4 5 (v1 ; v2 ; v3 ; xa ) 4 0 12 5 6 1 4

14 3 4 5 16 3 4 0 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xa ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) 4 5 5 0 4 5 6 (v1 ; v2 ; v3 ; xa ) 4 0 15 5 3 1 4

4 4

5 5

5 17

6 20

See footnote of Table 2.2 for the nomenclature of structural parameters

5.2.2 Topologies with Complex Limbs In the redundantly actuated topologies of PMs with uncoupled Schönflies motions F / G1-G2-G3-G4 presented in this section, the moving platform n : nGi (i = 1, 2, 3, 4) is connected to the reference platform 1 : 1Gi : 0 by four spatial complex limbs with four or five degrees of connectivity. Two linear actuators are combined in the two first joints of G4-limb. Limbs G1, G2 and G3 are actuated by one linear motor mounted on the fixed base.

5.2 Redundantly Actuated Topologies

555

Fig. 5.99 Complex limbs for redundantly actuated PMs with uncoupled Schönflies motions, combining one planar closed loop, defined by ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, MG = 5, SG = 4 (a–d, k) and ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa Þ; xd , MG = 6, SG = 5 (e–j, l)

556

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.99 (continued)

The various types of complex kinematic chains with four and five degrees of connectivity used in G4-limb of the redundantly actuated topologies illustrated in this section are presented in Figs. 5.99 and 5.100. These complex limbs combine one

5.2 Redundantly Actuated Topologies

Fig. 5.99 (continued)

557

558

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.100 Complex limbs for redundantly actuated PMs with uncoupled Schönflies motions, combining two planar closed loops, defined by ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, MG = 5, SG = 4 (a) and ðRG Þ ¼ ðv1 ; v2 ; v3 ; xa ; xd Þ, MG = 6, SG = 5 (b, c)

5.2 Redundantly Actuated Topologies

559

Fig. 5.101 3PPPaR-1PPPPaR type redundantly actuated PM with uncoupled Schönflies motions defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 18, limb topology P\P\kPa||R, P\P\kPa\kR, P||P\P\kPa\kR

560

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.102 3PPPaR-1PPPPaR type redundantly actuated PM with uncoupled Schönflies motions defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 18, limb topology P\P\kPa||R, P\P\kPa\\R, P||P\P\kPa\\R

5.2 Redundantly Actuated Topologies

561

Fig. 5.103 3PPaPR-1PPPaPR type redundantly actuated PM with uncoupled Schönflies motions defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 18, limb topology P||Pa\P\kR, P||Pa\P\\R, P||P||Pa\P\\R

562

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.104 3PPaPR-1PPPaPR type redundantly actuated PM with uncoupled Schönflies motions defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 18, limb topology P||Pa\P\kR, P||Pa\P||R, P||P||Pa\P||R

5.2 Redundantly Actuated Topologies

563

Fig. 5.105 1PPaRR-2PPaRRR-1PPPaRRR type redundantly actuated PM with uncoupled Schönflies motions defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 15, limb topology P||Pa||R||R, P||Pa||R||R\R, P||P||Pa||R||R\R

564

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.106 1PRRbR-2PRRbRR-1PPRRbRR type redundantly actuated PM with uncoupled Schönflies motions defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 15, limb topology P||R||Rb||R, P||R||Rb||R\R, P||P||R||Rb||R\R

5.2 Redundantly Actuated Topologies

565

Fig. 5.107 1PPn2R-2PPn2RR-1PPPn2RR type redundantly actuated PM with uncoupled Schönflies motions defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 15, limb topology P||Pn2||R, P||Pn2||R\R, P||P||Pn2||R\R

566

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.108 1PPn2R-2PPn2RR-1PPPn2RR type redundantly actuated PM with uncoupled Schönflies motions defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 15, limb topology P||Pn2||R, P||Pn2||R\R, P||P||Pn2||R\R

5.2 Redundantly Actuated Topologies

567

Fig. 5.109 1PPn3-2PPn3R-1PPPn3R type redundantly actuated PM with uncoupled Schönflies motions defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 15, limb topology P||Pn3, P||Pn3\R, P||P||Pn3\R

568

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.110 1PPn3-2PPn3R-1PPPn3R type redundantly actuated PM with uncoupled Schönflies motions defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 15, limb topology P||Pn3, P||Pn3\R, P||P||Pn3\R

5.2 Redundantly Actuated Topologies

569

Fig. 5.111 1PPaPR-2PPaC-1PPPaC type redundantly actuated PM with uncoupled Schönflies motions defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 18, limb topology P||Pa\P\kR, P||Pa\C, P||P||Pa\C

570

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.112 1CRbR-2CRbRR-1PCRbRR type redundantly actuated PM with uncoupled Schönflies motions defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 15, limb topology C||Rb||R, C||Rb||R\R, P||C||Rb||R\R

5.2 Redundantly Actuated Topologies

571

Fig. 5.113 3PPaPaR-1PPPaPaR type redundantly actuated PM with uncoupled Schönflies motions defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 30, limb topology P||Pa||Pa||R, P||Pa||Pa\R, P||P||Pa||Pa\R

572

5

Topologies with Uncoupled Schönflies Motions

Fig. 5.114 1PRRbRbR-2PRRbRbRR-1PPRRbRbRR type redundantly actuated PM with uncoupled Schönflies motions defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 27, limb topology P||R||Rb||Rb||R, P||R||Rb||Rb||R\R, P||P||R||Rb||Rb||R\R

5.2 Redundantly Actuated Topologies

573

Fig. 5.115 1CRbRbR-2CRbRbRR-1PCRbRbRR type redundantly actuated PM with uncoupled Schönflies motions defined by MF = 5, SF = 4, ðRF Þ ¼ ðv1 ; v2 ; v3 ; xa Þ, TF = 1, NF = 27, limb topology C||Rb||Rb||R, C||Rb||Rb||R\R, P||C||Rb||Rb||R\R

574

5

Topologies with Uncoupled Schönflies Motions

Table 5.16 Limb topology and connecting conditions of the fully-parallel solutions with no idle mobilities presented in Figs. 5.101, 5.102, 5.103, 5.104, 5.105, 5.106, 5.107, 5.108, 5.109, 5.110, 5.111, 5.112, 5.113, 5.114, 5.115 No. PM type

Limb topology

Connecting conditions

1.

3PPPaR-1PPPPaR (Fig. 5.101)

P\P\kPa||R (Fig. 4.8c) P\P\kPa\kR (Fig. 4.8b) P||P\P\kPa\kR (Fig. 5.99a)

2.

3PPPaR-1PPPPaR (Fig. 5.102)

3.

3PPaPR-1PPPaPR (Fig. 5.103)

4.

3PPaPR-1PPPaPR (Fig. 5.104)

5.

1PPaRR-2PPaRRR-1PPPaRRR (Fig. 5.105)

6.

1PRRbR-2PRRbRR-1PPRRbRR (Fig. 5.106)

7.

1PPn2R-2PPn2RR-1PPPn2RR (Fig. 5.107)

8.

1PPn2R-2PPn2RR-1PPPn2RR (Fig. 5.108)

9.

1PPn3-2PPn3R-1PPPn3R (Fig. 5.109)

10.

1PPn3-2PPn3R-1PPPn3R (Fig. 5.110)

11.

1PPaPR-2PPaC-1PPPaC (Fig. 5.111)

12.

1CRbR-2CRbRR-1PCRbRR (Fig. 5.112)

13.

3PPaPaR-1PPPaPaR (Fig. 5.113)

14.

1PRRbRbR-2PRRbRbRR1PPRRbRbRR (Fig. 5.114)

15.

1CRbRbR-2CRbRbRR1PCRbRbRR (Fig. 5.115)

P\P\kPa||R (Fig. 4.8c) P\P\kPa\\R (Fig. 4.8a) P||P\P\kPa\\R (Fig. 5.99b) P||Pa\P\kR (Fig. 4.8e) P||Pa\P\\R (Fig. 4.8d) P||P||Pa\P\R (Fig. 5.99c) P||Pa\P\kR (Fig. 4.8e) P||Pa\P||R (Fig. 4.8f) P||P||Pa\P||R (Fig. 5.99d) P||Pa||R||R (Fig. 4.8g) P||Pa||R||R\R (Fig. 4.9a) P||P||Pa||R||R\R (Fig. 5.99e) P||R||Rb||R (Fig. 4.1i) P||R||Rb||R\R (Fig. 4.9c) P||P||R||Rb||R\R (Fig. 5.99f) P||Pn2||R (Fig. 4.8j) P||Pn2||R\R (Fig. 4.9d) P||P||Pn2||R\R (Fig. 5.99g) P||Pn2||R (Fig. 4.8k) P||Pn2||R\R (Fig. 4.9e) P||P||Pn2||R\R (Fig. 5.99h) P||Pn3 (Fig. 4.8l) P||Pn3\R (Fig. 4.9f) P||P||Pn3\R (Fig. 5.99i) P||Pn3 (Fig. 4.8m) P||Pn3\R (Fig. 4.9g) P||P||Pn3\R (Fig. 5.99j) P||Pa\P\kR (Fig. 4.8e) P||Pa\C (Fig. 4.8o) P||P||Pa\C (Fig. 5.99k) C||Rb||R (Fig. 4.8n) C||Rb||R\R (Fig. 4.9 h) P||C||Rb||R\R (Fig. 5.99l) P||Pa||Pa||R (Fig. 4.10b) P||Pa||Pa\R (Fig. Fig. 4.10a) P||P||Pa||Pa\R (Fig. 5.100a) P||R||Rb||Rb||R (Fig. Fig. 4.10e) P||R||Rb||Rb||R\R (Fig. 4.11c) P||P||R||Rb||Rb||R\R (Fig. 5.100b) C||Rb||Rb||R (Fig. Fig. 4.10p) C||Rb||Rb||R\R (Fig. 4.11n) P||C||Rb||Rb||R\R (Fig. 5.100c)

The last revolute joints of the four limbs have parallel axes. The actuated prismatic joints of limbs G1, G2 and G3 hvave orthogonal directions. The actuated prismatic joints of limbs G3 and G4 have parallel directions. Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

5.2 Redundantly Actuated Topologies

575

Table 5.17 Structural parametersa of parallel mechanisms in Figs. 5.101, 5.102, 5.103, 5.104, 5.105, 5.106, 5.107, 5.108, 5.109, 5.110, 5.111 No. Structural Solution parameter Figures 5.105, 5.106, 5.107, Figures 5.111 Figures 5.101, 5.102, 5.103, 5.104 5.108, 5.109, 5.110 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 3 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. a

m p1 pi (i = 2, 3) p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SG1 SGi (i = 2, 3) SG4 rGi

23 7 7 8 29 7 0 4 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) 4 4 4 (i = 1,…,4)

26 7 8 9 32 7 0 4 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xa ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) 4 5 5 3

20 7 6 7 26 7 0 4 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) 4 4 4 3

MG1 MGi (i = 2, 3) MG4 (RF) SF rl rF MF NF T PF p1 fj Ppj¼1 2 j¼1 fj Pp3 fj Ppj¼1 4 f j Ppj¼1 j¼1 fj

4 4 5 (v1 ; v2 ; v3 ; xa ) 4 12 24 5 18 1 7

4 5 6 (v1 ; v2 ; v3 ; xa ) 4 12 27 5 15 1 7

4 4 5 (v1 ; v2 ; v3 ; xa ) 4 12 24 5 18 1 7

7 7

8 8

7 7

8 29

9 32

8 29

See footnote of Table2.2 for the nomenclature of structural parameters

or two planar loops. Three joint parameters loose their independence in each planar loop of types Pa, Pn2 and Pn3 used in the topologies illustrated in this section. We recall that the parallelogram planar loop Pa has on degree of mobility and the planar loops denoted by Pn2 and Pn3 has two and, respectively, three degress of mobility.

576

5

Topologies with Uncoupled Schönflies Motions

Table 5.18 Structural parametersa of parallel mechanisms in Figs. 5.112 and 5.113 No. Structural parameter Solution 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. a

m p1 pi (i = 2, 3) p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SG1 SGi (i = 2, 3) SG4 rGi (i = 1,…,4) MG1 MGi (i = 2, 3) MG4 (RF) SF rl rF MF NF T PF p1 fj Ppj¼1 2 f j Ppj¼1 3 f j Ppj¼1 4 j¼1 fj Pp j¼1 fj

Figure 5.112

Figure 5.113

22 6 7 8 28 7 0 4 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xa ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) 4 5 5 3 4 5 6 (v1 ; v2 ; v3 ; xa ) 4 12 27 5 15 1 7

31 10 10 11 41 11 0 4 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ) 4 4 4 6 4 4 5 (v1 ; v2 ; v3 ; xa ) 4 24 36 5 30 1 10

8 8

10 10

9 32

11 41

See footnote of Table 2.2 for the nomenclature of structural parameters

Various topologies of redundantly actuated PMs with uncoupled Schönflies motions and no idle mobilities can be obtained by using G1, G2 and G3 limbs, with the limb: topology presented in Figs. 4.8, 4.9, 4.10, 4.11, and G4-limb presented in Figs. 5.99 and 5.100. Only topologies with at least two identical limb types and linear actuators are illustrated in Figs. 5.101, 5.102, 5.103, 5.104, 5.105, 5.106, 5.107, 5.108, 5.109, 5.110, 5.111, 5.112, 5.113, 5.114, 5.115.

5.2 Redundantly Actuated Topologies

577

Table 5.19 Structural parametersa of parallel mechanisms in Figs. 5.114 and 5.115 No. Structural parameter Solution 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. a

m p1 pi (i = 2, 3) p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SG1 SGi (i = 2, 3) SG4 rGi (i = 1,…,4) MG1 MGi (i = 2, 3) MG4 (RF) SF rl rF MF NF T PF p1 fj Ppj¼1 2 f j Ppj¼1 3 f j Ppj¼1 4 j¼1 fj Pp j¼1 fj

Figure 5.114

Figure 5.115

34 10 11 12 44 11 0 4 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xa ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) 4 5 5 6 4 5 6 (v1 ; v2 ; v3 ; xa ) 4 24 39 5 27 1 10

30 9 10 11 40 11 0 4 4 (v1 ; v2 ; v3 ; xa ) (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xa ; xd ) (v1 ; v2 ; v3 ; xa ; xd ) 4 5 5 6 4 5 6 (v1 ; v2 ; v3 ; xa ) 4 24 39 5 27 1 10

11 11

11 11

12 44

12 44

See footnote of Table 2.2 for the nomenclature of structural parameters

The limb topology and connecting conditions of the solutions in Figs. 5.101, 5.102, 5.103, 5.104, 5.105, 5.106, 5.107, 5.108, 5.109, 5.110, 5.111, 5.112, 5.113, 5.114, 5.115 are systematized in Table 5.16 and their structural parameters in Tables 5.17, 5.18, 5.19.

578

5

Topologies with Uncoupled Schönflies Motions

References 1. Gogu G (2008) Structural synthesis of parallel robots: part 1-methodology. Springer, Dordrecht 2. Gogu G (2009) Structural synthesis of parallel robots: part 2-translational topologies with two and three degrees of freedom. Springer, Dordrecht 3. Gogu G (2010) Structural synthesis of parallel robots: part 3-topologies with planar motion of the moving platform. Springer, Dordrecht 4. Gogu G (2012) Structural synthesis of parallel robots: part 4-other topologies with two and three degrees of freedom. Springer, Dordrecht

Chapter 6

Maximally Regular Topologies with Schönflies Motions

Maximally regular parallel robotic manipulators with Schönflies motions are actuated by three linear and one rotating actuators and can have various degrees of over constraint. In these solutions, the four operational velocities are equal to their corresponding actuated joint velocities: v1 ¼ q_ 1 , v2 ¼ q_ 2 , v3 ¼ q_ 3 and xd ¼ q_ 4 . The Jacobian matrix in Eq. (1.18) is the identity matrix. We call Isoglide4-T3R1 with Schönflies motions of the moving platform the parallel mechanisms of this family. The limbs can be simple or complex kinematic chains and the actuators can be mounted on the fixed base or on a moving link. The solutions presented in this section are obtained by using the methodology of structural synthesis proposed in Part 1 [1] and also used in Parts 2–4 of this work [2–4]. This original methodology combines new formulae for mobility connectivity, redundancy and overconstraints, and the evolutionary morphology in a unified approach of structural synthesis of parallel robotic manipulators.

6.1 Fully-Parallel Topologies with Simple Limbs In the fully-parallel and maximally regular topologies of PMs with Schönflies motions F G1-G2-G3-G4 presented in this section, the moving platform n : nGi (i = 1, 2, 3, 4) is connected to the reference platform 1 : 1Gi : 0 by four simple limbs with four, five or six degrees of connectivity. One linear actuator is combined in the first prismatic or cylindrical pair of limbs G1, G2 and G3, and one rotary actuator in the first revolute pair of limb G4. Limbs G1, G2 and G3 are used for positioning the moving platform and limb G4 for orienting it. Various maximally regular topologies of PMs with Schönflies motions of the moving platform and no idle mobilities can be obtained by using various limb topologies presented in Figs. 4.1, 4.2 and 5.1. Fully-parallel topologies with at least two identical limbs are illustrated in Figs. 6.1, 6.2, 6.3, 6.4, 6.5, 6.6. The limb topology and connecting conditions of these solutions are systematized in Table 6.1, as are their structural parameters in Tables 6.2, 6.3, 6.4.

G. Gogu, Structural Synthesis of Parallel Robots, Solid Mechanics and Its Applications 206, DOI: 10.1007/978-94-007-7401-8_6,  Springer Science+Business Media Dordrecht 2014

579

580

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.1 Maximaly regular fully-parallel PMs with Schönflies motions of types 3PPPR-1RPPP (a) and 2PPRRR-1PPRR-1RPPP (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 6 (a), NF = 4 (b), limb topology R\P\\P\\P and P\P\\P\||R, P\P\\P\\R (a), P\P\||R||R\R, P\P\||R||R (b)

6.1 Fully-Parallel Topologies with Simple Limbs

581

Fig. 6.2 Maximaly regular fully-parallel PMs with Schönflies motions of types 2PRPRR1PRPR-1RPPP (a) and 2PRRRR-1PRRR-1RPPP (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 4, limb topology R\P\\P\\P and P||R\P\||R\R, P||R\P\||R (a), P||R||R||R\R, P||R||R||R (b)

582

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.3 Maximaly regular fully-parallel PMs with Schönflies motions of types 3PPPR-1RUPU (a) and 2PPRRR-1PPRR-1RUPU (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 4 (a), NF = 2 (b), limb topology R\R\R\P\||R\R and P\P\\P\||R, P\P\\P\\R (a), P\P\||R||R\R, P\P\||R||R (b)

6.1 Fully-Parallel Topologies with Simple Limbs

583

Fig. 6.4 Maximaly regular fully-parallel PMs with Schönflies motions of types 2PRPRR1PRPR-1RUPU (a) and 2PRRRR-1PRRR-1RUPU (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 2, limb topology R\R\R\P\||R\R and P||R\P\||R\R, P||R\P\||R (a), P||R||R||R\R, P||R||R||R (b)

584

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.5 Maximaly regular fully-parallel PMs with Schönflies motions of types 2CPRR-1CPR1RPPP (a) and 2CRRR-1CRR-1RPPP (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 4, limb topology R\P\\P ?? P and C\P\||R\R, C\P\||R (a), C||R||R\R, C||R||R (b)

6.1 Fully-Parallel Topologies with Simple Limbs

585

Fig. 6.6 Maximaly regular fully-parallel PMs with Schönflies motions of types 2CPRR-1CPR1RUPU (a) and 2CRRR-1CRR-1RUPU (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 2, limb topology R\R\R\P\||R\R and C\P\||R\R, C\P\||R (a), C||R||R\R, C||R||R (b)

P\P ?? P\||R (Fig. 4.1a) P\P ?? P ?? R (Fig. 4.1b) R ?P ?? P ?? P (Fig. 5.1a) P ?P ?||R||R ?R (Fig. 4.2a) P ?P ?||R||R (Fig. 4.1d) R ?P ?? P ?? P (Fig. 5.1a) P||R ?P ?||R ?R (Fig. 4.2b) P||R ?P ?||R (Fig. 4.1e) R ?P ?? P ?? P (Fig. 5.1a) P||R||R||R ?R (Fig. 4.2d) P||R||R||R (Fig. 4.1g) R ?P ?? P ?? P (Fig. 5.1a) P ?P ?? P ?||R (Fig. 4.1a) P ?P ?? P ?? R (Fig. 4.1b) R ?R ?R ?P ?||R ?R (Fig. 5.1b)

P ?P ?||R||R ?R (Fig. 4.2a) P ?P ?||R||R (Fig. 4.1d) R ?R ?R ?P ?||R ?R (Fig. 5.1b) P||R ?P ?||R ?R (Fig. 4.2b) P||R ?P ?||R (Fig. 4.1e) R ?R ?R ?P ?||R ?R (Fig. 5.1b) P||R||R||R ?R (Fig. 4.2d) P||R||R||R (Fig. 4.1g) R ?R ?R ?P ?||R ?R (Fig. 5.1b) C ?P ?||R ?R (Fig. 4.2e) C ?P ?||R (Fig. 4.1i) R ?P ?? P ?? P (Fig. 5.1a)

3PPPR-1RPPP (Fig. 6.1a)

2PPRRR-1PPRR-1RPPP (Fig. 6.1b)

2PRPRR-1PRPR-1RPPP (Fig. 6.2a)

2PRRRR-1PRRR-1RPPP (Fig. 6.2b)

3PPPR-1RUPU (Fig. 6.3a)

2PPRRR-1PPRR-1RUPU (Fig. 6.3b)

2PRPRR-1PRPR-1RUPU (Fig. 6.4a)

2PRRRR-1PRRR-1RUPU (Fig. 6.4b)

2CPRR-1CPR-1RPPP (Fig. 6.5a)

1.

2.

3.

4.

5.

6.

7.

8.

9.

(continued)

The last joints of the four limbs have superposed axes/directions. The cylindrical joints of limbs G1, G2 and G3 have orthogonal axes

Idem no. 5

Idem no. 5

The first revolute joint of G4-limb and the last last joints of limbs G1, G2 and G3 have parallel axes The last revolute joints of limbs G1, G2 and G3 have superposed axes. The actuated prismatic joints of limbs G1, G2 and G3 have orthogonal directions Idem no. 5

Idem no. 1

Idem no. 1

The last joints of the four limbs have superposed axes/directions. The actuated prismatic joints of limbs G1, G2 and G3 have orthogonal directions Idem no. 1

Table 6.1 Limb topology and connecting conditions of the fully-parallel solutions with no idle mobilities presented in Figs. 6.1, 6.2, 6.3, 6.4, 6.5, 6.6 No. PM type Limb topology Connecting conditions

586 6 Maximally Regular Topologies with Schönflies Motions

2CRRR-1CRR-1RPPP (Fig. 6.5b)

2CPRR-1CPR-1RUPU (Fig. 6.6a)

2CRRR-1CRR-1RUPU (Fig. 6.6b)

10.

11.

12.

Table 6.1 (continued) No. PM type Limb topology

C||R||R ?R (Fig. 4.2g) C||R||R (Fig. 4.1k) R ?R ?R ?P ?||R ?R (Fig. 5.1b)

C||R||R ?R (Fig. 4.2g) C||R||R (Fig. 4.1k) R ?P ?? P ?? P (Fig. 5.1a) C ?P ?||R ?R (Fig. 4.2e) C ?P ?||R (Fig. 4.1i) R ?R ?R ?P ?||R ?R (Fig. 5.1b)

Connecting conditions

The first revolute joint of G4-limb and the last last joints of limbs G1, G2 and G3 have parallel axes The last revolute joints of limbs G1, G2 and G3 have superposed axes. The cylindrical joints of limbs G1, G2 and G3 have orthogonal axes Idem no. 11

Idem no. 9

6.1 Fully-Parallel Topologies with Simple Limbs 587

588

6 Maximally Regular Topologies with Schönflies Motions

Table 6.2 Structural parametersa of parallel mechanisms in Figs. 6.1 and 6.2 No. Structural parameter Solution 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. a

m pi (i = 1, 3) p2 p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1, 3) SG2 SG4 rGi (i = 1, 2, 3) rG4 MGi (i = 1, 3) MG2 MG4 (RF) SF rl rF MF NF TF Pp1 fj Pj¼1 p2 fj Pj¼1 p3 fj Pj¼1 p4 fj Pj¼1 p j¼1 fj

Figure 6.1a

Figures 6.1b and 6.2

14 4 4 4 16 3 4 0 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) 4 4 4 0 0 4 4 4 (v1 ; v2 ; v3 ; xb ) 4 0 12 4 6 0 4

16 5 4 4 18 3 4 0 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) 5 4 4 0 0 5 4 4 (v1 ; v2 ; v3 ; xb ) 4 0 14 4 4 0 5

4 4

4 5

4 16

4 18

See footnote of Table 2.2 for the nomenclature of structural parameters

6.1 Fully-Parallel Topologies with Simple Limbs

589

Table 6.3 Structural parametersa of parallel mechanisms in Figs. 6.3 and 6.4 No. Structural parameter Solution 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. a

m pi (i = 1,3) p2 p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1, 3) SG2 SG4 rGi (i = 1, 2, 3) rG4 MGi (i = 1, 3) MG2 MG4 (RF) SF rl rF MF NF TF Pp1 fj Pj¼1 p2 fj Pj¼1 p3 fj Pj¼1 p4 fj Pj¼1 p j¼1 fj

Figure 6.3a

Figures 6.3b and 6.4

16 4 4 6 18 3 4 0 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xa ; xb ; xd ) 4 4 6 0 0 4 4 6 (v1 ; v2 ; v3 ; xb ) 4 0 14 4 4 0 4

18 5 4 6 20 3 4 0 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xa ; xb ; xd ) 5 4 6 0 0 5 4 6 (v1 ; v2 ; v3 ; xb ) 4 0 16 4 2 0 5

4 4

4 5

6 18

6 20

See footnote of Table 2.2 for the nomenclature of structural parameters

590

6 Maximally Regular Topologies with Schönflies Motions

Table 6.4 Structural parametersa of parallel mechanisms in Figs. 6.5 and 6.6 No. Structural parameter Solution 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. a

m pi (i = 1,3) p2 p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1, 3) SG2 SG4 rGi (i = 1, 2, 3) rG4 MGi (i = 1, 3) MG2 MG4 (RF) SF rl rF MF NF TF Pp1 fj Pj¼1 p2 fj Pj¼1 p3 fj Pj¼1 p4 fj Pj¼1 p j¼1 fj

Figure 6.5

Figure 6.6

13 4 3 4 15 3 4 0 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xb ) 5 4 4 0 0 5 4 4 (v1 ; v2 ; v3 ; xb ) 4 0 14 4 4 0 5

15 4 3 6 17 3 4 0 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xa ; xb ; xd ) 5 4 6 0 0 5 4 6 (v1 ; v2 ; v3 ; xb ) 4 0 16 4 2 0 5

4 5

4 5

4 18

6 20

See footnote of Table 2.2 for the nomenclature of structural parameters

6.2 Fully-Parallel Topologies with Simple and Complex Limbs

591

6.2 Fully-Parallel Topologies with Simple and Complex Limbs In the fully-parallel and maximally regular topologies of PMs with Schönflies motions F G1-G2-G3-G4 presented in this section, the moving platform n : nGi (i = 1, 2, 3, 4) is connected to the reference platform 1 : 1Gi : 0 by four simple and complex limbs with four, five or six degrees of connectivity. One linear actuator is combined in the first prismatic or cylindrical pair of limbs G1, G2 and G3, and one rotary actuator in the first revolute or cylindrical pair of limb G4. In the cylindrical joint denoted by C just one motion is actuated. This can be the translational or the rotational motion and is indicated in the structural diagram by a linear or circular arrow. Limbs G1, G2 and G3 are used for positioning the moving platform and limb G4 for orienting it. There are no idle mobilities in these basic topologies. Various maximally regular topologies of PMs with Schönflies motions of the moving platform and no idle mobilities can be obtained by using different limb topologies presented in Figs. 4.1a, b, d, e, g, i, k, 4.2a, b, d, e, g, 4.8a–e, g, 4.9a, 4.10a, b, 5.1a, b and 5.4k–p. The simple limbs in Figs 4.1a, b, d, e, g, 4.2a, b, d, e, g and 5.1 combine only revolute and prismatic pairs. A cylindrical pair is combined in the simple limbs in Figs. 4.1i, k and 4.2e, g and in the complex limbs in Fig. 5.4o to replace a group of revolute and prismatic pairs with coincident axis/direction. One (Figs. 4.8a–e, g and 4.9a) or two (Figs. 4.10a, b and 5.4k, m, o) planar parallelogram loops are combined in the complex limbs. A planar telescopic parallelogram loop is combined in the limbs in Fig. 5.4l, n and p. Three joint parameters loose their independence in each parallelogram loop. Fully-parallel topologies with at least two identical limbs are illustrated in Figs. 6.7, 6.8, 6.9, 6.10, 6.11, 6.12, 6.13, 6.14, 6.15, 6.16, 6.17, 6.18, 6.19, 6.20, 6.21, 6.22, 6.23, 6.24, 6.25, 6.26. The limb topology and connecting conditions of these solutions are systematized in Tables 6.5 and 6.6, as are their structural parameters in Tables 6.7, 6.8, 6.9, 6.10, 6.11, 6.12.

592

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.7 Maximaly regular fully-parallel PMs with Schönflies motions of types 3PPPR1RPaPaP (a) and 3PPPR-1RPaPatP (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 12, limb topology P\P ?? P\||R, P\P ?? P ?? R and R||Pa||Pa||P (a), R||Pa||Pat||P (b)

6.2 Fully-Parallel Topologies with Simple and Complex Limbs

593

Fig. 6.8 Maximaly regular fully-parallel PMs with Schönflies motions of types 2PPRRR1PPRR-1RPaPaP (a) and 2PPRRR-1PPRR-1RPaPatP (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 10, limb topology P\P\||R||R\R, P\P\||R||R and R||Pa||Pa||P (a), R||Pa||Pat||P (b)

594

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.9 Maximaly regular fully-parallel PMs with Schönflies motions of types 2PRPRR1PRPR-1RPaPaP (a) and 2PRPRR-1PRPR-1RPaPatP (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 10, limb topology P||R\P\||R\R, P||R\P\||R and R||Pa||Pa||P (a), R||Pa||Pat||P (b)

6.2 Fully-Parallel Topologies with Simple and Complex Limbs

595

Fig. 6.10 Maximaly regular fully-parallel PMs with Schönflies motions of types 2PRRRR1PRRR-1RPaPaP (a) and 2PRRRR-1PRRR-1RPaPatP (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 10, limb topology P||R||R||R\R, P||R||R||R and R||Pa||Pa||P (a), R||Pa||Pat||P (b)

596

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.11 Maximaly regular fully-parallel PMs with Schönflies motions of types 3PPPR1RPPaPa (a) and 3PPPR-1RPPaPat (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 12, limb topology P\P ?? P\||R, P\P ?? P ?? R and R||P||Pa||Pa (a), R||P||Pa||Pat (b)

6.2 Fully-Parallel Topologies with Simple and Complex Limbs

597

Fig. 6.12 Maximaly regular fully-parallel PMs with Schönflies motions of types 2PPRRR1PPRR-1RPPaPa (a) and 2PPRRR-1PPRR-1RPPaPat (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 10, limb topology P\P\||R||R\R, P\P\||R||R and R||P||Pa||Pa (a), R||P||Pa||Pat (b)

598

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.13 Maximaly regular fully-parallel PMs with Schönflies motions of types 2PRPRR1PRPR-1RPPaPa (a) and 2PRPRR-1PRPR-1RPPaPat (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 10, limb topology P||R\P\||R\R, P||R\P\||R and R||P||Pa||Pa (a), R||P||Pa||Pat (b)

6.2 Fully-Parallel Topologies with Simple and Complex Limbs

599

Fig. 6.14 Maximaly regular fully-parallel PMs with Schönflies motions of types 2PRRRR1PRRR-1RPPaPa (a) and 2PRRRR-1PRRR-1RPPaPat (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 10, limb topology P||R||R||R\R, P||R||R||R and R||P||Pa||Pa (a), R||P||Pa||Pat (b)

600

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.15 Maximaly regular fully-parallel PMs with Schönflies motions of types 2CPRR-1CPR1RPaPaP (a) and 2CPRR-1CPR-1RPaPatP (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 10, limb topology C\P\||R\R, C\P\||R and R||Pa||Pa||P (a), R||Pa||Pat||P (b)

6.2 Fully-Parallel Topologies with Simple and Complex Limbs

601

Fig. 6.16 Maximaly regular fully-parallel PMs with Schönflies motions of types 2CRRR-1CRR1RPaPaP (a) and 2CRRR-1CRR-1RPaPatP (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 10, limb topology C||R||R\R, C||R||R and R||Pa||Pa||P (a), R||Pa||Pat||P (b)

602

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.17 Maximaly regular fully-parallel PMs with Schönflies motions of types 3PPPR1CPaPa (a) and 3PPPR-1CPaPat (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 12, limb topology P\P ?? P\||R, P\P ?? P ?? R and C||Pa||Pa (a), C||Pa||Pat (b)

6.2 Fully-Parallel Topologies with Simple and Complex Limbs

603

Fig. 6.18 Maximaly regular fully-parallel PMs with Schönflies motions of types 2PPRRR1PPRR-1CPaPa (a) and 2PPRRR-1PPRR-1CPaPat (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 10, limb topology P\P\||R||R\R, P\P\||R||R and C||Pa||Pa (a), C||Pa||Pat (b)

604

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.19 Maximaly regular fully-parallel PMs with Schönflies motions of types 2CPRR-1CPR1CPaPa (a) and 2CPRR-1CPR-1CPaPat (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 10, limb topology C\P\||R\R, C\P\||R and C||Pa||Pa (a), C||Pa||Pat (b)

6.2 Fully-Parallel Topologies with Simple and Complex Limbs

605

Fig. 6.20 Maximaly regular fully-parallel PMs with Schönflies motions of types 2CRRR-1CRR1CPaPa (a) and 2CRRR-1CRR-1CPaPat (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 10, limb topology C||R||R\R, C||R||R and C||Pa||Pa (a), C||Pa||Pat (b)

606

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.21 3PPPaR-1RPPP-type maximaly regular fully-parallel PMs with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 15, limb topology P\P\||Pa||R, R\P ?? P ?? P and P\P\||Pa\||R (a), P\P\||Pa ?? R (b)

6.2 Fully-Parallel Topologies with Simple and Complex Limbs

607

Fig. 6.22 Maximaly regular fully-parallel PMs with Schönflies motions of types 3PPaPR1RPPP (a) and 2PPaRRR-1PPaRR-1RPPP (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 15 (a), NF = 13 (b), limb topology R\P ?? P ?? P and P||Pa\P ?? R, P||Pa\P\||R (a), P||Pa||R||R\R, P||Pa||R||R (b)

608

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.23 3PPPaR-1RUPU-type maximaly regular fully-parallel PMs with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 13, limb topology P\P\||Pa||R, R\R\R\P\||R\R and P\P\||Pa\||R (a), P\P\||Pa ?? R (b)

6.2 Fully-Parallel Topologies with Simple and Complex Limbs

609

Fig. 6.24 Maximaly regular fully-parallel PMs with Schönflies motions of types 3PPaPR(a) and 2PPaRRR-1PPaRR-1RUPU (b) defined by MF = SF = 4, 1RUPU (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 13 (a), NF = 11 (b), limb topology R\R\R\P\||R\R and P||Pa\P ?? R, P||Pa\P\||R (a), P||Pa||R||R\R, P||Pa||R||R (b)

610

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.25 3PPaPaR-1RPPP-type maximaly regular fully-parallel PM with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 24, limb topology P||Pa||Pa\R, P||Pa||Pa||R and R\P ?? P ?? P

6.2 Fully-Parallel Topologies with Simple and Complex Limbs

611

Fig. 6.26 3PPaPaR-1RUPU-type maximaly regular fully-parallel PM with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 22, limb topology P||Pa||Pa\R, P||Pa||Pa||R and R\R\R\P\||R\R

612

6 Maximally Regular Topologies with Schönflies Motions

Table 6.5 Limb topology and connecting conditions of the fully-parallel solutions with no idle mobilities presented in Figs. 6.7, 6.8, 6.9, 6.10, 6.11, 6.12, 6.13, 6.14, 6.15, 6.16 No. PM type Limb topology Connecting conditions The last joints of the four P\P ?? P\||R (Fig. 4.1a) limbs have P\P ?? P ?? R (Fig. 4.1b) superposed axes/ R||Pa||Pa||P (Fig. 5.4k) directions. The actuated prismatic joints of limbs G1, G2 and G3 have orthogonal directions Idem no. 1 2. 3PPPR-1RPaPatP (Fig. 6.7b) P\P ?? P\||R (Fig. 4.1a) P\P ?? P ?? R (Fig. 4.1b) R||Pa||Pat||P (Fig. 5.4l) 3. 2PPRRR-1PPRR-1RPaPaP Idem no. 1 P ?P\||R||R\R (Fig. 4.2a) P\P\||R||R (Fig. 4.1d) (Fig. 6.8a) R||Pa||Pa||P (Fig. 5.4k) Idem no. 1 P\P\||R||R\R (Fig. 4.2a) 4. 2PPRRR-1PPRRP\P\||R||R (Fig. 4.1d) 1RPaPatP (Fig. 6.8b) R||Pa||Pat||P (Fig. 5.4l) 5. 2PRPRR-1PRPR-1RPaPaP Idem no. 1 P||R\P\||R\R (Fig. 4.1b) P||R\P\||R (Fig. 4.1e) (Fig. 6.9a) R||Pa||Pa||P (Fig. 5.4k) P||R\P\||R\R (Fig. 4.1b) 6. 2PRPRR-1PRPRIdem no. 1 1RPaPatP (Fig. 6.9b) P||R\P\||R (Fig. 4.1e) R||Pa||Pat||P (Fig. 5.4l) P||R||R||R\R (Fig. 4.2d) Idem no. 1 7. 2PRRRR-1PRRR-1RPaPaP (Fig. 6.10a) P||R||R||R (Fig. 4.1g) R||Pa||Pa||P (Fig. 5.4k) 8. 2PRRRR-1PRRRP||R||R||R\R (Fig. 4.2d) Idem no. 1 P||R||R||R (Fig. 4.1g) 1RPaPatP (Fig. 6.10b) R||Pa||Pat||P (Fig. 5.4l) The last revolute joints of 9. 3PPPR-1RPPaPa (Fig. 6.11a) P\P ?? P\||R (Fig. 4.1a) limbs G1, G2 and G3 P\P ?? P ?? R (Fig. 4.1b) and the revolute joint R||P||Pa||Pa (Fig. 5.4m) between links 7D and 9 have superposed axes. The actuated prismatic joints of limbs G1, G2 and G3 have orthogonal directions Idem no. 9 10. 3PPPR-1RPPaPat (Fig. 6.11b) P\P ?? P\||R (Fig. 4.1a) P\P ?? P ?? R (Fig. 4.1b) R||P||Pa||Pat (Fig. 5.4n) 11. 2PPRRR-1PPRR-1RPPaPa P\P\||R||R\R (Fig. 4.2a) Idem no. 9 (Fig. 6.12a) P\P\||R||R (Fig. 4.1d) R||P||Pa||Pa (Fig. 5.4m)

1.

3PPPR-1RPaPaP (Fig. 6.7a)

(continued)

6.2 Fully-Parallel Topologies with Simple and Complex Limbs Table 6.5 (continued) No. PM type 12. 2PPRRR-1PPRR-1RPPaPat (Fig. 6.12b) 13. 2PRPRR-1PRPR-1RPPaPa (Fig. 6.13a) 14. 2PRPRR-1PRPR-1RPPaPat (Fig. 6.13b) 15. 2PRRRR-1PRRR-1RPPaPa (Fig. 6.14a) 16. 2PRRRR-1PRRR1RPaPatP (Fig. 6.14b) 17. 2CPRR-1CPR-1RPaPaP (Fig. 6.15a) 18. 2CPRR-1CPR1RPaPatP (Fig. 6.15b) 19. 2CRRR-1CRR-1RPaPaP (Fig. 6.16a) 20. 2CRRR-1CRR1RPaPatP (Fig. 6.16b)

613

Limb topology

Connecting conditions

P\P\||R||R\R (Fig. 4.2a) P\P\||R||R (Fig. 4.1d) R||P||Pa||Pat (Fig. 5.4n) P||R\P\||R\R (Fig. 4.2b) P||R\P\||R (Fig. 4.1e) R||P||Pa||Pa (Fig. 5.4m) P||R\P\||R\R (Fig. 4.2b) P||R\P\||R (Fig. 4.1e) R||P||Pa||Pat (Fig. 5.4n) P||R||R||R\R (Fig. 4.2d) P||R||R||R (Fig. 4.1g) R||P||Pa||Pa (Fig. 5.4m) P||R||R||R\R (Fig. 4.2d) P||R||R||R (Fig. 4.1g) R||P||Pa||Pat (Fig. 5.4n) C\P\||R\R (Fig. 4.2e) C ?P ?||R (Fig. 4.1i) R||Pa||Pa||P (Fig. 5.4k) C ?P ?||R ?R (Fig. 4.2e) C ?P ?||R (Fig. 4.1i) R||Pa||Pat||P (Fig. 5.4l) C||R||R ?R (Fig. 4.2g) C||R||R (Fig. 4.1k) R||Pa||Pa||P (Fig. 5.4k) C||R||R ?R (Fig. 4.2g) C||R||R (Fig. 4.1k) R||Pa||Pat||P (Fig. 5.4l)

Idem no. 9

Idem no. 9

Idem no. 9

Idem no. 9

Idem no. 9

Idem no. 1

Idem no. 1

Idem no. 1

Idem no. 1

Table 6.6 Limb topology and connecting conditions of the fully-parallel solutions with no idle mobilities presented in Figs. 6.17, 6.18, 6.19, 6.20, 6.21, 6.22, 6.23, 6.24, 6.25, 6.26 No. PM type Limb topology Connecting conditions 1.

2.

3.

The last revolute joints of 3PPPR-1CPaPaP (Fig. 6.17a) P ?P\\P ?||R (Fig. 4.1a) limbs G1, G2 and G3 P ?P\\P\\R (Fig. 4.1b) C||Pa||Pa (Fig. 5.4o) and the revolute joint between links 6D and 8 have superposed axes. The actuated prismatic joints of limbs G1, G2 and G3 have orthogonal directions Idem no. 1 3PPPR-1CPaPatP (Fig. 6.17b) P ?P\\P ?||R (Fig. 4.1a) P ?P\\P\\R (Fig. 4.1b) CPa||Pat (Fig. 5.4p) Idem no. 1 2PPRRR-1PPRR-1CPaPa P ?P ?||R||R ?R (Fig. 4.2a) P ?P ?||R||R (Fig. 4.1d) (Fig. 6.18a) C||Pa||Pa (Fig. 5.4o) (continued)

614

6 Maximally Regular Topologies with Schönflies Motions

Table 6.6 (continued) No. PM type 4.

2PPRRR-1PPRR-1CPaPat (Fig. 6.18b)

5.

2CPRR-1CPR-1CPaPa (Fig. 6.19a)

6.

2CPRR-1CPR-1CPaPat (Fig. 6.19b)

7.

2CRRR-1CRR-1CPaPa (Fig. 6.20a)

8.

2CRRR-1CRR-1CPaPat (Fig. 6.20b)

9.

3PPPaR-1RPPP (Fig. 6.21a)

10. 3PPPaR-1RPPP (Fig. 6.21b)

11. 3PPaPR-1RPPP (Fig. 6.22a)

12. 2PPaRRR-1PPaRR-1RPPP (Fig. 6.22b) 13. 3PPPaR-1RUPU (Fig. 6.23a)

Limb topology

Connecting conditions

P ?P ?||R||R ?R (Fig. 4.2a) P ?P ?||R||R (Fig. 4.1d) C||Pa||Pat (Fig. 5.4p) C ?P ?||R ?R (Fig. 4.2e) C ?P ?||R (Fig. 4.1i) C||Pa||Pa (Fig. 5.4o) C ?P ?||R ?R (Fig. 4.2e) C ?P ?||R (Fig. 4.1i) C||Pa||Pat (Fig. 5.4p) C||R||R ?R (Fig. 4.2g) C||R||R (Fig. 4.1k) C||Pa||Pa (Fig. 5.4o) C||R||R ?R (Fig. 4.2g) C||R||R (Fig. 4.1k) C||Pa||Pat (Fig. 5.4p) P ?P ?||Pa ?||R (Fig. 4.8b) P ?P ?||Pa||R, (Fig. 4.8c) R ?P\\P\\P (Fig. 5.1a)

Idem no. 1

Idem no. 1

Idem no. 1

Idem no. 1

Idem no. 1

The last joints of the four limbs have superposed axes/ directions. The actuated prismatic joints of limbs G1, G2 and G3 have orthogonal directions Idem no. 9

P ?P ?||Pa\\R (Fig. 4.8a) P ?P ?||Pa||R, (Fig. 4.8c) R\\P\\P\\P (Fig. 5.1a) Idem no. 9 P||Pa ?P\\R (Fig. 4.8d) P||Pa ?P ?||R (Fig. 4.8e) R ?P\\P\\P (Fig. 5.1a) P||Pa||R||R ?R (Fig. 4.9a) Idem no. 9 P||Pa||R||R (Fig. 4.8g) R ?P\\P\\P (Fig. 5.1a) The first revolute joint of P ?P ?||Pa ?||R (Fig. 4.8b) G4-limb and the last P ?P ?||Pa||R, (Fig. 4.8c) R ?R ?R ?P ?||R ?R last joints of limbs (Fig. 5.1b) G1, G2 and G3 have parallel axes. The last revolute joints of limbs G1, G2 and G3 have superposed axes. The actuated prismatic joints of limbs G1, G2 and G3 have orthogonal directions (continued)

6.2 Fully-Parallel Topologies with Simple and Complex Limbs Table 6.6 (continued) No. PM type 14. 3PPPaR-1RUPU (Fig. 6.23b)

15. 3PPaPR-1RUPU (Fig. 6.24a)

16. 2PPaRRR-1PPaRR-1RUPU (Fig. 6.24b)

17. 3PPaPaR-1RPPP (Fig. 6.25)

18. 3PPaPaR-1RUPU (Fig. 6.26)

615

Limb topology

Connecting conditions

P ?P ?||Pa\\R (Fig. 4.8a) P ?P ?||Pa||R, (Fig. 4.8c) R ?R ?R ?P ?||R ?R (Fig. 5.1b) P||Pa ?P\\R (Fig. 4.8d) P||Pa ?P ?||R (Fig. 4.8e) R ?R ?R ?P ?||R ?R (Fig. 5.1b) P||Pa||R||R ?R (Fig. 4.9a) P||Pa||R||R (Fig. 4.8g) R ?R ?R ?P ?||R ?R (Fig. 5.1b) P||Pa||Pa ?R (Fig. 4.10a) P||Pa||Pa||R (Fig. 4.10b) R ?P\\P\\P (Fig. 5.1a) P||Pa||Pa ?R (Fig. 4.10a) P||Pa||Pa||R (Fig. 4.10b) R ?R ?R ?P ?||R ?R (Fig. 5.1b)

Idem no. 13

Idem no. 13

Idem no. 13

Idem no. 9

Idem no. 13

Table 6.7 Structural parametersa of parallel mechanisms in Figs. 6.7, 6.8, 6.9, 6.10, 6.11, 6.12, 6.13, 6.14 No. Structural parameter Solution 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

m pi (i = 1, 3) p2 p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1, 3) SG2 SG4 rGi (i = 1, 2, 3) rG4 MGi (i = 1, 3)

Figures 6.7 and 6.11

Figures 6.8, 6.9, 6.10, 6.12, 6.13, 6.14

18 4 4 10 22 5 3 1 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) 4 4 4 0 6 4

20 5 4 10 24 5 3 1 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xb ) 5 4 4 0 6 5 (continued)

616

6 Maximally Regular Topologies with Schönflies Motions

Table 6.7 (continued) No. Structural parameter

20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33 a

MG2 MG4 (RF) SF rl rF MF NF TF Pp1

Ppj¼1 2

fj

fj Ppj¼1 3 fj Ppj¼1 4 j¼1 fj Pp j¼1 fj

Solution Figures 6.7 and 6.11

Figures 6.8, 6.9, 6.10, 6.12, 6.13, 6.14

4 4 (v1 ; v2 ; v3 ; xb ) 4 6 18 4 12 0 4 4

4 4 (v1 ; v2 ; v3 ; xb ) 4 6 20 4 10 0 5 4

4 10

5 10

22

24

See footnote of Table 2.2 for the nomenclature of structural parameters

Table 6.8 Structural parametersa of parallel mechanisms in Figs. 6.15, 6.16, 6.17 No. Structural parameter Solution 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

m pi (i = 1, 3) p2 p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1, 3) SG2 SG4 rGi (i = 1, 2, 3) rG4 MGi (i = 1, 3) MG2

Figures 6.15 and 6.16

Figure 6.17

17 4 3 10 21 5 3 1 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xb ) 5 4 4 0 6 5 4

17 4 4 9 21 5 3 1 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) 4 4 4 0 6 4 4 (continued)

6.2 Fully-Parallel Topologies with Simple and Complex Limbs Table 6.8 (continued) No. Structural parameter

21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. a

MG4 (RF) SF rl rF MF NF TF Pp1

fj Pj¼1 p2 fj Pj¼1 p3 j¼1 fj Pp4 fj Pj¼1 p j¼1 fj

617

Solution Figures 6.15 and 6.16

Figure 6.17

4 (v1 ; v2 ; v3 ; xb ) 4 6 20 4 10 0 5

4 (v1 ; v2 ; v3 ; xb ) 4 6 18 4 12 0 4

4 5

4 4

10 24

10 22

See footnote of Table 2.2 for the nomenclature of structural parameters

Table 6.9 Structural parametersa of parallel mechanisms in Figs. 6.18, 6.19, 6.20 No. Structural parameter Solution 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

m pi (i = 1, 3) p2 p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1, 3) SG2 SG4 rGi (i = 1, 2, 3) rG4 MGi (i = 1, 3) MG2 MG4 (RF)

Figure 6.18

Figures 6.19 and 6.20

19 5 4 9 23 5 3 1 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xb ) 5 4 4 0 6 5 4 4 (v1 ; v2 ; v3 ; xb )

16 4 3 9 20 5 3 1 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xb ) 5 4 4 0 6 5 4 4 (v1 ; v2 ; v3 ; xb ) (continued)

618

6 Maximally Regular Topologies with Schönflies Motions

Table 6.9 (continued) No. Structural parameter

23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. a

SF rl rF MF NF TF Pp1 fj Pj¼1 p2 fj Pj¼1 p3 fj Pj¼1 p4 fj Pj¼1 p j¼1 fj

Solution Figure 6.18

Figures 6.19 and 6.20

4 6 20 4 10 0 5 4

4 6 20 4 10 0 5 4

5 10

5 10

24

24

See footnote of Table 2.2 for the nomenclature of structural parameters

Table 6.10 Structural parametersa of parallel mechanisms in Figs. 6.21 and 6.22 No. Structural parameter Solution 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

m pi (i = 1, 3) p2 p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1, 3) SG2 SG4 rGi (i = 1, 2, 3) rG4 MGi (i = 1, 3) MG2 MG4 (RF) SF

Figures 6.21 and 6.22a

Figure 6.22b

20 7 7 4 25 6 1 3 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) 4 4 4 3 0 4 4 4 (v1 ; v2 ; v3 ; xb ) 4

22 8 7 4 27 6 1 3 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xb ) 5 4 4 3 0 5 4 4 (v1 ; v2 ; v3 ; xb ) 4 (continued)

6.2 Fully-Parallel Topologies with Simple and Complex Limbs Table 6.10 (continued) No. Structural parameter

24. 25. 26. 27. 28. 29. 30. 31. 32. 33. a

rl rF MF NF TF Pp1

fj Pj¼1 p2 fj Pj¼1 p3 fj Pj¼1 p4 fj Pj¼1 p j¼1 fj

619

Solution Figures 6.21 and 6.22a

Figure 6.22b

9 21 4 15 0 7

9 23 4 13 0 8

7

7

7 4

8 4

25

27

See footnote of Table 2.2 for the nomenclature of structural parameters

Table 6.11 Structural parametersa of parallel mechanisms in Figs. 6.23 and 6.24 No. Structural parameter Solution 1. 2. 3. 4. 5. 6. 7. 8. 9. 10 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

m pi (i = 1, 3) p2 p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1, 3) SG2 SG4 rGi (i = 1, 2, 3) rG4 MGi (i = 1, 3) MG2 MG4 (RF) SF rl

Figures 6.23 and 6.24a

Figure 6.24b

22 7 7 6 27 6 1 3 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xa ; xb ; xd ) 4 4 6 3 0 4 4 6 (v1 ; v2 ; v3 ; xb ) 4 9

24 8 7 6 29 6 1 3 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xa ; xb ; xd ) 5 4 6 3 0 5 4 6 (v1 ; v2 ; v3 ; xb ) 4 9 (continued)

620

6 Maximally Regular Topologies with Schönflies Motions

Table 6.11 (continued) No. Structural parameter

25. 26. 27. 28. 29. 30. 31. 32. 33. a

rF MF NF TF Pp1

fj Pj¼1 p2 fj Pj¼1 p3 j¼1 fj Pp4 fj Pj¼1 p j¼1 fj

Solution Figures 6.23 and 6.24a

Figure 6.24b

23 4 13 0 7

25 4 11 0 8

7 7

7 8

6 27

6 29

See footnote of Table 2.2 for the nomenclature of structural parameters

Table 6.12 Structural parametersa of parallel mechanisms in Figs. 6.25 and 6.26 No. Structural parameter Solution 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

m pi (i = 1, 3) p2 p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1, 3) SG2 SG4 rGi (i = 1, 2, 3) rG4 MGi (i = 1, 3) MG2 MG4 (RF) SF rl rF

Figure 6.25

Figure 6.26

26 10 10 4 34 9 1 3 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) 4 4 4 6 0 4 4 4 (v1 ; v2 ; v3 ; xb ) 4 18 30

28 10 10 6 36 9 1 3 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xa ; xb ; xd ) 4 4 6 6 0 4 4 6 (v1 ; v2 ; v3 ; xb ) 4 18 32 (continued)

6.3 Fully-Parallel Topologies with Complex Limbs Table 6.12 (continued) No. Structural parameter

26. 27. 28. 29. 30. 31. 32. 33. a

MF NF TF Pp1 fj Ppj¼1 2 j¼1 fj Pp3 fj Ppj¼1 4 f j Ppj¼1 f j¼1 j

621

Solution Figure 6.25

Figure 6.26

4 24 0 10 10

4 22 0 10 10

10 4

10 6

34

36

See footnote of Table 2.2 for the nomenclature of structural parameters

6.3 Fully-Parallel Topologies with Complex Limbs In the fully-parallel and maximally regular topologies of PMs with Schönflies motions F G1-G2-G3-G4 presented in this section, the moving platform n : nGi (i = 1, 2, 3, 4) is connected to the reference platform 1 : 1Gi : 0 by four complex limbs with four or five degrees of connectivity. One linear actuator is combined in the first prismatic pair of limbs G1, G2 and G3, and one rotary actuator the first revolute or cylindrical pair of limb G4. In the cylindrical joint denoted by C just the rotation motion is actuated. Limbs G1, G2 and G3 are used for positioning the moving platform and limb G4 for orienting it. There are no idle mobilities in these basic topologies. Various maximally regular topologies of PMs with Schönflies motions of the moving platform and no idle mobilities can be obtained by using different limb topologies presented in Figs. 4.8a–e, g, 4.9a, 4.10a, b, and 5.4k–p. One (Figs. 4.8a–e, g and 4.9a) or two (Figs. 4.10a, b and 5.4k, m, o) planar parallelogram loops are combined in these complex limbs. A planar telescopic parallelogram loop is combined in the limbs in Fig. 5.4l, n and p. Three joint parameters loose their independence in each parallelogram loop. A cylindrical pair is combined in the complex limbs in Fig. 5.4o and p to replace a group of revolute and prismatic pairs with coincident axis/direction. Fully-parallel topologies with at least two identical limbs are illustrated in Figs. 6.27, 6.28, 6.29, 6.30, 6.31, 6.32, 6.33, 6.34, 6.35, 6.36, 6.37, 6.38, 6.39, 6.40, 6.41, 6.42, 6.43, 6.44, 6.45, 6.46. The limb:topology and connecting conditions of these solutions are systematized in Tables 6.13 and 6.14, as are their structural parameters in Tables 6.15 and 6.16.

622

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.27 3PPPaR-1RPaPaP-type maximaly regular fully-parallel PMs with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 21, limb topology P\P\||Pa||R, R||Pa||Pa||P and P\P\||Pa\||R (a), P\P\||Pa ?? R (b)

6.3 Fully-Parallel Topologies with Complex Limbs

623

Fig. 6.28 3PPPaR-1RPaPatP-type maximaly regular fully-parallel PMs with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 21, limb topology P\P\||Pa||R, R||Pa||Pat||P and P\P\||Pa\||R (a), P\P\||Pa\R (b)

624

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.29 Maximaly regular fully-parallel PMs with Schönflies motions of types 3PPaPR1RPaPaP (a) and 3PPaPR-1RPaPatP (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 15, limb topology P||Pa\P ?? R, P||Pa\P\||R and R||Pa||Pa||P (a), R||Pa||Pat||P (b)

6.3 Fully-Parallel Topologies with Complex Limbs

625

Fig. 6.30 2PPaRRR-1PPaRR-1RPaPaP-type maximaly regular fully-parallel PM with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 19, limb topology P||Pa||R||R\R, P||Pa||R||R\R and R||Pa||Pa||P

626

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.31 2PPaRRR-1PPaRR-1RPaPatP-type maximaly regular fully-parallel PM with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 19, limb topology P||Pa||R||R\R, P||Pa||R||R\R and R||Pa||Pat||P

6.3 Fully-Parallel Topologies with Complex Limbs

627

Fig. 6.32 3PPPaR-1RPPaPa-type maximaly regular fully-parallel PMs with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 21, limb topology P\P\||Pa||R, R||P||Pa||Pa and P\P\||Pa\||R (a), P\P\||Pa ?? R (b)

628

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.33 3PPPaR-1RPPaPat-type maximaly regular fully-parallel PMs with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 21, limb topology P\P\||Pa||R, R||P||Pa||Pat and P\P\||Pa\||R (a), P\P\||Pa ?? R (b)

6.3 Fully-Parallel Topologies with Complex Limbs

629

Fig. 6.34 3PPaPR-1RPPaPa-type maximaly regular fully-parallel PM with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 21, limb topology P||Pa\P ?? R, P||Pa\P\||R and R||P||Pa||Pa

630

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.35 3PPaPR-1RPPaPat-type maximaly regular fully-parallel PM with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 21, limb topology P||Pa\P ?? R, P||Pa\P\||R and R||P||Pa||Pat

6.3 Fully-Parallel Topologies with Complex Limbs

631

Fig. 6.36 2PPaRRR-1PPaRR-1RPPaPa-type maximaly regular fully-parallel PM with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 19, limb topology P||Pa||R||R\R, P||Pa||R||R and R||P||Pa||Pa

632

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.37 2PPaRRR-1PPaRR-1RPPaPat-type maximaly regular fully-parallel PM with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 19, limb topology P||Pa||R||R\R, P||Pa||R||R and R||P||Pa||Pat

6.3 Fully-Parallel Topologies with Complex Limbs

633

Fig. 6.38 3PPaPaR-1RPaPaP-type maximaly regular fully-parallel PM with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 30, limb topology P||Pa||Pa\R, P||Pa||Pa||R and R||Pa||Pa||P

634

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.39 3PPaPaR-1RPaPatP-type maximaly regular fully-parallel PM with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 30, limb topology P||Pa||Pa\R, P||Pa||Pa||R and R||Pa||Pat||P

6.3 Fully-Parallel Topologies with Complex Limbs

635

Fig. 6.40 3PPaPaR-1RPPaPa-type maximaly regular fully-parallel PM with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 30, limb topology P||Pa||Pa\R, P||Pa||Pa||R and R||P||Pa||Pa

636

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.41 3PPaPaR-1RPPaPat-type maximaly regular fully-parallel PM with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 30, limb topology P||Pa||Pa\R, P||Pa||Pa||R and R||P||Pa||Pat

6.3 Fully-Parallel Topologies with Complex Limbs

637

Fig. 6.42 3PPPaR-1CPaPa-type maximaly regular fully-parallel PMs with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 21, limb topology P\P\||Pa||R, C||Pa||Pa and P\P\||Pa\||R (a), P\P\||Pa ?? R (b)

638

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.43 3PPPaR-1CPaPat-type maximaly regular fully-parallel PMs with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 21, limb topology P\P\||Pa||R, C||Pa||Pat and P\P\||Pa\||R (a), P\P\||Pa ?? R (b)

6.3 Fully-Parallel Topologies with Complex Limbs

639

Fig. 6.44 Maximaly regular fully-parallel PMs with Schönflies motions of types 3PPaPR1CPaPa (a) and 3PPaPR-1CPaPat (b) defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 21, limb topology P||Pa\P ?? R, P||Pa\P\||R and C||Pa||Pa (a), C||Pa||Pat (b)

640

6 Maximally Regular Topologies with Schönflies Motions

Fig. 6.45 2PPaRRR-1PPaRR-1CPaPa-type maximaly regular fully-parallel PM with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 19, limb topology P||Pa||R||R\R, P||Pa||R||R and C||Pa||Pa

6.3 Fully-Parallel Topologies with Complex Limbs

641

Fig. 6.46 2PPaRRR-1PPaRR-1CPaPat-type maximaly regular fully-parallel PM with Schönflies motions defined by MF = SF = 4, (RF) = (v1 ; v2 ; v3 ; xb ), TF = 0, NF = 19, limb topology P||Pa||R||R\R, P||Pa||R||R and C||Pa||Pat

642

6 Maximally Regular Topologies with Schönflies Motions

Table 6.13 Limb topology and connecting conditions of the fully-parallel solutions with no idle mobilities presented in Figs. 6.27, 6.28, 6.29, 6.30, 6.31, 6.32, 6.33, 6.34, 6.35, 6.36, 6.37, 6.38, 6.39, 6.40, 6.41 No. PM type Limb topology Connecting conditions P\P\||Pa\||R (Fig. 4.8b) The last joints of the four limbs have superposed P\P\||Pa||R, axes/directions. The (Fig. 4.8c) R||Pa||Pa||P (Fig. 5.4k) actuated prismatic joints of limbs G1, G2 and G3 have orthogonal directions Idem no. 1 2. 3PPPaR-1RPaPaP (Fig. 6.27b) P\P\||Pa\R (Fig. 4.8a) P\P\||Pa||R, (Fig. 4.8c) R||Pa||Pa||P (Fig. 5.4k) 3. 3PPPaR-1RPaPatP (Fig. 6.28a) P\P\||Pa\||R (Fig. 4.8b) Idem no. 1 P\P\||Pa||R, (Fig. 4.8c) R||Pa||Pat||P (Fig. 5.4l) 4. 3PPPaR-1RPaPatP (Fig. 6.28b) P\P\||Pa\R (Fig. 4.8a) Idem no. 1 P\P\||Pa||R, (Fig. 4.8c) R||Pa||Pat||P (Fig. 5.4l) 5. 3PPaPR-1RPaPaP (Fig. 6.29a) P||Pa\P\\R (Fig. 4.8d) Idem no. 1 P||Pa\P\||R (Fig. 4.8e) R||Pa||Pa||P (Fig. 5.4k) Idem no. 1 6. 3PPaPR-1RPaPatP (Fig. 6.29b) P||Pa\P\\R (Fig. 4.8d) P||Pa\P\||R (Fig. 4.8e) R||Pa||Pat||P (Fig. 5.4l) 7. 2PPaRRR-1PPaRR-1RPaPaP P||Pa||R||R\R (Fig. 4.9a) Idem no. 1 P||Pa||R||R (Fig. 4.8g) (Fig. 6.30) R||Pa||Pa||P (Fig. 5.4k) 8. 2PPaRRR-1PPaRRP||Pa||R||R\R (Fig. 4.9a) Idem no. 1 P||Pa||R||R (Fig. 4.8g) 1RPaPatP (Fig. 6.31) R||Pa||Pat||P (Fig. 5.4l) 9. 3PPPaR-1RPPaPa (Fig. 6.32a) P\P\||Pa\||R (Fig. 4.8b) The last revolute joints of limbs G1, G2 and G3 P\P\||Pa||R, and the revolute joint (Fig. 4.8c) R||P||Pa||Pa (Fig. 5.4m) between links 7D and 9 have superposed axes. The actuated prismatic joints of limbs G1, G2 and G3 have orthogonal directions Idem no. 9 10. 3PPPaR-1PRPaPa (Fig. 6.32b) P\P\||Pa\R (Fig. 4.8a) P\P\||Pa||R, (Fig. 4.8c) R||P||Pa||Pa (Fig. 5.4m) 11. 3PPPaR-1RPPaPat (Fig. 6.33a) P\P\||Pa\||R (Fig. 4.8b) Idem no. 9 P\P\||Pa||R, (Fig. 4.8c) R||P||Pa||Pat (Fig. 5.4n)

1.

3PPPaR-1RPaPaP (Fig. 6.27a)

(continued)

6.3 Fully-Parallel Topologies with Complex Limbs Table 6.13 (continued) No. PM type

643

Limb topology

Connecting conditions

12. 3PPPaR-1RPPaPat (Fig. 6.33b) P\P\||Pa\\R (Fig. 4.8a) P\P\||Pa||R, (Fig. 4.8c) R||P||Pa||Pat (Fig. 5.4n) 13. 3PPaPR-1RPPaPa (Fig. 6.34) P||Pa\P\R (Fig. 4.8d) P||Pa\P\||R (Fig. 4.8e) R||P||Pa||Pa (Fig. 5.4m) 14. 3PPaPR-1RPPatPa (Fig. 6.35) P||Pa\P\\R (Fig. 4.8d) P||Pa\P\||R (Fig. 4.8e) R||P||Pa||Pat (Fig. 5.4n) 15. 2PPaRRR-1PPaRR-1RPPaPa P||Pa||R||R\R (Fig. 4.9a) P||Pa||R||R (Fig. 4.8g) (Fig. 6.36) R||P||Pa||Pa (Fig. 5.4m) 16. 2PPaRRR-1PPaRR-1RPPaPat P||Pa||R||R\R (Fig. 4.9a) (Fig. 6.37) P||Pa||R||R (Fig. 4.8g) R||P||Pa||Pat (Fig. 5.4n) P||Pa||Pa\R (Fig. 4.10a) 17. 3PPaPaR-1RPaPaP P||Pa||Pa||R (Fig. 6.38) (Fig. 4.10b) R||Pa||Pa||P (Fig. 5.4k) 18. 3PPaPaR-1RPaPatP P||Pa||Pa\R (Fig. 4.10a) P||Pa||Pa||R (Fig. 6.39) (Fig. 4.10b) R||Pa||Pat||P (Fig. 5.4l) 19. 3PPaPaR-1RPPaPa P||Pa||Pa\R (Fig. 4.10a) P||Pa||Pa||R (Fig. 6.40) (Fig. 4.10b) R||P||Pa||Pa (Fig. 5.4m) P||Pa||Pa\R (Fig. 4.10a) 20. 3PPaPaR-1RPaPatP P||Pa||Pa||R (Fig. 6.41) (Fig. 4.10b) R||P||Pa||Pat (Fig. 5.4n)

Idem no. 9

Idem no. 9

Idem no. 9

Idem no. 9

Idem no. 9

Idem no. 1

Idem no. 1

Idem no. 9

Idem no. 9

Table 6.14 Limb topology and connecting conditions of the fully-parallel solutions with no idle mobilities presented in Figs. 6.42, 6.43, 6.44, 6.45, 6.46 No. PM type Limb topology Connecting conditions 1.

3PPPaR-1CPaPa (Fig. 6.42a)

P\P\||Pa\||R (Fig. 4.8b) P\P\||Pa||R, (Fig. 4.8c) C||Pa||Pa (Fig. 5.4o)

2.

3PPPaR-1CPaPa (Fig. 6.42b)

P\P\||Pa\\R (Fig. 4.8a) P\P\||Pa||R, (Fig. 4.8c) C||Pa||Pa (Fig. 5.4o)

The last revolute joints of limbs G1, G2 and G3 and the revolute joint between links 6D and 8 have superposed axes. The actuated prismatic joints of limbs G1, G2 and G3 have orthogonal directions Idem no. 1

(continued)

644

6 Maximally Regular Topologies with Schönflies Motions

Table 6.14 (continued) No. PM type 3.

4.

5.

6.

7.

8.

Limb topology

3PPPaR-1CPaPat (Fig. 6.43a)

P\P\||Pa\||R (Fig. 4.8b) P\P\||Pa||R, (Fig. 4.8c) C||Pa||Pat (Fig. 5.4p) t P\P\||Pa\R (Fig. 4.8a) 3PPPaR-1CPaPa (Fig. 6.43b) P\P\||Pa||R, (Fig. 4.8c) C||Pa||Pat (Fig. 5.4p) 3PPaPR-1CPaPa P||Pa\P\\R (Fig. 4.8d) P||Pa\P\||R (Fig. 4.8e) (Fig. 6.44a) C||Pa||Pa (Fig. 5.4o) 3PPaPR-1CPaPat P||Pa\P\\R (Fig. 4.8d) (Fig. 6.44b) P||Pa\P\||R (Fig. 4.8e) C||Pa||Pat (Fig. 5.4p) P||Pa||R||R\R (Fig. 4.9a) 2PPaRRR-1PPaRR1CPaPa (Fig. 6.45) P||Pa||R||R (Fig. 4.8g) C||Pa||Pa (Fig. 5.4o) 2PPaRRR-1PPaRRP||Pa||R||R\R (Fig. 4.9a) P||Pa||R||R (Fig. 4.8g) 1CPaPat (Fig. 6.46) C||Pa||Pat (Fig. 5.4p)

Connecting conditions Idem no. 1

Idem no. 1

Idem no. 1

Idem no. 1

Idem no. 1

Idem no. 1

Table 6.15 Structural parametersa of parallel mechanisms in Figs. 6.27, 6.28, 6.29, 6.30, 6.31, 6.32, 6.33, 6.34, 6.35, 6.36, 6.37 No. Structural Solution parameter Figures 6.27, 6.28, 6.29, 6.32, 6.33, 6.34, Figures 6.30, 6.31, 6.36, 6.35 6.37 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

m pi (i = 1, 3) p2 p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1, 3) SG2 SG4 rGi (i = 1, 2, 3) rG4 MGi (i = 1, 3) MG2

24 7 7 10 31 8 0 4 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) 4 4 4 3 6 4 4

26 8 7 10 33 8 0 4 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xb ) 5 4 4 3 6 5 4 (continued)

6.3 Fully-Parallel Topologies with Complex Limbs

645

Table 6.15 (continued) No. Structural Solution parameter Figures 6.27, 6.28, 6.29, 6.32, 6.33, 6.34, Figures 6.30, 6.31, 6.36, 6.35 6.37 MG4 (RF) SF rl rF MF NF TF Pp1 fj Pj¼1 p2 fj Pj¼1 p3 31. fj Pj¼1 p4 32. fj Ppj¼1 33. j¼1 fj

21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

a

4 (v1 ; v2 ; v3 ; xb ) 4 15 27 4 21 0 7 7

4 (v1 ; v2 ; v3 ; xb ) 4 15 29 4 19 0 8 7

7

8

10 31

10 33

See footnote of Table 2.2 for the nomenclature of structural parameters

Table 6.16 Structural parametersa of parallel mechanisms in Figs. 6.38, 6.39, 6.40, 6.41, 6.42, 6.42, 6.43, 6.44, 6.45, 6.46 No. Structural Solution parameter Figures 6.38, 6.39, 6.40, Figures 6.42, 6.43, Figures 6.45 and 6.41 6.44 6.46 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

m pi (i = 1, 3) p2 p4 p q k1 k2 k (RG1) (RG2) (RG3) (RG4) SGi (i = 1, 3) SG2 SG4 rGi (i = 1, 2, 3) rG4

30 10 10 10 40 11 0 4 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) 4 4 4 6 6

23 7 7 9 30 8 0 4 4 (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ) 4 4 4 3 6

25 8 7 9 32 8 0 4 4 (v1 ; v2 ; v3 ; xa ; xb ) (v1 ; v2 ; v3 ; xb ) (v1 ; v2 ; v3 ; xb ; xd ) (v1 ; v2 ; v3 ; xb ) 5 4 4 3 6 (continued)

646

6 Maximally Regular Topologies with Schönflies Motions

Table 6.16 (continued) No. Structural Solution parameter Figures 6.38, 6.39, 6.40, 6.41 MGi (i = 1, 3) MG2 MG4 (RF) SF rl rF MF NF TF Pp1 fj Pj¼1 p2 fj Pj¼1 p3 31. j¼1 fj Pp4 32. fj Pj¼1 p 33. j¼1 fj

19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

a

Figures 6.42, 6.43, 6.44

Figures 6.45 and 6.46

4 4 4 (v1 ; v2 ; v3 ; xb ) 4 24 36 4 30 0 10 10

4 4 4 (v1 ; v2 ; v3 ; xb ) 4 15 27 4 21 0 7 7

5 4 4 (v1 ; v2 ; v3 ; xb ) 4 15 29 4 19 0 8 7

10 10

7 10

8 10

40

31

33

See footnote of Table 2.2 for the nomenclature of structural parameters

References 1. Gogu G (2008) Structural synthesis of parallel robots: Part 1-methodology. Springer, Dordrecht 2. Gogu G (2009) Structural synthesis of parallel robots: Part 2-translational topologies with two and three degrees of freedom. Springer, Dordrecht 3. Gogu G (2010) Structural synthesis of parallel robots: Part 3-topologies with planar motion of the moving platform. Springer, Dordrecht Heidelberg London New York 4. Gogu G (2012) Structural synthesis of parallel robots: Part 4-other topologies with two and three degrees of freedom. Springer, Dordrecht Heidelberg London New York

Index

A Actuator, 2 Algorithm evolutionary, 17 Approach systematic, 11

B Base, 2 Fixed, 2, 3 Basis, 15 of operational velocity space, 19 of vector space, 15 Branch mobility, 12

C Characteristic point, 15 Condition number, 19 Connectivity, 13 Connecting conditions, 36, 62, 186, 259, 372, 434, 441, 546, 556, 581, 593, 623 Constraint equation, 12 Coupled motions, 18, 185 CPM, 20

D Decoupled motions, 18 Degree of freedom, 11 Design objectives, 17 Dimension vector space, 14

E Element, 3 pairing, 3 reference, 3 End-effector, 10 Equation constraint, 12 Evolutionary morphology, 17

F Frame, 3 Fully-isotropic, 18 Fully-parallel, 23, 36, 369, 372, 581, 586, 613 overconstrained topologies, 35, 369

G Graph, 8 structural, 8

H HELIA, 4 Hexapod, 2 H4, 4

I Idle mobility, 6 IFMA, 19 Independent motion, 13 I4, 4 Isoglide4-T3R1, 4, 581 Isoglide3-T3, 19 Isotropy, 18

G. Gogu, Structural Synthesis of Parallel Robots, Solid Mechanics and Its Applications 206, DOI: 10.1007/978-94-007-7401-8,  Springer Science+Business Media Dordrecht 2014

647

648 J Jacobian matrix, 18 Joint, 3 Cardan, 6 heterokinetic, 6 homokinetic, 5 idle, 6 universal, 5 velocity space, 18

K Kanuk, 4 Kinematic pair, 3, 5 Kinematic chain, 2, 3 closed, 5 complex, 5 open, 5 serial, 2 simple, 5 Kinematic model, 21 direct, 21

L Limb, 2 complex, 5 simple, 5 topology, 36, 186, 259, 372, 434, 546, 554, 581, 587, 613 Link, 3 binary, 5 distal, 5 monary, 5 polinary, 5 Loop parallel concatenated, 17 parallelogram, 6 serial concatenated, 17

M Manta, 4 McGill, 4 Schönflies motion generator, 4 Mechanism, 2, 3 element, 3 kinematotropic, 12 parallel, 2, 6, 13 Mobility, 11, 13 external, 11 full-cycle, 12

Index general, 12 idle, 6 instantaneous, 12 internal, 11 Model kinematic, 21 direct, 21 Morphological operator, 17 Motion coupling, 18

N Number of overconstraints, 13 of limbs, 8

O Operational vector space, 13 velocity space, 13, 18 Orthogonal Tripteron, 19 Overactuated solutions, 19, 185, 249 Overactuation, 19 Overconstraint, 13

P Paire cylindrical, 5 helical, 5 kinematic, 5 lower, 5 passive, 8 planar, 5 revolute, 5 spherical, 5 Pairing element, 3 PAMINSA, 4 Pantopteron, 19 Pantopteron-4, 4 Parallel mechanism, 6, 13 maximally regular, 581, 593, 623 with coupled Schönflies motions, 35, 36, 185, 249 with decoupled Schönflies motions, 369 with uncoupled Schönflies motions, 433, 546 Parallel robot, 2, 14 non overconstrained, 15 non-redundant, 14 overconstrained, 15

Index redundant, 14 redundantly-actuated, 19, 433, 544 Parallel robotic manipulator, 18 fully-isotropic, 18 maximally regular, 18 non overconstrained, 15 non redundant, 15 overconstrained, 15 redundant, 15 redundantly-actuated, 2 R2-type, 4 R3-type, 4 T1R2-type, 4 T1R3-type, 4 T2R1-type, 4 T2R2-type, 4 T2R3-type, 4 T3-type, 4 T3R1-type, 4 T3R2-type, 4 T3R3-type, 4 with coupled motions, 18 with decoupled motions, 18 with uncoupled motions, 18 Par4, 4 Performance index, 18 Platform, 2 fixed, 2 moving, 2, 5 reference, 5 Point characteristic, 15 Protoelement, 17

Q Quadriglide, 4 Quadrupteron, 4

R Rank, 12 Redundancy, 13, 546 Robot, 1 fully parallel, 10

649 hexapod, 2 hybrid, 10 non fully-parallel, 10 parallel, 2, 10 SCARA, 21 serial, 10 Robotics, 2

S Schönflies motions, 21 Singular configuration, 12 Structural diagram, 8 graph, 8 parameters, 11 redundancy, 13 synthesis, 10 Synthesis structural, 10 Systematic approach, 11

T Theory of linear transformations, 13 Topology, 11 fully-parallel redundantly-actuated, 546

U Uncoupled motions, 18 Universal joint, 5

V Vector space, 13 Velocity, 13 Velocity vector space, 13 joint, 16 operational, 13