Structural Synthesis of Parallel Robots

Structural Synthesis of Parallel Robots SOLID MECHANICS AND ITS APPLICATIONS Volume 159 Series Editor: G.M.L. GLADW...

Author: Grigore Gogu

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Structural Synthesis of Parallel Robots

SOLID MECHANICS AND ITS APPLICATIONS Volume 159

Series Editor:

G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI

Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the ﬁrst year graduate student. Some texts are monographs deﬁning the current state of the ﬁeld; others are accessible to ﬁnal year undergraduates; but essentially the emphasis is on readability and clarity.

For other titles published in this series, go to www.springer.com/series/6557

Grigore Gogu

Structural Synthesis of Parallel Robots Part 2: Translational Topologies with Two and Three Degrees of Freedom

Grigore Gogu Mechanical Engineering Research Group French Institute of Advanced Mechanics and Blaise Pascal University Clermont-Ferrand, France [email protected]

ISBN: 978-1-4020-9793-5

e-ISBN: 978-1-4020-9794-2

Library of Congress Control Number: 2008942995 © Springer Science + Business Media B.V. 2009 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microﬁlming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied speciﬁcally for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 987654321 springer.com

Contents

Preface ......................................................................................................vii Acknowledgements .............................................................................xiii List of abbreviations and notations.......................................................xv 1 Introduction............................................................................................. 1 1.1 Terminology ....................................................................................... 1 1.2 Methodology of structural synthesis................................................... 7 1.2.1 New formulae for mobility, connectivity, redundancy and overconstraint of parallel robots ......................................... 8 1.2.2 Evolutionary morphology approach ........................................ 13 1.2.3 Types of parallel robots with respect to motion coupling ....... 14 1.3 Translational parallel robots ............................................................. 15 2 Translational parallel robots with two degrees of freedom .............. 23 2.1 T2-type translational parallel robots with coupled motions.............. 23 2.1.1 Overconstrained solutions ....................................................... 23 2.1.2 Non overconstrained solutions ................................................ 54 2.2 T2-type translational parallel robots with decoupled motions .......... 66 2.2.1 Overconstrained solutions ....................................................... 66 2.2.2 Non overconstrained solutions ................................................ 78 2.3 T2-type translational parallel robots with uncoupled motions.......... 85 2.3.1 Overconstrained solutions ....................................................... 86 2.3.2 Non overconstrained solutions ................................................ 89 2.4 Maximally regular T2-type translational parallel robots .................. 93 2.4.1 Overconstrained solutions ....................................................... 93 2.4.2 Non overconstrained solutions ................................................ 95 2.5 Other T2-type translational parallel robots ....................................... 97 2.5.1 Overconstrained solutions ....................................................... 98 2.5.2 Non overconstrained solutions .............................................. 105 3 Overconstrained T3-type TPMs with coupled motions ................... 107 3.1 Basic solutions with linear actuators .............................................. 108 3.2 Derived solutions with linear actuators .......................................... 138 v

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Contents

3.3 Basic solutions with rotating actuators ........................................... 194 3.4 Derived solutions with rotating actuators ....................................... 266 4 Non overconstrained T3-type TPMs with coupled motions ............ 365 4.1 Basic solutions with linear actuators .............................................. 365 4.2 Derived solutions with linear actuators .......................................... 375 4.3 Basic solutions with rotating actuators ........................................... 401 4.4 Derived solutions with rotating actuators ....................................... 412 5 Overconstrained T3-type TPMs with uncoupled motions .............. 471 5.1 Basic solutions with rotating actuators ........................................... 471 5.2 Derived solutions with rotating actuators ....................................... 537 6 Non overconstrained T3-type TPMs with uncoupled motions........ 615 6.1 Basic solutions with rotating actuators ........................................... 615 6.2 Derived solutions with rotating actuators ....................................... 621 7 Maximally regular T3-type translational parallel robots................ 687 7.1 Overconstrained solutions .............................................................. 687 7.1.1 Basic solutions with no idle mobilities.................................. 687 7.1.2 Derived solutions with idle mobilities................................... 706 7.2 Non overconstrained solutions ....................................................... 731 References............................................................................................... 749 Index........................................................................................................ 759

Preface

“The mathematical investigations referred to bring the whole apparatus of a great science to the examination of the properties of a given mechanism, and have accumulated in this direction rich material, of enduring and increasing value. What is left unexamined is however the other, immensely deeper part of the problem, the question: How did the mechanism, or the elements of which it is composed, originate? What laws govern its building up? Is it indeed formed according to any laws whatever? Or have we simply to accept as data what invention gives us, the analysis of what is thus obtained being the only scientific problem left – as in the case of natural history?” 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 second part of a larger work dedicated to the structural synthesis of parallel robots. Part 1 already published in 2008 (Gogu 2008a) has presented the methodology proposed for structural synthesis. This book focuses on various topologies of translational parallel robots 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 feed-back, 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 vii

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has concentrated primarily on six degrees of freedom (DoFs) GoughStewart-type PMs introduced by Gough for a tire-testing device, and by Stewart for flight simulators. In the last decade, PMs with fewer than 6DoFs attracted researchers’ attention. Lower mobility PMs are suitable for many tasks requiring less than six DoFs. The motion freedoms of the end-effector are usually coupled together due to the multi-loop kinematic structure of the parallel manipulator. Hence, motion planning and control of the end-effector for PMs usually become very complicated. With respect to serial manipulators, such mechanisms can offer advantages in terms of stiffness, accuracy, load-to-weight ratio, dynamic performances. Their disadvantages include a smaller workspace, complex command and lower dexterity due to a high motion coupling, and multiplicity of singularities inside their workspace. Uncoupled, fullyisotropic 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 end-effector velocity space, and (ii) the static load on the end-effector and the actuated joint forces or torques. The isotropic design aims at ideal kinematic and dynamic performance of the manipulator. We distinguish five types of PMs (i) maximally regular PMs, if the Jacobian J is an identity matrix throughout the entire workspace, (ii) fullyisotropic PMs, if the Jacobian J is a diagonal matrix with identical diagonal elements throughout the entire workspace, (iii) PMs with uncoupled motions if J is a diagonal matrix with different diagonal elements, (iv) PMs with decoupled motions, if J is a triangular matrix and (v) PMs with coupled motions if J is neither a triangular nor a diagonal matrix. Maximally regular and fully-isotropic PMs give a one-to-one mapping between the actuated joint velocity space and the external velocity space. The first solution for a fully-isotropic T3-type translational parallel robot was developed at the same time and independently by Carricato and ParentiCastelli at University of Genoa, Kim and Tsai at University of California, Kong and Gosselin at University of Laval, and the author of this work at the French Institute of Advanced Mechanics. In 2002, the four groups published the first results of their works. The general methods used for structural synthesis of parallel mechanisms can be divided into three approaches: the method based on displacement group theory, the methods based on screw algebra, and the method based

Preface

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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 decoupled, uncoupled, fully-isotropic and maximally regular PMs with two to six DoFs in a systematic way. The new formulae for mobility, connectivity (spatiality), redundancy and overconstraint of PMs proposed recently by the author are integrated into the synthesis approach developed in this work. Various solutions of TaRb-type PMs are known today. In this notation, a = 1,2,3 indicates the number of independent translations and b = 1,2,3 the number of independent rotations of the moving platform. The parallel robots actually proposed by the robot industry have coupled and decoupled motions and just some isotropic positions in their workspace. As far as we are aware, this is the first work on robotics presenting solutions of uncoupled, fully-isotropic and maximally regular PMs obtained by a systematic approach of structural synthesis. Non-redundant/redundant, overconstrained/isostatic solutions of uncoupled and fully-isotropic/maximally regular PMs with elementary/complex limbs actuated by linear/rotary actuators with/without idle mobilities and two to six DoFs are present in a systematic approach of structural synthesis. A serial kinematic chain is associated with each elementary limb and at least one closed loop is integrated in each complex limb. The synthesis methodology and the solutions of PMs presented in this work represent the outcome of some recent research developed by the author in the last years 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 3 years. In these works the author has proposed the following for the first time in the literature: (a) New formulae for calculating the degree of mobility, the degree of connectivity(spatiality), the degree of redundancy and the number of overconstraints of parallel robotic manipulators that overcome the drawbacks of the classical Chebychev-Grübler-Kutzbach formulae. (b) A new approach to systematic innovation in engineering design called evolutionary morphology. (c) Solutions of TaRb-type fully-isotropic and maximally regular PMs for any combination of a independent translations and b independent rotations of the moving platform. The various solutions of maximally regular PMs proposed by the author belong to a modular family called Isogliden-TaRb with a + b = n with 2 d n d 6, a = 1,2,3 and b = 1,2,3. The mobile platform of these robots can have any combination of n independent translations (T) and rotations (R).

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The Isogliden-TaRb modular family was developed by the author and his research team of the Mechanical Engineering Research Group (LaMI), Blaise Pascal University and French Institute of Advanced Mechanics (IFMA) in Clermont-Ferrand. Part 1 of this work (Gogu 2008a) was organized in ten chapters. The first chapter 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-GrüblerKutzbach mobility formulae. The drawbacks and the limitations of these formulae are discussed, and the new formulae for mobility, connectivity, redundancy and overconstraint are demonstrated via an original approach based on the theory of linear transformations. These formulae are applied in Chapter 3 for the structural analysis of parallel robots with simple and complex limbs. The new formulae are also applied to calculate the mobility and other structural parameters of single and multi-loop mechanisms that do not obey the classical Chebychev-Grübler-Kutzbach formulae, such as the mechanisms proposed by De Roberval, Sarrus, Bennett, Bricard and other so called “paradoxical mechanisms”. We have shown that these mechanisms completely obey the definitions, the theorems and the formulae proposed in the previous chapter. There is no reason to continue to consider them as “paradoxical”. Chapter 4 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 the second chapter are integrated in the structural synthesis approach founded on the evolutionary morphology (EM) presented in Chapter 5. The main paradigms of EM are presented in a closed relation with the biological background of morphological approaches and the synthetic theory of evolution. The main difference between the evolutionary algorithms and the EM are also discussed. The evolutionary algorithms are methods for solving optimization-oriented problems, and are not suited to solving conceptual design-oriented problems. They always start from a given initial population of solutions and do not solve the problem of creating these solutions. The first stage in structural synthesis of parallel robots is the generation of the kinematic chains called limbs used to give some constrained or unconstrained motion to the moving platform. The constrained motion of the mobile platform is obtained by using limbs with less than six degrees of connectivity. The various solutions of simple and complex limbs with two to six degrees of connectivity are systematically generated by the structural

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synthesis approach and presented in Chapters 6–10. We focused on the solutions with a unique basis of the operational velocity space that are useful for generating various topologies of decoupled, uncoupled, fullyisotropic and maximally regular parallel robots presented in Part 2. Limbs with multiple bases of the operational velocity space and redundant limbs are also presented in these chapters. These limb solutions are systematized with respect to various combinations of independent motions of the distal link. They are defined by symbolic notations and illustrated in about 250 figures containing more than 1,500 structural diagrams. The kinematic chains presented in Chapters 6–10 are useful as innovative solutions of limbs in parallel, serial and hybrid robots. In fact, serial and hybrid robots may be considered as a particular case of parallel robots with only one limb which can be a simple, complex or hybrid kinematic chain. Many serial robots actually combine closed loops in their kinematic structure. The various types of kinematic chains generated in Chapters 6–10 of Part 1 are combined in Part 2 and the following parts to set up innovative solutions of parallel robots with two to six degrees of mobility and various sets of independent motions of the moving platform. This book representing Part 2 is organised in seven chapters. The first chapter recalls the main concepts, the new formulae used to calculate the main structural parameters of PMs, and the original approach of structural synthesis. Chapter 2 focuses 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 presents 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. Idle mobilities 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 Chapter 4. Basic and derived solutions with linear or rotating actuators are on hand. Chapters 5 and 6 present 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 focuses 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.

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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 you are trying to disseminate the result of the structural synthesis of kinematic chains. The following parts of this work will present the structural synthesis of other PMs with two and three degrees of freedom (Part 3) and PMs with four, five and six degrees of freedom (Part 4). The writing of Parts 3 and 4 is still in progress and will soon be finalized. 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.

Acknowledgements

The scientific environment of the projects ROBEA-MAX and ROBEAMP2 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, Deputy Director of LIRMM and coordinator of both ROBEA projects, and also to all colleagues involved in these projects from the research laboratories LIRMM, INRIA, IRCCyN LASMEA and LaMI for the valuable scientific exchanges during the joint work 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, Professor 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 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.

xiii

List of abbreviations and notations

C – cylindrical joint C* – cylindrical joint with one or two idle mobilities CNRS – Centre National de la Recherche Scientifique (National Center for Scientific Research) DoF – degree-of-freedom eA and eG1 – link of G1-limb (e = 1,2,3,…,n) eB and eG2 – link of G2-limb (e = 1,2,3,…,n) eC and eG3 – link of G3-limb (e = 1,2,3,…,n) eD and eG4 – link of G4-limb (e = 1,2,3,…,n) EM – evolutionary morphology fi – degree of mobility of the ith joint F m G1–G2–…Gk general notation for the kinematic chain associated to a parallel mechanism with k simple and/or complex limbs Gi (i = 1, 2,…,k) FR TIMS – Fédération de Recherche Technologies de l’Information et de la Mobilité, de la Sûreté Gi (1Gi–2Gi–…nGi) the kinematic chain associated to the ith limb H – characteristic point of the distal link/end-effector IFMA – Institut Français de Mécanique Avancée (French Institute of Advanced Mechanics) IFToMM – International Federation for the Promotion of Mechanism and Machine Science INRIA – Institut National de Recherche en Informatique et en Automatique (The French National Institute for Research in Computer Science and Control) IRCCyN – Institut de Recherche en Communications et Cybernétique de Nantes I nun – n × n identity matrix J – Jacobian matrix k – total number of limbs in the parallel manipulator k1 – number of simple limbs in the parallel manipulator k2 – number of complex limbs in the parallel manipulator LaMI – Laboratoire de Mécanique et Ingénieries (Mechanical Engineering Research Group) xv

xvi

List of abbreviations and notations

LASMEA – Laboratoire des Sciences et Matériaux pour l’Electronique, et d’Automatique (Laboratory of Sciences and Materials for Electronic, and of Automatic) LIRMM – Laboratoire d’Informatique, de Robotique et de Microélectronique de Montpellier (Montpellier Laboratory of Computer Science, Robotics, and Microelectronics) m – total number of links including the fixed base MF – mobility of parallel mechanism F MGi – mobility of the kinematic chain associated with limb Gi NF – number of overconstraints in the parallel mechanism F n Ł nGi – moving platform in the parallel mechanism F m G1–G2–…Gk, O0x0y0z0 – reference frame p – total number of joints in the parallel mechanism pGi – number of joints in Gi-limb P – prismatic joint P – actuated prismatic joint P* – prismatic joint with idle mobility Pa – R||R||R||R-type planar parallelogram loop Pa – R||R||R||R-type parallelogram loop with an actuated revolute joint Pa* or Pacs – R||R||C-S-type parallelogram loop with three idle mobilities combined in a cylindrical and a spherical joint c Pa – R||R||R||C-type parallelogram loop with one idle mobility combined in a cylindrical joint Pacc – R||R||C||C-type parallelogram loop combining two cylindrical joints Paccs and Pascc – R||C||C-S-type parallelogram loop combining one spherical and two cylindrical joints s Pa – R||R||R-S-type parallelogram loop with two idle mobilities combined in a spherical joint Pass – R||R-S-S-type parallelogram loop with idle mobilities combined in two spherical joints adjacent to the same link Pat – R A P A ||R||R A P A ||R-type telescopic planar parallelogram loop Patcs – telescopic parallelogram loop with three idle mobilities combined in a cylindrical and a spherical pair u Pa – parallelogram loop with one idle mobility combined in a universal joint Pauu – parallelogram loop with two idle mobilities combined in two universal joints PM – parallel manipulator Pn2 – planar close loop with two degrees of mobility Pn2* or Pn2cs – close loop with two degrees of mobility and three idle mobilities combined in a cylindrical and a spherical pair Pn3 – planar close loop with three degrees of mobility

List of abbreviations and notations

xvii

Pn3* or Pn3cs – close loop with three degrees of mobility and three idle mobilities combined in a cylindrical and a spherical pair Pr – prism mechanism Prs – prism mechanism with one idle mobility combined in a spherical joint Prss – prism mechanism with two idle mobilities combined in two spherical joints q – number of independent closed loops in the parallel mechanism &q – joint velocity vector qi – finite displacement in the ith actuated joint rF – total number of joint parameters that lose their independence in the closed loops combined in parallel mechanism F rl – total number of joint parameters that lose their independence in the closed loops combined in the k limbs rGi – number of joint parameters that lost their independence in the closed loops combined in Gi-limb, R – revolute joint R – actuated revolute joint R* – revolute joint with idle mobility Rb – rhombus loop Rb* or Rbcs – planar rhombus loop with three idle mobilities combined in a cylindrical and a spherical joint RF – the vector space of relative velocities between the mobile and the reference platforms in the parallel mechanism F m G1–G2–…Gk, (RF) – the basis of vector space RF RGi – the vector space of relative velocities between the mobile and the reference platforms in the kinematic chain Gi disconnected from the parallel mechanism F m G1–G2–…Gk, (RGi) – the basis of vector space RGi S – spherical joint S* – spherical joint with idle mobilities SF – the connectivity between the mobile and the reference platforms in the parallel mechanism F m G1–G2–…Gk. SGi – the connectivity between the mobile and the reference platforms in the kinematic chain Gi disconnected from the parallel mechanism F m G1–G2–…Gk. TF – degree of structural redundancy of parallel mechanism F TPM – translational parallel manipulator U – universal joint U* – universal joint with an idle mobility v , v1 , v2 , v3 – translational velocity vectors x, y, z – coordinates of characteristic point H

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List of abbreviations and notations

&x,&y,z& – time derivatives of coordinates D , E ,G – rotation angles D& , E& ,G& – time derivatives of the rotation angles Ȧ , ȦD , ȦE , ȦG – angular velocity vectors 0 – fixed base of a kinematic chain/mechanism 1 Ł 1Gi – fixed platform in the parallel mechanism F m G1–G2–…Gk, 1Gi–2Gi–…–nGi – links of limb Gi 1A–2A–…–nA – links of limb G1 1B–2B–…–nB – links of limb G2 1C–2C–…–nC – links of limb G3 1D–2D–…–nD – links of limb G4 2 in the notation PPRRR*-2PPRR – the parallel mechanism has one limb of type PPRRR* and two limbs of type PPRR ' – planar star configuration of three coplanar and intersecting axes of the joints adjacent to the moving platform; C – planar star configuration of three coplanar axes of the joints adjacent to the moving platform; two joint axes are parallel and perpendicular on the third axis || – 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 A – perpendicular position of joint axes/directions; for example the notation P A Pa indicates the fact that the axes of revolute joints in the parallelogram loop are perpendicular to the direction of the prismatic joint || || A in the notation R A P A C – 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 || || A in the notation P A Pa A Pa – the revolute axes of the second parallelogram loop are perpendicular to the revolute axes of the first parallelogram loop and parallel to the direction of the actuated prismatic joint A A in the notation P A Pa A A Pa – the revolute axes of the second parallelogram loop are perpendicular to the revolute axes of the first parallelogram loop and also to the direction of the actuated prismatic joint A A in the notation Pass A R||R A A Pa – the revolute axes of parallelogram loop Pa are perpendicular to the axes of the parallel revolute joints RŒR and also to the axes of the revolute joints of parallelogram loop Pass

1 Introduction

This book represents Part 2 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 (Gogu 2008a) 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 of this work focuses on the structural solutions of translational parallel robotic manipulators (TPMs) with two and three degrees of mobility. 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 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 1 G. Gogu, Structural Synthesis of Parallel Robots: Part 2: Translational Topologies with Two and Three Degrees of Freedom, Solid Mechanics and Its Applications 159, 1–21. © Springer Science + Business Media B.V. 2009

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1 Introduction

studies related to robotics. 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. 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 redundantly-actuated parallel robots. The paradigm of parallel robots is the hexapod-type robot, which has six degrees of freedom, but recently, the machine industry has discovered the potential applications of lower-mobility parallel robots with just two, three, four or five 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 or uncoupled motions, 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 (Ionescu 2003). The main terms used in this book concerning kinematic pairs (joints), kinematic chains and robot kinematics are defined in Tables 1.1–1.3 in Part 1 of this work. They are completed by some complementary remarks, notations and symbols used in this book. IFToMM terminology (Ionescu 2003) defines a link as a mechanism element (component) carrying kinematic pairing elements and a joint is a

1.1 Terminology

3

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 “Ł” is used between the links to indicate that they are welded together in the same link. For example, the notation 1G Ł 0 is used to indicate that the first link 1G is the fixed base. A kinematic chain G is denoted by the sequence of its links. The notation G (1G Ł 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, 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

4

1 Introduction

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 ternary 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 extreme elements are the mobile 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 (Ionescu 2003) 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 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 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. (DudiĠӽ et al. 2001a, b). 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 that is not used by the mechanism and does not influence a mechanism’s

1.1 Terminology

5

mobility in the hypothesis of perfect manufacturing and assembling precisions. In theoretical conditions, when no errors exist with respect to parallel and perpendicular positions of joint axes, motion amplitude in an idle mobility is zero. Real life manufacturing and assembling processes introduce errors in the relative positions of the joint axes and, in this case, the idle mobilities become effective mobilities usually with small amplitudes, depending on the precision of the parallel robot. 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 1.2 gives an example of the various representations of a GoughStewart type parallel robot largely used today in industrial applications. The physical implementation in Fig. 1.2a is a photograph of the parallel robot built by Deltalab (http://www.deltalab.fr/). In a CAD model (Fig. 1.2b) the links and the joints are represented as being as close as possible to the physical implementation (Fig. 1.2a). In a structural diagram (Fig. 1.2c) they are represented by simplified symbols, such as those introduced in Fig. 1.1, respecting the geometric relations defined by the relative positions of joint axes. A structural graph (Fig. 1.2d) is a network of vertices or nodes connected by edges or arcs with no geometric relations. The links are noted in the nodes and the joints on the edges. We can see that the Gough–Stewart type parallel robot has six identical limbs denoted in Fig. 1.2c by A, B, C, D, E and F. The final link is the mobile platform 4 Ł 4A Ł 4B Ł 4C Ł 4D Ł 4E Ł 4F and the reference member is the fixed platform 1A Ł 1B Ł 1C Ł 1D Ł 1E Ł 1F Ł 0. Each limb is connected to both platforms by spherical pairs.

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

6

1 Introduction

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

1.2 Methodology of structural synthesis

7

A prismatic pair is actuated in each limb. The spherical pairs are not actuated and are called passive pairs. The two platforms are polinary links, the other two links of each limb are binary links. The mechanism associated with the Gough–Stewart type parallel robot is a complex mechanism with a multi-loop associated graph (Fig. 1.2d). The simple open kinematic chain associated with A-limb is denoted by A (1A Ł 0–2A–3A–4A Ł 4) – Fig. 1.2e and its associated graph is tree-type (Fig. 1.2f). We consider the general case of a robot in which the end-effector is connected to the reference link by k 1 kinematic chains. The end-effector is a 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 m 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 Ł nGi and the reference link 1 Ł 1Gi. If the reference link is the fixed base, it is denoted by 1 Ł 1Gi Ł 0. The total number of robot joints is denoted by p. A serial robot F m G1 is a robot in which the end-effector n Ł nG1 is connected to the reference link 1 Ł 1G1 by just one simple open kinematic chain Gi (1Gi–2Gi–…nGi) called a serial kinematic chain. A parallel robot F m G1–G2–…Gk is a robot in which the end-effector n Ł nGi is connected in parallel to the reference link 1 Ł 1Gi by k 2 kinematic chains Gi (1Gi–2Gi–…nGi) called limbs or legs. A hybrid serial-parallel robot F m G1 is a robot in which end-effector n Ł nG1 is connected to reference link 1 Ł 1G1 by just one complex kinematic chain Gi (1Gi–2Gi–…nGi) called complex limb or complex leg. A fully-parallel robot F m G1–G2–…Gk is a parallel robot in which the number of limbs is equal to the robot mobility (k = M 2), and just one actuator exists in each limb.

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,

8

1 Introduction

motion platforms, machine tools, medical applications, planetary and underwater exploration. Therefore, the need for methodologies devoted to the systematic design of highly performing parallel robots is continually increasing. Structural synthesis is directly related to the conceptual phase of robot design, and represents one of the highly challenging subjects in recent robotics research. One of the most important activities in the invention and the design of parallel robots is to propose the most suitable solutions to increase the performance characteristics. The challenging and difficult objective of structural synthesis is to find a method to set up the mechanical architecture to achieve 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 have been proposed in Part 1 (Gogu 2008a). The approach integrates the new formulae for mobility, connectivity, redundancy and overconstraint of parallel manipulators (Gogu 2005d, e) and a new method of systematic innovation (Gogu 2005a). 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 (Gogu 2008a). 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

1.2 Methodology of structural synthesis

9

the last few years. Mobility calculation still remains a central subject in the theory of mechanisms. In Part 1 (Gogu 2008a) we have shown that the various methods proposed in the literature for mobility calculation of the closed loop mechanisms fall into two basic categories: (a) Approaches for mobility calculation based on setting up the kinematic constraint equations and calculating their rank for a given position of the mechanism with specific joint locations. (b) Formulae for a quick calculation of mobility without the need to develop the set of constraint equations. The approaches used for mobility calculation based on setting up the kinematic constraint equations and their rank calculation are valid without exception. The major drawback of these approaches is that the mobility cannot be determined quickly without setting up the kinematic model of the mechanism. Usually this model is expressed by the closure equations that must be analyzed for dependency. 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, fullcycle mobility). The general mobility represents the minimum value of the instantaneous mobility. For a given mechanism, general mobility has a unique value. It is a global parameter characterizing the mechanism in all its configurations except its singular ones. Instantaneous mobility is a local parameter characterizing the mechanism in a given configuration including singular ones. In a singular configuration the instantaneous mobility could be different from the general mobility. In this book, unless otherwise stated, general mobility is simply called mobility. Note 1. In a kinematotropic mechanism with branching singularities, fullcycle mobility is associated with each branch. 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 branch. As each branch has its own mobility, a single value for global mobility cannot be associated with the kinematotropic mechanisms. The term kinematotropic was coined by K. Wohlhart (1996) to define the linkages that permanently change their global mobility when passing by a singularity in which a certain transitory infinitesimal mobility is attained. Various

10

1 Introduction

kinematotropic parallel mechanisms have been recently presented (Fanghela et al. 2006; Gogu 2008b, c). 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-GrüblerKutzbach (CGK) formula in its original or extended forms. These formulae have been critically reviewed (Gogu 2005b) and a criterion governing mechanisms to which this formula can be applied has been set up in (Gogu 2005c). 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 (Gogu 2005d) 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 m G1–G2–…Gk in which the mobile platform n Ł nGi is connected to the reference platform 1 Ł 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 m 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 Ł nGi and 1 Ł 1Gi, in the parallel mechanism F m G1– G2–…Gk, whose basis is

1.2 Methodology of structural synthesis

(RF)=( RG1 RG 2 ... RGk ),

11

(1.1)

SG – 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 Ł nGi and 1 Ł 1Gi, in the parallel mechanism F m 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 Ł 1Gi Ł 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 Chapter 2 – Part 1 (Gogu 2008a) for mobility MF, connectivity SF, number of overconstraints NF and redundancy TF of the parallel mechanism F m G1–G2–…Gk are used in structural synthesis of parallel robotic manipulators: p

¦f

MF

rF ,

i

(1.2)

i 1

NF=6q-rF ,

(1.3)

TF=MF-SF ,

(1.4)

where dim( RGi ) ,

(1.5)

dim( RF ) dim( RG1 RG 2 ... RGk ) ,

(1.6)

SGi

SF

k

rF

¦S

Gi

S F rl ,

(1.7)

i 1

k

p

¦p

,

Gi

(1.8)

i 1

q=p-m+1,

(1.9)

and k

rl

¦r

Gi

i 1

.

(1.10)

12

1 Introduction

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 limbs of mechanism F. In Eqs. (1.5) and (1.6), 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). Eq. (1.8) indicates that the limbs of the parallel mechanism F m 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. In Chapter 5 – Part 1 the following structural conditions have been established: (a) For the non redundant parallel robots (TF = 0) SF=MFMGi

(i=1,…,k),

(1.11)

MGi=SGi6

(i=1,…,k),

(1.12)

(b) For the redundant parallel robots with TF > 0 SF<MFMGi

(i=1,…,k),

(1.13)

MGi>SGi6

(i=1,…,k),

(1.14)

(c) For the non overconstrained parallel robots (NF = 0) p

MF= ¦ f i 6q ,

(1.15)

i 1

(d) For the overconstrained parallel robots with NF > 0 p

MF> ¦ f i 6q .

(1.16)

i 1

We recall that pGi

M Gi

¦f i 1

i

rGi .

(1.17)

1.2 Methodology of structural synthesis

13

We note that the intersection in Eqs. (1.1) and (1.6) 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 Ł nGi in the mechanism F m 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 m G1–G2–…Gk must be valid for any point of the moving platform n Ł nGi. Note 2. When there are various ways to choose the bases of the vector spaces RGi in Eqs. (1.1) and (1.6), the bases (RGi) are selected such that the minimum value of S F is obtained by Eq. (1.6). By this choice, the result of Eq. (1.2) fits in with the definition of general mobility as the minimum value of the instantaneous mobility. The parameters used in the new formulae (1.1)–(1.17) can be easily obtained by inspection without calculating the rank of the homogeneous linear set of constraint equations associated with loop closure or without calculating 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. These formulae have been successfully applied in Part 1 to structural analysis of various mechanisms including so called “paradoxical” mechanisms. These formulae are useful for the structural synthesis of various types of parallel mechanisms with 2 MF 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 (Gogu 2005a). 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 overconstrained and redundancy and the level of motion coupling. The protoelements are the revolute and prismatic joints. The morphological

14

1 Introduction

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 Chapter 5 – Part 1. 1.2.3 Types of parallel robots with respect to motion coupling Various levels of motion coupling have been introduced in Chapter 4 – Part 1 in relation with the Jacobian matrix of the robotic manipulator which is the matrix mapping (i) the actuated joint velocity space and the end-effector velocity space, and (ii) 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: (I) maximally regular PMs, if the Jacobian J is an identity matrix throughout the entire workspace, (ii) fully-isotropic PMs, if 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. The term maximally regular parallel robot was recently coined by Merlet (2006) 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

1.3 Translational parallel robots

15

number of the Jacobian matrix was first used by Salisbury and Craig (1982) to design mechanical fingers and developed by Angeles (1997) as a kinetostatic performance index of the robotic mechanical systems. The isotropic design aims at ideal kinematic and dynamic performance of the manipulator (Fattah and Ghasemi 2002). 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 (Zanganeh and Angeles 1997; Tsai and Huang 2003). 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.

1.3 Translational parallel robots This book focuses on the structural synthesis of translational parallel robotic manipulators (TPMs) with two and three degrees of freedom. In such a robot, the moving platform can undergo two or three independent translational motions and its orientation remains unchanged. The translational parallel robots with two degrees of freedom are also called T2-type parallel robots and give two translational velocities v1 and v2 in the basis of the operational velocity vector space (RF) = (v1,v2) along with a constant orientation of the moving platform. We consider the xyplane as the plane of motion of the moving platform (v1 = vx and v2 = vy). T2-type parallel robots have mobility MF = 2 and the connectivity between the moving and fixed platforms is SF = 2. These kinds of parallel robots are useful in pick-and-place operations when the end-effector only needs to undergo purely translational motion in one plane. Pick and place parallel robot mechanisms are typically used in light industries such as the electronics industry and packaging industries. They have to repeat accurately a simple transfer operation many times over with relatively high speed in two degrees of freedom planar motion without altering the orientation of the moving platform (Vainstock 1990; Hesselbach and Frindt 1999; Dagefoerde et al. 2001; Bergmeyer 2002; Huang et al. 2003, 2004, 2005, 2006).

16

1 Introduction

The direct kinematic model of the T2-type parallel robots becomes ª v1 º «v » ¬ 2¼

ª&q º

> J @2u2 «&q1 » ¬ 2¼

(1.18)

where: v1 = vx = &x and v2 = vy = &y are the translational velocities of the characteristic point H of the moving platform, &q1 and &q2 are the velocities of the actuated joints, J 2u2 is the Jacobian matrix. As presented in Chapter 4 – Part 1, for translational parallel robots, the design and conventional Jacobians have the same expression and they are simply called Jacobians or Jacobian matrices. To obtain a non redundant T2-type translational parallel robot, a basic limb presented in Figs. 6.1 and 6.2 – Part 1 is associated with at least one simple or complex limb with 2 d MGi = SGi d 6 that integrates velocities v1 and v2 in the basis of its operational space. In this way, a large set of solutions can be obtained. We recall that the basic legs in Figs. 6.1 and 6.2 – Part 1 give rise to two independent translations along with a constant orientation of the moving platform. The translational parallel robotic manipulators with three degrees of mobility are also called T3-type parallel robots and give three translational velocities v1, v2 and v3 in the basis of the operational velocity vector space (RF) = (v1,v2,v3) along with a constant orientation of the moving platform. T3-type parallel robots have mobility MF = 3 and the connectivity between the moving and fixed platforms is SF = 3. This kind of parallel robots are useful in pick-and-place operations when the end-effector only needs to undergo purely translational motion in space. 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 without altering the orientation of the moving platform. The direct kinematic model of the T3-type parallel robots becomes ª v1 º «v » « 2» «¬v3 »¼

ª&q1 º > J @3u3 ««&q2 »» «¬&q3 »¼

(1.19)

1.3 Translational parallel robots

17

where: v1 = vx = &x , v2 = vy = &y and v3 = vz = &z are the translational velocities of the characteristic point H of the moving platform, &q1 , &q2 and &q3 are the velocities of the actuated joints, J 3u3 is the Jacobian matrix. To obtain a non redundant T3-type translational parallel robot of type F m G1–G2–G3, a basic limb presented in Figs. 7.1–7.11 – Part 1 is associated with other two simple or complex limbs with 3 d MGi = SGi d 6 that integrate velocities v1, v2 and v3 in the basis of their operational velocity spaces. We recall that the basic limbs in Figs. 7.1–7.11 – Part 1 give rise to three independent translations along with a constant orientation of the moving platform. In this way, a large set of solutions with coupled, decoupled, uncoupled motions along with maximally regular solutions can be obtained by using three simple or complex limbs with 3 d MGi = SGi d 6 that respect the condition (RF) = ( RG1 RG 2 RG 3 ) = (v1,v2,v3). Translational parallel robots are largely used in classical manipulating processes, jiggle mechanisms (Li et al. 2005) and micro and nanomanipulation (Jensen et al. 2006; Xu and Li 2006) or MEMS fabrication (Bamberger et al. 2007). The various architectures of TPM 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 Table 1.1. Examples of implemented translational parallel robotic manipulators No. Robot name

Type

References

1

DELTA

Clavel (1988, 1990, 1991)

2

University of Maryland manipulator NUWAR Urane Sx Orthoglide

3-RRPaR, 3-RUU, 3-RPass, 3-PRPaR 3-RRPaR 3-RPass 3-PUU 3-PRPaR

Miller (1999, 2001) Company and Pierrot (2002) Wenger and Chablat (2000) Chablat and Wenger (2002) Kim and Tsai (2002, 2003)

3 4 5 6

Cartesian Parallel Manipulator

3-PRRR

7

Tripteron

3-CRR, 3-PRRR

8

Isoglide3-T3

3-PRRR

Tsai and Stamper (1996)

Gosselin and Kong (2002) Kong and Gosselin (2002b, c) Gosselin et al. (2004) Gogu (2002)

18

1 Introduction

Table 1.2. Literature dedicated to the study of the parallel robots of DELTA topology No. Type of study

References

1

Dimensional synthesis and optimization

2 3 4

Dynamic modelling Isotropic conditions Kinematic analysis

5 6 7 8

Kinematic calibration Kinematic and dynamic modelling Modelling and control Singularities

9

Stiffness

10

Workspace

Company and Pierrot (2002) Bruzzone et al. (2002) Kosinska et al. (2003) Stock and Miller (2003) Chablat et al. (2004a, b) Johannesson et al. (2004) Yoon et al. (2004) Lou and Li (2006) Laribi et al. (2007) Stamper and Tsai (1998) Baron et al. (2002) Company and Pierrot (2002) Bruzzone et al. (2002) Yoon et al. (2004) Lee et al. (2005) Laribi et al. (2007) Lou and Li (2006) Vischer and Clavel (1998) Pierrot et al. (1990, 1991) Pierrot et al. (1991) Di Gregorio (2004) Liu et al. (2003) Liu et al. (2003) Yoon et al. (2004) Miller (1999, 2002) Di Gregorio and Zanforlin (2003) Liu et al. (2003) Chablat et al. (2004a, b)

parallelogram loop Pa which can be considered as a complex pair of circular translation (Huang and Li 2003; Liu and Wang 2003; Hervé 2004; Liu et al. 2004). Examples of implemented translational parallel robotic manipulators are presented Table 1.1. As a matter of fact, some architectures of TPMs are quite popular already, for instance the DELTA robots of types 3-RRPaR, 3-RUU, 3-PRPaR proposed by Clavel (1988, 1990). Much literature is dedicated to the study of the parallel robots of DELTA topologies (Table 1.2). and industrial implementation of these solutions has already been reached by Demareux, ABB, Hitachi, Mikron, Renault-Automation, Comau and other.

1.3 Translational parallel robots

19

Various solutions are derived from this topology by integrating parallelogram loops in different configurations: 3-RHPaR Y-star, the 3-RPPR PrismRobot, the 2-RHPaR + 1PRPaR H-robot (Hervé and Sparacino 1991, 1992, 1993; Hervé 1995), 3-RRPaR University of Maryland manipulator (Tsai and Stamper 1996), 3-PUU Urane Sx (Company and Pierrot 2002). A special case of 3-PRPaR-type is Orthoglide having the linear actuators on three orthogonal directions. (Wenger and Chablat 2000; Chablat and Wenger 2002). The literature dedicated to the study of this parallel robot is presented in Table 1.3. Various other architectures have been proposed to achieve three pure translational motions of the platform by using limbs with three, four and five degrees of freedom (Hervé and Sparacino 1991, 1992, 1993; Tsai 1998; Frisoli et al. 2000; Tsai and Joshi 2000; Zhao and Huang 2000; Di Gregorio 2001; Carricato and Parenti-Castelli 2001, 2003a, 2004b; Huang and Li 2002a, b, 2003; Gao et al. 2002; Zlatanov and Gosselin 2004; Kong and Gosselin 2004a, b, 2007; Li et al. 2005; Alizade et al. 2007; Lee and Hervé 2006.). Many studies have been dedicated to these various architectures of translational parallel robots (Table 1.4). The first solutions of maximally regular and implicitly fully-isotropic T3-type translational parallel robots 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 of Advanced Mechanics (IFMA). In 2002, the four groups published the first results of their works (Carricato and Parenti-Castelli 2002; Kim and Tsai 2002; Gosselin and Kong 2002; Kong and Gosselin 2002a, b, c; Gogu 2002). Each of the last Table 1.3. Literature dedicated to the study of the Orthoglide parallel robot No. Type of study

References

1 2

Chablat and Wenger (2002, 2003) Chablat et al. (2004a, b)

3

Architecture optimization Dextrous workspace and design parameters Design and geometric synthesis

4 5 6 7

Dynamic modelling Kinematics and workspace Sensitivity analysis Stiffness analysis

Majou et al. (2002a, b) Pashkevich et al. (2005) Guégan an Khalil (2002) Pashkevich et al. (2006) Caro et al. (2006) Majou et al. (2005)

20

1 Introduction

Table 1.4. Literature dedicated to the study of translational parallel manipulators No. 1 2 3 4

Type of study Calibration Dynamic performances Dynamic balancing Dimensional synthesis

5

Kinematic analysis

6 7

Kinetostatic indices Mobility analysis

8

Position accuracy

9 10

Optimal design and modelling Workspace analysis and optimization

11

Singularity analysis

References Bleicher and Günther (2004) Di Gregorio and Parenti-Castelli (2004) Wu and Gosselin (2005) Callegari and Marzetti (2003) Wolf and Shoham (2006) Joshi and Tsai (2002) Tsai and Joshi (2002) Carricato and Parenti-Castelli (2003b, c) Ji and Wu (2003) Kim and Chung (2003) Li et al. (2004b, 2005) Shen et al. (2005) Zeng et al. (2006) Zhao et al. (2006) Gogu (2007) Li and Huang (2004) Rico et al. (2005) Han et al. (2002) Frisoli et al. (2007) Xu and Li (2007) Miller (2004) Badescu et al. (2002) Zhao et al. (2008) Tsai and Joshi (2002) Wolf and Shohan (2003) Liu et al. (2003) Zhao et al. (2005) Di Gregorio and Parenti-Castelli (2002 Li et al. (2004) Callegari and Tarantini (2003)

three groups has built a prototype of this robot in their research laboratories and has called this robot CPM (Kim and Tsai 2002), Orthogonal Tripteron (Gosselin et al. 2004) or Isoglide3-T3 (Gogu 2004a). The first practical implementation of this robot was the CPM developed at University of California by Kim and Tsai (2002). The various methods used in structural synthesis of TPM are systematized in Table 1.5.

1.3 Translational parallel robots

21

Table 1.5. Approaches used in the structural synthesis of translational parallel manipulators No 1

Approach Additional passive limb

2 3 4 5 6

Algebraic methods CAD functionalities Constraint method Evolutionary morphology Group theory

7

Mobility formulae

8 9

Plücker coordinates Screw theory

10 11

Structural parameters Theory of linear Gogu (2002, 2004a, b, 2008a) transformations Units for single-opened-chains Jin and Yang (2004) Velocity-loop equations Di Gregorio and Parenti-Castelli (1998); Di Gregorio (2002), Carricato and Parenti-Castelli (2001, 2002, 2003a, b, c, 2004b)

11 12

References Brogårdh (2002) Hess-Coelho (2007) Danescu (1995) Lu (2004) Huang and Li (2002a, b, 2003) Gogu (2002, 2004a, b, 2008a) Hervé (1995, 2004) Hervé and Sparacino (1991, 1992, 1993); Lee and Hervé (2006); Rico et al. (2006). Kong and Gosselin (2007) Alizade and Bayram (2004); Alizade et al. (2007) Tsai (1998, 1999) Yu et al. (2006) Gao et al. (2002, 2005) Tsai (1999); Frisoli et al. (2000); Kong and Gosselin (2001, 2002a, b, c, 2004a, b), Huang and Li (2002a, b, 2003); Li and Huang (2004) Fang and Tsai (2004) Gogu (2002, 2004a, b, 2008a)

2 Translational parallel robots with two degrees of freedom

The translational parallel robots with two degrees of freedom can be actuated by linear and/or rotating actuators. Topologies with coupled, decoupled and uncoupled motions along with maximally regular solutions are presented in this section. They give rise to two independent translations along with a constant orientation of the moving platform.

2.1 T2-type translational parallel robots with coupled motions T2-type translational parallel robots (TPMs) with coupled motions and linear or rotating actuators with various degrees of overconstraint can be obtained by using the basic limb topologies presented in Figs. 6.1b and 6.2b, c – Part 1. In these solutions the two operational velocities depend on & 2 ) and v2 v2 ( &q1 ,q & 2 ) . We the two actuated joint velocities: v1 v1 ( &q1 ,q consider v1 = vx and v2 = vy. 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 is more compact. 2.1.1 Overconstrained solutions Equation (1.16) indicates that overconstrained solutions of T2-type translational parallel robots with coupled motions and q independent loops p meet the condition ¦ 1 f i 2 6q . Various solutions fulfil this condition along with MF = SF = 2 and (RF) = (vx,vy). They may have identical limbs or limbs with different structures and may be actuated by linear or rotating motors. The simplest solutions have just two limbs. Solutions with an additional unactuated limb are also presented in this section.

23 G. Gogu, Structural Synthesis of Parallel Robots: Part 2: Translational Topologies with Two and Three Degrees of Freedom, Solid Mechanics and Its Applications 159, 23–106. © Springer Science + Business Media B.V. 2009

24

2 Translational parallel robots with two degrees of freedom

Solutions with linear actuators

In the basic solutions with linear actuators and coupled motions F m G1– G2, the moving platform n Ł nGi (i = 1 and 2) is connected to the reference platform 1 Ł 1Gi Ł 0 by two limbs of type P A Pa (Fig. 6.1b – Part 1). One prismatic and four revolute joints exist in each limb (pGi = 5). This topology is denoted by 2PPa-type and has the two linear actuators situated on the fixed base. They may have any direction parallel to the plane of motion of the moving platform. The axes of the revolute joints in the parallelogram loop are perpendicular on the plane of motion of the moving platform. The example in Fig. 2.1 has the moving platform 5 and the following arrangement of joints: Px A Paz in G1-limb and Py A Paz in G2-limb. Indices x, y and z, associated with the joint symbol denote the direction of the joint axis. Two consecutive joints with the same index have parallel axes. Two consecutive joints with different indices have perpendicular axes. The index of the planar parallelogram closed loop indicates the direction of the axes of revolute pairs. The actuated joint of each limb Gi (i = 1,2) is underlined. To simplify link notation eAi (e = 1,…,n and i = 1,2) by avoiding the double index in Fig. 2.1 and the following figures we have denoted by eA the links of G1-limb (eA { eG1) and by eB the links of G2-limb (eB { eG2).

Fig. 2.1. 2PPa-type translational parallel mechanism with ten overconstraints, limb topology P A Pa

2.1 T2-type translational parallel robots with coupled motions

25

Fig. 2.2. Translational parallel mechanisms of types PR*Pa-PR*C*Pa with six overconstraints (a) and 2PPa* with four overconstraints (b), limb topologies P||R* A Pa and P||R* A C*||Pa (a) and P A Pa* (b)

26

2 Translational parallel robots with two degrees of freedom

Fig. 2.3. PPass-PR*Pass-type translational parallel mechanism with one overconstraint, limb topologies P A Pass and P A R*||Pass

The vector spaces of relative velocities between the moving and the reference platforms in the kinematic chains G i (i = 1 and 2) disconnected from mechanism F have the same basis (RG1) = (RG2) = (vx,vy). This is also the basis of the vector space of relative velocities between the moving and the reference platforms in mechanism F, given by Eq. (1.1), that is (RF) = (vx,vy). This solution with two prismatic and eight revolute joints has p = 10 10 and ¦ i 1 f i = 10. The structural diagram in Fig. 2.1b and Eq. (1.9) indicate that the mechanism has three independent closed loops (q = 3). Three joint parameters lost their independence in each closed planar parallelogram loop combined in a complex limb ( rlGi = 3). For the solution in Fig. 2.1, Eqs. (1.2)–(1.8) and (1.17) give the following structural parameters: MGi = SGi = SF = 2, rF = 8, MF = 2, NF = 10 and TF = 0 (see Table 2.1). A large diversity of overconstrained solutions with coupled motions and linear motors combined in the two complex limbs with parallelogram loops can be derived from the solution in Fig. 2.1. They have 1 NF < 10

2.1 T2-type translational parallel robots with coupled motions

27

and could integrate up to three idle mobilities in each parallelogram loop and up to four idle mobilities outside the parallelogram loops. Structural solutions using parallelogram loops with up to three idle mobilities are presented in Part 1 (Fig. 6.3b–h). The idle mobilities integrated outside the parallelogram loops are three orthogonal rotations and one translation perpendicular to the motion plane of the moving platform. For example, four idle mobilities are introduced outside the parallelogram loop in Fig. 2.2a and six idle mobilities are introduced in the parallelogram loops in Fig. 2.2b. In the example in Fig. 2.3, three idle mobilities are introduced outside the parallelogram loops and six in the parallelogram loops. The structural parameters of the solutions illustrated in Figs. 2.1–2.3 are presented in Table 2.1. We recall that C denotes a cylindrical pair and S a spherical pair. The notations Pa* or Pacs are associated with a parallelogram loop with three idle mobilities combined in a cylindrical and a spherical pair, and Pass with four idle mobilities combined in two spherical joints adjacent to the moving platform. We note that in the parallelogram loop Pass-type, three idle mobilities are introduced in the loop and one outside the loop. The number of overconstraints can be further reduced by replacing one limb by a simple open kinematic chain with 2 < MGi = SGi < 6 that integrates velocities vx and vy in the basis of its operational space. In this way, a large diversity of overconstrained translational parallel robots with linear actuators and 0 < NF d 6 may be set up. They have just two independent loops (q = 2). For example, the solution in Fig. 2.4 is obtained by replacing G2-limb in Fig. 2.1 by a planar kinematic chain PRR-type (Py A Rz||Rz). For the solution in Fig. 2.4, Eqs. (1.2)–(1.8) and (1.17) give the following structural parameters: MG1 = SG1 = 2, MG2 = SG2 = 3, (RG1) = (vx,vy), (RG2) = ( v x , v y , ȦG ), (RF) = (vx,vy), SF = 2, rF = 6, MF = 2, NF = 6 and TF = 0 (see Table 2.2). Overconstrained solutions with 0 < NF < 6 can be derived from the solution in Fig. 2.4 by introducing up to three idle mobilities in the parallelogram loop and up to three idle mobilities outside the parallelogram loop. The idle mobilities integrated inside and outside the parallelogram loop are two orthogonal rotations and one translation perpendicular to the motion plane of the moving platform. For example, in Fig. 2.5 two idle mobilities are introduced outside the parallelogram loops and three inside the parallelogram loop. Three idle mobilities are introduced outside the parallelogram loop in Fig. 2.6a and three idle mobilities are introduced in the parallelogram loop in Fig. 2.6b.

28

2 Translational parallel robots with two degrees of freedom

The structural parameters of the solutions illustrated in Figs. 2.4–2.6 are presented in Table 2.2. A solution with linear actuators non adjacent to the fixed base can be derived from the solution in Fig. 2.4 by using, in G2-limb, a planar kinematic chain RPR-type (see Fig. 2.7a). This solution also has six overconstraints. Overconstrained solutions with 0 < NF < 6 can be derived from the solution in Fig. 2.7a by introducing up to three idle mobilities outside the parallelogram loop (Fig. 2.7b) and up to three idle mobilities inside the parallelogram loop (Fig. 2.8a) The example in Fig. 2.8b has one overconstraint with two idle mobilities outside the parallelogram loop and three idle mobilities in the parallelogram loop. The solutions in Figs. 2.7 and 2.8 have the same structural parameters as their counterparts with linear motors mounted on the fixed base in Figs. 2.4–2.6 (see Table 2.2). Solutions of type F m G1–G2–G3 with an additional unactuated limb can also be derived by using two simple actuated limbs with 2 < MGi = SGi < 6 and an unactuated limb P A P-type. The directions of the two prismatic joints are parallel to the motion plane x–y of the moving platform. The actuated limbs G1 and G2 must integrate velocities vx and vy in the basis of their operational space. The three limbs form q = 2 independent loops and must meet the conditions: MF = SF = 2 and (RF) = (RG1 RG2 RG2) = (vx,vy). Equation (1.16) indicates that the overconstrained solutions of these T2-type translational parallel robots with coupled motions and an additional p unactuated limb have ¦ 1 f i 14 . Various solutions with identical or different limb architectures exist. The linear actuators can be mounted on the fixed base or on a moving link. For example, the solutions in Fig. 2.9a have two actuated planar limbs of type PRR and that in Fig. 2.10a two actuated planar limbs of type RPR. The axes of the revolute joints are parallel to the z-axis. The solutions in Figs. 2.9a and 2.10a have six overconstraints and that in Figs. 2.9b and 2.10b two overconstraints. The structural parameters of these solutions are presented in Table 2.3.

2.1 T2-type translational parallel robots with coupled motions

29

Table 2.1. Structural parameters of translational parallel mechanisms in Figs. 2.1–2.3 No. Structural parameter 1 2 3 4 5 6 7 8 9

m p1 p2 p q k1 k2 K (RG1)

Solution 2PPa Fig. 2.1 8 5 5 10 3 0 2 2 ( vx , v y )

10

(RG2)

( vx , v y )

( v x , v y , vz , ȦE , ȦG ) ( v x , v y )

( v x , v y , ȦD , ȦG )

11 12 13 14 15 16 17

SG1 SG2 rG1 rG2 MG1 MG2 (RF)

2 2 3 3 2 2 ( vx , v y )

3 5 3 3 3 5 ( vx , v y )

2 2 6 6 2 2 ( vx , v y )

3 4 6 6 3 4 ( vx , v y )

18 19 20 21 22 23 24

SF rl rF MF NF TF

2 6 8 2 10 0 5

2 6 12 2 6 0 6

2 12 14 2 4 0 8

2 12 17 2 1 0 9

fj

5

8

8

10

fj

10

14

16

19

25 26

¦ ¦ ¦

p1 j 1 p2 j 1 p j 1

fj

PR*Pa-PR*C*Pa Fig. 2.2a 11 6 7 13 3 0 2 2 ( v x , v y , ȦD )

2PPa* Fig. 2.2b 8 5 5 10 3 0 2 2 ( vx , v y )

PPass-PR*Pass Fig. 2.3 9 5 6 11 3 0 2 2 ( v x , v y , ȦE )

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

30

2 Translational parallel robots with two degrees of freedom

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 overconstraints in the parallel mechanismk, TF degree of structural redundancy of the parallel mechanisml, fj mobility of jth joint. a

p=

¦

k

pGi ,

i 1

b

q = p-m+1, k = k1+k2, d SGi = dim(RGi) , i = 1,2,...,k, c

e

MGi =

pG i

¦

j 1

f j rGi , i = 1,2,...,k,

f

(RF) = (RG1) ... (RGk), SF = dim(RF) ,

g h i

rF

j

MF

k l

rl =

k

¦ r , ¦ S S ¦ f r i 1 Gi k

i 1

Gi

F

rl ,

F

,

p

i 1

i

N F 6q rF , TF M F S F .

2.1 T2-type translational parallel robots with coupled motions

31

Table 2.2. Structural parametersa of translational parallel mechanisms in Figs. 2.4–2.8

No. Structural parameter

1 2 3 4 5 6 7 8 9

m p1 p2 p q k1 k2 k (RG1)

Solution PPa-PRR (Fig. 2.4), PPa-RPR (Fig. 2.7a) 7 5 3 8 2 1 1 2 ( vx , v y )

10

(RG2)

( v x , v y , ȦG ) ( v x , v y , vz , ȦD , ȦE , ȦG ) ( v x , v y , ȦG ) ( v x , v y , vz , ȦG )

11 12 13 14 15 16 17

SG1 SG2 rG1 rG2 MG1 MG2 (RF)

2 3 3 0 2 3 ( vx , v y )

2 6 3 0 2 6 ( vx , v y )

2 3 6 0 2 3 ( vx , v y )

3 4 6 0 3 4 ( vx , v y )

18 19 20 21 22 23 24

SF rl rF MF NF TF

2 3 6 2 6 0 5

2 3 9 2 3 0 5

2 6 9 2 3 0 8

2 6 11 2 1 0 9

fj

3

6

3

4

fj

8

11

11

13

25 26 a

¦ ¦ ¦

p1 j 1 p2 j 1 p j 1

fj

PPa-PC*S* (Fig. 2.6a), PPa-C*PS* (Fig. 2.7b) 7 5 3 8 2 1 1 2 ( vx , v y )

PPa*-PRR (Fig. 2.6b), PPa*-RPR (Fig. 2.8a), 7 5 3 8 2 1 1 2 ( vx , v y )

PPass-PRC* (Fig. 2.5) PPass-RPC* (Fig. 2.8b) 7 5 3 8 2 1 1 2 ( v x , v y , ȦE )

See footnote of Table 2.1 for the nomenclature of structural parameters

32

2 Translational parallel robots with two degrees of freedom

Table 2.3. Structural parametersa of translational parallel mechanisms in Figs. 2.9–2.10

No. Structural parameter 1 2 3 4 5 6 7 8 9 10

m p1 p2 p3 p q k1 k2 k (RG1)

Solution 2PRR-PP (Fig. 2.9a) 2RPR-PP (Fig. 2.10a) 7 3 3 2 8 2 3 0 3 ( v x , v y , ȦG )

11

(RG2)

( v x , v y , ȦG )

( v x , v y , ȦD , ȦE , ȦG )

12

(RG3)

( vx , v y )

( vx , v y )

13 14 15 16 17 18 19 20 21 22

SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

3 3 2 0 0 0 3 3 2 ( vx , v y )

5 5 2 0 0 0 5 5 2 ( vx , v y )

23 24 25 26 27 28 29

SF rl rF MF NF TF

2 0 6 2 6 0 3

2 0 10 2 2 0 5

fj

3

5

fj

2

2

fj

8

12

30 31 32 a

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

2PRS*-PP (Fig. 2.9b) 2RPS*-PP (Fig. 2.10b) 7 3 3 2 8 2 3 0 3 ( v x , v y , ȦD , ȦE , ȦG )

See footnote of Table 2.1 for the nomenclature of structural parameters

2.1 T2-type translational parallel robots with coupled motions

33

Fig. 2.4. PPa-PRR-type translational parallel mechanism with six overconstraints and linear motors on the fixed base, limb topologies P A Pa and P A R||R

Fig. 2.5. PPass-PRC*-type translational parallel mechanism with one overconstraint and linear motors on the fixed base, limb topologies P A Pass and P A R||C*

34

2 Translational parallel robots with two degrees of freedom

Fig. 2.6. Translational parallel mechanisms with three overconstraints and linear motors on the fixed base of types PPa-PC*S* (a) and PPa*-PRR (b), limb topologies P A Pa and P A C*S* (a), P A Pa* and P A R||R (b)

2.1 T2-type translational parallel robots with coupled motions

35

Fig. 2.7. Translational parallel mechanisms of types: PPa-RPR with six overconstraints (a) and PPa-C*PS* with three overconstraints (b), limb topologies P A Pa and R A P A ||R (a), P A Pa and C* A PS* (b)

36

2 Translational parallel robots with two degrees of freedom

Fig. 2.8. Translational parallel mechanisms of types PPa*-RPR with three overconstraints (a) and PPass-RPC* with one overconstraint (b), limb topologies P A Pa* and R A P A ||R (a), P A Pass and R A P A ||C* (b)

2.1 T2-type translational parallel robots with coupled motions

37

Fig. 2.9. Translational parallel mechanisms with an additional unactuated limb of types 2PRR-PP (a) and 2PRS*-PP (b), limb topologies P A R||R and P A P (a), P A R-S* and P A P (b)

38

2 Translational parallel robots with two degrees of freedom

Fig. 2.10. Translational parallel mechanisms with an additional unactuated limb of types 2RPR-PP (a) and 2RPS*-PP (b), limb topologies R A P A ||R and P A P (a), R A P-S* and P A P (b)

2.1 T2-type translational parallel robots with coupled motions

39

Solutions with rotating actuators

In the basic solution with rotating actuators and coupled motions F m G1–G2, the moving platform n Ł nGi is connected to the reference link 1 Ł 1Gi Ł 0 by two limbs of type Pa||Pa. Two planar parallelogram closed loops are combined in each limb. The solution is denoted by 2PaPa-type and has the two rotating actuators mounted on the fixed base. The axes of the revolute pairs of the parallelogram loops are perpendicular to the motion plane of the moving platform. The example in Fig. 2.11 has the same arrangement of joints Paz||Paz in both limbs. The structural diagram in Fig. 2.11 and Eq. (1.9) indicates that the mechanism has five independent closed loops (q = 5). For this solution, Eqs. (1.2)–(1.8) and (1.17) give the following structural parameters: MGi = SGi = 2, (RG1) = (vx,vy), (RG2) = (vx,vy), (RF) = (vx,vy), SF = 2, MF = 2, NF = 16 and TF = 0. (see Table 2.4). A wide range of overconstrained solutions with coupled motions and linear motors combined in two complex limbs with parallelogram loops can be derived from the solution in Fig. 2.11. They have 1 NF < 16 and may integrate up to three idle mobilities in each parallelogram loop and up to four idle mobilities outside the parallelogram loop. The idle mobilities integrated outside the parallelogram loops are three orthogonal rotations and one translation perpendicular to the motion plane of the moving platform. For example, four idle mobilities are introduced outside the parallelogram loops in Fig. 2.12a and three idle mobilities are introduced in each parallelogram loop in Fig. 2.12b.

Fig. 2.11. 2PaPa-type translational parallel mechanism with coupled motions and sixteen overconstraints, limb topology Pa||Pa

40

2 Translational parallel robots with two degrees of freedom

Fig. 2.12. Translational parallel mechanisms of types PaPaP*-PaPaS* with twelve overconstraints (a), 2Pa*Pa* with four overconstraints (b) and Pa*Pa*C*Pa*Pa*R* with one overconstraint (c), limb topologies Pa||Pa||P*-Pa||PaS* (a), Pa*||Pa* (b), Pa*||Pa*||C* and Pa*||Pa* A R* (c)

2.1 T2-type translational parallel robots with coupled motions

41

Table 2.4. Structural parametersa of translational parallel mechanisms in Figs. 2.11 and 2.12

No. Structural Solution parameter 2PaPaFig. 2.11 1 2 3 4 5 6 7 8 9

m p1 p2 p q k1 k2 k (RG1)

12 8 8 16 5 0 2 2 ( vx , v y )

PaPaP*PaPaS* Fig. 2.12a 14 9 9 18 5 0 2 2 ( vx , v y , vz )

10

(RG2)

( vx , v y )

( v x , v y , ȦD , ȦE , ȦG ) ( v x , v y )

( v x , v y , ȦE )

11 12 13 14 15 16 17

SG1 SG2 rG1 rG2 MG1 MG2 (RF)

2 2 6 6 2 2 ( vx , v y )

3 5 6 6 3 5 ( vx , v y )

2 2 12 12 2 2 ( vx , v y )

4 3 12 12 4 3 ( vx , v y )

18 19 20 21 22 23 24

SF rl rF MF NF TF

2 12 14 2 16 0 8

2 12 18 2 12 0 9

2 24 26 2 4 0 14

2 24 29 2 1 0 16

fj

8

11

14

15

fj

16

20

28

31

25 26

¦ ¦ ¦

p1 j 1 p2 j 1 p j 1

fj

2Pa*Pa* Fig. 2.12b 12 8 8 16 5 0 2 2 ( vx , v y )

Pa*Pa*C*Pa*Pa*R* Fig. 2.12c 14 9 9 18 5 0 2 2 ( v x , v y , vz , ȦG )

a

See footnote of Table 2.1 for the nomenclature of structural parameters

In the example in Fig. 2.12c three idle mobilities are introduced outside the parallelogram loops and twelve inside the parallelogram loops. The structural parameters of the solutions illustrated in Figs. 2.11 and 2.12 are presented in Table 2.4. As we have shown, the number of overconstraints could be further reduced by replacing one limb by a simple open kinematic chain with 2 < MGi = SGi < 6 that integrates velocities vx and vy as the basis of its operational space.

42

2 Translational parallel robots with two degrees of freedom

In this way, a wide range of overconstrained translational parallel robots with rotating actuators and 0 < NF d 9 could be obtained. They have three independent loops (q = 3). For example, the solution in Fig. 2.13 is obtained by replacing G2-limb in Fig. 2.11 by a planar kinematic chain RRR-type (Rz||Rz||Rz). Equations (1.2)–(1.8) and (1.17) give the following structural parameters for the parallel mechanisms in Fig. 2.13: MG1 = SG1 = 2, MG2 = SG2 = 3, (RG1) = (vx,vy), (RG2) = ( v x , v y , ȦG ), (RF) = (vx,vy), SF = 2,

MF = 2, NF = 9 and TF = 0 (see Table 2.5). Overconstrained solutions with 0 < NF < 9 can be derived from the solution in Fig. 2.13 by introducing up to three idle mobilities in each parallelogram loop and up to three idle mobilities outside the parallelogram loops. The idle mobilities integrated inside and outside the parallelogram loops could be two orthogonal rotations and one translation perpendicular to the motion plane of the moving platform. For example, three idle mobilities are introduced outside the parallelogram loops in Fig. 2.14a and six idle mobilities are introduced in the parallelogram loops in Fig. 2.14b. In the example in Fig. 2.14c two idle mobilities are introduced outside the parallelogram loops and six in the parallelogram loops. The structural parameters of the solutions illustrated in Figs. 2.13 and 2.14 are presented in Table 2.5.

Fig. 2.13. PaPa-RRR-type translational parallel mechanism with coupled motions and nine overconstraints, limb topologies Pa||Pa and R||R||R

2.1 T2-type translational parallel robots with coupled motions

43

Fig. 2.14. Translational parallel mechanisms of types PaPa-RC*S* with six overconstraints (a), Pa*Pa*-RRR with three overconstraints (b) and Pa*Pa*RRS* with one overconstraint (c), limb topologies Pa||Pa and R||C*-S* (a), Pa*||Pa* and R||R||R (b), Pa*||Pa* and R||R-S* (c)

44

2 Translational parallel robots with two degrees of freedom

Table 2.5. Structural parametersa of translational parallel mechanisms in Figs. 2.13 and 2.14

No. Structural Solution parameter PaPa-RRR PaPa- RC*S* Fig. 2.13 Fig. 2.14a 1 m 9 9 2 p1 8 8 3 p2 3 3 4 p 11 11 5 q 3 3 6 k1 1 1 7 k2 1 1 8 k 2 2 9 (RG1) ( vx , v y ) ( vx , v y ) 10 (RG2)

Pa*Pa*-RRR Fig. 2.14b 9 8 3 11 3 1 1 2 ( vx , v y )

Pa*Pa*- RRS* Fig. 2.14c 9 8 3 11 3 1 1 2 ( vx , v y )

( v x , v y , ȦG ) ( v x , v y , v z , ȦD , ȦE , ȦG ) ( v x , v y , ȦG )

( v x , v y , ȦD , ȦE , ȦG )

11 12 13 14 15 16 17

SG1 SG2 rG1 rG2 MG1 MG2 (RF)

2 3 6 0 2 3 ( vx , v y )

2 6 6 0 2 6 ( vx , v y )

2 3 12 0 2 3 ( vx , v y )

2 5 12 10 2 5 ( vx , v y )

18 19 20 21 22 23 24

SF rl rF MF NF TF

2 6 9 2 9 0 8

2 6 12 2 6 0 8

2 12 15 2 3 0 14

2 12 17 2 1 0 14

fj 3

6

3

5

f j 11

14

17

19

25 26

¦ ¦ ¦

p1 j 1 p2 j 1 p j 1

fj

a

See footnote of Table 2.1 for the nomenclature of structural parameters

Solutions of type F m G1–G2–G3 with an additional unactuated limb can also be derived by using two simple actuated limbs with 2 < MGi = SGi < 6 and an unactuated limb P A P-type. The directions of the two prismatic pairs are parallel to the plane of motion of the moving platform. Actuated limbs G1 and G2 must integrate velocities vx and vy in the bases of their operational spaces. The three limbs form q = 2 independent loops and must meet the following conditions: MF = SF = 2 and (RF) = (RG1 RG2 RG2) = (vx,vy).

2.1 T2-type translational parallel robots with coupled motions

45

Fig. 2.15. Translational parallel mechanisms with an additional unactuated limb of types 2RRR-PP (a) and 2RRS*-PP (b), limb topologies R||R||R and P A P (a), R||R-S* and P A P (b)

46

2 Translational parallel robots with two degrees of freedom

Equation (1.16) indicates that the overconstrained solutions of these T2type translational parallel robots with coupled motions and an additional p unactuated limb have ¦ 1 f i 14 . Various solutions with identical or different limb architectures can be generated. For example the solution in Fig. 2.15a has two actuated limbs of type RRR and six overconstraints. The solution in Fig. 2.15b has two overconstraints. These two solutions have the same structural parameters as their counterparts with linear motors (see Table 2.3). Solutions with linear and rotating actuators

The basic solution with linear and rotating actuators and coupled motions F m G1–G2 has a moving platform n Ł nGi connected to a reference link 1 Ł 1Gi Ł 0 by two limbs of types P A Pa and Pa||Pa. This solution is denoted by PPa-PaPa and has one linear and one rotating actuator situated on the fixed base. The linear actuator may have any direction parallel to the plane of motion of the moving platform. The axis of the actuated revolute pair is perpendicular to the motion plane of the moving platform. The example in Fig. 2.16 has the following arrangement of joints: Px A Paz in G1-limb and Paz||Paz in G2-limb. The structural diagram in Fig. 2.16 and Eq. (1.9) indicates that the mechanism has four independent closed loops (q = 4). Equations (1.2)–(1.8) and (1.17) give the following structural parameters for the parallel mechanisms in Fig. 2.16: MGi = SGi = 2, (RG1) = (vx,vy), (RG2) = (vx,vy), (RF) = (vx,vy), SF = 2, MF = 2, NF = 13 and TF = 0 (see Table 2.6).

Fig. 2.16. PPa-PaPa-type translational parallel mechanism with coupled motions and thirteen overconstraints, limb topologies P A Pa and Pa||Pa

2.1 T2-type translational parallel robots with coupled motions

47

Fig. 2.17. Translational parallel mechanisms of types PPaP*-PaPaS* with nine overconstraints (a), PPa*-Pa*Pa* with four overconstraints (b) and PPa*C*Pa*Pa*R* with one overconstraint (c), limb topologies P A Pa||P* and Pa||Pa-S* (a), P A Pa* and Pa*||Pa* (b), P A Pa*||C* and Pa*||Pa* A R* (c)

48

2 Translational parallel robots with two degrees of freedom

Table 2.6. Structural parametersa of translational parallel mechanisms in Figs. 2.16 and 2.17

No. Structural Solution parameter PPaPaPa Fig. 2.16 1 m 10 2 p1 5 3 p2 8 4 p 13 5 q 4 6 k1 0 7 k2 2 8 k 2 9 (RG1) ( vx , v y )

PPaP*PaPaS* Fig. 2.17a 12 6 9 15 4 0 2 2 ( vx , v y , vz )

PPa*Pa*Pa* Fig. 2.17b 10 5 8 13 4 0 2 2 ( vx , v y )

PPa*C*Pa*Pa*R* Fig. 2.17c 12 6 9 15 4 0 2 2 ( v x , v y , vz , ȦG )

10

(RG2)

( vx , v y )

( v x , v y , ȦD , ȦE , ȦG ) ( v x , v y )

( v x , v y , ȦE )

11 12 13 14 15 16 17

SG1 SG2 rG1 rG2 MG1 MG2 (RF)

2 2 3 6 2 2 ( vx , v y )

3 5 3 6 3 5 ( vx , v y )

2 2 6 12 2 2 ( vx , v y )

4 3 6 12 4 3 ( vx , v y )

18 19 20 21 22 23 24

SF rl rF MF NF TF

2 9 11 2 13 0 5

2 9 15 2 9 0 6

2 18 20 2 4 0 8

2 18 23 2 1 0 10

fj

8

11

14

15

fj

13

17

22

25

25 26

¦ ¦ ¦

p1 j 1 p2 j 1 p j 1

fj

a

See footnote of Table 2.1 for the nomenclature of structural parameters

A wide range of overconstrained solutions with coupled motions and linear and rotating actuators integrated in two complex limbs with parallelogram loops can be derived from the solution in Fig. 2.16. They have 0 < NF < 13 and could integrate up to three idle mobilities in each parallelogram loop and up to four idle mobilities outside the parallelogram loop as presented in the previous sections. For example, four idle mobilities are introduced outside the parallelogram loops in Fig. 2.17a and three idle

2.1 T2-type translational parallel robots with coupled motions

49

mobilities are introduced in each parallelogram loop in Fig. 2.17b. In the example in Fig. 2.17c three idle mobilities are introduced outside the parallelogram loops and nine inside the parallelogram loops. The structural parameters of the solutions illustrated in Figs. 2.16 and 2.17 are presented in Table 2.6. One limb in Fig. 2.16 can be replaced by a simple open kinematic chain with 2 < MGi = SGi < 6 that integrates the velocities vx and vy in the basis of its operational space. In this way, a large diversity of overconstrained translational parallel robots with 0 < NF < 10 actuated by linear and rotating motors can be obtained. They have three independent closed loops (q = 3) when G1-limb is replaced (Fig. 2.18) and q = 2 when G2-limb is replaced (Fig. 2.19).

Fig. 2.18. PRR-PaPa-type translational parallel mechanism with nine overconstraints, limb topologies P A R||R and Pa||Pa

Fig. 2.19. PPa-RRR-type translational parallel mechanism with six overconstraints, limb topologies P A Pa-R||R||R

50

2 Translational parallel robots with two degrees of freedom

Fig. 2.20. Translational parallel mechanisms of types PC*S*-PaPa with six overconstraints (a), PRR-Pa*Pa* with three overconstraints (b) and PRC*Pa*Pass with one overconstraint (c), limb topologies P A C*-S* and Pa||Pa (a), P A R||R and Pa*||Pa* (b), P A R||C* and Pa*||Pass (c)

2.1 T2-type translational parallel robots with coupled motions

51

Fig. 2.21. Translational parallel mechanisms of types PPa-RC*S* with three overconstraints (a), PPa*-RRR with three overconstraints (b) and PPa*-RRS* with one overconstraint (c), limb topologies P A Pa and R||C*-S* (a), P A Pa* and R||R||R (b), P A Pa* and R||R-S*(c)

52

2 Translational parallel robots with two degrees of freedom

For example, the solution in Fig. 2.18 is obtained by replacing G1-limb in Fig. 2.16 by a planar kinematic chain PyRzRz-type. Equations (1.2)–(1.8) and (1.17) give the following structural parameters for the solution in Fig. 2.18: MG1 = SG1 = 3, MG2 = SG2 = 2, (RG1) = ( v x , v y , ȦG ), (RG2 ) = ( v x , v y ), (RF) = ( v x , v y ), SF = 2, MF = 2, NF = 9 and TF = 0. The solution in Fig. 2.19 is obtained by replacing G2-limb in Fig. 2.16 by a planar kinematic chain RzRzRz-type. The solution in Fig. 2.19 has the following structural parameters: MG1 = SG1 = 2, MG2 = SG2 = 3, (RG1) = ( v x , v y ), (RG2) = ( v x , v y , ȦG ), (RF) = ( v x , v y ), SF = 2, MF = 2, NF = 6 and TF = 0. A wide range of overconstrained solutions with coupled motions and linear and rotating actuators can be derived from solutions in Figs. 2.18 and 2.19. They have 0 < NF < 9 and could integrate up to three idle mobilities in each parallelogram loop and up to three idle mobilities outside the parallelogram loops. For example, three idle mobilities are introduced outside the parallelogram loops in Figs. 2.20a and 2.21a, and three idle mobilities inside each parallelogram loop in Figs. 2.20b and 2.21b.

Fig. 2.22. RRR-PRR-PP-type translational parallel mechanism with an additional unactuated limb and six overconstraints, limb topologies R||R||R, P A R||R and PA P

2.1 T2-type translational parallel robots with coupled motions

53

Fig. 2.23. RRS*-PRS*-PP-type translational parallel mechanism with an additional unactuated limb and two overconstraints, limb topologies R||R-S*, P A R-S* and P A P

Two idle mobilities are introduced outside the parallelogram loops and three inside each parallelogram loop in Figs. 2.20c and 2.21c. The structural parameters of the solutions illustrated in Figs. 2.18 and 2.20 are obtained by similarity with their counterparts using rotating actuators (see Table 2.5). The solutions in Figs. 2.19 and 2.21 have the same structural parameters as their counterparts with linear actuators (see Table 2.2). Solutions of type F m G1–G2–G3 with an additional unactuated limb and two limbs actuated by linear and rotating motors can also be generated. For example, the solution in Fig. 2.22 with six overconstraints has the actuated limbs of types RzRzRz and PyRz Rz . The solution in Fig. 2.23 with two overconstraints has the actuated limbs of type RRS and PRS. These solutions have the same structural parameters as their counterparts with linear motors (see Table 2.3).

54

2 Translational parallel robots with two degrees of freedom

2.1.2 Non overconstrained solutions Equation (1.15) indicates that non overconstrained solutions of T2-type translational parallel robots with coupled motions and q independent loops p meet the condition ¦ 1 f i 2 6q . Various solutions fulfil this condition along with MF = SF = 2 and (RF) = (vx,vy). They can have identical limbs or limbs with different structures and may be actuated by linear or rotating motors. Solutions with linear actuators

Non overconstrained solutions F m G1–G2 with linear actuators and coupled motions have

¦

p 1

fi

20 when two complex limbs are used. They can be

derived from the solution in Fig. 2.1 by introducing ten idle mobilities: four outside the parallelogram loops and six inside the parallelogram loops. For example the solution in Fig. 2.24 with G1-limb PPass-type and G2-limb PC*Pass-type has MG1 = SG1 = 3, MG2 = SG2 = 5, (RG1) = ( v x , v y , ȦE ), (RG2) = ( v x , v y , v z , ȦD , ȦG ), (RF) = ( v x , v y ), MF = SF = 2, NF =

0 and TF = 0.

Fig. 2.24. PPass-PC*Pass-type non overconstrained translational parallel mechanism, limb topologies P A Pass and P A C*||Pass

2.1 T2-type translational parallel robots with coupled motions

55

Fig. 2.25. Non overconstrained translational parallel mechanisms of types PPa*PC*S* (a) and PPa*-PU*S* (b), limb topologies P A Pa* and P A C*-S* (a), P A Pa* and P A U*S* (b)

56

2 Translational parallel robots with two degrees of freedom

Fig. 2.26. Non overconstrained translational parallel mechanisms of types PPa*C*PS* (a) and PPa*-U*PS* (b), limb topology P A Pa* and C* A P-S* (a), P A Pa* and U* A P-S* (b)

2.1 T2-type translational parallel robots with coupled motions

57

Table 2.7. Structural parametersa of translational parallel mechanisms in Figs. 2.24–2.26

No. Structural Solution parameter PPass-PC*Pass (Fig. 2.24)

PPa*-PC*S* PPa*-C*PS* (Figs. 2.25a, 2.26a) 7 5 3 8 2 1 1 2 ( vx , v y )

PPa*-PU*S* PPa*-U*PS* (Figs. 2.25b, 2.26b) 8 5 4 9 2 1 1 2 ( vx , v y )

1 2 3 4 5 6 7 8 9

m p1 p2 p q k1 k2 k (RG1)

9 5 6 11 3 0 2 2 ( v x , v y , ȦE )

10

(RG2)

( v x , v y , v z , ȦD , ȦG ) ( v x , v y , v z , ȦD , ȦE , ȦG ) ( v x , v y , v z , ȦD , ȦE , ȦG )

11 12 13 14 15 16 17

SG1 SG2 rG1 rG2 MG1 MG2 (RF)

3 5 6 6 3 5 ( vx , v y )

2 6 6 0 2 6 ( vx , v y )

2 6 6 0 2 6 ( vx , v y )

18 19 20 21 22 23 24

SF rl rF MF NF TF

2 12 18 2 0 0 9

2 6 12 2 0 0 8

2 6 12 2 0 0 8

fj

11

6

6

fj

20

14

14

25 26

¦ ¦ ¦

p1 j 1 p2 j 1 p j 1

fj

a

See footnote of Table 2.1 for the nomenclature of structural parameters

Non overconstrained solutions F m G1–G2 with linear actuators and coupled motions have

¦

p 1

fi

14 when one simple and one complex limb are used.

They can be derived from the solution in Fig. 2.4 by introducing six idle mobilities: three outside and three inside the parallelogram loop. For example, the solution in Fig. 2.25a has G1-limb of type PPa* and G2-limb

58

2 Translational parallel robots with two degrees of freedom

Fig. 2.27. Non overconstrained translational parallel mechanisms with an additional unactuated limb of types 2PC*S*-PP (a) and 2C*PS*-PP (b), limb topologies P A C*-S* and P A P (a), C* A P-S* and P A P (b)

2.1 T2-type translational parallel robots with coupled motions

59

of type PC*S*. The solution in Fig. 2.25b has G1-limb of type PPa* and G2-limb of type U*PS*. Both solutions have the following structural parameters MG1 = SG1 = 2, MG2 = SG2 = 6, (RF) = ( v x , v y ), (RG1) = ( v x , v y ), (RG2) = ( v x , v y , v z , ȦD , ȦE , ȦG ), MF = SF = 2, NF = 0 and TF = 0. Non overconstrained solutions with linear actuators non adjacent to the fixed base can be derived from the overconstrained solutions presented in Fig. 2.7a by introducing three idle mobilities inside and three outside the parallelogram loop. For example, the solution in Fig. 2.26a has G1-limb of type PPa* and G 2 -limb of type C*PS*. The solution in Fig. 2.26b has G1limb of type PPa* and G 2 -limb of type U*PS*. Both solutions have MG1 = SG1 = 2, MG2 = SG2 = 6, (RG1) = ( v x , v y ), (RG2) = ( v x , v y , v z , ȦD , ȦE , ȦG ), (RF) = ( v x , v y ), MF = SF = 2, NF = 0 and TF = 0. The structural parameters of the solutions illustrated in Figs. 2.24–2.26 are presented in Table 2.7. Non overconstrained solutions of type F m G1–G2–G3 with an additional unactuated limb can be derived from the overconstrained solutions presented in Figs. 2.9a and 2.10a. The solutions in Fig. 2.27 have an unactuated simple limb PP-type and two actuated simple limbs PC*S*-type (Fig. 2.27a) and C*PS*-type (Fig. 2.27b). For both solutions have MG1 = SG1 = 6, MG2 = SG2 = 6, MG3 = SG3 = 2, (RG1) = (RG2) = ( v x , v y , v z , ȦD , ȦE , ȦG ), (RG3) = ( v x , v y ), (RF) = ( v x , v y ), MF = SF = 2, NF = 0 and TF = 0. Solutions with rotating actuators

Non overconstrained solutions F m G1–G2 with rotating actuators and coupled motions have

¦

p 1

fi

32 when two complex limbs are used.

Fig. 2.28. Pa*Pa*P*-Pa*Pa*S*-type non overconstrained translational parallel mechanism, limb topologies Pa*||Pa*||P*-Pa*||Pa*S*

60

2 Translational parallel robots with two degrees of freedom

Fig. 2.29. Pa*Pa*-RC*S*-type non overconstrained translational parallel mechanism, limb topologies Pa*||Pa* and R||C*-S*

Fig. 2.30. 2RC*S*-PP-type non overconstrained translational parallel mechanism with an additional unactuated limb, limb topologies R||C*-S* and P A P

2.1 T2-type translational parallel robots with coupled motions

61

Table 2.8. Structural parametersa of translational parallel mechanisms in Figs. 2.28–2.30

No. Structural Solution parameter Pa*Pa*P*Pa*Pa*S* Fig. 2.28 1 m 14 2 p1 9 3 p2 9 4 p3 – 5 p 18 6 q 5 7 k1 0 8 k2 2 9 k 2 10 (RG1) ( vx , v y , vz )

Pa*Pa*-RC*S* Fig. 2.29

2RC*S*-PP Fig. 2.30

9 8 3 – 11 3 1 1 2 ( vx , v y )

7 3 3 2 8 2 3 0 3 ( v x , v y , v z , ȦD , ȦE , ȦG )

11

(RG2)

( v x , v y , ȦD , ȦE , ȦG )

12

(RG3)

–

( v x , v y , v z , ȦD , ȦE , ȦG ) ( v x , v y , v z , ȦD , ȦE , ȦG ) – ( vx , v y )

13 14 15 16 17 18 19 20 21 22

SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

3 5 – 12 12 – 3 5 – ( vx , v y )

2 6 – 12 0 – 2 6 – ( vx , v y )

6 6 2 0 0 0 6 6 2 ( vx , v y )

23 24 25 26 27 28 29

SF rl rF MF NF TF

2 24 30 2 0 0 15

2 12 18 2 0 0 14

2 0 12 2 0 0 6

fj

17

6

6

fj

–

–

2

fj

32

20

14

30 31 32 a

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

62

2 Translational parallel robots with two degrees of freedom

They can be derived from the solution in Fig. 2.11 by introducing sixteen idle mobilities: four outside and twelve inside the parallelogram loops. For example the solution in Fig. 2.28 has G1-limb of type Pa*Pa*P* and G2-limb of type Pa*Pa*S* with MG1 = SG1 = 3, MG2 = SG2 = 5, (RG1) = ( v x , v y , v z ), (RG2) = ( v x , v y , ȦD , ȦE , ȦG ), (RF) = ( v x , v y ), MF = SF = 2, NF = 0 and TF = 0. Non overconstrained solutions F m G1–G2 with rotating actuators and coupled motions have

¦

p 1

fi

20 when a simple and a complex limb are

used. They can be derived from the solution in Fig. 2.13 by introducing nine idle mobilities: three outside and six inside the parallelogram loops. For example, the solution in Fig. 2.29 with G1-limb of type Pa*Pa* and G2-limb of type RC*S* has the following structural parameters: MG1 = SG1 = 2, MG2 = SG2 = 6, (RG1) = ( v x , v y ), (RG2) = ( v x , v y , v z , ȦD , ȦE , ȦG ), (RF) = ( v x , v y ), MF = SF = 2, NF = 0 and TF = 0. Non overconstrained solutions of type F m G1–G2–G3 with an additional unactuated limb and rotating actuators can be derived from the overconstrained solution in Fig. 2.15a. For example, the solution in Fig. 2.30 has an unactuated simple limb PP-type and two actuated simple limbs RC*S*-type. This solution has the following structural parameters: MG1 = SG1 = 6, MG2 = SG2 = 6, MG3 = SG3 = 2, (RG1) = (RG2) = ( v x , v y , v z , ȦD , ȦE , ȦG ), (RG3) = ( v x , v y ), (RF) = ( v x , v y ), MF = SF = 2, NF = 0 and TF = 0. The structural parameters of the solutions illustrated in Figs. 2.28–2.30 are presented in Table 2.8. Solutions with linear and rotating actuators

Non overconstrained solutions F m G1–G2 with linear and rotating actuators and coupled motions have

¦

p 1

fi

26 when two complex limbs are used.

They can be derived from the solution in Fig. 2.16 by introducing thirteen idle mobilities: four outside and nine inside the parallelogram loops. For example, the solution in Fig. 2.31 has G1-limb of type PPa*P* and G2limb of type Pa*Pa*S* with MG1 = SG1 = 3, MG2 = SG2 = 5, (RG1) = ( v x , v y , v z ), (RG2) = ( v x , v y , ȦD , ȦE , ȦG ), (RF) = ( v x , v y ), MF = SF = 2, NF = 0 and TF = 0. Non overconstrained solutions F m G1–G2 with linear and rotating actuators and coupled motions have

¦

p 1

fi

20 when a simple limb is actuated by

2.1 T2-type translational parallel robots with coupled motions

63

Fig. 2.31. PPa*P*-Pa*Pa*S*-type non overconstrained translational parallel mechanism, limb topologies P A Pa*||P* and Pa* A Pa*S*

Fig. 2.32. PC*S*-Pa*Pa*-type non overconstrained translational parallel mechanism, limb topologies P A C*-S* and Pa*||Pa*

a linear motor and a complex limb by a rotating motor. They can be derived from the solution in Fig. 2.18 by introducing nine idle mobilities: three outside and six inside the parallelogram loops. For example, the solution in Fig. 2.32 has G1-limb of type PC*S* and G2-limb of type Pa*Pa* with MG1 = SG1 = 6, MG2 = SG2 = 2, (RG1) = ( v x , v y , v z , ȦD , ȦE , ȦG ), (RG2) = ( v x , v y ), (RF) = ( v x , v y ), MF = SF = 2, NF = 0 and TF = 0.

64

2 Translational parallel robots with two degrees of freedom

Fig. 2.33. PPa*-RC*S*-type non overconstrained translational parallel mechanism, limb topologies P A Pa* and R||C*-S*

Fig. 2.34. RC*S*-PC*S*-PP-type non overconstrained translational parallel mechanism with an additional unactuated limb, limb topologies R||C*-S*, P AC*S* and P A P

2.1 T2-type translational parallel robots with coupled motions

65

Table 2.9. Structural parametersa of translational parallel mechanisms in Figs. 2.31–2.33

No. Structural Solution parameter PPa*P*-Pa*Pa*S* Fig. 2.31 1 m 12 2 p1 6 3 p2 9 4 15 p 5 4 q 6 k1 0 7 k2 2 8 2 k 9 (RG1) ( vx , v y , vz )

PC*S*-Pa*Pa* Fig. 2.32 9 3 8 11 3 1 1 2 ( v x , v y , v z , ȦD , ȦE , ȦG )

PPa*-RC*S* Fig. 2.33 7 5 3 8 2 1 1 2 ( vx , v y )

10

(RG2)

( v x , v y , ȦD , ȦE , ȦG )

( vx , v y )

( v x , v y , v z , ȦD , ȦE , ȦG )

11 12 13 14 15 16 17

SG1 SG2 rG1 rG2 MG1 MG2 (RF)

3 5 6 12 3 5 ( vx , v y )

6 2 0 12 6 2 ( vx , v y )

2 6 6 0 2 6 ( vx , v y )

18 19 20 20 21 22 23

SF rl rF MF NF TF

2 18 24 2 0 0 9

2 12 18 2 0 0 6

2 6 12 2 0 0 8

fj

17

14

6

fj

26

20

14

24 25

¦ ¦ ¦

p1 j 1 p2 j 1 p j 1

fj

a

See footnote of Table 2.1 for the nomenclature of structural parameters

Non overconstrained solutions F m G1–G2 with linear and rotating actuators and coupled motions have

¦

p 1

fi

14 when a complex limb is

actuated by a linear motor and a simple limb by a rotating motor. They can be derived from the solution in Fig. 2.19 by introducing six idle mobilities: three outside and three inside the parallelogram loop. For example, the solution in Fig. 2.33 has G1-limb of type PPa* and G2-limb of type RC*S* with the following structural parameters: MG1 = SG1 = 2, MG2 = SG2 = 6,

66

2 Translational parallel robots with two degrees of freedom

(RG1) = ( v x , v y ), (RG2) = ( v x , v y , v z , ȦD , ȦE , ȦG ), (RF) = ( v x , v y ), MF = SF =

2, NF = 0 and TF = 0. The structural parameters of the solutions illustrated in Figs. 2.31–2.33 are presented in Table 2.9. Non overconstrained solutions of type F m G1–G2–G3 with an additional unactuated limb and linear and rotating actuators can be derived from the overconstrained solution in Fig. 2.22. For example, the solution in Fig. 2.34 has an unactuated simple limb PP-type and two simple actuated limbs RC*S*- and PC*S*-type. The solution in Fig. 2.34 is characterised by the following structural parameters: MG1 = SG1 = 6, MG2 = SG2 = 6, MG3 = SG3 = 2, (RG1) = (RG2) = ( v x , v y , v z , ȦD , ȦE , ȦG ), (RG3) = ( v x , v y ), (RF) = ( v x , v y ), MF = SF = 2, NF = 0 and TF = 0. The solution in Figs. 2.30 and 2.34 have the same structural parameters (see Table 2.8).

2.2 T2-type translational parallel robots with decoupled motions T2-type translational parallel robots with decoupled motions and linear or rotating actuators with various degrees of overconstraint can be generated. In these solutions one of the operational velocities depends on just one actuated joint velocity and the other depends of both actuated joint & 2 ) . We consider v1 = vx and v2 = vy. velocities: v1 v1 ( &q1 ) and v2 v2 ( &q1 ,q 2.2.1 Overconstrained solutions The overconstrained solutions of T2-type translational parallel robots with decoupled motions and q independent loops meet the same condition p ¦ 1 fi 2 6q as their counterparts with coupled motions. Solutions with linear actuators

T2-type translational parallel robots with decoupled motions and linear actuators can be derived from their counterparts with coupled motions by replacing a PPa-limb by a PP-limb as illustrated in Figs. 2.35–2.37. These solutions have two limbs which could be one simple and one complex (Fig. 2.35) or both simple limbs (Figs. 2.36 and 2.37). The complex limb G2 combines a closed parallelogram loop. The linear actuators can be mounted on the fixed base (Figs. 2.35–2.37) or on a moving link (Fig. 2.37).

2.2 T2-type translational parallel robots with decoupled motions

67

Fig. 2.35. Translational parallel mechanisms of types PP-PPa with seven overconstraints (a), PP-PPa* with four overconstraints (b) and PR*P-PR*PaC* with three overconstraints (c), limb topologies P A P and P A Pa (a), P A P and P A Pa* (b), P||R* A P and P||R* A Pa||C* (c)

68

2 Translational parallel robots with two degrees of freedom

Table 2.10. Structural parameters a of translational parallel mechanisms in Fig. 2.35

No.

1 2 3 4 5 6 7 8 9

m p1 p2 p q k1 k2 k (RG1)

Solution PP-PPa Fig. 2.35a 6 2 5 7 2 1 1 2 ( vx , v y )

10

(RG2)

( vx , v y )

( vx , v y )

( v x , v y , v z , ȦE , ȦG )

11 12 13 14 15 16 17

SG1 SG2 rG1 rG2 MG1 MG2 (RF)

2 2 0 3 2 2 ( vx , v y )

2 2 0 6 2 2 ( vx , v y )

3 5 0 3 3 5 ( vx , v y )

18 19 20 21 22 23 24

SF rl rF MF NF TF

2 3 5 2 7 0 2

2 6 8 2 4 0 2

2 3 9 2 3 0 3

fj

5

8

8

fj

7

10

11

25 26

Structural parameter

¦ ¦ ¦

p1 j 1 p2 j 1 p j 1

fj

PP-PPa* Fig. 2.35b 6 2 5 7 2 1 1 2 ( vx , v y )

PR*P-PR*PaC* Fig. 2.35c 9 3 7 10 2 1 1 2 ( v x , v y , ZD )

a

See footnote of Table 2.1 for the nomenclature of structural parameters

The overconstrained solutions with a simple and a complex limb integrating a parallelogram loop have 1 NF 7. These solutions could combine up to three idle mobilities in the parallelogram loop and up to four idle mobilities outside it. The structural parameters of the solutions illustrated in Fig. 2.35 are presented in Table 2.10. A large variety of T2-type translational parallel robots with decoupled motions and linear actuators can be obtained by associating a simple limb G1 of type PP with any simple G2 - limb that integrates velocities vx and vy in

2.2 T2-type translational parallel robots with decoupled motions

69

Fig. 2.36. Translational parallel mechanisms of types PP-PRR with three overconstraints (a) and PP-PRS* with one overconstraint (b), limb topologies P A P and P A R||R (a) P A P and P A R-S* (b)

the basis of its operational velocity vector space. G2-limb can have 2 < MGi = SGi < 6. The overconstrained solutions with decoupled motions and two simple limbs have 1 NF 3 and may integrate up to three idle mobilities. For the solutions with two simple limbs in Figs. 2.36a and 2.37a, Eqs. (1.2)–(1.8) and (1.17) give the following structural parameters: MG1 = SG1 = 2, MG2 = SG2 = 3, (RG1) = ( v x , v y ), (RG2) = ( v x , v y , ȦG ), (RF) = ( v x , v y ),

MF = SF = 2, NF = 3 and TF = 0 (see Table 2.11).

70

2 Translational parallel robots with two degrees of freedom

Fig. 2.37. Translational parallel mechanisms of types PP-RPR with three overconstraints (a) and PP-RPS* with one overconstraint (b), limb topologies: P A P-R A P A ||R (a) and P A P-R A PS (b)

For the solutions with two simple limbs in Figs. 2.36b and 2.37b, Eqs. (1.2)–(1.8) and (1.17) give the following parameters: MG1 = SG1 = 2, MG2 = SG2 = 5, (RG1) = ( v x , v y ), (RG2) = ( v x , v y , ȦD , ȦE , ȦG ), (RF) = ( v x , v y ), MF =

SF = 2, NF = 1 and TF = 0 (see Table 2.11).

2.2 T2-type translational parallel robots with decoupled motions

71

Table 2.11. Structural parameters a of translational parallel mechanisms in Figs. 2.36–2.37

No.

1 2 3 4 5 6 7 8 9

m p1 p2 p q k1 k2 k (RG1)

Solution PP-PRR (Fig. 2.36a) PP-RPR (Fig. 2.37a) 5 2 3 5 1 2 0 2 ( vx , v y )

10

(RG2)

( v x , v y , ZG )

( v x , v y , ZD , Z E , ZG )

11 12 13 14 15 16 17

SG1 SG2 rG1 rG2 MG1 MG2 (RF)

2 3 0 0 2 3 ( vx , v y )

2 5 0 0 2 5 ( vx , v y )

18 19 20 21 22 23 24

SF rl rF MF NF TF

2 0 3 2 3 0 2

2 0 5 2 1 0 2

fj

3

5

fj

5

7

25 26

Structural parameter

¦ ¦ ¦

p1 j 1 p2 j 1 p j 1

fj

PP-PRS* (Fig. 2.36b) PP-RPS* (Fig. 2.37b) 5 2 3 5 1 2 0 2 ( vx , v y )

a

See footnote of Table 2.1 for the nomenclature of structural parameters

Solutions with rotating actuators

T2-type translational parallel robots with decoupled motions and rotating actuators can be derived from their counterparts with coupled motions by replacing a PaPa-limb by a PaP-limb as illustrated in Figs. 2.38 and 2.39. These solutions have two complex limbs (Fig. 2.38) or a complex and a simple limb (Figs. 2.39).

72

2 Translational parallel robots with two degrees of freedom

Fig. 2.38. Translational parallel mechanisms of types PaP-PaPa with thirteen overconstraints (a), PaPP*-PaPaS* with nine overconstraints (b) and Pa*PPa*Pa* with four overconstraints (c), limb topologies Pa A P and Pa||Pa (a), Pa A P A P* and Pa||PaS* (b), Pa* A P and Pa*||Pa* (c)

2.2 T2-type translational parallel robots with decoupled motions

73

Fig. 2.39. Translational parallel mechanisms of types PaP-RRR with six overconstraints (a) PaP-RC*S* with three overconstraints (b) and Pa*P-RRR with three overconstraints (c), limb topologies Pa A P and R||R||R (a) Pa A P and R||C*S* (b), Pa* A P and R||R||R (c)

74

2 Translational parallel robots with two degrees of freedom

The overconstrained solutions with two complex limbs have 1 NF 13. The complex limb G1 combines one closed parallelogram loop and complex limb G2 two parallelogram loops (Fig. 2.38a). These solutions could combine up to three idle mobilities in each parallelogram loop (Fig. 2.38c) and up to four idle mobilities outside them (Fig. 2.38b). A large variety of T2-type translational parallel robots with decoupled motions and rotating actuators can be generated by associating the complex limb G1 of type PaP with any simple G2-limb that integrates velocities vx and vy in the basis of its operational velocity vector space along with a constant orientation of the moving platform. G2-limb can have 2 < MGi = SGi < 6. For example, in Fig. 2.39a, a planar simple limb RRR-type with MGi = SGi = 3 is used. The overconstrained solutions with decoupled motions and a complex and a simple limb have 1 NF 6 and may integrate up to three idle mobilities inside and three others outside the parallelogram loop (Fig. 2.39b and c). The solutions illustrated in Fig. 2.38 have the same structural parameters as those in Figs. 2.16, 2.17a and b (see Table 2.6). The solutions in Fig. 2.39 have the same structural parameters as those in Figs. 2.4, 2.5a and b (see Table 2.2). Solutions with rotating and linear actuators

Basic solutions of overconstrained T2-type translational parallel robots with decoupled motions actuated by rotating and linear motors have two complex limbs of types PaP and PPa (Fig. 2.40a). They have 1 NF 10. Each complex limb combines one closed parallelogram loop. These solutions may integrate up to three idle mobilities in each parallelogram loop (Fig. 2.40c) and up to four idle mobilities outside them (Fig. 2.40b). The solutions illustrated in Figs. 2.40a and c have the same structural parameters as those in Figs. 2.1 and 2.2b (see Table 2.1). The solution in Fig. 2.40b has the following structural parameters: MG1 = SG1 = 3, MG2 = SG2 = 5, (RG1) = ( v x , v y , v z ), (RG2) = ( v x , v y , ȦD , ȦE , ȦG ), (RF) = ( v x , v y ), MF

= SF = 2, NF = 6 and TF = 0. A large variety of overconstrained T2-type translational parallel robots with decoupled motions actuated by rotating and linear motors can be set up by associating a complex limb G1 of type PaP with any simple G2-limb that integrates velocities vx and vy in the basis of its operational velocity space. G2-limb can have 2 < MGi = SGi < 6. For example, simple planar limbs with MGi = SGi = 3 of type PRR is used in Fig. 2.41a and RPR in Fig. 2.42a.

2.2 T2-type translational parallel robots with decoupled motions

75

Fig. 2.40. Translational parallel mechanisms of types PaP-PPa with ten overconstraints (a) PaPP*-PPaS* with six overconstraints (b) and Pa*P-PPa* with four overconstraints (c), limb topologies Pa A P and P A Pa (a), Pa A P A P* and P A PaS* (b), Pa* A P and P A Pa* (c)

76

2 Translational parallel robots with two degrees of freedom

Fig. 2.41. Translational parallel mechanisms of types PaP-PRR with six overconstraints (a), PaP-PC*S* with three overconstraints (b) and Pa*P-PRR with three overconstraints (c), limb topologies Pa A P and P A R||R (a), Pa A P and P A C*-S* (b), Pa* A P and P A R||R (c)

2.2 T2-type translational parallel robots with decoupled motions

77

Fig. 2.42. Translational parallel mechanisms of types PaP-RPR with six overconstraints (a), PaP-C*PS* with three overconstraints (b ) and Pa*P-RPR with three overconstraints (c), limb topologies Pa A P and R A P A ||R (a), Pa A P and C* A P-S* (b), Pa* A P and R A P A ||R (c)

78

2 Translational parallel robots with two degrees of freedom

These overconstrained solutions with decoupled motions and a complex and a simple limb have 1 NF 6 and could combine up to three idle mobilities outside the parallelogram loop (Figs. 2.41b and 2.42b) and other three idle mobilities inside the parallelogram loop (Figs. 2.41c and 2.42c). The solutions in Figs. 2.41 and 2.42 have the same structural parameters as those in Figs. 2.4, 2.5a and b (see Table 2.2). 2.2.2 Non overconstrained solutions

Non overconstrained solutions of T2-type translational parallel robots with decoupled motions and q independent loops meet the same condition p ¦ 1 fi 2 6q as their counterparts with coupled motions. Solutions with linear actuators

Non overconstrained solutions F m G1–G2 with linear actuators and decoupled motions have

¦

p 1

fi

14 when one complex limb is used.

They are derived from the solution in Fig. 2.35a by introducing ten idle mobilities: four outside and three inside the parallelogram loop. Structural parameters of solutions in Figs. 2.43 and 2.44 are presented in Table 2.12. Non overconstrained solutions F m G1–G2 with linear actuators and decoupled motions have

¦

p 1

fi

8 when two simple limbs are used. They

Fig. 2.43. PPR*-PC*Pass-type non overconstrained translational parallel mechanism, limb topologies P A P||R* and P A C*||Pass

2.2 T2-type translational parallel robots with decoupled motions

79

Fig. 2.44. Non overconstrained translational parallel mechanisms of types PPPR*Pa*C*R* (a) and PPS*-PPa*P* (b), limb topologies P A P and P A R* A A Pa*||C* A R* (a), P A PS* and P A Pa*||P* (b)

are derived from the solutions in Figs. 2.36a and 2.37a by using three idle mobilities. For example, the solutions in Fig. 2.45 have G1-limb of type PP and G2-limb of type PC*S* (Fig. 2.45a), PU*S* (Fig. 2.45b) and U*PS* (Fig. 2.45c). These solutions have MG1 = SG1 = 2, MG2 = SG2 = 6, (RG1) = ( v x , v y ), (RG2) = ( v x , v y , v z , ȦD , ȦE , ȦG ), (RF) = ( v x , v y ), MF = SF = 2, NF = 0 and TF = 0.

80

2 Translational parallel robots with two degrees of freedom

Fig. 2.45. Non overconstrained translational parallel mechanisms of types PPPC*S* (a), PP-PU*S* (b) and PP-U*PS* (c), limb topologies P A P and P A C*S* (a), P A P and P A U*-S* (b), P A P and U* A P-S* (c)

2.2 T2-type translational parallel robots with decoupled motions

81

Table 2.12. Structural parameters a of translational parallel mechanisms in Figs. 2.43 and 2.44

No. Structural Solution parameter PPR*-PC*Pass Fig. 2.43 1 m 8 2 p1 3 p2 6 3 p 9 4 q 2 5 k1 1 6 k2 1 7 k 2 8 9 (RG1) ( vx , v y ,ZE )

PP-PR*Pa*C*R* Fig. 2.44a 9 2 8 10 2 1 1 2 ( vx , v y )

PPS*-PPa*P* Fig. 2.44b 8 3 6 9 2 1 1 2 ( v x , v y , ZD , Z E , ZG )

10

(RG2)

( v x , v y , v z , ȦD , ȦG )

( v x , v y , v z , ȦD , ȦE , ȦG ) ( v x , v y , v z )

11 12 13 14 15 16 17

SG1 SG2 rG1 rG2 MG1 MG2 (RF)

3 5 0 6 3 5 ( vx , v y )

2 6 0 6 2 6 ( vx , v y )

5 3 0 6 5 3 ( vx , v y )

18 19 20 21 22 23 24

SF rl rF MF NF TF

2 6 12 2 0 0 3

2 6 12 2 0 0 2

2 6 12 2 0 0 5

fj

11

12

9

fj

14

14

14

25 26

¦ ¦ ¦

p1 j 1 p2 j 1 p j 1

fj

a

See footnote of Table 2.1 for the nomenclature of structural parameters

Solutions with rotating actuators

Non overconstrained solutions F m G1–G2 with rotating actuators and decoupled motions have

¦

p 1

fi

26 when two complex limbs are used.

They are derived from the solution in Fig. 2.38a by introducing thirteen idle mobilities: four outside the parallelogram loops and three inside each

82

2 Translational parallel robots with two degrees of freedom

Fig. 2.46. Non overconstrained translational parallel mechanisms of types Pa*PP*-Pa*Pa*S* (a) and Pa*P-RC*S* (b), limb topologies Pa* A P A ||P* and Pa*||Pa*S* (a), Pa* A P and R||C*S* (b)

parallelogram loop. For example, the solution in Fig. 2.46a has G1-limb of type Pa*PP* and G2-limb of type Pa*Pa*S*. Equations (1.2)–(1.8) and (1.17) give the following structural parameters for this solution: MG1 = SG1 = 3, MG2 = SG2 = 5, (RG1) = ( v x , v y , v z ), (RG2) = ( v x , v y , ȦD , ȦE , ȦG ), (RF) = ( v x , v y ), MF = SF = 2, NF = 0 and TF = 0 (see Table 2.13). Non overconstrained solutions F m G1–G2 with rotating actuators and decoupled motions have

¦

p 1

fi

14 when a complex and a simple limb

are used. They are derived from the solution in Fig. 2.39a by introducing three idle mobilities inside and four outside the parallelogram loop. For example, the solution in Fig. 2.46b has G1-limb of type Pa*P and G2-limb

2.2 T2-type translational parallel robots with decoupled motions

83

Table 2.13. Structural parametersa of translational parallel mechanisms in Fig. 2.46

No.

1 2 3 4 5 6 7 8 9

Structural Solution parameter Pa*PP*-Pa*Pa*S Fig. 2.46a m 12 p1 6 p2 9 p 15 q 4 k1 0 k2 2 k 2 (RG1) ( vx , v y , vz )

10

(RG2)

( v x , v y , ȦD , ȦE , ȦG )

( v x , v y , v z , ȦD , ȦE , ȦG )

11 12 13 14 15 16 17

SG1 SG2 rG1 rG2 MG1 MG2 (RF)

3 5 6 12 3 5 ( vx , v y )

2 6 6 0 2 6 ( vx , v y )

18 19 20 21 22 23 24

SF rl rF MF NF TF

2 18 24 2 0 0 9

2 6 12 2 0 0 8

fj

17

6

fj

26

14

25 26

¦ ¦ ¦

p1 j 1 p2 j 1 p j 1

fj

Pa*P-RC*S* Fig. 2.46b 7 5 3 8 2 1 1 2 ( vx , v y )

a

See footnote of Table 2.1 for the nomenclature of structural parameters

of type RC*S*. Equations (1.2)–(1.8) and (1.17) give the following structural parameters for this parallel mechanism: MG1 = SG1 = 2, MG2 = SG2 = 6, (RG2) = ( v x , v y , v z , ȦD , ȦE , ȦG ), (RG1) = ( v x , v y ), (RF) = ( v x , v y ), MF = SF =

2, NF = 0 and TF = 0 (see Table 2.13).

84

2 Translational parallel robots with two degrees of freedom

Solutions with rotating and linear actuators

Non overconstrained solutions F m G1–G2 with rotating and linear actuators and decoupled motions have

¦

p 1

fi

20 when two complex limbs are

used. They are derived from the solution in Fig. 2.40a by introducing ten idle mobilities: four outside the parallelogram loops and three inside each parallelogram loop. For example, the solution in Fig. 2.47 has G1-limb of type Pa*PP* and G2-limb of type Pa*PaS*. For this solution, Eqs. (1.2)–(1.8) and (1.17) give the following structural parameters: MG1 = SG1 = 3, MG2 = SG2 = 5, (RG1) = ( v x , v y , v z ), (RG2) = ( v x , v y , ȦD , ȦE , ȦG ), (RF) = ( v x , v y ), MF = SF =

2, NF = 0 and TF = 0. The solutions in Figs. 2.46a and 2.47 have the same structural parameters (see Table 2.13). Non overconstrained solutions F m G1–G2 with rotating and linear actuators and decoupled motions have

¦

p 1

fi

14 when a complex and a

simple limb are used. They are derived from the solutions in Figs. 2.41a and 2.42a by introducing three idle mobilities in the parallelogram loop and four outside the parallelogram loop. For example, the solutions in Fig. 2.48 have G1-limb of type Pa*P and G2-limb of type PC*S* (Fig. 2.48a) or C*PS* (Fig. 2.48b). Equations (1.2)–(1.8) and (1.17) give the following structural parameters for both solutions: MG1 = SG1 = 2, MG2 = SG2 = 6, (RG1) = ( v x , v y ), (RG2) = ( v x , v y , v z , ȦD , ȦE , ȦG ), (RF) = ( v x , v y ), MF = SF =

2, NF = 0 and TF = 0. The solutions in Figs. 2.46b and 2.48 have the same structural parameters (see Table 2.13).

Fig. 2.47. Pa*PP*-PPa*S*-type non overconstrained translational parallel mechanism, limb topologies Pa* A P A ||P* and P A Pa*-S*

2.3 T2-type translational parallel robots with uncoupled motions

85

Fig. 2.48. Non overconstrained translational parallel mechanisms of types Pa*PPC*S* (a) and Pa*P-C*PS* (b), limb topologies Pa* A P and P A C*-S* (a), Pa* A P and C* A PS* (b)

2.3 T2-type translational parallel robots with uncoupled motions T2-type translational parallel robots with uncoupled motions and various degrees of overconstraint can be actuated by linear or rotating motors. In these solutions, both operational velocities depend on just one actuated joint velocity: v1 v1 ( &q1 ) and v2 v2 ( &q2 ) .

86

2 Translational parallel robots with two degrees of freedom

2.3.1 Overconstrained solutions The overconstrained solutions of T2-type translational parallel robots with uncoupled motions and q independent loops meet the same condition as their counterparts with coupled and decoupled motions, that is p ¦ 1 fi 2 6q . Solutions with rotating actuators

Overconstrained T2-type translational parallel robots with uncoupled motions and rotating actuators can be obtained by using two limbs PaPtype (Fig. 2.49a). Each limb integrates a parallelogram loop and p ¦ 1 fi 20 .

Fig. 2.49. Translational parallel mechanisms of types 2PaP with ten overconstraints (a) PaPP*-PaPS* with six overconstraints (b), limb topologies Pa A P (a) Pa A P A ||P* and Pa A P-S* (b)

2.3 T2-type translational parallel robots with uncoupled motions

87

Fig. 2.50. 2Pa*P-type translational parallel mechanism with four overconstraints, limb topology Pa* A P

These overconstrained solutions have 1 NF 10 and may integrate up to nine idle mobilities. The example presented in Fig. 2.49b integrates four idle mobilities outside the parallelogram loops and that in Fig. 2.50 has six idle mobilities inside the parallelogram loops. The solutions in Fig. 2.49a and b have the same structural parameters as their counterparts with reciprocal limb structures in Figs. 2.1 and 2.2b (see Table 2.1). Equations (1.2)–(1.8) and (1.17) give the following structural parameters for the solution 2Pa*P-type in Fig. 2.50: MG1 = SG1 = 3, MG2 = SG2 = 3, (RG1) = (RG2) = ( v x , v y ), (RF) = ( v x , v y ), MF = SF = 2, NF = 4 and TF = 0. Solutions with rotating and linear actuators

Overconstrained T2-type translational parallel robots with uncoupled motions actuated by rotating and linear motors can be obtained by using a complex limb PaP-type and a simple one PP-type (Fig. 2.51a). p The complex limb integrates a parallelogram loop and ¦ 1 f i 14 . The overconstrained solutions have 1 NF 7 and may integrate up to six idle mobilities. The example presented in Fig. 2.51b combines three idle mobilities in the parallelogram loop and that in Fig. 2.51c has four idle mobilities outside the parallelogram loop. The solutions in Fig. 2.51a and b have the same structural parameters as their counterparts with reciprocal limb structures in Fig. 2.35a and b (see Table 2.10). Equations (1.2)–(1.8) and (1.17) give the following structural parameters for the solution PaPP*-PPS*-type

88

2 Translational parallel robots with two degrees of freedom

Fig. 2.51. Translational parallel mechanisms of types PaP-PP with seven overconstraints (a), Pa*P-PP with four overconstraints (b) and PaPP*- PPS* with three overconstraints (c), limb topologies Pa A P and P A P (a), Pa* A P and P A P (b), Pa A P A ||P* and P A P-S* (c)

in Fig. 2.51c: MG1 = SG1 = 3, MG2 = SG2 = 5, MF = SF = 2, (RG1) = ( v x , v y , v z ), (RG2) = ( v x , v y , ȦD , ȦE , ȦG ), (RF) = ( v x , v y ), NF = 0 and TF = 0.

2.3 T2-type translational parallel robots with uncoupled motions

89

2.3.2 Non overconstrained solutions

Non overconstrained solutions of T2-type translational parallel robots with uncoupled motions and q independent loops meet the same condition p ¦ 1 fi 2 6q as their counterparts with coupled and decoupled motions. Solutions with rotating actuators

Non overconstrained solutions F m G1–G2 with rotating actuators and uncoupled motions have

¦

p 1

fi

20 . They are derived from the solution in

Fig. 2.52. Non overconstrained translational parallel mechanisms of types Pa*PP*-Pa*PS*(a) and PassPP*-PassPR* (b), limb topologies Pa* A P A ||P* and Pa* A P-S*(a), Pass A P A ||P* and Pass A P A ||R* (b)

90

2 Translational parallel robots with two degrees of freedom

Table 2.14. Structural parametersa of translational parallel mechanisms in Fig. 2.52

No.

1 2 3 4 5 6 7 8 9

Structural Solution parameter Pa*PP*-Pa*PS* Fig. 2.52a m 10 p1 6 p2 6 p 12 q 3 k1 0 k2 2 k 2 (RG1) ( vx , v y , vz )

10

(RG2)

( v x , v y , ȦD , ȦE , ȦG )

( v x , v y , ȦD , ȦG )

11 12 13 14 15 16 17

SG1 SG2 rG1 rG2 MG1 MG2 (RF)

3 5 6 6 3 5 ( vx , v y )

4 4 6 6 4 4 ( vx , v y )

18 19 20 21 22 23 24

SF rl rF MF NF TF

2 12 18 2 0 0 9

2 12 18 2 0 0 10

fj

11

10

fj

20

20

25 26

¦ ¦ ¦

p1 j 1 p2 j 1 p j 1

fj

PassPP*-PassPR* Fig. 2.52b 10 6 6 12 3 0 2 2 ( v x , v y , v z , ȦE )

a

See footnote of Table 2.1 for the nomenclature of structural parameters

Fig. 2.49a by introducing ten idle mobilities: four outside the parallelogram loops and three inside each parallelogram loop. For example, the solution in Fig. 2.52a has G1-limb of type Pa*PP* and G2-limb of type Pa*PS*. The solution in Fig. 2.52b has G1-limb of type PassPP* and G2-limb of type PassPR*. The structural parameters of these solutions are presented in Table 2.14.

2.3 T2-type translational parallel robots with uncoupled motions

91

Solutions with rotating and linear actuators

Non overconstrained solutions F m G1–G2 with uncoupled motions actuated by rotating and linear motors have

¦

p 1

fi

14 . They are derived from the

solution in Fig. 2.51a by introducing seven idle mobilities: four outside the parallelogram loops and three inside the parallelogram loop. For example, the solution in Fig. 2.53a has G1-limb of type Pa*PP* and G2-limb of type PPS*. The solution in Fig. 2.53b has G1-limb of type PassPC* and G2-limb of type PPR*. The structural parameters of these solutions are presented in Table 2.15.

Fig. 2.53. Non overconstrained translational parallel mechanisms of types Pa*PP*-PPS* (a) and PassPC*-PPR* (b), limb topologies Pa*P A ||P* and P A P-S* (a), Pass A P A ||C* and P A P A ||R* (b)

92

2 Translational parallel robots with two degrees of freedom

Table 2.15. Structural parametersa of translational parallel mechanisms in Fig. 2.53

No.

1 2 3 4 5 6 7 8 9

Structural Solution parameter Pa*PP*-PPS* Fig. 2.53a m 8 p1 6 p2 3 p 9 q 2 k1 1 k2 1 k 2 (RG1) ( vx , v y , vz )

10

(RG2)

( v x , v y , ȦD , ȦE , ȦG )

( v x , v y , ȦD )

11 12 13 14 15 16 17

SG1 SG2 rG1 rG2 MG1 MG2 (RF)

3 5 6 0 3 5 ( vx , v y )

5 3 6 0 5 3 ( vx , v y )

18 19 20 21 22 23 24

SF rl rF MF NF TF

2 6 12 2 0 0 9

2 6 12 2 0 0 11

fj

5

3

fj

14

14

25 26 a

¦ ¦ ¦

p1 j 1 p2 j 1 p j 1

fj

PassPC*-PPR* Fig. 2.53b 8 6 3 9 2 1 1 2 ( v x , v y , v z , ȦE , ȦG )

See footnote of Table 2.1 for the nomenclature of structural parameters

2.4 Maximally regular T2-type translational parallel robots

93

2.4 Maximally regular T2-type translational parallel robots Maximally regular T2-type translational parallel robots are actuated by linear motors and may have various degrees of overconstraint. In these solutions, both operational velocities are equal to their corresponding actuated joint velocities: v1 &q1 and v2 &q2 . We call Isoglide2-T2 the translational parallel mechanisms of this family. 2.4.1 Overconstrained solutions The overconstrained solutions of Isoglide2-T2 parallel robots have just one p closed loop and 4 d ¦ 1 f i 8 . These solutions have 1 NF 4 and could integrate up to four idle mobilities. The basic solution has two identical limbs P A P-type (Fig. 2.54) and no idle mobilities. The translations in the two limbs have orthogonal directions: PyPx-type. This solution has the following structural parameters: MG1 = SG1 = 2, MG2 = SG2 = 2, (RG1) = (RG2) = (RF) = ( v x , v y ), MF = SF = 2, NF = 4 and TF = 0. There are various possibilities to introduce up to four idle mobilities. For example, solution PPR*-PP-type in Fig. 2.55a is characterized by MG1 = SG1 = 3, MG2 = SG2 = 2, (RG1) = ( v x , v y , ȦG ), (RG2) = ( v x , v y ), NF = 3. Solution 2PC*-type in Fig. 2.55b is characterized by MG1 = SG1 = 3, MG2 = SG2 = 3, (RG1) = ( v x , v y , ȦD ), (RG2) = ( v x , v y ,ȦE ), NF = 2 and solution

PPC*-PPR*-type in Fig. 2.55c by MG1 = SG1 = 4, MG2 = SG2 = 3, (RG1) = ( v x , v y , v z , ȦG ), (RG2) = ( v x , v y ,ȦE ), NF = 1. The structural parameters of the solutions in Figs. 2.54 and 2.55 are presented in Table 2.16.

Fig. 2.54. Maximally regular 2PP-type translational parallel mechanism with four overconstraints, limb topology P A P

94

2 Translational parallel robots with two degrees of freedom

Fig. 2.55. Maximally regular parallel mechanisms of types PPR*-PP with three overconstraints (a), 2PC* with two overconstraints (b) and PPC*-PPR* with one overconstraint (c), limb topologies P A P A A R* and P A P (a), P A C (b) P A P A A C* and P A P||R* (c)

2.4 Maximally regular T2-type translational parallel robots

95

Table 2.16. Structural parametersa of translational parallel mechanisms in Figs. 2.54–2.55

No. Structural parameter 1 2 3 4 5 6 7 8 9

m p1 p2 p q k1 k2 k (RG1)

Solution 2PP Fig. 2.54 4 2 2 4 1 1 1 2 ( vx , v y )

10

(RG2)

( vx , v y )

( vx , v y )

( v x , v y , ȦD )

( v x , v y , ȦE )

11 12 13 14 15 16 17

SG1 SG2 rG1 rG2 MG1 MG2 (RF)

2 2 0 0 2 2 ( vx , v y )

3 2 0 0 3 2 ( vx , v y )

3 3 0 0 3 3 ( vx , v y )

4 3 0 0 4 3 ( vx , v y )

18 19 20 21 22 23 24

SF rl rF MF NF TF

2 0 2 2 4 0 2

2 0 3 2 3 0 3

2 0 4 2 2 0 3

2 0 5 2 1 0 4

fj

2

2

3

3

fj

4

5

6

7

25 26

¦ ¦ ¦

p1 j 1 p2 j 1 p j 1

fj

PPR*-PP Fig. 2.55a 5 3 2 5 1 1 1 2 ( v x , v y , ȦG )

2PC* Fig. 2.55b 4 2 2 4 1 1 1 2 ( v x , v y , ȦE )

PPC*-PPR* Fig. 2.55c 6 4 3 7 1 1 1 2 ( v x , v y , v y , ȦG )

a

See footnote of Table 2.1 for the nomenclature of structural parameters

2.4.2 Non overconstrained solutions The non overconstrained solutions of Isoglide2-T2 parallel robots have p ¦ 1 fi 8 . They are derived from the solution in Fig. 2.54 by introducing four idle mobilities as follows: one translation perpendicular to the motion

96

2 Translational parallel robots with two degrees of freedom

plane of the moving platform and three rotations. Various solutions exist for introducing these idle mobilities. For example, the solution in Fig. 2.56a has G1-limb of type PPP* and G2-limb of type PPS* with MG1 = SG1 = 3, MG2 = SG2 = 5, (RG1) = ( v x , v y , v z ), (RG2) = ( v x , v y , ȦD , ȦE , ȦG ). The solution in Fig. 2.56b has G1limb of type PC*C* and G2-limb of type PC* with MG1 = SG1 = 5, MG2 = SG2 = 3, (RG1) = ( v x , v y , v z , ȦE , ȦG ), (RG2) = ( v x , v y , ȦD ). Both solutions are characterized by (RF) = ( v x , v y ), MF = SF = 2, NF = 0 and TF = 0 (see Table 2.17).

Fig. 2.56. Non overconstrained maximally regular parallel mechanisms of types PPP*-PPS* (a) and PC*C*-PC* (b), limb topologies P A P A A P* and P A P-S* (a), P A C*C*-P A C* (b)

2.5 Other T2-type translational parallel robots

97

Table 2.17. Structural parameters a of translational parallel mechanisms in Fig. 2.56

No.

1 2 3 4 5 6 7 8 9

Structural Solution parameter PPP*-PPS* Fig. 2.56a m 6 p1 3 p2 3 p 6 q 1 k1 1 k2 1 k 2 (RG1) ( vx , v y , vz )

10

(RG2)

( v x , v y , ȦD , ȦE , ȦG )

( v x , v y , ȦD )

11 12 13 14 15 16 17

SG1 SG2 rG1 rG2 MG1 MG2 (RF)

3 5 0 0 3 5 ( vx , v y )

5 3 0 0 5 3 ( vx , v y )

18 19 20 21 22 23 24

SF rl rF MF NF TF

2 0 6 2 0 0 3

2 0 6 2 0 0 6

fj

5

3

fj

8

8

25 26

¦ ¦ ¦

p1 j 1 p2 j 1 p j 1

fj

PC*C*-PC* Fig. 2.56b 5 3 2 5 1 1 1 2 ( v x , v y , v z , ȦE , ȦG )

a

See footnote of Table 2.1 for the nomenclature of structural parameters

2.5 Other T2-type translational parallel robots In the previous sections of this chapter, we have presented translational parallel robots with two degrees of freedom and a unique base of the operational velocity space for any position of the characteristic point H on the moving platform. In this section, we present translational parallel mechanisms for parallel robots with two degrees of freedom and different

98

2 Translational parallel robots with two degrees of freedom

bases of their operational velocity space for any position of the characteristic point H on the moving platform. These parallel mechanisms have mobility MF = 2 and connectivity between the moving and fixed platforms SF = 2. We note that, in this case, the moving platform performs three translations but just two of them are independent (Gao et al. 2002). The solutions presented in this section have uncoupled motions and three legs. They are of type F m G1–G2–G3 in which the first two legs are actuated by rotating motors and the last leg is unactuated. The basis of the operational velocity space contains any combination of two independent translations. 2.5.1 Overconstrained solutions The basic overconstrained solutions have three identical limbs each of them combining two universal joints. The distance between the centres of the two universal joints is the same in the three limbs and the fixed and the moving platforms are identical. The centers of the universal joints connecting the limbs with the fixed and the moving platforms form two identical triangles. The spatial form of this mechanism is a triangular prism. For this reason we call this solution prism mechanism and we denote it by Pr. The limbs G1 and G2 are actuated by rotating motors. The solution in Fig. 2.57a has one actuator on the fixed base and the second actuator is mounted on a moving link. In this solution, the axes of the first revolute joint in the actuated limbs are coplanar. Both actuators are mounted on the fixed base in Fig. 2.57b. In this case, the axes of the first revolute joints in the actuated limbs are perpendicular. The three complex limbs form q = 14 independent loops and fulfil the conditions: SF = 2 and (RF) = (RG1 RG2 RG3) = ( v1 ,v2 ). Two closed loops composed of two revolute joints with the same axis exist in each universal joint in Fig. 2.57. Both solutions in Fig. 2.57 have the following structural parameters: MG1 = SG1 = 4, MG2 = SG2 = 4, MG3 = SG3 = 4, (RG1) = ( v1 ,v2 ,ȦD ,ȦE ), (RG2) = ( v1 ,v2 ,v3 ,ȦD ), (RG3) = ( v1 ,v2 ,v3 ,ȦE ), (RF) = ( v1 ,v2 ), MF = SF = 2,

2.5 Other T2-type translational parallel robots

99

Fig. 2.57. Overconstrained T2 parallel mechanisms with an additional unactuated limb and two revolute joints on each axis of types: (RRRR)2-(RRRR)2-(RRRR)2 (a), (RRRR)2-(RRRR)2-(RRRR)2 (b)

100

2 Translational parallel robots with two degrees of freedom

Fig. 2.58. Overconstrained T2 parallel mechanisms with an additional unactuated limb of types: RRRR-RRRR-RRRR (a), RRRR-RRRR-RRRR (b)

2.5 Other T2-type translational parallel robots

101

Fig. 2.59. Overconstrained T2 parallel mechanisms with an additional unactuated limb of types: RRRR-RRRR-RRS (a), RRRR-RRRR-RRS (b)

102

2 Translational parallel robots with two degrees of freedom

Fig. 2.60. Simplified representation of the overconstrained T2 parallel mechanisms with an additional unactuated limb of type Pr (a) and Prs (b)

2.5 Other T2-type translational parallel robots

103

Table 2.18. Structural parametersa of translational parallel mechanisms in Figs. 2.57 and 2.58

No. Structural parameter

1 2 3 4 5 6 7 8 9 10

m p1 p2 p3 p q k1 k2 k (RG1)

Solution (RRRR)2-(RRRR)2-(RRRR)2 (Fig. 2.57a) (RRRR)2-(RRRR)2-(RRRR)2 (Fig. 2.57b) 11 8 8 8 24 14 0 3 3 ( v1 , v2 , ȦD , ȦE )

11

(RG2)

( v1 ,v2 , v3 ,ȦD )

12

(RG3)

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦD ) ( v1 , v2 , v3 , ȦE )

13 14 15 16 17 18 19 20 21 22

SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

23 24 25 26 27 28 29

SF rl rF MF NF TF

4 4 4 4 4 4 4 4 4 ( v1 , v2 ) 2 12 22 2 62 0 8

4 4 4 0 0 0 4 4 4 ( v1 , v2 ) 2 0 10 2 2 0 4

fj

8

4

fj

8

4

fj

24

12

30 31 32 a

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

RRRR-RRRR-RRRR (Fig. 2.58a) RRRR-RRRR-RRRR (Fig. 2.58b) 11 4 4 4 12 2 3 0 3 ( v1 , v2 , ȦD , ȦE )

See footnote of Table 2.1 for the nomenclature of structural parameters

104

2 Translational parallel robots with two degrees of freedom

Table 2.19. Structural parametersa of translational parallel mechanisms in Figs. 2.59 and 2.61

No. Structural parameter

1 2 3 4 5 6 7 8 9 10

m p1 p2 p3 p q k1 k2 k (RG1)

Solution RRRR-RRRR-RRS (Fig. 2.59a) RRRR-RRRR-RRS (Fig. 2.59b) 10 4 4 3 11 2 3 0 3 ( v1 , v2 , ȦD , ȦE )

11

(RG2)

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦD , ȦG )

12

(RG3)

( v1 , v2 , v3 , ȦE , ȦG )

( v1 , v2 , v3 , ȦE , ȦG )

13 14 15 16 17 18 19 20 21 22

SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

23 24 25 26 27 28 29

SF rl rF MF NF TF

4 4 5 0 0 0 4 4 5 ( v1 , v2 ) 2 0 11 2 1 0 4

4 5 5 0 0 0 4 5 5 ( v1 , v2 ) 2 0 12 2 0 0 4

fj

4

5

fj

5

5

fj

13

14

30 31 32 a

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

RRRR-RRS-RRS (Fig. 2.61a) RRRR-RRS-RRS (Fig. 2.61b) 9 4 3 3 10 2 3 0 3 ( v1 , v2 , ȦD , ȦE )

See footnote of Table 2.1 for the nomenclature of structural parameters

2.5 Other T2-type translational parallel robots

105

NF = 62 and TF = 0 (see Table 2.18). Each closed loop in the universal joints introduces five overconstraints. If we consider just two revolute pairs in each universal joint, (Fig. 2.58), the number of overconstraint becomes NF = 2 (see Table 2.18). Note 1 (see Chapter 1) must be taken into account when establishing the bases of the operational velocity spaces associated with the solutions in Figs. 2.57–2.61. In accordance with this note, the bases of the vector spaces RGi in Table 2.18 are selected such as the minimum value of S F to be obtained by Eq. (1.6). By this choice, the result of Eq. (1.2) fits in with the definition of general mobility as the minimum value of the instantaneous mobility. Overconstrained solution with NF = 1 can be derived from the basic solutions in Fig. 2.58 by replacing a universal joint by a spherical joint that combines an idle mobility (see Fig. 2.59). The solutions in Fig. 2.59 have the following structural parameters: MG1 = SG1 = 4, MG2 = SG2 = 4, MG3 = SG3 = 5, (RG1) = ( v1 ,v2 ,ȦD ,ȦE ), (RG2) = ( v1 ,v2 ,v3 ,ȦD ), (RG3) = ( v1 ,v2 ,v3 ,ȦE ,ȦG ), (RF) = ( v1 ,v2 ), MF = SF = 2, NF

= 1 and TF = 0 (see Table 2.19). Figure 2.60 represents the simplified structural diagrams of the solutions in Figs. 2.58b and 2.59b. 2.5.2 Non overconstrained solutions

Non overconstrained solutions can be derived from the solutions in Fig. 2.59 by replacing a second universal joint by a spherical joint. The two spherical joint must be introduced in two distinct legs. The solutions in Fig. 2.61 have the following structural parameters: MG1 = SG1 = 4, MG2 = SG2 = 5, MG3 = SG3 = 5, (RG1) = ( v1 ,v2 ,ȦD ,ȦE ), (RG2) = ( v1 ,v2 ,v3 ,ȦD ,ȦG ), (RG3) = ( v1 ,v2 ,v3 ,ȦE ,ȦG ), (RF) = ( v1 ,v2 ), MF = SF = 2, NF = 0 and TF = 0 (see Table 2.19). Figure 2.61c represents a simplified structural diagram of the solution in Fig. 2.61b.

106

2 Translational parallel robots with two degrees of freedom

Fig. 2.61. Non overconstrained T2 parallel mechanisms with an additional unactuated limb of types: RRRR-RRS-RRS (a), RRRR-RRS-RRS (b) and Prss (c) – a simplified representation of the solution (b)

3 Overconstrained T3-type TPMs with coupled motions

T3-type TPMs are translational parallel robotic manipulators with three degrees of connectivity between the moving and fixed platforms SF = 3. They give three translational velocities v1, v2 and v3 in the basis of the operational velocity vector space (RF) = (v1,v2,v3) along with a constant orientation of the moving platform. Equation (1.16) indicates that overconstrained solutions of T3-type translational parallel robots with coupled motions and q independent loops p meet the condition ¦ 1 f i 3 6q . Various solutions fulfil this condition along with SF = 3, (RF) = (v1,v2,v3) and the number of overconstraints NF 1. T3-type translational parallel robots 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 combine idle mobilities. In these solutions, the three operational velocities given by Eq. (1.19) depend, in the general case, on the three actuated joint velocities: & 2 ,q & 3 ) , i = 1,2,3. In some specific solutions, each operational vi vi ( &q1 ,q velocity depends on at least two actuated joints. We note that, in this particular case, the Jacobian matrix in Eq. (1.19) is not triangular and the parallel robot always has coupled motions. They have just a few partially decoupled motions. 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. Basic and derived solutions are presented in this section. No idle mobilities exist in the basic solutions. The derived solutions are obtained from the basic solutions by combining various idle mobilities. We limit our presentation in this section to the solutions with just three limbs. A large set of solutions with an additional unactuated limb can also be obtained by combining an unactuated limb presented in Figs. 7.1–7.11 – Part 1 with other three limbs with 4 d MGi = SGi d 6 that integrate velocities v1, v2 and v3 in the basis of their operational space. 107 G. Gogu, Structural Synthesis of Parallel Robots: Part 2: Translational Topologies with Two and Three Degrees of Freedom, Solid Mechanics and Its Applications 159, 107–363. © Springer Science + Business Media B.V. 2009

108

3 Overconstrained T3-type TPMs with coupled motions

3.1 Basic solutions with linear actuators In the basic solutions of the T3-type TPMs with linear actuators and coupled motions F m G1–G2–G3, the moving platform n Ł nGi (i = 1, 2, 3) is connected to the reference platform 1 Ł 1Gi Ł 0 by three limbs with three or four degrees of connectivity. No idle mobilities exist in these basic solutions and the linear actuators are combined in a prismatic pair of each limb. The various types of limbs with three degrees of connectivity are systematized in Fig. 3.1. They integrate one (Fig. 3.1a–d) or two (Fig. 3.1e–g) parallelogram loops or a prism mechanism (Fig. 3.1h). These limbs are actuated by linear motors mounted on the fixed base. We recall that the parallelogram loop Pacc and the prism mechanism Pr have two degrees of mobility. The various types of limbs with four degrees of connectivity are systematized in Figs. 3.2 and 3.3. They are simple (Fig. 3.2) or complex (Fig. 3.3) kinematic chains and can be actuated by linear motors mounted on the fixed base. Examples of simple or complex limbs with four degrees of connectivity that can be actuated by linear actuators mounted on a moving link are presented in Figs. 3.4 and 3.5. The prismatic joints between links 2 and 3 (Fig. 3.4a, b and e–g) and 3 and 4 (Fig. 3.4c, d) must be actuated to obtain solutions with coupled motions. If the prismatic joint adjacent to the fixed base (Fig. 3.4c, d) or between links 2 and 3 (Fig. 3.4d) is actuated, solutions with uncoupled motions can be obtained as shown in the following sections. The same is true when the translation in the cylindrical joint is actuated in Fig. 3.4f. The prismatic joint can be actuated in Fig. 3.5a–d. The cylindrical joint in Figs. 3.2e, f, 3.4e, f and 3.5e replaces the combination of two successive revolute and the prismatic joints with the same axis/direction of types R||P and P||R (see Figs. 3.2a–d, 3.4a–d and 3.5c, d). The limbs in Figs. 3.3 and 3.5 combine a Pa-type parallelogram loop. Various solutions of translational parallel robots with coupled motions and no idle mobilities can be obtained by using three limbs with identical or different topology presented in Figs. 3.1–3.5. We only show solutions with identical limb type as illustrated in Figs. 3.6–3.23. The limb topology and connecting conditions in these solutions are systematized in Tables 3.1 and 3.2. The actuated prismatic joints adjacent to the fixed base in the three limbs have orthogonal directions (Figs. 3.6–3.11) in the solutions using the limbs systematized in Fig. 3.1. The structural parameters of the solutions in Figs. 3.6–3.11 are systematized in Table 3.3.

3.1 Basic solutions with linear actuators

109

The directions of the three actuated prismatic joints adjacent to the fixed base can be orthogonal (Fig. 3.12) or parallel (Figs. 3.13 and 3.14) when the limbs in Fig. 3.2 are used. They can be coplanar (Fig. 3.15a) or orthogonal in space (Fig. 3.15b) when the limbs in Fig. 3.3a, b are used. The axes of the three unactuated revolute or cylindrical joints adjacent to the moving platform can be orthogonal in space (Figs. 3.12 and 3.15b) or coplanar (Figs. 3.13, 3.14 and 3.15a). The three coplanar axes can form a planar star (Fig. 3.15a), a triangle (Figs. 3.13a, 3.14a and 3.15a), or can be situated on three sides of a rectangle (Figs. 3.13b and 3.14b). The axes of the unactuated joints adjacent to the moving platform form a configuration called a spatial star, if they are orthogonal in space, and planar star, ' or C if they are coplanar. The structural parameters of the solutions in Figs. 3.12–3.17 are systematized in Tables 3.4 and 3.5. The solutions based on the use of limb topologies presented in Figs. 3.4 and 3.5 have the linear actuators non adjacent to the fixed base. In these solutions, the axes/directions of the first unactuated joint in the three limbs can be orthogonal in space (Figs. 3.18, 3.21–3.23) or coplanar (Figs. 3.19 and 3.20). The three coplanar axes can also be in a ' (Figs. 3.19a and 3.20) or a C (Fig. 3.19b) configuration. The first unactuated joint can be a revolute (Figs. 3.18a and 3.19), a prismatic (Fig. 3.21) or a cylindrical joint (Figs. 3.18b and 3.20). The structural parameters of the solutions in Figs. 3.18–3.23 are systematized in Table 3.6.

110

3 Overconstrained T3-type TPMs with coupled motions

Table 3.1. Limb topology and connecting conditions of the TPM with no idle mobilities and linear actuators mounted on the fixed base presented in Figs. 3.6– 3.17 No. 1 2 3 4 5 6 7 8 9 10 11 12

13 14 15 16 17

TPM type 3-PPaP (Fig. 3.6) 3-PPPa (Fig. 3.7) 3-PPacc (Figs. 3.8 and 3.9) 3-PPaPa (Fig. 3.10a) 3-PPaPa (Fig. 3.10b) 3-PPaPa (Fig. 3.11a) 3-PPr (Fig. 3.11b) 3-PRC (Fig. 3.12a) 3-PCR (Fig. 3.12b) 3-PRC (Fig. 3.13a, b) 3-PCR (Fig. 3.14a, b) 3-PRPaR (Fig. 3.15a)

Limb topology P A Pa||P (Fig. 3.1a) P A P||Pa (Fig. 3.1b) P A Pacc (Fig. 3.1c, d) P A Pa A A Pa (Fig. 3.1e) P A Pa A ||Pa (Fig. 3.1f) P||Pa A Pa (Fig. 3.1g) P-Pr (Fig. 3.1h) P A R||C (Fig. 3.2e) P A C||R (Fig. 3.2f) P A R||C (Fig. 3.2e) P A C||R (Fig. 3.2f) P||R A Pa A ||R (Fig. 3.3a)

3-PRPaR (Fig. 3.15b) 3-PRRPa (Fig. 3.16a) 3-PRRPa (Fig. 3.16b) 3-PPaRR (Fig. 3.17a) 3-PPaRR (Fig. 3.17b)

P A R A Pa A ||R (Fig. 3.3b) P A R||R A ||Pa (Fig. 3.3c) P||R||R A Pa (Fig. 3.3d) P A Pa A A R||R (Fig. 3.3e) P||Pa A R||R (Fig. 3.3f)

Connecting conditions Actuated P joints with orthogonal 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 Actuated P joints with parallel directions Idem No. 10 Actuated P joints with coplanar directions (planar star configuration) Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1

3.1 Basic solutions with linear actuators

111

Table 3.2. Limb topology and connecting conditions of the TPM with no idle mobilities and linear actuators mounted on a moving link presented in Figs. 3.18– 3.23 No.

TPM type 3-RPC (Fig. 3.18a)

Limb topology R A P A ||C (Fig. 3.4e)

3-CPR (Fig. 3.18b) 3-RPC (Fig. 3.19a)

C A P A ||R (Fig. 3.4f) R A P A ||C (Fig. 3.4e)

4

3-RPC (Fig. 3.19b)

R A P A ||C (Fig. 3.4e)

5

3-CPR (Fig. 3.20a) 3-CPR (Fig. 3.20b) 3-PPRR (Fig. 3.21) 3-RPaPR (Fig. 3.22a) 3-RPaPR (Fig. 3.22b) 3-RPaRP (Fig. 3.23a) 3-RPaRP (Fig. 3.23b)

C A P A ||R (Fig. 3.4f) C A P A ||R (Fig. 3.4f) P A P A ||R||R (Fig. 3.4g) R A Pa A A P A A R (Fig. 3.5a) R A Pa A ||P||R (Fig. 3.5c) R A Pa A ||R A A P (Fig. 3.5b) R A Pa A ||R||P (Fig. 3.5d)

1

2 3

6 7 8 9 10 11

Connecting conditions Actuated P joints non adjacent to the fixed base and the moving platform in a spatial star configuration Idem No. 1 Actuated P joints non adjacent to the fixed base and the moving platform in a ' configuration Actuated P joints non adjacent to the fixed base and the moving platform in a C configuration Idem 3 Idem 4 Idem 1 Idem 1 Idem 1 Idem 1 Idem 1

112

3 Overconstrained T3-type TPMs with coupled motions

Table 3.3. Structural parametersa of translational parallel mechanisms in Figs. 3.6–3.11 No. Structural parameter

1 2 3 4 5 6 7 8 9 10

m p1 p2 p3 p q k1 k2 k (RG1)

Solution 3-PPaP (Fig. 3.6a, b) 3-PPPa (Fig. 3.7a, b) 14 6 6 6 18 5 0 3 3 ( v1 ,v2 ,v3 )

11

(RG2)

( v1 ,v2 ,v3 )

( v1 ,v2 ,v3 )

( v1 ,v2 ,v3 )

( v1 ,v2 ,v3 )

12

(RG3)

13 14 15 16 17 18 19 20 21 22

SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

23 24 25 26 27 28 29

SF rl rF MF NF TF

( v1 ,v2 ,v3 ) 3 3 3 3 3 3 3 3 3 ( v1 ,v2 ,v3 ) 3 9 15 3 15 0 6

( v1 ,v2 ,v3 ) 3 3 3 4 4 4 3 3 3 ( v1 ,v2 ,v3 ) 3 12 18 3 12 0 7

( v1 ,v2 ,v3 ) 3 3 3 6 6 6 3 3 3 ( v1 ,v2 ,v3 ) 3 18 24 3 24 0 9

( v1 ,v2 ,v3 ) 3 3 3 10 10 10 3 3 3 ( v1 ,v2 ,v3 ) 3 30 36 3 12 0 13

fj

6

7

9

13

fj

6

7

9

13

fj

18

21

27

39

30 31 32 a

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

3-PPacc (Fig. 3.8a, b) (Fig. 3.9a, b)

3-PPr 3-PPaPa (Fig. 3.10a, b) (Fig. 3.11b) (Fig. 3.11a)

11 5 5 5 15 5 0 3 3 ( v1 ,v2 ,v3 )

20 9 9 9 27 8 0 3 3 ( v1 ,v2 ,v3 )

14 7 7 7 21 8 0 3 3 ( v1 ,v2 ,v3 )

See footnote of Table 2.1 for the nomenclature of structural parameters

3.1 Basic solutions with linear actuators

113

Table 3.4. Structural parametersa of translational parallel mechanisms in Figs. 3.12–3.17 No. Structural parameter

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

8 3 3 3 9 2 3 0 3 See Table 3.5

3-PRPaR (Fig. 3.15) 3-PRRPa (Fig. 3.16) 3-PPaRR (Fig. 3.17) 17 7 7 7 21 5 0 3 3 See Table 3.5

4 4 4 0 0 0 4 4 4 ( v1 ,v2 ,v3 ) 3 0 9 3 3 0 4

4 4 4 3 3 3 4 4 4 ( v1 ,v2 ,v3 ) 3 9 18 3 12 0 7

fj

4

7

fj

4

7

fj

12

21

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

Solution 3-PRC (Figs. 3.12a, 3.13, 3.14) 3-PCR (Fig. 3.12b)

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

114

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.1. Complex limbs for TPMs with coupled motions defined by MG = SG = 3, (RG) = (v1,v2,v3) and actuated by linear motors mounted on the fixed base

3.1 Basic solutions with linear actuators

115

Fig. 3.2. Simple limbs for TPMs with coupled motions defined by MG = SG = 4, (RG) = ( v1 , v2 , v3 ,Ȧ1 ) and actuated by linear motors mounted on the fixed base

116

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.3. Complex limbs for TPMs with coupled motions defined by MG = SG = 4, (RG) = ( v1 , v2 , v3 ,Ȧ1 ) and actuated by linear motors mounted on the fixed base

3.1 Basic solutions with linear actuators

117

Fig. 3.4. Simple limbs for TPMs with coupled motions defined by MG = SG = 4, (RG) = ( v1 , v2 , v3 ,Ȧ1 ) and actuated by linear motors mounted in the prismatic joint between links 2, 3 (a, b, e–g) and 3, 4 (c, d)

118

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.5. Complex limbs for TPMs with coupled motions defined by MG = SG = 4, (RG) = ( v1 , v2 , v3 ,Ȧ1 ) and actuated by linear motors mounted in the prismatic or cylindrical joint non adjacent to the fixed base

3.1 Basic solutions with linear actuators

119

Fig. 3.6. 3-PPaP-type overconstrained TPMs with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 15, limb topology P A Pa||P

120

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.7. 3-PPPa-type overconstrained TPMs with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 15, limb topology P A P||Pa

3.1 Basic solutions with linear actuators

121

Fig. 3.8. 3-PPacc-type overconstrained TPMs with coupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, linear actuators on the fixed base and six cylindrical joints adjacent to the moving platform, limb topology P A Pacc

122

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.9. 3-PPacc-type overconstrained TPMs with coupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, linear actuators on the fixed base and six revolute joints adjacent to the moving platform, limb topology P A Pacc

3.1 Basic solutions with linear actuators

123

Fig. 3.10. 3-PPaPa-type overconstrained TPMs with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 24, limb topology P A Pa A A Pa (a) and P A Pa A ||Pa (b)

124

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.11. Overconstrained TPMs of types 3-PPaPa (a) and 3-PPr (b) with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0 and NF = 24 (a), NF = 12 (b), limb topology P||Pa A Pa (a) and P A Pr (b)

3.1 Basic solutions with linear actuators

125

Fig. 3.12. Overconstrained TPMs of types 3-PRC (a) and 3-PCR (b) with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology P A R||C (a) and P A C||R (b)

126

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.13. 3-PRC-type overconstrained TPMs with coupled motions and linear actuators of parallel directions mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology P A R||C

3.1 Basic solutions with linear actuators

127

Fig. 3.14. 3-PCR-type overconstrained TPMs with coupled motions and linear actuators of parallel directions mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology P A C||R

128

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.15. 3-PRPaR-type overconstrained TPMs with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology P||R A Pa A ||R (a) and P A R A Pa A ||R (b)

3.1 Basic solutions with linear actuators

129

Fig. 3.16. 3-PRRPa-type overconstrained TPMs with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology P A R||R A ||Pa (a) and P||R||R A Pa (b)

130

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.17. 3-PPaRR-type overconstrained TPMs with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology P A Pa A A R||R (a) and P||Pa A R||R (b)

3.1 Basic solutions with linear actuators

131

Fig. 3.18. Overconstrained TPMs of types 3-RPC (a) and 3-CPR (b) with coupled motions and linear actuators non adjacent to the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology R A P A ||C (a) and C A P A ||R (b)

132

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.19. 3-RPC-type overconstrained TPMs with coupled motions and linear actuators non adjacent to the fixed base in ' (a) and C (b) configurations defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology R A P A ||C

3.1 Basic solutions with linear actuators

133

Fig. 3.20. 3-CPR-type overconstrained TPMs with coupled motions and linear actuators non adjacent to the fixed base in ' (a) and C (b) configurations defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology C A P A ||R

134

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.21. 3-PPRR-type overconstrained TPMs with coupled motions and linear actuators non adjacent to the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology P A P A ||R||R Table 3.5. Bases of the operational velocities spaces of the limbs isolated from the parallel mechanisms presented in Figs. 3.12–3.23 No. Parallel mechanism 1 Fig. 3.12

Basis (RG1) ( v1 , v2 , v3 , ȦE )

(RG2) ( v1 ,v2 , v3 ,ȦG )

(RG3) ( v1 ,v2 , v3 ,ȦD )

2

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦD )

( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦG )

3

4 5

Figs. 3.13, 3.14, 3.15a, 3.19, 3.20 Figs. 3.15b, 3.17b, 3.18, 3.21, 3.22a, 3.23a Figs. 3.16a, 3.17a, 3.22b, 3.23b Fig. 3.16b

3.1 Basic solutions with linear actuators

135

Fig. 3.22. 3-RPaPR-type overconstrained TPMs with coupled motions and linear actuators non adjacent to the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology R A Pa A A P A A R (a) and R A Pa A ||P||R (b)

136

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.23. 3-RPaRP-type overconstrained TPMs with coupled motions and linear actuators non adjacent to the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology R A Pa A ||R A A P (a) and R A Pa A ||R||P (b)

3.1 Basic solutions with linear actuators

137

Table 3.6. Structural parametersa of translational parallel mechanisms in Figs. 3.18–3.23 No.

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

Structural parameter

Solution 3-RPC 3-PPRR (Figs. 3.18a, 3.19) (Fig. 3.21) 3-CPR (Figs. 3.18b, 3.20)

3-RPaPR (Fig. 3.22) 3-RPaRP (Fig. 3.23)

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

8 3 3 3 9 2 3 0 3 See Table 3.5

11 4 4 4 12 2 3 0 3 See Table 3.5

17 7 7 7 21 5 0 3 3 See Table 3.5

4 4 4 0 0 0 4 4 4 ( v1 ,v 2 ,v3 ) 3 0 9 3 3 0 4

4 4 4 0 0 0 4 4 4 ( v1 ,v 2 ,v3 ) 3 0 9 3 3 0 4

4 4 4 3 3 3 4 4 4 ( v1 ,v 2 ,v3 ) 3 9 18 3 12 0 7

fj

4

4

7

fj

4

4

7

fj

12

12

21

SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

138

3 Overconstrained T3-type TPMs with coupled motions

3.2 Derived solutions with linear actuators Solutions with lower degrees of overconstraint can be derived from the basic solutions in Figs. 3.6–3.23 by using joints with idle mobilities. A large set of solutions can be obtained by introducing one or two rotational idle mobilities outside the parallelogram loops and up to three idle mobilities (two rotations and one translation) in the parallelogram loop. The joints combining idle mobilities are denoted by an asterisk. The idle mobilities combined in a parallelogram loop (see Fig. 6.3 – Part 1) are systematized in Table 3.7. For example, the solutions of overconstrained TPMs with identical limbs derived from the basic solution 3-PPaP in Fig. 3.6a are systematized in Tables 3.8–3.10. One rotational idle mobility is combined in each cylindrical joint C* outside the parallelogram loop. The rotational mobility of the revolute joint denoted by R* is an idle mobility. In the same way, a large set of overconstrained solutions can be derived from each solution in Figs. 3.6–3.11. Examples of solutions with identical limbs and three to twelve degrees of overconstraint derived from the basic solutions in Figs. 3.6–3.11, 3.15– 3.17, 3.22 and 3.23 are illustrated in Figs. 3.24–3.42. The limb topology and the number of overconstraints of these solutions are systematized in Tables 3.11 and 3.12, and the structural parameters in Tables 3.13–3.17. Two idle rotational mobilities are introduced in the spherical joint of the parallelogram loops denoted by Paccs and Pascc which combines two cylindrical, one revolute and one spherical joint (see Fig. 3.34). Solutions with non identical limbs can be also obtained by various combinations of the idle mobilities in the three limbs. Examples of overconstrained solutions with idle mobilities and non identical limbs derived from the basic solutions in Figs. 3.12–3.15 and 3.18–3.21 are presented in Figs. 3.43–3.59. The limb topology and the number of overconstraints of these solutions are systematized in Tables 3.18 and 3.19, and the structural parameters in Tables 3.20–3.24. The linear actuators are mounted on the fixed base in the solutions in Figs. 3.43–3.52 and on a moving link in the solutions in Figs. 3.53–3.59. In the cylindrical joint denoted by C* in Figs. 3.53–3.58, the translation is the actuated motion and the rotation is the idle mobility.

3.2 Derived solutions with linear actuators

139

Table 3.7. Parallelogram loops with idle mobilities No. Parallelogram loop 1 Pac (Fig. 6.3b – Part 1) 2 3 4 5

6

7

Idle mobilities One translational idle mobility combined in a cylindrical joint Pau (Fig. 6.3c – Part 1) One rotational idle mobility combined in a universal joint Pas (Fig. 6.3d – Part 1) Two rotational idle mobilities combined in a spherical joint Pauu (Fig. 6.3e – Part 1) Two rotational idle mobilities combined in two universal joints Pacu (Fig. 6.3f – Part 1) One translational idle mobility combined in a cylindrical joint and one rotational idle mobilities combined in a universal joint Pacs , Pa* (Fig. 6.3g – Part 1) One translational idle mobility combined in a cylindrical joint and two rotational idle mobilities combined in a spherical joint Pass (Fig. 6.3h – Part 1) Three idle mobilities combined in two spherical joints adjacent to the same link with a complementary rotational mobility

Table 3.8. Overconstrained TPMs with idle mobilities in the parallelogram loop derived from the basic solution 3-PPaP in Fig. 3.6a No. 1 2 3 4 5 6

Derived TPM 3-PPacP 3-PPauP 3-PPasP 3-PPauuP 3-PPacuP 3-PPacsP

Limb topology P A Pac||P P A Pau||P P A Pas||P P A Pauu||P P A Pacu||P P A Pacs||P

NF 12 12 9 9 9 6

140

3 Overconstrained T3-type TPMs with coupled motions

Table 3.9. Overconstrained TPMs with idle mobilities outside the parallelogram loop derived from the basic solution 3-PPaP in Fig. 3.6a No. Derived TPM 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

3-PPaPR*

3-PPaR*P

3-PPaC* (Fig. 3.24) 3-PR*PaP (Fig. 3.25a)

3-R*PPaP

3-PPaPR*R* 3-PPaR*PR* 3-PPaC*R* 3-PR*PaPR* 3-PR*PaC* 3-R*PPaPR* 3-R*PPaC* 3-PR*PaR*P 3-R*PPaR*P 3-PR*R*PaP 3-R*R*PPaP

Limb topology A

P A Pa||P A R (Fig. 8.13b – Part 1) P A Pa||P A ||R (Fig. 8.13c – Part 1) P A Pa||P||R (Fig. 8.13h – Part 1) P A Pa A A R A ||P (Fig. 8.11h – Part 1) P A Pa A ||R A ||P (Fig. 8.11g – Part 1) P A Pa||R||P (Fig. 8.12h – Part 1) P A Pa||C P A R A A Pa||P (Fig. 8.11c – Part 1) P A R||Pa||P (Fig. 8.11f – Part 1) P||R A Pa||P (Fig. 8.8d – Part 1) R A P A A Pa||P (Fig. 8.21f – Part 1) R A P A ||Pa||P (Fig. 8.21d – Part 1) R||P A Pa||P (Fig. 8.21e – Part 1) P A Pa||P A A R A ||R (Fig. 9.30f – Part 1) P A Pa||P||R A A R (Fig. 9.30i – Part 1) P A Pa||R||P A A R (Fig. 9.31h – Part 1) P A Pa||C A A R P A R A A Pa||P||R (Fig. 9.19n – Part 1) P A R A A Pa||C R A P A A Pa||P||R R A P A A Pa||C P A R A A Pa||R||P (Fig. 9.19q – Part 1) R A P A A Pa||R||P P A R A A R||Pa||P (Fig. 9.24g – Part 1) P A R A A R A ||Pa||P (Fig. 9.24b – Part 1) R A R A A P A A Pa||P (Fig. 9.56e – Part 1)

NF 12 12 12 12 12 12 12 12 12 12 12 12 12 9 9 9 9 9 9 9 9 9 9 9 9 9

3.2 Derived solutions with linear actuators

141

Table 3.10. Overconstrained TPMs with idle mobilities inside and outside the parallelogram loop derived from the basic solution 3-PPaP in Fig. 3.6a No. 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 34 35 36 37 38 39

Derived TPM 3-PPacPR* 3-PPauPR*

3-PPacR*P 3-PPauR*P 3-PPacC* 3-PPauC* 3-PR*PacP 3-PR*PauP

3-R*PPacP 3-R*PPauP 3-PPacPR*R* 3-PPauPR*R* 3-PPacR*PR* 3-PPauR*PR* 3-PPacC*R* 3-PPauC*R* 3-PR*PacPR* 3-PR*PauPR* 3-PR*PacC* 3-PR*PauC* 3-R*PPacPR*

Limb topology P A Pac||P A A R P A Pac||P A ||R P A Pac||P||R P A Pau||P A A R P A Pau||P A ||R P A Pau||P||R P A Pac A A R A ||P P A Pac A ||R A ||P P A Pac||R||P P A Pau A A R A ||P P A Pau A ||R A ||P P A Pau||R||P P A Pac||C P A Pau||C P A R A A Pac||P P A R||Pac||P P||R A Pac||P P A R A A Pau||P P A R||Pau||P P||R A Pau||P R A P A A Pac||P R A P A ||Pac||P R||P A Pac||P R A P A A Pau||P R A P A ||Pau||P R||P A Pau||P P A Pac||P A A R A ||R P A Pac||P||R A A R P A Pau||P A A R A ||R P A Pau||P||R A A R P A Pac||R||P A A R P A Pau||R||P A A R P A Pac||C A A R P A Pau||C A A R P A R A A Pac||P||R P A R A A Pau||P||R P A R A A Pac||C P A R A A Pau||C R A P A A Pac||P||R

NF 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 6 6 6 6 6 6 6 6 6 6 6 6 6

142

3 Overconstrained T3-type TPMs with coupled motions Table 3.10. (cont.)

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79

u

3-R*PPa PR* 3-R*PPacC* 3-R*PPauC* 3-PR*PacR*P 3-PR*PauR*P 3-R*PPacR*P 3-R*PPauR*P 3-PR*R*PacP 3-PR*R*PauP 3-R*R*PPacP 3-R*R*PPauP 3-PPasPR* 3-PPauuPR*

3-PPacuPR* 3-PPasR*P 3-PPauuR*P

3-PPacuR*P 3-PPasC* 3-PPauuC* 3-PPacuC* PR*PasP PR*PauuP

R A P A A Pau||P||R R A P A A Pac||C R A P A A Pau||C P A R A A Pac||R||P P A R A A Pau||R||P R A P A A Pac||R||P R A P A A Pau||R||P P A R A A R||Pac||P P A R A A R A ||Pac||P P A R A A R||Pau||P P A R A A R A ||Pau||P R A R A A P A A Pac||P R A R A A P A A Pau||P P A Pas||P A A R P A Pas||P A ||R P A Pas||P||R P A Pauu||P A A R P A Pauu||P A ||R P A Pauu||P||R P A Pacu||P A A R P A Pacu||P A ||R P A Pacu||P||R P A Pas A A R A ||P P A Pas A ||R A ||P P A Pas||R||P P A Pauu A A R A ||P P A Pauu A ||R A ||P P A Pauu||R||P P A Pacu A A R A ||P P A Pacu A ||R A ||P P A Pacu||R||P P A Pas||C P A Pauu||C P A Pacu||C P A R A A Pas||P P A R||Pas||P P||R A Pas||P P A R A A Pauu||P P A R||Pauu||P P||R A Pauu||P

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

3.2 Derived solutions with linear actuators Table 3.10. (cont.) 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

cu

PR*Pa P 3-R*PPasP

3-R*PPauuP 3-R*PPacuP 3-PPassP (Fig. 3.25a) 3-PPacsPR* 3-PPacsPR* 3-PPacsR*P

3-PPacsR*P 3-PPacsC* 3-PPacsC* 3-PR*PacsP 3-PR*PacsP

3-R*PPacsP 3-R*PPacsP 3-PPasPR*R*

P A R A A Pacu||P P A R||Pacu||P P||R A Pacu||P R A P A A Pas||P R A P A ||Pas||P R||P A Pas||P R A P A A Pauu||P R A P A ||Pauu||P R||P A Pauu||P R A P A A Pacu||P R A P A ||Pacu||P R||P A Pacu||P P A Pass||P P A Pacs||P A A R P A Pacs||P A ||R P A Pacs||P||R P A Pacs||P A A R P A Pacs||P A ||R P A Pacs||P||R P A Pacs A A R A ||P P A Pacs A ||R A ||P P A Pacs||R||P P A Pacs A A R A ||P P A Pacs A ||R A ||P P A Pacs||R||P P A Pacs||C P A Pacs||C P A R A A Pacs||P P A R||Pacs||P P||R A Pacs||P P A R A A Pacs||P P A R||Pacs||P P||R A Pacs||P R A P A A Pacs||P R A P A ||Pacs||P R||P A Pacs||P R A P A A Pacs||P R A P A ||Pacs||P R||P A Pacs||P P A Pas||P A A R A ||R P A Pas||P||R A A R

6 6 6 6 6 6 6 6 6 6 6 6 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

143

144

3 Overconstrained T3-type TPMs with coupled motions Table 3.10. (cont.)

121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157

uu

3-PPa PR*R* 3-PPacuPR*R* 3-PPasR*PR* 3-PPauuR*PR* 3-PPacuR*PR* 3-PPasC*R* 3-PPauuC*R* 3-PPacuC*R* 3-PR*PasPR* 3-PR*PauuPR* 3-PR*PacuPR* 3-PR*PasC* 3-PR*PauuC* 3-PR*PacuC* 3-R*PPasPR* 3-R*PPauuPR* 3-R*PPacuPR* 3-R*PPasC* 3-R*PPauuC* 3-R*PPacuC* 3-PR*PasR*P 3-PR*PauuR*P 3-PR*PacuR*P 3-R*PPasR*P 3-R*PPauuR*P 3-R*PPacuR*P 3-PR*R*PasP 3-PR*R*PauuP 3-PR*R*PacuP 3-R*R*PPasP 3-R*R*PPauuP 3-R*R*PPacuP

P A Pauu||P A A R A ||R P A Pauu||P||R A A R P A Pacu||P A A R A ||R P A Pacu||P||R A A R P A Pas||R||P A A R P A Pauu||R||P A A R P A Pacu||R||P A A R P A Pas||C A A R P A Pauu||C A A R P A Pacu||C A A R P A R A A Pas||P||R P A R A A Pauu||P||R P A R A A Pacu||P||R P A R A A Pas||C P A R A A Pauu||C P A R A A Pacu||C R A P A A Pas||P||R R A P A A Pauu||P||R R A P A A Pacu||P||R R A P A A Pas||C R A P A A Pauu||C R A P A A Pacu||C P A R A A Pas||R||P P A R A A Pauu||R||P P A R A A Pacu||R||P R A P A A Pas||R||P R A P A A Pauu||R||P R A P A A Pacu||R||P P A R A A R||Pas||P P A R A A R A ||Pas||P P A R A A R||Pauu||P P A R A A R A ||Pauu||P P A R A A R||Pacu||P P A R A A R A ||Pacu||P R A R A A P A A Pas||P R A R A A P A A Pauu||P R A R A A P A A Pacu||P

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

3.2 Derived solutions with linear actuators

145

Table 3.11. Limb topology and the number of overconstraints NF of the derived TPMs with idle mobilities and linear actuators mounted on the fixed base presented in Figs. 3.24–3.38 No. Basic TPM type 1 3-PPaP (Fig. 3.6a) 2

NF 15

3 4

3-PPPa (Fig. 3.7a)

15

5 6 7

3-PPacc (Fig. 3.8a)

12

8 9

3-PPacc (Fig. 3.9a)

12

10 11

3-PPaPa (Fig. 3.10a)

24

12 13

3-PPaPa (Fig. 3.10b)

24

14 15

3-PPaPa (Fig. 3.11a)

24

16 17

3-PPr (Fig. 3.11b)

12

18 19 20

3-PRPaR (Fig. 3.15a)

12

Derived TPM type 3-PPaC* (Fig. 3.24a) 3-PR*PaP (Fig. 3.25a) 3-PPassP (Fig. 3.26a) 3-PC*Pa (Fig. 3.24b) 3-PPR*Pa (Fig. 3.25b) 3-PPPass (Fig. 3.26b) 3-PR*Pacc (Fig. 3.28a) 3-PR*Pascc (Fig. 3.27a) 3-PR*Pacc (Fig. 3.28b) 3-PR*Paccs (Fig. 3.27b) 3-PPaPass (Fig. 3.29a) 3-PPassPa (Fig. 3.30a) 3-PPaPass (Fig. 3.29b) 3-PPassPa (Fig. 3.30b) 3-PPaPass (Fig. 3.31a) 3-PPassPa (Fig. 3.31b) 3-PPrss (Fig. 3.34a) 3-PPrssR* (Fig. 3.34b) 3-PR*RPaR (Fig. 3.32a) 3-PRPass (Fig. 3.33a)

NF 12

Limb topology P A Pa||C*

12

P A R* A A Pa||P

3

P A Pass||P

12

P A C*||Pa

12

P A P A A R* A ||Pa

3

P A P||Pass

9

P A R* A A Pacc

3

P A R* A A Pascc

9

P A R* A A Pacc

3

P A R* A A Paccs

12

P A Pa A A Pass

12

P A Pass A A Pa

12

P A Pa A ||Pass

12

P A Pa A ||Pass

12

P||Pa A Pass

12

P||Pass A Pa

6

P A Prss

3

P A Prss A ||R*

9

P A R* A ||R A Pa A ||R

3

P||R A Pass

146

3 Overconstrained T3-type TPMs with coupled motions Table 3.11. (cont.)

21

3-PRPaR (Fig. 3.15b)

12

22 23

3-PRRPa (Fig. 3.16a)

12

24 25

3-PRRPa (Fig. 3.16b)

12

26 27

3-PPaRR (Fig. 3.17a)

12

28 29

3-PPaRR (Fig. 3.17b)

12

30

3-PR*RPaR (Fig. 3.32b) 3-PRPass (Fig. 3.33b) 3-PRRPaR* (Fig. 3.35a) 3-PRPass (Fig. 3.36a) 3-PRRPaR* (Fig. 3.35b) 3-PRRPacs (Fig. 3.36b) 3-PR*PaRR (Fig. 3.37a) 3-PPassR (Fig. 3.38a) 3-PR*PaRR (Fig. 3.37b) 3-PPassR (Fig. 3.38b)

9

P||R* A R A Pa A ||R

3

P A R A Pass

9

P A R||R A ||Pa||R*

3

P A R A ||Pass

9

P||R||R A Pa||R*

3

P||R||R A Pacs

9

P A R*||Pa A A R||R

3

P A Pass A A R

9

P||R*||Pa A R||R

3

P||Pass A R

Table 3.12. Limb topology and the number of overconstraints NF of the derived TPMs with idle mobilities and linear actuators mounted on a moving link presented in Figs. 3.39–3.42 No. Basic TPM type 1 3-RPaPR (Fig. 3.22a) 2 3

3-RPaPR (Fig. 3.22b)

NF 12

12

4 5

3-RPaRP (Fig. 3.23a)

12

6 7 8

3-RPaRP (Fig. 3.23b)

12

Derived TPM type 3-RPaPRR* (Fig. 3.39a) 3-RPacsPR (Fig. 3.40a) 3-RPaPRR* (Fig. 3.39b) 3-RPacsPR (Fig. 3.40b) 3-RPaRR*P (Fig. 3.41a) 3-RPassP (Fig. 3.42a) 3-RPaRPR* (Fig. 3.41b) 3-RPacsRP (Fig. 3.42b)

NF 9

Limb topology R A Pa A A P A A R A R*

3

R A Pacs A A P A A R

9

R A Pa A ||P||R A R*

3

R A Pacs A ||P||R

9

R A Pa A ||R A R* A A P

3

R A Pass A A P

9

R A Pa A ||R||P A R*

3

R A Pacs A ||R||P

3.2 Derived solutions with linear actuators

147

Fig. 3.24. Overconstrained TPMs of types 3-PPaC* (a) and 3-PC*Pa (b) with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 12, limb topology P A Pa||C* (a) and P A C*||Pa (b)

148

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.25. Overconstrained TPMs of types 3-PR*PaP (a) and 3-PPR*Pa (b) with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 12, limb topology P A R* A A Pa||P (a) and P A P A A R* A ||Pa (b)

3.2 Derived solutions with linear actuators

149

Fig. 3.26. Overconstrained TPMs of types 3-PPassP (a) and 3-PPPass (b) with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 3, limb topology P A Pass||P (a) and P A P||Pass (b)

150

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.27. Overconstrained TPMs of types 3-PR*Pascc (a) and 3-PR*Paccs (b) with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ) TF = 0, NF = 3, limb topology P A R* A A Pascc (a) and P A R* A A Paccs (b)

3.2 Derived solutions with linear actuators

151

Fig. 3.28. 3-PR*Pacc-type overconstrained TPMs with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 9, limb topology P A R* A A Pacc

152

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.29. 3-PPaPass-type overconstrained TPMs with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 12, limb topology P A Pa A A Pass (a) and P A Pa A ||Pass (b)

3.2 Derived solutions with linear actuators

153

Fig. 3.30. 3-PPassPa-type overconstrained TPMs with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 12, limb topology P A Pass A A Pa (a) and P A Pass A ||Pa (b)

154

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.31. Overconstrained TPMs of types 3-PPaPass (a) and 3-PPassPa (b) with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 12, limb topology P||Pa A Pass (a) and P||Pass A Pa (b)

3.2 Derived solutions with linear actuators

155

Fig. 3.32. 3-PR*RPaR-type overconstrained TPMs with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 9, limb topology P A R* A ||R A Pa A ||R (a) and P||R* A R A Pa A ||R (b)

156

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.33. 3-PRPass-type overconstrained TPMs with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 3, limb topology P||R A Pass (a) and P A R A Pass (b)

3.2 Derived solutions with linear actuators

157

Fig. 3.34. Overconstrained TPMs of types 3-PPrss (a) and 3-PPrssR* (b) with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ) TF = 0, NF = 6 (a), NF = 3 (b), limb topology P A Prss (a) and P A Prss A ||R* (b)

158

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.35. 3-PRRPaR*-type overconstrained TPMs with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 9, limb topology P A R||R A ||Pa||R* (a) and P||R||R A Pa||R* (b)

3.2 Derived solutions with linear actuators

159

Fig. 3.36. Overconstrained TPMs of types 3-PRPass (a) and 3-PRRPacs (b) with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 3, limb topology P A R A ||Pass (a) and P||R||R A Pacs (b)

160

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.37. 3-PR*PaRR-type overconstrained TPMs with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 9, limb topology P A R*||Pa A A R||R (a) and P||R*||Pa A R||R (b)

3.2 Derived solutions with linear actuators

161

Fig. 3.38. 3-PPassR-type overconstrained TPMs with coupled motions and linear actuators on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 3, limb topology P A Pass A A R (a) and P||Pass A R (b)

162

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.39. 3-RPaPRR*-type overconstrained TPMs with coupled motions and linear actuators on a moving link, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 9, limb topology R A Pa A A P A A R A R* (a) and R A Pa A ||P||R A R* (b)

3.2 Derived solutions with linear actuators

163

Fig. 3.40. 3-RPacsPR-type overconstrained TPMs with coupled motions and linear actuators on a moving link, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 3, limb topology R A Pacs A A P A A R (a) and R A Pacs A ||P||R (b)

164

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.41. Overconstrained TPMs of types 3-RPaRR*P (a) and 3-RPaRPR* (b) with coupled motions and linear actuators mounted on a moving link, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 9, limb topology R A Pa A ||R A R* A A P (a) and R A Pa A ||R||P A R* (b)

3.2 Derived solutions with linear actuators

165

Fig. 3.42. Overconstrained TPMs of types 3-RPassP (a) and 3-RPacsRP (b) with coupled motions and linear actuators mounted on the moving platform, defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ) TF = 0, NF = 3, limb topology R A Pass A A P (a) and R A Pacs A ||R||P (b)

166

3 Overconstrained T3-type TPMs with coupled motions

Table 3.13. Bases of the operational velocities spaces of the limbs isolated from the parallel mechanisms presented in Figs. 3.24–3.42 No. Parallel mechanism 1 Fig. 3.24 2 3

4 5

6

Basis (RG1) ( v1 ,v2 , v3 ,ȦG )

Figs. 3.25–3.28, ( v1 , v2 , v3 , ȦE ) 3.30b Figs. 3.29a, 3.33b, ( v1 , v2 , v3 , ȦE ) 3.38b, 3.40a, 3.42a Figs. 3.29b, 3.31a, ( v1 ,v2 , v3 ,ȦG ) Figs. 3.30a, 3.36a, ( v1 ,v2 , v3 ,ȦG ) 3.38a, 3.40b, 3.42b Fig. 3.31b ( v1 , v2 , v3 , ȦE )

(RG2) ( v1 ,v2 , v3 ,ȦG )

(RG3) ( v1 ,v2 , v3 ,ȦD )

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦD )

( v1 ,v2 , v3 ,ȦG )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦD )

( v1 ,v2 , v3 ,ȦD )

8

Figs. 3.32a, 3.35a, ( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG ) 3.39b, 3.41b Figs. 3.32b, 3.35b ( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG )

9

Fig. 3.33a

7

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 , v2 , v3 , ȦE )

( v1 , v 2 , v 3 )

( v1 , v 2 , v 3 )

( v1 , v 2 , v 3 )

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦG )

10 Fig. 3.34a 11 Figs. 3.34b, 3.36b 12 Fig. 3.37a

( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE )

13 Figs. 3.37b, 3.39a, ( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG ) 3.41a

3.2 Derived solutions with linear actuators

167

Table 3.14. Structural parametersa of translational parallel mechanisms in Figs. 3.24–3.27 No.

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

Structural parameter

Solution 3-PPaC* (Fig. 3.24a) 3-PC*Pa (Fig. 3.24b) 14 6 6 6 18 5 0 3 3 See Table 3.13

3-PR*PaP (Fig. 3.25a) 3-PPR*Pa (Fig. 3.25b) 17 7 7 7 21 5 0 3 3 See Table 3.13

3-PPassP (Fig. 3.26a) 3-PPPass(Fig. 3.26b) 3-PR*Pascc (Fig. 3.27a) 3-PR*Paccs (Fig. 3.27b) 14 6 6 6 18 5 0 3 3 See Table 3.13

4 4 4 3 3 3 4 4 4 ( v1 ,v 2 ,v3 ) 3 9 18 3 12 0 7

4 4 4 3 3 3 4 4 4 ( v1 ,v 2 ,v3 ) 3 9 18 3 12 0 7

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 3 0 10

fj

7

7

10

fj

7

7

10

fj

21

21

30

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

168

3 Overconstrained T3-type TPMs with coupled motions

Table 3.15. Structural parametersa of translational parallel mechanisms in Figs. 3.28–3.32 No.

Structural parameter

1 2 3 4 5 6 7 8 9 10

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 a

4 4 4 4 4 4 4 4 4 ( v1 ,v 2 ,v3 ) 3 12 21 3 9 0 8

4 4 4 9 9 9 4 4 4 ( v1 ,v 2 ,v3 ) 3 27 36 3 12 0 13

5 5 5 3 3 3 5 5 5 ( v1 ,v 2 ,v3 ) 3 9 21 3 9 0 8

fj

8

13

8

fj

8

13

8

fj

24

39

24

SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

Solution 3-PPaPass 3-PR*RPaR 3-PR*Pacc (Fig. 3.28a, b) (Figs. 3.29a, b, 3.31a)(Fig. 3.32a, b) 3-PPassPa (Figs. 3.30a, b, 3.31b) 14 20 20 6 9 8 6 9 8 6 9 8 18 27 24 5 8 5 0 0 0 3 3 3 3 3 3 See Table 3.13 See Table 3.13 See Table 3.13

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

3.2 Derived solutions with linear actuators

169

Table 3.16. Structural parametersa of translational parallel mechanisms in Figs. 3.33 and 3.34 No.

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

Structural parameter

Solution 3-PRPass (Fig. 3.33a, b) 14 6 6 6 18 5 0 3 3 See Table 3.13

3-PPrss (Fig. 3.34a) 14 7 7 7 21 8 0 3 3 See Table 3.13

3-PPrssR* (Fig. 3.34b) 17 8 8 8 24 8 0 3 3 See Table 3.13

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 3 0 10

3 3 3 12 12 12 3 3 3 ( v1 ,v 2 ,v3 ) 3 36 42 3 6 0 15

4 4 4 12 12 12 4 4 4 ( v1 ,v 2 ,v3 ) 3 36 45 3 3 0 16

fj

10

15

16

fj

10

15

16

fj

30

45

48

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

170

3 Overconstrained T3-type TPMs with coupled motions

Table 3.17. Structural parametersa of translational parallel mechanisms in Figs. 3.35–3.42 No.

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

Structural Solution parameter 3-PRRPaR* (Fig. 3.35a, b) 3-PR*PaRR (Fig. 3.37a, b) 3-RPaPRR* (Fig. 3.39a, b) 3-RPaRR*P (Fig. 3.41a) 3-RPaRPR* (Fig. 3.41b) 20 8 8 8 24 5 0 3 3 See Table 3.13

3-PRPass (Fig. 3.36a) 3-PPassR (Fig. 3.38a, b) 3-RPassP (Fig. 3.42a) 14 6 6 6 18 5 0 3 3 See Table 3.13

3-PRRPacs (Fig. 3.36b) 3-RPacsPR (Fig. 3.40a, b) 3-RPacsRP (Fig. 3.42b) 17 7 7 7 21 5 0 3 3 See Table 3.13

5 5 5 3 3 3 5 5 5 ( v1 ,v 2 ,v3 ) 3 9 21 3 9 0 8

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 3 0 10

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 3 0 10

fj

8

10

10

fj

8

10

10

fj

24

30

30

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

3.2 Derived solutions with linear actuators

171

Table 3.18. Limb topology and the number of overconstraints NF of the derived TPMs with idle mobilities and linear actuators mounted on the fixed base presented in Figs. 3.43–3.52 No. Basic TPM type 1 3-PRC (Fig. 3.12a) 2 3

3-PCR (Fig. 3.12b)

NF 3

3

4 5

3-PRC 3 (Fig. 3.13a, b)

6 7

3-PCR 3 (Fig. 3.14a, b)

8 9

3-PRPaR (Fig. 3.15a)

12

10 11 12 13 14 15 16

3-PRPaR (Fig. 3.15b)

12

Derived TPM type PRCR*-2PRC (Fig. 3.43a) 2PRCR*-PRC (Fig. 3.44a) PCRR*-2PCR (Fig. 3.43b) 2PCRR*-PCR (Fig. 3.44b) PRR*C-2PRC (Fig. 3.45a, b) 2PRR*C-PRC (Fig. 3.47a, b) PCR*R-2PCR (Fig. 3.46a, b) 2PCR*R-PCR (Fig. 3.48a, b) PR*RPaR-2PRPaR (Fig. 3.49a) PRPass-2PRPaR (Fig. 3.50a) PR*RPass-2PRPass (Fig. 3.51a) 2PR*RPass-PRPass (Fig. 3.52a) PR*RPaR-2PRPaR (Fig. 3.49b) PRPass-2PRPaR (Fig. 3.50b) PR*RPass-2PRPass (Fig. 3.51b) 2PR*RPass-PRPass (Fig. 3.52b)

NF 2

Topology of the limbs P A R||C A ||R* P A R||C

1 2

P A C||R A ||R* P A C||R

1 2

P A R A R* A ||C P A R||C

1 2

P A C A R* A ||R P A C||R

1 11 9 2

P A R* A ||R A Pa A ||R P||R A Pa A ||R P||R A Pass P||R A Pa A ||R P A R* A ||R A Pass P||R A Pass

1 11 9 2 1

P||R* A R A Pa A ||R P A R A Pa A ||R P A R A Pass P A R A Pa A ||R P||R* A R A Pass P A R A Pass

172

3 Overconstrained T3-type TPMs with coupled motions

Table 3.19. Limb topology and the number of overconstraints NF of the derived TPMs with idle mobilities and linear actuators mounted on a moving link presented in Figs. 3.53–3.59 No. Basic TPM type 1 3-RPC (Fig. 3.18a 2 3

3-CPR (Fig. 3.18b)

NF 3

3

4 5

3-RPC 3 (Fig. 3.19a, b)

6 7

3-CPR 3 (Fig. 3.20a, b)

8 9 10

3-PPRR (Fig. 3.21)

3

Derived TPM type RC*C-2RPC (Fig. 3.53a) 2RC*C-RPC (Fig. 3.54a) CC*R-2CPR (Fig. 3.53b) 2CC*R-CPR (Fig. 3.54b) RC*C-2RPC (Fig. 3.55a, b) 2RC*C-RPC (Fig. 3.57a, b) CC*R-2CPR (Fig. 3.56a, b) 2CC*R-CPR (Fig. 3.58a, b) PPRRR*-2PPRR (Fig. 3.59a) 2PPRRR*-PPRR (Fig. 3.59b)

NF 2

Topology of the limbs R A C* A ||C R A P A ||C

1 2

C A C* A ||R C A P A ||R

1 2

R A C* A ||C R A P A ||C

1 2

C A C* A ||R C A P A ||R

1 2 1

P A P A ||R||R A ||R* P A P A ||R||R

3.2 Derived solutions with linear actuators

173

Fig. 3.43. Overconstrained TPMs of types PRCR*-2PRC (a) and PCRR*-2PCR (b) with coupled motions and linear actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 2, limb toplogies P A R||C A ||R* and P A R||C (a), P A C||R A ||R* and P A C||R (b)

174

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.44. Overconstrained TPMs of types 2PRCR*-PRC (a) and 2PCRR*-PCR (b) with coupled motions and linear actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ) TF = 0, NF = 1, limb topologies P A R||C A ||R* and P A R||C (a), P A C||R A ||R* and P A C||R (b)

3.2 Derived solutions with linear actuators

175

Fig. 3.45. PRR*C-2PRC-type overconstrained TPMs with coupled motions and parallel linear actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 2, limb topologies P A R A R* A ||C and P A R||C

176

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.46. PCR*R-2PCR-type overconstrained TPMs with coupled motions and parallel linear actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 2, limb topologies P A C A R* A ||R and P A C||R

3.2 Derived solutions with linear actuators

177

Fig. 3.47. 2PRR*C-PRC-type overconstrained TPMs with coupled motions and parallel linear actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 1, limb topologies P A R A R* A ||C and P A R||C

178

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.48. 2PCR*R-PCR-type overconstrained TPMs with coupled motions and parallel linear actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 1, limb topologies P A C A R* A ||R and P A C||R

3.2 Derived solutions with linear actuators

179

Fig. 3.49. PR*RPaR-2PRPaR-type overconstrained TPMs with coupled motions and linear actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 11, limb topologies P A R* A ||R A Pa A ||R and P||R A Pa A ||R (a), P||R* A R A Pa A ||R and P A R A Pa A ||R (b)

180

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.50. PRPass-2PRPaR-type overconstrained TPMs with coupled motions and linear actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topologies P||R A Pass and P||R A Pa A ||R (a), P A R A Pass and P A R A Pa A ||R (b)

3.2 Derived solutions with linear actuators

181

Fig. 3.51. PR*RPass-2PRPass-type overconstrained TPMs with coupled motions and linear actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 2, limb topologies P A R* A ||R A Pass and P||R A Pass (a) and P||R* A R A Pass and P A R A Pass (b)

182

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.52. 2PR*RPass-PRPass-type overconstrained TPMs with coupled motions and linear actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 1, limb topologies P A R* A ||R A Pass and P||R A Pass (a), P||R* A R A Pass and P A R A Pass (b)

3.2 Derived solutions with linear actuators

183

Fig. 3.53. Overconstrained TPMs of types RC*C-2RPC (a) and CC*R-2CPR (b) with coupled motions and linear actuators non adjacent to the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 2, limb topologies R A C* A ||C and R A P A ||C (a), C A C* A ||R and C A P A ||R (b)

184

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.54. Overconstrained TPMs of types 2RC*C-RPC (a) and 2CC*R-CPR (b) with coupled motions and linear actuators non adjacent to the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 1, limb topologies R A C* A ||C and R A P A ||C (a), C A C* A ||R and C A P A ||R (b)

3.2 Derived solutions with linear actuators

185

Fig. 3.55. RC*C-2RPC-type overconstrained TPMs with coupled motions and linear actuators non adjacent to the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 2, limb topologies R A C* A ||C and R A P A ||C

186

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.56. CC*R-2CPR-type overconstrained TPMs with coupled motions and linear actuators non adjacent to the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 2, limb topologies C A C* A ||R and C A P A ||R

3.2 Derived solutions with linear actuators

187

Fig. 3.57. 2RC*C-RPC-type overconstrained TPMs with coupled motions and linear actuators non adjacent to the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 1, limb topologies R A C* A ||C and R A P A ||C

188

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.58. 2CC*R-CPR-type overconstrained TPMs with coupled motions and linear actuators non adjacent to the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 1, limb topologies C A C* A ||R and C A P A ||R

3.2 Derived solutions with linear actuators

189

Fig. 3.59. Overconstrained TPMs of types PPRRR*-2PPRR (a) and 2PPRRR*PPRR (b) with coupled motions and linear actuators non adjacent to the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 2 (a), NF = 1 (b), limb topologies P A P A ||R||R A ||R* and P A P A ||R||R (a), P A P A ||R||R A ||R* and P A P A ||R||R (b)

190

3 Overconstrained T3-type TPMs with coupled motions

Table 3.20. Structural parametersa of translational parallel mechanisms in Figs. 3.43–3.48 No. Structural parameter

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

Solution PRCR*-2PRC (Fig. 3.43a) PCRR*-2PCR (Fig. 3.43b) PRR*C-2PRC (Fig. 3.45a, b) PCR*R-2PCR (Fig. 3.46a, b) 9 4 3 3 10 2 3 0 3 See Table 3.24

2PRCR*-PRC (Fig. 3.44a) 2PCRR*-PCR (Fig. 3.44b) 2PRR*C-PRC (Fig. 3.47a, b) 2PCR*R-PCR (Fig. 3.48a, b) 10 4 4 3 11 2 3 0 3 See Table 3.24

5 4 4 0 0 0 5 4 4 ( v1 ,v 2 ,v3 ) 3 0 10 3 2 0 5

5 5 4 0 0 0 5 5 4 ( v1 ,v 2 ,v3 ) 3 0 11 3 1 0 5

fj

4

5

fj

4

4

fj

13

14

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

3.2 Derived solutions with linear actuators

191

Table 3.21. Structural parametersa of translational parallel mechanisms in Figs. 3.49–3.51 No.

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

Structural Solution parameter PR*RPaR-2PRPaR (Fig. 3.49a, b) m 18 p1 8 p2 7 p3 7 p 22 q 5 0 k1 k2 3 k 3 See Table 3.24 (RGi) (i = 1,2,3) SG1 5 4 SG2 4 SG3 3 rG1 rG2 3 3 rG3 5 MG1 4 MG2 4 MG3 (RF) ( v1 ,v 2 ,v3 ) SF 3 9 rl rF 19 MF 3 11 NF 0 TF p1 8 f

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

PRPass-2PRPaR (Fig. 3.50a, b) 16 7 6 7 20 5 0 3 3 See Table 3.24

PR*RPass-2PRPass (Fig. 3.51a, b) 15 7 6 6 19 5 0 3 3 See Table 3.24

4 4 4 3 6 3 4 4 4 ( v1 ,v 2 ,v3 ) 3 12 21 3 9 0 7

5 4 4 6 6 6 5 4 4 ( v1 ,v 2 ,v3 ) 3 18 28 3 2 0 11

j

fj

7

10

10

fj

7

7

10

fj

22

24

31

See footnote of Table 2.1 for the nomenclature of structural parameters

192

3 Overconstrained T3-type TPMs with coupled motions

Table 3.22. Structural parametersa of translational parallel mechanisms in Figs. 3.52–3.58 No.

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

Structural parameter

Solution 2PR*RPass-PRPass (Fig. 3.52a, b)

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

16 7 7 6 20 5 0 3 3 See Table 3.24

RC*C-2RPC (Figs. 3.53a, 3.55a, b) CC*R-2CPR (Figs. 3.53b, 3.56a, b) 8 3 3 3 9 2 3 0 3 See Table 3.24

2RC*C-RPC (Figs. 3.54a, 3.57a, b) 2CC*R-CPR (Figs. 3.54b, 3.58a, b) 8 3 3 3 9 2 3 0 3 See Table 3.24

5 5 4 6 6 6 5 5 4 ( v1 ,v 2 ,v3 ) 3 18 29 3 1 0 11

5 4 4 0 0 0 5 4 4 ( v1 ,v 2 ,v3 ) 3 0 10 3 2 0 5

5 5 4 0 0 0 5 5 4 ( v1 ,v 2 ,v3 ) 3 0 11 3 1 0 5

fj

11

4

5

fj

10

4

4

fj

32

13

14

SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

3.2 Derived solutions with linear actuators

193

Table 3.23. Structural parametersa of translational parallel mechanisms in Fig. 3.59 No.

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

Structural parameter

Solution PPRRR*-2PPRR (Fig. 3.59a)

2PPRRR*-PPRR (Fig. 3.59b)

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

12 5 4 4 13 2 3 0 3 See Table 3.24

13 5 5 4 14 2 3 0 3 See Table 3.24

5 4 4 0 0 0 5 4 4 ( v1 ,v 2 ,v3 ) 3 0 10 3 2 0 5

5 5 4 0 0 0 5 5 4 ( v1 ,v 2 ,v3 ) 3 0 11 3 1 0 5

fj

4

5

fj

4

4

fj

13

14

SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

194

3 Overconstrained T3-type TPMs with coupled motions

Table 3.24. Bases of the operational velocities spaces of the limbs isolated from the parallel mechanisms presented in Figs. 3.43–3.59 No. Parallel mechanism 1 Figs. 3.43, 3.49b, 3.51b, 3.53, 3.59a 2 Figs. 3.44, 3.52b, 3.54, 3.59b 3 Figs. 3.45, 3.46, 3.55, 3.56 4 Figs. 3.47, 3.48

Basis (RG2) (RG1) ( v1 , v2 , v3 , ȦD , ȦE ) ( v1 ,v2 , v3 ,ȦG )

5

Figs. 3.49a, 3.51a

( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦD )

6

Fig. 3.50a

( v1 ,v2 , v3 ,ȦD )

( v1 ,v2 , v3 ,ȦD )

7

Fig. 3.50b

( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG )

8

Fig. 3.52a

( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE ) ( v1 ,v2 , v3 ,ȦD )

9

Figs. 3.57, 3.58

( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦE )

(RG3) ( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG ) ( v1 ,v2 , v3 ,ȦD ) ( v1 , v2 , v3 , ȦE , ȦG ) ( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦE ) ( v1 , v2 , v3 , ȦE )

3.3 Basic solutions with rotating actuators In the basic solutions with rotating actuators and coupled motions F m G1G2, the moving platform n Ł nGi (i = 1, 2, 3) is connected to the reference platform 1 Ł 1Gi Ł 0 by three limbs with three, four or five degrees of connectivity. No idle mobilities exist in these basic solutions and the rotating actuator is associated with a revolute pair in each limb. The various types of limbs with three degrees of connectivity are systematized in Figs. 3.60–3.62. They combine two (Figs. 3.60 and 3.61) or three (Fig. 3.62) parallelogram loops of types Pa (Figs. 3.60 and 3.62) or Pacc (Fig. 3.61). We recall that Pa-type parallelogram loop has one degree of mobility and Pacc-type parallelogram loop has two degrees of mobility. These limbs are actuated by rotating motors mounted on the fixed base and integrated in a parallelogram loop. The limbs with four degrees of connectivity are systematized in Figs. 3.63–3.68. They can be simple (Fig. 3.63) or complex (Figs. 3.64–3.68) kinematic chains. The complex limbs combine parallelogram loops of types Pa (Figs. 3.64, 3.65, 3.67 and 3.68) and Pacc (Fig. 3.66a–c) or a prism mechanism Pr (Fig. 3.66d). The complex limbs are actuated by rotating motors mounted on the fixed base and combined in the kinematic chain of the parallelogram loop (Figs. 3.64, 3.65 and 3.66a) or outside the parallelogram loop (Figs. 3.66b–d, 3.67 and 3.68).

3.3 Basic solutions with rotating actuators

195

The limbs with five degrees of connectivity are systematized in Figs. 3.69 and 3.70. They are complex kinematic chains combining a Pa-type parallelogram loop. These limbs are actuated by rotating motors mounted on the fixed base and combined in the parallelogram loop (Fig. 3.69a–e) or outside the parallelogram loop (Figs. 3.69f and 3.70). Various solutions of translational parallel robots with coupled motions and no idle mobilities can be obtained by using three limbs with identical or different topologies presented in Figs. 3.60–3.70. We only show solutions with identical limb type as illustrated in Figs. 3.71–3.118. The limb topology and connecting conditions in these solutions with no idle mobilities are systematized in Table 3.25. The axes of the three actuated revolute joints adjacent to the fixed base can be in one of the following configurations (see Table 3.25): (i) reciprocally orthogonal in space, (ii) parallel to two orthogonal directions or (iii) parallel to a plane. The axes of the unactuated revolute or cylindrical joints adjacent to the moving platform can be orthogonal in space or parallel to a plane. In the latter case, the axes can form a triangle or can be situated on three sides of a rectangle. These axes form a configuration called spatial star, if they are orthogonal in space, and ' or C -type if they are parallel to a plane (see Table 3.25). The structural parameters of the solutions presented in Figs. 3.71–3.118 are systematized in Tables 3.26–3.31.

196

3 Overconstrained T3-type TPMs with coupled motions

Table 3.25. Limb topology and connecting conditions of the TPMs with no idle mobilities and rotating actuators mounted on the fixed base presented in Figs. 3.71–3.118 No. TPM type

Limb topology

Connecting conditions Axes of the actuated joints are reciprocally orthogonal and the moving platform in a spatial star configuration Axes of the actuated joints are parallel to two orthogonal directions and the moving platform in a C configuration Idem No. 1

1

3-PaPaP (Fig. 3.71a)

Pa A Pa||P (Fig. 3.60a)

2

3-PaPaP (Fig. 3.71b)

Pa A Pa||P (Fig. 3.60a)

3

3-PaPPa (Fig. 3.72a) 3-PaPaPa (Fig. 3.72b) 3-PaPPa (Fig. 3.73a) 3-PaPPa (Fig. 3.73b)

Pa||Pa||P (Fig. 3.60b) Pa||Pa||P (Fig. 3.60b) Pa A P||Pa (Fig. 3.60c) Pa A P||Pa (Fig. 3.60c)

3-PaPPa (Fig. 3.74a) 3-PaPPa (Fig. 3.74b) 3-PaPacc (Fig. 3.75a) 3-PaPacc (Fig. 3.75b) 3-PaPacc (Fig. 3.76a) 3-PaPacc (Fig. 3.76b) 3-PaPacc (Fig. 3.77a) 3-PaPacc (Fig. 3.77b) 3-PaPacc (Fig. 3.78a) 3-PaPacc (Fig. 3.78b) 3-PaccPa (Fig. 3.79a) 3-PaccPa (Fig. 3.79b)

Pa||P||Pa (Fig. 3.60d) Pa||P||Pa (Fig. 3.60d) Pa A Pacc (Fig. 3.61a) Pa A Pacc (Fig. 3.61a) Pa A Pacc (Fig. 3.61b) Pa A Pacc (Fig. 3.61b) Pa||Pacc (Fig. 3.61c) Pa||Pacc (Fig. 3.61c) Pa||Pacc (Fig. 3.61d) Pa||Pacc (Fig. 3.61d) Pacc||Pa (Fig. 3.61e) Pacc||Pa (Fig. 3.61e)

3-PaPaPa (Fig. 3.80) 3-PaPaPa (Fig. 3.81)

Pa A Pa A ||Pa (Fig. 3.62a) Pa A Pa A A Pa (Fig. 3.62b)

4 5 6

7 8 9 10 11 12 13 14 15 16 17 18

19 20

Idem No. 2 Idem No. 1 Axes of the actuated joints are parallel to two orthogonal directions and the moving platform in a star configuration Idem No. 1 Idem No. 6 Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 6 Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 6 Idem No. 1 Idem No. 6

Idem No. 1 Idem No. 1

3.3 Basic solutions with rotating actuators

197

Table 3.25. (cont.) 21 22

23

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

3-PaPaPa (Fig. 3.82) 3-RRC (Fig. 3.83a)

Pa||Pa A Pa (Fig. 3.62c) R||R||C (Fig. 3.63c)

3-RRC (Fig. 3.83b, 3.84a) 3-RRC (Fig. 3.84b) 3-RCR (Fig. 3.85a) RPRR (Fig. 3.85b) 3-RPC (Fig. 3.86a) 3-RPPR (Fig. 3.86b) 3-PaRC (Fig. 3.87a) 3-PaRC (Fig. 3.87b) 3-PaCR (Fig. 3.88a) 3-PaCR (Fig. 3.88b) 3-PaPaRR (Fig. 3.89) 3-PaPaRR (Fig. 3.90) 3-PaPaRR (Fig. 3.91) 3-PaRRPa (Fig. 3.92) 3-PaRRPa (Fig. 3.93) 3-PaRRPa (Fig. 3.94) 3-PaRPaR (Fig. 3.95) 3-PaRPaR(Fig. 3.96) 3-RRPaP (Fig. 3.97a)

R||R||C (Fig. 3.63c) R||R||C (Fig. 3.63c) R||C||R (Fig. 3.63f) R||P||R||R (Fig. 3.63j) R A P A ||C (Fig. 3.63i) R||P A P A ||R (Fig. 3.63k) Pa||R||C (Fig. 3.64c) Pa||R||C (Fig. 3.64c) Pa||C||R (Fig. 3.64f) Pa||C||R (Fig. 3.64f) Pa||Pa A R||R (Fig. 3.65a) Pa A Pa A ||R||R (Fig. 3.65b) Pa A Pa A A R||R (Fig. 3.65c) Pa||R||R A Pa (Fig. 3.65d) Pa A R||R A A Pa (Fig. 3.65e) Pa A R||R A ||Pa (Fig. 3.65f) Pa A R A Pa A ||R (Fig. 3.65g) Pa||R A Pa A ||R (Fig. 3.65h) R||R||Pa||P (Fig. 3.67a)

Idem No. 1 Axes of the actuated joints parallel to a plane and the moving platform in a ' configuration Idem No. 2

Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 2 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

198

3 Overconstrained T3-type TPMs with coupled motions Table 3.25. (cont.)

42 3-RRPaP (Fig. 3.97b) 43 3-RPRPa (Fig. 3.98a) 44 3-RPRPa (Fig. 3.98b) 45 3-RCPa (Fig. 3.99a) 46 3-RCPa (Fig. 3.99b) 47 3-RPaRR (Fig. 3.100a) 48 3-RPaRR (Fig. 3.100b) 49 3-RRPaR (Fig. 3.101a) 50 3-RRPaR (Fig. 3.101b) 51 3-RPaPaR (Fig. 3.102a) 52 3-RPaPaR (Fig. 3.102b) 53 3-RPaRPa (Fig. 3.103a) 54 3-RPaRPa (Fig. 3.103b) 55 3-RPaRPa (Fig. 3.104a) 56 3-RPaRPa (Fig. 3.104b) 57 3-RPaPaR (Fig. 3.105a) 58 3-RPaPaR (Fig. 3.105b) 59 3-RRPacc (Fig. 3.106a) 60 3-RRPacc (Fig. 3.106b) 61 3-RRPacc (Fig. 3.107a) 62 3-RRPacc (Fig. 3.107b) 63 3-PaccRR (Fig. 3.108a)

R||R||Pa||P (Fig. 3.67a) R||P||R||Pa (Fig. 3.67b) R||P||R||Pa (Fig. 3.67b) R||C||Pa (Fig. 3.67e) R||C||Pa (Fig. 3.67e) R A Pa A ||R||R (Fig. 3.67f) R A Pa A ||R||R (Fig. 3.67f) R||R A Pa A ||R (Fig. 3.68a) R||R A Pa A ||R (Fig. 3.68a) R A Pa||Pa A ||R (Fig. 3.68b) R A Pa||Pa A ||R (Fig. 3.68b) R A Pa A ||R A Pa (Fig. 3.68c) R A Pa A ||R A Pa (Fig. 3.68c) R A Pa A ||R A Pa (Fig. 3.68d) R A Pa A ||R A Pa (Fig. 3.68d) R A Pa||Pa A ||R (Fig. 3.68e) R A Pa||Pa A ||R (Fig. 3.68e) R||R||Pacc (Fig. 3.66b) R||R||Pacc (Fig. 3.66b) R||R||Pacc (Fig. 3.66c) R||R||Pacc (Fig. 3.66c) Pacc||R||R (Fig. 3.66a)

Idem No. 2 Idem No. 1 Idem No. 6 Idem No. 1 Idem No. 6 Idem No. 1 Idem No. 22 Idem No. 1 Idem No. 22 Idem No. 1 Axes of the actuated joints are parallel to two orthogonal directions Idem No. 1 Idem No. 52 Idem No. 1 Idem No. 52 Idem No. 1 Idem No. 52 Idem No. 1 Idem No. 6 Idem No. 1 Idem No. 2 Idem No. 1

3.3 Basic solutions with rotating actuators Table 3.25. (cont.) cc

64 3-Pa RR (Fig. 3.108b) 65 3-PaPr (Fig. 3.109a) 66 3-RRPr (Fig. 3.109b) 67 3-PaRRRR (Fig. 3.110a) 68 3-PaRRRR (Fig. 3.110b) 69 3-PaRRRR (Fig. 3.111a) 70 3-PaRRRR (Fig. 3.111b) 71 3-PaRRRR (Fig. 3.112a) 72 3-RRPaRR (Fig. 3.112b) 73 3-RPaRRR (Fig. 3.113a) 74 3-RPaRRR (Fig. 3.113b) 75 3-RPaRRR (Fig. 3.114a) 76 3-RPaRRR (Fig. 3.114b) 77 3-RRPaRR (Fig. 3.115a) 78 3-RRPaRR (Fig. 3.115b) 79 3-RRRRPa (Fig. 3.116a) 80 3-RRRRPa (Fig. 3.116b) 81 3-RRRPaR (Fig. 3.117a) 82 3-RRRRPa (Fig. 3.117b) 83 3-RRRPaR (Fig. 3.118a, b)

cc

Pa ||R||R Idem No. 2 (Fig. 3.66a) Pa-Pr Idem No. 1 (Fig. 3.62d) R||R-Pr Idem No. 1 (Fig. 3.66d) Pa||R||R A R||R Idem No. 1 (Fig. 3.69a) Pa A R A R||R A R Idem No. 1 (Fig. 3.69b) Pa A R||R A ||R||R Idem No. 1 (Fig. 3.69c) Pa A R A R||R A ||R Idem No. 1 (Fig. 3.69d) Pa||R A R||R A ||R Idem No. 1 (Fig. 3.69e) R A R A ||Pa A ||R A R Idem No. 1 (Fig. 3.69f) R A Pa||R||R A ||R Idem No. 1 (Fig. 3.70a) R A Pa A A R||R A A R Idem No. 1 (Fig. 3.70b) R A Pa A ||R A R||R Idem No. 1 (Fig. 3.70c) R A Pa||R||R A ||R Idem No. 1 (Fig. 3.70d) R||R||Pa A R||R Idem No. 1 (Fig. 3.70g) R||R A Pa A A R||R Idem No. 1 (Fig. 3.70i) R||R A R||R A A Pa Idem No. 1 (Fig. 3.70h) R||R A R||R A A Pa Idem No. 1 (Fig. 3.70j) R||R A R A Pa A ||R Idem No. 1 (Fig. 3.70k) R A R||R A ||R A Pa Idem No. 1 (Fig. 3.70l) R A R||R||Pa A ||R Idem No. 1 (Fig. 3.69e, f)

199

200

3 Overconstrained T3-type TPMs with coupled motions

Table 3.26. Bases of the operational velocities spaces of the limbs isolated from the parallel mechanisms presented in Figs. 3.71–3.118 No. Parallel mechanism 1 Figs. 3.71–3.82, 3.109a 2 Figs. 3.83, 3.84a, 3.100b, 3.101b 3 Figs. 3.84b, 3.85, 3.86 4 Figs. 3.87a, 3.88a, 3.89, 3.92, 3.93, 3.102a, 3.104a, 3.109b, 5 Figs. 3.87b, 3.88b, 3.97b, 3.98b, 3.99b, 3.106b, 3.107b, 3.108b 6 Fig. 3.94

Basis (RG1) ( v1 , v 2 , v 3 )

(RG2) ( v1 , v 2 , v 3 )

(RG3) ( v1 , v 2 , v 3 )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦD )

( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦD )

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦD )

8

Figs. 3.90, 3.91, ( v1 , v2 , v3 , ȦE ) 3.95, 3.96, 3.97a, 3.98a, 3.99a, 3.100a, 3.101a, 3.103a, 3.105a, 3.106a, 3.107a, 3.108a Fig. 3.102b ( v1 ,v2 , v3 ,ȦD )

( v1 ,v2 , v3 ,ȦG )

( v1 , v2 , v3 , ȦE )

9

Fig. 3.103b, 3.105b ( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦD )

( v1 ,v2 , v3 ,ȦD )

( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦG )

( v1 , v2 , v3 , ȦE )

7

10 Fig. 3.104b

11 Figs. 3.110a, 3.111, ( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG ) 3.114b, 3.118b 12 Figs. 3.110b, 3.112ª, ( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE ) 3.113a, 3.114a, 3.115, 3.116a, 3.117a 13 Figs. 3.112b, ( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG ) 3.113b, 3.116b, 3.117b, 3.118a

3.3 Basic solutions with rotating actuators

201

Table 3.27. Structural parametersa of translational parallel mechanisms in Figs. 3.71–3.82 No.

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

Structural Solution parameter 3-PaPaP (Figs. 3.71, 3.72) 3-PaPPa (Figs. 3.73, 3.74) m 20 p1 9 p2 9 p3 9 p 27 q 8 k1 0 3 k2 k 3 (RGi) See Table 3.26 (i = 1,2,3) SG1 3 3 SG2 SG3 3 6 rG1 6 rG2 6 rG3 3 MG1 3 MG2 3 MG3 (RF) ( v1 ,v 2 ,v3 ) SF 3 rl 18 24 rF 3 MF 24 NF 0 TF p1 9 f

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

j

3-PaPacc (Figs. 3.75–3.78) 3-PaccPa (Fig. 3.79) 17 8 8 8 24 8 0 3 3 See Table 3.26

26 12 12 12 36 11 0 3 3 See Table 3.26

3 3 3 7 7 7 3 3 3 ( v1 ,v 2 ,v3 ) 3 21 27 3 21 0 10

3 3 3 9 9 9 3 3 3 ( v1 ,v 2 ,v3 ) 3 27 33 3 33 0 12

3-PaPaPa (Figs. 3.80–3.82)

fj

9

10

12

fj

9

10

12

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

202

3 Overconstrained T3-type TPMs with coupled motions

Table 3.28. Structural parametersa of translational parallel mechanisms in Figs. 3.83–3.88 No.

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

Structural Solution parameter 3-RRC (Figs. 3.83, 3.84) 3-RCR, 3-RPC (Figs. 3.85a, 3.86a) m 8 p1 3 p2 3 p3 3 p 9 q 2 k1 3 0 k2 k 3 (RGi) See Table 3.26 (i = 1,2,3) SG1 4 4 SG2 SG3 4 0 rG1 0 rG2 0 rG3 4 MG1 4 MG2 4 MG3 (RF) ( v1 ,v 2 ,v3 ) SF 3 rl 0 9 rF 3 MF 3 NF 0 TF p1 4 f

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

j

3-RPRR (Fig. 3.85b) 3-RPPR (Fig. 3.86b) 11 4 4 4 12 2 3 0 3 See Table 3.26

3-PaRC (Fig. 3.87) 3-PaCR (Fig. 3.88) 14 6 6 6 18 5 3 0 3 See Table 3.26

4 4 4 0 0 0 4 4 4 ( v1 ,v 2 ,v3 ) 3 0 9 3 3 0 4

4 4 4 3 3 3 4 4 4 ( v1 ,v 2 ,v3 ) 3 9 18 3 12 0 7

fj

4

4

7

fj

4

4

7

fj

12

12

21

See footnote of Table 2.1 for the nomenclature of structural parameters

3.3 Basic solutions with rotating actuators

203

Table 3.29. Structural parametersa of translational parallel mechanisms in Figs. 3.89–3.99 No.

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

Structural Solution parameter 3-PaPaRR (Figs. 3.89–3.91) 3-PaRRPa (Figs. 3.92–3.94) 3-PaRPaR (Figs. 3.95, 3.96) m 23 p1 10 p2 10 p3 10 p 30 q 8 0 k1 k2 3 k 3 See Table 3.26 (RGi) (i = 1,2,3) SG1 4 4 SG2 4 SG3 6 rG1 6 rG2 6 rG3 4 MG1 4 MG2 MG3 4 (RF) ( v1 ,v 2 ,v3 ) SF 3 18 rl 27 rF 3 MF NF 21 0 TF p1 10 f

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

3-RRPaP (Fig. 3.97) 3-RPRPa (Fig. 3.98)

3-RCPa (Fig. 3.99)

17 7 7 7 21 5 0 3 3 See Table 3.26

14 6 6 6 18 5 3 0 3 See Table 3.26

4 4 4 3 3 3 4 4 4 ( v1 ,v 2 ,v3 ) 3 9 18 3 12 0 7

4 4 4 3 3 3 4 4 4 ( v1 ,v 2 ,v3 ) 3 9 18 3 12 0 7

j

fj

10

7

7

fj

10

7

7

fj

30

21

21

See footnote of Table 2.1 for the nomenclature of structural parameters

204

3 Overconstrained T3-type TPMs with coupled motions

Table 3.30. Structural parametersa of translational parallel mechanisms in Figs. 3.100–3.108 No.

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

Structural Solution parameter 3-RPaRR (Fig. 3.100) 3-RRPaR (Fig. 3.101) m 17 p1 7 p2 7 p3 7 p 21 q 5 k1 0 3 k2 k 3 (RGi) See Table 3.26 (i = 1,2,3) SG1 4 4 SG2 SG3 4 3 rG1 3 rG2 3 rG3 4 MG1 4 MG2 4 MG3 (RF) ( v1 ,v 2 ,v3 ) SF 3 rl 9 18 rF 3 MF 12 NF 0 TF p1 7 f

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

j

3-RPaPaR (Figs. 3.102, 3.105) 3-RPaRPa (Figs. 3.103, 3.104) 23 10 10 10 30 8 0 3 3 See Table 3.26

3-RRPacc (Figs. 3.106, 3.107) 3-PaccRR (Fig. 3.108) 14 6 6 6 18 5 3 0 3 See Table 3.26

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 21 0 10

4 4 4 4 4 4 4 4 4 ( v1 ,v 2 ,v3 ) 3 12 21 3 9 0 8

fj

7

10

8

fj

7

10

8

fj

21

10

24

See footnote of Table 2.1 for the nomenclature of structural parameters

3.3 Basic solutions with rotating actuators

205

Table 3.31. Structural parametersa of translational parallel mechanisms in Figs. 3.109–3.118 No.

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

Structural parameter

Solution 3-PaPr (Fig. 3.109a)

3-RRPr (Fig. 3.109b)

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

20 10 10 10 30 11 0 3 3 See Table 3.26

17 8 8 8 24 8 0 3 3 See Table 3.26

3-PaRRRR, 3-RPaRRR, 3-RRPaRR, 3-RRRRPa, 3-RRRPaR Figs. 3.110–3.118 20 8 8 8 24 5 0 3 3 See Table 3.26

3 3 3 13 13 13 3 3 3 ( v1 ,v 2 ,v3 ) 3 39 45 3 21 0 16

4 4 4 10 10 10 4 4 4 ( v1 ,v 2 ,v3 ) 3 30 39 3 9 0 14

5 5 5 3 3 3 5 5 5 ( v1 ,v 2 ,v3 ) 3 9 21 3 9 0 8

fj

16

14

8

fj

16

14

8

fj

48

42

24

SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

206

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.60. Complex limbs for TPMs with coupled motions defined by MG = SG = 3, (RG) = (v1,v2,v3), combining two Pa-type parallelogram loops and actuated by rotating motors mounted on the fixed base

3.3 Basic solutions with rotating actuators

207

Fig. 3.61. Complex limbs for TPMs with coupled motions defined by MG = SG = 3, (RG) = (v1,v2,v3), combining parallelogram loops of types Pa and Pacc and actuated by rotating motors mounted on the fixed base

208

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.62. Complex limbs for TPMs with coupled motions defined by MG = SG = 3, (RG) = (v1,v2,v3), combining three Pa-type parallelogram loops (a–c) or one Patype parallelogram loop and one Pr-type prism mechanism (d) and actuated by rotating motors mounted on the fixed base

3.3 Basic solutions with rotating actuators

209

Fig. 3.63. Simple limbs for TPMs with coupled motions defined by MG = SG = 4, (RG) = ( v1 , v2 , v3 ,Ȧ1 ) and actuated by rotating motors mounted on the fixed base

210

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.64. Complex limbs for TPMs with coupled motions defined by MG = SG = 4, (RG) = ( v1 , v2 , v3 ,Ȧ1 ), combining one Pa-type parallelogram loop and actuated by rotating motors mounted on the fixed base in the parallelogram loop

3.3 Basic solutions with rotating actuators

211

Fig. 3.65. Complex limbs for TPMs with coupled motions defined by MG = SG = 4, (RG) = ( v1 , v2 , v3 ,Ȧ1 ), combining two Pa-type parallelogram loop and actuated by rotating motors mounted on the fixed base in a parallelogram loop

212

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.65. (cont.)

Fig. 3.66. Complex limbs for TPMs with coupled motions defined by MG = SG = 4, (RG) = ( v1 , v2 , v3 ,Ȧ1 ), combining a Pacc-type parallelogram loop (a–c) or a Pr-type prism mechanism (d) and actuated by rotating motors mounted on the fixed base

3.3 Basic solutions with rotating actuators

213

Fig. 3.67. Complex limbs for TPMs with coupled motions defined by MG = SG = 4, (RG) = ( v1 ,v2 , v3 , Ȧ1 ), combining one Pa-type parallelogram loop and actuated by rotating motors mounted on the fixed base

214

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.68. Complex limbs for TPMs with coupled motions defined by MG = SG = 4, (RG) = ( v1 , v2 , v3 ,Ȧ1 ), combining one (a) or two (b–e) Pa-type parallelogram loop and actuated by rotating motors mounted on the fixed base

3.3 Basic solutions with rotating actuators

215

Fig. 3.69. Complex limbs for TPMs with coupled motions defined by MG = SG = 5, (RG) = ( v1 , v2 , v3 , Ȧ1 , Ȧ2 ), combining one parallelogram loop and actuated by rotating motors mounted on the fixed base inside (a–e) or outside (f) the parallelogram loop

216

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.70. Complex limbs for TPMs with coupled motions combining one parallelogram loop, defined by MG = SG = 5, (RG) = ( v1 , v2 , v3 , Ȧ1 , Ȧ2 ) and actuated by rotating motors mounted on the fixed base

3.3 Basic solutions with rotating actuators

Fig. 3.70. (cont.)

217

218

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.71. 3-PaPaP-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 24, limb topology Pa A Pa||P

3.3 Basic solutions with rotating actuators

219

Fig. 3.72. 3-PaPaP-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 24, limb topology Pa||Pa||P

220

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.73. 3-PaPPa-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 24, limb topology Pa A P||Pa

3.3 Basic solutions with rotating actuators

221

Fig. 3.74. 3-PaPPa-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 24, limb topology Pa||P||Pa

222

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.75. 3-PaPacc overconstrained TPMs with coupled motions, rotating actuators on the fixed base and six cylindrical joints adjacent to the moving platform, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa A Pacc

3.3 Basic solutions with rotating actuators

223

Fig. 3.76. 3-PaPacc-type overconstrained TPMs with coupled motions, rotating actuators on the fixed base and six revolute joints adjacent to the moving platform, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa A Pacc

224

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.77. 3-PaPacc-type overconstrained TPMs with coupled motions, rotating actuators mounted on the fixed base and six cylindrical joints adjacent to the moving platform, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa||Pacc

3.3 Basic solutions with rotating actuators

225

Fig. 3.78. 3-PaPacc-type overconstrained TPMs with coupled motions, rotating actuators on the fixed base and six revolute joints adjacent to the moving platform, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa||Pacc

226

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.79. 3-PaccPa-type overconstrained TPMs with coupled motions, rotating actuators on the fixed base and six revolute joints adjacent to the moving platform, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pacc||Pa

3.3 Basic solutions with rotating actuators

227

Fig. 3.80. 3-PaPaPa-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 33, limb topology Pa A Pa A ||Pa

228

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.81. 3-PaPaPa-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 33, limb topology Pa A Pa A A Pa

3.3 Basic solutions with rotating actuators

229

Fig. 3.82. 3-PaPaPa-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 33, limb topology Pa||Pa A Pa

230

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.83. 3-RRC-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base (no isotropic position), defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ), TF = 0, NF = 3, limb topology R||R||C

3.3 Basic solutions with rotating actuators

231

Fig. 3.84. 3-RRC-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base (with an isotropic position), defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology R||R||C

232

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.85. Overconstrained TPMs with coupled motions of types 3-RCR (a) and 3RPRR (b) defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ), TF = 0, NF = 3, limb topology R||C||R (a) and R||P||R||R (b)

3.3 Basic solutions with rotating actuators

233

Fig. 3.86. Overconstrained TPMs with coupled motions of types 3-RPC (a) and 3RPPR (b) defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ), TF = 0, NF = 3, limb topology R A P A ||C (a) and R||P A P A ||R (b)

234

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.87. 3-PaRC-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by M F = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology Pa||R||C

3.3 Basic solutions with rotating actuators

235

Fig. 3.88. 3-PaCR-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology Pa||C||R

236

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.89. 3-PaPaRR-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa||Pa A R||R

3.3 Basic solutions with rotating actuators

237

Fig. 3.90. 3-PaPaRR-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa A Pa A ||R||R

238

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.91. 3-PaPaRR-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa A Pa A A R||R

3.3 Basic solutions with rotating actuators

239

Fig. 3.92. 3-PaRRPa-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa||R||R A Pa

240

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.93. 3-PaRRPa-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa A R||R A A Pa

3.3 Basic solutions with rotating actuators

241

Fig. 3.94. 3-PaRRPa-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa A R||R A ||Pa

242

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.95. 3-PaRPaR-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa A R A Pa A ||R

3.3 Basic solutions with rotating actuators

243

Fig. 3.96. 3-PaRPaR-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa||R A Pa A ||R

244

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.97. 3-RRPaP-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology R||R||Pa||P

3.3 Basic solutions with rotating actuators

245

Fig. 3.98. 3-RPRPa-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology R||P||R||Pa

246

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.99. 3-RCPa-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology R||C||Pa

3.3 Basic solutions with rotating actuators

247

Fig. 3.100. 3-RPaRR-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology R A Pa A ||R||R

248

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.101. 3-RRPaR-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology R||R A Pa A ||R

3.3 Basic solutions with rotating actuators

249

Fig. 3.102. 3-RPaPaR-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology R A Pa||Pa A ||R

250

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.103. 3-RPaRPa-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology R A Pa A ||R A Pa

3.3 Basic solutions with rotating actuators

251

Fig. 3.104. 3-RPaRPa-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology R A Pa A ||R A Pa

252

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.105. 3-RPaPaR-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology R A Pa||Pa A ||R

3.3 Basic solutions with rotating actuators

253

Fig. 3.106. 3-RRPacc-type overconstrained TPMs with coupled motions, rotating actuators on the fixed base and six revolute joints adjacent to the moving platform, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R||R||Pacc

254

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.107. 3-RRPacc-type overconstrained TPMs with coupled motions, rotating actuators mounted on the fixed base and six cylindrical joints adjacent to the moving platform, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R||R||Pacc

3.3 Basic solutions with rotating actuators

255

Fig. 3.108. 3-PaccRR-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology Pacc||R||R

256

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.109. Overconstrained TPMs with coupled motions of types 3-PaPr (a) and 3-RRPr (b) defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21 (a), NF = 9 (b), limb topology Pa-Pr (a) and 3-R||R-Pr (b)

3.3 Basic solutions with rotating actuators

257

Fig. 3.110. 3-PaRRRR -type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology Pa||R||R A R||R (a) and Pa A R A R||R A R (b)

258

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.111. 3-PaRRRR-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology Pa A R||R A ||R||R (a) and Pa A R A R||R A ||R (b)

3.3 Basic solutions with rotating actuators

259

Fig. 3.112. Overconstrained TPMs of types 3-PaRRRR (a) and 3-RRPaRR (b) with coupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology Pa||R A R||R A ||R (a) and R A R A ||Pa A ||R A R (b)

260

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.113. 3-RPaRRR-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R A Pa||R||R A ||R (a) and R A Pa A A R||R A A R (b)

3.3 Basic solutions with rotating actuators

261

Fig. 3.114. 3-RPaRRR -type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R A Pa A ||R A R||R (a) and R A Pa||R||R A ||R (b)

262

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.115. 3-RRPaRR -type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R||R||Pa A R||R (a) and R||R A Pa A A R||R (b)

3.3 Basic solutions with rotating actuators

263

Fig. 3.116. 3-RRRRPa-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R||R A R||R A A Pa

264

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.117. Overconstrained TPMs of types 3-RRRPaR (a) and 3-RRRRPa (b) with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R||R A R A Pa A ||R (a) and R A R||R A R A Pa (b)

3.3 Basic solutions with rotating actuators

265

Fig. 3.118. 3-RRRPaR-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R A R||R||Pa A ||R

266

3 Overconstrained T3-type TPMs with coupled motions

3.4 Derived solutions with rotating actuators Solutions with lower degrees of overconstraint can be derived from the basic solutions in Figs. 3.71–3.118 by using joints with idle mobilities. A large set of solutions can be obtained by introducing adequate idle mobilities inside and/or outside the parallelogram loops. For example, the solution 3-PaPaC*-type in Fig. 3.119 is derived from the solution 3-PaPaP-type in Fig. 3.71 by replacing in each leg the prismatic joint P by a cylindrical joint C* combining one rotational idle mobility. We recall that the joints combining idle mobilities are denoted by an asterisk. The idle mobilities that can be combined in a parallelogram loop (see Fig. 6.3 – Part 1) are systematized in Table 3.7. One rotational idle mobility is combined in each cylindrical joint C* outside the parallelogram loop. The rotational mobility of the revolute joint denoted by R* is an idle mobility. Two idle rotational mobilities are introduced in the spherical joint of the parallelogram loops denoted by Paccs and Pascc which combines two cylindrical, one revolute and one spherical joint (see Figs. 3.147–3.151 and 3.194–3.196). Examples of solutions with identical limbs and 3–30 degrees of overconstraint derived from the basic solutions in Figs. 3.71–3.82 and 3.87–3.109 are illustrated in Figs. 3.119–3.198. The limb topology and the number of overconstraints of these solutions are systematized in Table 3.32, and their structural parameters in Tables 3.33–3.44.

3.4 Derived solutions with rotating actuators

267

Table 3.32. Limb topology and the number of overconstraints NF of the derived TPMs with idle mobilities and rotating actuators mounted on the fixed base presented in Figs. 3.119–3.198 No. Basic TPM Type 1 3-PaPaP (Fig. 3.71) 2

NF 24

3 4

3-PaPaP (Fig. 3.72)

24

5 6 7

3-PaPPa (Fig. 3.73)

24

8 9 10

3-PaPPa (Fig. 3.74)

24

11 12 13

3-PaPacc (Fig. 3.75)

21

14 15 16

3-PaPacc (Fig. 3.76)

21

17 18 19 20

3-PaPacc (Fig. 3.77)

21

Derived TPM Type 3-PaPaC* (Fig. 3.119) 3-PaR*PaC* (Fig. 3.131) 3-Pa*PassP (Fig. 3.143) 3-PaPaC* (Fig. 3.120) 3-PaR*PaC* (Fig. 3.132) 3-Pa*PassP (Fig. 3.144) 3-PaC*Pa (Fig. 3.121) 3-PaC*PaR* (Fig. 3.133) 3-Pa*PPass (Fig. 3.145) 3-PaC*Pa (Fig. 3.122) 3-PaC*PaR* (Fig. 3.134) 3-Pa*PPass (Fig. 3.146) 3-PaR*Pacc (Fig. 3.123) 3-PaR*R*Pacc (Fig. 3.135) 3-Pa*Pascc (Fig. 3.147) 3-PaR*Pacc (Fig. 3.124) 3-PaR*PaccR* (Fig. 3.136) 3-Pa*Paccs (Fig. 3.148) 3-PaR*Pacc (Fig. 3.125) 3-PaR*R*Pacc (Fig. 3.137)

NF 21

Limb topology Pa A Pa||C*

18

Pa||R A Pa||C*

3

Pa* A Pass||P

21

Pa||Pa||C*

18

Pa A R* A ||Pa||C*

3

Pa*||Pass||P

21

Pa A C*||Pa

18

Pa A C*||Pa A ||R*

3

Pa* A P||Pass

21

Pa||C*||Pa

18

Pa||C*||Pa A R*

3

Pa*||P||Pass

18

Pa A R*||Pacc

15

Pa||R* A R*||Pacc

6

Pa* A Pascc

18

Pa A R* A A Pacc

15

Pa A R*||Pacc A ||R*

6

Pa* A Paccs

18

Pa||R*||Pacc

15

Pa A R* A ||R*||Pacc

268

3 Overconstrained T3-type TPMs with coupled motions Table 3.32. (cont.)

21 22

3-PaPacc (Fig. 3.78)

21

23 24 25

3-PaccPa (Fig. 3.79)

21

26 27 28

3-PaPaPa (Fig. 3.80)

33

29 30 31

3-PaPaPa (Fig. 3.81)

33

32 33 34

3-PaPaPa (Fig. 3.82)

33

35 36 37

3-PaRC (Fig. 3.87)

12

38 39

3-PaCR (Fig. 3.88)

12

40 41 42

3-PaPaRR (Fig. 3.89)

21

3-Pa*Pascc (Fig. 3.149) 3-PaR*Pacc (Fig. 3.126) 3-PaR*PaccR* (Fig. 3.138) 3-Pa*Paccs (Fig. 3.150) 3-PaccR*Pa (Fig. 3.127) 3-PaccR*PaR* (Fig. 3.139) 3-PasccPass (Fig. 3.151) 3-PaR*PaPa (Fig. 3.128) 3-PaR*PaPaR* (Fig. 3.140) 3-PassPaPass (Fig. 3.152) 3-PaR*PaPa (Fig. 3.129) 3-PaR*PaPaR* (Fig. 3.141) 3-PaPassPass (Fig. 3.153) 3-PaR*PaPa (Fig. 3.130) 3-PaR*PaPaR* (Fig. 3.142) 3-PassPaPass (Fig. 3.154) 3-PaR*RC (Fig. 3.155) 3-Pa*RC (Fig. 3.157) 3-PaR*CR (Fig. 3.156) 3-Pa*CR (Fig. 3.158) 3-PaPassR (Fig. 3.159) 3-PassPaRR (Fig. 3.166)

6

Pa*||Pascc

18

Pa A R* A ||Pacc

15

Pa||R*||Pacc A R*

6

Pa*||Paccs

18

Pacc A R* A ||Pa

15

Pacc A R* A ||Pa||R*

3

Pascc||Pass

30

Pa A R* A A Pa A A Pa

27

Pa A R* A A Pa A A Pa A ||R*

9

Pass A Pa A ||Pass

30

Pa A R* A A Pa A ||Pa

27

Pa A R* A A Pa A ||Pa A A R*

9

Pa A Pass A A Pass

30

Pa A R A ||Pa A A Pa

27

Pa A R* A ||Pa A A Pa A ||R*

9

Pass||Pa A Pass

9

Pa A R* A ||R||C

3

Pa*||R||C

9

Pa A R* A ||C||R

3

Pa*||C||R

12

Pa||Pass A R

9

Pass||Pa A R||R

3.4 Derived solutions with rotating actuators Table 3.32. (cont.) 43

3-PaPaRR (Fig. 3.90)

21

44 45

3-PaPaRR (Fig. 3.91)

21

46 47

3-PaRRPa (Fig. 3.92)

21

48 49

3-PaRRPa (Fig. 3.93)

21

50 51

3-PaRRPa (Fig. 3.94)

21

52 53

3-PaRPaR (Fig. 3.95)

21

54 55

3-PaRPaR (Fig. 3.96)

21

56 57

3-RRPaP (Fig. 3.97)

12

58 59

3-RPRPa (Fig. 3.98)

12

60 61

3-RCPa (Fig. 3.99)

12

62 63 64

3-RPaRR 12 (Fig. 3.100a)

3-PaPassR (Fig. 3.160) 3-PassPaRR (Fig. 3.167) 3-PaPassR (Fig. 3.161) 3-PassPaRR (Fig. 3.168) 3-PaRPass (Fig. 3.162) 3-PassRRPa (Fig. 3.169) 3-PaRPass (Fig. 3.163) 3-PassRRPa (Fig. 3.170) 3-PassRPa (Fig. 3.164) 3-PaRRPass (Fig. 3.171) 3-PaRPass (Fig. 3.165a) 3-PassRPaR (Fig. 3.172a) 3-PaRPass (Fig. 3.165b) 3-PassRPaR (Fig. 3.172b) 3-RR*RPaP (Fig. 3.173) 3-RRPa*P (Fig. 3.176) 3-RPR*RPa (Fig. 3.174) 3-RPRPa* (Fig. 3.177) 3-RR*CPa (Fig. 3.175) 3-RCPa* (Fig. 3.178) 3-RPaRRR* (Fig. 3.179a) 3-RPassR (Fig. 3.185a)

12

Pa A Pass A ||R

9

Pass A Pa A ||R||R

12

Pa A Pass A A R

9

Pass A Pa A A R||R

12

Pa||R A Pass

9

Pass||R||R A Pa

12

Pa A R A A Pass

9

Pass A R||R A A Pa

12

Pass A R A ||Pa

9

Pa A R||R A ||Pass

12

Pa A R A Pass

9

Pass A R A Pa A ||R

12

Pa||R A Pass

9

Pass||R A Pa A ||R

9

R A R* A ||R||Pa||R

3

R||R||Pa*||P

9

R||P A R* A ||R||Pa

3

R||P||R||Pa*

9

R A R* A ||C||Pa

3

R||C||Pa*

9

R A Pa A ||R||R A R*

3

R A Pass A ||R

269

270

3 Overconstrained T3-type TPMs with coupled motions Table 3.32. (cont.)

65

12 3-RPaRR (Fig. 3.100b)

66 67

3-RRPaR 12 (Fig. 3.101a)

68 69

3-RRPaR 12 (Fig. 3.101b)

70 71

3-RPaPaR (Fig. 3.102)

21

72 73

3-RPaRPa (Fig. 3.103)

21

73 75

3-RPaRPa (Fig. 3.104)

21

76 77

3-RPaPaR (Fig. 3.105)

21

78 79

3-RRPacc (Fig. 3.106)

9

80 81

3-RRPacc (Fig. 3.107)

9

82 83

3-PaccRR (Fig. 3.108)

9

84 85 86

3-PaPr 21 (Fig. 3.109a)

3-RPaRR*R (Fig. 3.179b) 3-RPassR (Fig. 3.185b) 3-RRPaRR* (Fig. 3.180a) 3-RRPass (Fig. 3.186a) 3-RR*RPaR (Fig. 3.180b) 3-RRPass (Fig. 3.186b) 3-RPaPass (Fig. 3.181) 3-RPassPaR (Fig. 3.187) 3-RPassPa (Fig. 3.182) 3-RPaRPass (Fig. 3.188) 3-RPaPass (Fig. 3.183) 3-RPassRPa (Fig. 3.189) 3-RPacsPaR* (Fig. 3.184) 3-RPaPassR (Fig. 3.190) 3-RR*RPacc (Fig. 3.191) 3-RRPaccs (Fig. 3.194) 3-RR*RPacc (Fig. 3.192) 3-RRPascc (Fig. 3.195) 3-PaccRRR*R (Fig. 3.193) 3-PasccRR (Fig. 3.196) 3-PaPr* (Fig. 3.197a) 3-PassPrR* (Fig. 3.198a)

9

R A Pa A ||R A R* A ||R

3

R A Pass A ||R

9

R||R A Pa A ||R A R*

3

R||R A Pass

9

R A R* A ||R A Pa A ||R

3

R||R A Pass

12

R A Pa||Pass

9

R A Pass||Pa A ||R

12

R A Pass||Pa

9

R A Pa A ||R A Pass

12

R A Pa A A Pass

9

R A Pass A ||R A ||Pa

9

R A Pacs||Pa A ||R*

9

R A Pa||Pass A ||R

6

R A R* A ||R||Pacc

3

R||R||Paccs

6

R A R* A ||R||Pacc

3

R||R||Pascc

6

Pacc||R A R* A ||R

3

Pascc||R||R

15

Pa-Pr*

6

Pass-Pr-R*

3.4 Derived solutions with rotating actuators

271

Table 3.32. (cont.) 87

9 3-RRPr (Fig. 3.109b)

88

3-RRPrR* (Fig. 3.197b) 3-RRPr* (Fig. 3.198b)

6

R||R-Pr-R*

3

R||R-Pr*

Table 3.33. Bases of the operational velocities spaces of the limbs isolated from the parallel mechanisms presented in Figs. 3.119–3.198 No. Parallel mechanism 1 Figs. 3.119a, 3.120a, 3.121a, 3.122a, 3.123a, 3.125a, 3.129, 3.160, 3.161, 3.165, 3.176a, 3.177a, 3.178a, 3.182a, 3.184a, 3.185a, 3.186a, 3.194a, 3.195a, 3.196a, 2 Figs. 3.119b, 3.120b, 3.121b, 3.122b, 3.123b, 3.125b, 3.157b, 3.158b, 3.176b, 3.177b, 3.178b, 3.181b, 3.194b, 3.195b, 3.196b 3 Figs. 3.124, 3.126, 3.128, 3.130, 3.164 4 Figs. 3.127a, 3.198b, 3.151a, 3.157a, 3.158a, 3.159, 3.162, 3.163, 3. 181a, 3.183a, 5 Figs. 3.127b, 3.151b, 3.185b, 3.186b, 3.198b

Basis (RG1) ( v1 , v2 , v3 , ȦE )

(RG2) ( v1 ,v2 , v3 ,ȦG )

(RG3) ( v1 ,v2 , v3 ,ȦD )

( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦD )

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

272

3 Overconstrained T3-type TPMs with coupled motions Table 3.33. (cont.)

6

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦG )

Fig. 3.131–3.139, 3.141, 3.153, 3.155, 3.156, 3.173–3.175 7 Fig. 3.140, 3.152, 3.167, 3.168, 3.172, 3.179, 3.180, 3.197b, 3.197b, 3.198a 8 Fig. 3.142, 3.154, 3.166, 3.169, 3.170, 3.171 9 Figs. 3.143a, 3.144a, 3.145a, 3.146a 10 Figs. 3.147–3.150, 3.197a 11 Figs. 3.182b, 3.184b 12 Fig. 3.183b 13 Figs. 3.187–3.193

( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE )

( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE )

( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG ) ( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 , v 2 , v 3 )

( v1 , v 2 , v 3 )

( v1 , v 2 , v 3 )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦD )

( v1 ,v2 , v3 ,ȦD )

( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦG )

( v1 , v2 , v3 , ȦE )

3.4 Derived solutions with rotating actuators

273

Table 3.34. Structural parametersa of translational parallel mechanisms in Figs. 3.119–3.130 No. Structural parameter

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

Solution 3-PaPaC* (Figs. 3.119, 3.120) 3-PaC*Pa (Figs. 3.121, 3.122) 20 9 9 9 27 8 0 3 3 See Table 3.33

3-PaR*Pacc (Figs. 3.123–3.126) 3-PaccR*Pa (Fig. 3.127) 20 9 9 9 27 8 0 3 3 See Table 3.33

29 13 13 13 39 11 0 3 3 See Table 3.33

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 21 0 10

4 4 4 7 7 7 4 4 4 ( v1 ,v 2 ,v3 ) 3 21 30 3 18 0 11

4 4 4 9 9 9 4 4 4 ( v1 ,v 2 ,v3 ) 3 27 36 3 30 0 13

fj

10

11

13

fj

10

11

13

fj

30

33

39

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

3-PaR*PaPa (Figs. 3.128– 3.130)

See footnote of Table 2.1 for the nomenclature of structural parameters

274

3 Overconstrained T3-type TPMs with coupled motions

Table 3.35. Structural parametersa of translational parallel mechanisms in Figs. 3.131–3.142 No. Structural parameter

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

23 10 10 10 30 8 0 3 3 See Table 3.33

3-PaR*R*Pacc (Figs. 3.135, 3.137) 3-PaR*PaccR* (Fig. 3.136, 3.138) 3-PaccR*PaR* (Fig. 3.139) 23 10 10 10 30 8 0 3 3 See Table 3.33

32 14 14 14 42 11 0 3 3 See Table 3.33

5 5 5 6 6 6 5 5 5 ( v1 ,v 2 ,v3 ) 3 18 30 3 18 0 11

5 5 5 7 7 7 5 5 5 ( v1 ,v 2 ,v3 ) 3 21 33 3 15 0 12

5 5 5 9 9 9 5 5 5 ( v1 ,v 2 ,v3 ) 3 27 39 3 27 0 14

fj

11

12

14

fj

11

12

14

fj

33

36

42

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

Solution 3-PaR*PaC* (Figs. 3.131, 3.132) 3-PaC*PaR* (Figs. 3.133, 3.134)

fj

3-PaR*PaPaR* (Figs. 3.140– 3.142)

See footnote of Table 2.1 for the nomenclature of structural parameters

3.4 Derived solutions with rotating actuators

275

Table 3.36. Structural parametersa of translational parallel mechanisms in Figs. 3.143–3.151 No. Structural parameter

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

Solution 3-Pa*PassP (Figs. 3.143, 3.144) 3-Pa*PPass (Figs. 3.145, 3.146) 20 9 9 9 27 8 0 3 3 See Table 3.33

3-Pa*Pascc (Figs. 3.147, 3.149) 3-Pa*Paccs (Figs. 3.148, 3.150) 17 8 8 8 24 8 0 3 3 See Table 3.33

17 8 8 8 24 8 0 3 3 See Table 3.33

4 4 4 12 12 12 4 4 4 ( v1 ,v 2 ,v3 ) 3 36 45 3 3 0 16

3 3 3 12 12 12 3 3 3 ( v1 ,v 2 ,v3 ) 3 36 42 3 6 0 15

4 4 4 12 12 12 4 4 4 ( v1 ,v 2 ,v3 ) 3 36 45 3 3 0 16

fj

16

15

16

fj

16

15

16

fj

48

45

48

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

3-PasccPass (Fig. 3.151)

See footnote of Table 2.1 for the nomenclature of structural parameters

276

3 Overconstrained T3-type TPMs with coupled motions

Table 3.37. Structural parametersa of translational parallel mechanisms in Figs. 3.152–3.158 No. Structural parameter

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

Solution 3-PassPaPass (Figs. 3.152, 3.154) 3-PaPassPass (Fig. 3.153) 26 12 12 12 36 11 0 3 3 See Table 3.33

3-PaR*RC (Fig. 3.155) 3-PaR*CR (Fig. 3.156) 17 7 7 7 21 5 0 3 3 See Table 3.33

3-Pa*RC (Fig. 3.157) 3-Pa*CR (Fig. 3.158) 14 6 6 6 18 5 0 3 3 See Table 3.33

5 5 5 15 15 15 5 5 5 ( v1 ,v 2 ,v3 ) 3 45 57 3 9 0 20

5 5 5 3 3 3 5 5 5 ( v1 ,v 2 ,v3 ) 3 9 21 3 9 0 8

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 3 0 10

fj

20

8

10

fj

20

8

10

fj

60

24

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

3.4 Derived solutions with rotating actuators

277

Table 3.38. Structural parametersa of translational parallel mechanisms in Figs. 3.159–3.172 No.

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

Structural parameter

Solution 3-PaPassR (Figs. 3.159–3.161) 3-PaRPass (Figs. 3.162, 3.163) 3-PassRPa (Fig. 3.164) 3-PaRPass (Fig. 3.165)

3-PassPaRR (Figs. 3.166–3.168) 3-PassRRPa (Figs. 3.169, 3.170) 3-PaRRPass (Fig. 3.171) 3-PassRPaR (Fig. 3.172)

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

20 9 9 9 27 8 0 3 3 See Table 3.33

23 10 10 10 30 8 0 3 3 See Table 3.33

4 4 4 9 9 9 4 4 4 ( v1 ,v 2 ,v3 ) 3 27 36 3 12 0 13

5 5 5 9 9 9 5 5 5 ( v1 ,v 2 ,v3 ) 3 27 39 3 9 0 14

fj

13

14

fj

13

14

fj

39

42

SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

278

3 Overconstrained T3-type TPMs with coupled motions

Table 3.39. Structural parametersa of translational parallel mechanisms in Figs. 3.173–3.177 No.

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

Structural parameter

Solution 3-RR*RPaP (Fig. 3.173) 3-RPR*RPa (Fig. 3.174) 20 8 8 8 24 5 0 3 3 See Table 3.33

17 7 7 7 21 5 0 3 3 See Table 3.33

3-RRPa*P (Fig. 3.176) 3-RPRPa* (Fig. 3.177) 17 7 7 7 21 5 0 3 3 See Table 3.33

5 5 5 3 3 3 5 5 5 ( v1 ,v 2 ,v3 ) 3 9 21 3 9 0 8

5 5 5 3 3 3 5 5 5 ( v1 ,v 2 ,v3 ) 3 9 21 3 9 0 8

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 3 0 10

fj

8

8

10

fj

8

8

10

fj

24

24

30

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

3-RR*CPa (Fig. 3.175)

See footnote of Table 2.1 for the nomenclature of structural parameters

3.4 Derived solutions with rotating actuators

279

Table 3.40. Structural parametersa of translational parallel mechanisms in Figs. 3.178–3.180 No.

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

Structural parameter

Solution 3-RCPa* (Fig. 3.178)

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

14 6 6 6 18 5 0 3 3 See Table 3.33

3-RPaRRR*,3-RPaRR*R (Fig. 3.179) 3-RRPaRR*, 3-RR*RPaR (Fig. 3.180) 20 8 8 8 24 5 0 3 3 See Table 3.33

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 3 0 10

5 5 5 3 3 3 5 5 5 ( v1 ,v 2 ,v3 ) 3 9 21 3 9 0 8

fj

10

8

fj

10

8

fj

30

24

SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

280

3 Overconstrained T3-type TPMs with coupled motions

Table 3.41. Structural parametersa of translational parallel mechanisms in Figs. 3.181–3.186 No.

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

Structural parameter

Solution 3-RPaPass (Figs. 3.181, 3.183) 3-RPassPa (Fig. 3.182) 20 9 9 9 27 8 0 3 3 See Table 3.33

23 10 10 10 30 8 0 3 3 See Table 3.33

3-RPassR (Fig. 3.185) 3-RRPass (Fig. 3.186) 14 6 6 6 18 5 0 3 3 See Table 3.33

4 4 4 9 9 9 4 4 4 ( v1 ,v 2 ,v3 ) 3 27 36 3 12 0 13

4 4 4 9 9 9 4 4 4 ( v1 ,v 2 ,v3 ) 3 27 36 3 12 0 13

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 3 0 10

fj

13

13

10

fj

13

13

10

fj

39

39

30

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

3-RPacsPaR* (Fig. 3.184)

See footnote of Table 2.1 for the nomenclature of structural parameters

3.4 Derived solutions with rotating actuators

281

Table 3.42. Structural parametersa of translational parallel mechanisms in Figs. 3.187–3.193 No.

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

Structural parameter

Solution 3-RPassPaR (Fig. 3.187) 3-RPaRPass (Fig. 3.188) 3-RPassRPa (Fig. 3.189) 3-RPaPassR (Fig. 3.190) 23 10 10 10 30 8 0 3 3 See Table 3.33

3-RR*RPacc (Figs. 3.191, 3.192) 3-PaccRR*R (Fig. 3.193) 17 7 7 7 21 5 0 3 3 See Table 3.33

5 5 5 9 9 9 5 5 5 ( v1 ,v 2 ,v3 ) 3 39 21 3 9 0 14

5 5 5 4 4 4 5 5 5 ( v1 ,v 2 ,v3 ) 3 12 24 3 6 0 9

fj

14

9

fj

14

9

fj

42

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

27

See footnote of Table 2.1 for the nomenclature of structural parameters

282

3 Overconstrained T3-type TPMs with coupled motions

Table 3.43. Structural parametersa of translational parallel mechanisms in Figs. 3.194–3.197 No. Structural parameter

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

3-RRPaccs (Fig. 3.194) 3-RRPascc (Fig. 3.195) 3-PasccRR (Fig. 3.196) 14 6 6 6 18 5 0 3 3 See Table 3.33

3-PaPr* (Fig. 3.197a)

3-RRPrR* (Fig. 3.197b)

20 10 10 10 30 11 0 3 3 See Table 3.33

20 9 9 9 27 8 0 3 3 See Table 3.33

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 3 0 10

3 3 3 15 15 15 3 3 3 ( v1 ,v 2 ,v3 ) 3 45 51 3 15 0 18

5 5 5 10 10 10 5 5 5 ( v1 ,v 2 ,v3 ) 3 30 42 3 6 0 15

fj

10

18

15

fj

10

18

15

fj

30

54

45

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

3.4 Derived solutions with rotating actuators

283

Table 3.44. Structural parametersa of translational parallel mechanisms in Fig. 3.198 No.

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

Structural parameter

Solution 3-PassPrR* (Fig. 3.198a) 23 11 11 11 33 11 0 3 3 See Table 3.33

3-RRPr* (Fig. 3.198b) 17 8 8 8 24 8 0 3 3 See Table 3.33

5 5 5 16 16 16 5 5 5 ( v1 ,v 2 ,v3 ) 3 48 60 3 6 0 21

4 4 4 12 12 12 4 4 4 ( v1 ,v 2 ,v3 ) 3 36 45 3 3 0 16

fj

21

16

fj

21

16

fj

63

48

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

284

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.119. 3-PaPaC*-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa A Pa||C*

3.4 Derived solutions with rotating actuators

285

Fig. 3.120. 3-PaPaC*-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa||Pa||C*

286

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.121. 3-PaC*Pa-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa A C*||Pa

3.4 Derived solutions with rotating actuators

287

Fig. 3.122. 3-PaC*Pa-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0 and NF = 21, limb topology Pa||C*||Pa

288

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.123. 3-PaR*Pacc overconstrained TPMs with coupled motions, rotating actuators mounted on the fixed base and six cylindrical joints adjacent to the moving platform, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 18, limb topology Pa A R*||Pacc

3.4 Derived solutions with rotating actuators

289

Fig. 3.124. 3-PaR*Pacc-type overconstrained TPMs with coupled motions, rotating actuators mounted on the fixed base and six revolute joints adjacent to the moving platform, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 18, limb topology Pa A R* A A Pacc

290

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.125. 3-PaR*Pacc-type overconstrained TPMs with coupled motions, rotating actuators mounted on the fixed base and six cylindrical joints adjacent to the moving platform, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 18, limb topology Pa||R*||Pacc

3.4 Derived solutions with rotating actuators

291

Fig. 3.126. 3-PaR*Pacc-type overconstrained TPMs with coupled motions, rotating actuators mounted on the fixed base and six revolute joints adjacent to the moving platform, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 18, limb topology Pa A R* A ||Pacc

292

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.127. 3-PaccR*Pa-type overconstrained TPMs with coupled motions, rotating actuators mounted on the fixed base and six revolute joints adjacent to the moving platform, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 18, limb topology Pacc A R* A ||Pa

3.4 Derived solutions with rotating actuators

293

Fig. 3.128. 3-PaR*PaPa-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 30, limb topology Pa A R* A A Pa A A Pa

294

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.129. 3-PaR*PaPa-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 30, limb topology Pa A R* A A Pa A ||Pa

3.4 Derived solutions with rotating actuators

295

Fig. 3.130. 3-PaR*PaPa-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 30, limb topology Pa A R* A ||Pa A A Pa

296

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.131. 3-PaR*PaC*-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 18, limb topology Pa||R* A Pa||C*

3.4 Derived solutions with rotating actuators

297

Fig. 3.132. 3-PaR*PaC*-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 18, limb topology Pa A R* A ||Pa||C*

298

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.133. 3-PaC*PaR*-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 18, limb topology Pa A C*||Pa A ||R*

3.4 Derived solutions with rotating actuators

299

Fig. 3.134. 3-PaC*PaR*-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 18, limb topology Pa||C*||Pa A R*

300

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.135. 3-PaR*R*Pacc-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 15, limb topology Pa||R* A R*||Pacc

3.4 Derived solutions with rotating actuators

301

Fig. 3.136. 3-PaR*PaccR*-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 15, limb topology Pa A R*||Pacc A ||R*

302

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.137. 3-PaR*R*Pacc-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 15, limb topology Pa A R* A ||R*||Pacc

3.4 Derived solutions with rotating actuators

303

Fig. 3.138. 3-PaR*PaccR*-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 15, limb topology Pa||R*||Pacc A R*

304

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.139. 3-PaccR*PaR*-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 15, limb topology Pacc A R* A ||Pa||R*

3.4 Derived solutions with rotating actuators

305

Fig. 3.140. 3-PaR*PaPaR*-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 27, limb topology Pa A R* A A Pa A A Pa A ||R*

306

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.141. 3-PaR*PaPaR*-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 27, limb topology Pa A R* A A Pa A ||Pa A A R*

3.4 Derived solutions with rotating actuators

307

Fig. 3.142. 3-PaR*PaPaR*-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 27, limb topology Pa A R* A ||Pa A A Pa A ||R*

308

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.143. 3-Pa*PassP-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pa* A Pass||P

3.4 Derived solutions with rotating actuators

309

Fig. 3.144. 3-Pa*PassP-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 3, limb topology Pa*||Pass||P

310

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.145. 3-Pa*PPass-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ), TF = 0, NF = 3, limb topology Pa* A P||Pass

3.4 Derived solutions with rotating actuators

311

Fig. 3.146. 3-Pa*PPass-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ), TF = 0, NF = 3, limb topology Pa*||P||Pass

312

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.147. 3-Pa*Pascc-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ), TF = 0, NF = 6, limb topology Pa* A Pascc

3.4 Derived solutions with rotating actuators

313

Fig. 3.148. 3-Pa*Paccs-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 6, limb topology Pa* A Paccs

314

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.149. 3-Pa*Pascc-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 6, limb topology Pa*||Pascc

3.4 Derived solutions with rotating actuators

315

Fig. 3.150. 3-Pa*Paccs-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 6, limb topology Pa*||Paccs

316

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.151. 3-PasccPass-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pascc||Pass

3.4 Derived solutions with rotating actuators

317

Fig. 3.152. 3-PassPaPass-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ), TF = 0, NF = 9, limb topology Pass A Pa A ||Pass

318

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.153. 3-PaPassPass-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology Pa A Pass A A Pass

3.4 Derived solutions with rotating actuators

319

Fig. 3.154. 3-PassPaPass-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology Pass||Pa A Pass

320

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.155. 3-PaR*RC-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology Pa A R* A ||R||C

3.4 Derived solutions with rotating actuators

321

Fig. 3.156. 3-PaR*CR-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology Pa A R* A ||C||R

322

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.157. 3-Pa*RC-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pa*||R||C

3.4 Derived solutions with rotating actuators

323

Fig. 3.158. 3-Pa*CR-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pa*||C||R

324

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.159. 3-PaPassR-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology Pa||Pass A R

3.4 Derived solutions with rotating actuators

325

Fig. 3.160. 3-PaPassR-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology Pa A Pass A ||R

326

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.161. 3-PaPassR-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology Pa A Pass A A R

3.4 Derived solutions with rotating actuators

327

Fig. 3.162. 3-PaRPass-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology Pa||R A Pass

328

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.163. 3-PaRPass-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology Pa A R A A Pass

3.4 Derived solutions with rotating actuators

329

Fig. 3.164. 3-PassRPa-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology Pass A R A ||Pa

330

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.165. 3-PaRPass-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 12, limb topology Pa A R A Pass (a) and Pa||R A Pass (b)

3.4 Derived solutions with rotating actuators

331

Fig. 3.166. 3-PassPaRR-type (Pass||Pa A R||R) overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0 and NF = 9

332

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.167. 3-PassPaRR-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology Pass A Pa A ||R||R

3.4 Derived solutions with rotating actuators

333

Fig. 3.168. 3-PassPaRR-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology Pass A Pa A A R||R

334

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.169. 3-PassRRPa-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology Pass||R||R A Pa

3.4 Derived solutions with rotating actuators

335

Fig. 3.170. 3-PassRRPa-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology Pass A R||R A A Pa

336

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.171. 3-PaRRPass-type overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ), TF = 0, NF = 9, limb topology Pa A R||R A ||Pass

3.4 Derived solutions with rotating actuators

337

Fig. 3.172. 3-PassRPaR-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ) TF = 0, NF = 9, limb topology Pass A R A Pa A ||R (a) and Pass||R A Pa A ||R (b)

338

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.173. 3-RR*RPaP-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R A R* A ||R||Pa||P

3.4 Derived solutions with rotating actuators

339

Fig. 3.174. 3-RPR*RPa-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R||P A R* A ||R||Pa

340

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.175. 3-RR*CPa-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R A R* A ||C||Pa

3.4 Derived solutions with rotating actuators

341

Fig. 3.176. 3-RRPa*P-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ), TF = 0, NF = 3, limb topology R||R||Pa*||P

342

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.177. 3-RPRPa*-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology R||P||R||Pa*

3.4 Derived solutions with rotating actuators

343

Fig. 3.178. 3-RCPa*-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ), TF = 0, NF = 3, limb topology R||C||Pa*

344

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.179. Overconstrained TPMs of types 3-RPaRRR* (a) and 3-RPaRR*R (b) with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R A Pa A ||R||R A R* (a) and R A Pa A ||R A R* A ||R (b)

3.4 Derived solutions with rotating actuators

345

Fig. 3.180. Overconstrained TPMs of types 3-RRPaRR* (a) and 3-RR*RPaR (b) with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ) TF = 0, NF = 9, limb topology R||R A Pa A ||R A R* (a) and R A R* A ||R A Pa A ||R (b)

346

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.181. 3-RPaPass-type TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ), TF = 0, NF = 12, limb topology R A Pa||Pass

3.4 Derived solutions with rotating actuators

347

Fig. 3.182. 3-RPassPa-type TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ), TF = 0, NF = 12, limb topology R A Pass||Pa

348

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.183. 3-RPaPass-type TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ), TF = 0, NF = 12, limb topology R A Pa A A Pass

3.4 Derived solutions with rotating actuators

349

Fig. 3.184. 3-RPacsPaR*-type TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R A Pacs||Pa A ||R*

350

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.185. 3-RPassR-type TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology R A Pass A ||R

3.4 Derived solutions with rotating actuators

351

Fig. 3.186. 3-RRPass-type TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology R||R A Pass

352

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.187. 3-RPassPaR-type TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R A Pass||Pa A ||R

3.4 Derived solutions with rotating actuators

353

Fig. 3.188. 3-RPaRPass-type TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R A Pa A ||R A Pass

354

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.189. 3-RPassRPa-type TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R A Pass A ||R A ||Pa

3.4 Derived solutions with rotating actuators

355

Fig. 3.190. 3-RPaPassR-type TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R A Pa||Pass A ||R

356

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.191. 3-RR*RPacc-type overconstrained TPMs with coupled motions, rotating actuators mounted on the fixed base and six revolute joints adjacent to the moving platform, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 6, limb topology R A R* A ||R||Pacc

3.4 Derived solutions with rotating actuators

357

Fig. 3.192. 3-RR*RPacc-type overconstrained TPMs with coupled motions, rotating actuators mounted on the fixed base and six cylindrical joints adjacent to the moving platform, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 6, limb topology R A R* A ||R||Pacc

358

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.193. 3-PaccRR*R-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 6, limb topology Pacc||R A R* A ||R

3.4 Derived solutions with rotating actuators

359

Fig. 3.194. 3-RRPaccs-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology R||R||Paccs

360

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.195. 3-RRPascc-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology R||R||Pascc

3.4 Derived solutions with rotating actuators

361

Fig. 3.196. 3-PasccRR-type overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pascc||R||R

362

3 Overconstrained T3-type TPMs with coupled motions

Fig. 3.197. Overconstrained TPMs with coupled motions of types 3-PaPr* (a) and 3-RRPrR* (b) defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 15 (a), NF = 6 (b), limb topology Pa-Pr* (a) and 3-R||RPr-R* (b)

3.4 Derived solutions with rotating actuators

363

Fig. 3.198. Overconstrained TPMs with coupled motions of types 3-PassPrR* (a) and 3-RRPr* (b) defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 6 (a), NF = 3 (b), limb topology Pass-Pr-R* (a) and R||R-Pr* (b)

4 Non overconstrained T3-type TPMs with coupled motions

Equation (1.15) indicates that non overconstrained solutions of T3-type TPMs with coupled motions and q independent loops meet the condition p ¦ 1 fi 3 6q . Various solutions fulfil this condition along with SF = 3, (RF) = (v1,v2,v3) and NF = 0. They can have identical limbs or limbs with different structures and may be actuated by linear or rotating motors.

4.1 Basic solutions with linear actuators In the basic non overconstrained TPMs with linear actuators and coupled motions F m G1–G2–G3, the moving platform n Ł nGi (i = 1, 2, 3) is connected to the reference platform 1 Ł 1Gi Ł 0 by three limbs with five degrees of connectivity. No idle mobilities exist in these basic solutions. The various types of limbs with five degrees of connectivity and no idle mobilities are systematized in Figs. 4.1 and 4.2. They are actuated by linear motors mounted on the fixed base (Fig. 4.1) or on a moving link (Fig. 4.2). The prismatic joints between links 2 and 3 (Fig. 4.2c–f) and 3 and 4 (Fig. 4.2a, b) are actuated in the solutions with the linear actuator non adjacent to the fixed base. Various solutions of TPMs with coupled motions and no idle mobilities can be obtained by using three limbs with identical or different topologies presented in Figs. 4.1 and 4.2. We only show solutions with identical limb type as illustrated in Figs. 4.3–4.9. The limb topology and connecting conditions in these solutions are systematized in Table 4.1. The actuated prismatic joints adjacent to the fixed base in the three limbs have orthogonal directions (Figs. 4.3–4.5) in the solutions using the limbs systematized in Fig. 4.1. The axes of the first unactuated revolute joints of the three limbs have orthogonal directions (Figs. 4.6, 4.7a, 4.8 and 4.9) or are parallel to one plane (Fig. 4.7b) in the solutions using the limbs systematized in Fig. 4.2. 365 G. Gogu, Structural Synthesis of Parallel Robots: Part 2: Translational Topologies with Two and Three Degrees of Freedom, Solid Mechanics and Its Applications 159, 365–469. © Springer Science + Business Media B.V. 2009

366

4 Non overconstrained T3-type TPMs with coupled motions

For the solutions in Figs. 4.3–4.9, Eqs. (1.2)–(1.8) and (1.17) give the following structural parameters: MGi = SGi = 3, (RG1) = ( v x , v y , v y , ȦE , ȦG ), (RG2) = ( v x , v y , v y , ȦD , ȦG ), (RG2) = ( v x , v y , v y , ȦD , ȦE ), (RF) = ( v x , v y , v y ), SF = 3, rF = 12, MF = 3, NF = 0 and TF = 0. Table 4.1. Limb topology and connecting conditions of the non overconstrained TPM with no idle mobilities and linear actuators presented in Figs. 4.3–4.9 No.

TPM type

Limb topology

Connecting conditions

1

3-PRRRR (Fig. 4.3a)

P A R||R A R||R (Fig. 4.1a)

2

3-PRRRR (Fig. 4.3b) 3-PRRRR (Fig. 4.4a) 3-PRRRR (Fig. 4.4b) 3-PRRRR (Fig. 4.5) 3-RRPRR (Fig. 4.6)

P A R A R||R A R (Fig. 4.1b) P||R||R A R||R (Fig. 4.1c) P||R A R||R A R (Fig. 4.1d) P A R A R||R A R (Fig. 4.1e) R A R||P||R A R (Fig. 4.2a)

Actuated P joints adjacent to the fixed base have orthogonal directions Idem No. 1

3-RRPRR (Fig. 4.7a) 3-RRPRR (Fig. 4.7b)

R A R A P A ||R A R (Fig. 4.2b) R A R A P A ||R A R (Fig. 4.2b)

3-RPRRR (Fig. 4.8a) 3-RPRRR (Fig. 4.8b) 3-RPRRR (Fig. 4.9a) 3-RPRRR (Fig. 4.9b)

R A P A A R||R A R (Fig. 4.2c) R A P||R||R A R (Fig. 4.2d) R A P A ||R A R||R (Fig. 4.2e) R||P A R||R A R (Fig. 4.2f)

3 4 5 6

7 8

9 10 11 12

Idem No. 1 Idem No. 1 Idem No. 1 Actuated P joints non adjacent to the fixed base and the first revolute joints of the three legs have orthogonal axes Idem No. 6 Actuated P joints non adjacent to the fixed base and the first revolute joints of the three legs are parallel to one plane Idem No. 6 Idem No. 6 Idem No. 6 Idem No. 6

4.1 Basic solutions with linear actuators

367

Fig. 4.1. Simple limbs for non overconstrained TPMs with coupled motions defined by MG = SG = 5, (RG) = ( v1 , v2 , v3 , Ȧ1 , Ȧ2 ) and actuated by linear motors mounted on the fixed base

368

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.2. Simple limbs for non overconstrained TPMs with coupled motions defined by MG = SG = 5, (RG) = ( v1 , v2 , v3 , Ȧ1 , Ȧ2 ) and actuated by linear motors non adjacent to the fixed base

4.1 Basic solutions with linear actuators

369

Fig. 4.3. 3-PRRRR-type non overconstrained TPMs with coupled motions and linear actuators mounted on the fixed base, limb topology P A R||R A R||R (a) and P A R A R||R A R (b)

370

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.4. 3-PRRRR-type non overconstrained TPMs with coupled motions and linear actuators mounted on the fixed base, limb topology P||R||R A R||R (a) and P||R A R||R A R (b)

4.1 Basic solutions with linear actuators

371

Fig. 4.5. 3-PRRRR-type non overconstrained TPM with coupled motions and linear actuators mounted on the fixed base, limb topology P A R A R||R A ||R

Fig. 4.6. 3-RRPRR-type non overconstrained TPM with coupled motions and linear actuators non adjacent to the fixed base, limb topology R A R||PŒR A ||R

372

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.7. 3-RRPRR-type non overconstrained TPMs with coupled motions and linear actuators non adjacent to the fixed base mounted on the fixed base, limb topology R A R A P A ||R A R

4.1 Basic solutions with linear actuators

373

Fig. 4.8. 3-RPRRR-type non overconstrained TPMs with coupled motions and linear actuators non adjacent to the fixed base, limb topology R A P A A R||R A ||R (a) and R A P||R||R A ||R (b)

374

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.9. 3-RPRRR-type non overconstrained TPMs with coupled motions and linear actuators non adjacent to the fixed base, limb topology R A P A ||R A R||R (a) and R||P A R||R A ||R (b)

4.2 Derived solutions with linear actuators

375

4.2 Derived solutions with linear actuators Non overconstrained solutions F m G1–G2–G2 with linear actuators and coupled motions can also be derived from the overconstrained solutions presented in Figs. 3.6–3.23 by introducing the required idle mobilities. They have the linear actuators mounted on the fixed base (Figs. 4.10–4.22) or between two moving links (Figs. 4.23–4.28). The limb topology of these non overconstrained solutions (NF = 0) are systematized in Tables 4.2 and 4.3 and the structural parameters in Tables 4.4–4.8. For example, the non overconstrained solutions in Fig. 4.10 are derived from the overconstrained solutions in Fig. 3.6 by introducing two rotational idle mobilities outside the parallelogram loop and one translational and two rotational idle mobilities in each parallelogram loop. They are introduced by replacing two revolute joints by spherical ones in each parallelogram loop and the prismatic joints by cylindrical ones. The prismatic joints in Fig. 3.6 are also replaced by cylindrical joints in Fig. 4.10. We may note that the two spherical joints adjacent to link 5 introduce one translational and two rotational idle mobilities in each parallelogram loop and also provide an idle rotational mobility of link 5. Attention must be paid when introducing the idle mobilities so as not to modify the mobility of the parallel mechanism and the connectivity of the moving platform.

376

4 Non overconstrained T3-type TPMs with coupled motions

Table 4.2. Limb topology of the derived non overconstrained TPMs with idle mobilities and linear actuators mounted on the fixed base presented in Figs. 4.10– 4.22 No. Basic TPM type 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

3-PPaP (Fig. 3.6) 3-PPPa (Fig. 3.7) 3-PPacc (Fig. 3.8) 3-PPacc (Fig. 3.9) 3-PPaPa (Fig. 3.10a) 3-PPaPa (Fig. 3.10b) 3-PPaPa (Fig. 3.11a) 3-PPr (Fig. 3.11b) 3-PRC (Fig. 3.12a) 3-PCR (Fig. 3.12b) 3-PRC (Fig. 3.13) 3-PCR (Fig. 3.14) 3-PRPaR (Fig. 3.15a) 3-PRPaR (Fig. 3.15b) 3-PRPaR (Fig. 3.15a, b) 3-PRRPa (Fig. 3.16a) 3-PRRPa (Fig. 3.16b) 3-PPaRR (Fig. 3.17a) 3-PPaRR (Fig. 3.17b)

NF 15 15 12 12 24 24 24 12 3 3 3 3 12 12 12 12 12 12 12

Derived TPM with NF = 0 type 3-PPassC* (Fig. 4.10) 3-PC*Pass (Fig. 4.11) 3-PR*R*Pascc (Fig. 4.12) 3-PR*PaccsR* (Fig. 4.13) 3-PPassPass (Fig. 4.14a) 3-PPassPass (Fig. 4.14b) 3-PPassPass (Fig. 4.15a) 3-PPr*R*R* (Fig. 4.15b) 3-PRCR* (Fig. 4.16a) 3-PCRR* (Fig. 4.16b) 3-PRR*C (Fig. 4.17) 3-PCR*R (Fig. 4.18) 3-PR*RPass (Fig. 4.19a) 3-PR*RPass (Fig. 4.19b) 3-PPa4s (Fig. 4.20a, b 3-PRPassR* (Fig. 4.21a) 3-PRRPass (Fig. 4.21b) 3-PR*PassR (Fig. 4.22a) 3-PR*PassR (Fig. 4.22b

Limb topology P A Pass||C* P A C*||Pass P A R* A A R* A ||Pascc Pa A R* A A Paccs||R* P A Pass A A Pass P A Pass A ||Pass P||Pass A Pass PPr*R* A R* P A R||C A ||R* P A C||R A ||R* P A R A R* A ||C P A C A R* A ||R P A R* A ||R A Pass P||R* A R A Pass P-Pa4s P A R A ||Pass||R* P||R||R A Pass P A R*||Pass A R P||R*||Pass A R

4.2 Derived solutions with linear actuators

377

Table 4.3. Limb topology of the derived TPMs with idle mobilities and linear actuators mounted on a moving link presented in Figs. 4.23–4.28 No. Basic TPM type 1 2 3 4 5 6 7 8 9

3-RPC (Fig. 3.18a) 3-CPR (Fig. 3.18b) 3-RPC (Fig. 3.19) 3-CPR (Fig. 3.20) 3-PPRR (Fig. 3.21) 3-RPaPR (Fig. 3.22a) 3-RPaPR (Fig. 3.22b) 3-RPaRP (Fig. 3.23a) 3-RPaRP (Fig. 3.23b)

NF 3 3 3 3 3 12 12 12 12

Derived TPM with NF = 0 type 3-RCC (Fig. 4.23a) 3-CCR (Fig. 4.23b) 3-RCC (Fig. 4.24) 3-CCR (Fig. 4.25) 3-PPRRR* (Fig. 4.26) 3-RPa*PRR* (Fig. 4.27a) 3-RPassPR (Fig. 4.28a) 3-RPassR*P (Fig. 4.27b) 3-RPassRP (Fig. 4.28b)

Limb topology R A C A ||C C A C A ||R R A C A ||C C A C A ||R P A P A ||R||R A ||R* R A Pa* A A P A R A R* R A Pass A ||P||R 3-R A Pass A R* A P 3-R A Pass A ||R||P

Table 4.4. Bases of the operational velocities spaces of the limbs isolated from the parallel mechanisms presented in Figs. 4.10–4.28 No. Parallel mechanism 1 Figs. 4.10–4.14, 4.15a, 4.24, 4.28 2 Figs. 4.15b, 4.16–4.18, 4.19b, 4.21b, 4.22, 4.23, 4.26–4.27 3 Fig. 4.19a

Basis (RG2) (RG3) (RG1) ( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE )

( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦD , ȦG )

4

Fig. 4.20

( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG )

5

Fig. 4.21a

( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE )

( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG )

378

4 Non overconstrained T3-type TPMs with coupled motions

Table 4.5. Structural parametersa of translational parallel mechanisms in Figs. 4.10–4.15 No. Structural parameter

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

Solution 3-PPassC* (Fig. 4.10) 3-PC*Pass (Fig. 4.11)

3-PR*R*Pascc (Fig. 4.12) 3-PR*PaccsR* (Fig. 4.13)

14 6 6 6 18 5 0 3 3 See Table 4.4

17 7 7 7 21 5 0 3 3 See Table 4.4

3-PPassPass (Fig. 4.14) 3-PPassPass (Fig. 4.15a) 3-PPr*R*R* (Fig. 4.15b) 20 9 9 9 27 8 0 3 3 See Table 4.4

5 5 5 6 6 6 5 5 5 ( v1 ,v 2 ,v3 ) 3 18 30 3 0 0 11

5 5 5 6 6 6 5 5 5 ( v1 ,v 2 ,v3 ) 3 18 30 3 0 0 11

5 5 5 12 12 12 5 5 5 ( v1 ,v 2 ,v3 ) 3 36 48 3 0 0 17

fj

11

11

17

fj

11

11

17

fj

33

33

51

M P1 P2 P3 P Q K1 K2 K (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

4.2 Derived solutions with linear actuators

379

Table 4.6. Structural parametersa of translational parallel mechanisms in Figs. 4.16–4.20 No. Structural Solution parameter 3-PRCR*(Fig. 4.16a) 3-PCRR*(Fig. 4.16b) 3-PRR*C(Fig. 4.17) 3-PCR*R(Fig. 4.18) 1 m 11 2 P1 4 3 P2 4 4 4 P3 5 P 12 6 Q 2 3 7 K1 0 8 K2 9 K 3 10 (RGi) See Table 4.4 (i = 1,2,3) 11 SG1 5 5 12 SG2 13 SG3 5 0 14 rG1 0 15 rG2 0 16 rG3 5 17 MG1 5 18 MG2 5 19 MG3 20 (RF) ( v1 ,v 2 ,v3 ) 21 SF 3 22 rl 0 12 23 rF 3 24 MF 0 25 NF 0 26 TF p1 5 27 f 28 29 30 a

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

j

3-PR*RPass (Fig. 4.19)

PPa4s (Fig. 4.20)

17 7 7 7 21 5 0 3 3 See Table 4.4

11 5 5 5 15 5 0 3 3 See Table 4.4

5 5 5 6 6 6 5 5 5 ( v1 ,v 2 ,v3 ) 3 18 30 3 0 0 11

5 5 5 6 6 6 7 7 7 ( v1 ,v 2 ,v3 ) 3 18 30 9 0 6 13

fj

5

11

13

fj

5

11

13

fj

15

33

39

See footnote of Table 2.1 for the nomenclature of structural parameters

380

4 Non overconstrained T3-type TPMs with coupled motions

Table 4.7. Structural parametersa of translational parallel mechanisms in Figs. 4.21–4.26 No. Structural Solution parameter 3-PRPassR* (Fig. 4.21a) 3-PRRPass (Fig. 4.21b) 3-PR*PassR (Fig. 4.22) 1 m 17 2 p1 7 p2 7 3 4 p3 7 p 21 5 6 q 5 0 7 k1 8 k2 3 9 k 3 See Table 4.4 10 (RGi) (i = 1,2,3) 11 SG1 5 5 12 SG2 5 13 SG3 6 14 rG1 6 15 rG2 6 16 rG3 5 17 MG1 5 18 MG2 19 MG3 5 20 (RF) ( v1 ,v 2 ,v3 ) 21 SF 3 18 22 rl 30 23 rF 3 24 MF 25 NF 0 0 26 TF p1 11 27 f 28 29 30 a

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

3-RCC (Figs. 4.23a, 4.24) 3-CCR (Figs. 4.23b, 4.25)

3-PPRRR* (Fig. 4.26)

8 3 3 3 9 2 3 0 3 See Table 4.4

14 5 5 5 15 2 3 0 3 See Table 4.4

5 5 5 0 0 0 5 5 5 ( v1 ,v 2 ,v3 ) 3 0 12 3 0 0 5

5 5 5 0 0 0 5 5 5 ( v1 ,v 2 ,v3 ) 3 0 12 3 0 0 5

j

fj

11

5

5

fj

11

5

5

fj

33

15

15

See footnote of Table 2.1 for the nomenclature of structural parameters

4.2 Derived solutions with linear actuators

381

Table 4.8. Structural parametersa of translational parallel mechanisms in Figs. 4.27 and 4.28 No. Structural parameter

Solution 3-RPa*PRR* (Fig. 4.27a)

M p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

20 8 8 8 24 5 0 3 3 See Table 4.4

3-RPassR*P (Fig. 4.27b) 3-RPassPR (Fig. 4.28a) 3-RPassRP (Fig. 4.28b) 17 7 7 7 21 5 0 3 3 See Table 4.4

5 5 5 6 6 6 5 5 5 ( v1 ,v 2 ,v3 ) 3 18 30 3 0 0 11

5 5 5 6 6 6 5 5 5 ( v1 ,v 2 ,v3 ) 3 18 3 3 0 0 11

fj

11

11

fj

11

11

fj

33

33

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

SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

382

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.10. 3-PPassC*-type non overconstrained TPMs with coupled motions and linear actuators mounted on the fixed base, limb topology P A Pass||C*

4.2 Derived solutions with linear actuators

383

Fig. 4.11. 3-PC*Pass-type non overconstrained TPMs with coupled motions and linear actuators mounted on the fixed base, limb topology P A C*||Pass

384

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.12. 3-PR*R*Pascc-type non overconstrained TPMs with coupled motions and linear actuators mounted on the fixed base, limb topology P A R* A A R* A ||Pascc

4.2 Derived solutions with linear actuators

385

Fig. 4.13. 3-PR*PaccsR*-type non overconstrained TPMs with coupled motions and linear actuators mounted on the fixed base, limb topology Pa A R* A A Paccs||R*

386

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.14. 3-PPassPass-type non overconstrained TPMs with coupled motions and linear actuators mounted on the fixed base, limb topology P A Pass A A Pass (a) and P A Pass A ||Pass (b)

4.2 Derived solutions with linear actuators

387

Fig. 4.15. Non overconstrained TPMs of types 3-PPassPass (a) and 3-PPr*R*R* (b) with coupled motions and linear actuators mounted on the fixed base, limb topology P||Pass A Pass (a) and PPr*R* A R* (b)

388

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.16. Non overconstrained TPMs of types 3-PRCR* (a) and 3-PCRR*(b) with coupled motions and linear actuators mounted on the fixed base, limb topology P A R||C A ||R* (a) and P A C||R A ||R*(b)

4.2 Derived solutions with linear actuators

389

Fig. 4.17. 3-PRR*C-type non overconstrained TPMs with coupled motions and linear actuators mounted on the fixed base, limb topology P A R A R* A ||C

390

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.18. 3-PCR*R-type non overconstrained TPMs with coupled motions and linear actuators mounted on the fixed base, limb topology P A C A R* A ||R

4.2 Derived solutions with linear actuators

391

Fig. 4.19. 3-PR*RPass-type non overconstrained TPMs with coupled motions and linear actuators mounted on the fixed base, limb topology P A R* A ||R A Pass (a) and P||R* A R A Pass (b)

392

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.20. 3-PPa4s-type non overconstrained TPMs with coupled motions and linear actuators mounted on the fixed base with 6 internal mobilities of links 3A, 4A, 3B, 4B, 3C, and 4C

4.2 Derived solutions with linear actuators

393

Fig. 4.21. Non overconstrained TPMs of types 3-PRPassR* (a) and 3-PRRPass (b) with coupled motions and linear actuators mounted on the fixed base, limb topology P A R A ||Pass||R* (a) and P||R||R A Pass (b)

394

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.22. 3-PR*PassR-type non overconstrained TPMs with coupled motions and linear actuators mounted on the fixed base, limb topology P A R*||Pass A R (a) and P||R*||Pass A R (b)

4.2 Derived solutions with linear actuators

395

Fig. 4.23. Non overconstrained TPMs of types 3-RCC (a) and 3-CCR (b) with coupled motions and linear actuators combined in cylindrical joints non adjacent to the fixed base, limb topology R A C A ||C (a) and C A C A ||R (b)

396

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.24. 3-RCC-type non overconstrained TPMs with coupled motions and linear actuators combined in cylindrical joints non adjacent to the fixed base, limb topology R A C A ||C

4.2 Derived solutions with linear actuators

397

Fig. 4.25. 3-CCR-type non overconstrained TPMs with coupled motions and linear actuators combined in cylindrical joints non adjacent to the fixed base, limb topology C A C A ||R

398

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.26. 3-PPRRR*-type non overconstrained TPM with coupled motions and linear actuators non adjacent to the fixed base, limb topology P A P A ||R||R A ||R*

4.2 Derived solutions with linear actuators

399

Fig. 4.27. Non overconstrained TPMs of types 3-RPa*PRR* (a) and 3-RPassR*P (b) with coupled motions and linear actuators non adjacent to the fixed base, limb topology R A Pa* A A P A R A R* (a) and R A Pass A R* A P (b)

400

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.28. Non overconstrained TPMs of types 3-RPassPR (a) and 3-RPassRP (b) with coupled motions and linear actuators non adjacent to the fixed base, limb topology R A Pass A ||P||R (a) and R A Pass A ||R||P (b)

4.3 Basic solutions with rotating actuators

401

4.3 Basic solutions with rotating actuators In the basic non overconstrained TPMs with rotating actuators and coupled motions F m G1–G2–G3, the moving platform n Ł nGi (i = 1, 2, 3) is connected to the reference platform 1 Ł 1Gi Ł 0 by three limbs with five degrees of connectivity. No idle mobilities exist in these basic solutions. The various types of limbs with five degrees of connectivity and no idle mobilities are systematized in Fig. 4.29. They are simple kinematic chains that can be actuated by rotating motors mounted on the fixed base. The limbs presented in Fig. 4.2 can also be used when the first revolute joint is actuated instead of the prismatic joint (see Figs. 4.6–4.9). Various solutions of TPMs with coupled motions and no idle mobilities can be obtained by using three limbs with identical or different topologies presented in Figs. 4.2 and 4.29. We only show solutions with identical limb type as illustrated in Figs. 4.30–4.35. The limb topology and connecting conditions in these solutions are systematized in Table 4.9 and the structural parameters of the solutions are presented in Table 4.10. The actuated revolute joints adjacent to the fixed base in the three limbs have orthogonal directions (Figs. 4.30b, 4.31–4.35) or are parallel to one plane (Fig. 4.30a).

402

4 Non overconstrained T3-type TPMs with coupled motions

Table 4.9. Limb topology and connecting conditions of the non overconstrained TPM with no idle mobilities and rotating actuators presented in Figs. 4.30–4.35 No.

TPM type

Limb topology

Connecting conditions

1

3-RRRRR (Fig. 4.30a)

R||R A R||R A R (Fig. 4.29a)

2

3-RRRRR (Fig. 4.30b)

R||R A R||R A R (Fig. 4.29a)

3

3-RRRRR (Fig. 4.31a) 3-RRRRR (Fig. 4.31b) 3-RRRRR (Fig. 4.32a) 3-RRRRP (Fig. 4.32b) 3-RRPRR (Fig. 4.33a) 3-RRRPR (Fig. 4.33b) 3-RCRR (Fig. 4.34a) 3-RRCR (Fig. 4.34b) 3-RRRC (Fig. 4.35a) 3-RRRC (Fig. 4.35b)

R||R||R A R||R (Fig. 4.29b) R A R||R A R||R (Fig. 4.29c) R A R||R||R A R (Fig. 4.29d) R A R||R A R A P (Fig. 4.29e) R A R A ||P A R A R (Fig. 4.29f) R A R||R A R A P (Fig. 4.29g) R A C||R A R (Fig. 4.29j) R A R||C A R (Fig. 4.29m) R||R A R||C (Fig. 4.29p) R A R||R A C (Fig. 4.29s)

Actuated R joints adjacent to the fixed base and their axes are parallel to one plane Actuated R joints adjacent to the fixed base and their axes are reciprocally orthogonal Idem No. 2

4 5 6 7 8 9 10 11 12

Idem No. 2 Idem No. 2 Idem No. 2 Idem No. 2 Idem No. 2 Idem No. 2 Idem No. 2 Idem No. 2 Idem No. 2

4.3 Basic solutions with rotating actuators

403

Table 4.10. Structural parametersa of translational parallel mechanisms in Figs. 4.30–4.35 No. Structural parameter

1 2 3 4 5 6 7 8 9 10

m p1 p2 p3 p q k1 k2 k (RG1)

Solution 3-RRRRR, 3-RRRRP 3-RRRRR (Figs. 4.30, 4.31) (Fig. 4.32) 3-RRPRR, 3-RRRPR (Fig. 4.33) 14 14 5 5 5 5 5 5 15 15 2 2 3 3 0 0 3 3 ( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE )

11

(RG2)

( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦE , ȦG )

12

(RG3)

( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦD , ȦG )

( v1 , v2 , v3 , ȦD , ȦE )

13 14 15 16 17 18 19 20 21 22

SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

23 24 25 26 27 28 29

SF rl rF MF NF TF

5 5 5 0 0 0 5 5 5 ( v1 ,v 2 ,v3 ) 3 0 12 3 0 0 5

5 5 5 0 0 0 5 5 5 ( v1 ,v 2 ,v3 ) 3 0 12 3 0 0 5

5 5 5 0 0 0 5 5 5 ( v1 ,v 2 ,v3 ) 3 0 12 3 0 0 5

fj

5

5

5

fj

5

5

5

fj

15

15

15

30 31 32 a

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

3-RCRR, 3-RRCR (Fig. 4.34) 3-RRRC (Fig. 4.35) 11 4 4 4 12 2 3 0 3 ( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG )

See footnote of Table 2.1 for the nomenclature of structural parameters

404

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.29. Simple limbs for non overconstrained TPMs with coupled motions defined by MG = SG = 5, (RG) = ( v1 , v2 , v3 , Ȧ1 , Ȧ2 ) and actuated by rotating motors mounted on the fixed base

4.3 Basic solutions with rotating actuators

Fig. 4.29. (cont.)

405

406

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.30. 3-RRRRR-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology R||R A R||R A ||R

4.3 Basic solutions with rotating actuators

407

Fig. 4.31. 3-RRRRR-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology R||R||R A R||R (a) and R A R||R A R||R (b)

408

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.32. Non overconstrained TPMs of types 3-RRRRR (a) and 3-RRRRP (b) with coupled motions and rotating actuators mounted on the fixed base, limb topology R A R||R||R A R (a) and R A R||R A R A P (b)

4.3 Basic solutions with rotating actuators

409

Fig. 4.33. Non overconstrained TPMs of types 3-RRPRR (a) and 3-RRRPR (b) with coupled motions and rotating actuators mounted on the fixed base, limb topology R A R A P A R A R (a) and R A R||R A R A P (b)

410

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.34. Non overconstrained TPMs of types 3-RCRR (a) and 3-RRCR (b) with coupled motions and rotating actuators mounted on the fixed base: R A C||R A ||R (a) and R A R||C A ||R (b)

4.3 Basic solutions with rotating actuators

411

Fig. 4.35. 3-RRRC-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base limb topology R||R A R||C (a) and R A R||R A ||C (b)

412

4 Non overconstrained T3-type TPMs with coupled motions

4.4 Derived solutions with rotating actuators Non overconstrained solutions F m G1G2–G2 with rotating actuators and coupled motions can be derived from the overconstrained solutions presented in Figs. 3.71–3.118 by introducing the required idle mobilities. They have the rotating actuators mounted on the fixed base (Figs. 4.36– 4.82). For example, the non overconstrained solutions in Fig. 4.36 are derived from the overconstrained solutions in Fig. 3.71 by introducing two rotational idle mobilities outside the parallelogram loop and one translational and two rotational idle mobilities in each parallelogram loop. They are introduced by replacing two revolute joints by one spherical and one cylindrical joint in the first parallelogram loop and by two spherical joints in the second parallelogram loop of each limb. The prismatic joints in Fig. 3.71 are also replaced by cylindrical ones in Fig. 4.36. We note that the two spherical joints adjacent to link 7 introduce one translational and two rotational idle mobilities in each parallelogram loop and also provide an idle rotational mobility of link 5. An idle mobility of rotation is combined in each cylindrical joint denoted by C*. The limb topology and connecting conditions of the solutions in Figs. 4.36–4.82 are systematized in Table 4.11 and the structural parameters of the solutions are presented in Tables 4.12–4.18.

4.4 Derived solutions with rotating actuators

413

Table 4.11. Limb topology of the non overconstrained solutions (NF = 0) of the derived TPMs with idle mobilities and rotating actuators mounted on the fixed base presented in Figs. 4.36–4.82 No. Basic TPM type 1 3-PaPaP (Fig. 3.71) 2 3-PaPaP (Fig. 3.72) 3 3-PaPPa (Fig. 3.73) 4 3-PaPPa (Fig. 3.74) 5 3-PaPacc (Fig. 3.75) 6 3-PaPacc (Fig. 3.76 7 3-PaPacc (Fig. 3.77 8 3-PaPacc (Fig. 3.78) 9 3-PaccPa (Fig. 3.79) 10 3-PaPaPa (Fig. 3.80) 11 3-PaPaPa (Fig. 3.81) 12 3-PaPaPa (Fig. 3.82 13 3-RRC (Fig. 3.83) 14 3-RRC (Fig. 3.84) 15 3-RCR (Fig. 3.85a) 16 3-RPPR (Fig. 3.85b) 17 3-RPC (Fig. 3.86a) 18 3-RPPR (Fig. 3.86b) 19 3-PaRC (Fig. 3.87) 20 3-PaCR (Fig. 3.88)

NF 24 24 24 24 21 21 21 21 21 33 33 33 3 3 3 3 3 3 12 12

Derived TPM with NF = 0 type 3-PacsPassC* (Fig. 4.36) 3-PacsPassC* (Fig. 4.37) 3-PacsC*Pass (Fig. 4.38) 3-PacsC*Pass (Fig. 4.39) 3-PacsR*R*Pascc (Fig. 4.40) 3-PacsR*PaccsR* (Fig. 4.41 3-PacsR*R*Pascc (Fig. 4.42 3-PacsR*PaccsR* (Fig. 4.43) 3-PasccPassR* (Fig. 4.44) 3-PassPacsPass (Fig. 4.45) 3-PacsPassPass (Fig. 4.46) 3-PassPacsPass (Fig. 4.47 3-RRR*C (Fig. 4.48) 3-RR*RC (Fig. 4.49 3-RR*CR (Fig. 4.50a) 3-RPR*RR (Fig. 4.50b) 3-RC*C (Fig. 4.51a) 3-RPC*R (Fig. 4.51b) 3-PacsR*RC (Fig. 4.52) 3-PacsR*CR (Fig. 4.53)

Limb topology Pacs A Pass||C* Pacs||Pass||C* Pacs A C*||Pass Pacs||C*||Pass Pacs||R* A R*||Pascc Pacs A R*||Paccs A ||R* Pacs A R* A ||R*||Pascc Pacs||R*||Paccs A R* Pascc||Pass||R* Pass A Pacs A ||Pass Pacs A Pass A A Pass Pass||Pacs A Pass R||R A R* A ||C R A R* A ||R||C R A R* A ||C||R R||P A R* A ||R||R R A C* A ||C R||P A C* A ||R Pacs A R* A ||R||C Pacs A R* A ||C||R

414

4 Non overconstrained T3-type TPMs with coupled motions Table 4.11. (cont.)

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

3-PaPaRR (Fig. 3.89) 3-PaPaRR (Fig. 3.90) 3-PaPaRR (Fig. 3.91) 3-PaRRPa (Fig. 3.92) 3-PaRRPa (Fig. 3.93) 3-PaRRPa (Fig. 3.94) 3-PaRPaR (Fig. 3.95) 3-PaRPaR (Fig. 3.96) 3-RRPaP (Fig. 3.97) 3-RPRPa (Fig. 3.98) 3-RCPa (Fig. 3.99) 3-RPaRR (Fig. 3.100a) 3-RPaRR (Fig. 3.100b) 3-RRPaR (Fig. 3.101) 3-RPaPaR (Fig. 3.102) 3-RPaRPa (Fig. 3.103) 3-RPaRPa (Fig. 3.104) 3-RPaPaR (Fig. 3.105) 3-RRPacc (Fig. 3.106) 3-RRPacc (Fig. 3.107) 3-PaccRR (Fig. 3.108) 3-PaPr (Fig. 3.109a)

21 21 21 21 21 21 21 21 12 12 12 12 12 12 21 21 21 21 9 9 9 21

3-PassPassR (Fig. 4.54) 3-PassPassR (Fig. 4.55) 3-PassPassR (Fig. 4.56) 3-PassRPass (Fig. 4.57) 3-PassRPass (Fig. 4.58) 3-PassRPass (Fig. 4.59) 3-PassRPass (Fig. 4.60a) 3-PassRPass (Fig. 4.60b) 3-RRPassP (Fig. 4.61) 3-RPRPass (Fig. 4.62) 3-RCPas (Fig. 4.63) 3-RPassRR* (Fig. 4.64a) 3-RPassR*R (Fig. 4.64b) 3-RPa4s (Fig. 4.65) 3-RPassPass (Fig. 4.66) 3-RPassPass (Fig. 4.67) 3-RPassPass (Fig. 4.68) 3-RPa*PassR* (Fig. 4.69) 3-RRPaccsR* (Fig. 4.70) 3-RR*RPascc (Fig. 4.71) 3-PasccRR*R (Fig. 4.72) 3-PassPrssR* (Fig. 4.73a)

Pass A Pass A A R Pass A Pass A ||R Pass A Pass A A R Pass||R A Pass Pass A R A A Pass Pass A R A ||Pass Pass A R A Pass Pass A ||R A Pass R||R||Pass||P R||P||R||Pass R||C||Pass R A Pass A ||R A R R A Pass-R* A R R-Pa4s R A Pass||Pass R A Pass||Pass R A Pass||Pass R A Pa*||Pass A ||R* R||R||Paccs A R* R A R* A ||R||Pascc Pascc||R A R* A ||R Pass-Prss-R*

4.4 Derived solutions with rotating actuators Table 4.11. (cont.) 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61

3-RRPr (Fig. 3.109b) 3-PaRRRR (Fig. 3.110a) 3-PaRRRR (Fig. 3.110b) 3-PaRRRR (Fig. 3.111a) 3-PaRRRR (Fig. 3.111b) 3-PaRRRR (Fig. 3.112a) 3-RRPaRR (Fig. 3.112b) 3-RPaRRR (Fig. 3.113a) 3-RPaRRR (Fig. 3.113b) 3-RPaRRR (Fig. 3.114a) 3-RPaRRR (Fig. 3.114b) 3-RRPaRR (Fig. 3.115a) 3-RRPaRR (Fig. 3.115b) 3-RRRRPa (Fig. 3.116a) 3-RRRRPa (Fig. 3.116b) 3-RRRPaR (Fig. 3.117a) 3-RRRRPa (Fig. 3.117b) 3-RRRPaR (Fig. 3.118a) 3-RRRPaR (Fig. 3.118b)

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

3-RRPrssR* (Fig. 4.73b) 3-PacsRRRR (Fig. 4.74a) 3-PacsRRRR (Fig. 4.74b) 3-PassRRR (Fig. 4.75a) 3-PassRRR (Fig. 4.75b) 3-PacsRRRR (Fig. 4.76a) 3-RPassRR (Fig. 4.76b) 3-RPacsRRR (Fig. 4.77a) 3-RPassRR (Fig. 4.77b) 3-RPacsRRR (Fig. 4.78a) 3-RPacsRRR (Fig. 4.78b) 3-RRPassR (Fig. 4.79a) 3-RRPassR (Fig. 4.79b) 3-RRRPass (Fig. 4.80a) 3-RRRRPacs (Fig. 4.80b) 3-RRRPass (Fig. 4.81a) 3-RRRRPacs (Fig. 4.81b) 3-RRRPacsR (Fig. 4.82a) 3-RRRPass (Fig. 4.82b)

R||R-Prss-R* Pacs||R||R A R||R Pacs A R A R||R A R Pass A R A R||R Pass||R||R A R Pa||R A R||R A R R||Pass A R A ||R R A Pacs||R||R A ||R R A Pass A A R A A R R A Pacs A ||R A R||R R A Pacs||R||R A ||R R||R||Pass A R R||R A Pass A A R R||R A R A A Pass R||R A R||R A A Pacs R||R A R A Pass R A R||R A ||R A Pacs R A R||R||Pacs A ||R R A R||R||Pass

415

416

4 Non overconstrained T3-type TPMs with coupled motions

Table 4.12. Bases of the operational velocities spaces of the limbs isolated from the parallel mechanisms presented in Figs. 4.36–4.82 No. Parallel Basis mechanism (RG1) 1 Figs. 4.36–4.44, ( v1 , v2 , v3 , ȦE , ȦG ) 4.46, 4.48, 4.49a, 4.52, 4.53, 4.61–4.63, 4.64b, 4.65b, 4.66–4.72, 4.74b, 4.76a, 4.77a, 4.78a, 4.79, 4.80a, 4.81a 2 Figs. 4.45, 4.55, ( v1 , v2 , v3 , ȦD , ȦE ) 4.56, 4.60, 4.64a, 4.65a, 4.73, 4.74a, 4.75, 4.82b 3 Figs. 4.47, 4.54, ( v1 , v2 , v3 , ȦD , ȦG ) 4.57–4.59, 4.76b, 4.77b, 4.78b, 4.80b, 4.81b, 4.82a 4 Figs. 4.49b, ( v1 , v2 , v3 , ȦE , ȦG ) 4.50, 4.51

(RG2) (RG3) ( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE )

( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG )

( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG )

( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦD , ȦG )

4.4 Derived solutions with rotating actuators

417

Table 4.13. Structural parametersa of translational parallel mechanisms in Figs. 4.36–4.47 No. Structural Solution parameter 3-PacsPassC* (Figs. 4.36, 4.37) 3-PacsC*Pass (Figs. 4.38, 4.39) 3-PasccPassR* (Fig. 4.44) 1 m 20 2 p1 9 p2 9 3 4 p3 9 p 27 5 6 q 8 0 7 k1 8 k2 3 9 k 3 See Table 4.12 10 (RGi) (i = 1,2,3) 11 SG1 5 5 12 SG2 5 13 SG3 12 14 rG1 12 15 rG2 12 16 rG3 5 17 MG1 5 18 MG2 19 MG3 5 20 (RF) ( v1 ,v 2 ,v3 ) 21 SF 3 36 22 rl 48 23 rF 3 24 MF 25 NF 0 0 26 TF p1 17 27 f 28 29 30 a

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

3-PacsR*R*Pascc (Figs. 4.40, 4.42) 3-PacsR*PaccsR* (Figs. 4.41, 4.43 )

3-PassPacsPass (Figs. 4.45, 4.47) 3-PacsPassPass (Fig. 4.46)

23 10 10 10 30 8 0 3 3 See Table 4.12

26 12 12 12 36 11 0 3 3 See Table 4.12

5 5 5 12 12 12 5 5 5 ( v1 ,v 2 ,v3 ) 3 36 48 3 0 0 17

5 5 5 18 18 18 5 5 5 ( v1 ,v 2 ,v3 ) 3 54 66 3 0 0 23

j

fj

17

17

23

fj

17

17

23

fj

51

51

69

See footnote of Table 2.1 for the nomenclature of structural parameters

418

4 Non overconstrained T3-type TPMs with coupled motions

Table 4.14. Structural parametersa of translational parallel mechanisms in Figs. 4.48–4.51 No. Structural Solution parameter 3-RRR*C, 3-RR*RC (Figs. 4.48, 4.49) 3-RR*CR, 3-RPC*R (Figs. 4.50a, 4.51b) 1 m 11 2 p1 4 3 p2 4 p3 4 4 p 12 5 6 q 2 7 k1 3 0 8 k2 9 k 3 10 (RGi) See Table 4.12 (i = 1,2,3) 11 SG1 5 5 12 SG2 13 SG3 5 0 14 rG1 0 15 rG2 0 16 rG3 5 17 MG1 5 18 MG2 5 19 MG3 20 (RF) ( v1 ,v 2 ,v3 ) 21 SF 3 22 rl 0 12 23 rF 3 24 MF 0 25 NF 0 26 TF p1 5 27 f 28 29 30 a

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

j

3-RC*C (Fig. 4.51a)

3-RPR*RR (Fig. 4.50b)

8 3 3 3 9 2 3 0 3 See Table 4.12

14 5 5 5 15 2 3 0 3 See Table 4.12

5 5 5 0 0 0 5 5 5 ( v1 ,v 2 ,v3 ) 3 0 12 3 0 0 5

5 5 5 0 0 0 5 5 5 ( v1 ,v 2 ,v3 ) 3 0 12 3 0 0 5

fj

5

5

5

fj

5

5

5

fj

15

15

15

See footnote of Table 2.1 for the nomenclature of structural parameters

4.4 Derived solutions with rotating actuators

419

Table 4.15. Structural parametersa of translational parallel mechanisms in Figs. 4.52–4.64 No. Structural Solution parameter 3-PacsR*RC (Fig. 4.52) 3-PacsR*CR (Fig. 4.53) 3-RRPassP (Fig. 4.61) 3-RPRPass (Fig. 4.62) 3-RPassRR* (Fig. 4.64a) 3-RPassR*R (Fig. 4.64b) 1 m 17 2 p1 7 p2 7 3 4 p3 7 p 21 5 6 q 5 0 7 k1 8 k2 3 9 k 3 See Table 4.12 10 (RGi) (i = 1,2,3) 11 SG1 5 5 12 SG2 5 13 SG3 6 14 rG1 6 15 rG2 6 16 rG3 5 17 MG1 5 18 MG2 19 MG3 5 20 (RF) ( v1 ,v 2 ,v3 ) 21 SF 3 18 22 rl 30 23 rF 3 24 MF 25 NF 0 0 26 TF p1 11 27 f 28 29 30 a

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

3-PassPassR (Figs. 4.54–4.56) 3-PassRPass (Figs. 4.57–4.60)

3-RCPass (Fig. 4.63)

20 9 9 9 27 8 0 3 3 See Table 4.12

14 6 6 6 18 5 0 3 3 See Table 4.12

5 5 5 12 12 12 5 5 5 ( v1 ,v 2 ,v3 ) 3 26 48 3 0 0 17

5 5 5 6 6 6 5 5 5 ( v1 ,v 2 ,v3 ) 3 18 30 3 0 0 11 11

j

fj

11

17

fj

11

17

fj

33

51

See footnote of Table 2.1 for the nomenclature of structural parameters

420

4 Non overconstrained T3-type TPMs with coupled motions

Table 4.16. Structural parametersa of translational parallel mechanisms in Figs. 4.65–4.69 No. Structural Solution parameter 3-RPa4s (Fig. 4.65) 1 m 11 2 p1 5 p2 5 3 4 p3 5 5 p 15 6 q 5 0 7 k1 8 k2 3 9 k 3 See Table 4.12 10 (RGi) (i = 1,2,3) 11 SG1 5 5 12 SG2 5 13 SG3 6 14 rG1 15 rG2 6 6 16 rG3 7 17 MG1 7 18 MG2 7 19 MG3 20 (RF) ( v1 ,v 2 ,v3 ) 21 SF 3 18 22 rl 23 rF 30 24 MF 9 0 25 NF 6 26 TF p1 13 27 f 28 29 30 a

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

3-RPassPass (Figs. 4.66–4.68) 20 9 9 9 27 8 0 3 3 See Table 4.12

3-RPa*PassR (Fig. 4.69) 23 10 10 10 30 8 0 3 3 See Table 4.12

5 5 5 12 12 12 5 5 5 ( v1 ,v 2 ,v3 ) 3 36 48 3 0 0 17

5 5 5 12 12 12 5 5 5 ( v1 ,v 2 ,v3 ) 3 36 48 3 0 0 17

j

fj

13

17

17

fj

13

17

17

fj

39

51

51

See footnote of Table 2.1 for the nomenclature of structural parameters

4.4 Derived solutions with rotating actuators

421

Table 4.17. Structural parametersa of translational parallel mechanisms in Figs. 4.70–4.73 No. Structural Solution parameter 3-RRPaccsR* (Fig. 4.70) 3-RR*RPascc (Fig. 4.71) 3-PasccRR*R (Fig. 4.72) 1 m 17 2 p1 7 p2 7 3 4 p3 7 p 21 5 6 q 5 0 7 k1 8 k2 3 9 k 3 See Table 4.12 10 (RGi) (i = 1,2,3) 11 SG1 5 5 12 SG2 5 13 SG3 6 14 rG1 6 15 rG2 6 16 rG3 5 17 MG1 5 18 MG2 19 MG3 5 20 (RF) ( v1 ,v 2 ,v3 ) 21 SF 3 18 22 rl 30 23 rF 3 24 MF 25 NF 0 0 26 TF p1 11 27 f 28 29 30 a

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

3-PassPrssR* (Fig. 4.73a)

3-RRPrssR* (Fig. 4.73b)

23 11 11 11 33 11 0 3 3 See Table 4.12

20 9 9 9 27 8 0 3 3 See Table 4.12

5 5 5 18 18 18 5 5 5 ( v1 ,v 2 ,v3 ) 3 54 66 3 0 0 23

5 5 5 12 12 12 5 5 5 ( v1 ,v 2 ,v3 ) 3 36 48 3 0 0 17

j

fj

11

23

17

fj

11

23

17

fj

33

69

51

See footnote of Table 2.1 for the nomenclature of structural parameters

4 Non overconstrained T3-type TPMs with coupled motions

422

Table 4.18. Structural parametersa of translational parallel mechanisms in Figs. 4.74–4.82 No. Structural Solution parameter 3-PacsRRRR (Figs. 4.74, 4.76a) 3-RPacsRRR (Figs. 4.77a, 4.78) 3-RRRRPacs (Figs. 4.80b, 4.81b) 3-RRRPacsR (Fig. 4.82a)

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

20 8 8 8 24 5 0 3 3 See Table 4.12

3-PassRRR (Fig. 4.75) 3-RPassRR (Figs. 4.76b, 4.77b) 3-RRPassR (Fig. 4.79) 3-RRRPass (Figs. 4.80a, 4.81a, 4.82b) 17 7 7 7 21 5 0 3 3 See Table 4.12

5 5 5 6 6 6 5 5 5 ( v1 ,v 2 ,v3 ) 3 18 30 3 0 0 11

5 5 5 6 6 6 5 5 5 ( v1 ,v 2 ,v3 ) 3 18 30 3 0 0 11

fj

11

11

fj

11

11

fj

33

33

M p1 p2 p3 P Q k1 k2 K (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

4.4 Derived solutions with rotating actuators

423

Fig. 4.36. 3-PacsPassC*-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology Pacs A Pass||C*

424

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.37. 3-PacsPassC*-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology Pacs||Pass||C*

4.4 Derived solutions with rotating actuators

425

Fig. 4.38. 3-PacsC*Pass-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology Pacs A C*||Pass

426

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.39. 3-PacsC*Pass-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology Pacs||C*||Pass

4.4 Derived solutions with rotating actuators

427

Fig. 4.40. 3-PacsR*R*Pascc-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology Pacs||R* A R*||Pascc

428

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.41. 3-PacsR*PaccsR*-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology Pacs A R*||Paccs A ||R*

4.4 Derived solutions with rotating actuators

429

Fig. 4.42. 3-PacsR*R*Pascc-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology Pacs A R* A ||R*||Pascc

430

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.43. 3-PacsR*PaccsR*-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology Pacs||R*||Paccs A R*

4.4 Derived solutions with rotating actuators

431

Fig. 4.44. 3-PasccPassR*-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology Pascc||Pass||R*

432

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.45. 3-PassPacsPass-type non overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, limb topology Pass A Pacs A ||Pass

4.4 Derived solutions with rotating actuators

433

Fig. 4.46. 3-PacsPassPass-type non overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, limb topology Pacs A Pass A A Pass

434

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.47. 3-PassPacsPass-type non overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, limb topology Pass||Pacs A Pass

4.4 Derived solutions with rotating actuators

435

Fig. 4.48. 3-RRR*C-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology R||R A R* A ||C

436

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.49. 3-RR*RC-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology R A R* A ||R||C

4.4 Derived solutions with rotating actuators

437

Fig. 4.50. Non overconstrained TPMs of types 3-RR*CR (a) and 3-RPR*RR (b) with coupled motions and rotating actuators mounted on the fixed base, limb toplogy R A R* A ||C||R (a) and RŒP A R* A ||R||R (b)

438

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.51. Non overconstrained TPMs with coupled motions of types 3-RC*C (a) and 3-RPC*R (b) and rotating actuators mounted on the fixed base, limb topology R A C* A ||C (a) and RŒP A C* A ||R (b)

4.4 Derived solutions with rotating actuators

439

Fig. 4.52. 3-PacsR*RC-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology Pacs A R* A ||R||C

440

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.53. 3-PacsR*CR-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology Pacs A R* A ||C||R

4.4 Derived solutions with rotating actuators

441

Fig. 4.54. 3-PassPassR-type non overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, limb topology Pass A Pass A A R

442

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.55. 3-PassPassR-type non overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, limb topology Pass A Pass A ||R

4.4 Derived solutions with rotating actuators

443

Fig. 4.56. 3-PassPassR-type non overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, limb topology Pass A Pass A A R

444

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.57. 3-PassRPass-type non overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, limb topology Pass||R A Pass

4.4 Derived solutions with rotating actuators

445

Fig. 4.58. 3-PassRPass-type non overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, limb topology Pass A R A A Pass

446

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.59. 3-PassRPass-type non overconstrained TPM with coupled motions and rotating actuators mounted on the fixed base, limb topology Pass A R A ||Pass

4.4 Derived solutions with rotating actuators

447

Fig. 4.60. 3-PassRPass-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology Pass A R A Pass (a) and Pass A ||R A Pass (b)

448

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.61. 3-RRPassP-type non overconstrained TPMs with co-upled motions and rotating actuators mounted on the fixed base, limb topology R||R||Pass||P

4.4 Derived solutions with rotating actuators

449

Fig. 4.62. 3-RPRPass-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology R||P||R||Pass

450

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.63. 3-RCPass-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology R||C||Pass

4.4 Derived solutions with rotating actuators

451

Fig. 4.64. Non overconstrained TPMs of types 3-RPassRR* (a) and 3-RPassR*R (b) with coupled motions and rotating actuators mounted on the fixed base, limb topology R A Pass A ||R A R (a) and R A Pass-R* A R (b)

452

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.65. 3-RPa4s-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base and six internal rotational mobilities of links 2A, 3A, 2B, 3B, 2C, 3C

4.4 Derived solutions with rotating actuators

453

Fig. 4.66. 3-RPassPass-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology R A Pass||Pass

454

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.67. 3-RPassPass-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology R A Pass||Pass

4.4 Derived solutions with rotating actuators

455

Fig. 4.68. 3-RPassPass-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology R A Pass||Pass

456

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.69. 3-RPa*PassR-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology R A Pa*||Pass A ||R

4.4 Derived solutions with rotating actuators

457

Fig. 4.70. 3-RRPaccsR*-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology R||R||Paccs A R*

458

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.71. 3-RR*RPascc-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology R A R* A ||R||Pascc

4.4 Derived solutions with rotating actuators

459

Fig. 4.72. 3-PasccRR*R-type non overconstrained TPMs with coupled motions and rotating actuators mounted on the fixed base, limb topology Pascc||R A R* A ||R

460

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.73. Non overconstrained TPMs of types 3-PassPrssR* (a) and 3-RRPrssR* (b) with coupled motions and rotating actuators mounted on the fixed base

4.4 Derived solutions with rotating actuators

461

Fig. 4.74. 3-PacsRRRR -type non overconstrained TPMs with coupled motions and rotating actuators on the fixed base, limb topology Pacs||R||R A R||R (a) and Pacs A R A R||R A R (b)

462

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.75. 3-PassRRR -type non overconstrained TPMs with coupled motions and rotating actuators on the fixed base, limb topology Pass A R A R||R (a) and Pass||R||R A R (b)

4.4 Derived solutions with rotating actuators

463

Fig. 4.76. Non overconstrained TPMs of types 3-PacsRRRR (a) and 3-RPassRR (b) with coupled motions and rotating actuators on the fixed base, limb topology Pacs||R A R||R A R (a) and R||Pass A R A ||R (b)

464

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.77. Non overconstrained TPMs of types 3-RPacsRRR (a) and 3-RPassRR (b) non overconstrained TPMs with coupled motions and rotating actuators on the fixed base, limb topology R A Pacs||R||R A ||R (a) and R A Pass A A R A A R (b)

4.4 Derived solutions with rotating actuators

465

Fig. 4.78. 3-RPacsRRR -type non overconstrained TPMs with coupled motions and rotating actuators on the fixed base, limb topology R A Pacs A ||R A R||R (a) and R A Pacs||R||R A ||R (b)

466

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.79. 3-RRPassR-type overconstrained TPMs with coupled motions and rotating actuators on the fixed base, limb topology R||R||Pass A R (a) and R||R A Pass A A R (b)

4.4 Derived solutions with rotating actuators

467

Fig. 4.80. Non overconstrained TPMs of types 3-RRRPass (a) and 3-RRRRPacs (b) with coupled motions and rotating actuators on the fixed base, limb topology R||R A R A A Pass (a) and R||R A R||R A A Pacs (b)

468

4 Non overconstrained T3-type TPMs with coupled motions

Fig. 4.81. Non overconstrained TPMs of types 3-RRRPass (a) and 3-RRRRPacs (b) with coupled motions and rotating actuators on the fixed base, limb topology R||R A R A Pass (a) and R A R||R A ||R A Pacs (b)

4.4 Derived solutions with rotating actuators

469

Fig. 4.82. Non overconstrained TPMs of types 3-RRRPacsR (a) and 3-RRRPass (b) with coupled motions and rotating actuators on the fixed base, limb topology R A R||R||Pacs A ||R (a) and R A R||R||Pass (b)

5 Overconstrained T3-type TPMs with uncoupled motions

T3-type translational parallel robots with uncoupled motions with various degrees of overconstraint may be obtained by using three simple or complex limbs. In these solutions, each operational velocity given by Eq. (1.19) depends, in the general case, on just one actuated joint velocity: vi vi ( &qi ) , i = 1,2,3. The Jacobian matrix in Eq. (1.19) is a diagonal matrix. They can be actuated by linear or rotating actuators which can be mounted on the fixed base or on a moving link. In the solutions presented in this section, the actuators are associated with a revolute joint mounted on the fixed base. Equation (1.16) indicates that overconstrained solutions of T3 translational parallel robots with coupled motions and q independent loops meet the p condition ¦ 1 f i 3 6q . Various solutions fulfil this condition along with SF = 3, (RF) = (v1,v2,v3) and NF 1. They may have identical limbs or limbs with different structures. We limit our presentation in this section to the solutions with just three identical limbs. A large set of solutions with an additional unactuated limb can also be obtained by combining an unactuated limb presented in Figs. 7.1–7.11 – Part 1 with three other limbs with 4 d MGi = SGi d 6 that integrate velocities v1, v2 and v3 in the basis of their operational space and fulfil the uncoupled motions condition.

5.1 Basic solutions with rotating actuators In the basic solutions of overconstrained TPMs with rotating actuators and uncoupled motions F m G1–G2–G3, the moving platform n Ł nGi (i = 1, 2, 3) is connected to the reference platform 1 Ł 1Gi Ł 0 by three limbs with three, four or five degrees of connectivity. No idle mobilities exist in these basic solutions. 471 G. Gogu, Structural Synthesis of Parallel Robots: Part 2: Translational Topologies with Two and Three Degrees of Freedom, Solid Mechanics and Its Applications 159, 471–614. © Springer Science + Business Media B.V. 2009

472

5 Overconstrained T3-type TPMs with uncoupled motions

The various types of limbs with three degrees of connectivity are systematized in Fig. 5.1. They combine one (Fig. 5.1a, b), two (Fig. 5.1c– f) or three (Fig. 5.1g) Pa-type parallelogram loops. Other parallelogram loops of types Pat (Fig. 5.1h), Patcc (Fig. 5.1i) or Pacc (Fig. 5.1j, k) may also be combined in these limbs. We recall that the parallelogram loop Pacc-type has two degrees of mobility, Pat-type has three degrees of mobility and Patcc-type has four degrees of mobility. The various types of limbs with four degrees of connectivity are systematized in Figs. 5.2–5.4. They are simple (Fig. 5.2) or complex (Figs. 5.3 and 5.4) kinematic chains. The following types of closed loops are integrated in the complex limbs: one (Figs. 5.3a–f and 5.4a–d) or two (Figs. 5.3g, h and 5.4e, f) Pa-type parallelogram loops, one (Fig. 5.3i) or two (Fig. 5.3j) Rb-type rhombus loops, one Pacc-type parallelogram loop (Fig. 5.3o) or various other types of Pn2 or Pn3 planar loops with two (Fig. 5.3k, l) or three degrees of mobility (Fig. 5.3m, n). The planar loops with two degrees of freedom Pn2-type illustrated in Fig. 5.3k, l are of types R||R||R||R||R and R A P A ||R A P A ||R. The planar loops with three degrees of freedom Pn3-type illustrated in Fig. 5.3m, n are of types R||R||R||R||R||R and R A P A ||R||R A P A ||R. Other planar loops of types Pn2 and Pn3 can also be used (see Table 5.1). The various types of limbs with five degrees of connectivity are systematized in Fig. 5.5. They are complex limbs which combine one (Fig. 5.5a) or two (Fig. 5.5b) Rb-type rhombus loops, one Pn2 (Fig. 5.5c, d) or Pn3-type (Fig. 5.5e, f) planar loop or one Pa-type parallelogram loop (Fig. 5.5g–i). Any planar loops of types Pn2 and Pn3 presented in Table 5.1 can be used. Various solutions of translational parallel robots with uncoupled motions and no idle mobilities can be obtained by using three limbs with identical or different topologies presented in Figs. 5.1–5.5. We only show solutions with identical limb type as illustrated in Figs. 5.6–5.48. The limb topology and connecting conditions in these solutions are systematized in Tables 5.2–5.5 and their structural parameters in Tables 5.6–5.12. The directions of the three actuated revolute joints adjacent to the fixed base can be reciprocally orthogonal or parallel to two coplanar and perpendicular directions. Basic solutions of T3-type TPMs with decoupled motions and different limb topologies can be obtained by using two limbs from Figs. 5.1–5.5 and one limb from Figs. 3.1–3.5, Figs. 3.60–3.70 and Figs. 4.1, 4.2 or 4.29. Basic solutions of T3-type TPMs with coupled motions and different limb topologies can be obtained by using one limb from Figs. 5.1–5.5 and two limbs from Figs. 3.1–3.5, Figs. 3.60–3.70 and Figs. 4.1, 4.2 or 4.29.

5.1 Basic solutions with rotating actuators

473

Table 5.1. Topology of the planar loops with two and three degrees of mobility that can be used in the limbs of TPM with uncoupled motions No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Pn2-type loops with two degrees of freedom R||R||R||R||R P A R||R||R||R R A P A ||R||R||R R||R A P A ||R||R P A P A ŏR||R||R R A P A ŏP A ŏR||R P A R A ŏP A R||R R A P A ||R A P A ||R P A R||R A P A ||R P A R||R||R A P P A P A ŏR||R A P

Pn3-type loops with three degrees of freedom R||R||R||R||R||R P A R||R||R||R||R R A P A ||R||R||R||R R||R A P A ||R||R||R P A P A ŏR||R||R||R R A P A ŏP A ŏR||R||R P A R A ŏP A ||R||R||R R A P A ||R A P A ||R||R P A R||R A P A ||R||R P A R||R||R A P A ||R R||R A P A ŏP A ŏR||R P A R||R||R||R A P R A P A ||R||R A P A ||R P A P A ŏR||R||R A P

474

5 Overconstrained T3-type TPMs with uncoupled motions

Table 5.2. Limb topology and connecting conditions of the overconstrained TPM with uncoupled motions and no idle mobilities presented in Figs. 5.6–5.15 No.

TPM type

Limb topology

Connecting conditions

1

3-PaPP (Fig. 5.6a)

Pa||P A A P (Fig. 5.1a)

2

3-PaPP (Fig. 5.6b)

Pa||P A A P (Fig. 5.1a)

3

3-PaPP (Fig. 5.7a) 3-PaPP (Fig. 5.7b) 3-PaPaP (Fig. 5.8a) 3-PaPaP (Fig. 5.8b) 3-PaPaP (Fig. 5.9a) 3-PaPaP (Fig. 5.9b) 3-PaPPa (Fig. 5.10a) 3-PaPPa (Fig. 5.10b) 3-PaPPa (Fig. 5.11a) 3-PaPPa (Fig. 5.11b) 3-PaPaPa (Fig. 5.12) 3-PaPaPa (Fig. 5.13) 3-PaPatP (Fig. 5.14a) 3-PaPatP (Fig. 5.14b) 3-PaPatcc (Fig. 5.15a) 3-PaPatcc (Fig. 5.15b)

Pa A P A ||P (Fig. 5.1b) Pa A P A ||P (Fig. 5.1b) Pa A Pa A ||P (Fig. 5.1c) Pa A Pa A ||P (Fig. 5.1c) Pa A Pa A A P (Fig. 5.1d) Pa A Pa A A P (Fig. 5.1d) Pa||P A Pa (Fig. 5.1e) Pa||P A Pa (Fig. 5.1e) Pa A P A A Pa (Fig. 5.1f) Pa A P A A Pa (Fig. 5.1f) Pa A Pa||Pa (Fig. 5.1g) Pa A Pa||Pa (Fig. 5.1g) Pa||Pat||P (Fig. 5.1h) Pa||Pat||P (Fig. 5.1h) Pa||Patcc (Fig. 5.1i) Pa||Patcc (Fig. 5.1i)

The rotation axes of the actuated revolute joints are reciprocally orthogonal The rotation axes of the actuated revolute joints are parallel to two orthogonal directions Idem No. 1

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Idem No. 2 Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 2

5.1 Basic solutions with rotating actuators

475

Table 5.3. Limb topology and connecting conditions of the overconstrained TPM with uncoupled motions and no idle mobilities presented in Figs. 5.16–5.24 No.

TPM type 3-PaccP (Fig. 5.16a)

Limb topology Pacc A P (Fig. 5.1j)

2

3-PaccP (Fig. 5.16b)

Pacc A P (Fig. 5.1j)

3

3-PaccPa (Fig. 5.17a) 3-PaccPa (Fig. 5.17b) 3-RRPP (Fig. 5.18a) 3-RRPP (Fig. 5.18b) 3-RCP (Fig. 5.19a) 3-RCP (Fig. 5.19b) 3-PaRRP (Fig. 5.20a) 3-PaRRP (Fig. 5.20b) 3-PaRRP (Fig. 5.21a) 3-PaRRP (Fig. 5.21b) 3-PaRPR (Fig. 5.22a) 3-PaRPR (Fig. 5.22b) 3-PaRRR (Fig. 5.23a) 3-PaRRR (Fig. 5.23b) 3-PaPRR (Fig. 5.24a) 3-PaPRR (Fig. 5.24b)

Pacc A Pa (Fig. 5.1k) Pacc A Pa (Fig. 5.1k) R||R A P A ||P (Fig. 5.2a) R||R A P A ||P (Fig. 5.2a) R||C A P (Fig. 5.2c) R||C A P (Fig. 5.2c) Pa A R||R A ||P (Fig. 5.3a) Pa A R||R A ||P (Fig. 5.3a) Pa A R||R A A P (Fig. 5.3b) Pa A R||R A A P (Fig. 5.3b) Pa A R A P A ||R (Fig. 5.3c) Pa A R A P A ||R (Fig. 5.3c) Pa A R||R||R (Fig. 5.3d) Pa A R||R||R (Fig. 5.3d) Pa||P A R||R (Fig. 5.3e) Pa||P A R||R (Fig. 5.3e)

1

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Connecting conditions The rotation axes of the actuated revolute joints are reciprocally orthogonal The rotation axes of the actuated revolute joints are parallel to two orthogonal coplanar directions Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 2

476

5 Overconstrained T3-type TPMs with uncoupled motions

Table 5.4. Limb topology and connecting conditions of the overconstrained TPM with uncoupled motions and no idle mobilities presented in Figs. 5.25–5.37 No.

TPM type 3-PaPRR (Fig. 5.25a)

Limb topology Pa A P A A R||R (Fig. 5.3f)

2

3-PaPRR (Fig. 5.25b)

Pa A P A A R||R (Fig. 5.3f)

3

3-PaPaRR (Fig. 5.26a) 3-PaPaRR (Fig. 5.26b) 3-PaRRPa (Fig. 5.27a) 3-PaRRPa (Fig. 5.27b) 3-PaRRbR (Fig. 5.28) 3-PaRRbRbR (Fig. 5.29) 3-PaPn2R (Figs. 5.30, 5.31) 3-PaPn3 (Figs. 5.32, 5.33) 3-PaccRR (Fig. 5.34a) 3-PaccRR (Fig. 5.34b) 3-RRPaP (Fig. 5.35a) 3-RRPaP (Fig. 5.35b) 3-RRPaP (Fig. 5.36a) 3-RRPaP (Fig. 5.36b) 3-RCPa (Fig. 5.37a) 3-RCPa (Fig. 5.37b)

Pa A Pa||R||R (Fig. 5.3g) Pa A Pa||R||R (Fig. 5.3g) Pa A R||R||Pa (Fig. 5.3h) Pa A R||R||Pa (Fig. 5.3h) Pa A R||Rb||R (Fig. 5.3i) Pa A R||Rb||Rb||R (Fig. 5.3j) Pa A Pn2||R (Fig. 5.3k, l) Pa A Pn3 (Fig. 5.3m, n) Pacc A R||R (Fig. 5.3o) Pacc A R||R (Fig. 5.3o) R||R A Pa A A P (Fig. 5.4a) R||R A Pa A A P (Fig. 5.4a) R||R A Pa A ||P (Fig. 5.4b) R||R A Pa A ||P (Fig. 5.4b) R||C A Pa (Fig. 5.4c) R||C A Pa (Fig. 5.4c)

1

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Connecting conditions The rotation axes of the actuated revolute joints are reciprocally orthogonal The rotation axes of the actuated revolute joints are parallel to two orthogonal coplanar directions Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 2 Idem No. 2 Idem No. 2 Idem No. 2 Idem No. 2 Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 2

5.1 Basic solutions with rotating actuators

477

Table 5.5. Limb topology and connecting conditions of the overconstrained TPM with uncoupled motions and no idle mobilities presented in Figs. 5.38–5.48 No.

TPM type 3-RRPPa (Fig. 5.38a)

Limb topology R||R A P A A Pa (Fig. 5.4d)

2

3-RRPPa (Fig. 5.38b)

R||R A P A A Pa (Fig. 5.4d)

3

3-RRPaPa (Fig. 5.39a) 3-RRPaPa (Fig. 5.39b) 3-RRPaPa (Fig. 5.40a) 3-RRPaPa (Fig. 5.40a) 3-RRRRbR (Fig. 5.41) 3-RRRRbRbR (Fig. 5.42) 3-RRPn2R (Figs. 5.43, 5.44) 3-RRPn3 (Figs. 5.45, 5.46) 3-RRRRPa (Fig. 5.47a) 3-RRPaRR (Fig. 5.48a) 3-RRPaRR (Fig. 5.48b)

R||R A Pa||Pa (Fig. 5.4e) R||R A Pa||Pa (Fig. 5.4e) R||R A Pa||Pa (Fig. 5.4f) R||R A Pa||Pa (Fig. 5.4f) R||R A R||Rb||R (Fig. 5.5a) R||R A R||Rb||Rb||R (Fig. 5.5b) R||R A Pn2||R (Fig. 5.5c, d) R||R A Pn3 (Fig. 5.5e, f) R||R A R||R||Pa (Fig. 5.5g) R||R A Pa||R||R (Fig. 5.5h) R||R A Pa||R||R (Fig. 5.5i)

1

4 5 6 7 8 9 10 11 13 13

Connecting conditions The rotation axes of the actuated revolute joints are reciprocally orthogonal The rotation axes of the actuated revolute joints are parallel to two orthogonal coplanar directions Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 2 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 2

478

5 Overconstrained T3-type TPMs with uncoupled motions

Table 5.6. Bases of the operational velocities spaces of the limbs isolated from the parallel mechanisms presented in Figs. 5.6–5.48 No. Parallel mechanism 1 Figs. 5.6–5.17 2

3

4 5 6 7

8

Basis (RG1) ( v1 , v2 , v3 )

Figs. 5.18a, ( v1 ,v2 , v3 ,ȦG ) 5.36a, 5.37a, 5.38a, 5.39a, Figs. 5.18b, ( v1 ,v2 , v3 ,ȦG ) 5.36b, 5.37b, 5.38b, 5.39b, Figs. 5.19a, ( v1 , v2 , v3 , ȦE ) 5.35a, 5.40a Figs. 5.19b, ( v1 , v2 , v3 , ȦE ) 5.35b, 5.40b Figs. 5.20–5.34 ( v1 ,v2 , v3 ,ȦD ) Figs. 5.41, 5.42, 5.47, 5.48a Figs. 5.43, 5.44, 5.45, 5.46, 5.48b

(RG2) ( v1 , v2 , v3 )

(RG3) ( v1 , v2 , v3 )

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦG )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦD )

( v1 ,v2 , v3 ,ȦD )

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦG )

( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG )

5.1 Basic solutions with rotating actuators

479

Table 5.7. Structural parametersa of translational parallel mechanisms in Figs. 5.6–5.14 No. Structural Solution parameter 3-PaPP (Figs. 5.6, 5.7)

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 30 a

14 6 6 6 18 5 0 3 3 See Table 5.6

3-PaPaP (Figs. 5.8, 5.9) 3-PaPPa (Figs. 5.10, 5.11) 3-PaPatP (Figs. 5.14) 20 9 9 9 27 8 0 3 3 See Table 5.6

26 12 12 12 36 11 0 3 3 See Table 5.6

3 3 3 3 3 3 3 3 3 ( v1 ,v 2 ,v3 ) 3 9 15 3 15 0 6

3 3 3 6 6 6 3 3 3 ( v1 ,v 2 ,v3 ) 3 18 24 3 24 0 9

3 3 3 9 9 9 3 3 3 ( v1 ,v 2 ,v3 ) 3 27 33 3 33 0 12

fj

6

9

12

fj

6

9

12

fj

18

27

36

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

3-PaPaPa (Figs. 5.12, 5.13)

See footnote of Table 2.1 for the nomenclature of structural parameters

480

5 Overconstrained T3-type TPMs with uncoupled motions

Table 5.8. Structural parametersa of translational parallel mechanisms in Figs. 5.15–5.18 No. Structural Solution parameter 3-PaPatcc (Fig. 5.15) 3-PaccPa (Fig. 5.17) 1 m 17 2 p1 8 3 p2 8 p3 8 4 p 24 5 6 q 8 7 k1 0 3 8 k2 9 k 3 10 (RGi) See Table 5.6 (i = 1,2,3) 11 SG1 3 3 12 SG2 13 SG3 3 7 14 rG1 7 15 rG2 7 16 rG3 3 17 MG1 3 18 MG2 3 19 MG3 20 (RF) ( v1 ,v 2 ,v3 ) 21 SF 3 22 rl 21 27 23 rF 3 24 MF 21 25 NF 0 26 TF p1 10 27 f 28 29 30 a

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

j

3-PaccP (Fig. 5.16)

3-RRPP (Fig. 5.18)

11 5 5 5 15 5 0 3 3 See Table 5.6

11 4 4 4 12 2 3 0 3 See Table 5.6

3 3 3 4 4 4 3 3 3 ( v1 ,v 2 ,v3 ) 3 12 18 3 12 0 7

4 4 4 0 0 0 4 4 4 ( v1 ,v 2 ,v3 ) 3 0 9 3 3 0 4

fj

10

7

4

fj

10

7

4

fj

30

21

12

See footnote of Table 2.1 for the nomenclature of structural parameters

5.1 Basic solutions with rotating actuators

481

Table 5.9. Structural parametersa of translational parallel mechanisms in Figs. 5.19–5.27 No. Structural Solution parameter 3-RCP (Fig. 5.19)

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

8 3 3 3 9 2 3 0 3 See Table 5.6

3-PaRRP (Figs. 5.20, 5.21) 3-PaRPR (Fig. 5.22) 3-PaRRR (Fig. 5.23) 3-PaPRR (Figs. 5.24, 5.25) 17 7 7 7 21 5 0 3 3 See Table 5.6

23 10 10 10 30 8 0 3 3 See Table 5.6

4 4 4 0 0 0 4 4 4 ( v1 ,v 2 ,v3 ) 3 0 9 3 3 0 4

4 4 4 3 3 3 4 4 4 ( v1 ,v 2 ,v3 ) 3 9 18 3 12 0 7

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 21 0 10

fj

4

7

10

fj

4

7

10

fj

12

21

30

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

3-PaPaRR (Fig. 5.26) 3-PaRRPa (Fig. 5.27)

See footnote of Table 2.1 for the nomenclature of structural parameters

482

5 Overconstrained T3-type TPMs with uncoupled motions

Table 5.10. Structural parametersa of translational parallel mechanisms in Figs. 5.28–5.34 No. Structural Solution parameter 3-PaRRbR (Fig. 5.28) 3-PaPn2R (Figs. 5.30, 5.31) 3-PaPn3 (Figs. 5.32, 5.33) 1 m 23 2 p1 10 p2 10 3 4 p3 10 p 30 5 6 q 8 0 7 k1 8 k2 3 9 k 3 See Table 5.6 10 (RGi) (i = 1,2,3) 11 SG1 4 4 12 SG2 4 13 SG3 6 14 rG1 6 15 rG2 6 16 rG3 4 17 MG1 4 18 MG2 19 MG3 4 20 (RF) ( v1 ,v 2 ,v3 ) 21 SF 3 18 22 rl 27 23 rF 3 24 MF 25 NF 21 0 26 TF p1 10 27 f 28 29 30 a

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

3-PaRRbRbR (Fig. 5.29)

3-PaccRR (Fig. 5.34)

29 13 13 13 39 11 0 3 3 See Table 5.6

14 6 6 6 18 5 0 3 3 See Table 5.6

4 4 4 9 9 9 4 4 4 ( v1 ,v 2 ,v3 ) 3 27 36 3 30 0 13

4 4 4 4 4 4 4 4 4 ( v1 ,v 2 ,v3 ) 3 12 21 3 9 0 8

j

fj

10

13

8

fj

10

13

8

fj

30

39

24

See footnote of Table 2.1 for the nomenclature of structural parameters

5.1 Basic solutions with rotating actuators

483

Table 5.11. Structural parametersa of translational parallel mechanisms in Figs. 5.35–5.40 No. Structural Solution parameter 3-RRPaP (Figs. 5.35, 5.36) 3-RRPPa (Fig. 5.38) 1 m 17 2 p1 7 3 p2 7 p3 7 4 p 21 5 6 q 5 7 k1 0 3 8 k2 9 k 3 10 (RGi) See Table 5.6 (i = 1,2,3) 11 SG1 4 4 12 SG2 13 SG3 4 3 14 rG1 3 15 rG2 3 16 rG3 4 17 MG1 4 18 MG2 4 19 MG3 20 (RF) ( v1 ,v 2 ,v3 ) 21 SF 3 22 rl 9 18 23 rF 3 24 MF 12 25 NF 0 26 TF p1 8 27 f 28 29 30 a

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

j

3-RCPa (Fig. 5.37)

3-RRPaPa (Figs. 5.39, 5.40)

14 6 6 6 18 5 0 3 3 See Table 5.6

23 10 10 10 30 8 0 3 3 See Table 5.6

4 4 4 3 3 3 4 4 4 ( v1 ,v 2 ,v3 ) 3 9 18 3 12 0 7

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 21 0 10

fj

8

7

10

fj

8

7

10

fj

24

21

30

See footnote of Table 2.1 for the nomenclature of structural parameters

484

5 Overconstrained T3-type TPMs with uncoupled motions

Table 5.12. Structural parametersa of translational parallel mechanisms in Figs. 5.41–5.48 No. Structural parameter

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

Solution 3-RRRRbR (Fig. 5.41) 3-RRPn2R (Figs. 5.43, 5.44) 3-RRPn3 (Figs. 5.45, 5.46) 3-RRRRPa (Fig. 5.47) 3-RRPaRR (Fig. 5.48) 20 8 8 8 24 5 0 3 3 See Table 5.6

26 11 11 11 33 8 0 3 3 See Table 5.6

5 5 5 3 3 3 5 5 5 ( v1 ,v 2 ,v3 ) 3 9 21 3 9 0 8

5 5 5 6 6 6 5 5 5 ( v1 ,v 2 ,v3 ) 3 18 30 3 18 0 11

fj

8

11

fj

8

11

fj

24

33

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

3-RRRRbRbR (Fig. 5.42)

See footnote of Table 2.1 for the nomenclature of structural parameters

5.1 Basic solutions with rotating actuators

485

Fig. 5.1. Complex limbs for overconstrained TPMs with uncoupled motions defined by MG = SG = 3, (RG) = (v1,v2,v3) and actuated by rotating motors mounted on the fixed base and combined in a parallelogram loop of type Pa (a–i) or Pacc (j and k)

486

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.1. (cont.)

5.1 Basic solutions with rotating actuators

487

Fig. 5.2. Simple limbs for overconstrained TPMs with uncoupled motions defined by MG = SG = 4, (RG) = ( v1 , v2 , v3 ,Ȧ1 ) and actuated by rotating motors mounted on the fixed base

488

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.3. Complex limbs for overconstrained TPMs with uncoupled motions defined by MG = SG = 4, (RG) = ( v1 , v2 , v3 ,Ȧ1 ) and actuated by rotating motors mounted on the fixed base and combined in a parallelogram loop of type Pa (a–n) or Pacc (o)

5.1 Basic solutions with rotating actuators

Fig. 5.3. (cont.)

489

490

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.3. (cont.)

5.1 Basic solutions with rotating actuators

491

Fig. 5.4. Complex limbs for overconstrained TPMs with uncoupled motions defined by MG = SG = 4, (RG) = ( v1 , v2 , v3 ,Ȧ1 ) and actuated by rotating motors mounted on the fixed base

492

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.5. Complex limbs for overconstrained TPMs with uncoupled motions defined by MG = SG = 5, (RG) = ( v1 , v2 , v3 , Ȧ1 , Ȧ2 ) and actuated by rotating motors mounted on the fixed base

5.1 Basic solutions with rotating actuators

Fig. 5.5. (cont)

493

494

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.6. 3-PaPP-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 15, limb topology Pa||P A A P

5.1 Basic solutions with rotating actuators

495

Fig. 5.7. 3-PaPP-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 15, limb topology Pa A P A ||P

496

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.8. 3-PaPaP-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 24, limb topology Pa A Pa A ||P

5.1 Basic solutions with rotating actuators

497

Fig. 5.9. 3-PaPaP-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 24, limb topology Pa A Pa A A P

498

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.10. 3-PaPPa-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 24, limb topology Pa||P A Pa

5.1 Basic solutions with rotating actuators

499

Fig. 5.11. 3-PaPPa-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 24, limb topology Pa A P A A Pa

500

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.12. 3-PaPaPa-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base with the axes parallel to three reciprocally orthogonal directions, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 33, limb topology Pa A Pa||Pa

5.1 Basic solutions with rotating actuators

501

Fig. 5.13. 3-PaPaPa-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base with the axes parallel to two planar orthogonal directions, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 33, limb topology Pa A Pa||Pa

502

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.14. 3-PaPatP-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 24, limb topology Pa||Pat||P

5.1 Basic solutions with rotating actuators

503

Fig. 5.15. 3-PaPatcc-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa||Patcc

504

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.16. 3-PaccP-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology Pacc A P

5.1 Basic solutions with rotating actuators

505

Fig. 5.17. 3-PaccPa-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pacc A Pa

506

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.18. 3-RRPP-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology R||R A P A ||P

5.1 Basic solutions with rotating actuators

507

Fig. 5.19. 3-RCP-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology R||C A P

508

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.20. 3-PaRRP-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology Pa A R||R A ||P

5.1 Basic solutions with rotating actuators

509

Fig. 5.21. 3-PaRRP-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology Pa A R||R A A P

510

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.22. 3-PaRPR-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology Pa A R A P A ||R

5.1 Basic solutions with rotating actuators

511

Fig. 5.23. 3-PaRRR-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology Pa A R||R||R

512

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.24. 3-PaPRR-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology Pa||P A R||R

5.1 Basic solutions with rotating actuators

513

Fig. 5.25. 3-PaPRR-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology Pa A P A A R||R

514

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.26. 3-PaPaRR-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa A Pa||R||R

5.1 Basic solutions with rotating actuators

515

Fig. 5.27. 3-PaRRPa-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa A R||R||Pa

516

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.28. 3-PaRRbR-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa A R||Rb||R

5.1 Basic solutions with rotating actuators

517

Fig. 5.29. 3-PaRRbRbR-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 30, limb topology Pa A R||Rb||Rb||R

518

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.30. 3-PaPn2R-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa A Pn2||R

5.1 Basic solutions with rotating actuators

519

Fig. 5.31. 3-PaPn2R-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa A Pn2||R

520

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.32. 3-PaPn3-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa A Pn3

5.1 Basic solutions with rotating actuators

521

Fig. 5.33. 3-PaPn3-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa A Pn3

522

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.34. 3-PaccRR-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology Pacc A R||R

5.1 Basic solutions with rotating actuators

523

Fig. 5.35. 3-RRPaP-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology R||R A Pa A A P

524

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.36. 3-RRPaP-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology R||R A Pa A ||P

5.1 Basic solutions with rotating actuators

525

Fig. 5.37. 3-RCPa-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology R||C A Pa

526

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.38. 3-RRPPa-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology R||R A P A A Pa

5.1 Basic solutions with rotating actuators

527

Fig. 5.39. 3-RRPaPa-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology R||R A Pa||Pa

528

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.40. 3-RRPaPa-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology R||R A Pa||Pa

5.1 Basic solutions with rotating actuators

529

Fig. 5.41. 3-RRRRbR-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9 limb topology R||R A R||Rb||R

530

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.42. 3-RRRRbRbR-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 18, limb topology R||R A R||Rb||Rb||R

5.1 Basic solutions with rotating actuators

531

Fig. 5.43. 3-RRPn2R-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R||R A Pn2||R

532

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.44. 3-RRPn2R-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R||R A Pn2||R

5.1 Basic solutions with rotating actuators

533

Fig. 5.45. 3-RRPn3-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R||R A Pn3

534

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.46. 3-RRPn3-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topologies R||R A Pn3

5.1 Basic solutions with rotating actuators

535

Fig. 5.47. 3-RRRRPa-type overconstrained TPM with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R||R A R||R||Pa

536

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.48. 3-RRPaRR-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R||R A Pa||R||R

5.2 Derived solutions with rotating actuators

537

5.2 Derived solutions with rotating actuators Solutions with lower degrees of overconstraint can be derived from the basic solutions in Figs. 5.6–5.48 by using joints with idle mobilities. A large set of solutions can be obtained by introducing one or two rotational idle mobilities outside the closed loops that can be integrated in the limbs and up to three idle mobilities (two rotations and one translation) in each planar loop. The joints combining idle mobilities are denoted by an asterisk. We recall that the idle mobilities combined in a parallelogram loop (see Fig. 6.3 – Part 1) are systematized in Table 3.12. Two idle rotational mobilities are introduced in the spherical joint of the parallelogram loops denoted by Paccs and Pascc which combines two cylindrical, one revolute and one spherical joint. In the cylindrical joint denoted by C*, the rotation is the idle mobility. For example, the solution 3-PaPC*-type in Fig. 5.49 is derived from the basic solution 3-PaPP in Fig. 5.6 by replacing the last prismatic joint P in each limb by a cylindrical joint C* which combines a rotational idle mobility. Examples of solutions with identical limbs and three to twenty one degrees of overconstraint derived from the basic solutions in Figs. 5.6– 5.48 are illustrated in Figs. 5.49–5.108. The limb topology and the number of overconstraints in these solutions are systematized in Table 5.13 and the structural parameters in Tables 5.14–5.26.

538

5 Overconstrained T3-type TPMs with uncoupled motions

Table 5.13. Limb topology and the number of overconstraints NF of the derived TPMs with idle mobilities and rotating actuators mounted on the fixed base presented in Figs. 5.49–5.108 No. Basic TPM type 1

3-PaPP (Fig. 5.6)

NF 15

2 3

3-PaPP (Fig. 5.7)

15

4

5

3-PaPaP (Fig. 5.8a)

24

6 7 8 9

3-PaPaP (Fig. 5.9)

24

10 11 12 13

3-PaPPa (Fig. 5.10)

24

14 15 16 17 18 19

3-PaPPa (Fig. 5.11)

24

Derived TPM type 3-PaPC* (Fig. 5.49) 3-PassPP (Fig. 5.50) 3-PaPC* (Fig. 5.51) 3-PassPP Pass A P A ||P (Fig. 5.52) 3-PaPassP (Fig. 5.53a) 3-PaPacsP (Fig. 5.53b) 3-PassPaP (Fig. 5.54a) 3-PaPacsPR*R* (Fig. 5.54b) 3-PaPassP (Fig. 5.55a) 3-PaPacsP (Fig. 5.55b) 3-PassPaP (Fig. 5.56a) 3-PaPacsPR*R* (Fig. 5.56b) 3-PaPPass (Fig. 5.57a) 3-PacsPPa (Fig. 5.57b) 3-PassPPa (Fig. 5.58a) 3-PaPPacsR*R* (Fig. 5.58b) 3-PaPPass (Fig. 5.59a) 3-PacsPPa (Fig. 5.59b) 3-PassPPa (Fig. 5.60a)

NF

Limb topology

12

Pa||P A C*

3

Pass||P A P

12

Pa A P A ||C*

3

Pass A P A ||P

12

Pa A Pass A ||P

15

Pa A Pacs A ||P

12

Pass A Pa A ||P

9

Pa A Pacs A ||P A A R* A ||R*

12

Pa A Pass A A P

15

Pa A Pacs A A P

12

Pass A Pa A A P

9

Pa A Pacs A A P A A R* A ||R*

12

Pa||P A Pass

15

Pacs||P A Pa

12

Pass||P A Pa

9

Pa||P A Pacs A ||R* A A R*

12

Pa A P A A Pass

15

Pacs A P A A Pa

12

Pass A P A A Pa

5.2 Derived solutions with rotating actuators

539

Table 5.13. (cont.) 20 21

3-PaPaPa (Fig. 5.12)

33

22 23 24 25

3-PaPatP (Fig. 5.14)

24

26 27 28

3-PaPatcc (Fig. 5.15)

21

29 30 31 32

3-PaccP (Fig. 5.16)

12

33 34

3-PaccPa (Fig. 5.17)

21

35 36 37

3-PaRRP (Fig. 5.20) 3-PaRRP (Fig. 5.21)

12 12

38 39

3-PaRPR (Fig. 5.22)

12

40 41

3-PaRRR (Fig. 5.23)

12

3-PaPPacsR*R* 9 (Fig. 5.60b) 3-PaPaPass 21 (Fig. 5.61) 3-PassPaPass 9 (Fig. 5.62) 3-PaPaPacs 24 (Fig. 5.63) 3-PacsPaPacsR*R* 9 (Fig. 5.64) 3-PaPatC* 21 (Fig. 5.65) 3-PaPatssP 12 (Fig. 5.66a) 3-PaPatcsPR* 12 (Fig. 5.66b) 3-PassPatcc 12 (Fig. 5.67a) 3-PacsPatcc 12 (Fig. 5.67b) 3-PassPatccR*R* 6 (Fig. 5.68a) 3-PacsPatccR*R* 6 (Fig. 5.68b) 3-PaccC*R* 6 (Fig. 5.69) 3-PaccsP 6 (Fig. 5.70) 3-PaccsPa 15 (Fig. 5.71) 3-PaccPassR* 6 (Fig. 5.72) 3-PassRP 3 (Fig. 5.73 3-PassRP 3 (Fig. 5.74) 3-PaRRC* 9 (Fig. 5.75) 3-PaRPRR* 9 (Fig. 5.76) 3-PassPR 3 (Fig. 5.77) 3-PaRRRR* 9 (Fig. 5.78)

Pa A P A A Pacs A ||R* A A R* Pa A Pa||Pass Pass A Pa||Pass Pa A Pa||Pacs Pacs A Pa||Pacs A ||R* A A R* Pa||Pat||C* Pa||Patss||P Pa||Patcs||P A A R* Pass||Patcc Pacs||Patcc Pass||Patcc A R* A ||R* Pacs||Patcc A R* A ||R* Pacc A C* A R* Paccs A P Paccs A Pa Pacc A Pass||R* Pass A R A ||P Pass A R A A P Pa A R||R A A C* Pa A R A P A ||R A R* Pass-P A A R Pa A R||R||R A R*

540

5 Overconstrained T3-type TPMs with uncoupled motions Table 5.13. (cont.)

42 43

3-PaPRR (Fig. 5.24)

12

44 45

3-PaPRR (Fig. 5.25)

12

46 47

3-PaPaRR (Fig. 5.26)

21

48 49

3-PaRRPa (Fig. 5.27)

21

50 51

3-PaRRbR (Fig. 5.28)

21

52 53

3-PaRRbRbR 30 (Fig. 5.29)

54 55

3-PaPn2R (Fig. 5.30)

21

56 57

3-PaPn2R (Fig. 5.31)

21

58 59

3-PaPn3 (Fig. 5.32)

21

60 61

3-PaPn3 (Fig. 5.33)

21

62 63

3-PaccRR (Fig. 5.34)

9

3-PassRR (Fig. 5.79) 3-PaPRRR* (Fig. 5.80) 3-PacsPRR (Fig. 5.81) 3-PaPRRR (Fig. 5.82) 3-PacsPRR (Fig. 5.83) 3-PaPaRRR* (Fig. 5.84) 3-PacsPacsRR (Fig. 5.85) 3-PassRPa (Fig. 5.86) 3-PassRPacs (Fig. 5.87) 3-PassRbR (Fig. 5.88) 3-PassRbcsR (Fig. 5.89) 3-PassRbRbR (Fig. 5.90) 3-PassRbcsRbcsR (Fig. 5.91) 3-PaPn2RR* (Fig. 5.92) 3-PacsPn2csR (Fig. 5.93) 3-PaPn2RR* (Fig. 5.94) 3-PacsPn2csR (Fig. 5.95) 3-PaPn3R* (Fig. 5.96) 3-PacsPn3cs (Fig. 5.97) 3-PaPn3R* (Fig. 5.98) 3-PacsPn3cs (Fig. 5.99) 3-PaccRRR* (Fig. 5.100)

3

Pass A R||R

9

Pa||P A R||R A R*

3

Pacs||P A R||R

9

Pa A P A A R||R A R*

3

Pacs A P A A R||R

18

Pa A Pa||R||R A A R*

3

Pacs A Pacs||R||R

12

Pass A R||Pa

3

Pass A R||Pacs

12

Pass A Rb||R

3

Pass A Rbcs||R

21

Pass A Rb||Rb||R

3

Pass A Rbcs||Rbcs||R

18

Pa A Pn2||R A R*

3

Pacs A Pn2cs||R

18

Pa A Pn2||R A R*

3

Pacs A Pn2cs||R

18

Pa A Pn3 A R*

3

Pacs A Pn3cs

18

Pa A Pn3 A R*

3

Pacs A Pn3cs

6

Pacc A R||R A R*

5.2 Derived solutions with rotating actuators Table 5.13. (cont.) 64 65 66

3-RRPaP (Fig. 5.35) 3-RRPPa (Fig. 5.38)

12 12

67 68 69 70 71

3-RRPaPa (Fig. 5.39)

21

3-PasccRR (Fig. 5.101) 3-RPassP (Fig. 5.102) 3-RRC*Pa (Fig. 5.103) 3-RRPPacs (Fig. 5.104) 3-RRPaPacs (Fig. 5.105) 3-RRPassPa (Fig. 5.106) 3-RPassPa (Fig. 5.107) 3-RRPaPass (Fig. 5.108)

3

Pascc A R||R

3

R A Pass A P

9

R||R A C* A A Pa

3

R||R A P A A Pacs

12

R||R A Pa||Pacs

9

R||R A Pass||Pa

12

R A Pass||Pa

9

R||R A Pa||Pass

541

542

5 Overconstrained T3-type TPMs with uncoupled motions

Table 5.14. Bases of the operational velocities spaces of the limbs isolated from the parallel mechanisms presented in Figs. 5.49–5.108 No. Parallel mechanism 1 Figs. 5.49a, 5.51a, 5.57a, 5.59a, 5.61, 5.104a, 5.105a 2 Figs. 5.49b, 5.51b, 5.104b, 5.105b 3 Figs. 5.50, 5.52, 5.54a, 5.56a, 5.58a, 5.60a, 5.66a, 5.73, 5.74, 5.77, 5.79, 5.81, 5.83, 5.85, 5.86, 5.87, 5.88-5.91, 5.93, 5.95, 5.97, 5.99, 5.101 4 Figs. 5.53a, 5.55a, 5.65a, 5.102a, 5.107a 5 Figs. 5.53b, 5.55b, 5.57b, 5.59b, 5.63 5.67, 5.70, 5.71 6 Figs. 5.54b, 5.56b, 5.58b, 5.60b, 5.64 5.68, 5.69b, 5.103 5.106, 5.108 7 Figs. 5.62, 5.69a 8 Fig. 5.65b 9

Fig. 5.66b

10 Figs. 5.72, 5.75, 5.76, 5.78, 5.80, 5.82, 5.84, 5.92, 5.94, 5.96, 5.98, 5.100 11 Figs. 5.102b, 5.107b

Basis (RG1) ( v1 ,v2 , v3 ,ȦG )

(RG2) ( v1 ,v2 , v3 ,ȦD )

(RG3) ( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦG )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦG )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦD )

( v1 , v 2 , v 3 )

( v1 , v 2 , v 3 )

( v1 , v 2 , v 3 )

( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE )

( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG ) ( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦG )

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦD )

( v1 ,v2 , v3 ,ȦD )

5.2 Derived solutions with rotating actuators

543

Table 5.15. Structural parametersa of translational parallel mechanisms in Figs. 5.49–5.52 No.

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

Structural parameter

Solution 3-PaPC* (Figs. 5.49, 5.51) 14 6 6 6 18 5 0 3 3 See Table 5.14

3-PassPP (Figs. 5.50, 5.52) 14 6 6 6 18 5 0 3 3 See Table 5.14

4 4 4 3 3 3 4 4 4 ( v1 ,v 2 ,v3 ) 3 9 18 3 12 0 7

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 3 0 10

fj

7

10

fj

7

10

fj

21

30

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

544

5 Overconstrained T3-type TPMs with uncoupled motions

Table 5.16. Structural parametersa of translational parallel mechanisms in Figs. 5.53–5.60 No. Structural Solution parameter 3-PaPassP (Figs. 5.53a, 5.55a) 3-PassPaP (Figs. 5.54a, 5.56a) 3-PaPPass (Figs. 5.57a, 5.59a) 3-PassPPa (Figs. 5.58a, 5.60a)

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

m 20 p1 9 p2 9 p3 9 p 27 q 8 0 k1 k2 3 k 3 See Table 5.14 (RGi) (i = 1,2,3) SG1 4 4 SG2 4 SG3 9 rG1 9 rG2 9 rG3 4 MG1 4 MG2 MG3 4 (RF) ( v1 ,v 2 ,v3 ) SF 3 27 rl 36 rF 3 MF NF 12 0 TF p1 13 f

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

3-PaPacsP 3-PaPacsPR*R* (Figs. 5.53b, (Figs. 5.54b, 5.55b) 5.56b) 3-PacsPPa 3-PaPPacsR*R* (Figs. 5.57b, (Figs. 5.58b, 5.59b) 5.60b) 20 26 9 11 9 11 9 11 27 33 8 8 0 0 3 3 3 3 See Table 5.14See Table 5.14 3 3 3 9 9 9 3 3 3 ( v1 ,v 2 ,v3 ) 3 27 33 3 15 0 12

5 5 5 9 9 9 5 5 5 ( v1 ,v 2 ,v3 ) 3 27 39 3 9 0 14

j

fj

13

12

14

fj

13

12

14

fj

39

36

42

See footnote of Table 2.1 for the nomenclature of structural parameters

5.2 Derived solutions with rotating actuators

545

Table 5.17. Structural parametersa of translational parallel mechanisms in Figs. 5.61–5.63 No.

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

Structural parameter

Solution 3-PaPaPass (Fig. 5.61) 26 12 12 12 36 11 0 3 3 See Table 5.14

3-PassPaPass (Fig. 5.62) 26 12 12 12 36 11 0 3 3 See Table 5.14

3-PaPaPacs (Fig. 5.63) 26 12 12 12 36 11 0 3 3 See Table 5.14

4 4 4 12 12 12 4 4 4 ( v1 ,v 2 ,v3 ) 3 36 45 3 21 0 16

5 5 5 15 15 15 5 5 5 ( v1 ,v 2 ,v3 ) 3 45 57 3 9 0 20

3 3 3 12 12 12 3 3 3 ( v1 ,v 2 ,v3 ) 3 36 42 3 24 0 15

fj

16

20

15

fj

16

20

15

fj

48

60

45

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

546

5 Overconstrained T3-type TPMs with uncoupled motions

Table 5.18. Structural parametersa of translational parallel mechanisms in Figs. 5.64–5.66 No. Structural Solution parameter 3-PacsPaPacsR*R* 3-PaPatC* (Fig. 5.64) (Fig. 5.65) 1 m 32 20 2 p1 14 9 p2 14 9 3 4 p3 14 9 5 p 42 27 6 q 11 8 0 0 7 k1 8 k2 3 3 9 k 3 3 See Table 5.14 See Table 5.14 10 (RGi) (i = 1,2,3) 11 SG1 5 4 5 4 12 SG2 5 4 13 SG3 15 6 14 rG1 15 rG2 15 6 15 6 16 rG3 5 4 17 MG1 5 4 18 MG2 5 4 19 MG3 20 (RF) ( v1 ,v 2 ,v3 ) ( v1 ,v 2 ,v3 ) 21 SF 3 3 45 18 22 rl 23 rF 57 27 24 MF 3 3 9 21 25 NF 0 0 26 TF p1 20 10 27 f 28 29 30 a

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

3-PaPatssP 3-PaPatcsPR* (Fig. 5.66a) (Fig. 5.66b) 20 23 9 10 9 10 9 10 27 30 8 8 0 0 3 3 3 3 See Table 5.14See Table 5.14 4 4 4 9 9 9 4 4 4 ( v1 ,v 2 ,v3 ) 3 27 36 3 12 0 13

4 4 4 9 9 9 4 4 4 ( v1 ,v 2 ,v3 ) 3 27 36 3 12 0 13

j

fj

20

10

13

13

fj

20

10

13

13

fj

60

30

39

39

See footnote of Table 2.1 for the nomenclature of structural parameters

5.2 Derived solutions with rotating actuators

547

Table 5.19. Structural parametersa of translational parallel mechanisms in Figs. 5.67 and 5.68 No. Structural Solution parameter 3-PassPatcc (Fig. 5.67a) 1 m 17 2 p1 8 p2 8 3 4 p3 8 5 p 24 6 q 8 0 7 k1 8 k2 3 9 k 3 See Table 5.14 10 (RGi) (i = 1,2,3) 11 SG1 3 3 12 SG2 3 13 SG3 10 14 rG1 15 rG2 10 10 16 rG3 4 17 MG1 4 18 MG2 4 19 MG3 20 (RF) ( v1 ,v 2 ,v3 ) 21 SF 3 30 22 rl 23 rF 36 24 MF 6 12 25 NF 3 26 TF p1 14 27 f 28 29 30 a

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

3-PacsPatcc PassPatccR*R* (Fig. 5.67b) (Fig. 5.68a) 17 23 8 10 8 10 8 10 24 30 8 8 0 0 3 3 3 3 See Table 5.14See Table 5.14

3-PacsPatccR*R* (Fig. 5.68b) 23 10 10 10 30 8 0 3 3 See Table 5.14

3 3 3 10 10 10 3 3 3 ( v1 ,v 2 ,v3 ) 3 30 36 3 12 0 13

5 5 5 10 10 10 6 6 6 ( v1 ,v 2 ,v3 ) 3 30 42 6 6 3 16

5 5 5 10 10 10 5 5 5 ( v1 ,v 2 ,v3 ) 3 30 42 3 6 0 15

j

fj

14

13

16

15

fj

14

13

16

15

fj

42

39

48

45

See footnote of Table 2.1 for the nomenclature of structural parameters

548

5 Overconstrained T3-type TPMs with uncoupled motions

Table 5.20. Structural parametersa of translational parallel mechanisms in Figs. 5.69–5.72 No. Structural Solution parameter 3-PaccC*R* (Fig. 5.69) 1 m 14 2 p1 6 p2 6 3 4 p3 6 5 p 18 6 q 5 0 7 k1 8 k2 3 9 k 3 See Table 5.14 10 (RGi) (i = 1,2,3) 11 SG1 5 5 12 SG2 5 13 SG3 4 14 rG1 15 rG2 4 4 16 rG3 5 17 MG1 5 18 MG2 5 19 MG3 20 (RF) ( v1 ,v 2 ,v3 ) 21 SF 3 12 22 rl 23 rF 24 24 MF 3 6 25 NF 0 26 TF p1 9 27 f 28 29 30 a

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

3-PaccsP 3-PaccsPa (Fig. 5.70) (Fig. 5.71) 11 17 5 8 5 8 5 8 15 24 5 8 0 0 3 3 3 3 See Table 5.14See Table 5.14

3-PaccPassR* (Fig. 5.72) 20 9 9 9 27 8 0 3 3 See Table 5.14

3 3 3 6 6 6 3 3 3 ( v1 ,v 2 ,v3 ) 3 18 24 3 6 0 9

3 3 3 9 9 9 3 3 3 ( v1 ,v 2 ,v3 ) 3 27 33 3 15 0 12

5 5 5 10 10 10 5 5 5 ( v1 ,v 2 ,v3 ) 3 30 42 3 6 0 15

j

fj

9

9

12

15

fj

9

9

12

15

fj

27

27

36

45

See footnote of Table 2.1 for the nomenclature of structural parameters

5.2 Derived solutions with rotating actuators

549

Table 5.21. Structural parametersa of translational parallel mechanisms in Figs. 5.73–5.80 No. Structural Solution parameter 3-PassRP (Figs. 5.73, 5.74) 3-PassPR (Fig. 5.77) 3-PassRR (Fig. 5.79) 1 m 14 2 p1 6 p2 6 3 4 p3 6 p 18 5 6 q 5 0 7 k1 8 k2 3 9 k 3 See Table 5.14 10 (RGi) (i = 1,2,3) 11 SG1 4 4 12 SG2 4 13 SG3 6 14 rG1 6 15 rG2 6 16 rG3 4 17 MG1 4 18 MG2 19 MG3 4 20 (RF) ( v1 ,v 2 ,v3 ) 21 SF 3 18 22 rl 27 23 rF 3 24 MF 25 NF 3 0 26 TF p1 10 27 f

17 7 7 7 21 5 0 3 3 See Table 5.14

3-PaRPRR* (Fig. 5.76) 3-PaRRRR* (Fig. 5.78) 3-PaPRRR* (Fig. 5.80) 20 8 8 8 24 5 0 3 3 See Table 5.14

5 5 5 3 3 3 5 5 5 ( v1 ,v 2 ,v3 ) 3 9 21 3 9 0 8

5 5 5 3 3 3 5 5 5 ( v1 ,v 2 ,v3 ) 3 9 21 3 9 0 8

28 29 30 a

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

3-PaRRC* (Fig. 5.75)

j

fj

10

8

8

fj

10

8

8

fj

30

24

24

See footnote of Table 2.1 for the nomenclature of structural parameters

550

5 Overconstrained T3-type TPMs with uncoupled motions

Table 5.22. Structural parametersa of translational parallel mechanisms in Figs. 5.81–5.85 No. Structural Solution parameter 3-PacsPRR (Figs. 5.81, 5.83) 1 m 17 2 p1 7 3 p2 7 4 p3 7 p 21 5 6 q 5 k1 0 7 k2 3 8 k 9 3 See Table 5.14 10 (RGi) (i = 1,2,3) 11 SG1 4 4 12 SG2 4 13 SG3 14 rG1 6 6 15 rG2 6 16 rG3 4 17 MG1 4 18 MG2 4 19 MG3 20 (RF) ( v1 ,v 2 ,v3 ) 21 SF 3 22 rl 18 23 rF 27 3 24 MF 3 25 NF 0 26 TF p1 10 27 f 28 29 30 a

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

3-PaPRRR* (Fig. 5.82)

3-PaPaRRR* (Fig. 5.84)

3-PacsPacsRR (Fig. 5.85)

20 26 8 11 8 11 8 11 24 33 5 8 0 0 3 3 3 3 See Table 5.14See Table 5.14

23 10 10 10 30 8 0 3 3 See Table 5.14

5 5 5 3 3 3 5 5 5 ( v1 ,v 2 ,v3 ) 3 9 21 3 9 0 8

5 5 5 6 6 6 5 5 5 ( v1 ,v 2 ,v3 ) 3 18 30 3 18 0 11

4 4 4 12 12 12 4 4 4 ( v1 ,v 2 ,v3 ) 3 36 45 3 3 0 16

j

fj

10

8

11

16

fj

10

8

11

16

fj

30

24

33

48

See footnote of Table 2.1 for the nomenclature of structural parameters

5.2 Derived solutions with rotating actuators

551

Table 5.23. Structural parametersa of translational parallel mechanisms in Figs. 5.86–5.91 No. Structural Solution parameter 3-PassRPa (Fig. 5.86) 3-PassRbR (Fig. 5.88) 1 m 20 2 p1 9 3 p2 9 p3 9 4 p 27 5 6 q 8 7 k1 0 3 8 k2 9 k 3 10 (RGi) See Table 5.14 (i = 1,2,3) 11 SG1 4 4 12 SG2 13 SG3 4 9 14 rG1 9 15 rG2 9 16 rG3 4 17 MG1 4 18 MG2 4 19 MG3 20 (RF) ( v1 ,v 2 ,v3 ) 21 SF 3 22 rl 27 36 23 rF 3 24 MF 12 25 NF 0 26 TF p1 13 27 f 28 29 30 a

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

j

3-PassRPacs 3-PassRbRbR (Fig. 5.87) (Fig. 5.90) 3-PassRbcsR (Fig. 5.89) 20 26 9 12 9 12 9 12 27 36 8 11 0 0 3 3 3 3 See Table 5.14See Table 5.14

3-PassRbcsRbcsR (Fig. 5.91)

26 12 12 12 36 11 0 3 3 See Table 5.14

4 4 4 12 12 12 4 4 4 ( v1 ,v 2 ,v3 ) 3 36 45 3 3 0 16

4 4 4 12 12 12 4 4 4 ( v1 ,v 2 ,v3 ) 3 36 45 3 21 0 16

4 4 4 18 18 18 4 4 4 ( v1 ,v 2 ,v3 ) 3 54 63 3 3 0 22

fj

13

16

16

22

fj

13

16

16

22

fj

39

48

48

66

See footnote of Table 2.1 for the nomenclature of structural parameters

552

5 Overconstrained T3-type TPMs with uncoupled motions

Table 5.24. Structural parametersa of translational parallel mechanisms in Figs. 5.92–5.101 No. Structural Solution parameter 3-PaPn2RR* (Figs. 5.92, 5.94) 3-PaPn3R* (Figs. 5.96, 5.98) 1 m 26 2 p1 11 p2 11 3 4 p3 11 p 33 5 6 q 8 0 7 k1 8 k2 3 9 k 3 See Table 5.14 10 (RGi) (i = 1,2,3) 11 SG1 5 5 12 SG2 5 13 SG3 6 14 rG1 6 15 rG2 6 16 rG3 5 17 MG1 5 18 MG2 19 MG3 5 20 (RF) ( v1 ,v 2 ,v3 ) 21 SF 3 18 22 rl 30 23 rF 3 24 MF 25 NF 18 0 26 TF p1 11 27 f 28 29 30 a

¦ ¦ ¦ ¦

j 1

p2 j 1 p3 j 1 p j 1

3-PacsPn2csR (Figs. 5.93, 5.95) 3-PacsPn3cs (Figs. 5.97, 5.99) 23 10 10 10 30 8 0 3 3 See Table 5.14

3-PaccRRR* (Fig. 5.100)

3-PasccRR (Fig. 5.101)

17 7 7 7 21 5 0 3 3 See Table 5.14

14 6 6 6 18 5 0 3 3 See Table 5.14

4 4 4 12 12 12 4 4 4 ( v1 ,v 2 ,v3 ) 3 36 45 3 3 0 16

5 5 5 4 4 4 5 5 5 ( v1 ,v 2 ,v3 ) 3 12 24 3 6 0 9

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 3 0 10

j

fj

11

16

9

10

fj

11

16

9

10

fj

33

48

27

30

See footnote of Table 2.1 for the nomenclature of structural parameters

5.2 Derived solutions with rotating actuators

553

Table 5.25. Structural parametersa of translational parallel mechanisms in Figs. 5.102–5.104 No.

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

Structural parameter

Solution 3-RPassP (Fig. 5.102) 14 6 6 6 18 5 0 3 3 See Table 5.14

3-RRC*Pa (Fig. 5.103) 17 7 7 7 21 5 0 3 3 See Table 5.14

3-RRPPacs (Fig. 5.104) 17 7 7 7 21 5 0 3 3 See Table 5.14

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 3 0 10

5 5 5 3 3 3 5 5 5 ( v1 ,v 2 ,v3 ) 3 9 21 3 9 0 8

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 3 0 10

fj

10

8

10

fj

10

8

10

fj

30

24

30

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

554

5 Overconstrained T3-type TPMs with uncoupled motions

Table 5.26. Structural parametersa of translational parallel mechanisms in Figs. 5.105–5.108 No.

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

Structural parameter

Solution 3-RRPaPacs (Fig. 5.105)

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

23 10 10 10 30 8 0 3 3 See Table 5.14

3-RRPassPa (Fig. 5.106) 3-RRPaPass (Fig. 5.108) 23 10 10 10 30 8 0 3 3 See Table 5.14

20 9 9 9 27 8 0 3 3 See Table 5.14

4 4 4 9 9 9 4 4 4 ( v1 ,v 2 ,v3 ) 3 27 36 3 12 0 13

5 5 5 9 9 9 5 5 5 ( v1 ,v 2 ,v3 ) 3 27 39 3 9 0 14

4 4 4 9 9 9 4 4 4 ( v1 ,v 2 ,v3 ) 3 27 36 3 12 0 13

fj

13

14

13

fj

13

14

13

fj

39

42

39

SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

3-RPassPa (Fig. 5.107)

See footnote of Table 2.1 for the nomenclature of structural parameters

5.2 Derived solutions with rotating actuators

555

Fig. 5.49. 3-PaPC*-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology Pa||P A C*

556

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.50. 3-PassPP-type (Pass||P A P) overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pass||P A P

5.2 Derived solutions with rotating actuators

557

Fig. 5.51. 3-PaPC*-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), || TF = 0, NF = 12, limb topology Pa A P A C*

558

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.52. 3-PassPP-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pass A P A ||P

5.2 Derived solutions with rotating actuators

559

Fig. 5.53. Overconstrained TPMs with uncoupled motions of types 3-PaPassP (a) and 3-PaPacsP (b) defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12 (a), NF = 15 (b), limb topology Pa A Pass A ||P (a) and Pa A Pacs A ||P (b)

560

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.54. Overconstrained TPMs with uncoupled motions of types 3-PassPaP (a) and 3-PaPacsPR*R* (b) defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12 (a), NF = 9 (b), limb topology Pass A Pa A ||P (a) and Pa A Pacs A ||P A A R* A ||R* (b)

5.2 Derived solutions with rotating actuators

561

Fig. 5.55. Overconstrained TPMs with uncoupled motions of types 3-PaPassP (a) and 3-PaPacsP (b) defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12 (a), NF = 15 (b), limb topology Pa A Pass A A P (a) and Pa A Pacs A A P (b)

562

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.56. Overconstrained TPMs with uncoupled motions of types 3-PassPaP (a) and 3-PaPacsPR*R* (b) defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12 (a), NF = 9 (b), limb topology Pass A Pa A A P (a) and Pa A Pacs A A P A A R* A || R*(b)

5.2 Derived solutions with rotating actuators

563

Fig. 5.57. Overconstrained TPMs with uncoupled motions of types 3-PaPPass (a) and 3-PacsPPa (b) defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12 (a), NF = 15 (b), limb topology Pa||P A Pass (a) and Pacs||P A Pa (b)

564

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.58. Overconstrained TPMs with uncoupled motions of types 3-PassPPa (a) and 3-PaPPacsR*R* (b) defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12 (a), NF = 9 (b), limb topology Pass||P A Pa (a) and Pa||P A Pacs A ||R* A A R* (b)

5.2 Derived solutions with rotating actuators

565

Fig. 5.59. Overconstrained TPMs with uncoupled motions of types 3-PaPPass (a) and 3-PacsPPa (b) defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12 (a), NF = 15 (b), limb topology Pa A P A A Pass (a) and Pacs A P A A Pa (b)

566

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.60. Overconstrained TPMs with uncoupled motions of types 3-PassPPa (a) and 3-PaPPacsR*R* (b) defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12 (a), NF = 9 (b), limb topology Pass A P A A Pa (a) and Pa A P A A Pacs A ||R* A A R* (b)

5.2 Derived solutions with rotating actuators

567

Fig. 5.61. 3-PaPaPass-type overconstrained TPM with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa A Pa||Pass

568

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.62. 3-PassPaPass-type overconstrained TPM with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology Pass A Pa||Pass

5.2 Derived solutions with rotating actuators

569

Fig. 5.63. 3-PaPaPacs-type overconstrained TPM with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 24, limb topology Pa A Pa||Pacs

570

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.64. 3-PacsPaPacsR*R*-type overconstrained TPM with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology Pacs A Pa||Pacs A ||R* A A R*

5.2 Derived solutions with rotating actuators

571

Fig. 5.65. 3-PaPatC*-type overconstrained TPMs with uncoupled motions and rotating actuators mounted on the fixed base, defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pa||Pat||C*

572

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.66. Overconstrained TPMs with uncoupled motions of types 3-PaPatssP (a) and 3-PaPatcsPR* (b) defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology Pa||Patss||P (a) and Pa||Patcs||P A A R* (b)

5.2 Derived solutions with rotating actuators

573

Fig. 5.67. Overconstrained TPMs with uncoupled motions of types 3-PassPatcc (a) and 3-PacsPatcc (b) defined by MF = 6 (a) MF = 3 (b) SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 3, TF = 0 (b), NF = 12, limb topology Pass||Patcc (a) and Pacs||Patcc (b)

574

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.68. Overconstrained TPMs with uncoupled motions of types 3-Pass PatccR*R* (a) and 3-PacsPatccR*R* (b) defined by MF = 6 (a) MF = 3 (b) SF = 3, (RF) = ( v1 , v 2 , v3 ), TF = 3 (a), TF = 0 (b), NF = 6, limb topology Pass|| Patcc A R* A ||R* (a) and Pacs||Patcc A R* A ||R* (b)

5.2 Derived solutions with rotating actuators

575

Fig. 5.69. 3-PaccC*R*-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ), TF = 0, NF = 6, limb topology Pacc A C* A R*

576

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.70. 3-PaccsP-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 6, limb topology Paccs A P

5.2 Derived solutions with rotating actuators

577

Fig. 5.71. 3-PaccsPa-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 15, limb topology Paccs A Pa

578

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.72. 3-PaccPassR*-type overconstrained TPM with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 6, limb topology Pacc A Pass||R*

5.2 Derived solutions with rotating actuators

579

Fig. 5.73. 3-PassRP-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pass A R A ||P

580

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.74. 3-PassRP-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pass A R A A P

5.2 Derived solutions with rotating actuators

581

Fig. 5.75. 3-PaRRC*-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology Pa A R||R A A C*

582

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.76. 3-PaRPRR*-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ), TF = 0, NF = 9, limb topology Pa A R A P A ||R A R*

5.2 Derived solutions with rotating actuators

583

Fig. 5.77. 3-PassPR-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pass-P A A R

584

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.78. 3-PaRRRR*-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology Pa A R||R||R A R*

5.2 Derived solutions with rotating actuators

585

Fig. 5.79. 3-PassRR-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pass A R||R

586

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.80. 3-PaPRRR*-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology Pa||P A R||R A R*

5.2 Derived solutions with rotating actuators

587

Fig. 5.81. 3-PacsPRR-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pacs||P A R||R

588

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.82. 3-PaPRRR*-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology Pa A P A A R||R A R*

5.2 Derived solutions with rotating actuators

589

Fig. 5.83. 3-PacsPRR-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pacs A P A A R||R

590

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.84. 3-PaPaRRR*-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 18, limb topology Pa A Pa||R||R A A R*

5.2 Derived solutions with rotating actuators

591

Fig. 5.85. 3-PacsPacsRR-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pacs A Pacs||R||R

592

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.86. 3-PassRPa-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology Pass A R||Pa

5.2 Derived solutions with rotating actuators

593

Fig. 5.87. 3-PassRPacs-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pass A R||Pacs

594

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.88. 3-PassRbR-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology Pass A Rb||R

5.2 Derived solutions with rotating actuators

595

Fig. 5.89. 3-PassRbcsR-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pass A Rbcs||R

596

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.90. 3-PassRbRbR-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 21, limb topology Pass A Rb||Rb||R

5.2 Derived solutions with rotating actuators

597

Fig. 5.91. 3-PassRbcsRbcsR-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pass A Rbcs||Rbcs||R

598

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.92. 3-PaPn2RR*-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 18, limb topology Pa A Pn2||R A R*

5.2 Derived solutions with rotating actuators

599

Fig. 5.93. 3-PacsPn2csR-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pacs A Pn2cs||R

600

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.94. 3-PaPn2RR*-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 18, limb topology Pa A Pn2||R A R*

5.2 Derived solutions with rotating actuators

601

Fig. 5.95. 3-PacsPn2csR-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pacs A Pn2cs||R

602

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.96. 3-PaPn3R*-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 18, limb topology Pa A Pn3 A R*

5.2 Derived solutions with rotating actuators

603

Fig. 5.97. 3-PacsPn3cs-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pacs A Pn3cs

604

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.98. 3-PaPn3R*-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 18, limb topology Pa A Pn3 A R*

5.2 Derived solutions with rotating actuators

605

Fig. 5.99. 3-PacsPn3cs-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pacs A Pn3cs

606

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.100. 3-PaccRRR*-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 6, limb topology Pacc A R||R A R*

5.2 Derived solutions with rotating actuators

607

Fig. 5.101. 3-PasccRR-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology Pascc A R||R

608

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.102. 3-RPassP-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology R A Pass A P

5.2 Derived solutions with rotating actuators

609

Fig. 5.103. 3-RRC*Pa-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R||R A C* A A Pa

610

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.104. 3-RRPPacs-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology R||R A P A A Pacs

5.2 Derived solutions with rotating actuators

611

Fig. 5.105. 3-RRPaPacs-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology R||R A Pa||Pacs

612

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.106. 3-RRPassPa-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology R||R A Pass||Pa

5.2 Derived solutions with rotating actuators

613

Fig. 5.107. 3-RPassPa-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology R A Pass||Pa

614

5 Overconstrained T3-type TPMs with uncoupled motions

Fig. 5.108. 3-RRPaPass-type overconstrained TPMs with uncoupled motions defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology R||R A Pa||Pass

6 Non overconstrained T3-type TPMs with uncoupled motions

Equation (1.15) indicates that non overconstrained solutions of T3-type TPMs with uncoupled motions and q independent loops meet the condition p ¦ 1 fi 3 6q along with SF = 3, (RF) = (v1,v2,v3) and NF = 0. They could have identical limbs or limbs with different structures and could be actuated by linear or rotating motors. Each operational velocity given by Eq. (1.19) depends on just one actuated joint velocity: vi vi ( &qi ) , i = 1,2,3. The Jacobian matrix in Eq. (1.19) is a diagonal matrix. They can be actuated by linear or rotating actuators which can be mounted on the fixed base or on a moving link. In the solutions presented in this section, the actuators are associated with a revolute joint mounted on the fixed base.

6.1 Basic solutions with rotating actuators In the basic non overconstrained TPMs with rotating actuators and uncoupled motions F m G1–G2–G3, the moving platform n Ł nGi (i = 1, 2, 3) is connected to the reference platform 1 Ł 1Gi Ł 0 by three limbs with five degrees of connectivity. No idle mobilities exist in these basic solutions. The various types of limbs with five degrees of connectivity and no idle mobilities are systematized in Fig. 6.1. They are simple kinematic chains actuated by rotating motors mounted on the fixed base. Various solutions of TPMs with uncoupled motions and no idle mobilities can be obtained by using three limbs with identical or different topologies presented in Fig. 6.1. We only show solutions with identical limb type as illustrated in Figs. 6.2–6.4. The actuated revolute joints adjacent to the fixed base in the three limbs have orthogonal directions (Figs. 6.2–6.4). The structural parameters of these solutions are presented in Table 6.1.

615 G. Gogu, Structural Synthesis of Parallel Robots: Part 2: Translational Topologies with Two and Three Degrees of Freedom, Solid Mechanics and Its Applications 159, 615–685. © Springer Science + Business Media B.V. 2009

616

6 Non overconstrained T3-type TPMs with uncoupled motions

Table 6.1. Structural parametersa of translational parallel mechanisms in Figs. 6.2–6.4 No. Structural Solution parameter 3-RRRRP, 3-RRPRR (Fig. 6.2a, b)

1 2 3 4 5 6 7 8 9 10

m p1 p2 p3 p q k1 k2 k (RG1)

14 5 5 5 15 2 3 0 3 ( v1 , v2 , v3 , ȦD , ȦE )

3-RRPRR (Fig. 6.3b) 3-RRRPR, 3-RRRRR (Fig. 6.4a, b) 11 14 4 5 4 5 4 5 12 15 2 2 3 3 0 0 3 3 ( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦD , ȦG )

11

(RG2)

( v1 , v2 , v3 , ȦE , ȦG )

( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE )

12

(RG3)

( v1 , v2 , v3 , ȦD , ȦG )

( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦE , ȦG )

13 14 15 16 17 18 19 20 21 22

SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

23 24 25 26 27 28 29

SF rl rF MF NF TF

5 5 5 0 0 0 5 5 5 ( v1 ,v2 ,v3 ) 3 0 12 3 0 0 5

5 5 5 0 0 0 5 5 5 ( v1 ,v2 ,v3 ) 3 0 12 3 0 0 5

5 5 5 0 0 0 5 5 5 ( v1 ,v2 ,v3 ) 3 0 12 3 0 0 5

fj

5

5

5

fj

5

5

5

fj

15

15

15

30 31 32 a

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

3-RCRR (Fig. 6.3a)

See footnote of Table 2.1 for the nomenclature of structural parameters

6.1 Basic solutions with rotating actuators

617

Fig. 6.1. Simple limbs for non overconstrained TPMs with uncoupled motions defined by MG = SG = 5, (RG) = ( v1 , v2 , v3 , Ȧ1 , Ȧ2 ) and actuated by rotating motors mounted on the fixed base

618

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.2. Non overconstrained TPMs with uncoupled motions of types 3-RRRRP (a) and 3-RRPRR (b), limb topology R||R A R||R A ||P (a) and R||R||P A R||R (b)

6.1 Basic solutions with rotating actuators

619

Fig. 6.3. Non overconstrained TPMs with uncoupled motions of types 3-RCRR (a) and 3-RRPRR (b), limb topology R||C A R||R (a) and R||R A P A A R||R (b)

620

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.4. Non overconstrained TPMs with uncoupled motions of types 3-RRRPR (a) and 3-RRRRR (b), limb topology R||R A R A P A ||R (a) and R||R A R||R||R (b)

6.2 Derived solutions with rotating actuators

621

6.2 Derived solutions with rotating actuators Non overconstrained solutions F m G1–G2–G2 with rotating actuators and uncoupled motions can be derived from the overconstrained solutions presented in Figs. 5.6–5.48 by introducing the required idle mobilities to obtain SF = 3, (RF) = (v1,v2,v3) and NF = 0. For example, the non overconstrained solution in Fig. 6.5a is derived from the overconstrained solution in Fig. 5.6a by replacing, in each limb, two revolute joints by spherical joints in the parallelogram loop and a prismatic joint by a cylindrical joint. We note that the two spherical joints adjacent to link 4 make the parallelogram loop non overconstrained and provide an idle rotational mobility of link 4. An idle mobility of rotation is combined in each cylindrical joint denoted by C*. The limb topology and connecting conditions of the solutions Figs. 6.5– 6.54 are systematized in Table 6.2 and the structural parameters of these solutions are presented in Tables 6.3–6.13.

622

6 Non overconstrained T3-type TPMs with uncoupled motions

Table 6.2. Limb topology of the derived non overconstrained TPMs with idle mobilities and linear actuators mounted on the fixed base presented in Figs. 6.5–6.54 No. Basic TPM type 1

3-PaPP (Fig. 5.6)

NF 15

2 3

3-PaPP (Fig. 5.7)

15

4 5

3-PaPaP (Fig. 24 5.8a)

6 7

3-PaPaP (Fig. 24 5.9)

8 9

3-PaPPa (Fig. 24 5.10)

10 11

3-PaPPa (Fig. 24 5.11)

12 13

3-PaPaPa (Fig. 5.12)

33

14 15

3-PaPatP (Fig. 5.14)

24

16 17

3-PaccP (Fig. 5.16)

12

18 19 20

3-PaccPa (Fig. 5.17)

21

Derived TPM with NF = 0 type 3-PassPC* (Fig. 6.5a) 3-PacsC*C* (Fig. 6.5b) PassPC* (Fig. 6.6a) 3-PacsC*C* (Fig. 6.6b) 3-PassPassP (Fig. 6.7a) 3-PacsPacsPR*R* (Fig. 6.7b) 3-PassPassP (Fig. 6.8a) 3-PacsPacsPR*R* (Fig. 6.8b) 3-PassPPass (Fig. 6.9a) 3-PacsPPacsR*R* (Fig. 6.9b) 3-PassPPass (Fig. 6.10a) 3-PacsPPacsR*R* (Fig. 6.10b) 3-PassPacsPass (Fig. 6.11) 3-PacsPacsPacsR*R* (Fig. 6.12) 3-PacsPatcsC* (Fig. 6.13a) 3-PacsPatcsC*R* (Fig. 6.13b) 3-PasccC*R* (Fig. 6.14a) 3-PasccC*R* (Fig. 6.14b) 3-PasccPassR* (Fig. 6.15) 3-PasccPacsR*R* (Fig. 6.16)

Limb topology Pass||P A C* Pacs||C* A A C* Pass A P A ||C* Pacs A C* A ||C* Pass A Pass A ||P Pacs A Pacs A ||P A A R A ||R Pass A Pass A A P Pacs A Pacs A A P A A R* A ||R* Pass||P A A Pass Pacs||P A Pacs A ||R* A A R* Pass A P A A Pass Pacs A P A A Pacs A ||R* A A R* Pass A Pacs||Pass Pacs A Pacs||Pacs A ||R* A A R* Pacs||Patss||C* 3 Pacs||Patcs||C* A R* Pascc A C* A A R* 3-Pascc A C* A ||R* Pascc A Pass||R* Pascc A Pacs||R* A A R*

6.2 Derived solutions with rotating actuators Table 6.2. (cont.) 21 22 23

3-RRPP (Fig. 5.18a) 3-RCP (Fig. 5.19a) 3-PaRRP (Fig. 5.20)

3 3 12

24 25

3-PaRRP (Fig. 12 5.21)

26 27

3-PaRPR (Fig. 5.22)

12

28 29 30

3-PaRRR (Fig. 5.23)

12

31 32

3-PaPRR (Fig. 5.24)

12

33 34

3-PaPRR (Fig. 5.25)

12

35 36

3-PaPaRR (Fig. 5.26)

21

37 38

3-PaRRPa (Fig. 5.27)

21

39 40 41

3-PaRRbR (Fig. 5.28)

21

3-RRC*P (Fig. 6.17) 3-RCC* (Fig. 6.18) 3-PacsRRC* (Fig. 6.19a) 3-PacsRRC* (Fig. 6.19b) 3-PassRC* (Fig. 6.20a) 3-PassRC* (Fig. 6.20b) 3-PacsRC*R (Fig. 6.21a) 3-PacsRC*R (Fig. 6.21b) 3-PacsRPRR* (Fig. 6.22) 3-PacsRRRR* (Fig. 6.23) 3-PassRRR* (Fig. 6.24) 3-PacsC*RR (Fig. 6.25a) 3-PacsPRRR* (Fig. 6.26) 3-Pa csC*RR (Fig. 6.25b) 3-PacsPRRR* (Fig. 6.27) 3-PacsPacsRRR* (Fig. 6.28) 3-PacsPacsRRR* (Fig. 6.29) 3-PacsRRPass (Fig. 6.30a) 3-PassRPass (Fig. 6.30b) 3-PassRbcsRR* (Fig. 6.31) 3-Pa csR*RRbcsR (Fig. 6.32)

R||R A C* A ||P R||C A C* Pacs A R||R A ||C* Pacs A R||R A A C* Pass A R A ||C* Pass A R A A C* Pacs A R A C* A ||R Pass||C* A R Pacs A R A P A ||R A R* Pacs A R||R||R A R* Pass A R||R A R* Pacs||C* A R||R Pacs||P A R||R A R* Pacs A C* A A R||R Pacs A P A A R||R A R* Pacs A Pacs||R||R A ||R* Pacs A Pacs||R||R A R* Pacs A R||R||Pass Pass A R||Pass Pass A Rbcs||R A R* Pacs||R* A R||Rbcs||R

623

624

6 Non overconstrained T3-type TPMs with uncoupled motions Table 6.2. (cont.)

42

3-PaRRbRbR 30 (Fig. 5.29)

43 44 45 46 47 48 49

3-PaPn2R (Fig. 5.30) 3-PaPn2R (Fig. 5.31) 3-PaPn3 (Fig. 5.32) 3-PaPn3 (Fig. 5.32) 3-PaccRR (Fig. 5.34) 3-RRPaP (Fig. 5.35)

21 21 21 21 9 12

50 51

3-RRPPa (Fig. 5.38)

12

52 53 54

3-RRPaPa (Fig. 5.39) 3-RRPaPa (Fig. 5.40)

21 21

55 56 57 58

59

60 61 62

3-RRRRbR (Fig. 5.41) 3-RRRRbRbR (Fig. 5.42) 3-RRPn2R (Figs. 5.43, 5.44) 3-RRPn3 (Figs. 5.45, 5.46) 3-RRRRPa (Fig. 5.47) 3-RRPaRR (Fig. 5.48a) 3-RRPaRR (Fig. 5.48b)

9 18 9

9

9 9 9

3-PassRbcsRbcsRR* (Fig. 6.33) 3-PacsR*RRbcsRbcsR (Fig. 6.34) 3-PacsPn2csRR* (Fig. 6.35) 3-PacsPn2csRR* (Fig. 6.36) 3-PacsPn3csR* (Fig. 6.37) 3-PacsPn3csR* (Fig. 6.38) 3-PasccRRR* (Fig. 6.39) 3-RPassC* (Fig. 6.40) 3-RRPassP (Fig. 6.41) 3-RCPass (Fig. 6.42) 3-RRC*Pacs (Fig. 6.43) 3-RRPassPacs (Fig. 6.44) 3-RRPacsPass (Fig. 6.45) 3-RPassPass (Fig. 6.46) 3-RRRRbcsR (Fig. 6.47) 3-RRRRbcsRbcsR (Fig. 6.48) 3-RRPn2csR (Figs. 6.49, 6.50) 3-RRPn3cs (Figs. 6.51, 6.52) 3-RRRRPacs (Fig. 6.53) 3-RRPacsRR (Fig. 6.54a) 3-RPassRR (Fig. 6.54b)

Pass A Rbcs||Rbcs||R A R* Pacs||R A R||Rbcs||Rbcs||R Pacs A Pn2cs||R A R* Pacs A Pn2cs||R A R* Pacs A Pn3cs A R* Pacs A Pn3cs A R* Pascc A R||R A R* R A Pass A A C* R||R A Pass A ||P R||C A Pass R||R A C* A A Pacs R||R A Pass||Pacs R||R A Pacs||Pass R A Pass||Pass R||R A R||Rbcs||R R||R A R||Rbcs||Rbcs||R R||R A Pn2cs||R R||R A Pn3cs R||R A R||R||Pacs R||R A Pacs||R||R R A Pass||R||R

6.2 Derived solutions with rotating actuators

625

Table 6.3. Bases of the operational velocities spaces of the limbs isolated from the parallel mechanisms presented in Figs. 6.5–6.54 No. Parallel mechanism 1 Figs. 6.5a, 6.6a, 6.9a, 6.10a, 6.11, 6.14a, 6.21, 6.30, 6.47, 6.48, 6.53, 6.54a 2 Figs. 6.5b, 6.6b, 6.7b, 6.8b, 6.9b, 6.10b, 6.12, 6.13b, 6.14b, 6.17, 6.18, 6.40–6.46 3 Fig. 6.7a, 6.8a, 6.13a, 6.15, 6.16, 6.19, 6.20, 6.22–6.29, 6.31–6.39, 6.49–6.52, 6.54b

Basis (RG2) (RG3) (RG1) ( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG )

( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE )

( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG )

626

6 Non overconstrained T3-type TPMs with uncoupled motions

Table 6.4. Structural parametersa of translational parallel mechanisms in Figs. 6.5–6.10 No. Structural parameter

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

Solution 3-PassPC* 3-PacsC*C* (Fig. 6.5a, b) 3-PassPC* 3-PacsC*C* (Fig. 6.6a, b) 14 6 6 6 18 5 0 3 3 See Table 6.3

3-PassPassP (Figs. 6.7a, 6.8a) 3-PassPPass (Figs. 6.9a, 6.10a) 20 9 9 9 27 8 0 3 3 See Table 6.3

26 11 11 11 33 8 0 3 3 See Table 6.3

5 5 5 6 6 6 5 5 5 ( v1 ,v2 ,v3 ) 3 18 30 3 0 0 11

5 5 5 12 12 12 5 5 5 ( v1 ,v2 ,v3 ) 3 36 48 3 0 0 17

5 5 5 12 12 12 5 5 5 ( v1 ,v2 ,v3 ) 3 36 48 3 0 0 17

fj

11

17

17

fj

11

17

17

fj

33

51

51

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

3-PacsPacsPR*R* (Figs. 6.7b, 6.8b) 3-PacsPPacsR*R* (Figs. 6.9b, 6.10b)

See footnote of Table 2.1 for the nomenclature of structural parameters

6.2 Derived solutions with rotating actuators

627

Table 6.5. Structural parametersa of translational parallel mechanisms in Figs. 6.11 and 6.12 No.

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

Structural parameter

Solution 3-PassPacsPass (Fig. 6.11)

3-PacsPacsPacsR*R* (Fig. 6.12)

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

26 12 12 12 36 11 0 3 3 See Table 6.3

32 14 14 14 42 11 0 3 3 See Table 6.3

5 5 5 18 18 18 5 5 5 ( v1 ,v2 ,v3 ) 3 54 66 3 0 0 23

5 5 5 18 18 18 5 5 5 ( v1 ,v2 ,v3 ) 3 54 66 3 0 0 23

fj

23

23

fj

23

23

fj

69

69

SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

628

6 Non overconstrained T3-type TPMs with uncoupled motions

Table 6.6. Structural parametersa of translational parallel mechanisms in Figs. 6.13–6.16 No. Structural parameter

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

Solution PacsPatssC* (Fig. 6.13a) 3-PasccPassR* (Fig. 6.15) 20 9 9 9 27 8 0 3 3 See Table 6.3

3-PacsPatssC*R* (Fig. 6.13b) 3-PasccPacsR*R* (Fig. 6.16) 23 10 10 10 30 8 0 3 3 See Table 6.3

14 6 6 6 18 5 0 3 3 See Table 6.3

5 5 5 12 12 12 5 5 5 ( v1 ,v2 ,v3 ) 3 36 48 3 0 0 17

5 5 5 12 12 12 5 5 5 ( v1 ,v2 ,v3 ) 3 36 48 3 0 0 17

5 5 5 6 6 6 5 5 5 ( v1 ,v2 ,v3 ) 3 18 30 3 0 0 11

fj

17

17

11

fj

17

17

11

fj

51

51

33

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

3-PasccC*R* (Fig. 6.14)

See footnote of Table 2.1 for the nomenclature of structural parameters

6.2 Derived solutions with rotating actuators

629

Table 6.7. Structural parametersa of translational parallel mechanisms in Figs. 6.17–6.19 No. Structural parameter 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

Solution 3-RRC*P (Fig. 6.17) 11 4 4 4 12 2 3 0 3 See Table 6.3

3-RCC* (Fig. 6.18) 8 3 3 3 9 2 3 0 3 See Table 6.3

3-PacsRRC* (Fig. 6.19) 17 7 7 7 21 5 0 3 3 See Table 6.3

5 5 5 0 0 0 5 5 5 ( v1 ,v2 ,v3 ) 3 0 12 3 0 0 5

5 5 5 0 0 0 5 5 5 ( v1 ,v2 ,v3 ) 3 0 12 3 0 0 5

5 5 5 6 6 6 5 5 5 ( v1 ,v2 ,v3 ) 3 18 30 3 0 0 11

fj

5

5

11

fj

5

5

11

fj

15

15

33

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

630

6 Non overconstrained T3-type TPMs with uncoupled motions

Table 6.8. Structural parametersa of translational parallel mechanisms in Figs. 6.20–6.27 No. Structural parameter

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

14 6 6 6 18 5 0 3 3 See Table 6.3

3-PacsRC*R (Fig. 6.21a) 3-PassRRR* (Fig. 6.24) 3-PacsC*RR (Fig. 6.25) 17 7 7 7 21 5 0 3 3 See Table 6.3

3-PacsRPRR* (Fig. 6.22) 3-PacsRRRR* (Fig. 6.23) 3-PacsPRRR* (Figs. 6.26, 6.27) 20 8 8 8 24 5 0 3 3 See Table 6.3

5 5 5 6 6 6 5 5 5 ( v1 ,v2 ,v3 ) 3 18 30 3 0 0 11

5 5 5 6 6 6 5 5 5 ( v1 ,v2 ,v3 ) 3 18 30 3 0 0 11

5 5 5 6 6 6 5 5 5 ( v1 ,v2 ,v3 ) 3 18 30 3 0 0 11

fj

11

11

11

fj

11

11

11

fj

33

33

33

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

Solution 3-PassRC* (Fig. 6.20) 3-PassC*R (Fig. 6.21b)

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

6.2 Derived solutions with rotating actuators

631

Table 6.9. Structural parametersa of translational parallel mechanisms in Figs. 6.28–6.32 No. Structural parameter

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

Solution 3-PacsPacsRRR* (Figs. 6.28, 6.29) 3-PassR*RRbcsR (Fig. 6.32) 26 11 11 11 33 8 0 3 3 See Table 6.3

3-PacsRRPass (Fig. 6.30a) 3-PassRbcsRR* (Fig. 6.31) 23 10 10 10 30 8 0 3 3 See Table 6.3

20 9 9 9 27 8 0 3 3 See Table 6.3

5 5 5 12 12 12 5 5 5 ( v1 ,v2 ,v3 ) 3 36 48 3 0 0 17

5 5 5 12 12 12 5 5 5 ( v1 ,v2 ,v3 ) 3 36 48 3 0 0 17

5 5 5 12 12 12 5 5 5 ( v1 ,v2 ,v3 ) 3 36 48 3 0 0 17

fj

17

17

17

fj

17

17

17

fj

51

51

51

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

3-PassRPass (Fig. 6.30b)

See footnote of Table 2.1 for the nomenclature of structural parameters

632

6 Non overconstrained T3-type TPMs with uncoupled motions

Table 6.10. Structural parametersa of translational parallel mechanisms in Figs. 6.33–6.38 No. Structural parameter

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

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

5 5 5 18 18 18 5 5 5 ( v1 ,v2 ,v3 ) 3 54 66 3 0 0 23

5 5 5 18 18 18 5 5 5 ( v1 ,v2 ,v3 ) 3 54 66 3 0 0 23

5 5 5 12 12 12 5 5 5 ( v1 ,v2 ,v3 ) 3 36 48 3 0 0 17

fj

23

23

17

fj

23

23

17

fj

69

69

51

SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

Solution 3-PassRbcsRbcsRR* 3-PacsR*RRbcsRbcsR 3-PacsPn2csRR* (Fig. 6.33) (Fig. 6.34) (Figs. 6.35, 6.36) 3-PacsPn3csR* (Figs. 6.37, 6.38) 29 32 26 13 14 11 13 14 11 13 14 11 39 42 33 11 11 8 0 0 0 3 3 3 3 3 3 See Table 6.3 See Table 6.3 See Table 6.3

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

6.2 Derived solutions with rotating actuators

633

Table 6.11. Structural parametersa of translational parallel mechanisms in Figs. 6.39–6.45 No. Structural parameter

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

Solution 3-PasccRRR* (Fig. 6.39) 3-RRPassP (Fig. 6.41) 3-RRC*Pacs (Fig. 6.43) 17 7 7 7 21 5 0 3 3 See Table 6.3

3-RPassC* (Fig. 6.40) 3-RCPass (Fig. 6.42)

3-RRPassPacs (Fig. 6.44) 3-RRPacsPass (Fig. 6.45)

14 6 6 6 18 5 0 3 3 See Table 6.3

23 10 10 10 30 8 0 3 3 See Table 6.3

5 5 5 6 6 6 5 5 5 ( v1 ,v2 ,v3 ) 3 18 30 3 0 0 11

5 5 5 6 6 6 5 5 5 ( v1 ,v2 ,v3 ) 3 18 30 3 0 0 11

5 5 5 12 12 12 5 5 5 ( v1 ,v2 ,v3 ) 3 36 48 3 0 0 17

fj

11

11

17

fj

11

11

17

fj

33

33

51

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

634

6 Non overconstrained T3-type TPMs with uncoupled motions

Table 6.12. Structural parametersa of translational parallel mechanisms in Figs. 6.46–6.52 No. Structural parameter

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

20 9 9 9 27 8 0 3 3 See Table 6.3

3-RRRRbcsR (Fig. 6.47) 3-RRPn2csR (Figs. 6.49, 6.50) 3-RRPn3cs (Figs. 6.51, 6.52) 20 8 8 8 24 5 0 3 3 See Table 6.3

26 11 11 11 33 8 0 3 3 See Table 6.3

5 5 5 12 12 12 5 5 5 ( v1 ,v2 ,v3 ) 3 36 48 3 0 0 17

5 5 5 6 6 6 5 5 5 ( v1 ,v2 ,v3 ) 3 18 30 3 0 0 11

5 5 5 12 12 12 5 5 5 ( v1 ,v2 ,v3 ) 3 36 48 3 0 0 17

fj

17

11

17

fj

17

11

17

fj

51

33

51

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

Solution 3-RPassPass (Fig. 6.46)

fj

3-RRRRbcsRbcsR (Fig. 6.48)

See footnote of Table 2.1 for the nomenclature of structural parameters

6.2 Derived solutions with rotating actuators

635

Table 6.13. Structural parametersa of translational parallel mechanisms in Figs. 6.53 and 6.54 No.

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

Structural parameter

Solution 3-RRRRPacs (Fig. 6.53) 3-RR PacsRR (Fig. 6.54a) 20 8 8 8 24 5 0 3 3 See Table 6.3

17 7 7 7 21 5 0 3 3 See Table 6.3

5 5 5 6 6 6 5 5 5 ( v1 ,v2 ,v3 ) 3 18 30 3 0 0 11

5 5 5 6 6 6 5 5 5 ( v1 ,v2 ,v3 ) 3 18 30 3 0 0 11

fj

11

11

fj

11

11

fj

33

33

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

3-RPassRR (Fig. 6.54b)

See footnote of Table 2.1 for the nomenclature of structural parameters

636

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.5. Non overconstrained TPMs with uncoupled motions of types 3-PassPC* (a) and 3-PacsC*C* (b), limb topology Pass||P A A C* (a) and Pacs||C* A A C* (b)

6.2 Derived solutions with rotating actuators

637

Fig. 6.6. Non overconstrained TPMs with uncoupled motions of types 3-PassPC* (a) and 3-PacsC*C* (b), limb topology Pass A P A ||C* (a) and Pacs A C* A ||C* (b)

638

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.7. Non overconstrained TPMs with uncoupled motions of types 3-PassPassP (a) and 3-PacsPacsPR*R* (b), limb topology Pass A Pass A ||P (a) and Pacs A Pacs A ||P A A R A ||R (b)

6.2 Derived solutions with rotating actuators

639

Fig. 6.8. Non overconstrained TPMs with uncoupled motions of types 3PassPassP (a) and 3-PacsPacsPR*R* (b), limb topology Pass A Pass A A P (a) and Pacs A Pacs A A P A A R* A ||R* (b)

640

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.9. Non overconstrained TPMs with uncoupled motions of types 3PassPPass (a) and 3-PacsPPacsR*R* (b), limb topology Pass||P A Pass (a) and Pacs||P A Pacs A ||R* A A R* (b)

6.2 Derived solutions with rotating actuators

641

Fig. 6.10. Non overconstrained TPMs with uncoupled motions of types 3PassPPass (a) and 3-PacsPPacsR*R* (b), limb topology Pass A P A A Pass (a) and Pacs A P A A Pacs A ||R* A A R* (b)

642

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.11. 3-PassPacsPass-type non overconstrained TPM with uncoupled motions, limb topology Pass A Pacs||Pass

6.2 Derived solutions with rotating actuators

643

Fig. 6.12. 3-PacsPacsPacsR*R*-type non overconstrained TPM with uncoupled motions, limb topology Pacs A Pacs||Pacs A ||R* A A R*

644

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.13. Non overconstrained TPMs with uncoupled motions of types 3-Pacs PatssC* (a) and 3-PacsPatcsC*R* (b), limb topology Pacs||Patss||C* (a) and Pacs|| Patcs||C* A R* (b)

6.2 Derived solutions with rotating actuators

645

Fig. 6.14. 3-PasccC*R*-type non overconstrained TPMs with uncoupled motions, limb topology Pascc A C* A A R* (a) and Pascc A C* A ||R* (b)

646

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.15. 3-PasccPassR*-type non overconstrained TPM with uncoupled motions, limb topology Pascc A Pass||R*

6.2 Derived solutions with rotating actuators

647

Fig. 6.16. 3-PasccPacsR*R*-type non overconstrained TPM with uncoupled motions, limb topology Pascc A Pacs||R* A A R*

648

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.17. 3-RRC*P-type non overconstrained TPMs with uncoupled motions, limb topology R||R A C* A ||P

6.2 Derived solutions with rotating actuators

649

Fig. 6.18. 3-RCC*-type non overconstrained TPMs with uncoupled motions, limb topology R||C A C*

650

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.19. 3-PacsRRC*-type non overconstrained TPMs with uncoupled motions, limb topology Pacs A R||R A ||C* (a) and Pacs A R||R A A C* (b)

6.2 Derived solutions with rotating actuators

651

Fig. 6.20. 3-PassRC*-type non overconstrained TPMs with uncoupled motions, limb topology Pass A R A ||C* (a) and Pass A R A A C* (b)

652

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.21. Non overconstrained TPMs with uncoupled motions of types 3-PacsRC*R (a) and 3-PassC*R (b), limb topology Pacs A R A C* A ||R (a) and Pass||C* A R (b)

6.2 Derived solutions with rotating actuators

653

Fig. 6.22. 3-PacsRPRR*-type non overconstrained TPMs with uncoupled motions, limb topology Pacs A R A P A ||R A R*

654

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.23. 3-PacsRRRR*-type non overconstrained TPMs with uncoupled motions, limb topology Pacs A R||R||R A R*

6.2 Derived solutions with rotating actuators

655

Fig. 6.24. 3-PassRRR*-type non overconstrained TPMs with uncoupled motions, limb topology Pass A R||R A R*

656

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.25. 3-PacsC*RR-type non overconstrained TPMs with uncoupled motions, limb topology Pacs||C* A R||R (a) and Pacs A C* A A R||R (b)

6.2 Derived solutions with rotating actuators

657

Fig. 6.26. 3-PacsPRRR*-type non overconstrained TPMs with uncoupled motions, limb topology Pacs||P A R||R A R*

658

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.27. 3-PacsPRRR*-type non overconstrained TPMs with uncoupled motions, limb topology Pacs || P A R||R A R*

6.2 Derived solutions with rotating actuators

659

Fig. 6.28. 3-PacsPacsRRR*-type non overconstrained TPM with uncoupled motions, limb topology Pacs A Pacs||R||R A ||R* and the actuated joints with orthogonal axes

660

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.29. 3-PacsPacsRRR*-type non overconstrained TPM with uncoupled motions, limb topology Pacs A Pacs||R||R A R* and the axes of the actuated joints parallel to two orthogonal directions

6.2 Derived solutions with rotating actuators

661

Fig. 6.30 Non overconstrained TPMs with uncoupled motions of types 3PacsRRPass (a) and 3-PassRPass (b), limb topology Pacs A R||R||Pass (a) and Pass A R||Pass (b)

662

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.31. 3-PassRbcsRR*-type non overconstrained TPM with uncoupled motions, limb topology Pass A Rbcs||R A R*

6.2 Derived solutions with rotating actuators

663

Fig. 6.32. 3-PacsR*RRbcsR-type non overconstrained TPM with uncoupled motions, limb topology Pacs||R* A R||Rbcs||R

664

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.33. 3-PassRbcsRbcsRR*-type non overconstrained TPM with uncoupled motions, limb topology Pass A Rbcs||Rbcs||R A R*

6.2 Derived solutions with rotating actuators

665

Fig. 6.34. 3-PacsR*RRbcsRbcsR-type non overconstrained TPM with uncoupled motions, limb topology Pacs||R A R||Rbcs||Rbcs||R

666

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.35. 3-PacsPn2csRR*-type non overconstrained TPM with uncoupled motions, limb topology Pacs A Pn2cs||R A R*

6.2 Derived solutions with rotating actuators

667

Fig. 6.36. 3-PacsPn2csRR*-type non overconstrained TPM with uncoupled motions, limb topology Pacs A Pn2cs||R A R*

668

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.37. 3-PacsPn3csR*-type non overconstrained TPM with uncoupled motions, limb topology Pacs A Pn3cs A R*

6.2 Derived solutions with rotating actuators

669

Fig. 6.38. 3-PacsPn3csR*-type non overconstrained TPM with uncoupled motions, limb topology Pacs A Pn3cs A R*

670

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.39. 3-PasccRRR*-type non overconstrained TPMs with uncoupled motions, limb topology PacsA Pn3csA R *

6.2 Derived solutions with rotating actuators

671

Fig. 6.40. 3-RPassC*-type non overconstrained TPMs with uncoupled motions, limb topology R A Pass A A C*

672

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.41. 3-RRPassP-type non overconstrained TPMs with uncoupled motions, limb topology R||R A Pass A ||P

6.2 Derived solutions with rotating actuators

673

Fig. 6.42. 3-RCPass-type non overconstrained TPMs with uncoupled motions, limb topology R||C A Pass

674

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.43. 3-RRC*Pacs-type non overconstrained TPMs with uncoupled motions, limb topology R||R A C* A A Pacs

6.2 Derived solutions with rotating actuators

675

Fig. 6.44. 3-RRPassPacs-type non overconstrained TPMs with uncoupled motions, limb topology R||R A Pass||Pacs

676

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.45 3-RRPacsPass-type non overconstrained TPMs with uncoupled motions, limb topology R||R A Pacs||Pass

6.2 Derived solutions with rotating actuators

677

Fig. 6.46 3-RPassPass-type non overconstrained TPMs with uncoupled motions, limb topology R A Pass||Pass

678

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.47. 3-RRRRbcsR-type non overconstrained TPM with uncoupled motions, limb topology R||R A R||Rbcs||R

6.2 Derived solutions with rotating actuators

679

Fig. 6.48. 3-RRRRbcsRbcsR-type non overconstrained TPM with uncoupled motions, limb topology R||R A R||Rbcs||Rbcs||R

680

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.49. 3-RRPn2csR-type non overconstrained TPM with uncoupled motions, limb topology R||R A Pn2cs||R

6.2 Derived solutions with rotating actuators

681

Fig. 6.50. 3-RRPn2csR-type non overconstrained TPM with uncoupled motions, limb topology R||R A Pn3cs||R

682

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.51. 3-RRPn3cs-type non overconstrained TPM with uncoupled motions, limb topology R||R A Pn3cs

6.2 Derived solutions with rotating actuators

683

Fig. 6.52. 3-RRPn3cs-type non overconstrained TPM with uncoupled motions, limb topology R||R A Pn3cs

684

6 Non overconstrained T3-type TPMs with uncoupled motions

Fig. 6.53. 3-RRRRPacs-type non overconstrained TPM with uncoupled motions, limb topology R||R A R||R ||Pacs

6.2 Derived solutions with rotating actuators

685

Fig. 6.54. Non overconstrained TPM with uncoupled motions of types 3-RRPacsRR (a) and 3-RPassRR (b), limb topology R||R A Pacs||R||R (a) and R A Pass||R||R (b)

7 Maximally regular T3-type translational parallel robots

Maximally regular T3-type translational parallel robots are actuated by linear motors and can have various degrees of overconstraint. In these solutions, the three operational velocities are equal to their corresponding actuated joint velocities: v1 &q1 , v2 &q2 and v3 &q3 . The Jacobian matrix in Eq. (1.19) is the identity matrix. We call Isoglide3-T3 the translational parallel mechanisms of this family.

7.1 Overconstrained solutions Equation (1.16) indicates that overconstrained solutions of maximally regular T3-type translational parallel robots with q independent loops meet p the condition ¦ 1 f i 3 6q . Various solutions fulfil this condition along with MF = SF = 3 and (RF) = (v1,v2,v3). They may have identical limbs or limbs with different topologies. We limit our presentation in this section to the solutions with just three identical limbs. 7.1.1 Basic solutions with no idle mobilities In the basic solutions of overconstrained maximally regular T3-type translational parallel robots, F m G1–G2–G3, the moving platform n Ł nGi (i = 1, 2, 3) is connected to the reference platform 1 Ł 1Gi Ł 0 by three simple or complex limbs with three (Figs. 7.1a, 7.2a–c) or four (Figs. 7.1b–e, 7.2d–k) degrees of connectivity. The complex limbs combine one (Fig. 7.2a, b, d–f, h–k) or two (Fig. 7.2c, g) planar closed loops of types Pa (Fig. 7.2a–e), Rb (Fig. 7.2f, g) Pn2 (Fig. 7.2h, i) and Pn3 (Fig. 7.2j, k). No idle mobilities exist in these basic solutions. The planar loops with two and three degrees of freedom Pn2-type illustrated in Fig. 7.2h, i are of types R||R||R||R||R and R A P A ||R A P A ||R. The planar loops with three degrees of freedom Pn3-type illustrated in 687 G. Gogu, Structural Synthesis of Parallel Robots: Part 2: Translational Topologies with Two and Three Degrees of Freedom, Solid Mechanics and Its Applications 159, 687–748. © Springer Science + Business Media B.V. 2009

688

7 Maximally regular T3-type translational parallel robots

Fig. 7.2j, k are of types R||R||R||R||R||R and R A P A ||R||R A P A ||R. Other planar loops of types Pn2 and Pn3 can also be used (see Table 5.1). Various solutions of maximally regular T3-type translational parallel robots with no idle mobilities can be obtained by using three limbs with identical or different topologies presented in Figs. 7.1 and 7.2. We only show solutions with identical limb type as illustrated in Figs. 7.3–7.13. The limb topologies and connecting conditions in these solutions are systematized in Table 7.1 and their structural parameters in Tables 7.2 and 7.3. The limbs presented in Figs. 7.1 and 7.2 can also be used to generate basic solutions of overconstrained T3-type TPMs with uncoupled, decoupled and coupled motions. Basic solutions of T3-type TPMs with uncoupled motions and different limb topologies can be obtained by using the following combinations of limbs: (i) one limb from Fig. 7.1 or 7.2 and two limbs from Figs. 5.1–5.5 or 6.1, (ii) two limbs from Fig. 7.2 or 7.3 and one limb from Figs. 5.1–5.5 or 6.1. Basic solutions of T3-type TPMs with decoupled motions and different limb topologies can be obtained by using two limbs from Figs. 7.1 and 7.2 and one limb from Figs. 3.1–3.5, Figs. 3.60–3.70 and Figs. 4.1, 4.2 or 4.29. Basic solutions of T3-type TPMs with coupled motions and different limb topologies can be obtained by using one limb from Fig. 7.1 or 7.2 and two limbs from Figs. 3.1–3.5, Figs. 3.60–3.70 and Figs. 4.1, 4.2 or 4.29.

7.1 Overconstrained solutions

689

Table 7.1. Limb topology and connecting conditions of the overconstrained maximally regular TPM with no idle mobilities presented in Figs. 7.3–7.13 No. TPM type 1 3-PPP (Fig. 7.3a)

Limb topology P A P AA P (Fig. 7.1a)

2

3-PPP (Fig. 7.3b, c)

P A P AA P (Fig. 7.1a)

3

3-PRRP (Fig. 7.4a) 3-PRPR (Fig. 7.4b)

P||R||R A P (Fig. 7.1b) P||R A P A ||R (Fig. 7.1c)

3-PPRR (Fig. 7.5a) 3-PRRR (Fig. 7.5b) 3-PPaP (Fig. 7.6a) 3-PPaP (Fig. 7.6b) 3-PPPa (Fig. 7.7a, b) 3-PPaPa (Fig. 7.8a, b) 3-PRRPa (Fig. 7.9a, b) 3-PPaRR (Fig. 7.10a, b) 3-PRRbR (Fig. 7.11a) 3-PRRbRbR (Fig. 7.11b) 3-PPn2R (Fig. 7.12a, b) 3-PPn3 (Fig. 7.13a, b)

Pa A Pa A ||P (Fig. 7.1d) P||R||R||R (Fig. 7.1e) P||Pa A P (Fig. 7.2a) P||Pa A P (Fig. 7.2a) P A P A ||Pa (Fig. 7.2b) P||Pa||Pa (Fig. 7.2c) P||R||R||Pa (Fig. 7.2d) P||Pa||R||R (Fig. 7.2e) P||R||Rb||R (Fig. 7.2f) P||R||Rb||Rb||R (Fig. 7.2g) P||Pn2||R (Fig. 7.2h, i) P||Pn3 (Fig. 7.2j, k)

4

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

Connecting conditions The directions of the actuated prismatic joints are reciprocally orthogonal. The directions of the prismatic joints adjacent to the moving platform are also reciprocally orthogonal. The directions of the actuated prismatic joints are reciprocally orthogonal. The directions of the prismatic joints adjacent to the moving platform are parallel to two orthogonal lines. Idem No. 1 The directions of the actuated prismatic joints are reciprocally orthogonal. Idem No. 4 Idem No. 4 Idem No. 1 Idem No. 2 Idem No. 4 Idem No. 4 Idem No. 4 Idem No. 4 Idem No. 4 Idem No. 4 Idem No. 4 Idem No. 4

690

7 Maximally regular T3-type translational parallel robots

Table 7.2. Structural parametersa of translational parallel mechanisms in Figs. 7.3–7.7 No.

Structural parameter

Solution 3-PPP (Fig. 7.3)

1 2 3 4 5 6 7 8 9 10

m p1 p2 p3 p q k1 k2 k (RG1)

11

8 3 3 3 9 2 3 0 3 ( v1 ,v2 ,v3 )

3-PRRP, 3-PRPR (Fig. 7.4) 3-PPRR, 3-PRRR (Fig. 7.5) 11 4 4 4 12 2 3 0 3 ( v1 ,v2 ,v3 ,ZD )

3-PPaP (Fig. 7.6) 3-PPPa (Fig. 7.7) 14 6 6 6 18 5 0 3 3 ( v1 ,v2 ,v3 )

(RG2)

( v1 ,v2 ,v3 )

( v1 ,v 2 ,v3 ,Z E )

( v1 ,v 2 ,v3 )

12

(RG3)

13 14 15 16 17 18 19 20 21 22

SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

23 24 25 26 27 28 29

SF rl rF MF NF TF

( v1 ,v 2 ,v3 ) 3 3 3 0 0 0 3 3 3 ( v1 ,v 2 ,v3 ) 3 0 6 3 6 0 3

( v1 ,v2 ,v3 ,ZG ) 4 4 4 0 0 0 4 4 4 ( v1 ,v 2 ,v3 ) 3 0 9 3 3 0 4

( v1 ,v 2 ,v3 ) 3 3 3 3 3 3 3 3 3 ( v1 ,v 2 ,v3 ) 3 9 15 3 15 0 6

fj

3

4

6

fj

3

4

6

fj

9

12

18

30 31 32 a

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

7.1 Overconstrained solutions

691

Table 7.3. Structural parametersa of translational parallel mechanisms in Figs. 7.8–7.13 No. Structural parameter

Solution 3-PPaPa (Fig. 7.8)

1 2 3 4 5 6 7 8 9 10

m p1 p2 p3 p q k1 k2 k (RG1)

11

(RG2)

20 9 9 9 27 8 0 3 3 ( v1 ,v 2 ,v3 ) ( v1 ,v 2 ,v3 )

12

(RG3)

13 14 15 16 17 18 19 20 21 22

SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

23 24 25 26 27 28 29

SF rl rF MF NF TF

30 31 32

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

3-PRRPa (Fig. 7.9) 3-PPaRR (Fig. 7.10) 3-PRRbR (Fig. 7.11a) 3-PPn2R (Fig. 7.12a, b) 3-PPn3 (Fig. 7.13a, b) 17 7 7 7 21 5 0 3 3 ( v1 ,v2 ,v3 ,ZD )

3-PRRbRbR (Fig. 7.11b)

( v1 ,v 2 ,v3 ,Z E )

23 10 10 10 30 8 0 3 3 ( v1 ,v2 ,v3 ,ZD ) ( v1 ,v 2 ,v3 ,Z E )

( v1 ,v 2 ,v3 ) 3 3 3 6 6 6 3 3 3 ( v1 ,v 2 ,v3 ) 3 18 24 3 24 0 9

( v1 ,v2 ,v3 ,ZG ) 4 4 4 3 3 3 4 4 4 ( v1 ,v 2 ,v3 ) 3 9 18 3 12 0 7

( v1 ,v2 ,v3 ,ZG ) 4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 21 0 10

fj

9

7

10

fj

9

7

10

fj

27

21

30

fj

692

7 Maximally regular T3-type translational parallel robots

Fig. 7.1. Simple limbs for overconstrained maximally regular TPMs with no idle mobilities actuated by linear motors mounted on the fixed base, defined by MG = SG = 3, (RG) = (v1,v2,v3) – (a) and MG = SG = 4, (RG) = ( v1 ,v2 , v3 ,ȦD ) – (b–e)

7.1 Overconstrained solutions

693

Fig. 7.2. Complex limbs for overconstrained maximally regular TPMs with no idle mobilities actuated by linear motors mounted on the fixed base, defined by MG = SG = 3, (RG) = (v1,v2,v3) – (a–c) and MG = SG = 4, (RG) = ( v1 ,v2 , v3 ,ȦD ) – (d–k)

694

7 Maximally regular T3-type translational parallel robots

Fig. 7.2. (cont.)

7.1 Overconstrained solutions

695

Fig. 7.3. 3-PPP-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 6, limb topology P A P A A P

696

7 Maximally regular T3-type translational parallel robots

Fig. 7.4. Overconstrained maximally-regular TPMs of types 3-PRRP (a) and 3PRPR (b) defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ), TF = 0, NF = 3, limb topology P||R||R A P (a) and P||R A P A ||R (b)

7.1 Overconstrained solutions

697

Fig. 7.5. Overconstrained maximally-regular TPMs of types 3-PPRR (a) and 3PRRR (b) defined by MF = SF = 3, (RF) = ( v1 , v 2 , v3 ), TF = 0, NF = 3, limb topology P A P A ||R||R (a) and P||R||R||R (b)

698

7 Maximally regular T3-type translational parallel robots

Fig. 7.6. 3-PPaP-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 15, limb topology P||Pa A P

7.1 Overconstrained solutions

699

Fig. 7.7. 3-PPPa-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 15, limb topology P A P A ||Pa

700

7 Maximally regular T3-type translational parallel robots

Fig. 7.8. 3-PPaPa-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 24, limb topology P||Pa||Pa

7.1 Overconstrained solutions

701

Fig. 7.9. 3-PRRPa-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology P||R||R||Pa

702

7 Maximally regular T3-type translational parallel robots

Fig. 7.10. 3-PPaRR-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology P||Pa||R||R

7.1 Overconstrained solutions

703

Fig. 7.11. Overconstrained maximally-regular TPMs of types 3-PRRbR (a) and 3-PRRbRbR (b) defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12 (a), NF = 21 (b), limb topology P||R||Rb||R (a) and P||R||Rb||Rb||R (b)

704

7 Maximally regular T3-type translational parallel robots

Fig. 7.12. 3-PPn2R-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology P||Pn2||R

7.1 Overconstrained solutions

705

Fig. 7.13. 3-PPn3-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology P||Pn3

706

7 Maximally regular T3-type translational parallel robots

7.1.2 Derived solutions with idle mobilities Solutions with lower degrees of overconstraint can be derived from the basic solutions in Figs. 7.3–7.13 by using joints with idle mobilities. A large set of solutions can be obtained by introducing one or two rotational idle mobilities outside the closed loops that can be integrated in the limbs and up to three idle mobilities (two rotations and one translation) in each planar loop. The joints combining idle mobilities are denoted by an asterisk. Examples of solutions with identical limbs and three to twelve degrees of overconstraint derived from the basic solutions in Figs. 7.3–7.13 are illustrated in Figs. 7.14–7.32. The limb topology and the number of overconstraints in these solutions are systematized in Table 7.5 and the structural parameters in Tables 7.4 and 7.6–7.9. We recall that two idle rotational mobilities are introduced in the spherical joint of the parallelogram loops denoted by Paccs and Pascc which combine two cylindrical, one revolute and one spherical joint. In the cylindrical joint denoted by C*, the rotation is an idle mobility. One idle translational mobility and two idle rotational mobilities are introduced in the prismatic and the spherical joints of each planar loop denoted by Pacs, Pbcs, Pn2cs or Pn3cs. Table 7.4. Bases of the operational velocities spaces of the limbs isolated from the parallel mechanisms presented in Figs. 7.14–7.32 No. Parallel mechanism 1 Figs. 7.14a, 7.16a, 7.19a, 7.20a 2 Fig. 7.14b 3 4 5 6 7

8

Basis (RG1) ( v1 ,v2 , v3 ,ȦG )

(RG2) ( v1 ,v2 , v3 ,ȦD )

(RG3) ( v1 , v2 , v3 , ȦE )

( v1 , v2 , v3 , ȦE )

( v1 ,v2 , v3 ,ȦD )

( v1 , v2 , v3 , ȦE )

Figs. 7.14c, 7.16b, ( v1 ,v2 , v3 ,ȦG ) ( v1 ,v2 , v3 ,ȦG ) 7.19b, 7.20b Figs. 7.15a, 7.17a, ( v1 , v2 , v3 , ȦE ) ( v1 ,v2 , v3 ,ȦG ) 7.18a, 7.21a Fig. 7.15b ( v1 ,v2 , v3 ,ȦG ) ( v1 ,v2 , v3 ,ȦG ) Figs. 7.15c, 7.17b, ( v1 , v2 , v3 , ȦE ) ( v1 ,v2 , v3 ,ȦD ) 7.18b, 7.21b Figs. 7.22, 7.24, ( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG ) 7.26b, 7.27, 7.29a, 7.30a, 7.31 Figs. 7.23, 7.25, ( v1 ,v2 , v3 ,ȦD ) ( v1 , v2 , v3 , ȦE ) 7.26b, 7.28, .471b, 7.30b, 7.32

( v1 , v2 , v3 , ȦE ) ( v1 ,v2 , v3 ,ȦD ) ( v1 ,v2 , v3 ,ȦD ) ( v1 ,v2 , v3 ,ȦD ) ( v1 , v2 , v3 , ȦD , ȦG ) ( v1 ,v2 , v3 ,ȦG )

7.1 Overconstrained solutions

707

Table 7.5. Limb topology and the number of overconstraints NF in the derived maximally regular TPMs presented in Figs. 7.14–7.32 No. Basic TPM type 1 3-PPP (Fig. 7.3) 2 3

3-PPaP (Fig. 7.6)

NF 6

15

4 5

3-PPPa (Fig. 7.7)

15

6 7

3-PPaPa (Fig. 7.8)

24

8 9

3-PRRPa (Fig. 7.9)

12

10 11

3-PPaRR (Fig. 7.10)

12

12 13

3-PRRbR (Fig. 7.11a)

12

14 15

3-PRRbRbR (Fig. 7.11b)

21

16 17

3-PPn2R 12 (Fig. 7.12a, b)

18 19 20

3-PPn3 12 (Fig. 7.13a, b)

Derived TPM type 3-PPC* (Fig. 7.14) 3-PC*P (Fig. 7.15) 3-PPaC* (Fig. 7.16) 3-PPassP (Fig. 7.17) 3-PC*Pa (Fig. 7.18) 3-PPPass (Fig. 7.19) 3-PPaPass (Fig. 7.20) 3-PPassPa (Fig. 7.21) 3-PR*RRPa (Fig. 7.22) 3-PRRPacs (Fig. 7.23) 3-PPaRRR* (Fig. 7.24) 3-PPacsRR (Fig. 7.25) 3-PR*RRbR (Fig. 7.26a) 3-PRRbcsR (Fig. 7.26b) 3-PR*RRbRbR (Fig. 7.27) 3-PRRbcsRbcsR (Fig. 7.28) 3-PPn2RR* (Figs. 7.29a, 7.30a) 3-PPn2csR (Figs. 7.29b, 7.30b) 3-PPn3R* (Fig. 7.31a, b) 3-PPn3cs (Fig. 7.32a, b)

NF 3

Limb topology P A P A A C*

3

P A C* A A P

12

P||Pa A C*

3

P||Pass A P

12

P A C* A ||Pa

3

P A P A ||Pass

12

P||Pa||Pass

12

P||Pass||Pa

9

P A R* A ||R||R||Pa

3

P||R||R||Pacs

9

P||Pa||R||R A R*

3

P||Pacs||R||R

9

P A R* A ||R||Rb||R

3

P||R||Rbcs||R

18

P A R* A ||R||Rb||Rb||R

3

P||R||Rbcs||Rbcs||R

9

P||Pn2||R A R*

3

P||Pn2cs||R

9

P||Pn3 A R*

3

P||Pn3cs

708

7 Maximally regular T3-type translational parallel robots

Table 7.6. Structural parametersa of translational parallel mechanisms in Figs. 7.14–7.19 No. Structural parameter

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

Solution 3-PPC* (Fig. 7.14) 3-PC*P (Fig. 7.15) 8 3 3 3 9 2 3 0 3 See Table 7.4

3-PPaC* (Fig. 7.16) 3-PC*Pa (Fig. 7.18) 14 6 6 6 18 5 0 3 3 See Table 7.4

3-PPassP (Fig. 7.17) 3-PPPass (Fig. 7.19) 14 6 6 6 18 5 0 3 3 See Table 7.4

4 4 4 0 0 0 4 4 4 ( v1 ,v 2 ,v3 ) 3 0 9 3 3 0 4

4 4 4 3 3 3 4 4 4 ( v1 ,v 2 ,v3 ) 3 9 18 3 12 0 7

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 3 0 10

fj

4

7

10

fj

4

7

10

fj

12

21

30

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

7.1 Overconstrained solutions

709

Table 7.7. Structural parametersa of translational parallel mechanisms in Figs. 7.20–7.26 No. Structural parameter

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

20 9 9 9 27 8 0 3 3 See Table 7.4

3-PR*RRPa (Fig. 7.22) 3-PPaRRR* (Fig. 7.24) 3-PR*RRbR (Fig. 7.26a) 20 8 8 8 24 5 0 3 3 See Table 7.4

3-PRRPacs (Fig. 7.23) 3-PPacsRR (Fig. 7.25) 3-PRRbcsR (Fig. 7.26b) 17 7 7 7 21 5 0 3 3 See Table 7.4

4 4 4 9 9 9 4 4 4 ( v1 ,v 2 ,v3 ) 3 27 36 3 12 0 13

5 5 5 3 3 3 5 5 5 ( v1 ,v 2 ,v3 ) 3 9 21 3 9 0 8

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 3 0 10

fj

13

8

10

fj

13

8

10

fj

39

24

30

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

Solution 3-PPaPass (Fig. 7.20) 3-PPassPa (Fig. 7.21)

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

710

7 Maximally regular T3-type translational parallel robots

Table 7.8. Structural parametersa of translational parallel mechanisms in Figs. 7.27 and 7.28 No.

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

Structural parameter

Solution 3-PR*RRbRbR (Fig. 7.27) 26 11 11 11 33 8 0 3 3 See Table 7.4

3-PRRbcsRbcsR (Fig. 7.28) 23 10 10 10 30 8 0 3 3 See Table 7.4

5 5 5 6 6 6 5 5 5 ( v1 ,v 2 ,v3 ) 3 18 30 3 18 0 11

4 4 4 12 12 12 4 4 4 ( v1 ,v 2 ,v3 ) 3 36 45 3 3 0 16

fj

11

16

fj

11

16

fj

33

48

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

7.1 Overconstrained solutions

711

Table 7.9. Structural parametersa of translational parallel mechanisms in Figs. 7.29–7.32 No.

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

Structural parameter

Solution 3-PPn2RR* (Figs. 7.29a, 7.30a) 3-PPn3R* (Fig. 7.31a, b) 20 8 8 8 24 5 0 3 3 See Table 7.4

3-PPn2csR (Figs. 7.29b, 7.30b) 3-PPn3cs (Fig. 7.32a, b) 17 7 7 7 21 5 0 3 3 See Table 7.4

5 5 5 3 3 3 5 5 5 ( v1 ,v 2 ,v3 ) 3 9 21 3 9 0 8

4 4 4 6 6 6 4 4 4 ( v1 ,v 2 ,v3 ) 3 18 27 3 3 0 10

fj

8

10

fj

8

10

fj

24

30

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

712

7 Maximally regular T3-type translational parallel robots

Fig. 7.14. 3-PPC*-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology P A P A A C*

7.1 Overconstrained solutions

713

Fig. 7.15. 3-PC*P-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology P A C* A A P

714

7 Maximally regular T3-type translational parallel robots

Fig. 7.16. 3-PPaC*-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology P||Pa A C*

7.1 Overconstrained solutions

715

Fig. 7.17. 3-PPassP-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology P||Pass A P

716

7 Maximally regular T3-type translational parallel robots

Fig. 7.18. 3-PC*Pa-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology P A C* A ||Pa

7.1 Overconstrained solutions

717

Fig. 7.19. 3-PPPass-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology P A P A ||Pass

718

7 Maximally regular T3-type translational parallel robots

Fig. 7.20. 3-PPaPass-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology P||Pa||Pass

7.1 Overconstrained solutions

719

Fig. 7.21. 3-PPassPa-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 12, limb topology P||Pass||Pa

720

7 Maximally regular T3-type translational parallel robots

Fig. 7.22. 3-PR*RRPa-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology P A R* A ||R||R||Pa

7.1 Overconstrained solutions

721

Fig. 7.23. 3-PRRPacs-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology P||R||R||Pacs

722

7 Maximally regular T3-type translational parallel robots

Fig. 7.24. 3-PPaRRR*-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology P||Pa||R||R A R*

7.1 Overconstrained solutions

723

Fig. 7.25. 3-PPacsRR-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology P||Pacs||R||R

724

7 Maximally regular T3-type translational parallel robots

Fig. 7.26. Overconstrained maximally-regular TPMs of types 3-PR*RRbR (a) and 3-PRRbcsR (b) defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9 (a), NF = 3 (b), limb topology P A R* A ||R||Rb||R (a) and P||R||Rbcs||R (b)

7.1 Overconstrained solutions

725

Fig. 7.27. 3-PR*RRbRbR-type overconstrained maximally-regular TPM defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 18, limb topology P A R* A ||R||Rb||Rb||R

726

7 Maximally regular T3-type translational parallel robots

Fig. 7.28. 3-PRRbcsRbcsR-type overconstrained maximally-regular TPM defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology P||R||Rbcs||Rbcs||R

7.1 Overconstrained solutions

727

Fig. 7.29. Overconstrained maximally-regular TPMs of types 3-PPn2RR* (a) and 3-PPn2csR (b) defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9 (a), NF = 3 (b), limb topology P||Pn2||R A R* (a) and P||Pn2cs||R (b)

728

7 Maximally regular T3-type translational parallel robots

Fig. 7.30. Overconstrained maximally-regular TPMs of types 3-PPn2RR* (a) and 3-PPn2csR (b) defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9 (a), NF = 3 (b), limb topology P||Pn2||R A R* (a) and P||Pn2cs||R (b)

7.1 Overconstrained solutions

729

Fig. 7.31. 3-PPn3R*-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 9, limb topology P||Pn3 A R*

730

7 Maximally regular T3-type translational parallel robots

Fig. 7.32. 3-PPn3cs-type overconstrained maximally-regular TPMs defined by MF = SF = 3, (RF) = ( v1 , v2 , v3 ), TF = 0, NF = 3, limb topology P||Pn3cs

7.2 Non overconstrained solutions

731

7.2 Non overconstrained solutions Equation (1.15) indicates that non overconstrained solutions of the maximally p regular TPMs with q independent loops meet the condition ¦ 1 f i 3 6q along with MF = SF = 3 and (RF) = (v1,v2,v3). They could have identical limbs or limbs with different topologies that may be actuated by linear motors. In the non overconstrained maximally regular TPMs F m G1–G2–G3, the moving platform n Ł nGi (i = 1, 2, 3) is connected to the reference platform 1 Ł 1Gi Ł 0 by three limbs with five degrees of connectivity. These solutions are derived from the overconstrained solutions presented in Figs. 7.3–7.32 by introducing the required idle mobilities. For example, the non overconstrained solution in Fig. 7.33a is derived from the overconstrained solution in Fig. 7.3a by replacing, in each limb, two prismatic joints by cylindrical joints. The bases of the operational velocities spaces of the limbs isolated from the parallel mechanisms presented in Figs. 7.33–7.45 are given in Table 7.10. The limb topology and connecting conditions of these solutions are systematized in Table 7.11 and the structural parameters of these solutions are presented in Tables 7.12–7.14. Table 7.10. Bases of the operational velocities spaces of the limbs isolated from the parallel mechanisms presented in Figs. 7.33–7.45 No. Parallel mechanism 1 Figs. 7.33, 7.36–7.38 2 Figs. 7.34a, 7.35, 7.39b, 7.40–7.45 3 Fig. 7.39a

Basis (RG1) (RG2) (RG3) ( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦG ) ( v1 , v2 , v3 , ȦD , ȦE ) ( v1 , v2 , v3 , ȦE , ȦG )

732

7 Maximally regular T3-type translational parallel robots

Table 7.11. Limb topology of the non overconstrained maximally regular TPMs presented in Figs. 7.33–7.45 No. Basic TPM Type 1 3-PPP (Fig. 7.3a) 2 3-PRRP (Fig. 7.4a) 3 3-PRPR (Fig. 7.4b) 4 3-PPRR (Fig. 7.5a) 5 3-PRRR (Fig. 7.5b) 6 3-PPaP (Fig. 7.6) 7 3-PPPa (Fig. 7.7) 8 3-PPaPa (Fig. 7.8) 9 3-PRRPa (Fig. 7.9) 10 11

3-PPaRR (Fig. 7.10)

NF 6 3 3 3 3 15 15 24 12

12

12 13 14 15 16

3-PRRbR (Fig. 7.11a) 3-PRRbRbR (Fig. 7.11b) 3-PPn2R (Fig. 7.12a, b) 3-PPn3 (Fig. 7.13a, b)

12 21 12 12

Derived TPM with NF=0 Type 3-PC*C* (Fig. 7.33) 3-PRRC* (Fig. 7.34a) 3-PRC*R (Fig. 7.34b) 3-PC*RR (Fig. 7.35a) 3-PR*RRR (Fig. 7.35b) 3-PPassC* (Fig. 7.36) 3-PC*Pass (Fig. 7.37) 3-PPassPass (Fig. 7.38) 3-PRRPass (Fig. 7.39a) 3-PR*RRPacs (Fig. 7.39b) 3-PPassRR (Fig. 7.40a) 3-PPacsRRR* (Fig. 7.40b) 3-PR*RRbcsR (Fig. 7.41) 3-PR*RRbcsRbcsR (Fig. 7.42) 3-PPn2csRR* (Figs. 7.43, 7.44) 3-PPn3csRR* (Fig. 7.45a, b)

Limb topology P A C* A A C* P||R||R A C* P||R A C* A ||R P A C* A ||R||R P A R* A ||R||R||R P||Pass A C* P A C* A ||Pass P||Pass||Pass P||R||R||Pass P A R* A ||R||R||Pacs P||Pass||R||R P||Pacs||R||R A R* P A R A ||Rbcs||R P A R A ||Rbcs||Rbcs||R P||Pn2cs||R A R* P||Pn3cs A R*

7.2 Non overconstrained solutions

733

Table 7.12. Structural parametersa of translational parallel mechanisms in Figs. 7.33–7.37 No. Structural parameter

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

8 3 3 3 9 2 3 0 3 See Table 7.10

3-PRRC* (Fig. 7.34a) 3-PRC*R (Fig. 7.34b) 3-PC*RR (Fig. 7.35a) 3-PR*RRR (Fig. 7.35b) 11 4 4 4 12 2 3 0 3 See Table 7.10

3-PPassC* (Fig. 7.36) 3-PC*Pass (Fig. 7.37) 14 6 6 6 18 5 0 3 3 See Table 7.10

5 5 5 0 0 0 5 5 5 ( v1 ,v 2 ,v3 ) 3 0 12 3 0 0 5

5 5 5 0 0 0 5 5 5 ( v1 ,v 2 ,v3 ) 3 0 12 3 0 0 5

5 5 5 6 6 6 5 5 5 ( v1 ,v 2 ,v3 ) 3 18 30 3 0 0 11

fj

5

5

11

fj

5

5

11

fj

15

15

33

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

Solution 3-PC*C* (Fig. 7.33)

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

734

7 Maximally regular T3-type translational parallel robots

Table 7.13. Structural parametersa of translational parallel mechanisms in Figs. 7.38–7.41 No. Structural parameter

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

20 9 9 9 27 8 0 3 3 See Table 7.10

17 7 7 7 21 5 0 3 3 See Table 7.10

3-PR*RRPacs (Fig. 7.39b) 3-PPacsRRR* (Fig. 7.40b) 3-PR*RRbcsR (Fig. 7.41) 20 8 8 8 24 5 0 3 3 See Table 7.10

5 5 5 12 12 12 5 5 5 ( v1 ,v 2 ,v3 ) 3 36 48 3 0 0 17

5 5 5 6 6 6 5 5 5 ( v1 ,v 2 ,v3 ) 3 18 30 3 0 0 11

5 5 5 6 6 6 5 5 5 ( v1 ,v 2 ,v3 ) 3 18 30 3 0 0 11

fj

17

11

11

fj

17

11

11

fj

51

33

33

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF) SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

Solution 3-PPassPass (Fig. 7.38)

fj

3-PRRPass (Fig. 7.39a) 3-PPassRR (Fig. 7.40a)

See footnote of Table 2.1 for the nomenclature of structural parameters

7.2 Non overconstrained solutions

735

Table 7.14. Structural parametersa of translational parallel mechanisms in Figs. 7.42–7.45 No.

Structural parameter

Solution 3-PR*RRbcsRbcsR (Fig. 7.42)

1 2 3 4 5 6 7 8 9 10

m p1 p2 p3 p q k1 k2 k (RGi) (i = 1,2,3) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

26 11 11 11 33 8 0 3 3 See Table 7.10

3-PPn2csRR* (Figs. 7.43, 7.44) 3-PPn3csRR* (Fig. 7.45) 20 8 8 8 24 5 0 3 3 See Table 7.10

5 5 5 12 12 12 5 5 5 ( v1 ,v 2 ,v3 ) 3 36 48 3 0 0 17

5 5 5 6 6 6 5 5 5 ( v1 ,v 2 ,v3 ) 3 18 30 3 0 0 11

fj

17

11

fj

17

11

fj

51

33

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 a

SF rl rF MF NF TF

¦ ¦ ¦ ¦

p1 j 1 p2 j 1 p3 j 1 p j 1

fj

See footnote of Table 2.1 for the nomenclature of structural parameters

736

7 Maximally regular T3-type translational parallel robots

Fig. 7.33. 3-PC*C*-type non overconstrained maximally regular TPMs, limb topology P A C* A A C*

7.2 Non overconstrained solutions

737

Fig. 7.34. Non overconstrained maximally regular TPMs of types 3-PRRC* (a) and 3-PRC*R (b), limb topology P||R||R A C* (a) and P||R A C* A ||R (b)

738

7 Maximally regular T3-type translational parallel robots

Fig. 7.35. Non overconstrained maximally regular TPMs of types 3-PC*RR (a) and 3-PR*RRR (b), limb topology P A C* A ||R||R (a) and P A R* A ||R||R||R (b)

7.2 Non overconstrained solutions

739

Fig. 7.36. 3-PPassC*-type non overconstrained maximally-regular TPMs, limb topology P||Pass A C*

740

7 Maximally regular T3-type translational parallel robots

Fig. 7.37. 3-PC*Pass-type non overconstrained maximally-regular TPMs, limb topology P A C* A ||Pass

7.2 Non overconstrained solutions

741

Fig. 7.38. 3-PPassPass-type non overconstrained maximally-regular TPMs, limb topology P||Pass||Pass

742

7 Maximally regular T3-type translational parallel robots

Fig. 7.39. Non overconstrained maximally-regular TPMs of types 3-PRRPass (a) and 3-PR*RRPacs (b), limb topology P||R||R||Pass (a) and P A R* A ||R||R||Pacs (b)

7.2 Non overconstrained solutions

743

Fig. 7.40. Non overconstrained maximally-regular TPMs of types 3-PPassRR (a) and 3-PPacsRRR* (b), limb topology P||Pass||R||R (a) and P||Pacs||R||R A R* (b)

744

7 Maximally regular T3-type translational parallel robots

Fig. 7.41. 3-PR*RRbcsR-type non overconstrained maximally-regular TPM, limb topology P A R A ||Rbcs||R

7.2 Non overconstrained solutions

745

Fig. 7.42. 3-PR*RRbcsRbcsR-type non overconstrained maximally-regular TPM, limb topology P A R A ||Rbcs||Rbcs||R

746

7 Maximally regular T3-type translational parallel robots

Fig. 7.43. 3-PPn2csRR*-type non overconstrained maximally-regular TPM, limb topology P||Pn2cs||R A R*

7.2 Non overconstrained solutions

747

Fig. 7.44. 3-PPn2csRR*-type non overconstrained maximally-regular TPM, limb topology P||Pn2cs||R A R*

748

7 Maximally regular T3-type translational parallel robots

Fig. 7.45. 3-PPn3csRR*-type non overconstrained maximally-regular TPMs, limb topology P||Pn3cs A R*

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Index

A

E

actuator, 2 linear, 24 rotating, 39 algorithm evolutionary, 14 approach systematic, 8

element pairing, 3 reference, 3 end-effector, 7 equation constraint, 9 evolutionary morphology, 13

B

F

base, 2 fixed, 3 basic solution, 107 basis, 13

frame, 3 fully-isotropic, 14 G graph, 5 structural, 5

C characteristic point, 13 condition number, 14 connectivity, 10 constraint equation, 9 coupled motions, 14 CPM, 20

H hexapod, 2 I IFMA, 19 independent motion, 11 Isoglide2-T2, 93 Isoglide3-T3, 20, 687 isotropy, 14

D decoupled motions, 66, 688 degree of freedom, 8 DELTA, 18 design objectives, 13 dimension vector space, 12

J Jacobian matrix, 14 joint, 2, 4 Cardan, 4 759

760

Index

heterokinetic, 4 homokinetic, 4

operator, 13 motion coupling, 14

K kinematic pair, 2, 4 kinematic chain, 2 closed, 4 complex, 4 open, 4 serial, 2 simple, 4 L limb, 2 actuated, 28 complex, 4 simple, 4 topology, 23 unactuated, 28 link, 2, 3 binary, 2 distal, 4 monary, 2 polinary, 2 loop parallelogram, 24 M manipulator parallel, 14 mechanism, 2, 3 parallel, 2 kinematotropic, 9 prism, 98 mobility, 8 full-cycle, 9 general, 9 idle, 5, 375, 421, 537, 621, 706 instantaneous, 9 model kinematic, 16, 17 direct, 16, 17 morphological

N number of overconstraints, 10 O operational space, 81 vector space, 107 velocity, 107 velocity space, 11 orientation, 107 Orthoglide, 19 Orthogonal Tripteron, 20 overconstraint, 11 P paire, 2 cylindrical, 4 helical, 4 kinematic, 4 lower, 4 passive, 7 planar, 4 revolute, 4 spherical, 4 pairing element, 2 parallel mechanism, 10 parallel robotic manipulator, 2 fully-isotropic, 14 maximally regular, 14 non overconstrained, 12 non redundant, 12 overconstrained, 12 redundant, 12 translational, 15 T2-type, 15 T3-type, 16 with coupled motions, 14 with decoupled motions, 14 with uncoupled motions, 14

Index performance index, 15 pick-and-place, 15 platform, 2 fixed, 2, 4 moving, 2, 4 point characteristic, 13 protoelement, 14 R rank, 9 redundancy, 11 robot, 1 fully parallel, 7 hexapod, 2 hybrid, 7 parallel, 7 serial, 7 robotics, 1 S singular configuration, 9 structural diagram, 5 graph, 5 parameters, 27 redundancy, 10 synthesis, 8 synthesis structural, 8 systematic approach, 8 T theory of linear transformations, 10 topology, 8, 24 TPM, 1 T2-type TPMs, 23

761

basic solutions, 24, 98 coupled motions, 23 decoupled motions, 66 linear actuators, 24, 54, 66, 85 maximally regular, 93 non overconstrained, 54, 85, 95, 98 overconstrained, 23, 66, 85, 93, 98 rotating actuators, 24, 59, 81, 85, 89 uncoupled motions, 85 with unactuated limb, 59 T3-type TPMs, 107 basic solutions, 107, 108, 194, 385, 471, 615 coupled motions, 107, 365, 412, 688 decoupled motions, 688 derived solutions, 107, 138 linear actuators, 108, 365, 375 maximally regular, 687 non overconstrained, 365, 412, 615, 731 overconstrained, 107, 471, 687 rotating actuators, 194, 401, 412, 471, 615 uncoupled motions, 615, 688 U uncoupled motions, 85 V vector space, 10 velocity operational, 107 velocity vector space joint, 15 operational, 15