Structural Synthesis of Parallel Robots: Part 3: Topologies with Planar Motion of the Moving Platform [1 ed.] 9048198305, 9789048198306

Citation preview

Structural Synthesis of Parallel Robots

SOLID MECHANICS AND ITS APPLICATIONS Volume 173

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 3: Topologies with planar motion of the moving platform

Grigore Gogu Clermont University IFMA, EA 3867, Mechanical Engineering Research Group F-63000 Clermont-Ferrand, France [email protected]

ISSN 0925-0042 ISBN 978-90-481-9830-6 e-ISBN 978-90-481-9831-3 DOI 10.1007/978-90-481-9831-3 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2010935808 © Springer Science+Business Media B.V. 2010 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. Cover Design: SPI Publisher Services Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

Contents

Preface.................................................................................................... VII Acknowledgements ............................................................................XIII List of abbreviations and notations ................................................... XIV 1 Introduction ............................................................................................ 1 1.1 Terminology..................................................................................... 1 1.1.1 Links, joints and kinematic chains ............................................ 2 1.1.2 Serial, parallel and hybrid robots .............................................. 9 1.2 Methodology of structural synthesis .............................................. 10 1.2.1 New formulae for mobility, connectivity, redundancy and overconstraint of parallel robots ...................................................... 10 1.2.2 Evolutionary morphology approach........................................ 16 1.2.3 Types of parallel robots with respect to motion coupling....... 16 1.3 Parallel robots with planar motion of the moving platform ........... 19 2 Overconstrained planar parallel robots with coupled motions........ 27 2.1 Basic solutions ............................................................................... 27 2.1.1 Fully-parallel solutions ........................................................... 27 2.1.2 Non fully-parallel solutions .................................................... 77 2.2 Derived solutions ........................................................................... 87 3 Non overconstrained planar parallel robots with coupled motions183 3.1 Fully-parallel solutions................................................................. 183 3.2 Non fully-parallel solutions.......................................................... 228 4 Planar parallel robots with uncoupled motions............................... 239 4.1 Overconstrained solutions............................................................ 239 4.1.1 Basic solutions ...................................................................... 239 4.1.2 Derived solutions .................................................................. 251 4.2 Non overconstrained solutions..................................................... 271 5 Maximally regular planar parallel robots........................................ 283 5.1 Overconstrained solutions............................................................ 283

VI

Contents

5.1.1 Basic solutions ...................................................................... 283 5.1.2 Derived solutions .................................................................. 289 5.2 Non overconstrained solutions..................................................... 300 6 Spatial PMs with coupled planar motion of the moving platform. 307 6.1 Overconstrained solutions............................................................ 307 6.1.1 Basic solutions ...................................................................... 307 6.1.2 Derived solutions .................................................................. 327 6.2 Non overconstrained solutions..................................................... 346 7 Spatial PMs with uncoupled planar motion of the moving platform.................................................................................................. 367 7.1 Overconstrained solutions............................................................ 367 7.1.1 Basic solutions ...................................................................... 367 7.1.2 Derived solutions .................................................................. 420 7.2 Non overconstrained solutions..................................................... 472 8 Maximally regular SPMs with planar motion of the moving platform.................................................................................................. 529 8.1 Overconstrained solutions............................................................ 529 8.1.1 Basic solutions ...................................................................... 529 8.1.2 Derived solutions .................................................................. 578 8.2 Non overconstrained solutions..................................................... 621 References .............................................................................................. 665 Index ....................................................................................................... 683

Preface

“In other words, the invention of a mechanism will be to the scientific kinematist a synthetic problem, - which he can solve by the use of systematic, if also difficult, methods.” 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 third part of a larger work dedicated to the structural synthesis of 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 (Gogu 2009a) presented structural solutions of translational parallel robotic manipulators with two and three degrees of mobility. This book focuses on various topologies of parallel robotic manipulators with planar motion of the moving platform systematically generated by using the structural synthesis approach proposed in Part 1. The originality of this work resides in the fact that it combines the new formulae for mobility connectivity, redundancy and overconstraints, and the evolutionary morphology in a unified approach of structural synthesis giving interesting innovative solutions for parallel mechanisms. Parallel robotic manipulators can be considered a well-established option for many different applications of manipulation, machining, guiding, testing, control, tracking, haptic force 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 has concentrated primarily on six degrees of freedom (DoFs) Gough-

VIII

Preface

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

Preface

IX

method based on the theory of linear transformations. The method proposed in this work is based on the theory of linear transformations and the evolutionary morphology and allows us to obtain the structural solutions of 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 along with coupled solutions 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 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 ≤ n ≤ 6, a=1,2,3 and b=1,2,3. The mobile platform of these robots can have any combination of n independent translations (T) and rotations (R).

X

Preface

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 inputoutput 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

Preface

XI

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, fully-isotropic and maximally regular parallel robots presented in Parts 2 and 3. 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 1500 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 Parts 2 and 3 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. Part 2 of this work (Gogu, 2009a) was organised in 7 chapters. The first chapter recalled the main concepts, the new formulae used to calculate the main structural parameters of PMs, and the original approach of structural synthesis. Chapter 2 focused on the structural synthesis of T2-type translational parallel manipulators (TPMs) with two degrees of freedom used in pick-and-place operations. Overconstrained/isostatic solutions of coupled, decoupled, uncoupled and fully-isotropic/maximally regular PMs with elementary/complex limbs actuated by linear/rotary actuators with/without idle mobilities are presented. Chapter 3 presented the structural synthesis of overconstrained T3-type translational parallel manipulators with three degrees of freedom and coupled motions. Basic and derived solutions with linear or rotating actuators are presented. The basic solutions do not combine idle mobilities. 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 presented the structural synthesis of overconstrained and non-overconstrained T3-type TPMs with uncoupled motions. Basic and derived solutions with rotating actuators and identical limbs are presented. Chapter 7 focused on the structural synthesis of overconstrained and nonoverconstrained maximally regular T3-type TPMs. Basic and derived solutions with linear actuators and identical limbs are on hand. About 1000 solutions of TPMs are illustrated in 550 figures. The structural parameters of these solutions are systematized in 134 tables.

XII

Preface

This book representing Part 3 is organised in 8 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 applied to parallel robots with planar motion of the moving platform. In such a robot, the moving platform can undergo two independent translational motions T2 and one rotational motion R1 around an axis perpendicular to the plane of translations. This motion can be obtained by using planar or spatial parallel mechanisms. Chapters 2 and 3 present the structural synthesis of overconstrained and non-overconstrained planar parallel robots with coupled motions. Basic and derived fully-parallel and non fully-parallel solutions are on hand. The structural synthesis of overconstrained and non-overconstrained planar parallel robots with uncoupled motions is presented in Chapter 4. Chapter 5 focuses on the structural synthesis of overconstrained and non-overconstrained maximally regular planar parallel robots. Chapters 6 and 7 present the structural synthesis of basic and derived solutions of overconstrained and non-overconstrained spatial parallel robots with coupled and uncoupled planar motions of the moving platform. Chapter 8 focuses on the structural synthesis of overconstrained and non-overconstrained maximally regular spatial parallel robots with planar motion of the moving platform. About 750 solutions are illustrated in 400 figures. The structural parameters of these solutions are systematized in 150 tables. 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 4) and PMs with four, five and six degrees of freedom (Part 5). The writing of Parts 4 and 5 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.

Preface

XIII

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, Prof. Graham M.L. Gladwell, the series editor of Solids Mechanics and Its Applications, and Mrs. Nathalie Jacobs, Springer Dordrecht Engineering Editor are gratefully acknowledged for their availability and encouragement in pursuing this publishing project. Ms. Sarah Davey is also gratefully acknowledged for the linguistic reviewing of this manuscript. May I also acknowledge the excellent facilities and research environment provided by LaMI 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.

XIV

Preface

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 ← 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, de la Mobilité et de la Sûreté Gi (1Gi-2Gi-…nGi) - 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 n×n - 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) LASMEA - Laboratoire des Sciences et Matériaux pour l’Electronique, et d’Automatique (Laboratory of Sciences and Materials for Electronic, and of Automatic)

Preface

XV

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 nznGi - moving platform in the parallel mechanism F ← 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 Pac - R||R||R||C-type parallelogram loop with one idle mobility combined in a cylindrical joint Pasu - parallelogram loop with three idle mobilities combined in a spherical and a revolute joint Pas - 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 ⊥ P ⊥ ||R||R ⊥ P ⊥ ||R-type telescopic planar parallelogram loop Patcs - telescopic parallelogram loop with three idle mobilities combined in a cylindrical and a spherical pair Pau - 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 Pn3* or Pn3cs - close loop with three degrees of mobility and three idle mobilities combined in a cylindrical and a spherical pair PPM - planar parallel manipulator q - number of independent closed loops in the parallel mechanism

XVI

Preface

&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 ← 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 ← 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 ← G1-G2-…-Gk. SGi - the connectivity between the mobile and the reference platforms in the kinematic chain Gi disconnected from the parallel mechanism F ← G1-G2-…-Gk. SPM - spatial parallel manipulator 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 &x,&y,z& - time derivatives of coordinates α , β ,δ - rotation angles

α& , β& ,δ& - time derivatives of the rotation angles ,

α

,

β

,

δ

- angular velocity vectors

0 - fixed base of a kinematic chain/mechanism

Preface

XVII

1z1Gi– fixed platform in the parallel mechanism F ← 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 1 and 2 in the notation 2PRR-1RPaPa - the parallel mechanism has two limb of type PRR and one limb of type RPaPa || - parallel position of joint axes/directions; for example the notation Pa||Pass indicates the fact that the axes of the revolute joints of the parallelogram loops Pa and Pass are parallel ⊥ - perpendicular position of joint axes/directions; for example the notation P ⊥ Pa indicates the fact that the axes of revolute joints in the parallelogram loop are perpendicular to the direction of the prismatic joint || || ⊥ in the notation R ⊥ P ⊥ 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 || || ⊥ in the notation R ⊥ Pa ⊥ Pa - the revolute axes of the second parallelogram loop are perpendicular to the revolute axes of the first parallelogram loop and parallel to the axis of the actuated revolute joint ⊥ ⊥ in the notation R ⊥ Pa ⊥ ⊥ Pa - the revolute axes of the second parallelogram loop are perpendicular to the revolute axes of the first parallelogram loop and also to the axis of the actuated prismatic joint ⊥ ⊥ in the notation Pass ⊥ R||R ⊥ ⊥ Pa - the revolute axes of parallelogram loop Pa are perpendicular to the axes of the parallel revolute joints RｬR and also to the axes of the revolute joints of parallelogram loop Pass

1 Introduction

This book represents Part 3 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 (Gogu 2009a) presented structural solutions of translational parallel robotic manipulators (TPMs) with two and three degrees of mobility. Part 3 of this work focuses on the structural solutions of parallel robotic manipulators with planar motion of the moving platform. This section recalls the terminology, the new formulae for the main structural parameters of parallel robots (mobility, connectivity, redundancy and overconstraint) and the main features of the methodology of structural synthesis based on the evolutionary morphology presented in Part 1.

1.1 Terminology Robots can be found today in the manufacturing industry, agricultural, military and domestic applications, space exploration, medicine, education, information and communication technologies, entertainment, etc. We have presented in Part 1 various definitions of the word robot and we have seen that it is mainly used to refer to a wide range of mechanical devices or mechanisms, the common feature of which is that they are all capable of movement and can be used to perform physical tasks. Robots take on many different forms, ranging from humanoid, which mimic the human form and mode of movement, to industrial, whose appearance is dictated by the function they are to 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 G. Gogu, Structural Synthesis of Parallel Robots, Solid Mechanics and Its Applications 173, DOI 10.1007/978-90-481-9831-3_1, © Springer Science + Business Media B.V. 2010

1

2

1 Introduction

which are the counterparts to the serial robots. The various definitions of robotics converge towards the integration of the design and the end use in the studies related to 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. 1.1.1 Links, joints and kinematic chains Serial or parallel kinematic chains are concatenated in the robot mechanism. The serial kinematic chain is formed by links connected sequentially by joints. Links are connected in series as well as in parallel making one or more closed-loops in a parallel mechanism. The mechanical architecture of parallel robots is based on parallel mechanisms in which a member called a moving platform is connected to a reference member by at least two limbs that can be simple or complex. The robot actuators are integrated in the limbs (also called legs) usually closed to the fixed member, also called the base or the fixed platform. The moving platform positions the robot end-effector in space and may have anything between two and six degrees of freedom. Usually, the number of actuators coincides with the degrees of freedom of the mobile platform, exceeding them only in the case of redundantly-actuated parallel robots. The paradigm of parallel robots is the hexapod-type robot, which has six degrees of freedom, but recently, the machine industry has discovered the potential applications of lower-mobility parallel robots with just 2, 3, 4 or 5 degrees of freedom. Indeed, the study of this type of parallel manipulator is very important. They exhibit interesting features when compared to hexapods, such as a simpler architecture, a simpler control system, highspeed 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

1.1 Terminology

3

(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 physical realization of a kinematic pair. The pairing element represents the assembly of surfaces, lines or points of a solid body through which it may contact with another solid body. The kinematic pair is the mechanical model of the connection of two pairing elements having relative motion of a certain type and degree of freedom. In the standard terminology, a kinematic chain is an assembly of links (mechanism elements) and joints, and a mechanism is a kinematic chain in which one of its links is taken as a “frame”. In this definition, the “frame” is a mechanism element deemed to be fixed. In this book, we use the notion of reference element to define the “frame” element. The reference element can be fixed or may merely be deemed to be fixed with respect to other mobile elements. The fixed base is denoted in this book by 0. A mobile element in a kinematic chain G is denoted by nG (n=1, 2, …). Two or more links connected together in the same link such that no relative motion can occur between them are considered as one link. The identity symbol “ ” is used between the links to indicate that they are welded together in the same link. For example, the notation 1G 0 is used to indicate that the first link 1G of the kinematic chain G is the fixed base. A kinematic chain G is denoted by the sequence of its links. The notation G (1G 0-2G-…-nG) indicates a kinematic chain in which the first link is fixed and the notation G (1G-2G-…nG) a kinematic chain with no fixed link. We will use the notion of mechanism to qualify the whole mechanical system, and the notion of kinematic chain to qualify the sub-systems of a mechanism. So, 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),

4

1 Introduction

b) binary link - a mechanism element connected in the kinematic chain by two joints (a link connected to two other links), c) polinary link - a mechanism element connected in the kinematic chain by more than two joints (ternary link - if the link is connected by three joints, quaternary link if the link is connected by four joints). The IFToMM terminology defines open/closed kinematic chains and mechanisms, but it does not introduce the notions of simple (elementary) and complex kinematic chains and mechanisms. A closed kinematic chain is a kinematic chain in which each link is connected with at least two other links, and an open kinematic chain is a kinematic chain in which there is at least one link which is connected in the kinematic chain by just one joint. In a simple open kinematic chain (open-loop mechanism) only monary and binary links are connected. In a complex kinematic chain at least one 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 distal links are the moving and the reference platform. The two platforms are connected by at least two simple or complex kinematic chains called limbs. Each limb contains at least one joint. A simple limb is composed of a simple open kinematic chain in which the final element is the mobile platform. A complex limb is composed of a complex kinematic chain in which the final element is also the mobile platform. IFToMM terminology (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. 1.1(a)-(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 (HJ) are known today: Tracta, Weiss, Bendix, Dunlop, Rzeppa, Birfield, Glaenzer,

1.1 Terminology

5

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. (Dudi et al. 1989, 2001a, b). The graphical representations used in this book for the universal homokinetic joints are presented in Fig. 1.1(g)-(h). Joints with idle mobilities are commonly used to reduce the number of overconstraints in a mechanism. The idle mobility is a potential mobility of a joint that is not used by the mechanism and does not influence mechanism’s mobility in the hypothesis of perfect manufacturing and assembling precision. In theoretical conditions, when no errors exist with respect to parallel, perpendicular or intersecting positions of joint axes, motion amplitude of an idle mobility is zero. Real life manufacturing and assembling processes introduce errors in the relative positions of the joint axes and, in this case, the idle mobilities become effective mobilities usually with small amplitudes, depending on the precision of the mechanism. For example, the idle mobilities which can be combined in the parallelogram loop in Fig. 1.2 are systematized in Table 1.1 along with the number r of parameters that lose their independence in the closed loop and the number of overconstraints N of the corresponding linkage. A parallel mechanism is a single or multi-loop linkage in which a moving link called characteristic link or platform is connected to a reference link (fixed base) by at least two non interconnected kinematic chains called limbs.

Fig. 1.1. Symbols used to represent the lower kinematic pairs and the kinematic joints: (a) revolute pair, (b) prismatic pair, (c) helical pair, (d) cylindrical pair, (e) spherical pair, (f) planar contact pair, (g) universal joint, (h) homokinetic joint, (i) two superposed revolute joints (1-2) and (2-3) with the same axis, (j) superposed cylindrical (1-2) and revolute (2-3) joints with the same axis, (k) superposed revolute (1-2) and cylindrical (2-3) joints with the same axis, and (l) two superposed cylindrical joints (1-2) and (2-3) with the same axis

6

1 Introduction

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

1.1 Terminology

7

Table 1.1. Parallelogram loops with idle mobilities and their corresponding number of overconstraints N No. Parallelogram loop 1 Pa (Fig. 1.2a) 2 Pac (Fig. 1.2b)

N 3 2

3

Pau (Fig. 1.2c)

2

4

Pas (Fig. 1.2d)

1

5

Pauu (Fig. 1.2e)

1

6

Pacu (Fig. 1.2f)

1

7

Pacs, Pa* (Fig. 1.2g) 0

8

Pasu (Fig. 1.2h)

0

9

Pass (Fig. 1.2i)

0

Idle mobilities No idle mobilities One translational idle mobility combined in a cylindrical joint One rotational idle mobility combined in a universal joint Two rotational idle mobilities combined in a spherical joint Two rotational idle mobilities combined in two universal joints One translational idle mobility combined in a cylindrical joint and one rotational idle mobilities combined in a universal joint One translational idle mobility combined in a cylindrical joint and two rotational idle mobilities combined in a spherical joint Three rotational idle mobilities combined in one revolute joint and one spherical joint Three idle mobilities combined in two spherical joints adjacent to the same link with a complementary internal rotational mobility of the link adjacent to the two spherical joints.

A parallel robot can be illustrated by a physical implementation or by an abstract representation. The physical implementation is usually illustrated by robot photography and the abstract representation by a CAD model, structural diagram and structural graph. Figure 1.3 gives an example of the various representations of a Gough-Stewart type parallel robot largely used today in industrial applications. The physical implementation in Fig. 1.3a is a photograph of the parallel robot built by Deltalab (http: //www.deltalab.fr/). In a CAD model (Fig. 1.3b) the links and the joints are represented as being as close as possible to the physical implementation (Fig. 1.3a). In a structural diagram (Fig. 1.3c) they are represented by simplified symbols, such as those introduced in Fig. 1.1, respecting the geometric relations defined by the relative positions of joint axes. A structural graph (Fig. 1.3d) is a network of vertices or nodes connected by edges or arcs with no geometric relations. The links are noted in the nodes and the joints on the edges. We can see that the Gough-Stewart type parallel robot has six identical limbs denoted in Fig. 1.3c by A, B, C, D , E and F. The final link is the mobile platform 4 4A 4B 4C 4D 4E 4F

8

1 Introduction

Fig. 1.3. Various representations of a Gough-Stewart type parallel robot: physical implementation (a), CAD model (b), structural diagram (c) and its associated graph (d), A-limb (e) and its associated graph (f)

1.1 Terminology

9

and the reference member is the fixed platform 1A 1B 1C 1D 1E 1F 0. Each limb is connected to both platforms by spherical pairs. A prismatic pair is actuated in each limb. The spherical pairs are not actuated and are called passive pairs. The two platforms are polinary links, the other two links of each limb are binary links. The parallel mechanism 6-SPS-type associated with the Gough-Stewart type parallel robot is a complex mechanism with a multi-loop associated graph (Fig. 1.3d). It has six simple limbs of type SPS. The actuated pair is underlined. The simple open kinematic chain associated with A-limb is denoted by A (1A 0-2A-3A-4A 4) – Fig. 1.3e and its associated graph is tree-type (Fig. 1.3f). 1.1.2 Serial, parallel and hybrid robots We consider the general case of a robot in which the end-effector is connected to the reference link by k≥1 kinematic chains. The end-effector is a binary or polynary link called a mobile platform in the case of parallel robots, and a monary link for serial robots. The reference link may either be the fixed base or may be deemed to be fixed. The kinematic chains connecting the end-effector to the reference link can be simple or complex. They are called limbs or legs in the case of parallel robots. A serial robot can be considered to be a parallel robot with just one simple limb, and a hybrid robot a parallel robot with just one complex limb. We denote by F ← G1-G2-…-Gk the kinematic chain associated with a general serial, parallel or hybrid robot, and by Gi (1Gi-2Gi-…-nGi) the kinematic chain associated with the ith limb (i=1,2,…,k). The end effector is n nGi and the reference link 1 1Gi. If the reference link is the fixed base, it is denoted by 1 1Gi 0. The total number of robot joints is denoted by p. A serial robot F ← G1 is a robot in which the end-effector n nG1 is connected to the reference link 1 1G1 by just one simple open kinematic chain Gi (1Gi-2Gi-…nGi) called a serial kinematic chain. A parallel robot F ← G1-G2-…-Gk is a robot in which the end-effector n nGi is connected in parallel to the reference link 1 1Gi by k≥2 kinematic chains Gi (1Gi-2Gi-…-nGi) called limbs or legs. A hybrid serial-parallel robot F ← G1 is a robot in which end-effector n nG1 is connected to reference link 1 1G1 by just one complex kinematic chain Gi (1Gi-2Gi-…nGi) called complex limb or complex leg. A fully-parallel robot F ← G1-G2-…Gk is a parallel robot in which the number of limbs is equal to the robot mobility (k=M≥2), and just one actuator exist in each limb.

10

1 Introduction

A non fully-parallel robot F ← G1-G2-…Gk is a parallel robot with fewer number of limbs than the robot mobility (k0), to indicate the number of independent parameters in robot modelling and to determine the number of inputs needed to drive the mechanism. Earlier works on the mobility of mechanisms go back to the second half of the nineteenth century. During the twentieth century, sustained efforts were made to find general methods for the determination of the mobility of any rigid body mechanism. Various formulae and approaches were derived and presented in the literature. Contributions have continued to emerge in the last few years. Mobility calculation still remains a central subject in the theory of mechanisms. In Part 1 (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, full-cycle mobility). The general mobility represents the minimum value of the instantaneous mobility in a free-of-singularity

12

1 Introduction

workspace. For a given mechanism, general mobility has a unique value for a free-of-singularity workspace. It is a global parameter characterizing the mechanism in all its configurations of the workspace except its singular ones. Instantaneous mobility is a local parameter characterizing the mechanism in a given configuration including singular ones. In a singular configuration the instantaneous mobility could be different from the general mobility. In this book, unless otherwise stated, general mobility is simply called mobility. Note 1. In a kinematotropic mechanism with branching singularities, full-cycle mobility is associated with each branch. In this case, the fullcycle mobility (global mobility) is replaced by the branch mobility which represents the minimum value of the instantaneous mobility inside the same free-of-singularity branch. As each branch has its own mobility, a single value for global mobility cannot be associated with the kinematotropic mechanisms (Gogu 2008b, c, d, 2009b, c). The term kinematotropic mechanism was coined by K. Wohlhart (1996) to define the linkages that permanently change their full-cycle mobility when passing by an instantaneous singularity from one branch to another. Various single and multiloop kinematotropic mechanisms have been presented in the literature (Wohlhart 1996, Dai and Jones 1999, Galletti and Fanghella 2001, Fanghella et al. 2006). 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

1.2 Methodology of structural synthesis

13

could lose their independence in the closed loops, and the number of joint kinematic parameters that actually lose their independence in the closed loops. The structural redundancy of a kinematic chain represents the difference between the mobility of the kinematic chain and connectivity between its distal links. Let us consider the case of the parallel mechanism F ← G1-G2-…-Gk in which the mobile platform n nGi is connected to the reference platform 1 1Gi by k simple and/or complex kinematic chains Gi (1Gi-2Gi-…-nGi) called limbs. In Part 1, the following parameters have been associated with the parallel mechanism F ← G1-G2-…-Gk : RGi - the vector space of relative velocities between the mobile and the reference platforms, nGi and 1Gi, in the kinematic chain Gi disconnected from the parallel mechanism F, RF - the vector space of relative velocities between the mobile and the reference platforms, n nGi and 1 1Gi, in the parallel mechanism F ← G1G2-…-Gk, whose basis is (RF)=( RG1 ∩ RG 2 ∩ ... ∩ RGk ),

(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 nGi and 1 1Gi, in the parallel mechanism F ← G1-G2-…Gk. We recall that the connectivity is defined by the number of independent motions between the mobile and the reference platforms. The notation 1 1Gi 0 is used when the reference platform is the fixed base. The vector spaces of relative velocities between the mobile and the reference platforms are also called operational velocity spaces. The following formulae demonstrated in Chapter 2-Part 1 (Gogu 2008a) for mobility MF, connectivity SF, number of overconstraints NF and redundancy TF of the parallel mechanism F ← G1-G2-…-Gk are used in structural synthesis of parallel robotic manipulators: M F = ∑ f i −rF , p

i =1

where

(1.2)

NF=6q-rF ,

(1.3)

TF=MF-SF ,

(1.4)

14

1 Introduction

SGi = dim( RGi ) ,

(1.5)

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

(1.6)

rF = ∑ SGi − S F + rl ,

(1.7)

p = ∑ pGi ,

(1.8)

q=p-m+1,

(1.9)

rl = ∑ rGi .

(1.10)

k

i =1

k

i =1

and

k

i =1

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 ← 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=MF≤MGi

(i=1,…,k),

(1.11)

MGi=SGi≤6

(i=1,…,k),

(1.12)

b) for the redundant parallel robots with TF>0

1.2 Methodology of structural synthesis

15

SFSGi≤6

(i=1,…,k),

(1.14)

c) for the non overconstrained parallel robots (NF=0) MF= ∑ f i − 6q , p

(1.15)

i =1

d) for the overconstrained parallel robots with NF>0 MF> ∑ f i − 6q .

(1.16)

M Gi = ∑ f i −rGi .

(1.17)

p

i =1

We recall that

pGi

i =1

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 nGi in the mechanism F ← G1-G2-…-Gk is less than or equal to the mobility MF of mechanism F. The basis of the vector space RF of relative velocities between the moving and reference platforms in the mechanism F ← G1-G2-…-Gk must be valid for any point of the moving platform n nGi. Note 2. 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 with no need to calculate the rank of the homogeneous linear set of constraint equations associated with loop closure or with the rank of the complete screw system associated to the joints of the mechanism. An analytical method to compute these parameters has also been developed in Part 1 just for verification and for a better understanding of the meaning of these parameters. 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

16

1 Introduction

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 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 endeffector 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

1.2 Methodology of structural synthesis

17

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 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. The various kinetostatic performance indices introduced in section 4.5-Part 1 have optimal values for fully-isotropic PMs (Gogu 2007f, 2008a, j) The first solutions of maximally regular and implicitly fully-isotropic parallel robot were developed at the same time and independently by Carricato and Parenti-Castelli at University of Genoa, Kim and Tsai at University of California, Gosselin and Kong at University of Laval, and the author at the French Institute 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 three groups has

18

1 Introduction

built a prototype of this T3-type translational parallel 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 physical implementation of this robot was the CPM developed at University of California by Kim and Tsai (2002). An innovative solution of fully-isotropic T3-type translational parallel robot called Pantopteron was recently proposed by Briot and Bonev (2009). In this solution based on pantograph linkages, the moving platform moves several times faster than its linear actuators. Table 1.2. Literature dedicated to maximally-regular and implicitly fully-isotropic parallel robotic manipulators No. Type of parallel robotic manipulator 1 T3-type

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

References

Carricato and Parenti-Castelli 2002 Gogu 2002, 2004a Gosselin and Kong 2002 Kim and Tsai 2002 Kong and Gosselin 2002a,b,c Rizk et al 2006 Stan et al. 2008 Wu et al. 2007 R2-type parallel wrist Gogu 2005f R3-type parallel wrist Gogu 2007b R3-type redundantly-actuated parallel Gogu 2007e wrists Planar T2R1-type Gogu 2004c Spatial T2R1-type Gogu 2008g Zhang et al. 2009 Spatial T2R1-type with planar motion Gogu 2008i of the moving platform T1R2-type Gogu 2005i T3R1-type with Schönflies Carricato 2005 motions Gogu 2004b, 2005g, 2006c, 2007a T2R2-type Gogu 2005h T1R3-type Gogu 2008h T1R3-type redundantly-actuated Gogu 2008e T2R3-type redundantly-actuated Gogu 2007d T3R2-type Gogu 2006b, d, 2009d T3R2-type redundantly-actuated Gogu 2006a T3R3-type hexapod Gogu 2006e

1.3 Parallel robots with planar motion of the moving platform

19

Various other types of maximally regular and implicitly fully-isotropic parallel robotic manipulators have been proposed in the last years (see Table 1.2). These solutions can be applied in machining applications (Gogu 2007c) or haptic devices (Gogu 2008f).

1.3 Parallel robots with planar motion of the moving platform This book focuses on the structural synthesis of three-degrees-of-freedom parallel robots with planar motion of the moving platform T2R1-type. In such a robot, the moving platform can undergo two independent translational motions T2 and one rotational motion R1 around an axis perpendicular to the plane of translations. This motion can be obtained by using planar or spatial parallel mechanisms. Overconstrained and non overconstrained solutions of parallel manipulators with coupled, decoupled and uncoupled motions of the moving platform along with maximally regular solutions are presented in the following sections of this book. These solutions are actuated by linear and/or rotating motors situated on the fixed base or on a moving link. Planar and spatial solutions of fully-parallel and non-fully parallel mechanisms are presented in this book. Basic and derived solutions are presented in each chapter. There are no idle mobilities in the basic solutions. To reduce the number of overconstraints in the parallel robot, derived solutions are used. These are obtained from the basic solutions by combining various idle mobilities. In a planar parallel mechanism, the links have a motion parallel to the same plane (planar motion). The rotation axes of revolute joints are parallel, and the directions of the translations in prismatic joints are parallel to a plane perpendicular to the rotation axes in the basic solutions. Just planar limbs with SGi=3 and (RGi)=( v1 ,v2 ,ωδ ) are used in the basic planar parallel mechanisms. Idle mobilities with other axis directions may also exist in the joints of the derived architectures. The basic solutions of spatial parallel mechanism with planar motion of the moving platform has at least one spatial limb with no idle mobilities and SGi>3. The parallel robots with planar motion of the moving platform presented in this book give two translational velocities v1 , v2 and one rotation velocity ωδ in the basis of the operational velocity vector space (RF)= ( v1 ,v2 ,ωδ ). The motion of the moving platform is parallel to the x0y0plane. The directions of velocities v1 and v2 are parallel to the x0- and

20

1 Introduction

y0-axes and the direction of ωδ is parallel to the z0- axis. These parallel robots have mobility MF=3 and the connectivity between the moving and fixed platforms is SF=3. The direct kinematic model of the parallel robots with planar motion of the moving platform becomes ⎡ v1 ⎤ ⎡&q1 ⎤ ⎢v ⎥ = J ⎢& ⎥ ⎢ 2 ⎥ [ ]3×3 ⎢ q2 ⎥ ⎢⎣ωδ ⎥⎦ ⎢⎣&q3 ⎥⎦

(1.18)

where: v1= &x and v2= &y are the translational velocities of the characteristic point H of the moving platform, ω = δ& is the rotation velocity of the moving δ

platform, &q1 , &q2 and &q3 are the velocities of the actuated joints, J 3×3 is the design Jacobian matrix. To obtain a non redundant solution of type F ← G1-G2-G3, a basic limb presented in Figs. 7.12-7.18-Part 1 is associated with two other simple or complex limbs with 3 ≤ MGi=SGi ≤ 6 that integrate velocities v1 , v2 and ωδ in the basis of their operational velocity spaces. We recall that the basic limbs in Figs. 7.12-7.18-Part 1 give rise to one rotational and two translational independent motions. 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 ≤ MGi=SGi ≤ 6 that respect the condition (RF)=( RG1 ∩ RG 2 ∩ RG 3 )= ( v1 ,v2 ,ωδ ). Parallel robots with planar motion of the moving platform are used in classical manipulation processes, machining, locomotion interfaces, compliance devices, precision positioning tables, lithographic apparatus, micro and nano manipulations (see Table 1.3). Various architectures are used in the literature to obtain three-legged parallel manipulators based on planar mechanisms (Merlet 1997). The possible combinations of revolute, R, and prismatic, P, joints which connect the moving platform to the fixed base in a three-legged PPM are: RRR, PRR, RPR, RRP, PPR, RPP and PRP. The successions of the three joints start from the fixed base to the moving platform. Since any one of the three joints in any of the seven kinematic chains may be actuated there are 21 possible limb architectures. The various combinations of these limb architectures give 1653 possible PPMs with only lower pairs possessing three DoFs (Hayes et al. 2004). We note that these architectures have coupled motions and some of them have been extensively studied in the literature.

1.3 Parallel robots with planar motion of the moving platform

21

This is the case of 3-PRR, 3-RPR and 3-RRR architectures (see Tables 1.4-1.6). Some few studies are also dedicated to the architectures of types 3-PPR (Choi 2003; Chung and Choi 2004) and 3-PRP (Chablat and Staicu 2009; Ronchi et al. 2004). As a matter of fact, the general literature dedicated to planar parallel robots is extremely rich (see Table 1.7). Spatial solutions with planar motion of the moving platform have been less thoroughly investigated. Table 1.3. Examples of practical applications of planar parallel robots No. Practical application 1 2

3

4 5 6 7

8

References

Lithographic apparatus Machining

Kwan et al. (2003) Du Plesis and Snyman (2002, 2006a) Long et al. (2003) Snyman and Smit (2002) Micro manipulation Alici and Shirinzadeh (2003) Balan et al. (2005) Bamberger et al. (2006) Movable tables Matsumoto (1992) Sakai et al. (1999) Nano manipulation Mukhopadhyay et al. (2008) Omni-directional locomotion interface Yoon and Ryu (2004) Positioning devices Bonev (2008) Burton and Burton (1996) Scheidegger and Liechti (2002) Remote centre of compliance device Kim et al. (1996a, b)

Table 1.4. Literature dedicated to the study of the 3-PRR-type planar parallel robots No. Type of study

References

1 2 3

Control Isotropy Kinematic analysis

4 5

Singularities Workspace

Sun et al. (2006) Caro et al. (2003) Gosselin et al. (1996) Staicu (2009) Staicu et al. (2007) Masouleh and Gosselin (2007) Gosselin et al. (1996)

22

1 Introduction

Table 1.5. Literature dedicated to the study of the 3-RPR-type planar parallel robots No. Type of study 1

2 3

4 5

6 7

8 9 10

References

Bonev et al. (2008) Fried et al. (2008) Husty (2009) Macho et al. (2007) Zein et al. (2008) Wenger et Chablat (2009) Cusp points Zein et al. (2006a, b, 2007a) Dimensional synthesis and Du Plessis and Snyman (2006b) Gallant and Boudreau (2002) optimization Hay and Snyman (2002) Jiang and Gosselin (2008) Lee et al. (1999) Murray and Pierrot (1998) Staicu (2008) Dynamics Collins (2002) Kinematics Kong (2008, 2009) Kong and Gosselin (2001, 2008) Merlet (1996b, 2000) Murray et al. (1997) Takeda (2005) Wenger et Chablat (2009) Wenger et al. (2007) Williams and Joshi (1999) Binaud et al. (2009a, b) Sensitivity analysis Briot et al. (2008) Singularities Chablat et al. (2006) Collins and McCarthy (1998) Husty and Gosselin (2008) Jiang and Gosselin (2008) Kong and Gosselin (2000) Sefrioui and Gosselin (1995) Wenger et Chablat (2009) Wenger et al. (2007) Yang and O'Brien (2007) Zein et al. (2007b) Static analysis Duffy (1980) Tsai (1999) Stiffness Li and Gosselin (2008a) Workspace analysis and optimization Gallant and Boudreau (2000, 2002) Jiang and Gosselin (2006, 2008) Yang and O'Brien (2007) Assembly modes

1.3 Parallel robots with planar motion of the moving platform

23

Table 1.6. Literature dedicated to the study of the 3-RRR-type planar parallel robots No. Type of study

References

1 2

Balancing Control

3

Dimensional synthesis and optimization Dynamics Kinematic performance

Arakelian and Smith (2008) Balan et al. (2005) Castillo-Castaneda et al. (2007) Yoon and Ryu (2004) Bouzgarrou et al. (2000) Geike and McPhee (2002, 2003) Guo et al. (2004) Chablat and Wenger (2001) Tsai (1999) Alba-Gomes et al. (2005) Rooney and Earl (1983) Rooney and Tanev (2002) Alba-Gomes et al. (2007) Cha et al. (2007) Binaud et al. (2009a, b) Bonev and Gosselin (2001) Chablat and Wenger (2004) Gosselin and Wang (1997) Duffy (1980) Li and Gosselin (2007) Hunt (1978, 1982, 1983) Arsenault and Boudreau (2004a, b) Liu et al. (2000)

4 5

6 7 8

Kinetostatic indices Postures and kinematic assembly configurations Redundancy

9 10

Sensitivity and dexterity Singularities

11 12 13 14

Statics Stiffness Structural kinematics Workspace analysis and optimisation

24

1 Introduction

Table 1.7. Literature dedicated to the general study of planar parallel robots No. Type of study

References

1

Accuracy and clearances

2

Balancing

3 4 5

Calibration Compliance Control and motion planning

6

Dimensional synthesis and optimization

7

Dynamics

8

Kinematics

Bamberger et al. (2006) Briot and Bonev (2008) Yu et al. (2008) Alici and Shirinzadeh (2004a, b, 2006) Briot et al. (2009) Fattah and Agrawal (2006) Foucault and Gosselin (2002, 2004) Jean and Gosselin (1996) Laliberte et al. (1999) Leblond and Gosselin (1998) Shirinzadeh and Alici (2004) Last et al. (2007) Kim et al (1996a, b) Hahn et al. (1999) Harms et al. (1991) Kang and Mills (2003, 2005) Kang et al. (2001, 2002) Li et al. (2007) Ren et al. (2004, 2005, 2006) Shao et al. (2009) Shvalb et al. (2007) Slutski (1996) Wang and Mills (2005a, b) Wu et al. (2009) Boudreau and Gosselin (1999) Du Plessis and Snyman (2002, 2006a) Gallant and Boudreau (2000, 2003) Gosselin and Angeles (1988) Hay and Snyman (2000) Du and Yu (2006) Fattah et al. (1994) Fu and Mills (2005) Kang and Mills (2002) Khan et al. (2005) Ma and Angeles (1989) Piras et al. (2005) Staicu (2008) Wang et al. (2003) Wang and Mills (2006) Wu et al. (2008) Chablat and Staicu (2009) Collins (2002)

1.3 Parallel robots with planar motion of the moving platform

25

Table 1.7. (cont.)

9

Isotropy

10

Kinetostatics

11

Optimal design and modelling

12

Reconfigurability

13

Redundancy

Gao et al. (1996) Gosselin and Merlet (1994) Gosselin et al. (1992, 1996) Hayes and Husty (2003) Hayes and Zsombor-Murray (1996, 1998) Hayes et al. (2004) Jeanneau et al. (2004) Ji (2003) Ji and Wu (2002) Kong and Gosselin (2002d) Ma and Angeles (1989) Merlet (1996a) Mohamadi Daniali et al. (1993) Mohamed and Duffy (1985) Murray and Hanchak (2000) Rolland (2006) Rooney and Tanev (2002) Staicu et al. (2007) Urizar et al. (2009) Wang et al. (2003) Wenger and Chablat (2004) Williams and Sheley (1997) Zsombor-Murray et al. (2002) Alici and Shirinzadeh (2004a) Briot and Bonev (2009) Gogu (2004c) Mohamadi-Daniali and Zsombor-Murray (1994, 1999) Mohamadi-Daniali et al. (1995c) Company et al. (2007) Rakotomanga et al. (2008) Gosselin and Angeles (1988) Long et al. (2003) Ridgeway et al. (1992) Snyman and Smit (2002) Yoon and Ryu (2004) Choi et al. (2004a, b) Fisher et al. (2001, 2004) Constantinescu et al. (2000) Cha et al (2009) Dasgupta and Mruthyunjaya (1998) Ebrahimi et al. (2007a, b, 2008) Firmani and Podhorodeski (2004, 2005, 2007)

26

1 Introduction

Table 1.7. (cont)

14 15

Sensitivity Singularities

16

Statics

17

Stiffness

18

Workspace analysis and optimization

Firmani et al. (2007) Hahn et al. (1999) Lösch (1995) Marquet et al. (2001a,b) Müller A (2005) Nokleby et al. (2007a, b) Wu et al. (2008) Yi et al. (2002) Zibil et al. (2007) Alici and Shirinzadeh (2004b, 2006) Arsenault and Boudreau (2006) Bonev et al. (2001, 2003) Cha et al. (2009) Collins and McCarthy (1998) Degani and Wolf (2006a, b) Di Gregorio (2009) Firmani and Podhorodeski (2009) Gosselin and Wang (1995, 1997) Heerah et al. (2002, 2003) Li and Gosselin (2008b) Li et al. (2006) Mohamadi-Daniali (2005) Mohamadi-Daniali et al. (1995a, b) Sefrioui and Gosselin (1992) Theingi et al. (2004) Yang et al (2002) Duffy (1980) Weiwei and Shuang (2006) Alici and Shirinzadeh (2003) Arsenault and Boudreau (2004b, 2006) Kim et al. (1997) Simaan and Shoham (2002) Arsenault and Boudreau (2004a) Gao et al. (2001) Gosselin and Jean (1996) Hay and Snyman (2005, 2006) Hayes and Husty (2000) Husty (1996) Li et al. (2006) Merlet et al. (1998) Snyman and Hay (2000) Stachera (2005) Urizar et al. (2009)

2 Overconstrained planar parallel robots with coupled motions

In the general case, in a planar parallel robotic manipulator (PPM) with coupled motions each operational velocity depends in the general case on & 2 ,q & 3 ) , v2 = v2 ( &q1 ,q & 2 ,q & 3 ) and three actuated joint velocities: v1 = v1 ( &q1 ,q & 2 ,q & 3 ) . In some specific solutions, one or two operational ωδ = ωδ ( &q1 ,q velocities depend on just one or two actuator velocities. We note that, in this particular case, the Jacobian matrix in Eq. (6.1) is not triangular and the parallel robot always has coupled motions. They have just a few partially decoupled motions. The overconstrained solutions of PPMs with coupled motions and q p independent loops meet the condition ∑ 1 f i < 3 + 6q . They may have

identical limbs or limbs with different structures and could be actuated by linear or rotating motors. The limbs can be simple or complex kinematic chains and can also combine idle mobilities. The actuators can be mounted on the fixed base or on a moving link. The first solution has the advantage of reducing the moving masses and large workspace. The second solution would be more compact.

2.1 Basic solutions The basic solutions presented in this section are either fully-parallel and non fully-parallel solutions. There are no idle mobilities in the basic solutions. 2.1.1 Fully-parallel solutions In the fully-parallel solutions of PPMs with coupled motions F ← G1-G2-G3 presented in this section, the moving platform n nGi (i=1, 2, 3) is connected to the reference platform 1 1Gi 0 by three planar limbs with three G. Gogu, Structural Synthesis of Parallel Robots, Solid Mechanics and Its Applications 173, DOI 10.1007/978-90-481-9831-3_2, © Springer Science + Business Media B.V. 2010

27

28

2 Overconstrained planar parallel robots with coupled motions

degrees of connectivity. One actuator is combined in a revolute or prismatic pair of each limb. The various types of planar simple and complex limbs with three degrees of connectivity used in the basic solutions illustrated in this section are presented in Figs. 2.1-2.3. The simple limbs combine only revolute and prismatic joints. One (Fig. 2.2a, c-g) or two (Figs. 2.2b, h and 2.3a) planar parallelogram loops are combined in the complex limbs. One telescopic parallelogram loop Pat is combined in the complex limb in Fig. 2.3b. Various solutions of PPMs with coupled motions and no idle mobilities can be obtained by using three limbs with identical or different topology presented in Figs. 2.1-2.3. Only solutions with three identical limb types are illustrated in Figs. 2.4-2.9. The revolute joints in the three limbs have parallel axes and the directions of the prismatic joints are parallel to a plane perpendicular to the rotation axes of the revolute joints. The actuator is mounted on the fixed base in the solutions illustrated in Figs. 2.4-2.9 excepting the solutions in Fig.2.5a and Fig. 2.9a in which the linear actuator is mounted on a moving link. All motions are coupled in the solutions presented in Figs. 2.4-2.9 and & 2 ,q & 3 ) , v2 = v2 ( &q1 ,q & 2 ,q & 3 ) and ωδ = ωδ ( &q1 ,q & 2 ,q & 3 ) - see Table v1 = v1 ( &q1 ,q 2.1. The prismatic joints between links 4 and 5 (Fig. 2.9a) is actuated to obtain solutions with coupled motions. If a revolute joint mounted on the fixed base and combined in the parallelogram loops is actuated, solutions with one uncoupled translational motion can be obtained and v1 = v1 ( &q1 ) . The particular case with one uncoupled translation v1 = v1 ( &q1 ) can also be obtained by using the solution in Figs. 2.10. Other solutions with this partially decoupled translation can be obtained (Figs. 2.11-2.15) by combining G1-limb in Fig. 2.1f or 2.2e with other two identical or different limbs in Figs. 2.1a-e and 2.2a-d. The particular case with partially decoupled translational motions & 2 ) and v2 = v2 ( &q1 ,q & 2 ) in Figs. 2.16-2.18 are obtained from the v1 = v1 ( &q1 ,q solutions in Figs. 2.4a and 2.5a by superposing the axes of the revolute joints connecting limbs G1 and G2 to the moving platform. The axes of the three actuated revolute joints are also superposed in the solutions presented in Figs. 2.16b and 2.17b. In this way, the workspace can be significantly increased. The limb topology and connecting conditions of the solutions in Figs. 2.4-2.17 are systematized in Table 2.2, as are their structural parameters in Tables 2.4-2.6. The particular case with decoupled rotational motion ωδ = ωδ ( &q3 ) can be obtained by using three limbs with identical or different topology

2.1 Basic solutions

29

presented in Figs. 2.1g and 2.2f-h. Solutions with three identical limb types but different actuation selection are illustrated in Figs. 2.18-2.20. One actuator is mounted on the fixed base and the other two are mounted on a moving link in these solutions. The limb topology and connecting conditions of the solutions in Figs. 2.18-2.20 are systematized in Table 2.3, as are their structural parameters in Table 2.6. The first revolute joint of the three limbs have the same rotation axis. The rotation axis of the moving platform has a fixed position in these solutions. Solutions with decoupled and unlimited rotational motion of the moving platform with different limb topologies are illustrated in Figs. 2.21-2.36. These solutions use two identical limbs (Figs. 2.1a-c and 2.2a-c) for positioning the moving platform and a different limb (Figs. 2.1g, 2.2f-g and 2.3) for rotating it. Their limb topology and connecting conditions are systematized in Table 2.3. The rotation axis of the moving platform has a fixed position in the solutions in Figs. 2.21-2.30 and a variable position in the solutions in Figs. 2.31-2.36. The solutions using G3-limb can provide an unlimited angle of rotation of the moving platform with variable position of the rotation axis. The structural parameters of the solutions in Figs. 2.21-2.36 are systematized in Tables 2.6-2.10.

30

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.1. Simple limbs for PPMs with coupled motions defined by MG=SG=3, (RG)=( v1 , v2 , δ )

2.1 Basic solutions

31

Fig. 2.2. Complex limbs for PPMs with coupled motions defined by MG=SG=3, (RG)=( v1 , v2 , δ ), and combining one or two planar parallelogram loops

32

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.3. Complex limbs for PPMs with coupled motions defined by MG=SG=3, (RG)=( v1 , v2 , δ ), and combining planar parallelogram loops (a) or a telescopic parallelogram loop (b)

Table 2.1. Motion coupling in overconstrained PPM with no idle mobilities presented in Figs. 2.4-2.36 No. Motion coupling 1

2

3

4

5

Examples

Note

The two translational and one & 2 ,q & 3 ) Figs. 2.4-2.9 v1 = v1 ( &q1 ,q rotational motions are coupled & 2 ,q &3 ) v2 = v2 ( &q1 ,q & 2 ,q &3 ) ωδ = ωδ ( &q1 ,q Figs. 2.10-2.15 One decoupled translational v1 = v1 ( &q1 ) motion & 2 ,q &3 ) v2 = v2 ( &q1 ,q & 2 ,q &3 ) ωδ = ωδ ( &q1 ,q

Figs. 2.16-2.17 Partially decoupled translational &2 ) v1 = v1 ( &q1 ,q motions &2 ) v2 = v2 ( &q1 ,q & 2 ,q &3 ) ωδ = ωδ ( &q1 ,q & 2 ,q & 3 ) Figs. 2.18-2.20 Decoupled rotational motion v1 = v1 ( &q1 ,q & 2 ,q &3 ) v2 = v2 ( &q1 ,q ωδ = ωδ ( &q1 ) Figs. 2.21-2.36 Uncoupled rotational motion and &2 ) v1 = v1 ( &q1 ,q partially decoupled translational &2 ) v2 = v2 ( &q1 ,q motions ωδ = ωδ ( &q3 )

2.1 Basic solutions

33

Table 2.2. Limb topology and connecting conditions of the overconstrained PPM with no idle mobilities presented in Figs. 2.4-2.17 No. PPM type

Limb topology

Connecting conditions

1

R||R||R (Fig. 2.1a) R||R ⊥ P (Fig. 2.1d)

The directions of the revolute joints of the three limbs are parallel. The directions of the revolute joints of the three limbs are parallel. The directions of the prismatic joints are parallel to a plane perpendicular to the axis of the revolute joints. Idem No. 2

2

3 4 5 6 7 8 9 10 11 12

3-RRR (Fig. 2.4a) 3-RRP (Fig. 2.4b)

3-RPR (Fig. 2.5a) 3-PRR (Fig. 2.5b) 3-PRP (Fig. 2.6) 3-PaPaR (Fig. 2.7) 3-PaRR (Fig. 2.8a) 3-PaRP (Fig. 2.8b) 3-PaPR (Fig. 2.9a) 3-PPaR (Fig. 2.9b) 3-PPR (Fig. 2.10) 1PPR-2RRR (Fig. 2.11)

13

1PPR-2RRP (Fig. 2.12a)

14

1PPR-2RPR (Fig. 2.12b)

R ⊥ P ⊥ ||R (Fig. 2.1b) P ⊥ R||R (Fig. 2.1c) P⊥ R⊥ P (Fig. 2.1e) Pa||Pa||R (Fig. 2.2b) Pa||R||R (Fig. 2.2a) Pa||R ⊥ P (Fig. 2.2d) Pa ⊥ P ⊥ ||R (Fig. 2.2e) P ⊥ Pa ||R (Fig. 2.2c) P ⊥ P ⊥⊥ R (Fig. 2.1f) P ⊥ P ⊥⊥ R (Fig. 2.1f) R||R||R (Fig. 2.1a) P ⊥ P ⊥⊥ R (Fig. 2.1f) R||R ⊥ P (Fig. 2.1d) P ⊥ P ⊥⊥ R (Fig. 2.1f) R ⊥ P ⊥ ||R (Fig. 2.1b)

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

Idem No. 2

Idem No. 2

34

2 Overconstrained planar parallel robots with coupled motions

Table 2.2. (cont.) 15

1PPR-2PRR (Fig. 2.13a)

16

1PPR-2PRP (Fig. 2.13b)

17

1PaPR-2PaRR (Fig. 2.14a)

18

1PaPR-2PPaR (Fig. 2.14b)

19

1PaPR-2PaPaR (Fig. 2.15a)

20

1PaPR-2PaRP (Fig. 2.15b)

21

3RRR (Fig. 2.16a)

P ⊥ P ⊥⊥ R (Fig. 2.1f) P ⊥ R||R (Fig. 2.1c) P ⊥ P ⊥⊥ R (Fig. 2.1f) P⊥ R⊥ P (Fig. 2.1e) Pa ⊥ P ⊥ ||R (Fig. 2.2e) Pa||R||R (Fig. 2.2a) Pa ⊥ P ⊥ ||R (Fig. 2.2e) P ⊥ Pa||R (Fig. 2.2c) Pa ⊥ P ⊥ ||R (Fig. 2.2e) Pa||Pa||R (Fig. 2.2b) Pa ⊥ P ⊥ ||R (Fig. 2.2e) Pa||R ⊥ P (Fig. 2.2d) R||R||R (Fig. 2.1a)

22

3RRR (Fig. 2.16b)

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

23

3RPR (Fig. 2.17a)

R ⊥ P ⊥ ||R (Fig. 2.1b)

24

2RPR-1RPR (Fig. 2.17b)

R ⊥ P ⊥ ||R R ⊥ P ⊥ ||R (Fig. 2.1b)

Idem No. 2

Idem No. 2

Idem No. 2

Idem No. 2

Idem No. 2

Idem No. 2

Idem No. 1 The axes of the revolute joints connecting limbs G1 and G2 to the moving platform are superposed. Idem No. 21 The axes of the three actuated revolute joints are superposed. Idem 2 The axes of the revolute joints connecting limbs G1 and G2 to the moving platform are superposed. Idem 23 The axes of the first revolute joints of the three limbs are superposed.

2.1 Basic solutions

35

Table 2.3. Limb topology and connecting conditions of the overconstrained PPM with no idle mobilities presented in Figs. 2.18-2.36 No. PPM type

Limb topology

1

2RPP-1RPP (Fig. 2.18)

R ⊥ P ⊥⊥ P (Fig. 2.1g)

2

2RPPa-1RPPa (Fig. 2.19) 2RPaP-1RPaP (Fig. 2.20a) 2RPaPa-1RPaPa (Fig. 2.20b) 2PRR-1RPP (Fig. 2.21a)

3 4 5

6

2RRR-1RPP (Fig. 2.21b)

7

2RPR-1RPP (Fig. 2.21c)

8

2PPaR-1RPP (Fig. 2.22a)

9

2PaRR-1RPP (Fig. 2.22b)

10

2PaPaR-1RPP (Fig. 2.22c)

11

2PRR-1RPPa (Fig. 2.23a)

R ⊥ P ⊥ ||Pa (Fig. 2.2f) R||Pa ⊥ P (Fig. 2.2g) R||Pa||Pa (Fig. 2.2h) P ⊥ R||R (Fig. 2.1c) R ⊥ P ⊥⊥ P (Fig. 2.1g) R||R||R (Fig. 2.1a) R ⊥ P ⊥⊥ P (Fig. 2.1g) R ⊥ P ⊥ ||R (Fig. 2.1b) R ⊥ P ⊥⊥ P (Fig. 2.1g) P ⊥ Pa||R (Fig. 2.2c) R ⊥ P ⊥⊥ P (Fig. 2.1g) Pa||R||R (Fig. 2.2a) R ⊥ P ⊥⊥ P (Fig. 2.1g) Pa||Pa||R (Fig. 2.2b) R ⊥ P ⊥⊥ P (Fig. 2.1g) P ⊥ R||R (Fig. 2.1c) R ⊥ P ⊥ ||Pa (Fig. 2.2f)

Connecting conditions The directions of the revolute joints of the three limbs are parallel. The directions of the prismatic joints are parallel to a plane perpendicular to the axis of the revolute joints. Idem No. 1 Idem No. 1 Idem No. 1 Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

36

2 Overconstrained planar parallel robots with coupled motions

Table 2.3. (cont.) 12

2RRR-1RPPa (Fig. 2.23b)

13

2RPR-1RPPa (Fig. 2.23c)

14

2PRR-1RPaP (Fig. 2.24a)

15

2RRR-1RPaP (Fig. 2.24b)

16

2RPR-1RPaP (Fig. 2.24c)

17

2PRR-1RPaPa (Fig. 2.25a)

18

2RRR-1RPaPa (Fig. 2.25b)

19

2RPR-1RPaPa (Fig. 2.25c)

20

2PPaR-1RPPa (Fig. 2.26a)

21

2PaRR-1RPPa (Fig. 2.26b)

22

2PPaR-1RPaP (Fig. 2.27a)

R||R||R (Fig. 2.1a) R ⊥ P ⊥ ||Pa (Fig. 2.2f) R ⊥ P ⊥ ||R (Fig. 2.1b) R ⊥ P ⊥ ||Pa (Fig. 2.2f) P ⊥ R||R (Fig. 2.1c) R||Pa ⊥ P (Fig. 2.2g) R||R||R (Fig. 2.1a) R||Pa ⊥ P (Fig. 2.2g) R ⊥ P ⊥ ||R (Fig. 2.1b) R||Pa ⊥ P (Fig. 2.2g) P ⊥ R||R (Fig. 2.1c) R||Pa||Pa (Fig. 2.2h) R||R||R (Fig. 2.1a) R||Pa||Pa (Fig. 2.2h) R ⊥ P ⊥ ||R (Fig. 2.1b) R||Pa||Pa (Fig. 2.2h) P ⊥ Pa||R (Fig. 2.2c) R ⊥ P ⊥ ||Pa (Fig. 2.2f) Pa||R||R (Fig. 2.2a) R ⊥ P ⊥ ||Pa (Fig. 2.2f) P ⊥ Pa||R (Fig. 2.2c) R||Pa ⊥ P (Fig. 2.2g)

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

2.1 Basic solutions

37

Table 2.3. (cont) 23

24

25

26

27

28

29

30

31

32

33

Pa||R||R (Fig. 2.2a) R||Pa ⊥ P (Fig. 2.2g) 2PaPaR-1RPPa Pa||Pa||R (Fig. 2.28a) (Fig. 2.2b) R ⊥ P ⊥ ||Pa (Fig. 2.2f) 2PaPaR-1RPaP Pa||Pa||R (Fig. 2.28b) (Fig. 2.2b) R||Pa ⊥ P (Fig. 2.2g) 2PPaR-1RPaPa P ⊥ Pa||R (Fig. 2.29a) (Fig. 2.2c) R||Pa||Pa (Fig. 2.2h) 2PaRR-1RPaPa Pa||R||R (Fig. 2.29b) (Fig. 2.2a) R||Pa||Pa (Fig. 2.2h) 2PaPaR-1RPaPa Pa||Pa||R (Fig. 2.30) (Fig. 2.2b) R||Pa||Pa (Fig. 2.2h) 2PRR-1RPaPa P ⊥ R||R (Fig. 2.31a) (Fig. 2.1c) R||Pa||Pa (Fig. 2.3a) 2PRR-1RPaPat P ⊥ R||R (Fig. 2.1c) (Fig. 2.31b) R||Pa||Pat (Fig. 2.3b) 2RRR-1RPaPa R||R||R (Fig. 2.32a) (Fig. 2.1a) R||Pa||Pa (Fig. 2.3a) 2RRR-1RPaPat R||R||R (Fig. 2.32b) (Fig. 2.1a) R||Pa||Pat (Fig. 2.3b) 2RPR-1RPaPa R ⊥ P ⊥ ||R (Fig. 2.33a) (Fig. 2.1b) R||Pa||Pa (Fig. 2.3a) 2PaRR-1RPaP (Fig. 2.27b)

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

The directions of the revolute joints of the three limbs are parallel.

Idem No. 27

Idem No. 1

Idem No. 1

Idem No. 27

Idem No. 1

Idem No. 1

38

2 Overconstrained planar parallel robots with coupled motions

Table 2.3. (cont.) 34

35

36

37

38

39

40

R ⊥ P ⊥ ||R (Fig. 2.1b) R||Pa||Pat (Fig. 2.3b) 2PPaR-1RPaPa P ⊥ Pa||R (Fig. 2.34a) (Fig. 2.2c) R||Pa||Pa (Fig. 2.3a) 2PPaR-1RPaPat P ⊥ Pa||R (Fig. 2.2c) (Fig. 2.34b) R||Pa||Pat (Fig. 2.3b) 2PaRR-1RPaPa Pa||R||R (Fig. 2.35a) (Fig. 2.2a) R||Pa||Pa (Fig. 2.3a) 2PaRR-1RPaPat Pa||R||R (Fig. 2.35b) (Fig. 2.2a) R||Pa||Pat (Fig. 2.3b) 2PaPaR-1RPaPa Pa||Pa||R (Fig. 2.36a) (Fig. 2.2b) R||Pa||Pa (Fig. 2.3a) 2PaPaR-1RPaPat Pa||Pa||R (Fig. 2.36b) (Fig. 2.2b) R||Pa||Pat (Fig. 2.3b) 2RPR-1RPaPat (Fig. 2.33b)

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 27

Idem No. 1

Idem No. 27

Idem No. 1

2.1 Basic solutions

39

Table 2.4. Structural parameters of planar parallel mechanisms in Figs. 2.4-2.10 No. Structural Solution parameter 3-RRR , 3-RRP (Fig. 2.4a,b) 3-RPR, 3-PRR (Fig. 2.5a,b) 3-PRP (Fig. 2.6) 3-PPR (Fig. 2.10) 1 m 8 2 p1 3 3 p2 3 4 p3 3 5 p 9 6 q 2 7 k1 3 8 k2 0 9 k 3 10 (RG1) ( v1 , v 2 , δ ) 11 (RG2) ( v1 , v2 , δ ) 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

fj

( v1 , v2 , 3 3 3 0 0 0 3 3 3 ( v1 , v2 , 3 0 6 3 6 0 3

δ

)

δ

)

3-PaPaR (Fig. 2.7)

20 9 9 9 27 8 0 3 3 ( v1 , v2 , ( v1 , v2 , ( v1 , v2 , 3 3 3 6 6 6 3 3 3 ( v1 , v2 , 3 18 24 3 24 0 9

)

3-PaRR (Fig. 2.8a) 3-PaRP (Fig. 2.8b) 3-PaPR (Fig. 2.9a) 3-PPaR (Fig. 2.9b) 14 6 6 6 18 5 0 3 3 ( v1 , v 2 , δ )

)

( v1 , v 2 ,

δ

)

δ

)

( v1 , v 2 , 3 3 3 3 3 3 3 3 3 ( v1 , v 2 , 3 9 15 3 15 0 6

δ δ

fj

3

9

6

fj

3

9

6

fj

9

27

18

δ

)

δ

)

δ

)

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

40

2 Overconstrained planar parallel robots with coupled motions

loops in the parallel mechanismb, k1 number of simple limbs, k2 number of complex limbs, k total number of limbsc, (RGi) basis of the vector space of relative velocities between the moving and reference platforms in Gi-limb disconnected from the parallel mechanism, SGi connectivity between the moving and reference platforms in Gi-limb disconnected from the parallel mechanismd, rGi number of joint parameters that lost their independence in the closed loops combined in Gi-limb, MGi mobility of Gi-limbe, (RF) basis of the vector space of relative velocities between the moving and reference platforms in the parallel mechanismf, SF connectivity between the mobile and reference platforms in the parallel mechanismg, rl total number of joint parameters that lose their independence in the closed loops combined in the k limbsh, rF total number of joint parameters that lose their independence in the closed loops combined in the parallel mechanismi, MF mobility of the parallel mechanismj, NF number of overconstraints in the parallel mechanismk, TF degree of structural redundancy of the parallel mechanisml, fj mobility of jth joint. p= ∑ i =1 pGi , k

a

b

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

c

MGi= ∑ j =Gi1 f j − rGi , i=1,2,...,k, p

e

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

f

g h

rF = ∑ i =1 SGi − S F + rl , M F = ∑ i =1 f i − rF , p

N F = 6q − rF , TF = M F − S F .

k l

k

k

i j

rl= ∑ i =1 rGi ,

2.1 Basic solutions

41

Table 2.5. Structural parametersa of planar parallel mechanisms in Figs. 2.11-2.15 No. Structural Solution parameter 1PPR-2RRR (Fig. 2.11) 1PPR-2RRP (Fig. 2.12a) 1PPR-2RPR (Fig. 2.12b) 1PPR-2PRR (Fig. 2.13a) 1PPR-2PRP (Fig. 2.13b) 1 m 8 2 p1 3 3 p2 3 4 p3 3 5 p 9 6 q 2 7 k1 3 8 k2 0 9 k 3 10 (RG1) ( v1 , v 2 , δ ) 11 (RG2) ( v1 , v 2 , δ ) 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 a

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

( v1 , v 2 , 3 3 3 0 0 0 3 3 3 ( v1 , v 2 , 3 0 6 3 6 0 3

δ

)

δ

)

1PaPR-2PaRR 1PaPR-2PPaR (Fig. 2.14a, b) 1PaPR-2PaRP (Fig. 2.15b) 14 6 6 6 18 5 0 3 3 ( v1 , v 2 , δ ) ( v1 , v 2 , ( v1 , v 2 , 3 3 3 3 3 3 3 3 3 ( v1 , v 2 , 3 9 15 3 15 0 6

δ

)

δ

)

δ

)

1PaPR-2PaPaR (Fig. 2.15a)

18 6 9 9 24 7 0 3 3 ( v1 , v 2 , ( v1 , v 2 , ( v1 , v 2 , 3 3 3 3 6 6 3 3 3 ( v1 , v 2 , 3 15 21 3 21 0 6

fj

3

6

9

fj

3

6

9

fj

9

18

24

See footnote of Table 2.4 for the nomenclature of structural parameters

δ δ

) )

δ

)

δ

)

42

2 Overconstrained planar parallel robots with coupled motions

Table 2.6. Structural parametersa of planar parallel mechanisms in Figs. 2.16-2.21 No. Structural Solution parameter 3RRR (Fig. 2.16) 3RPR, 2RPR-1RPR (Fig. 2.17) 2RPP-1RPP (Fig. 2.18) 2PRR-1RPP, 2RRR-1RPP, 2RPR-1RPP (Fig. 2.21) 1 m 8 2 p1 3 3 p2 3 4 p3 3 5 p 9 6 q 2 7 k1 3 8 k2 0 9 k 3 10 (RG1) ( v1 , v 2 , δ ) 11 (RG2) ( v1 , v2 , δ ) 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 a

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

( v1 , v2 , 3 3 3 0 0 0 3 3 3 ( v1 , v2 , 3 0 6 3 6 0 3

δ

)

δ

)

2RPPa-1RPPa 2RPaPa-1RPaPa (Fig. 2.19) (Fig. 2.20b) 2RPaP-1RPaP (Fig. 2.20a) 14 6 6 6 18 5 0 3 3 ( v1 , v2 , ( v1 , v2 , ( v1 , v2 , 3 3 3 3 3 3 3 3 3 ( v1 , v2 , 3 9 15 3 15 0 6

δ δ

) )

δ

)

δ

)

20 9 9 9 27 8 0 3 3 ( v1 , v2 , ( v1 , v2 , ( v1 , v2 , 3 3 3 6 6 6 3 3 3 ( v1 , v2 , 3 18 24 3 24 0 9

fj

3

6

9

fj

3

6

9

fj

9

18

27

See footnote of Table 2.4 for the nomenclature of structural parameters

δ δ

) )

δ

)

δ

)

2.1 Basic solutions

43

Table 2.7. Structural parametersa of planar parallel mechanisms in Figs. 2.22-2.24 No. Structural Solution parameter 2PPaR-1RPP (Fig. 2.22a) 2PaRR-1RPP (Fig. 2.22b) 1 m 12 2 p1 6 3 p2 6 4 p3 3 5 p 15 6 q 4 7 k1 1 8 k2 2 9 k 3 10 (RG1) ( v1 , v 2 , δ ) 11 (RG2) ( v1 , v 2 , δ ) 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 a

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

( v1 , v 2 , 3 3 3 3 3 0 3 3 3 ( v1 , v 2 , 3 6 12 3 12 0 6

δ

)

δ

)

2PaPaR-1RPP 2PRR-1RPPa, 2RRR-1RPPa (Fig. 2.22c) 2RPR-1RPPa (Fig. 2.23a,b,c) 2PRR-1RPaP, 2RRR-1RPaP 2RPR-1RPaP (Fig. 2.24a,b,c) 16 10 9 3 9 3 3 6 21 12 6 3 1 2 2 1 3 3 ( v1 , v2 , δ ) ( v1 , v 2 , δ ) ( v1 , v2 , ( v1 , v2 , 3 3 3 6 6 0 3 3 3 ( v1 , v2 , 3 12 18 3 18 0 9

δ

)

( v1 , v 2 ,

δ

)

δ

)

( v1 , v 2 , 3 3 3 0 0 3 3 3 3 ( v1 , v 2 , 3 3 9 3 9 0 3

fj

6

9

3

fj

3

3

6

fj

15

21

12

δ

)

δ

)

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

44

2 Overconstrained planar parallel robots with coupled motions

Table 2.8. Structural parametersa of planar parallel mechanisms in Figs. 2.25-2.28 No. Structural Solution parameter 2PRR-1RPaPa 2RRR-1RPaPa 2RPR-1RPaPa (Fig. 2.25a,b,c) 1 m 12 2 p1 3 3 p2 3 4 p3 9 5 p 15 6 q 4 7 k1 2 8 k2 1 9 k 3 10 (RG1) ( v1 , v 2 , δ ) 11 (RG2) ( v1 , v2 , δ ) 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 a

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

( v1 , v2 , 3 3 3 0 0 6 3 3 3 ( v1 , v2 , 3 6 12 3 12 0 3

δ

)

δ

)

2PPaR-1RPPa (Fig. 2.26a) 2PaRR-1RPPa (Fig. 2.26b) 2PPaR-1RPaP (Fig. 2.27a) 2PaRR-1RPaP (Fig. 2.27b) 14 6 6 6 18 5 0 3 3 ( v1 , v2 , δ )

2PaPaR-1RPPa (Fig. 2.28a) 2PaPaR-1RPaP (Fig. 2.28b) 18 9 9 6 24 7 0 3 3 ( v1 , v2 , δ )

( v1 , v2 ,

)

( v1 , v2 ,

δ

)

δ

)

( v1 , v2 , 3 3 3 6 6 3 3 3 3 ( v1 , v2 , 3 15 21 3 21 0 9

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

δ

fj

3

6

9

fj

9

6

6

fj

15

18

24

See footnote of Table 2.4 for the nomenclature of structural parameters

δ

)

δ

)

δ

)

2.1 Basic solutions

45

Table 2.9. Structural parametersa of planar parallel mechanisms in Figs. 2.29-2.33 No. Structural Solution parameter 2PPaR-1RPaPa 2PaPaR-1RPaPa 2PRR-1RPaPa (Fig. 2.31a) (Fig. 2.29a) (Fig. 2.30) 2PRR-1RPaPat (Fig. 2.31b) 2PaRR-1RPaPa 2RRR-1RPaPa (Fig. 2.32a) (Fig. 2.29b) 2RRR-1RPaPat (Fig. 2.32b) 2RPR-1RPaPa (Fig. 2.33a) 2RPR-1RPaPat (Fig. 2.33b) 1 m 16 20 12 2 p1 6 9 3 3 p2 6 9 3 4 p3 9 9 9 5 p 21 27 15 6 q 6 8 4 7 k1 0 0 2 8 k2 3 3 1 9 k 3 3 3 10 (RGi) ( v1 , v2 , δ ) ( v1 , v2 , δ ) ( v1 , v 2 , δ ) i=1,2,3 11 SG1 3 3 3 12 SG2 3 3 3 13 SG3 3 3 3 14 rG1 3 6 0 15 rG2 3 6 0 16 rG3 6 6 6 17 MG1 3 3 3 18 MG2 3 3 3 19 MG3 3 3 3 20 (RF) ( v1 , v2 , δ ) ( v1 , v2 , δ ) ( v1 , v2 , δ ) 21 SF 3 3 3 22 rl 12 18 6 23 rF 18 24 12 24 MF 3 3 3 25 NF 18 24 12 26 TF 0 0 0 p1 27 6 9 3 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

fj

6

9

3

fj

9

9

9

fj

21

27

15

See footnote of Table 2.4 for the nomenclature of structural parameters

46

2 Overconstrained planar parallel robots with coupled motions

Table 2.10. Structural parametersa of planar parallel mechanisms in Figs. 2.342.36 No. Structural parameter

1 2 3 4 5 6 7 8 9 10

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

11

(RG2)

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 a

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

Solution 2PPaR-1RPaPa (Fig. 2.34a) 2PPaR-1RPaPat (Fig. 2.34b) 2PaRR-1RPaPa (Fig. 2.35a) 2PaRR-1RPaPat (Fig. 2.35b) 16 6 6 9 21 6 0 3 3 ( v1 , v2 , δ ) ( v1 , v2 , ( v1 , v2 , 3 3 3 3 3 6 3 3 3 ( v1 , v2 , 3 12 18 3 18 0 6

δ

)

δ

)

δ

)

2PaPaR-1RPaPa (Fig. 2.36a) 2PaPaR-1RPaPat (Fig. 2.36b)

20 9 9 9 27 8 0 3 3 ( v1 , v2 , ( v1 , v2 , ( v1 , v2 , 3 3 3 6 6 6 3 3 3 ( v1 , v2 , 3 18 24 3 24 0 9

fj

6

9

fj

9

9

fj

21

27

δ δ

) )

δ

)

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

2.1 Basic solutions

47

Fig. 2.4. Overconstrained PPMs with coupled motions of types 3-RRR (a) and 3RRP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=6, limb topology R||R||R (a) and R||R ⊥ P (b)

48

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.5. Overconstrained PPMs with coupled motions of types 3-RPR (a) and 3PRR (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=6, limb topology R ⊥ P ⊥ ||R (a) and P ⊥ R||R (b)

2.1 Basic solutions

49

Fig. 2.6. Overconstrained PPM with coupled motions of type 3-PRP defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=6, limb topology P ⊥ R ⊥ P

Fig. 2.7. Overconstrained PPM with coupled motions of type 3-PaPaR defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=24, limb topology Pa||Pa||R

50

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.8. Overconstrained PPMs with coupled motions of types 3-PaRR (a) and 3PaRP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=15, limb topology Pa||R||R (a) and Pa||R ⊥ P (b)

2.1 Basic solutions

51

Fig. 2.9. Overconstrained PPMs with coupled motions of types 3-PaPR (a) and 3PPaR (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=15, limb topology Pa ⊥ P ⊥ ||R (a) and P ⊥ Pa||R (b)

52

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.10. Overconstrained PPM with one decoupled translation of type 3-PPR defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=6, limb topology P ⊥ P ⊥ ⊥ R

Fig. 2.11. Overconstrained PPM with one decoupled translation of type 1PPR2RRR defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=6, limb topology P ⊥ P ⊥ ⊥ R and R||R||R

2.1 Basic solutions

53

Fig. 2.12. Overconstrained PPMs with one decoupled translation of types 1PPR2RRP (a) and 1PPR-2RPR (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=6, limb topology P ⊥ P ⊥ ⊥ R and R||R ⊥ P (a), P ⊥ P ⊥ ⊥ R and R ⊥ P ⊥ ||R (b)

54

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.13. Overconstrained PPMs with one decoupled translation of types 1PPR2PRR (a) and 1PPR-2PRP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=6, limb topology P ⊥ P ⊥ ⊥ R and P ⊥ R||R (a), P ⊥ P ⊥ ⊥ R and P ⊥ R ⊥ P (b)

2.1 Basic solutions

55

Fig. 2.14. Overconstrained PPMs with one decoupled translation of types 1PaPR2PaRR (a) and 1PaPR-2PPaR (b) defined by MF=SF=3, (RF)= ( v1 , v2 , δ ), TF=0, NF=15, limb topology Pa ⊥ P ⊥ ||R and Pa||R||R (a), Pa ⊥ P ⊥ ||R and P ⊥ Pa||R (b)

56

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.15. Overconstrained PPMs with one decoupled translation of types 1PaPR2PaPaR (a) and 1PaPR-2PaRP (b) defined by MF=SF=3, (RF)= ( v1 , v2 , δ ), TF=0, NF=21 (a), NF=15 (b), limb topology Pa ⊥ P ⊥ ||R and Pa||Pa||R (a), Pa ⊥ P ⊥ ||R and Pa||R ⊥ P (b)

2.1 Basic solutions

57

Fig. 2.16. 3RRR-type overconstrained PPMs with partially decoupled translations defined by MF=SF=3, (RF)= ( v1 , v2 , δ ), TF=0, NF=6, limb topology R||R||R

58

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.17. Overconstrained PPMs with partially decoupled translations of types 3RPR (a) and 2RPR-1RPR (b) defined by MF=SF=3, (RF)= ( v1 , v2 , δ ), TF=0, NF=6, limb topology R ⊥ P ⊥ ||R (a), R ⊥ P ⊥ ||R and R ⊥ P ⊥ ||R (b)

2.1 Basic solutions

59

Fig. 2.18. Overconstrained PPM with decoupled rotation of type 2RPP-1RPP defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=6, limb topology R ⊥ P ⊥ ⊥ P

Fig. 2.19. Overconstrained PPM with decoupled rotation of type 2RPPa-1RPPa, NF=21 defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=15, limb topology R ⊥ P ⊥ ||Pa

60

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.20. Overconstrained PPMs with decoupled rotation of types 2RPaP-1RPaP (a) and 2RPaPa-1RPaPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=15, (a), NF=24 (b), limb topology R||Pa ⊥ P (a) and R||Pa||Pa (b)

2.1 Basic solutions

61

Fig. 2.21. Overconstrained PPMs with uncoupled rotation of types 2PRR-1RPP (a), 2RRR-1RPP (b) and 2RPR-1RPP (c) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=6, limb topology R ⊥ P ⊥ ⊥ P and P ⊥ R||R (a), R||R||R (b), R ⊥ P ⊥ ||R (c)

62

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.22. Overconstrained PPMs with uncoupled rotation of types 2PPaR-1RPP (a), 2PaRR-1RPP (b) and 2PaPaR-1RPP (c) defined by MF=SF=3, (RF)= ( v1 , v2 , δ ), TF=0, NF=12 (a) and (b), NF=18 (c), limb topology R ⊥ P ⊥ ⊥ P and P ⊥ Pa||R (a), Pa||R||R (b), Pa||Pa||R (c)

2.1 Basic solutions

63

Fig. 2.23. Overconstrained PPMs with uncoupled rotation of types 2PRR-1RPPa (a), 2RRR-1RPPa (b) and 2RPR-1RPPa (c) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=9, limb topology R ⊥ P ⊥ ||Pa and P ⊥ R||R (a), R||R||R (b), R ⊥ P ⊥ ||R (c)

64

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.24. Overconstrained PPMs with uncoupled rotation of types 2PRR-1RPaP (a), 2RRR-1RPaP (b) and 2RPR-1RPaP (c) defined by MF=SF=3, (RF)= ( v1 , v2 , δ ), TF=0, NF=9, limb topology R||Pa ⊥ P and P ⊥ R||R (a), R||R||R (b), R ⊥ P ⊥ ||R (c)

2.1 Basic solutions

65

Fig. 2.25. Overconstrained PPMs with uncoupled rotation of types 2PRR-1RPaPa (a), 2RRR-1RPaPa (b) and 2RPR-1RPaPa (c) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=12, limb topology R||Pa||Pa and P ⊥ R||R (a), R||R||R (b), R ⊥ P ⊥ ||R (c)

66

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.26. Overconstrained PPMs with uncoupled rotation of types 2PPaR-1RPPa (a) and 2PaRR-1RPPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=15, limb topology R ⊥ P ⊥ ||Pa and P ⊥ Pa||R (a), Pa||R||R (b)

2.1 Basic solutions

67

Fig. 2.27. Overconstrained PPMs with uncoupled rotation of types 2PPaR-1RPaP (a) and 2PaRR-1RPaP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=15, limb topology R||Pa ⊥ P and P ⊥ Pa||R (a), Pa||R||R (b)

68

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.28. Overconstrained PPMs with uncoupled rotation of types 2PaPaR1RPPa (a) and 2PaPaR-1RPaP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=21, limb topology Pa||Pa||R and R ⊥ P ⊥ ||Pa (a), R||Pa ⊥ P (b)

2.1 Basic solutions

69

Fig. 2.29. Overconstrained PPMs with uncoupled rotation of types 2PPaR1RPaPa (a) and 2PaRR-1RPaPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=18, limb topology R||Pa||Pa and P ⊥ Pa||R (a), Pa||R||R (b)

70

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.30. 2PaPaR-1RPaPa-type overconstrained PPM with uncoupled rotation defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=24, limb topology Pa||Pa||R and R||Pa||Pa

2.1 Basic solutions

71

Fig. 2.31. Overconstrained PPMs with uncoupled rotation of types 2PRR-1RPaPa (a) and 2PRR-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=12, limb topology P ⊥ R||R and R||Pa||Pa (a), R||Pa||Pat (b)

72

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.32. Overconstrained PPMs with uncoupled rotation of types 2RRR-1RPaPa (a) and 2RRR-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=12, limb topology R||R||R and R||Pa||Pa (a), R||Pa||Pat (b)

2.1 Basic solutions

73

Fig. 2.33. Overconstrained PPMs with uncoupled rotation of types 2RPR-1RPaPa (a) and 2RPR-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=12, limb topology R ⊥ P ⊥ ||R and R||Pa||Pa (a), R||Pa||Pat (b)

74

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.34. Overconstrained PPMs with uncoupled rotation of types 2PPaR1RPaPa (a) and 2PPaR-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=18, limb topology P ⊥ Pa||R and R||Pa||Pa (a), R||Pa||Pat (b)

2.1 Basic solutions

75

Fig. 2.35. Overconstrained PPMs with uncoupled rotation of types 2PaRR1RPaPa (a) and 2PaRR-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=18, limb topology Pa||R||R and R||Pa||Pa (a), R||Pa||Pat (b)

76

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.36. Overconstrained PPMs with uncoupled rotation of types 2PaPaR1RPaPa (a) and 2PaPaR-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=24, limb topology Pa||Pa||R and R||Pa||Pa (a), R||Pa||Pat (b)

2.1 Basic solutions

77

2.1.2 Non fully-parallel solutions The non fully-parallel solutions presented in Figs. 2.37-2.43 are obtained from the fully-parallel solutions in Figs. 2.21, 2.23-2.25, 2.31-2.33 by superposing the last revolute joints of limbs G1 and G2. In this way, these two limbs compose just one complex limb denoted by G1-G2 in which two actuators are combined. For example, the solutions in Fig. 2.37 combine one simple and one complex limb. Simple limb G3 is of type RPP and complex limb G1-G2 is of types (PRRRP)R (Fig. 2.37a), (RRRRR)R (Fig. 2.37b) and (RPRPR)R (Fig. 2.37c). One closed loop defined in the brackets is combined in each complex limb G1-G2. The solutions in Figs. 2.38-2.43 combine two complex limbs G1-G2 and G3. The rotation axis of the moving platform has a fixed position in the solutions in Figs. 2.38-2.40 and a variable position in Figs. 2.41-2.43. The structural parameters of the solutions presented in Figs. 2.37-2.43 are systematized in Tables 2.10 and 2.11. The basis of the vector space of the relative velocities between the moving and the fixed platforms in the complex limb G1-G2 isolated from the parallel mechanism is denoted by (RG1-G2) in Tables 2.10 and 2.11. The connectivity between the moving and the fixed platforms of this complex limb isolated from the parallel mechanism is denoted by SG1-G2 (see Table 2.10-2.11). The non fully-parallel solutions presented in Figs. 2.37-2.43 have the same number of overconstraints as their fully-parallel counterparts in Figs. 2.21, 2.23-2.25, 2.31-2.33.

78

2 Overconstrained planar parallel robots with coupled motions

Table 2.11. Structural parametersa of planar parallel mechanisms in Figs. 2.372.39 No. Structural parameter

Solution (PRRRP)R-RPP (Fig. 2.37a) (RRRRR)R-RPP (Fig. 2.37b) (RPRPR)R-RPP (Fig. 2.37c)

1 2 3 4 5 6 7 8 9

m p1 p2 p q k1 k2 k (RG1-G2)

10

(RG3)

11 12 13 14 15 16 17

SG1-G2 SG3 rG1-G2 rG3 MG1-G2 MG3 (RF)

18 19 20 21 22 23 24

SF rl rF MF NF TF

8 6 3 9 2 1 1 2 ( v1 , v 2 , ( v1 , v 2 , 3 3 3 0 3 3 ( v1 , v 2 , 3 3 6 3 6 0 6

25 26 a

∑ ∑ ∑

p1

j =1

p2

j =1

p

j =1

fj

δ

)

δ

)

δ

)

(PRRRP)R-RPPa (Fig. 2.38a) (RRRRR)R-RPPa (Fig. 2.38b) (RPRPR)R-RPPa (Fig. 2.38c) (PRRRP)R-RpaP (Fig. 2.39a) (RRRRR)R-RPaP (Fig. 2.39b) (RPRPR)R-RPaP (Fig. 2.39c) 10 6 6 12 3 0 2 2 ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) 3 3 3 3 3 3 ( v1 , v 2 , δ ) 3 6 9 3 9 0 6

fj

3

6

fj

9

12

See footnote of Table 2.4 for the nomenclature of structural parameters

2.1 Basic solutions

79

Table 2.12. Structural parametersa of planar parallel mechanisms in Figs. 2.402.43 No.

Structural parameter

1 2 3 4 5 6 7 8 9

m p1 p2 p q k1 k2 k (RG1-G2)

10

(RG3)

11 12 13 14 15 16 17

SG1-G2 SG3 rG1-G2 rG3 MG1-G2 MG3 (RF)

18 19 20 21 22 23 24

SF rl rF MF NF TF

25 26 a

∑ ∑ ∑

p1

j =1

p2

j =1

p

j =1

fj

Solution (PRRRP)R-RpaPa (Fig. 2.40a) (RRRRR)R-RPaPa (Fig. 2.40b) (RPRPR)R-RPaPa (Fig. 2.40c) (PRRRP)R-RPaPa (Fig. 2.41a) (PRRRP)R-RPaPat (Fig. 2.41b) (RRRRR)R-RPaPa (Fig. 2.42a) (RRRRR)R-RPaPat (Fig. 2.42b) (RPRPR)R-RPaPa (Fig. 2.43a) (RPRPR)R-RPaPat (Fig. 2.43b) 12 6 9 15 4 0 2 2 ( v1 , v 2 , δ ) ( v1 , v2 , δ ) 3 3 3 6 3 3 ( v1 , v2 , δ ) 3 9 12 3 12 0 6

fj

9

fj

15

See footnote of Table 2.4 for the nomenclature of structural parameters

80

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.37. Overconstrained PPMs with uncoupled rotation of types (PRRRP)RRPP (a), (RRRRR)R-RPP (b) and (RPRPR)R-RPP (c), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=6

2.1 Basic solutions

81

Fig. 2.38. Overconstrained PPMs with uncoupled rotation of types (PRRRP)RRPPa (a), (RRRRR)R-RPPa (b) and (RPRPR)R-RPPa (c), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=9

82

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.39. Overconstrained PPMs with uncoupled rotation of types (PRRRP)RRPaP (a), (RRRRR)R-RPaP (b) and (RPRPR)R-RPaP (c), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=9

2.1 Basic solutions

83

Fig. 2.40. Overconstrained PPMs with uncoupled rotation of types (PRRRP)RRPaPa (a), (RRRRR)R-RPaPa (b) and (RPRPR)R-RPaPa (c), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=12

84

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.41. Overconstrained PPMs with uncoupled rotation of types (PRRRP)RRPaPa (a) and (PRRRP)R-RPaPat (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=12

2.1 Basic solutions

85

Fig. 2.42. Overconstrained PPMs with uncoupled rotation of types (RRRRR)RRPaPa (a) and (RRRRR)R-RPaPat (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=12

86

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.43. Overconstrained PPMs with uncoupled rotation of types (RPRPR)RRPaPa (a) and (RPRPR)R-RPaPat (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=12

2.2 Derived solutions

87

2.2 Derived solutions Solutions with lower degrees of overconstraint can be derived from the basic solutions in Figs. 2.4-2.43 by using joints with idle mobilities. A large set of solutions can be obtained by introducing idle mobilities in the limbs. The joints combining idle mobilities are denoted by an asterisk. The idle mobilities which can be combined in a parallelogram loop are systematized in Fig. 1.2 and Table 1.1. One idle mobility is combined in each cylindrical joint C* and two idle mobilities in each spherical joint S*. These idle mobilities can be introduced outside or inside the loops combined in the limbs. The rotational mobility of the revolute joint denoted by R* is an idle mobility. In the cylindrical joint denoted by C* in Figs. 2.45, 2.46b, 2.54b, 2.55a, 2.61, 2.62a the translation is the actuated motion and the rotation is the idle mobility. The rotation is also the idle mobility in the cylindrical joint denoted by C* in Figs 2.65a, 2.70, 2.71 and 2.72a. We recall that the notation Pacs is associated with a parallelogram loop with three idle mobilities combined in a cylindrical and a spherical joint, and Pass with four idle mobilities combined in two spherical joints adjacent to the same link. In the cylindrical joints of the Pacs-type parallelogram loops (Figs. 2.84-2.95 and 2.104-2.109) the translational motion is an idle mobility. We note that in the Pass-type parallelogram loop, three idle mobilities are introduced in the loop and one outside the loop. If the link adjacent to the two spherical joints is a binary link than the idle mobility introduced outside the loop becomes an internal rotational mobility of the binary link around the axis passing by the centre of the two spherical joints. This internal mobility gives one degree of structural redundancy (see Table 2.13). If the link adjacent to the two spherical joints is connected in the limb by three or more joints (polinary link) than the rotational motion around the axis passing by the centre of the two spherical joints is an idle (potential) mobility of the limb. This mobility is restricted by the constraints of the parallel mechanism and remains just a potential mobility. For example in Fig. 2.68c, this rotational motion is internal mobility of links 3A and 3B, and idle mobility for the ternary links 7A and 7B. Examples of solutions with identical limbs and 3-18 overconstraints derived from the basic solutions in Figs. 2.4-2.10 are illustrated in Figs. 2.44-2.52. The solutions with one decoupled translation v1 = v1 ( &q1 ) and 3-15 overconstraints illustrated in Figs. 2.53-2.59 are derived from the basic solutions in Figs. 2.11-2.15. The solutions with partially decoupled translational motions and three overconstraints illustrated in Figs. 2.60-2.61 are derived from the solutions in Figs. 2.16 and 2.17. The solutions with decoupled

88

2 Overconstrained planar parallel robots with coupled motions

rotational motion ωδ = ωδ ( &q3 ) and 2-22 overconstraints illustrated in Figs. 2.62-2.95 are derived from the solutions in Figs. 2.18-2.36. The non fully-parallel solutions with 2-8 overconstraints illustrated in Figs. 2.96-2.109 are derived from the solutions in Figs. 2.37-2.43. The bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms in Figs. 2.44-109 are given in Tables 2.14 and 2.15. The limb topology and connecting conditions of these solutions are systematized in Tables 2.16-2.21, as are their structural parameters in Tables 2.22-2.40.

Table 2.13. Links with internal mobilities and the degree of structural redundancy TF of overconstrained SPMs with uncoupled planar motion of the moving platform No. Parallel mechanism Figure

TF

Link with internal rotational mobility in limb G1 G2 G3

1

3

3A

3B

3C

6 3 2

3A, 6 A 4A 3A

3 B, 6B 4B 3B

3 C, 6C 4C -

1 1 2

-

-

5C 4C 3 C, 6C

2 3 4 5 6 7

Figs. 2.47b, 2.50b, 2.51a Fig. 2.50a Fig.2.51b Figs. 2.68c, 2.79b, 2.82 Figs. 2.72, 2.98 Figs. 2.73, 2.99 Figs. 2.74, 2.100

2.2 Derived solutions

89

Table 2.14. Bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 2.44-2.95 No. Parallel mechanism 1 Figs. 2.44-2.46, 2.60 2 Figs. 2.47a, 2.48, 2.49, 2.56, 2.57 3 Figs. 2.47b, 2.50, 2.51, 2.72-2.74 4 Figs. 2.52, 2.53, 2.54a, 2.55b 5 Figs. 2.54b, 2.61b 6 Fig. 2.55a

Basis (RG1) ( v1 , v 2 ,

α,

δ

)

( v1 , v 2 ,

α

,

β

,

( v1 , v 2 ,

δ

)

( v1 , v 2 ,

β

,

δ

( v1 , v 2 ,

β

,

( v1 , v 2 ,

α

7

Fig. 2.58

( v1 , v 2 ,

Figs. 2.59, 2.61a, 2.63, 2.64, 2.80b, 2.81b 9 Figs. 2.62, 2.80a, 2.81a, 2.82 10 Figs. 2.65, 2.69-2.71, 2.75-2.78, 2.79a, 2.84-2.86 11 Figs. 2.66, 2.87-2.89 12 Figs. 2.67

8

δ

)

β,

α

( v1 , v 2 , v 3 , )

)

( v1 , v 2 ,

α

,

δ

δ

)

( v1 , v 2 ,

β

,

,

δ

)

( v1 , v 2 ,

β

α

,

δ

)

( v1 , v 2 ,

( v1 , v 2 ,

α

,

δ

)

( v1 , v 2 ,

β

,

δ

)

δ

α

,

( v1 , v 2 ,

δ

)

13 Figs. 2.68a,c ( v1 , v2 ,

β

,

δ

14 Figs. 2.68b

( v1 , v 2 ,

α

,

15 Figs. 2.79b, 2.83a 16 Figs. 2.83b

( v1 , v 2 ,

β

( v1 , v 2 ,

α

( v1 , v 2 , v 3 ,

β

,

δ

β

( v1 , v 2 , v 3 ,

) δ

,

)

δ

)

)

( v1 , v 2 ,

α

,

δ

)

δ

)

( v1 , v 2 ,

α

,

δ

)

,

δ

)

( v1 , v 2 ,

β

,

δ

)

α

,

δ

)

( v1 , v 2 ,

β

,

δ

)

( v1 , v 2 ,

β

,

δ

)

( v1 , v 2 ,

α

,

δ

)

( v1 , v 2 ,

α

,

δ

)

( v1 , v 2 ,

β

,

δ

)

( v1 , v 2 ,

δ

)

( v1 , v 2 ,

δ

)

( v1 , v 2 ,

α

,

β

,

δ

)

δ

)

( v1 , v 2 ,

α

,

( v1 , v 2 ,

δ

)

)

( v1 , v 2 ,

α

,

δ

)

( v1 , v 2 ,

δ

)

δ

)

( v1 , v 2 ,

β

,

δ

)

( v1 , v 2 ,

δ

)

,

δ

)

( v1 , v 2 ,

α

,

δ

)

( v1 , v 2 ,

α

,

β

,

δ

)

,

δ

)

( v1 , v 2 ,

β

,

δ

)

( v1 , v 2 ,

α

,

β

,

δ

)

( v1 , v 2 ,

δ

)

β

,

δ

)

β

( v1 , v 2 ,

( v1 , v 2 , v 3 ,

)

δ

,

δ

( v1 , v 2 , v 3 ,

(RG3) ( v1 , v 2 ,

δ )

( v1 , v 2 ,

( v1 , v 2 ,

17 Figs. 2.90, 2.93

(RG2) ( v1 , v 2 ,

,

δ

)

)

( v1 , v 2 , v 3 ,

β

α

,

δ

,

δ

)

)

90

2 Overconstrained planar parallel robots with coupled motions

Table 2.14. (cont.) 18 Figs. 2.91, 2.94 19 Figs. 2.92, 2.95

( v1 , v 2 , v 3 ,

α

,

δ

)

( v1 , v 2 , v 3 ,

α

,

β

,

δ

( v1 , v 2 , v 3 ,

β

,

δ

)

) ( v1 , v 2 , v 3 ,

α

,

β

,

δ

( v1 , v 2 ,

δ

)

) ( v1 , v 2 ,

δ

)

Table 2.15. Bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 2.96-2.109 No. Parallel mechanism 1 Figs. 2.96 2

Figs. 2.97, 2.101-2.103

3

Figs. 2.98, 2.99, 2.100, 2.104-2.106 Figs. 2.107-109

4

Basis (RG1-G2) ( v1 , v 2 , v 3 , δ ) ( v1 , v 2 , β , δ ) ( v1 , v 2 ,

δ

( v1 , v 2 , v 3 ,

) β

,

δ

)

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

δ δ

)

( v1 , v 2 ,

δ

)

( v1 , v 2 ,

δ

)

)

Table 2.16. Limb topology and the number of overconstraints NF of the derived PPMs with idle mobilities presented in Figs. 2.44-2.46 No. Basic PPM Type 1 3-RRR (Fig. 2.4a) 2 3-RRP (Fig. 2.4b) 3 3-RPR (Fig. 2.5a) 4 3-PRR (Fig. 2.5b) 5 3-PRP (Fig. 2.6) 6

NF 6 6 6 6 6

Derived PPM Type 3-RR*RR (Fig. 2.44a) 3-RRC* (Fig. 2.44b) 3-RC*R (Fig. 2.45a) 3-C*RR (Fig. 2.45b) 3-PRC* (Fig. 2.46a) 3-C*RP (Fig. 2.46b)

NF 3 3 3 3 3 3

Limb topology R ⊥ R* ⊥ ||R||R R||R ⊥ C*

R ⊥ C* ⊥ ||R C* ⊥ R||R

P ⊥ R ⊥ C*

C* ⊥ R ⊥ P

2.2 Derived solutions

91

Table 2.17. Limb topology and the number of overconstraints NF of the derived PPMs with idle mobilities presented in Figs. 2.47-2.54 No. Basic PPM Type 1 3-PaPaR (Fig. 2.7) 2 3

3-PaRR (Fig. 2.8a)

NF 24

15

4 5

3-PaRP (Fig. 2.8b)

15

6 7

3-PaPR (Fig. 2.9a)

15

8 9

3-PPaR (Fig. 2.9b)

15

10 11 12 13 14

3-PPR (Fig. 2.10) 1PPR-2RRR (Fig. 2.11) 1PPR-2RRP (Fig. 2.12a) 1PPR-2RPR (Fig. 2.12b)

6 6 6 6

Derived PPM Type 3-PaPaS* (Fig. 2.48a) 3-PassPassR (Fig. 2.50a) 3-PaRS* (Fig. 2.47a) 3-PassRR (Fig. 2.47b) 3-PaS*P (Fig. 2.48b) 3-PassRP (Fig. 2.50b) 3-PaPS* (Fig. 2.49a) 3-PassPR (Fig. 2.51a) 3-PPaS* (Fig. 2.49b) 3-PPassR (Fig. 2.51b) 3-PC*R (Fig. 2.52) 1PC*R-2RR*RR (Fig. 2.53) 1PC*R-2RRC* (Fig. 2.54a) 1PC*R-2RC*R (Fig. 2.54b)

NF 18

Limb topology Pa||PaS*

6

Pass||Pass||R

9

Pa||RS*

6

Pass||R||R

9

PaS*P

6

Pass||R ⊥ P

9 6 9 6 3 3 3 3

Pa ⊥ PS*

Pass ⊥ P ⊥ ||R P ⊥ PaS*

P ⊥ Pass||R

P ⊥ C* ⊥ ⊥ R

P ⊥ C* ⊥ ⊥ R R ⊥ R* ⊥ ||R||R P ⊥ C* ⊥ ⊥ R R||R ⊥ C* P ⊥ C* ⊥ ⊥ R R ⊥ C* ⊥ ||R

92

2 Overconstrained planar parallel robots with coupled motions

Table 2.18. Limb topology and the number of overconstraints NF of the derived PPMs with idle mobilities presented in Figs. 2.55-2.64 No. Basic PPM Type 1 1PPR-2PRR (Fig. 2.13a) 2 1PPR-2PRP (Fig. 2.13b) 3 1PaPR-2PaRR (Fig. 2.14a) 4 5

1PaPR-2PPaR (Fig. 2.14b)

NF 6 6 15

15

6 7

1PaPR-2PaPaR (Fig. 2.15a)

21

8 9

1PaPR-2PaRP (Fig. 2.15b)

15

10 11 12 13 14 15 16 17

3RRR (Fig. 2.16a,b) 3RPR (Fig. 2.17a) 2RPR-1RPR (Fig. 2.17b) 2RPP-1RPP (Fig. 2.18) 2RPPa-1RPPa (Fig. 2.19) 2RPaP-1RPaP (Fig. 2.20a) 2RPaPa-1RPaPa (Fig. 2.20b)

6 6 6 6 15 15 24

Derived PPM Type 1C*PR-2C*RR (Fig. 2.55a) 1PC*R-2PRC* (Fig. 2.55b) 1PaPS*-2PaRS* (Fig. 2.56a) 1PassPR-2PassRR (Fig. 2.58a) 1PaPS*-2PPaS* (Fig. 2.56b) 1PassPR-2PPassR (Fig. 2.58b) 1PaPS*-2PaPaS* (Fig. 2.57a) 1PassPR-2PaPassR (Fig. 2.59a) 1PaPS*-2PaS*P (Fig. 2.57b) 1PassPR-2PassRP (Fig. 2.59b) 3RR*RR (Fig. 2.60a,b) 3RC*R (Fig. 2.61a) 2RC*R-1RC*R (Fig. 2.61b) 2RPC*-1RPC* (Fig. 2.62) 2RC*Pa-1RC*Pa (Fig. 2.63) 2RPaC*-1RPaC* (Fig. 2.64a) 2RPaPass-1RPaPass (Fig. 2.64b)

NF 3 3 9 3 9 3 15 9 9 3 3 3 3 3 12 12 12

Limb topology C* ⊥ P ⊥ ⊥ R C* ⊥ R||R P ⊥ C* ⊥ ⊥ R P ⊥ R ⊥ C* Pa ⊥ PS* Pa||RS* Pass ⊥ P ⊥ ||R Pass||R||R 1Pa ⊥ PS* P ⊥ PaS* Pass ⊥ P ⊥ ||R P ⊥ Pass||R Pa ⊥ PS* Pa||PaS* Pass ⊥ P ⊥ ||R Pass||R ⊥ P Pa ⊥ PS* PaS*P Pass ⊥ P ⊥ ||R Pass||R ⊥ P R ⊥ R* ⊥ ||R||R R ⊥ C* ⊥ ||R

R ⊥ C* ⊥ ||R R ⊥ C* ⊥ ||R R ⊥ P ⊥ ⊥ C*

R ⊥ C* ⊥ ||Pa R||Pa ⊥ C* R||Pa ⊥ C* R||Pa||Pass R||Pa||Pass

2.2 Derived solutions

93

Table 2.19. Limb topology and the number of overconstraints NF of the derived PPMs with idle mobilities presented in Figs. 2.65-2.89 No. Basic PPM Type 1 2PRR-1RPP (Fig. 2.21a) 2

6

3

6

2RRR-1RPP (Fig. 2.21b)

4 5

6 2RPR-1RPP (Fig. 2.21c)

6 7

NF 6

6 6

2PPaR-1RPP (Fig. 2.22a)

12

8 9

2PaRR-1RPP (Fig. 2.22b)

12

10 11

2PaPaR-1RPP (Fig. 2.22c)

18

12 13

2PRR-1RPPa (Fig. 2.23a)

9

14 15

2RRR-1RPPa (Fig. 2.23b)

9

16 17

2RPR-1RPPa (Fig. 2.23c)

9

18 19

2PRR-1RPaP (Fig. 2.24a)

9

Derived PPM Type 2PRC*-1RPP (Fig. 2.65a) 2PS*R-1RPP (Fig. 2.66a) 2RRC*-1RPP (Fig. 2.65b) 2RS*R-1RPP (Fig. 2.66b) 2RPC*-1RPP (Fig. 2.65c) 2S*PR-1RPP (Fig. 2.66c) 2PPaR-1RC*C* (Fig. 2.67a) 2PPassR-1RPP (Fig. 2.68a) 2PaRR-1RC*C* (Fig. 2.67b) 2PassRR-1RPP (Fig. 2.68b) 2PaPaR-1RC*C* (Fig. 2.67c) 2PassPassR-1RPP (Fig. 2.68c) 2PRC*-1RPPa (Fig. 2.69a) 2PRR-1RPPass (Fig. 2.72a) 2RRC*-1RPPa (Fig. 2.69b) 2RRR-1RPPass (Fig. 2.72b) 2RPC*-1RPPa Fig. 2.69c) 2RPR-1RPPass (Fig. 2.72c) 2PRC*-1RPaP (Fig. 2.70a)

NF 4 2 4 2 4 2 10 4 10 4 16 4 7 6 7 6 7 6 7

Limb topology P ⊥ R||C* R ⊥ P ⊥⊥ P PS*R R ⊥ P ⊥⊥ P R||R||C* R ⊥ P ⊥⊥ P RS*R R ⊥ P ⊥⊥ P R ⊥ P ⊥ ||C* R ⊥ P ⊥⊥ P SP ⊥ R R ⊥ P ⊥⊥ P P ⊥ Pa||R R ⊥ C* ⊥ ⊥ C* P ⊥ Pass||R R ⊥ P ⊥⊥ P Pa||R||R R ⊥ C* ⊥ ⊥ C* Pass||R||R R ⊥ P ⊥⊥ P Pa||Pa||R Pass||Pass||R R ⊥ P ⊥⊥ P P ⊥ R||C* R ⊥ P ⊥ ||Pa P ⊥ R||R R ⊥ P ⊥ ||Pass R||R||C* R ⊥ P ⊥ ||Pa R||R||R R ⊥ P ⊥ ||Pass R ⊥ P ⊥ ||C* R ⊥ P ⊥ ||Pa R ⊥ P ⊥ ||R R ⊥ P ⊥ ||Pass P ⊥ R||C* R||Pa ⊥ P

94

2 Overconstrained planar parallel robots with coupled motions

Table 2.19. (cont.) 20 21

2RRR-1RPaP (Fig. 2.24b)

9

22 23

2RPR-1RPaP (Fig. 2.24c)

9

24 25

2PRR-1RPaPa (Fig. 2.25a)

12

26 27

2RRR-1RPaPa (Fig. 2.25b)

12

28 29

2RPR-1RPaPa (Fig. 2.25c)

12

30 31

2PPaR-1RPPa (Fig. 2.26a)

15

32 33

2PaRR-1RPPa (Fig. 2.26b)

15

34 35

2PPaR-1RPaP (Fig. 2.27a)

15

36 37

2PaRR-1RPaP (Fig. 2.27b)

15

38 39

2PaPaR-1RPPa (Fig. 2.28a)

21

40 41

2PaPaR-1RPaP (Fig. 2.28b)

21

2PRR-1RPassP (Fig. 2.73a) 2RRC*-1RPaP (Fig. 2.70b) 2RRR-1RPassP (Fig. 2.73b) 2RPC*-1RPaP (Fig. 2.70c) 2RPR-1RPassP (Fig. 2.73c) 2PRC*-1RPaPa (Fig. 2.71a) 2PRR-1RPassPass (Fig. 2.74a) 2RRC*-1RPaPa (Fig. 2.71b) 2RRR-1RPassPass (Fig. 2.74b) 2RPC*-1RPaPa (Fig. 2.71c) 2RPR-1RPassPass (Fig. 2.74c) 2PPaC*-1RPPa (Fig. 2.75a) 2PPassR-1RPPass (Fig. 2.80a) 2PaRC*-1RPPa (Fig. 2.75b) 2PassRR-1RPPass (Fig. 2.80b) 2PPaC*-1RPaP (Fig. 2.76a) 2PPassR-1RPassP (Fig. 2.81a) 2PaRC*-1RPaP (Fig. 2.76b) 2PassRR-1RPassP (Fig. 2.81b) 2PaPaC*-1RPPa (Fig. 2.77a) 2PassPassR-1RPPass (Fig. 2.82a) 2PaPaC*-1RPaP (Fig. 2.77b)

6 7 6 7 6 10 6 10 6 10 6 13 3 13 3 13 3 13 3 19 3 19

P ⊥ R||R R||Pass ⊥ P R||R||C* R||Pa ⊥ P R||R||R R||Pass ⊥ P R ⊥ P ⊥ ||C* R||Pa ⊥ P R ⊥ P ⊥ ||R R||Pass ⊥ P P ⊥ R||C* R||Pa||Pa P ⊥ R||R R||Pass||Pass R||R||C* R||Pa||Pa R||R||R R||Pass||Pass R ⊥ P ⊥ ||C* R||Pa||Pa R ⊥ P ⊥ ||R R||Pass||Pass P ⊥ Pa||C* R ⊥ P ⊥ ||Pa P ⊥ Pass||R R ⊥ P ⊥ ||Pass Pa||R||C* R ⊥ P ⊥ ||Pa Pass||R||R R ⊥ P ⊥ ||Pass P ⊥ Pa||C* R||Pa ⊥ P P ⊥ Pass||R R ⊥ Pass ⊥ ||P Pa||R||C* R||Pa ⊥ P Pass||R||R R ⊥ Pass ⊥ ||P Pa||Pa||C* R ⊥ P ⊥ ||Pa Pass||Pass||R R ⊥ P ⊥ ||Pass Pa||Pa||C* R||Pa ⊥ P

2.2 Derived solutions Table 2.19. (cont.) 42 43

2PPaR-1RPaPa (Fig. 2.29a)

18

44 45

2PaRR-1RPaPa (Fig. 2.29b)

18

46 47

2PaPaR-1RPaPa (Fig. 2.30)

24

48 49

2PRR-1RPaPa (Fig. 2.31a)

12

50 51

2PRR-1RPaPat (Fig. 2.31b)

12

52 53

2RRR-1RPaPa (Fig. 2.32a)

12

54 55

2RRR-1RPaPat (Fig. 2.32b)

12

56 57

2RPR-1RPaPa (Fig. 2.33a)

12

58 59 60

2RPR-1RPaPat (Fig. 2.33b)

12

2PassPassR-1RPassP (Fig. 2.82b) 2PPaC*-1RPaPa (Fig. 2.78a) 2PPassR-1RPassPass (Fig. 2.83a) 2PaRC*-1RPaPa (Fig. 2.78b) 2PassRR-1RPassPass (Fig. 2.83b) 2PaPaC*-1RPaPa (Fig. 2.79a) 2PassPassR-1RPassPass (Fig. 2.79b) 2PRC*-1RPacsPacs (Fig. 2.84a) 2PS*R-1RPacsPacs (Fig. 2.87a) 2PRC*-1RPacsPatcs (Fig. 2.84b) 2PS*R-1RPacsPatcs (Fig. 2.87b) 2RRC*-1RPacsPacs (Fig. 2.85a) 2RS*R-1RPacsPacs (Fig. 2.88a) 2RRC*-1RPacsPatcs (Fig. 2.85b) 2RS*R-1RPacsPatcs (Fig. 2.88b) 2RPC*-1RPacsPacs (Fig. 2.86a) 2S*PR-1RPacsPacs (Fig. 2.89a) 2RPC*-1RPacsPatcs (Fig. 2.86b) 2S*PR-1RPacsPatcs (Fig. 2.89b)

3 16 2 16 2 22 2 4 2 4 2 4 2 4 2 4 2 4 2

Pass||Pass||R R||Pass ⊥ P P ⊥ Pa||C* R||Pa||Pa P ⊥ Pass||R R||Pass||Pass Pa||R||C* R||Pa||Pa Pass||R||R R||Pass||Pass Pa||Pa||C* R||Pa||Pa Pass||Pass||R R||Pass||Pass P ⊥ R||C* R||Pacs||Pacs PS*R R||Pacs||Pacs P ⊥ R||C* R||Pacs||Pat cs PS*R R||Pacs||Patcs R||R||C* R||Pacs||Pacs RS*R R||Pacs||Pacs R||R||C* R||Pacs||Patcs R||R||C* R||Pacs||Patcs R ⊥ P ⊥ ||C* R||Pacs||Pacs S*P ⊥ R R||Pacs||Pacs R ⊥ P ⊥ ||C* R||Pacs||Patcs S*P ⊥ R R||Pacs||Patcs

95

96

2 Overconstrained planar parallel robots with coupled motions

Table 2.20. Limb topology and the number of overconstraints NF of the derived PPMs with idle mobilities presented in Figs. 2.90-2.95 No. Basic PPM Type 61 2PPaR-1RPaPa (Fig. 2.34a) 62 63

NF 18

2PPaR-1RPaPat (Fig. 2.34b)

18

2PaRR-1RPaPa (Fig. 2.35a)

18

2PaRR-1RPaPat (Fig. 2.35b)

18

2PaPaR-1RPaPa (Fig. 2.36a)

24

2PaPaR-1RPaPat (Fig. 2.36b)

24

64 65 66 67 68 69 70 71 72

Derived PPM Type NF 2PPassC*-1RPaPa 8 (Fig. 2.90a) 2PPassC*-1RPacsPacs 2 (Fig. 2.93a) 2PPassC*-1RPaPat 8 (Fig. 2.90b) 2PPassC*-1RPacsPatcs 2 (Fig. 2.93b) 2PassRC*-1RPaPa 8 (Fig. 2.91a) 2PassRC*-1RPacsPacs 2 (Fig. 2.94a) 2PassRC*-1RPaPat 8 (Fig. 2.91b) 2PassRC*-1RPacsPatcs 2 (Fig. 2.94b) 2PassPassC*-1RPaPa 6 (Fig. 2.92a) 2PassPassC*-1RPacsPacs 3 (Fig. 2.95a) 2PassPassC*-1RPaPat 6 (Fig. 2.92b) 2PassPassC*-1RPacsPatcs 3 (Fig. 2.95b)

Limb topology P ⊥ Pass||C* R||Pa||Pa P||Pass||C* R||Pacs||Pacs P ⊥ Pass||C* R||Pa||Pat P||Pass||C* R||Pacs||Patcs Pass||R||C* R||Pa||Pa Pass||R||C* R||Pacs||Pacs Pass||R||C* R||Pa||Pat Pass||R||C* R||Pacs||Patcs Pass||Pass||C* R||Pa||Pa Pass||Pass||C* R||Pacs||Pacs Pass||Pass||C* R||Pa||Pat Pass||Pass||C* R||Pacs||Patcs

2.2 Derived solutions

97

Table 2.21. Limb topology and the number of overconstraints NF of the derived PPMs with idle mobilities presented in Figs. 2.96-2.109 No. Basic PPM Type 1 (PRRRP)R-RPP (Fig. 2.37a) 2 3

(RRRRR)R-RPP (Fig. 2.37b)

NF 6

6

4 5

(RPRPR)R-RPP (Fig. 2.37c)

6

6 7

(PRRRP)R-RPPa (Fig. 2.38a)

9

8 9

(RRRRR)R-RPPa (Fig. 2.38b)

9

10 11

(RPRPR)R-RPPa (Fig. 2.38c)

9

12 13

(PRRRP)R-RPaP (Fig. 2.39a)

9

14 15

(RRRRR)R-RPaP (Fig. 2.39b)

9

16 17

(RPRPR)R-RPaP (Fig. 2.39c)

9

18 19 20

(PRRRP)R-RPaPa 12 (Fig. 2.40a)

Derived PPM Type (PRRRP)C*-RPP (Fig. 2.96a) (PS*RS*P)R-RPP (Fig. 2.97a) (RRRRR)C*-RPP (Fig. 2.96b) (RS*RS*R)R-RPP (Fig. 2.97b) (RPRPR)C*-RPP (Fig. 2.96c) (S*PRPS*)R-RPP (Fig. 2.97c) (PRRRP)R-RPPass (Fig. 2.98a) (PS*RS*P)R-RPPa (Fig. 2.101a) (RRRRR)R-RPPass (Fig. 2.98b) (RS*RS*R)R-RPPa (Fig. 2.101b) (RPRPR)R-RPPass (Fig. 2.98c) (S*PRPS*)R-RPPa (Fig. 2.101c) (PRRRP)R-RPassP (Fig. 2.99a) (PS*RS*P)R-RPaP (Fig. 2.102a) (RRRRR)R-RPassP (Fig. 2.99b) (RS*RS*R)R-RPaP (Fig. 2.102b) (RPRPR)R-RPassP (Fig. 2.99c) (S*PRPS*)R-RPaP (Fig. 2.102c) (PRRRP)R-RPassPass (Fig. 2.100a) (PS*RS*P)R-RPaPa (Fig. 2.103a)

NF 5

Limb topology (PRRRP)C* R ⊥ P ⊥⊥ P

2 5

(RRRRR)C* R ⊥ P ⊥⊥ P

2 5 2 6 5 6 5 6 5 6 5 6 5 6 5 6 8

(RPRPR)C* R ⊥ P ⊥⊥ P (S*PRPS*)R R ⊥ P ⊥⊥ P (PRRRP)R R ⊥ P ⊥ ||Pass (PS*RS*P) R ⊥ P ⊥ ||Pa (RRRRR)R R ⊥ P ⊥ ||Pass (RS*RS*R)R R ⊥ P ⊥ ||Pa (RPRPR)R R ⊥ P ⊥ ||Pass (S*PRPS*)R R ⊥ P ⊥ ||Pa PRRRP)R R||Pass ⊥ P (PS*RS*P)R R||Pa ⊥ P (RRRRR)R R||Pass ⊥ P (RS*RS*R)R R||Pa ⊥ P (RPRPR)R R||Pass ⊥ P (S*PRPS*)R R||Pa ⊥ P (PRRRP)R R||Pass||Pass (PS*RS*P)R R||Pa||Pa

98

2 Overconstrained planar parallel robots with coupled motions

Table 2.21. (cont.) 21

(RRRRR)R-RPaPa 12 (Fig. 2.40b)

22 23

(RPRPR)R-RPaPa 12 (Fig. 2.40c)

24 25

(PRRRP)R-RPaPa 12 (Fig. 2.41a)

26 27

(PRRRP)R-RPaPat 12 (Fig. 2.41b)

28 29

(RRRRR)R-RPaPa 12 (Fig. 2.42a)

30 31

(RRRRR)R-RPaPat 12 (Fig. 2.42b)

32 33

(RPRPR)R-RPaPa 12 (Fig. 2.43a)

34 35 36

(RPRPR)R-RPaPat 12 (Fig. 2.43b)

(RRRRR)R-RPassPass (Fig. 2.100b) (RS*RS*R)R-RPaPa (Fig. 2.103b) (RPRPR)R-RPassPass (Fig. 2.100c) (S*PRPS*)R-RPaPa (Fig. 2.103c) (PRRRP)R-RPacsPacs (Fig. 2.104a) (PS*RS*P)R-RPacsPacs (Fig. 2.107a) (PRRRP)R-RPacsPatcs (Fig. 2.104b) (PS*RS*P)R-RPacsPatcs (Fig. 2.107b) (RRRRR)R-RPacsPacs (Fig. 2.105a) (RS*RS*R)R-RPacsPacs (Fig. 2.108a) (RRRRR)R-RPacsPatcs (Fig. 2.105b) (RS*RS*R)R-RPacsPatcs (Fig. 2.108b) (RPRPR)R-RPacsPacs (Fig. 2.106a) (S*PRPS*)R-RPacsPacs (Fig. 2.109a) (RPRPR)R-RPacsPatcs (Fig. 2.106b) (S*PRPS*)R-RPacsPatcs (Fig. 2.109b)

6 8 6 8 6 2 6 2 6 2 6 2 6 2 6 2

RRRRR)R R||Pass||Pass (RS*RS*R)R R||Pa||Pa (RPRPR)R R||Pass||Pass (S*PRPS*)R R||Pa||Pa (PRRRP)R R||Pacs||Pacs (PS*RS*P)R R||Pacs||Pacs (PRRRP)R R||Pacs||Patcs (PS*RS*P)R R||Pacs||Patcs (RRRRR)R R||Pacs||Pacs (RS*RS*R)R R||Pacs||Pacs (RRRRR)R R||Pacs||Patcs (RS*RS*R)R R||Pacs||Patcs (RPRPR)R R||Pacs||Pacs (S*PRPS*)R R||Pacs||Pacs (RPRPR)R R||Pacs||Patcs (S*PRPS*)R R||Pacs||Patcs

2.2 Derived solutions

99

Table 2.22. Structural parametersa of planar parallel mechanisms in Figs. 2.442.46 No. Structural parameter

Solution 3-RR*RR (Fig. 2.44a)

1 2 3 4 5 6 7 8 9 10

11 4 4 4 12 2 3 0 3 See Table 2.14

3-RRC* (Fig. 2.44b) 3-RC*R (Fig. 2.45a) 3-C*RR (Fig. 2.45b) 3-PRC* (Fig. 2.46a) 3-C*RP (Fig. 2.46b) 8 3 3 3 9 2 3 0 3 See Table 2.14

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

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

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

fj

4

4

fj

4

4

fj

12

12

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

100

2 Overconstrained planar parallel robots with coupled motions

Table 2.23. Structural parametersa of planar parallel mechanisms in Figs. 2.472.49 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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

Solution 3-PaRS* (Fig. 2.47a) 3-PaS*P (Fig. 2.48b) 3-PaPS* (Fig. 2.49a) 3-PPaS* (Fig. 2.49b) 14 6 6 6 18 5 0 3 3 See Table 2.14

3-PassRR (Fig. 2.47b)

3-PaPaS* (Fig. 2.48a)

14 6 6 6 18 5 0 3 3 See Table 2.14

20 9 9 9 27 8 0 3 3 See Table 2.14

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

3 3 3 6 6 6 4 4 4 ( v1 , v 2 , 3 18 24 6 6 3 10

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

δ

)

δ

)

fj

8

10

11

fj

8

10

11

fj

24

30

33

See footnote of Table 2.4 for the nomenclature of structural parameters

δ

)

2.2 Derived solutions

101

Table 2.24. Structural parametersa of planar parallel mechanisms in Figs. 2.502.52 No. Structural parameter

Solution 3-PassPassR (Fig. 2.50a)

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

20 9 9 9 27 8 0 3 3 See Table 2.14

3-PassRP (Fig. 2.50b) 3-PassPR (Fig. 2.51a) 3-PPassR (Fig. 2.51b) 14 6 6 6 18 5 0 3 3 See Table 2.14

8 3 3 3 9 2 3 0 3 See Table 2.14

3 3 3 12 12 12 5 5 5 ( v1 , v 2 , 3 36 42 9 6 6 17

3 3 3 6 6 6 4 4 4 ( v1 , v 2 , 3 18 24 6 6 3 10

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

δ

)

δ

)

3-PC*R (Fig. 2.52)

fj

17

10

4

fj

17

10

4

fj

51

30

12

See footnote of Table 2.4 for the nomenclature of structural parameters

δ

)

102

2 Overconstrained planar parallel robots with coupled motions

Table 2.25. Structural parametersa of planar parallel mechanisms in Figs. 2.532.56 No. Structural Solution parameter 1PC*R-2RR*RR 1PC*R-2RRC* (Fig. 2.54a) (Fig. 2.53) 1PC*R-2RC*R (Fig. 2.54b) 1C*PR-2C*RR (Fig. 2.55a) 1PC*R-2PRC* (Fig. 2.55b) 1 m 10 8 2 p1 3 3 3 p2 4 3 4 p3 4 3 5 p 11 9 6 q 2 2 7 k1 3 3 8 k2 0 0 9 k 3 3 10 (RGi) See Table 2.14 See Table 2.14 (i=1,2,3) 11 SG1 4 4 12 SG2 4 4 13 SG3 4 4 14 rG1 0 0 15 rG2 0 0 16 rG3 0 0 17 MG1 4 4 18 MG2 4 4 19 MG3 4 4 20 (RF) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) 21 SF 3 3 22 rl 0 0 23 rF 9 9 24 MF 3 3 25 NF 3 3 26 TF 0 0 p1 27 4 4 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

1PaPS*-2PaRS* (Fig. 2.56a) 1PaPS*-2PPaS* (Fig. 2.56b) 14 6 6 6 18 5 0 3 3 See Table 2.14 5 5 5 3 3 3 5 5 5 ( v1 , v 2 , 3 9 21 3 9 0 8

j

fj

4

4

8

fj

4

4

8

fj

12

12

24

See footnote of Table 2.4 for the nomenclature of structural parameters

δ

)

2.2 Derived solutions

103

Table 2.26. Structural parametersa of planar parallel mechanisms in Figs. 2.57 and 2.58 No. Structural Solution parameter 1PaPS*-2PaPaS* (Fig. 2.57a)

1PaPS*-2PaS*P (Fig. 2.57b)

1 2 3 4 5 6 7 8 9 10

18 6 9 9 24 7 0 3 3 See Table 2.14

14 6 6 6 18 5 0 3 3 See Table 2.14

1PassPR-2PassRR (Fig. 2.58a) 1PassPR-2PPassR (Fig. 2.58b) 14 6 6 6 18 5 0 3 3 See Table 2.14

5 5 5 3 6 6 5 5 5 ( v1 , v 2 , 3 15 27 3 15 0 8

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

4 4 4 6 6 6 4 4 4 ( v1 , v 2 , 3 18 27 3 3 0 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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

δ

)

fj

11

8

10

fj

11

8

10

fj

30

24

30

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

104

2 Overconstrained planar parallel robots with coupled motions

Table 2.27. Structural parametersa of planar parallel mechanisms in Figs. 2.59 and 2.60 No. Structural Solution parameter 1PassPR-2PaPassR (Fig. 2.59a) 1 m 18 2 p1 6 3 p2 9 4 p3 9 5 p 24 6 q 7 7 k1 0 8 k2 3 9 k 3 10 (RGi) See Table 2.14 (i=1,2,3) 11 SG1 4 12 SG2 4 13 SG3 4 14 rG1 6 15 rG2 9 16 rG3 9 17 MG1 4 18 MG2 4 19 MG3 4 20 (RF) ( v1 , v 2 , δ ) 21 SF 3 22 rl 24 23 rF 33 24 MF 3 25 NF 9 26 TF 0 p1 27 10 f

1PassPR-2PassRP (Fig. 2.59b) 14 6 6 6 18 5 0 3 3 See Table 2.14

3RR*RR (Fig. 2.60) 11 4 4 4 12 2 3 0 3 See Table 2.14

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

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

28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

δ

)

δ

)

j

fj

13

10

4

fj

13

10

4

fj

36

30

12

See footnote of Table 2.4 for the nomenclature of structural parameters

2.2 Derived solutions

105

Table 2.28. Structural parametersa of planar parallel mechanisms in Figs. 2.612.64 No. Structural Solution parameter 3RC*R (Fig. 2.61a) 2RC*R-1RC*R (Fig. 2.61b) 2RPC*-1RPC* (Fig. 2.62) 1 m 8 2 p1 3 3 p2 3 4 p3 3 5 p 9 6 q 2 7 k1 3 8 k2 0 9 k 3 10 (RGi) See Table 2.14 (i=1,2,3) 11 SG1 4 12 SG2 4 13 SG3 4 14 rG1 0 15 rG2 0 16 rG3 0 17 MG1 4 18 MG2 4 19 MG3 4 20 (RF) ( v1 , v 2 , δ ) 21 SF 3 22 rl 0 23 rF 9 24 MF 3 25 NF 3 26 TF 0 p1 27 4 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

2RC*Pa-1RC*Pa (Fig. 2.63) 2RPaC*-1RPaC* (Fig. 2.64a)

2RPaPass-1RPaPass (Fig. 2.64b)

14 6 6 6 18 5 0 3 3 See Table 2.14

20 9 9 9 27 8 3 0 3 See Table 2.14

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

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

δ

)

fj

4

7

13

fj

4

7

13

fj

12

21

39

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

106

2 Overconstrained planar parallel robots with coupled motions

Table 2.29. Structural parametersa of planar parallel mechanisms in Figs. 2.652.67 No. Structural Solution parameter 2PRC*-1RPP 2PS*R-1RPP 2PPaR-1RC*C* (Fig. 2.65a) (Fig. 2.66a) (Fig. 2.67a) 2RRC*-1RPP 2RS*R-1RPP 2PaRR-1RC*C* (Fig. 2.65b) (Fig. 2.66b) (Fig. 2.67b) 2RPC*-1RPP 2S*PR-1RPP (Fig. 2.65c) (Fig. 2.66c) 1 m 8 8 12 2 p1 3 3 6 3 p2 3 3 6 4 p3 3 3 3 5 p 9 9 15 6 q 2 2 4 7 k1 3 3 2 8 k2 0 0 1 9 k 3 3 3 10 (RGi) See Table 2.14See Table 2.14See Table 2.14 (i=1,2,3) 11 SG1 4 5 3 12 SG2 4 5 3 13 SG3 3 3 5 14 rG1 0 0 3 15 rG2 0 0 3 16 rG3 0 0 0 17 MG1 4 5 3 18 MG2 4 5 3 19 MG3 3 3 5 20 (RF) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) 21 SF 3 3 3 22 rl 0 0 6 23 rF 8 10 14 24 MF 3 3 3 25 NF 4 2 10 26 TF 0 0 0 p1 27 4 5 6 f

3 3 5 6 6 0 3 3 5 ( v1 , v 2 , 3 12 20 3 16 0 9

28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

2PaPaR-1RC*C* (Fig. 2.67c)

16 9 9 3 21 6 2 1 3 See Table 2.14

fj

4

5

6

9

fj

3

3

5

5

fj

11

13

17

23

See footnote of Table 2.4 for the nomenclature of structural parameters

δ

)

2.2 Derived solutions

107

Table 2.30. Structural parametersa of planar parallel mechanisms in Figs. 2.682.70 No. Structural Solution parameter 2PPassR-1RPP 2PassPassR-1RPP (Fig. 2.68a) (Fig. 2.68c) 2PassRR-1RPP (Fig. 2.68b)

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

12 6 6 3 15 4 1 2 3 See Table 2.14

16 9 9 3 21 6 1 2 3 See Table 2.14

2PRC*-1RPPa (Fig. 2.69a) 2RRC*-1RPPa (Fig. 2.69b) 2RPC*-1RPPa (Fig. 2.69c) 2PRC*-1RPaP (Fig. 2.70a) 2RRC*-1RPaP (Fig. 2.70b) 2RPC*-1RPaP (Fig. 2.70c) 10 3 3 6 12 3 2 1 3 See Table 2.14

4 4 3 6 6 0 4 4 3 ( v1 , v 2 , 3 12 20 3 4 0 10

4 4 3 12 12 0 5 5 3 ( v1 , v 2 , 3 24 32 5 4 2 17

4 4 3 0 0 3 4 4 3 ( v1 , v 2 , 3 3 11 3 7 0 4

δ

)

δ

)

fj

10

17

4

fj

3

3

6

fj

23

37

14

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

108

2 Overconstrained planar parallel robots with coupled motions

Table 2.31. Structural parametersa of planar parallel mechanisms in Figs. 2.712.74 No. Structural Solution parameter 2PRC*-1RPaPa (Fig. 2.71a) 2RRC*-1RPaPa (Fig. 2.71b) 2RPC*-1RPaPa (Fig. 2.71c) 1 m 12 2 p1 3 3 p2 3 4 p3 9 5 p 15 6 q 4 7 k1 2 8 k2 1 9 k 3 10 (RGi) See Table 2.14 (i=1,2,3) 11 SG1 4 12 SG2 4 13 SG3 3 14 rG1 0 15 rG2 0 16 rG3 6 17 MG1 4 18 MG2 4 19 MG3 3 20 (RF) ( v1 , v 2 , δ ) 21 SF 3 22 rl 6 23 rF 14 24 MF 3 25 NF 10 26 TF 0 p1 27 4 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

2PRR-1RPPass (Fig. 2.72a) 2RRR-1RPPass (Fig. 2.72b) 2RPR-1RPPass (Fig. 2.72c) 2PRR-1RPassP (Fig. 2.73a) 2RRR-1RPassP (Fig. 2.73b) 2RPR-1RPassP (Fig. 2.73c) 10 3 3 6 12 3 2 1 3 See Table 2.14

2PRR-1RPassPass (Fig. 2.74a) 2RRR-1RPassPass (Fig. 2.74b) 2RPR-1RPassPass (Fig. 2.74c) 12 3 3 9 15 4 2 1 3 See Table 2.14

3 3 3 0 0 6 3 3 4 ( v1 , v 2 , 3 6 12 4 6 1 3

3 3 3 0 0 12 3 3 5 ( v1 , v 2 , 3 12 18 5 6 2 3

δ

)

fj

4

3

3

fj

9

10

17

fj

17

16

23

See footnote of Table 2.4 for the nomenclature of structural parameters

δ

)

2.2 Derived solutions

109

Table 2.32. Structural parametersa of planar parallel mechanisms in Figs. 2.752.78 No. Structural Solution parameter 2PPaC*-1RPPa (Fig. 2.75a) 2PaRC*-1RPPa (Fig. 2.75b) 2PPaC*-1RpaP (Fig. 2.76a) 2PaRC*-1RPaP (Fig. 2.76b) 1 m 14 2 p1 6 3 p2 6 4 p3 6 5 p 18 6 q 5 7 k1 0 8 k2 3 9 k 3 10 (RGi) See Table 2.14 (i=1,2,3) 11 SG1 4 12 SG2 4 13 SG3 3 14 rG1 3 15 rG2 3 16 rG3 3 17 MG1 4 18 MG2 4 19 MG3 3 20 (RF) ( v1 , v 2 , δ ) 21 SF 3 22 rl 9 23 rF 17 24 MF 3 25 NF 13 26 TF 0 p1 27 7 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

2PaPaC*-1RPPa 2PPaC*-1RPaPa (Fig. 2.77a) (Fig. 2.78a) 2PaPaC*-1RPaP 2PaRC*-1RPaPa (Fig. 2.77b) (Fig. 2.78b) 18 16 9 6 9 6 6 9 24 21 7 6 0 0 3 3 3 3 See Table 2.14 See Table 2.14 4 4 3 6 6 3 4 4 3 ( v1 , v 2 , 3 15 23 3 19 0 10

δ

)

4 4 3 3 3 6 4 4 3 ( v1 , v 2 , 3 12 20 3 16 0 7

j

fj

7

10

7

fj

6

6

9

fj

20

26

23

See footnote of Table 2.4 for the nomenclature of structural parameters

δ

)

110

2 Overconstrained planar parallel robots with coupled motions

Table 2.33. Structural parametersa of planar parallel mechanisms in Figs. 2.792.81 No. Structural Solution parameter 2PaPaC*-1RPaPa 2PassPassR-1RPassPass 2PPassR-1RPPass (Fig. 2.79a) 2PassRR-1RPPass (Fig. 2.79b) (Fig. 2.80a,b) 2PPassR-1RPassP 2PassRR-1RPassP (Fig. 2.81a,b) 1 m 20 20 14 2 p1 9 9 6 3 p2 9 9 6 4 p3 9 9 6 5 p 27 27 18 6 q 8 8 5 7 k1 0 0 0 8 k2 3 3 3 9 k 3 3 3 10 (RGi) See Table 2.14 See Table 2.14 See Table 2.14 (i=1,2,3) 11 SG1 4 4 4 12 SG2 4 4 4 13 SG3 3 5 4 14 rG1 6 12 6 15 rG2 6 12 6 16 rG3 6 12 6 17 MG1 4 5 4 18 MG2 4 5 4 19 MG3 3 5 4 20 (RF) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) 21 SF 3 3 3 22 rl 18 36 18 23 rF 26 46 27 24 MF 3 5 3 25 NF 22 2 3 26 TF 0 2 0 p1 27 10 17 10 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

fj

10

17

10

fj

9

17

10

fj

29

51

30

See footnote of Table 2.4 for the nomenclature of structural parameters

2.2 Derived solutions

111

Table 2.34. Structural parametersa of planar parallel mechanisms in Figs. 2.822.86 No. Structural Solution parameter 2PassPassR1RPPass (Fig. 2.82a) 2PassPassR1RPassP (Fig. 2.82b) 1 m 18 2 p1 9 3 p2 9 4 p3 6 5 p 24 6 q 7 7 k1 0 8 k2 3 9 k 3 10 (RGi) See Table 2.14 (i=1,2,3) 11 SG1 4 12 SG2 4 13 SG3 4 14 rG1 12 15 rG2 12 16 rG3 6 17 MG1 5 18 MG2 5 19 MG3 4 20 (RF) ( v1 , v 2 , δ ) 21 SF 3 22 rl 30 23 rF 39 24 MF 5 25 NF 3 26 TF 2 p1 27 17 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

2PPassR1RPassPass (Fig. 2.83a) 2PassRR1RPassPass (Fig. 2.83b) 16 6 6 9 21 6 0 3 3 See Table 2.14

2PRC*-1RPacsPacs (Fig. 2.84a) 2PRC*-1RPacsPatcs (Fig. 2.84b) 2RRC*-1RPacsPacs (Fig. 2.85a) 2RRC*-1RPacsPatcs (Fig. 2.85b) 2RPC*-1RPacsPacs (Fig. 2.86a) 2RPC*-1RPacsPatcs (Fig. 2.86b) 12 3 3 9 15 4 2 1 3 See Table 2.14

4 4 5 6 6 12 4 4 5 ( v1 , v 2 , 3 24 34 3 2 0 10

4 4 3 0 0 12 4 4 3 ( v1 , v 2 , 3 12 20 3 4 0 4

δ

)

fj

17

10

4

fj

10

17

15

fj

44

37

23

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

112

2 Overconstrained planar parallel robots with coupled motions

Table 2.35. Structural parametersa of planar parallel mechanisms in Figs. 2.872.91 No. Structural Solution parameter 2PS*R-1RPacsPacs (Fig. 2.87a) 2PS*R-1RPacsPatcs (Fig. 2.87b) 2RS*R-1RPacsPacs (Fig. 2.88a) 2RS*R-1RPacsPatcs (Fig. 2.88b) 2S*PR-1RPacsPacs (Fig. 2.89a) 2S*PR-1RPacsPatcs (Fig. 2.89b) 1 m 12 2 p1 3 3 p2 3 4 p3 9 5 p 15 6 q 4 7 k1 2 8 k2 1 9 k 3 10 (RGi) See Table 2.14 (i=1,2,3) 11 SG1 5 12 SG2 5 13 SG3 3 14 rG1 0 15 rG2 0 16 rG3 12 17 MG1 5 18 MG2 5 19 MG3 3 20 (RF) ( v1 , v 2 , δ ) 21 SF 3 22 rl 12 23 rF 22 24 MF 3 25 NF 2 26 TF 0 p1 27 5 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

2PPassC*-1RPaPa (Fig. 2.90a) 2PPassC*-1RPaPat (Fig. 2.90b) 2PassRC*-1RPaPa (Fig. 2.91a) 2PassRC*-1RPaPat (Fig. 2.91b)

16 6 6 9 21 6 0 3 3 See Table 2.14 5 5 3 6 6 6 5 5 3 ( v1 , v 2 , 3 18 28 3 8 0 11

fj

5

11

fj

15

9

fj

25

31

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

2.2 Derived solutions

113

Table 2.36. Structural parametersa of planar parallel mechanisms in Figs. 2.922.95 No. Structural Solution parameter 2PassPassC*- 2PPassC*-1RPacsPacs (Fig. 2.93a) 1RPaPa 2PPassC*-1RPacsPatcs (Fig. 2.93b) (Fig. 2.92a) 2PassRC*-1RPacsPacs (Fig. 2.94a) 2PassPassC*- 2PassRC*-1RPacsPatcs (Fig. 2.94b) 1RPaPat (Fig. 2.92b) 1 m 20 16 2 p1 9 6 3 p2 9 6 4 p3 9 9 5 p 27 21 6 q 8 6 7 k1 0 0 8 k2 3 3 9 k 3 3 10 (RGi) See Table 2.14See Table 2.14 (i=1,2,3) 11 SG1 6 5 12 SG2 6 5 13 SG3 3 3 14 rG1 12 6 15 rG2 12 6 16 rG3 6 12 17 MG1 6 5 18 MG2 6 5 19 MG3 3 3 20 (RF) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) 21 SF 3 3 22 rl 30 24 23 rF 42 34 24 MF 3 3 25 NF 6 2 26 TF 0 0 p1 27 18 11 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

2PassPassC*1RPacsPacs (Fig. 2.95a) 2PassPassC*1RPacsPatcs (Fig. 2.95b) 20 9 9 9 27 8 0 3 3 See Table 2.14 6 6 3 12 12 9 6 6 3 ( v1 , v 2 , 3 33 45 3 3 0 18

fj

18

11

18

fj

9

15

12

fj

45

37

48

See footnote of Table 2.4 for the nomenclature of structural parameters

δ

)

114

2 Overconstrained planar parallel robots with coupled motions

Table 2.37. Structural parametersa of planar parallel mechanisms in Figs. 2.96 and 2.97 No. Structural Solution parameter (PRRRP)C*-RPP (Fig. 2.96a) (RRRRR)C*-RPP (Fig. 2.96b) (RPRPR)C*-RPP (Fig. 2.96c) 1 m 8 2 p1 6 3 p2 3 4 p 9 5 q 2 6 k1 1 7 k2 1 8 k 2 9 (RG1-G2) ( v1 , v 2 , v 3 , δ ) 10

(RG3)

11 12 13 14 15 16 17

SG1-G2 SG3 rG1-G2 rG3 MG1-G2 MG3 (RF)

18 19 20 21 22 23 24

SF rl rF MF NF TF

25 26 a

∑ ∑ ∑

p1

j =1

p2

j =1

p

j =1

fj

( v1 , v 2 , 4 3 3 0 4 3 ( v1 , v 2 , 3 3 7 3 5 0 7

δ

)

δ

)

(PS*RS*P)R-RPP (Fig. 2.97a) (RS*RS*R)R-RPP (Fig. 2.97b) (S*PRPS*)R-RPP (Fig. 2.97c) 8 6 3 9 2 1 1 2 ( v1 , v 2 , β , δ ) ( v1 , v 2 , 4 3 6 0 4 3 ( v1 , v 2 , 3 6 10 3 2 0 10

fj

3

3

fj

10

13

δ

)

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

2.2 Derived solutions

115

Table 2.38. Structural parametersa of planar parallel mechanisms in Figs. 2.982.100 No. Structural Solution parameter (PRRRP)R-RPPass (Fig. 2.98a) (RRRRR)R-RPPass (Fig. 2.98b) (RPRPR)R-RPPass (Fig. 2.98c) PRRRP)R-RPassP (Fig. 2.99a) (RRRRR)R-RPassP (Fig. 2.99b) (RPRPR)R-RPassP (Fig. 2.99c) 1 m 10 2 p1 6 3 p2 6 4 p 12 5 q 3 6 k1 0 7 k2 2 8 k 2 9 (RG1-G2) ( v1 , v 2 , δ ) 10 (RG3) ( v1 , v 2 , δ ) 11 SG1-G2 3 12 SG3 3 13 rG1-G2 3 14 rG3 6 15 MG1-G2 3 16 MG3 4 17 (RF) ( v1 , v 2 , δ ) 18 SF 3 19 rl 9 20 rF 12 21 MF 4 22 NF 6 23 TF 1 p1 24 6 f

(PRRRP)R-RPassPass (Fig. 2.100a) (RRRRR)R-RPassPass (Fig. 2.100b) (RPRPR)R-RPassPass (Fig. 2.100c) 12 6 9 15 4 0 2 2 ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) 3 3 3 12 3 5 ( v1 , v 2 , δ ) 3 15 18 5 6 2 6

25 26 a

∑ ∑ ∑

j =1

p2

j =1

p

j =1

j

fj

10

17

fj

16

23

See footnote of Table 2.4 for the nomenclature of structural parameters

116

2 Overconstrained planar parallel robots with coupled motions

Table 2.39. Structural parametersa of planar parallel mechanisms in Figs. 2.1012.103 No. Structural Solution parameter (PS*RS*P)R-RPPa (Fig. 2.101a) (RS*RS*R)R-RPPa (Fig. 2.101b) (S*PRPS*)R-RPPa (Fig. 2.101c) (PS*RS*P)R-RPaP (Fig. 2.102a) (RS*RS*R)R-RPaP (Fig. 2.102b) (S*PRPS*)R-RPaP (Fig. 2.102c) 1 m 10 2 p1 6 3 p2 6 4 p 12 5 q 3 6 k1 0 7 k2 2 8 k 2 9 (RG1-G2) ( v1 , v 2 , β , δ ) 10

(RG3)

11 12 13 14 15 16 17

SG1-G2 SG3 rG1-G2 rG3 MG1-G2 MG3 (RF)

18 19 20 21 22 23 24

SF rl rF MF NF TF

25 26 a

∑ ∑ ∑

p1

j =1

p2

j =1

p

j =1

fj

( v1 , v 2 , 4 3 6 3 4 3 ( v1 , v 2 , 3 9 13 3 5 0 10

δ

)

δ

)

(PS*RS*P)R-RPaPa (Fig. 2.103a) (RS*RS*R)R-RPaPa (Fig. 2.103b) (S*PRPS*)R-RPaPa (Fig. 2.103c) 12 6 9 15 4 0 2 2 ( v1 , v 2 , β , δ ) ( v1 , v 2 , 4 3 6 6 4 3 ( v1 , v 2 , 3 12 16 3 8 0 10

fj

6

9

fj

16

19

δ

)

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

2.2 Derived solutions

117

Table 2.40. Structural parametersa of planar parallel mechanisms in Figs. 2.1042.109 No. Structural Solution parameter (PRRRP)R-RPacsPacs (Fig. 2.104a) (PRRRP)R-RPacsPatcs (Fig. 2.104b) (RRRRR)R-RPacsPacs (Fig. 2.105a) (RRRRR)R-RPacsPatcs (Fig. 2.105b) (RPRPR)R-RPacsPacs (Fig. 2.106a) (RPRPR)R-RPacsPatcs (Fig. 2.106b) 1 m 12 2 p1 6 3 p2 9 4 p 15 5 q 4 6 k1 0 7 k2 2 8 k 2 9 (RG1-G2) ( v1 , v 2 , δ ) 10

(RG3)

11 12 13 14 15 16 17

SG1-G2 SG3 rG1-G2 rG3 MG1-G2 MG3 (RF)

18 19 20 21 22 23 24

SF rl rF MF NF TF

25 26 a

∑ ∑ ∑

p1

j =1

p2

j =1

p

j =1

fj

( v1 , v 2 , 3 3 3 12 3 3 ( v1 , v 2 , 3 15 18 3 6 0 6

δ

)

δ

)

(PS*RS*P)R-RPacsPacs (Fig. 2.107a) (PS*RS*P)R-RPacsPatcs (Fig. 2.107b) (RS*RS*R)R-RPacsPacs (Fig. 2.108a) (RS*RS*R)R-RPacsPatcs (Fig. 2.108b) (S*PRPS*)R-RPacsPacs (Fig. 2.109a) (S*PRPS*)R-RPacsPatcs (Fig. 2.109b) 12 6 9 15 4 0 2 2 ( v1 , v 2 , β , δ ) ( v1 , v 2 , 4 3 6 12 4 3 ( v1 , v 2 , 3 18 22 3 2 0 10

fj

9

15

fj

15

25

δ

)

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

118

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.44. Overconstrained PPMs with coupled motions of types 3-RR*RR (a) and 3-RRC* (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3, limb topology R ⊥ R* ⊥ ||R||R (a) and R||R ⊥ C* (b)

2.2 Derived solutions

119

Fig. 2.45. Overconstrained PPMs with coupled motions of types 3-RC*R (a) and 3-C*RR (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3, limb topology R ⊥ C* ⊥ ||R (a) and C* ⊥ R||R (b)

120

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.46. Overconstrained PPMs with coupled motions of types 3-PRC* (a) and 3-C*RP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3, limb topology P ⊥ R ⊥ C* (a) and C* ⊥ R ⊥ P (b)

2.2 Derived solutions

121

Fig. 2.47. Overconstrained PPMs with coupled motions of types 3-PaRS* (a) and 3-PassRR (b) defined by SF=3, (RF)=( v1 , v2 , δ ) and MF=3, NF=9, TF=0 (a), MF=6, NF=6, TF=3 (b), limb topology Pa||RS* (a) and Pass||R||R (b)

122

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.48. Overconstrained PPMs with coupled motions of types 3-PaPaS* (a) and 3-PaS*P (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=18 (a), NF=9 (b), limb topology Pa||PaS* (a) and PaS*P (b)

2.2 Derived solutions

123

Fig. 2.49. Overconstrained PPMs with coupled motions of types 3-PaPS* (a) and 3-PPaS* (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=9, limb topology Pa ⊥ PS* (a) and P ⊥ PaS* (b)

124

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.50. Overconstrained PPMs with coupled motions of types 3-PassPassR (a) and 3-PassRP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=6 and MF=9, TF=6 (a) MF=6, TF=3 (a), limb topology Pass||Pass||R (a) and Pass||R ⊥ P (b)

2.2 Derived solutions

125

Fig. 2.51. Overconstrained PPMs with coupled motions of types 3-PassPR (a) and 3-PPassR (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=6, NF=6, TF=3, limb topology Pass ⊥ P ⊥ ||R (a) and P ⊥ Pass||R (b)

126

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.52. Overconstrained PPM with one decoupled translation of type 3-PC*R defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3, limb topology P ⊥ C* ⊥ ⊥ R

Fig. 2.53. Overconstrained PPM with one decoupled translation of type 1PC*R2RR*RR defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3, limb topology P ⊥ C* ⊥ ⊥ R and R ⊥ R* ⊥ ||R||R

2.2 Derived solutions

127

Fig. 2.54. Overconstrained PPMs with one decoupled translation of types 1PC*R2RRC* (a) and 1PC*R-2RC*R (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0,

NF=3, limb topology P ⊥ C* ⊥ ⊥ R and R||R ⊥ C* (a), P ⊥ C* ⊥ ⊥ R and R ⊥ C* ⊥ ||R (b)

128

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.55. Overconstrained PPMs with one decoupled translation of types 1C*PR2C*RR (a) and 1PC*R-2PRC* (b) defined by MF=SF=3, (RF)= ( v1 , v2 , δ ), TF=0,

NF=3, limb topology P ⊥ C* ⊥ ⊥ R and C* ⊥ R||R (a), P ⊥ C* ⊥ ⊥ R and P ⊥ R ⊥ C* (b)

2.2 Derived solutions

129

Fig. 2.56. Overconstrained PPMs with one decoupled translation of types 1PaPS*-2PaRS* (a) and 1PaPS*-2PPaS* (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=9, limb topology Pa ⊥ PS* and Pa||RS* (a), Pa ⊥ PS* and P ⊥ PaS* (b)

130

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.57. Overconstrained PPMs with one decoupled translation of types 1PaPS*-2PaPaS* (a) and 1PaPS*-2PaS*P (b) defined by MF=SF=3, (RF)= ( v1 , v2 , δ ), TF=0, NF=15 (a), NF=9 (b), limb topology Pa ⊥ PS* and Pa||PaS* (a), Pa ⊥ PS* and PaS*P (b)

2.2 Derived solutions

131

Fig. 2.58. Overconstrained PPMs with one decoupled translation of types 1PassPR-2PassRR (a) and 1PassPR-2PPassR (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3, limb topology Pass ⊥ P ⊥ ||R and Pass||R||R (a), Pass ⊥ P ⊥ ||R and P ⊥ Pass||R (b)

132

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.59. Overconstrained PPMs with one decoupled translation of types 1PassPR-2PaPassR (a) and 1PassPR-2PassRP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=9 (a), NF=3 (b), limb topology Pass ⊥ P ⊥ ||R and Pa||Pass||R (a), Pass ⊥ P ⊥ ||R and Pass||R ⊥ P (b)

2.2 Derived solutions

133

Fig. 2.60. 3RR*RR-type overconstrained PPMs with partially decoupled translations defined by MF=SF=3, (RF)= ( v1 , v2 , δ ), TF=0, NF=3, limb topology R ⊥ R* ⊥ ||R||R

134

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.61. Overconstrained PPMs with partially decoupled translations of types 3RC*R (a) and 2RC*R-1RC*R (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3, limb topology R ⊥ C* ⊥ ||R (a), R ⊥ C* ⊥ ||R and R ⊥ C* ⊥ ||R (b)

2.2 Derived solutions

135

Fig. 2.62. Overconstrained PPM with decoupled rotation of type 2RPC*-1RPC* defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3, limb topology R ⊥ P ⊥ ⊥ C*

Fig. 2.63. Overconstrained PPM with decoupled rotation of type 2RC*Pa1RC*Pa defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=12, limb topology R ⊥ C* ⊥ ||Pa

136

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.64. Overconstrained PPMs with decoupled rotation of types 2RPaC*1RPaC* (a) and 2RPaPass-1RPaPass (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=12, limb topology R||Pa ⊥ C* (a) and R||Pa||Pass (b)

2.2 Derived solutions

137

Fig. 2.65. Overconstrained PPMs with uncoupled rotation of types 2PRC*-1RPP (a), 2RRC*-1RPP (b) and 2RPC*-1RPP (c) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=4, limb topology R ⊥ P ⊥ ⊥ P and P ⊥ R||C* (a), R||R||C* (b), R ⊥ P ⊥ ||C* (c)

138

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.66. Overconstrained PPMs with uncoupled rotation of types 2PS*R-1RPP (a), 2RS*R-1RPP (b) and 2S*PR-1RPP (c) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=2, limb topology R ⊥ P ⊥ ⊥ P and PS*R (a), RS*R (b), SP ⊥ R (c)

2.2 Derived solutions

139

Fig. 2.67. Overconstrained PPMs with uncoupled rotation of types 2PPaR1RC*C* (a), 2PaRR-1RC*C* (b) and 2PaPaR-1RC*C* (c) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=10 (a), (b) NF=16 (c), limb topology R ⊥ C* ⊥ ⊥ C* and P ⊥ Pa||R (a), Pa||R||R (b), Pa||Pa||R (c)

140

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.68. Overconstrained PPMs with uncoupled rotation of types 2PPassR-1RPP (a), 2PassRR-1RPP (b) and 2PassPassR-1RPP (c) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=4 and MF=3, TF=0 (a and b), MF=5, TF=2 (c), limb

topology R ⊥ P ⊥ ⊥ P and P ⊥ Pass||R (a), Pass||R||R (b), Pass||Pass||R (c)

2.2 Derived solutions

141

Fig. 2.69. Overconstrained PPMs with uncoupled rotation of types 2PRC*-1RPPa (a), 2RRC*-1RPPa (b) and 2RPC*-1RPPa (c) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=7, limb topology R ⊥ P ⊥ ||Pa and P ⊥ R||C* (a), R||R||C* (b), R ⊥ P ⊥ ||C* (c)

142

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.70. Overconstrained PPMs with uncoupled rotation of types 2PRC*-1RPaP (a), 2RRC*-1RPaP (b) and 2RPC*-1RPaP (c) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=7, limb topology R||Pa ⊥ P and P ⊥ R||C* (a), R||R||C* (b), R ⊥ P ⊥ ||C* (c)

2.2 Derived solutions

143

Fig. 2.71. Overconstrained PPMs with uncoupled rotation of types 2PRC*1RPaPa (a), 2RRC*-1RPaPa (b) and 2RPC*-1RPaPa (c) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=10, limb topology R||Pa||Pa and P ⊥ R||C* (a), R||R||C* (b), R ⊥ P ⊥ ||C* (c)

144

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.72. Overconstrained PPMs with uncoupled rotation of types 2PRR-1RPPass (a), 2RRR-1RPPass (b) and 2RPR-1RPPass (c) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, NF=6, TF=1, limb topology R ⊥ P ⊥ ||Pass and P ⊥ R||R (a), R||R||R (b), R ⊥ P ⊥ ||R (c)

2.2 Derived solutions

145

Fig. 2.73. Overconstrained PPMs with uncoupled rotation of types 2PRR-1RPassP (a), 2RRR-1RPassP (b) and 2RPR-1RPassP (c) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4 NF=6, TF=1, limb topology R||Pass ⊥ P and P ⊥ R||R (a), R||R||R (b), R ⊥ P ⊥ ||R (c)

146

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.74. Overconstrained PPMs with uncoupled rotation of types 2PRR1RPassPass (a), 2RRR-1RPassPass (b) and 2RPR-1RPassPass (c) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, NF=6, TF=2, limb topology R||Pass||Pass and P ⊥ R||R (a), R||R||R (b), R ⊥ P ⊥ ||R (c)

2.2 Derived solutions

147

Fig. 2.75. Overconstrained PPMs with uncoupled rotation of types 2PPaC*1RPPa (a) and 2PaRC*-1RPPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=13, limb topology R ⊥ P ⊥ ||Pa and P ⊥ Pa||C* (a), Pa||R||C* (b)

148

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.76. Overconstrained PPMs with uncoupled rotation of types 2PPaC*1RPaP (a) and 2PaRC*-1RPaP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=13, limb topology R||Pa ⊥ P and P ⊥ Pa||C* (a), Pa||R||C* (b)

2.2 Derived solutions

149

Fig. 2.77. Overconstrained PPMs with uncoupled rotation of types 2PaPaC*1RPPa (a) and 2PaPaC*-1RPaP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=19, limb topology Pa||Pa||C* and R ⊥ P ⊥ ||Pa (a), R||Pa ⊥ P (b)

150

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.78. Overconstrained PPMs with uncoupled rotation of types 2PPaC*1RPaPa (a) and 2PaRC*-1RPaPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=16, limb topology R||Pa||Pa and P ⊥ Pa||C* (a), Pa||R||C* (b)

2.2 Derived solutions

151

Fig. 2.79. Overconstrained PPMs with uncoupled rotation of types 2PaPaC*1RPaPa (a) and 2PassPassR-1RPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ) and MF=3, NF=22, TF=0 (a), MF=5, NF=2, TF=2 (b), limb topology Pa||Pa||C* and R||Pa||Pa (a), Pass||Pass||R and R||Pass||Pass (b)

152

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.80. Overconstrained PPMs with uncoupled rotation of types 2PPassR1RPPass (a) and 2PassRR-1RPPass (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3, limb topology R ⊥ P ⊥ ||Pass and P ⊥ Pass||R (a), Pass||R||R (b)

2.2 Derived solutions

153

Fig. 2.81. Overconstrained PPMs with uncoupled rotation of types 2PPassR1RPassP (a) and 2PassRR-1RPassP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3, limb topology R ⊥ Pass ⊥ ||P and P ⊥ Pass||R (a), Pass||R||R (b)

154

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.82. Overconstrained PPMs with uncoupled rotation of types 2PassPassR 1RPPass (a) and 2PassPassR-1RPassP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, NF=3, TF=2, limb topology Pass||Pass||R and R ⊥ P ⊥ ||Pass (a), R||Pass ⊥ P (b)

2.2 Derived solutions

155

Fig. 2.83. Overconstrained PPMs with uncoupled rotation of types 2PPassR1RPassPass (a) and 2PassRR-1RPassPass (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=2, limb topology R||Pass||Pass and P ⊥ Pass||R (a), Pass||R||R (b)

156

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.84. Overconstrained PPMs with uncoupled rotation of types 2PRC*1RPacsPacs (a) and 2PRC*-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=4, limb topology P ⊥ R||C* and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

2.2 Derived solutions

157

Fig. 2.85. Overconstrained PPMs with uncoupled rotation of types 2RRC*1RPacsPacs (a) and 2RRC*-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=4, limb topology R||R||C* and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

158

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.86. Overconstrained PPMs with uncoupled rotation of types 2RPC*1RPacsPacs (a) and 2RPC*-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=4, limb topology R ⊥ P ⊥ ||C* and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

2.2 Derived solutions

159

Fig. 2.87. Overconstrained PPMs with uncoupled rotation of types 2PS*R1RPacsPacs (a) and 2PS*R-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=2, limb topology PS*R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

160

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.88. Overconstrained PPMs with uncoupled rotation of types 2RS*R1RPacsPacs (a) and 2RS*R-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=2, limb topology RS*R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

2.2 Derived solutions

161

Fig. 2.89. Overconstrained PPMs with uncoupled rotation of types 2S*PR1RPacsPacs (a) and 2S*PR-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=2, limb topology S*P ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

162

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.90. Overconstrained PPMs with uncoupled rotation of types 2PPassC*1RPaPa (a) and 2PPassC*-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=8, limb topology P ⊥ Pass||C* and R||Pa||Pa (a), R||Pa||Pat (b)

2.2 Derived solutions

163

Fig. 2.91. Overconstrained PPMs with uncoupled rotation of types 2PassRC*1RPaPa (a) and 2PassRC*-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=8, limb topology Pass||R||C* and R||Pa||Pa (a), R||Pa||Pat (b)

164

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.92. Overconstrained PPMs with uncoupled rotation of types 2PassPassC*1RPaPa (a) and 2PassPassC*-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=6, limb topology Pass||Pass||C* and R||Pa||Pa (a), R||Pa||Pat (b)

2.2 Derived solutions

165

Fig. 2.93. Overconstrained PPMs with uncoupled rotation of types 2PPassC*1RPacsPacs (a) and 2PPassC*-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=2, limb topology P||Pass||C* and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

166

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.94. Overconstrained PPMs with uncoupled rotation of types 2PassRC*1RPacsPacs (a) and 2PassRC*-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=2, limb topology Pass||R||C*and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

2.2 Derived solutions

167

Fig. 2.95. Overconstrained PPMs with uncoupled rotation of types 2PassPassC*1RPacsPacs (a) and 2PassPassC*-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3, limb topology Pass||Pass||C*and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

168

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.96. Overconstrained PPMs with uncoupled rotation of types (PRRRP)C*RPP (a), (RRRRR)C*-RPP (b) and (RPRPR)C*-RPP (c) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=5

2.2 Derived solutions

169

Fig. 2.97. Overconstrained PPMs with uncoupled rotation of types (PS*RS*P)RRPP (a), (RS*RS*R)R-RPP (b) and (S*PRPS*)R-RPP (c) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=2

170

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.98. Overconstrained PPMs with uncoupled rotation of types (PRRRP)RRPPass (a), (RRRRR)R-RPPass (b) and (RPRPR)R-RPPass (c) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, NF=6, TF=1

2.2 Derived solutions

171

Fig. 2.99. Overconstrained PPMs with uncoupled rotation of types (PRRRP)RRPassP (a), (RRRRR)R-RPassP (b) and (RPRPR)R-RPassP (c) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, NF=6, TF=1

172

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.100. Overconstrained PPMs with uncoupled rotation of types (PRRRP)RRPassPass (a), (RRRRR)R-RPassPass (b) and (RPRPR)R-RPassPass (c) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, NF=6, TF=2

2.2 Derived solutions

173

Fig. 2.101. Overconstrained PPMs with uncoupled rotation of types (PS*RS*P)RRPPa (a), (RS*RS*R)R-RPPa (b) and (S*PRPS*)R-RPPa (c) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=5

174

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.102. Overconstrained PPMs with uncoupled rotation of types (PS*RS*P)RRPaP (a), (RS*RS*R)R-RPaP (b) and (S*PRPS*)R-RPaP (c) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=5

2.2 Derived solutions

175

Fig. 2.103. Overconstrained PPMs with uncoupled rotation of types (PS*RS*P)RRPaPa (a), (RS*RS*R)R-RPaPa (b) and (S*PRPS*)R-RPaPa (c) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=8

176

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.104. Overconstrained PPMs with uncoupled rotation of types (PRRRP)RRPacsPacs (a) and (PRRRP)R-RPacsPatcs (b) defined by MF=SF=3, (RF)= ( v1 , v2 , δ ), TF=0, NF=6

2.2 Derived solutions

177

Fig. 2.105. Overconstrained PPMs with uncoupled rotation of types (RRRRR)RRPacsPacs (a) and (RRRRR)R-RPacsPatcs (b) defined by MF=SF=3, (RF)= ( v1 , v2 , δ ), TF=0, NF=6

178

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.106. Overconstrained PPMs with uncoupled rotation of types (RPRPR)RRPacsPacs (a) and (RPRPR)R-RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=6

2.2 Derived solutions

179

Fig. 2.107. Overconstrained PPMs with uncoupled rotation of types (PS*RS*P)RRPacsPacs (a) and (PS*RS*P)R-RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=2

180

2 Overconstrained planar parallel robots with coupled motions

Fig. 2.108. Overconstrained PPMs with uncoupled rotation of types (RS*RS*R)RRPacsPacs (a) and (RS*RS*R)R-RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=2

2.2 Derived solutions

181

Fig. 2.109. Overconstrained PPMs with uncoupled rotation of types (S*PRPS*)RRPacsPacs (a) and (S*PRPS*)R-RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=2

3 Non overconstrained planar parallel robots with coupled motions

Equation (1.15) indicates that non overconstrained solutions of planar 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 ,ωδ ) and NF=0. They can have identical limbs or limbs with different structures and may be actuated by linear or rotating motors. These solutions are derived from the overconstrained counterparts presented in Figs. 2.4-2.43 by introducing the required idle mobilities. They can be fully- or not fully-parallel solutions with the actuators mounted on the base or on a moving link.

3.1 Fully-parallel solutions The fully-parallel non overconstrained solutions presented in Figs. 3.13.32 are derived from the solutions in Figs. 2.4-2.36 by introducing the required idle mobilities. 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. The idle mobilities can be introduced outside or inside the loops combined in the limbs. For example, the non overconstrained solutions in Fig. 3.1 are derived from the overconstrained solutions in Fig. 2.4 by introducing two rotational idle mobilities in each spherical joint. They are introduced by replacing one revolute joint in each limb of the solutions in Fig. 2.4 by a spherical joint. We recall that the notation Pacs is associated with a parallelogram loop with three idle mobilities combined in a cylindrical and a spherical joint, and Pass with four idle mobilities combined in two spherical joints adjacent to the same link. In the cylindrical joints of the parallelogram loop denoted by Pacs (Figs. 3.28-3.33 and 3.38-3.40) the translational motion is an idle mobility. In the parallelogram loop Pass-type, three idle mobilities are introduced in the loop and one outside the loop. If the link adjacent to the G. Gogu, Structural Synthesis of Parallel Robots, Solid Mechanics and Its Applications 173, DOI 10.1007/978-90-481-9831-3_3, © Springer Science + Business Media B.V. 2010

183

184

3 Non overconstrained planar parallel robots with coupled motions

two spherical joints is a binary link than the idle mobility introduced outside the loop becomes an internal rotational mobility of the binary link around the axis passing by the centre of the two spherical joints. This internal mobility gives one degree of structural redundancy (see Table 3.1). If the link adjacent to the two spherical joints is connected in the limb by three or more joints (polinary link) than the rotational motion around the axis passing by the centre of the two spherical joints is an idle (potential) mobility of the limb. This mobility is restricted by the constraints of the parallel mechanism and remains just a potential mobility. For example in Fig. 3.27, this rotational motion is internal mobility of links 3C and 6C, and idle mobility for the ternary links 4A, 7A and 4B, 7B. The limb topology of the non overconstrained solutions (NF=0) in Figs. 3.1-3.32 are systematized in Tables 3.2-3.5, as are their structural parameters in Tables 3.7-3.14. The bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms in Figs. 3.1-3.32 are given in Table 3.6. Table 3.1. Links with internal mobilities and the degree of structural redundancy TF of overconstrained SPMs with uncoupled planar motion of the moving platform No. Parallel mechanism Example

TF

Link with internal rotational mobility in limb G1 G2 G3

1 2

6 3

3A, 6 A 3A

3 B, 6B 3B

3 C, 6C 3C

3 3 5 1 1 2 1

4A 3A 3A -

4B 4B 3 B, 6B -

4C 4C 3 C, 6C 5C 4C 3 C, 6C 3C

3 4 5 6 7 8 9

Fig. 3.4a Figs. 3.4b, 3.5a, 3.6, 3.11a, 3.12b Fig. 3.5b Fig. 3.11b Fig. 3.12a Fig. 3.20, 3.23 Fig. 3.21, 3.24a Fig. 3.22, 3.26, 3.27 Fig. 3.24b, 3.25

3.1 Fully-parallel solutions

185

Table 3.2. Limb topology of the non overconstrained PPMs with idle mobilities presented in Figs. 3.1-3.11 No. Basic PPM Type 1 3-RRR (Fig. 2.4a) 2 3-RRP (Fig. 2.4b) 3 3-RPR (Fig. 2.5a) 4 3-PRR (Fig. 2.5b) 5 3-PRP (Fig. 2.6) 6 7 8 9 10 11 12 13

3-PaPaR (Fig. 2.7) 3-PaRR (Fig. 2.8a) 3-PaRP (Fig. 2.8b) 3-PaPR (Fig. 2.9a) 3-PPaR (Fig. 2.9b) 3-PPR (Fig. 2.10) 1PPR-2RRR (Fig. 2.11)

NF 6 6 6 6 6

24 15 15 15 15 6 6

14 15 16 17 18 19 20

1PPR-2RRP (Fig. 2.12a) 1PPR-2RPR (Fig. 2.12b) 1PPR-2PRR (Fig. 2.13a) 1PPR-2PRP (Fig. 2.13b) 1PaPR-2PaRR (Fig. 2.14a) 1PaPR-2PPaR (Fig. 2.14b)

6 6 6 6 15 15

PPM with NF=0 Type 3-RRS* (Fig. 3.1a) 3-RS*P (Fig. 3.1b) 3-RPS* (Fig. 3.2a) 3-PRS* (Fig. 3.2b) 3-PS*P (Fig. 3.3a) 3-C*RC* (Fig. 3.3b) 3-PassPassS* (Fig. 3.4a) 3-PassRS* (Fig. 3.6) 3-PassS*P (Fig. 3.4b) 3-PassPS* (Fig. 3.5a) 3-PPassS* (Fig. 3.5b) 3-PPS* (Fig. 3.7) 1PPS*-2RRS* (Fig. 3.8a) 1C*C*R-2RRS* (Fig. 3.8b) 1PPS*-2RS*P (Fig. 3.9a) 1PPS*-2RPS* (Fig. 3.9b) 1PPS*-2PRS* (Fig. 3.10a) 1C*C*R-2C*RC* (Fig. 3.10b) 1PassPS*-2PassRS* (Fig. 3.11a) 1PassPS*-2PPassS* (Fig. 3.11b)

Limb topology R||RS* RS*P R ⊥ PS*

P ⊥ RS* PS*P C* ⊥ R ⊥ C* Pass||PassS* Pass||RS* PassS*P Pass ⊥ PS*

P ⊥ PassS* P ⊥ PS*

P ⊥ PS* R||RS* C* ⊥ C* ⊥ ⊥ R R||RS* P ⊥ PS* RS*P P ⊥ PS* R ⊥ PS* P ⊥ PS* P ⊥ RS* C* ⊥ C* ⊥ ⊥ R C* ⊥ R ⊥ C* Pass ⊥ PS* Pass||RS* Pass ⊥ PS* P ⊥ PassS*

186

3 Non overconstrained planar parallel robots with coupled motions

Table 3.3. Limb topology of the non overconstrained PPMs with idle mobilities presented in Figs. 3.12-3.20 No. Basic PPM Type 1 1PaPR-2PaPaR (Fig. 2.15a) 2 1PaPR-2PaRP (Fig. 2.15b) 3 3RRR (Fig. 2.16a,b) 4 3RPR (Fig. 2.17a) 5 2RPR-1RPR (Fig. 2.17b) 6 2RPP-1RPP (Fig. 2.18) 7 2RPPa-1RPPa (Fig. 2.19) 8 2RPaP-1RPaP (Fig. 2.20a) 9 2RPaPa-1RPaPa (Fig. 2.20b) 10 2PRR-1RPP (Fig. 2.21a) 11 2RRR-1RPP (Fig. 2.21b) 12 2RPR-1RPP (Fig. 2.21c) 13 2PPaR-1RPP (Fig. 2.22a) 14 2PaRR-1RPP (Fig. 2.22b) 15 2PaPaR-1RPP (Fig. 2.22c) 16 2PRR-1RPPa (Fig. 2.23a) 17 2RRR-1RPPa (Fig. 2.23b) 18 2RPR-1RPPa (Fig. 2.23c)

NF 21 15 6 6 6 6 15 15 24 6 6 6 12 12 18 9 9 9

PPM with NF=0 Type 1PassPS*-2PassPassS* (Fig. 3.12a) 1PassPS*-2PassS*P (Fig. 3.12b) 3RRS* (Fig. 3.13a,b) 3RPS* (Fig. 3.14a) 2RPS*-1RPS* (Fig. 3.14b) 2RC*C*-1RC*C* (Fig. 3.15) 2RC*Pass-1RC*Pass (Fig. 3.16) 2RPassC*-1RPassC* (Fig. 3.17a) 2RPassPass-1RPassPass (Fig. 3.17b) 2PS*C*-1RPP (Fig. 3.18a) 2RS*C*-1RPP (Fig. 3.18b) 2S*PC*-1RPP (Fig. 3.18c) 2PR*PassC*-1RPP (Fig. 3.19a) 2PassR*RC*-1RPP (Fig. 3.19b) 2PassPassC*-1RPP (Fig. 3.19c) 2PS*C*-1RPPass (Fig. 3.20a) 2RS*C*-1RPPass (Fig. 3.20b) 2S*PC*-1RPPass (Fig. 3.20c)

Limb topology Pass ⊥ PS* Pass||PassS* Pass ⊥ PS* PassS*P R||RS* R ⊥ PS*

R ⊥ PS* R ⊥ PS* R ⊥ C* ⊥ ⊥ C* R ⊥ C* ⊥ ⊥ C* R ⊥ C* ⊥ ||Pass R ⊥ C* ⊥ ||Pass R||Pass ⊥ C* R||Pass ⊥ C* R||Pass||Pass R||Pass||Pass PS*C* R ⊥ P ⊥⊥ P RS*C* R ⊥ P ⊥⊥ P S*PC* R ⊥ P ⊥⊥ P P||R ⊥ Pass||C* R ⊥ P ⊥⊥ P Pass ⊥ R* ⊥ ||R||C* R ⊥ P ⊥⊥ P Pass||Pass||C* R ⊥ P ⊥⊥ P PS*C* R ⊥ P ⊥ ||Pass RS*C* R ⊥ P ⊥ ||Pass S*PC* R ⊥ P ⊥ ||Pass

3.1 Fully-parallel solutions

187

Table 3.4. Limb topology of the non overconstrained PPMs with idle mobilities presented in Figs. 3.21-3.29 No. Basic PPM Type 1 2PRR-1RPaP (Fig. 2.24a) 2 2RRR-1RPaP (Fig. 2.24b) 3 2RPR-1RPaP (Fig. 2.24c) 4 2PRR-1RPaPa (Fig. 2.25a) 5 2RRR-1RPaPa (Fig. 2.25b) 6 2RPR-1RPaPa (Fig. 2.25c) 7 2PPaR-1RPPa (Fig. 2.26a) 8 2PaRR-1RPPa (Fig. 2.26b) 9 2PPaR-1RPaP (Fig. 2.27a) 10 2PaRR-1RPaP (Fig. 2.27b) 11 2PaPaR-1RPPa (Fig. 2.28a) 12 2PaPaR-1RPaP (Fig. 2.28b) 13 2PPaR-1RPaPa (Fig. 2.29a) 14 2PaRR-1RPaPa (Fig. 2.29b) 15 2PaPaR-1RPaPa (Fig. 2.30) 16 2PRR-1RPaPa (Fig. 2.31a) 17 2PRR-1RPaPat (Fig. 2.31b) 18 2RRR-1RPaPa (Fig. 2.32a) 19 2RRR-1RPaPat (Fig. 2.32b)

NF 9 9 9 12 12 12 15 15 15 15 21 21 18 18 24 12 12 12 12

PPM with NF=0 Type 2PS*C*-1RPassP (Fig. 3.21a) 2RS*C*-1RPassP (Fig. 3.21b) 2S*PC*-1RPassP (Fig. 3.21c) 2PS*C*-1RPassPass (Fig. 3.22a) 2RS*C*-1RPassPass (Fig. 3.22b) 2S*PC*-1RPassPass (Fig. 3.22c) 2PR*PassC*-1RPPass (Fig. 3.23a) 2PassR*RC*-1RPPass (Fig. 3.23b) 2PR*PassC*-1RPassP (Fig. 3.25a) 2PassR*RC*-1RPassP (Fig. 3.25b) 2PassPassC*-1RPPass (Fig. 3.24a) 2PassPassC*-1RPassP (Fig. 3.24b) 2PR*PassC*-1RPassPass (Fig. 3.26a) 2PassR*RC*-1RPassPass (Fig. 3.26b) 2PassPassC*-1RPassPass (Fig. 3.27) 2PS*C*-1RPacsPacs (Fig. 3.28a) 2PS*C*-1RPacsPatcs (Fig. 3.28b) 2RS*C*-1RPacsPacs (Fig. 3.29a) 2RS*C*-1RPacsPatcs (Fig. 3.29b)

Limb topology PS*C* R||Pass ⊥ P RS*C* R||Pass ⊥ P S*PC* R||Pass ⊥ P PS*C* R||Pass||Pass RS*C* R||Pass||Pass S*PC* R||Pass||Pass P||R* ⊥ Pass||C* R ⊥ P ⊥ ||Pass Pass ⊥ R* ⊥ ||R||C* R ⊥ P ⊥ ||Pass P||R* ⊥ Pass||C* R||Pass ⊥ P Pass ⊥ R* ⊥ ||R||C* R||Pass ⊥ P Pass||Pass||C* R ⊥ P ⊥ ||Pass Pass||Pass||C* R||Pass ⊥ P P||R* ⊥ Pass||C* R||Pass||Pass Pass ⊥ R* ⊥ ||R||C* R||Pass||Pass Pass||Pass||C* R||Pass||Pass PS*C* R||Pacs||Pacs PS*C* R||Pacs||Patcs RS*C* R||Pacs||Pacs RS*C* R||Pacs||Patcs

188

3 Non overconstrained planar parallel robots with coupled motions

Table 3.5. Limb topology of the non overconstrained PPMs with idle mobilities presented in Figs. 3.30-3.33 No. Basic PPM Type 1 2RPR-1RPaPa (Fig. 2.33a) 2 2RPR-1RPaPat (Fig. 2.33b) 3 2PPaR-1RPaPa (Fig. 2.34a) 4 2PPaR-1RPaPat (Fig. 2.34b) 5 2PaRR-1RPaPa (Fig. 2.35a) 6 2PaRR-1RPaPat (Fig. 2.35b) 7 2PaPaR-1RPaPa (Fig. 2.36a) 8 2PaPaR-1RPaPat (Fig. 2.36b)

PPM with NF=0 Type 2S*PC*-1RPacsPacs (Fig. 3.30a) 2S*PC*-1RPacsPatcs (Fig. 3.30b) 2PR*PassC*-1RPacsPacs (Fig. 3.31a) 2PR*PassC*-1RPacsPatcs (Fig. 3.31b) 2PassR*RC*-1RPacsPacs (Fig. 3.32a) 2PassR*RC*-1RPacsPatcs (Fig. 3.32b) 2PassPassC*-1RPacsPacs (Fig. 3.33a) 2PassPassC*-1RPacsPatcs (Fig. 3.33b)

NF 12 12 18 18 18 18 24 24

Limb topology S*PC* R||Pacs||Pacs S*PC* R||Pacs||Patcs P||R* ⊥ Pass||C* R||Pacs||Pacs P||R* ⊥ Pass||C* R||Pacs||Patcs Pass ⊥ R* ⊥ ||R||C* R||Pacs||Pacs Pass ⊥ R* ⊥ ||R||C* R||Pacs||Patcs Pass||Pass||C* R||Pacs||Pacs Pass||Pass||C* R||Pacs||Patcs

Table 3.6. Bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 3.1-3.33 No. Parallel mechanism 1 Figs. 3.1, 3.4-3.9, 3.13 2 Figs. 3.2, 3.3, 3.10-3.12, 3.14-3.17 3 Figs. 3.18, 3.19-3.33

Basis (RG1) ( v1 , v 2 ,

α

,

β

,

δ

)

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

β

,

δ

)

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

( v1 , v 2 ,

α

,

β

,

δ

)

( v1 , v 2 , v 3 ,

α

,

δ

)

( v1 , v 2 , v 3 ,

) ( v1 , v 2 , v 3 ,

α

,

β

,

( v1 , v 2 , v 3 ,

α

,

β

,

δ

δ

) ( v1 , v 2 ,

δ

α

β

)

,

,

δ

δ

)

)

3.1 Fully-parallel solutions

189

Table 3.7. Structural parametersa of non overconstrained PPMs in Figs. 3.1-3.7 No. Structural Solution parameter 3-RRS*, 3-RS*P (Fig. 3.1) 3-PassPassS* 3-PassS*P (Fig. 3.4b) 3-RPS*, PRS* (Fig. 3.2) (Fig. 3.4a) 3-PassPS* (Fig. 3.5a) 3-PS*P, 3-C*RC* (Fig. 3.3) 3-PPassS* (Fig. 3.5b) 3-PPS* (Fig. 3.7) 3-PassRS* (Fig. 3.6) 1 m 8 20 14 2 p1 3 9 6 3 p2 3 9 6 4 p3 3 9 6 5 p 9 27 18 6 q 2 8 5 7 k1 3 0 0 8 k2 0 3 3 9 k 3 3 3 10 (RGi) see Table 3.5 see Table 3.5 see Table 3.5 (i=1,2,3) 11 SG1 5 5 5 12 SG2 5 5 5 13 SG3 5 5 5 14 rG1 0 12 6 15 rG2 0 12 6 16 rG3 0 12 6 17 MG1 5 7 6 18 MG2 5 7 6 19 MG3 5 7 6 20 (RF) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) 21 SF 3 3 3 22 rl 0 36 18 23 rF 12 48 30 24 MF 3 9 6 25 NF 0 0 0 26 TF 0 6 3 p1 27 5 19 12 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

fj

5

19

12

fj

5

19

12

fj

15

57

36

See footnote of Table 2.4 for the nomenclature of structural parameters

190

3 Non overconstrained planar parallel robots with coupled motions

Table 3.8. Structural parametersa of non overconstrained PPMs in Figs. 3.8-3.12 No. Structural Solution parameter 1PPS*-2RRS* (Fig. 3.8a) 1C*C*R-2RRS* (Fig. 3.8b) 1PPS*-2RS*P (Fig. 3.9a) 1PPS*-2RPS* (Fig. 3.9b) 1PPS*-2PRS* (Fig. 3.10a) 1C*C*R-2C*RC* (Fig. 3.10b) 1 m 8 2 p1 3 3 p2 3 4 p3 3 5 p 9 6 q 2 7 k1 3 8 k2 0 9 k 3 10 (RGi) see Table 3.5 (i=1,2,3) 11 SG1 5 12 SG2 5 13 SG3 5 14 rG1 0 15 rG2 0 16 rG3 0 17 MG1 5 18 MG2 5 19 MG3 5 20 (RF) ( v1 , v 2 , δ ) 21 SF 3 22 rl 0 23 rF 12 24 MF 3 25 NF 0 26 TF 0 p1 27 5 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

1PassPS*-2PassRS* (Fig. 3.11a) 1PassPS*-2PPassS* (Fig. 3.11b) 1PassPS*-2PassS*P (Fig. 3.12b) 14 6 6 6 18 5 0 3 3 see Table 3.5

1PassPS*2PassPassS* (Fig. 3.12a)

5 5 5 6 6 6 6 6 6 ( v1 , v 2 , 3 18 30 6 0 3 12

5 5 5 6 12 12 6 7 7 ( v1 , v 2 , 3 30 42 8 0 5 12

δ

)

18 6 9 9 24 7 0 3 3 see Table 3.5

j

fj

5

12

19

fj

5

12

19

fj

15

36

50

See footnote of Table 2.4 for the nomenclature of structural parameters

δ

)

3.1 Fully-parallel solutions

191

Table 3.9. Structural parametersa of non overconstrained PPMs in Figs. 3.13-3.17 No. Structural Solution parameter 3RRS* (Fig. 3.13) 3RPS* (Fig. 3.14a) 2RPS*-1RPS* (Fig. 3.14b) 2RC*C*-1RC*C* (Fig. 3.15) 1 m 8 2 p1 3 3 p2 3 4 p3 3 5 p 9 6 q 2 7 k1 3 8 k2 0 9 k 3 10 (RGi) see Table 3.5 (i=1,2,3) 11 SG1 5 12 SG2 5 13 SG3 5 14 rG1 0 15 rG2 0 16 rG3 0 17 MG1 5 18 MG2 5 19 MG3 5 20 (RF) ( v1 , v 2 , δ ) 21 SF 3 22 rl 0 23 rF 12 24 MF 3 25 NF 0 26 TF 0 p1 27 5 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

2RC*Pass-1RC*Pass 2RPassPass(Fig. 3.16) 1RPassPass ss ss 2RPa C*-1RPa C* (Fig. 3.17b) (Fig. 3.17a) 14 20 6 9 6 9 6 9 18 27 5 8 0 0 3 3 3 3 see Table 3.5 see Table 3.5 5 5 5 6 6 6 5 5 5 ( v1 , v 2 , 3 18 30 3 0 0 11

δ

)

5 5 5 12 12 12 5 5 5 ( v1 , v 2 , 3 36 48 3 0 0 17

fj

5

11

17

fj

5

11

17

fj

15

33

51

See footnote of Table 2.4 for the nomenclature of structural parameters

δ

)

192

3 Non overconstrained planar parallel robots with coupled motions

Table 3.10. Structural parametersa of non overconstrained PPMs in Figs. 3.18 and 3.19 No. Structural Solution parameter 2PS*C*-1RPP (Fig. 3.18a) 2PR*PassC*-1RPP 2RS*C*-1RPP (Fig. 3.18b) (Fig. 3.19a) 2S*PC*-1RPP (Fig. 3.18c) 2PassR*RC*-1RPP (Fig. 3.19b) 1 m 8 14 2 p1 3 7 3 p2 3 7 4 p3 3 3 5 p 9 17 6 q 2 4 7 k1 3 1 8 k2 0 2 9 k 3 3 10 (RGi) see Table 3.5 see Table 3.5 (i=1,2,3) 11 SG1 6 6 12 SG2 6 6 13 SG3 3 3 14 rG1 0 6 15 rG2 0 6 16 rG3 0 0 17 MG1 6 6 18 MG2 6 6 19 MG3 3 3 20 (RF) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) 21 SF 3 3 22 rl 0 12 23 rF 12 24 24 MF 3 3 25 NF 0 0 26 TF 0 0 p1 27 6 12 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

2PassPassC*1RPP (Fig. 3.19c) 16 9 9 3 21 6 1 2 3 see Table 3.5 6 6 3 12 12 0 6 6 3 ( v1 , v 2 , 3 24 36 3 0 0 18

j

fj

6

12

18

fj

3

3

3

fj

15

27

39

See footnote of Table 2.4 for the nomenclature of structural parameters

δ

)

3.1 Fully-parallel solutions

193

Table 3.11. Structural parametersa of non overconstrained PPMs in Figs. 3.203.23 No. Structural Solution parameter 2PS*C*-1RPPass (Fig. 3.20a) 2RS*C*-1RPPass (Fig. 3.20b) 2S*PC*-1RPPass (Fig. 3.20c) 2PS*C*-1RPassP (Fig. 3.21a) 2RS*C*-1RPassP (Fig. 3.21b) 2S*PC*-1RPassP (Fig. 3.21c) 1 m 10 2 p1 3 3 p2 3 4 p3 6 5 p 12 6 q 3 7 k1 2 8 k2 1 9 k 3 10 (RGi) see Table 3.5 (i=1,2,3) 11 SG1 6 12 SG2 6 13 SG3 3 14 rG1 0 15 rG2 0 16 rG3 6 17 MG1 6 18 MG2 6 19 MG3 4 20 (RF) ( v1 , v 2 , δ ) 21 SF 3 22 rl 6 23 rF 18 24 MF 4 25 NF 0 26 TF 1 p1 27 6 f

6 6 3 0 0 12 6 6 5 ( v1 , v 2 , 3 12 24 5 0 2 6

28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

2PS*C*-1RPassPass 2PR*PassC*(Fig. 3.22a) 1RPPass ss ss 2RS*C*-1RPa Pa (Fig. 3.23a) (Fig. 3.22b) 2PassR*RC*ss ss 2S*PC*-1RPa Pa 1RPPass (Fig. 3.23b) (Fig. 3.22c) 12 16 3 7 3 7 9 6 15 20 4 5 2 0 1 3 3 3 see Table 3.5 see Table 3.5

δ

)

6 6 3 6 6 6 6 6 4 ( v1 , v 2 , 3 18 30 4 0 1 12

fj

6

6

12

fj

10

17

10

fj

22

29

34

See footnote of Table 2.4 for the nomenclature of structural parameters

δ

)

194

3 Non overconstrained planar parallel robots with coupled motions

Table 3.12. Structural parametersa of non overconstrained PPMs in Figs. 3.243.26 No. Structural Solution parameter 2PassPassC*-1RPPass (Fig. 3.24a) 2PassPassC*-1RPassP (Fig. 3.24b)

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

18 9 9 6 24 7 0 3 3 see Table 3.5

16 7 7 6 20 5 0 3 3 see Table 3.5

2PR*PassC*1RPassPass (Fig. 3.26a) 2PassR*RC*1RPassPass (Fig. 3.26b) 18 7 7 9 23 6 0 3 3 see Table 3.5

6 6 3 12 12 6 6 6 4 ( v1 , v 2 , 3 30 42 4 0 1 18

6 6 3 6 6 6 6 6 4 ( v1 , v 2 , 3 18 30 4 0 1 12

6 6 3 6 6 12 6 6 5 ( v1 , v 2 , 3 24 36 5 0 2 12

δ

)

2PR*PassC*-1RPassP (Fig. 3.25a) 2PassR*RC*-1RPassP (Fig. 3.25b)

δ

)

fj

18

12

12

fj

10

10

17

fj

46

34

41

See footnote of Table 2.4 for the nomenclature of structural parameters

δ

)

3.1 Fully-parallel solutions

195

Table 3.13. Structural parametersa of non overconstrained PPMs in Figs. 3.273.30 No. Structural Solution parameter 2PassPassC*-1RPassPass (Fig. 3.27)

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

20 9 9 9 27 8 0 3 3 see Table 3.5

2PS*C*-1RPacsPacs (Fig. 3.28a) 2PS*C*-1RPacsPatcs (Fig. 3.28b) 2RS*C*-1RPacsPacs (Fig. 3.29a) 2RS*C*-1RPacsPatcs (Fig. 3.29b) 2S*PC*-1RPacsPacs (Fig. 3.30a) 2S*PC*-1RPacsPatcs (Fig. 3.30b) 12 3 3 9 15 4 2 1 3 see Table 3.5

6 6 3 12 12 12 6 6 5 ( v1 , v 2 , 3 36 48 5 0 2 18

6 6 3 0 0 12 6 6 3 ( v1 , v 2 , 3 12 24 3 0 0 6

δ

)

fj

18

6

fj

17

15

fj

53

27

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

196

3 Non overconstrained planar parallel robots with coupled motions

Table 3.14. Structural parametersa of non overconstrained PPMs in Figs. 3.313.33 No. Structural Solution parameter 2PR*PassC*-1RPacsPacs (Fig. 3.31a) 2PR*PassC*-1RPacsPatcs (Fig. 3.31b) 2PassR*RC*-1RPacsPacs (Fig. 3.32a) 2PassR*RC*-1RPacsPatcs (Fig. 3.32b) 1 m 18 2 p1 7 3 p2 7 4 p3 9 5 p 23 6 q 6 7 k1 0 8 k2 3 9 k 3 10 (RGi) see Table 3.5 (i=1,2,3) 11 SG1 6 12 SG2 6 13 SG3 3 14 rG1 6 15 rG2 6 16 rG3 12 17 MG1 6 18 MG2 6 19 MG3 3 20 (RF) ( v1 , v 2 , δ ) 21 SF 3 22 rl 24 23 rF 36 24 MF 3 25 NF 0 26 TF 0 p1 27 12 f

6 6 3 12 12 12 6 6 3 ( v1 , v 2 , 3 36 48 3 0 0 18

28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

2PassPassC*-1RPacsPacs (Fig. 3.33a) 2PassPassC*-1RPacsPatcs (Fig. 3.33b) 20 9 9 9 27 8 0 3 3 see Table 3.5

δ

)

j

fj

12

18

fj

15

15

fj

39

51

See footnote of Table 2.4 for the nomenclature of structural parameters

3.1 Fully-parallel solutions

197

Fig. 3.1. Non overconstrained PPMs with coupled motions of types 3-RRS* (a) and 3-RS*P (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology R||RS* (a) and RS*P (b)

198

3 Non overconstrained planar parallel robots with coupled motions

Fig. 3.2. Non overconstrained PPMs with coupled motions of types 3-RPS* (a) and 3-PRS* (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology R ⊥ PS* (a) and P ⊥ RS* (b)

3.1 Fully-parallel solutions

199

Fig. 3.3. Non overconstrained PPMs with coupled motions of types 3-PS*P (a) and 3-C*RC* (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology PS*P (a) and C* ⊥ R ⊥ C* (b)

200

3 Non overconstrained planar parallel robots with coupled motions

Fig. 3.4. Non overconstrained PPMs with coupled motions of types 3-PassPassS* (a) and 3-PassS*P (b) defined by SF=3, (RF)=( v1 , v2 , δ ) and MF=9, NF=0, TF=6 (a), MF=6, NF=0, TF=3 (b), limb topology Pass||PassS* (a) and PassS*P (b)

3.1 Fully-parallel solutions

201

Fig. 3.5. Non overconstrained PPMs with coupled motions of types 3-PassPS* (a) and 3-PPassS* (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=6, NF=0, TF=3, limb topology Pass ⊥ PS* (a) and P ⊥ PassS* (b)

202

3 Non overconstrained planar parallel robots with coupled motions

Fig. 3.6. Non overconstrained PPM with coupled motions of type 3-PassRS* defined by SF=3, (RF)=( v1 , v2 , δ ), MF=6, NF=0, TF=3, limb topology Pass||RS*

Fig. 3.7. Non overconstrained PPM with one decoupled translation of type 3PPS* defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology P ⊥ PS*

3.1 Fully-parallel solutions

203

Fig. 3.8. Non overconstrained PPMs with one decoupled translation of types 1PPS*-2RRS* (a) and 1C*C*R-2RRS* (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology R||RS* and P ⊥ PS* (a) R||RS* and

C* ⊥ C* ⊥ ⊥ R (b)

204

3 Non overconstrained planar parallel robots with coupled motions

Fig. 3.9. Non overconstrained PPMs with one decoupled translation of types 1PPS*-2RS*P (a) and 1PPS*-2RPS* (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology P ⊥ PS* and RS*P (a) and P ⊥ PS* and R ⊥ PS* (b)

3.1 Fully-parallel solutions

205

Fig. 3.10. Non overconstrained PPMs with one decoupled translation of types 1PPS*-2PRS* (a) and 1C*C*R-2C*RC* (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology P ⊥ PS* and P ⊥ RS* (a),

C* ⊥ C* ⊥ ⊥ R and C* ⊥ R ⊥ C* (b)

206

3 Non overconstrained planar parallel robots with coupled motions

Fig. 3.11. Non overconstrained PPMs with one decoupled translation of types 1PassPS*-2PassRS* (a) and 1PassPS*-2PPassS* (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=6, NF=0, TF=3, limb topology Pass ⊥ PS* and Pass||RS* (a) and Pass ⊥ PS* and P ⊥ PassS* (b)

3.1 Fully-parallel solutions

207

Fig. 3.12. Non overconstrained PPMs with one decoupled translation of types 1PassPS*-2PassPassS* (a) and 1PassPS*-2PassS*P (b) defined by SF=3, (RF)=( v1 , v2 , δ ) and MF=8, NF=0, TF=5 (a), MF=6, NF=0, TF=3 (b), limb topology Pass ⊥ PS* and Pass||PassS* (a) and Pass ⊥ PS* and PassS*P (b)

208

3 Non overconstrained planar parallel robots with coupled motions

Fig. 3.13. 3RRS*-type non overconstrained PPMs with partially decoupled translations defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology R||RS*

3.1 Fully-parallel solutions

209

Fig. 3.14. Overconstrained PPMs with partially decoupled translations of types 3RPS* (a) and 2RPS*-1RPS* (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology R ⊥ PS* (a), R ⊥ PS* and R ⊥ PS* (b)

210

3 Non overconstrained planar parallel robots with coupled motions

Fig. 3.15. Non overconstrained PPM with partially decoupled rotation of type 2RC*C*-1RC*C* defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology R ⊥ C* ⊥ ⊥ C*

Fig. 3.16. Non overconstrained PPM with partially decoupled rotation of type 2RC*Pass-1RC*Pass defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology R ⊥ C* ⊥ ||Pass

3.1 Fully-parallel solutions

211

Fig. 3.17. Non overconstrained PPMs with partially decoupled rotation of types 2RPassC*-1RPassC* (a) and 2RPassPass-1RPassPass (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology R||Pass ⊥ C* (a) and R||Pass||Pass (b)

212

3 Non overconstrained planar parallel robots with coupled motions

Fig. 3.18. Non overconstrained PPMs with with uncoupled rotation of types 2PS*C*-1RPP (a), 2RS*C*-1RPP (b) and 2S*PC*-1RPP (c) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology R ⊥ P ⊥ ⊥ P and PS*C* (a), RS*C* (b), S*PC* (c)

3.1 Fully-parallel solutions

213

Fig. 3.19. Non overconstrained PPMs with with uncoupled rotation of types 2PR*PassC*-1RPP (a), 2PassR*RC*-1RPP (b) and 2PassPassC*-1RPP (c) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology R ⊥ P ⊥ ⊥ P and P||R ⊥ Pass||C* (a), Pass ⊥ R* ⊥ ||R||C* (b), Pass||Pass||C* (c)

214

3 Non overconstrained planar parallel robots with coupled motions

Fig. 3.20. Non overconstrained PPMs with with uncoupled rotation of types 2PS*C*-1RPPass (a), 2RS*C*-1RPPass (b) and 2S*PC*-1RPPass (c) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4 NF=0, TF=1, limb topology R ⊥ P ⊥ ||Pass and PS*C* (a), RS*C* (b), S*PC* (c)

3.1 Fully-parallel solutions

215

Fig. 3.21. Non overconstrained PPMs with with uncoupled rotation of types 2PS*C*-1RPassP (a), 2RS*C*-1RPassP (b) and 2S*PC*-1RPassP (c) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, NF=0, TF=1, limb topology R|| Pass ⊥ P and PS*C* (a), RS*C* (b), S*PC* (c)

216

3 Non overconstrained planar parallel robots with coupled motions

Fig. 3.22. Non overconstrained PPMs with with uncoupled rotation of types 2PS*C*-1RPassPass (a), 2RS*C*-1RPassPass (b) and 2S*PC*-1RPassPass (c) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, NF=0, TF=2, limb topology R||Pass||Pass and PS*C* (a), RS*C* (b), S*PC* (c)

3.1 Fully-parallel solutions

217

Fig. 3.23. Non overconstrained PPMs with with uncoupled rotation of types 2PR*PassC*-1RPPass (a) and 2PassR*RC*-1RPPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, NF=0, TF=1, limb topology R ⊥ P ⊥ ||Pass and ss ss || P||R* ⊥ Pa ||C* (a), Pa ⊥ R* ⊥ R||C* (b)

218

3 Non overconstrained planar parallel robots with coupled motions

Fig. 3.24. Non overconstrained PPMs with with uncoupled rotation of types 2PassPassC*-1RPPass (a) and 2PassPassC*-1RPassP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, NF=0, TF=1, limb topology Pass||Pass||C* and R ⊥ P ⊥ ||Pass (a), R||Pass ⊥ P (b)

3.1 Fully-parallel solutions

219

Fig. 3.25. Non overconstrained PPMs with with uncoupled rotation of types 2PR*PassC*-1RPassP (a) and 2PassR*RC*-1RPassP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, NF=0, TF=1, limb topology R||Pass ⊥ P and P||R* ⊥ Pass||C* (a), Pass ⊥ R* ⊥ ||R||C* (b)

220

3 Non overconstrained planar parallel robots with coupled motions

Fig. 3.26. Non overconstrained PPMs with with uncoupled rotation of types 2PR*PassC*-1RPassPass (a) and 2PassR*RC*-1RPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, NF=0, TF=2, limb topology R||Pass||Pass and P||R* ⊥ Pass||C* (a), Pass ⊥ R* ⊥ ||R||C* (b)

3.1 Fully-parallel solutions

221

Fig. 3.27. 2PassPassC*-1RPassPass-type non overconstrained PPM with uncoupled rotation SF=3, (RF)=( v1 , v2 , δ ), MF=5, NF=0, TF=2, limb topology Pass||Pass||C* and R||Pass||Pass

222

3 Non overconstrained planar parallel robots with coupled motions

Fig. 3.28. Non overconstrained PPMs with uncoupled rotation of types 2PS*C*1RPacsPacs (a) and 2PS*C*-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology PS*C* and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

3.1 Fully-parallel solutions

223

Fig. 3.29. Non overconstrained PPMs with uncoupled rotation of types 2RS*C*1RPacsPacs (a) and 2RS*C*-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology RS*C* and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

224

3 Non overconstrained planar parallel robots with coupled motions

Fig. 3.30. Non overconstrained PPMs with uncoupled rotation of types 2S*PC*1RPacsPacs (a) and 2S*PC*-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology S*PC* and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

3.1 Fully-parallel solutions

225

Fig. 3.31. Non overconstrained PPMs with uncoupled rotation of types 2PR*PassC*-1RPacsPacs (a) and 2PR*PassC*-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology P||R* ⊥ Pass||C* and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

226

3 Non overconstrained planar parallel robots with coupled motions

Fig. 3.32. Non overconstrained PPMs with uncoupled rotation of types 2PassR*RC*-1RPacsPacs (a) and 2PassR*RC*-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology Pass ⊥ R* ⊥ ||R||C* and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

3.1 Fully-parallel solutions

227

Fig. 3.33. Non overconstrained PPMs with uncoupled rotation of types 2PassPassC*-1RPacsPacs (a) and 2PassPassC*-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology Pass||Pass||C* and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

228

3 Non overconstrained planar parallel robots with coupled motions

3.2 Non fully-parallel solutions The non fully-parallel solutions presented in Figs. 3.34-3.40 are obtained from the fully-parallel solutions in Figs. 3.18, 3.20-3.22 and 3.28-3.30 by superposing the last revolute joints of limbs G1 and G2. In this way, these two limbs compose just one complex limb denoted by G1-G2 in which two actuators are combined. For example, the solutions in Fig. 3.34 combine one simple and one complex limb. Simple limb G3 is of type RPP and complex limb G1-G2 is of types (PS*C*S*P)C* (Fig. 3.34a), (RS*C*S*R)C* (Fig. 3.34b) and (S*PC*PS*)C* (Fig. 3.34c). One closed loop defined in the brackets is combined in each complex limb G1-G2 of these solutions. The solutions in Figs. 3.35-3.40 combine two complex limbs G1-G2 and G3. In these solutions, complex limb G3 is of types RPPass (Fig. 3.35), RPassP (Fig. 3.36), RPassPass (Fig. 3.37), RPacsPacs (Fig. 3.38a, 3.39a, 3.40a) and RPacsPatcs (Fig. 3.38b, 3.39b and 3.40b). We recall that the notations Pacs and Patcs are associated with a parallelogram and telescopic parallelogram loops with three idle mobilities combined in a cylindrical and a spherical joint, and Pass with four idle mobilities combined in two spherical joints adjacent to the same link. In the cylindrical joints of the parallelogram loops denoted by Pacs and Patcs (Figs. 3.38-3.40) the translational motion is an idle mobility. In the parallelogram loop Pass-type, three idle mobilities are introduced in the loop and one outside the loop. The idle mobility introduced outside the loop (Figs. 3.35-3.37) becomes an internal rotational mobility of the binary link adjacent to the two spherical joints. This is the case of the binary links 5C (Fig. 3.35), 4C (Fig. 3.36), 3C and 6C (Fig. 3.37). This internal mobility gives one degree of structural redundancy. The rotation axis of the moving platform has a fixed position in the solutions in Figs. 3.34-3.37 and a variable position in Figs. 3.38-3.40. The limb topology of these non overconstrained solutions (NF=0) are systematized in Table 3.15 and the structural parameters in Tables 3.16 and 3.17. The basis of the vector space of the relative velocities between the moving and the fixed platforms in the complex limb G1-G2 isolated from the parallel mechanism is denoted by (RG1-G2) in Tables 3.16 and 3.17. The connectivity between the moving and the fixed platforms of this complex limb isolated from the parallel mechanism is denoted by SG1-G2 (see Table 3.16 and 3.17).

3.2 Non fully-parallel solutions

229

Table 3.15. Limb topology of the non overconstrained PPMs with idle mobilities presented in Figs. 3.34-3.40 No. Basic PPM Type 1 (PRRRP)R-RPP (Fig. 2.37a) 2 (RRRRR)R-RPP (Fig. 2.37b) 3 (RPRPR)R-RPP (Fig. 2.37c) 4 (PRRRP)R-RPPa (Fig. 2.38a) 5 (RRRRR)R-RPPa (Fig. 2.38b) 6 (RPRPR)R-RPPa (Fig. 2.38c) 7 (PRRRP)R-RPaP (Fig. 2.39a) 8 (RRRRR)R-RPaP (Fig. 2.39b) 9 (RPRPR)R-RPaP (Fig. 2.39c) 10 (PRRRP)R-RPaPa (Fig. 2.40a) 11 (RRRRR)R-RPaPa (Fig. 2.40b) 12 (RPRPR)R-RPaPa (Fig. 2.40c) 13 (PRRRP)R-RPaPa (Fig. 2.41a) 14 (PRRRP)R-RPaPat (Fig. 2.41b) 15 (RRRRR)R-RPaPa (Fig. 2.42a) 16 (RRRRR)R-RPaPat (Fig. 2.42b) 17 (RPRPR)R-RPaPa (Fig. 2.43a) 18 (RPRPR)R-RPaPat (Fig. 2.43b)

NF 6 6 6 9 9 9 9

9 12 12 12 12 12 12 12 12 12

PPM with NF=0 Type Limb topology (PS*C*S*P)C*-RPP (PS*C*S*P)C* (Fig. 3.34a) R ⊥ P ⊥⊥ P (RS*C*S*R)C*-RPP (RS*C*S*R)C* (Fig. 3.34b) R ⊥ P ⊥⊥ P (S*PC*PS*)C*-RPP (S*PC*PS*)C* (Fig. 3.34c) R ⊥ P ⊥⊥ P (PS*C*S*P)C* (PS*C*S*P)C*-RPPass (Fig. 3.35a) R ⊥ P ⊥ ||Pass (RS*C*S*R)C*-RPPass (RS*C*S*R)C* (Fig. 3.35b) R ⊥ P ⊥ ||Pass (S*PC*PS*)C*-RPPass (S*PC*PS*)C* R ⊥ P ⊥ ||Pass (Fig. 3.35c) (PS*C*S*P)C*-RPassP (PS*C*S*P)C* (Fig. 3.36a) R||Pass ⊥ P (RS*C*S*R)C*-RPassP (RS*C*S*R)C* (Fig. 3.36b) R||Pass ⊥ P (S*PC*PS*)C* (S*PC*PS*)C*-RPassP (Fig. 3.36c) R||Pass ⊥ P (PS*C*S*P)C*-RPassPass (PS*C*S*P)C* (Fig. 3.37a) R||Pass||Pass ss ss (RS*C*S*R)C*-RPa Pa (RS*C*S*R)C* (Fig. 3.37b) R||Pass||Pass ss ss (S*PC*PS*)C*-R Pa Pa (S*PC*PS*)C* R||Pass||Pass (Fig. 3.37c) cs cs (PS*C*S*P)C*-RPa Pa (PS*C*S*P)C* R||Pacs||Pacs (Fig. 3.38a) cs tcs (PS*C*S*P)C*-RPa Pa (PS*C*S*P)C* (Fig. 3.38b) R||Pacs||Patcs cs cs (RS*C*S*R)C*-RPa Pa (RS*C*S*R)C* (Fig. 3.39a) R||Pacs||Pacs cs tcs (RS*C*S*R)C*-RPa Pa (RS*C*S*R)C* R||Pacs||Patcs (Fig. 3.39b) cs cs (S*PC*PS*)C*-RPa Pa (S*PC*PS*)C* (Fig. 3.40a) R||Pacs||Pacs cs tcs (S*PC*PS*)C*-RPa Pa (S*PC*PS*)C* (Fig. 3.40b) R||Pacs||Patcs

230

3 Non overconstrained planar parallel robots with coupled motions

Table 3.16. Structural parametersa of non overconstrained PPMs in Figs. 3.343.36 No. Structural parameter

1 2 3 4 5 6 7 8 9

m p1 p2 p q k1 k2 k (RG1-G2)

10

(RG3)

11 12 13 14 15 16 17

SG1-G2 SG3 rG1-G2 rG3 MG1-G2 MG3 (RF)

18 19 20 21 22 23 24

SF rl rF MF NF TF

25 26 a

∑ ∑ ∑

p1

j =1

p2

j =1

p

j =1

fj

Solution (PS*C*S*P)C*-RPP (Fig. 3.34a) (RS*C*S*R)C*-RPP (Fig. 3.34b) (S*PC*PS*)C*-RPP (Fig. 3.34b) 8 6 3 9 2 1 1 2 ( v1 , v 2 , v 3 , α , β , δ ) ( v1 , v 2 , 6 3 6 0 6 3 ( v1 , v 2 , 3 6 12 3 0 0 12

δ

)

δ

)

(PS*C*S*P)C*-RPPass (Fig. 3.35a) (RS*C*S*R)C*-RPPass (Fig. 3.35b) (S*PC*PS*)C*-RPPass (Fig. 3.35c) (PS*C*S*P)C*-RPassP (Fig. 3.36a) (RS*C*S*R)C*-RPassP (Fig. 3.36b) (S*PC*PS*)C*-R Pass P (Fig. 3.36c) 10 6 6 12 3 0 2 2 ( v1 , v 2 , v 3 , α , β , δ ) ( v1 , v 2 , 6 3 6 6 6 4 ( v1 , v 2 , 3 12 18 4 0 1 12

fj

3

10

fj

15

22

δ

)

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

3.2 Non fully-parallel solutions

231

Table 3.17. Structural parametersa of non overconstrained PPMs in Figs. 3.373.40 No. Structural parameter (PS*C*S*P)C*RPacsPacs (Fig. 3.37a) (RS*C*S*R)C*RPacsPacs (Fig. 3.37b) (S*PC*PS*)C*RPacs Pacs (Fig. 3.37c) 1 m 12 2 p1 6 3 p2 9 4 p 15 5 q 4 6 k1 0 7 k2 2 8 k 2 9 (RG1-G2) ( v1 , v2 , v3 , α , β , δ ) 10

(RG3)

11 12 13 14 15 16 17

SG1-G2 SG3 rG1-G2 rG3 MG1-G2 MG3 (RF)

18 19 20 21 22 23 24

SF rl rF MF NF TF

25 26 a

∑ ∑ ∑

p1

j =1

p2

j =1

p

j =1

fj

( v1 , v 2 , 6 3 6 12 6 5 ( v1 , v 2 , 3 18 24 5 0 2 12

δ

)

δ

)

Solution (PS*C*S*P)C*-RPacsPacs (Fig. 3.38a) (PS*C*S*P)C*-RPacsPatcs (Fig. 3.38b) (RS*C*S*R)C*-RPacsPacs (Fig. 3.39a) (RS*C*S*R)C*-RPacsPatcs (Fig. 3.39b) (S*PC*PS*)C*-RPacsPacs (Fig. 3.40a) (S*PC*PS*)C*-RPacsPatcs (Fig. 3.40b) 12 6 9 15 4 0 2 2 ( v1 , v 2 , v 3 , α , β , δ ) ( v1 , v 2 , 6 3 6 12 6 3 ( v1 , v 2 , 3 18 24 3 0 0 12

fj

17

15

fj

29

27

δ

)

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

232

3 Non overconstrained planar parallel robots with coupled motions

Fig. 3.34. Non overconstrained PPMs with uncoupled rotation of types (PS*C*S*P)C*-RPP (a), (RS*C*S*R)C*-RPP (b) and (S*PC*PS*)C*-RPP (c) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0

3.2 Non fully-parallel solutions

233

Fig. 3.35. Non overconstrained PPMs with uncoupled rotation of types (PS*C*S*P)C*-RPPass (a), (RS*C*S*R)C*-RPPass (b) and (S*PC*PS*)C*RPPass (c) defined by MF=SF=4, (RF)=( v1 , v2 , δ ), TF=0, NF=1

234

3 Non overconstrained planar parallel robots with coupled motions

Fig. 3.36. Non overconstrained PPMs with uncoupled rotation of types (PS*C*S*P)C*-RPassP (a), (RS*C*S*R)C*-RPassP (b) and (S*PC*PS*)C*RPassP (c) defined by MF=SF=4, (RF)=( v1 , v2 , δ ), TF=0, NF=1

3.2 Non fully-parallel solutions

235

Fig. 3.37. Non overconstrained PPMs with uncoupled rotation of types (PS*C*S*P)C*-RPassPass (a), (RS*C*S*R)C*-RPassPass (b) and (S*PC*PS*)C*RPass Pass (c) defined by MF=SF=5, (RF)=( v1 , v2 , δ ), TF=0, NF=2

236

3 Non overconstrained planar parallel robots with coupled motions

Fig. 3.38. Non overconstrained PPMs with uncoupled rotation of types (PS*C*S*P)C*-RPacsPacs (a) and (PS*C*S*P)C*-RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0

3.2 Non fully-parallel solutions

237

Fig. 3.39. Non overconstrained PPMs with uncoupled rotation of types (RS*C*S*R)C*-RPacsPacs (a) and (RS*C*S*R)C*-RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0

238

3 Non overconstrained planar parallel robots with coupled motions

Fig. 3.40. Non overconstrained PPMs with uncoupled rotation of types (S*PC*PS*)C*-RPacsPacs (a) and (S*PC*PS*)C*-RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0

4 Planar parallel robots with uncoupled motions

Planar parallel robotic manipulator (PPM) with uncoupled motions with various degrees of overconstraint may be obtained by using simple and/or complex limbs. In these solutions, each operational velocity given by Eq. (1.18) depends on just one actuated joint velocity: v1 = v1 ( &q1 ) , v2 = v2 ( &q2 ) and ωδ = ωδ ( &q3 ) . The Jacobian matrix in Eq. (1.18) is a diagonal matrix. They can be actuated by linear and 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 or prismatic joint mounted on the fixed base.

4.1 Overconstrained solutions Equation (1.16) indicates that overconstrained solutions of PPMs with uncoupled motions and q independent loops meet the condition p ∑ 1 fi < 3 + 6q . Various basic or derived solutions fulfil this condition along with MF=SF=3 and (RF)=( v1 ,v2 ,ωδ ). 4.1.1 Basic solutions In the basic solutions of overconstrained PPMs with uncoupled motions, F ← G1-G2-G3, the moving platform nznGi (i=1, 2, 3) is connected to the reference platform 1z1Giz0 by three limbs with three degrees of connectivity. The simple and complex limbs presented in Figs. 2.1f,g, 2.2e-h and 2.3a,b are used in the solutions illustrated in this section (Figs. 4.1-4.6). The complex limbs combine one (Fig. 2.2e-g) or two (Figs. 2.2h and 2.3a) planar parallelogram loops of types Pa. One planar telescopic parallelogram loop of type Pat (Fig. 2.3b) along with a Pa-type loop is combined in the complex limb in Fig. 2.3b. No idle mobilities exist in these basic solutions. G. Gogu, Structural Synthesis of Parallel Robots, Solid Mechanics and Its Applications 173, DOI 10.1007/978-90-481-9831-3_4, © Springer Science + Business Media B.V. 2010

239

240

4 Planar parallel robots with uncoupled motions

Basic solutions of PPMs with uncoupled motions and different limb topologies can be obtained by using the following combinations of limbs: (i) two identical limbs G1 and G2 from Fig. 2.2e and G3-limb from Figs. 2.1g, 2.2f-h or 2.3a,b, (ii) two different limbs G1 and G2 from Figs. 2.1f and 2.2e and G3 limb from Figs. 2.1g, 2.2f-h or 2.3a,b. The rotation axis of the moving platform can be in a fixed or variable position. G3-limb in Figs. 2.1g and 2.2f-h are used to give a fixed position of the rotation axis and G3-limb in Fig. 2.3a,b to obtain a variable position. The limb topologies and connecting conditions in the PPMs with uncoupled motions presented in Figs. 4.1-4.6 are systematized in Table 4.1 and their structural parameters in Tables 4.2 and 4.3. The solutions in Figs 4.3a and 4.6a can provide an unlimited angle of rotation of the moving platform. In the fully-parallel solutions in Figs. 4.1-4.6, the moving platform nznGi (i=1, 2, 3) is connected to the reference platform 1z1Giz0 by three planar limbs with three degrees of connectivity, and one actuator is combined in a revolute or prismatic pair of each limb. Non fully-parallel solutions can be obtained from the fully-parallel solutions in Figs. 4.1-4.6 by superposing the last revolute joints of limbs G1 and G2. In this way, these two limbs compose just one complex limb denoted by G1-G2 in which two actuators are combined, as we have presented in Chapters 2 and 3. Note 1: The limbs presented in Figs. 2.1-2.3 can also be used to generate basic solutions of overconstrained PPMs with decoupled motions. Basic solutions of PPMs with decoupled motions and different limb topologies can be obtained by using G1-limb from Figs. 2.1f and 2.2e, G2lumb from Figs. 2.1a-e and 2.2a-d and G3-limb from Figs. 2.1g, 2.2f-h or 2.3a,b. A large number of solutions with decoupled motions can be obtained by various combinations of the three limbs. These solutions are not presented in this book.

4.1 Overconstrained solutions

241

Table 4.1. Limb topology and connecting conditions of the overconstrained PPM with no idle mobilities presented in Figs. 4.1-4.6 No. PPM type 1

2PaPR-1RPP (Fig. 4.1a)

2

2PaPR-1RPPa (Fig. 4.1b)

3

2PaPR-1RPaP (Fig. 4.2a)

4

2PaPR-1RPaPa (Fig. 4.2b)

5

2PaPR-1RPaPa (Fig. 4.3a)

6

2PaPR-1RPaPat (Fig. 4.3b)

7

PPR-PaPR-RPP (Fig. 4.4a)

8

PPR-PaPR-RPPa (Fig. 4.4b)

Limb topology Pa ⊥ P ⊥ ||R (Fig. 2.2e) R ⊥ P ⊥⊥ P (Fig. 2.1g) Pa ⊥ P ⊥ ||R (Fig. 2.2e) R ⊥ P ⊥ ||Pa (Fig. 2.2f) Pa ⊥ P ⊥ ||R (Fig. 2.2e) R||Pa ⊥ P (Fig. 2.2g) Pa ⊥ P ⊥ ||R (Fig. 2.2e) R||Pa||Pa (Fig. 2.2h) Pa ⊥ P ⊥ ||R (Fig. 2.2e) R||Pa||Pa (Fig. 2.3a) Pa ⊥ P ⊥ ||R (Fig. 2.2e) R||Pa||Pat (Fig. 2.3b) P ⊥ P ⊥⊥ R (Fig. 2.1f) Pa ⊥ P ⊥ ||R (Fig. 2.2e) R ⊥ P ⊥⊥ P (Fig. 2.1g) P ⊥ P ⊥⊥ R (Fig. 2.1f) Pa ⊥ P ⊥ ||R (Fig. 2.2e) R ⊥ P ⊥ ||Pa (Fig. 2.2f)

Connecting conditions The directions of the revolute joints of the three limbs are parallel. The last revolute joint of limbs G1 and G2 have superposed axes. Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

242

4 Planar parallel robots with uncoupled motions

Table 4.1. (cont.) 9

PPR-PaPR-RPaP (Fig. 4.5a)

10

PPR-PaPR-RPaPa (Fig. 4.5b)

11

PPR-PaPR-RPaPa (Fig. 4.6a)

12

PPR-PaPR-RPaPat (Fig. 4.6b)

P ⊥ P ⊥⊥ R (Fig. 2.1f) Pa ⊥ P ⊥ ||R (Fig. 2.2e) R||Pa ⊥ P (Fig. 2.2g) P ⊥ P ⊥⊥ R (Fig. 2.1f) Pa ⊥ P ⊥ ||R (Fig. 2.2e) R||Pa||Pa (Fig. 2.2h) P ⊥ P ⊥⊥ R (Fig. 2.1f) Pa ⊥ P ⊥ ||R (Fig. 2.2e) R||Pa||Pa (Fig. 2.3a) P ⊥ P ⊥⊥ R (Fig. 2.1f) Pa ⊥ P ⊥ ||R (Fig. 2.2e) R||Pa||Pat (Fig. 2.3b)

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

4.1 Overconstrained solutions

243

Table 4.2. Structural parametersa of planar parallel mechanisms in Figs. 4.1-4.3 No. Structural Solution parameter 2PaPR-1RPP (Fig. 4.1a)

1 2 3 4 5 6 7 8 9 10

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

11

(RG2)

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 a

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

12 6 6 3 15 4 1 2 3 ( v1 , v 2 , ( v1 , v2 ,

fj

( v1 , v2 , 3 3 3 3 3 0 3 3 3 ( v1 , v 2 , 3 6 12 3 12 0 6

δ δ

) )

δ

)

δ

)

)

2PaPR-1RPaPa (Fig. 4.2b) 2PaPR-1RPaPa 2PaPR-1RPaPa (Fig. 4.3a,b) 16 6 6 9 21 6 0 3 3 ( v1 , v2 , δ )

)

( v1 , v2 ,

δ

)

δ

)

( v1 , v2 , 3 3 3 3 3 6 3 3 3 ( v1 , v2 , 3 12 18 3 18 0 6

2PaPR-1RPPa (Fig. 4.1b) 2PaPR-1RPaP (Fig. 4.2a) 14 6 6 6 18 5 0 3 3 ( v1 , v 2 , ( v1 , v 2 , ( v1 , v 2 , 3 3 3 3 3 3 3 3 3 ( v1 , v 2 , 3 9 15 3 15 0 6

δ δ

fj

6

6

6

fj

3

6

9

fj

15

18

21

δ

)

δ

)

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

244

4 Planar parallel robots with uncoupled motions

Table 4.3. Structural parametersa of planar parallel mechanisms in Figs. 4.4-4.6 No. Structural Solution parameter PPR-PaPR-RPP (Fig. 4.4a)

1 2 3 4 5 6 7 8 9 10

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

11

(RG2)

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 a

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

10 3 6 3 12 3 2 1 3 ( v1 , v 2 , ( v1 , v2 ,

fj

( v1 , v2 , 3 3 3 0 3 0 3 3 3 ( v1 , v 2 , 3 3 9 3 9 0 3

δ δ

) )

δ

)

δ

)

PPR-PaPR-RPPa (Fig. 4.4b) PPR-PaPR-RPaP (Fig. 4.5a) 12 3 6 6 15 4 1 2 3 ( v1 , v 2 , ( v1 , v 2 , ( v1 , v 2 , 3 3 3 0 3 3 3 3 3 ( v1 , v 2 , 3 6 12 3 12 0 3

)

PPR-PaPR-RPaPa (Fig. 4.5b) PPR-PaPR-RPaPa PPR-PaPR-RPaPat (Fig. 4.6a,b) 14 3 6 9 18 5 0 3 3 ( v1 , v2 , δ )

)

( v1 , v2 ,

δ

)

δ

)

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

δ δ

fj

6

6

6

fj

3

6

9

fj

12

15

18

δ

)

δ

)

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

4.1 Overconstrained solutions

245

Fig. 4.1. Overconstrained PPMs with uncoupled motions of types 2PaPR-1RPP (a) and 2PaPR-1RPPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=12 (a), NF=15 (b), limb topology Pa ⊥ P ⊥ ||R and R ⊥ P ⊥ ⊥ P (a), R ⊥ P ⊥ ||Pa (b)

246

4 Planar parallel robots with uncoupled motions

Fig. 4.2. Overconstrained PPMs with uncoupled motions of types 2PaPR-1RPaP (a) and 2PaPR-1RPaPa (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=15 (a), NF=18 (b), limb topology Pa ⊥ P ⊥ ||R and R||Pa ⊥ P (a), R||Pa||Pa (b)

4.1 Overconstrained solutions

247

Fig. 4.3. Overconstrained PPMs with uncoupled motions of types 2PaPR-1RPaPa (a) and 2PaPR-1RPaPat (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=18, limb topology Pa ⊥ P ⊥ ||R and R||Pa||Pa (a), R||Pa||Pat (b)

248

4 Planar parallel robots with uncoupled motions

Fig. 4.4. Overconstrained PPMs with uncoupled motions of types PPR-PaPRRPP (a) and PPR-PaPR-RPPa (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0,

NF=9 (a), NF=12 (b), limb topology P ⊥ P ⊥ ⊥ R, Pa ⊥ P ⊥ ||R and R ⊥ P ⊥ ⊥ P (a), R ⊥ P ⊥ ||Pa (b)

4.1 Overconstrained solutions

249

Fig. 4.5. Overconstrained PPMs with uncoupled motions of types PPR-PaPRRPaP (a) and PPR-PaPR-RPaPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ),

TF=0, NF=12 (a), NF=15 (b), limb topology P ⊥ P ⊥ ⊥ R, Pa ⊥ P ⊥ ||R and R||Pa ⊥ P (a), R||Pa||Pa (b)

250

4 Planar parallel robots with uncoupled motions

Fig. 4.6. Overconstrained PPMs with uncoupled motions of types PPR-PaPRRPaPa (a) and PPR-PaPR-RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ),

TF=0, NF=15, limb topology P ⊥ P ⊥ ⊥ R, Pa ⊥ P ⊥ ||R and R||Pa||Pa (a), R||Pa||Pat (b)

4.1 Overconstrained solutions

251

4.1.2 Derived solutions Solutions with lower degrees of overconstraint can be derived from the basic solutions in Figs. 4.1-4.6 by using joints with idle mobilities. A large set of solutions can be obtained by introducing one or two idle mobilities outside the planar loops and up to three idle mobilities in each planar loop combined in the limbs. We recall that the joints combining idle mobilities are denoted by an asterisk. The idle mobilities which can be combined in a parallelogram loop are systematized in Fig. 1.2 and Table 1.1. The rotational mobility of the revolute joint denoted by R* is an idle mobility. One idle mobility is combined in each cylindrical joint C* and two idle mobilities in each spherical joint S*. These idle mobilities can be introduced outside or inside the planar loops combined in the limbs. For example, in the cylindrical joint denoted by C*, the idle mobility is the rotational motion in Figs. 4.7, 4.8a, 4.10 and 4.11 and the translational motion in Fig. 4.13a. In the limbs with two cylindrical joints C* in Figs. 4.13b, 4.14-4.18, the idle mobility is the rotational motion is in the first cylindrical joint and the translational motion in the second one. The notation Pacs is associated with a parallelogram loop with three idle mobilities combined in a cylindrical and a spherical joint, and Pass with four idle mobilities combined in two spherical joints adjacent to the same coupler link. In the parallelogram loop Passtype, three idle mobilities are introduced in the loop and one outside the loop. The idle mobility introduced outside the parallelogram loop is the internal mobility of the coupler link adjacent to the two spherical joints. Examples of solutions with 1-12 overconstraints derived from the basic solutions in Figs. 4.1-4.6 are illustrated in Figs. 4.7-4.18. The limb topology and the number of overconstraints of these solutions are systematized in Table 4.4 and the structural parameters in Tables 4.5-4.10.

252

4 Planar parallel robots with uncoupled motions

Table 4.4. Limb topology and the number of overconstraints NF of the derived PPMs with idle mobilities presented in Figs. 4.7-4.18 No. Basic PPM Type 1 2PaPR-1RPP (Fig. 4.1a) 2 3

2PaPR-1RPPa (Fig. 4.1b)

4 5

2PaPR-1RPaP (Fig. 4.2a)

6 7

2PaPR-1RPaPa (Fig. 4.2b)

8 9

2PaPR-1RPaPa (Fig. 4.3a)

10 11 2PaPR-1RPaPat (Fig. 4.3b) 12 13 PPR-PaPR-RPP (Fig. 4.4a) 14

15 PPR-PaPR-RPPa (Fig. 4.4b) 16

17 PPR-PaPR-RPaP (Fig. 4.5a)

Derived PPM NF Type 12 2PaPR-1RC*C* (Fig. 4.7a) 2PassPC*-1RPP (Fig. 4.13a) 15 2PaPR-1RC*Pass (Fig. 4.7b) 2PassC*C*-1RPPa (Fig. 4.13b) 15 2PaPR-1RPassC* (Fig. 4.8a) 2PassC*C*-1RPaP (Fig. 4.14a) 18 2PaPR-1RPassPass (Fig. 4.8b) 2PassC*C*-1RPaPa (Fig. 4.14b) 18 2PaPR-1RPacsPacs (Fig. 4.9a) 2PassPC*-1RPaPa (Fig. 4.15a) 18 2PaPR-1RPacsPatcs (Fig. 4.9b) 2PassPC*-1RPaPat (Fig. 4.15b) 9 PPR-PaPR-RC*C* (Fig. 4.10a) PC*C*-PassC*C*-RPP (Fig. 4.16a) 12 PPR-PaPR-RC*Pass (Fig. 4.10b) PR*C*C*-PassC*C*-RPPa (Fig. 4.16b) 12 PPR-PaPR-RPassC* (Fig. 4.11a)

NF Limb topology 10 Pa ⊥ P ⊥ ||R R ⊥ C* ⊥ ⊥ C* 2 Pass ⊥ P ⊥ ||C* R ⊥ P ⊥⊥ P 10 Pa ⊥ P ⊥ ||R R ⊥ C* ⊥ ||Pass 3 Pass ⊥ C* ⊥ ||C* R ⊥ P ⊥ ||Pa 10 Pa ⊥ P ⊥ ||R R||Pass ⊥ C* 3 Pass ⊥ C* ⊥ ||C* R||Pa ⊥ P 10 Pa ⊥ P ⊥ ||R R||Pass||Pass 6 Pass ⊥ C* ⊥ ||C* R||Pa||Pa 12 R||Pacs||Pacs Pa ⊥ P ⊥ ||R 8 Pass ⊥ P ⊥ ||C* R||Pa||Pa 12 R||Pacs||Patcs Pa ⊥ P ⊥ ||R 8 Pass ⊥ P ⊥ ||C* R||Pa||Pat 7 P ⊥ P ⊥ ⊥ R, Pa ⊥ P ⊥ ||R R ⊥ C* ⊥ ⊥ C* 1 P ⊥ C* ⊥ ⊥ C* Pass ⊥ C* ⊥ ||C* R ⊥ P ⊥⊥ P 7 P ⊥ P ⊥⊥ R Pa ⊥ P ⊥ ||R R ⊥ C* ⊥ ||Pass 3 P||R* ⊥ C* ⊥ ⊥ C* Pass ⊥ C* ⊥ ||C* R ⊥ P ⊥ ||Pa 7 P ⊥ P ⊥⊥ R Pa ⊥ P ⊥ ||R R||Pass ⊥ C*

4.1 Overconstrained solutions Table 4.4. (cont.) PR*C*C*-PassC*C*-RPaP (Fig. 4.17a)

18

P||R* ⊥ C* ⊥ ⊥ C* Pass ⊥ C* ⊥ ||C* R||Pa ⊥ P P ⊥ P ⊥⊥ R Pa ⊥ P ⊥ ||R R||Pass||Pass P||R* ⊥ C* ⊥ ⊥ C* Pass ⊥ C* ⊥ ||C* R||Pa||Pa P ⊥ P ⊥⊥ R Pa ⊥ P ⊥ ||R R||Pacs||Pacs P||R* ⊥ C* ⊥ ⊥ C* Pass ⊥ C* ⊥ ||C* R||Pa||Pa P ⊥ P ⊥⊥ R Pa ⊥ P ⊥ ||R R||Pacs||Patcs P||R* ⊥ C* ⊥ ⊥ C* Pass ⊥ C* ⊥ ||C* R||Pa||Pat

3

19 PPR-PaPR-RPaPa 15 PPR-PaPR-RPassPass (Fig. 4.5b) (Fig. 4.11b)

7

PR*C*C*-PassC*C*-RPaPa 6 (Fig. 4.17b)

20

21 PPR-PaPR-RPaPa 15 PPR-PaPR-RPacsPacs (Fig. 4.6a) (Fig. 4.12a)

9

PR*C*C*-PassC*C*-RPaPa 6 (Fig. 4.18a)

22

23 PPR-PaPR-RPaPat 15 PPR-PaPR-RPacsPatcs (Fig. 4.6b) (Fig. 4.12b)

9

PR*C*C*-PassC*C*-RPaPat 6 (Fig. 4.18b)

24

253

Table 4.5. Bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 4.7-4.18 No. Parallel mechanism 1 Figs. 4.7, 4.8, 4.10, 4.11 2 Figs. 4.9, 4.12 3 Figs. 4.13a,

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

( v1 , v 2 , v 3 ,

α

,

δ

)

4

( v1 , v 2 , v 3 ,

α

,

β

,

( v1 , v 2 , v 3 ,

β

,

δ

)

5

Figs. 4.13b, 4.14, 4.15, 4.16b, 4.17, 4.18 Figs. 4.16a

δ

)

(RG2) ( v1 , v2 ,

δ

)

( v1 , v2 ,

δ

δ

)

(RG3) ( v1 , v 2 ,

α

,

δ

)

( v1 , v2 ,

δ

)

( v1 , v 2 ,

δ

)

( v1 , v 2 , v 3 ,

β

,

δ

)

) ( v1 , v 2 , v 3 ,

α

,

β

,

δ

) ( v1 , v 2 ,

δ

)

( v1 , v 2 , v 3 ,

α

,

β

,

δ

) ( v1 , v 2 ,

δ

)

β

,

δ

)

254

4 Planar parallel robots with uncoupled motions

Table 4.6. Structural parametersa of overconstrained PPMs in Figs. 4.7 and 4.8. No. Structural Solution parameter 2PaPR-1RC*C* (Fig. 4.7a)

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

12 6 6 3 15 4 1 2 3 see Table 4.5

2PaPR-1RC*Pass (Fig. 4.7b) 2PaPR-1RPassC* (Fig. 4.8a) 14 6 6 6 18 5 0 3 3 see Table 4.5

16 6 6 9 21 6 0 3 3 see Table 4.5

3 3 5 3 3 0 3 3 5 ( v1 , v 2 , 3 6 14 3 10 0 6

3 3 5 3 3 6 3 3 5 ( v1 , v 2 , 3 12 20 3 10 0 6

3 3 5 3 3 12 3 3 5 ( v1 , v 2 , 3 18 26 3 10 0 6

δ

)

δ

)

2PaPR-1RPassPass (Fig. 4.8b)

fj

6

6

6

fj

5

11

17

fj

17

23

29

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

4.1 Overconstrained solutions

255

Table 4.7. Structural parametersa of overconstrained PPMs in Figs. 4.9 and 4.10. No. Structural Solution parameter 2PaPR-1RPacsPacs (Fig. 4.9a) 2PaPR-1RPacsPatcs (Fig. 4.9b) 1 m 16 2 p1 6 3 p2 6 4 p3 9 5 p 21 6 q 6 7 k1 0 8 k2 3 9 k 3 10 (RGi) see Table 4.5 (i=1,2,3) 11 SG1 3 12 SG2 3 13 SG3 3 14 rG1 3 15 rG2 3 16 rG3 12 17 MG1 3 18 MG2 3 19 MG3 3 20 (RF) ( v1 , v 2 , δ ) 21 SF 3 22 rl 18 23 rF 24 24 MF 3 25 NF 12 26 TF 0 p1 27 6 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

PPR-PaPR-RC*C* (Fig. 4.10a)

PPR-PaPR-RC*Pass (Fig. 4.10b)

10 3 6 3 12 3 2 1 3 see Table 4.5

12 3 6 6 15 4 1 2 3 see Table 4.5

3 3 5 0 3 0 3 3 5 ( v1 , v 2 , 3 3 11 3 7 0 3

3 3 5 0 3 6 3 3 5 ( v1 , v 2 , 3 9 17 3 7 0 3

δ

)

fj

6

6

6

fj

15

5

11

fj

27

14

20

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

256

4 Planar parallel robots with uncoupled motions

Table 4.8. Structural parametersa of overconstrained PPMs in Figs. 4.11 and 4.12. No. Structural Solution parameter PPR-PaPR-RPassC* PPR-PaPR-RPassPass PPR-PaPR-RPacsPacs (Fig. 4.11a) (Fig. 4.12a) (Fig. 4.11b) PPR-PaPR-RPacsPatcs (Fig. 4.12b) 1 m 12 14 14 2 p1 3 3 3 3 p2 6 6 6 4 p3 6 9 9 5 p 15 18 18 6 q 4 5 5 7 k1 1 2 1 8 k2 2 1 2 9 k 3 3 3 10 (RGi) see Table 4.5 see Table 4.5 see Table 4.5 (i=1,2,3) 11 SG1 3 3 3 12 SG2 3 3 3 13 SG3 5 5 3 14 rG1 0 0 0 15 rG2 3 3 3 16 rG3 6 12 12 17 MG1 3 3 3 18 MG2 3 3 3 19 MG3 5 5 3 20 (RF) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) 21 SF 3 3 3 22 rl 9 15 15 23 rF 17 23 21 24 MF 3 3 3 25 NF 7 7 9 26 TF 0 0 0 p1 27 3 3 3 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

fj

6

6

6

fj

11

17

15

fj

20

26

24

See footnote of Table 2.4 for the nomenclature of structural parameters

4.1 Overconstrained solutions

257

Table 4.9. Structural parametersa of overconstrained PPMs in Figs. 4.13-4.15. No. Structural Solution parameter 2PassPC*-1RPP (Fig. 4.13a)

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

12 6 6 3 15 4 1 2 3 see Table 4.5

14 6 6 6 18 5 0 3 3 see Table 4.5

2PassC*C*-1RPaPa (Fig. 4.14b) 2PassPC*-1RPaPa (Fig. 4.15a) 2PassPC*-1RPaPat (Fig. 4.15b) 16 6 6 9 21 6 0 3 3 see Table 4.5

5 5 3 6 6 0 5 5 3 ( v1 , v 2 , 3 12 22 3 2 0 11

6 6 3 6 6 3 6 6 3 ( v1 , v 2 , 3 15 27 3 3 0 12

6 6 3 6 6 6 6 6 3 ( v1 , v 2 , 3 18 30 3 6 0 12

δ

)

2PassC*C*-1RPPa (Fig. 4.13b) 2PassC*C*-1RPaP (Fig. 4.14a)

δ

)

fj

11

12

12

fj

3

6

9

fj

25

30

33

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

258

4 Planar parallel robots with uncoupled motions

Table 4.10. Structural parametersa of overconstrained PPMs in Figs. 4.16-4.18. No. Structural Solution parameter PC*C*-PassC*C*RPP (Fig. 4.16a)

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

10 3 6 3 12 3 2 1 3 see Table 4.5

13 4 6 6 16 4 1 2 3 see Table 4.5

PR*C*C*-PassC*C*RPaPa (Fig. 4.17b) PR*C*C*-PassC*C*RPaPa (Fig. 4.18a) PR*C*C*-PassC*C*RPaPat (Fig. 4.18b) 15 4 6 9 19 5 1 2 3 see Table 4.5

5 6 3 0 6 0 5 6 3 ( v1 , v 2 , 3 9 17 3 1 0 5

6 6 3 0 6 3 6 6 3 ( v1 , v 2 , 3 0 21 3 3 0 6

6 6 3 0 6 6 6 6 3 ( v1 , v 2 , 3 12 24 3 6 0 6

δ

)

PR*C*C*-PassC*C*RPPa (Fig. 4.16b) PR*C*C*-PassC*C*RPaP (Fig. 4.17a)

δ

)

fj

12

12

12

fj

3

6

9

fj

20

24

27

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

4.1 Overconstrained solutions

259

Fig. 4.7. Overconstrained PPMs with uncoupled motions of types 2PaPR-1RC*C* (a) and 2PaPR-1RC*Pass (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0,

NF=10, limb topology Pa ⊥ P ⊥ ||R and R ⊥ C* ⊥ ⊥ C* (a), R ⊥ C* ⊥ ||Pass (b)

260

4 Planar parallel robots with uncoupled motions

Fig. 4.8. Overconstrained PPMs with uncoupled motions of types 2PaPR1RPassC* (a) and 2PaPR-1RPassPass (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=10, limb topology Pa ⊥ P ⊥ ||R and R||Pass ⊥ C* (a), R||Pass||Pass (b)

4.1 Overconstrained solutions

261

Fig. 4.9. Overconstrained PPMs with uncoupled motions of types 2PaPR1RPacsPacs (a) and 2PaPR-1RPacsPatcs (b), defined by MF=SF=3, (RF)= ( v1 , v2 , δ ), TF=0, NF=12, limb topology Pa ⊥ P ⊥ ||R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

262

4 Planar parallel robots with uncoupled motions

Fig. 4.10. Overconstrained PPMs with uncoupled motions of types PPR-PaPRRC*C* (a) and PPR-PaPR-RC*Pass (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ),

TF=0, NF=7, limb topology P ⊥ P ⊥ ⊥ R, Pa ⊥ P ⊥ ||R and R ⊥ C* ⊥ ⊥ C* (a), R ⊥ C* ⊥ ||Pass (b)

4.1 Overconstrained solutions

263

Fig. 4.11. Overconstrained PPMs with uncoupled motions of types PPR-PaPRRPassC* (a) and PPR-PaPR-RPassPass (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=7, limb topology P ⊥ P ⊥ ⊥ R, Pa ⊥ P ⊥ ||R and R||Pass ⊥ C* (a), R||Pass||Pass (b)

264

4 Planar parallel robots with uncoupled motions

Fig. 4.12. Overconstrained PPMs with uncoupled motions of types PPR-PaPRRPacsPacs (a) and PPR-PaPR-RPacsPatcs (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=9, limb topology P ⊥ P ⊥ ⊥ R, Pa ⊥ P ⊥ ||R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

4.1 Overconstrained solutions

265

Fig. 4.13. Overconstrained PPMs with uncoupled motions of types 2PassPC*1RPP (a) and 2PassC*C*-1RPPa (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ),

TF=0, NF=2 (a), NF=3 (b), limb topology Pass ⊥ P ⊥ ||C* and R ⊥ P ⊥ ⊥ P (a), Pass ⊥ C* ⊥ ||C* and R ⊥ P ⊥ ||Pa (b)

266

4 Planar parallel robots with uncoupled motions

Fig. 4.14. Overconstrained PPMs with uncoupled motions of types 2PassC*C*1RPaP (a) and 2PassC*C*-1RPaPa (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3 (a), NF=6 (b), limb topology Pass ⊥ C* ⊥ ||C* and R||Pa ⊥ P (a), R||Pa||Pa (b)

4.1 Overconstrained solutions

267

Fig. 4.15. Overconstrained PPMs with uncoupled motions of types 2PassPC*1RPaPa (a) and 2PassPC*-1RPaPat (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=6, limb topology Pass ⊥ P ⊥ ||C* and R||Pa||Pa (a), R||Pa||Pat (b)

268

4 Planar parallel robots with uncoupled motions

Fig. 4.16. Overconstrained PPMs with uncoupled motions of types PC*C*PassC*C*-RPP (a) and PR*C*C*-PassC*C*-RPPa (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=1 (a), NF=3 (b), limb topology P||R* ⊥ C* ⊥ ⊥ C*, Pass ⊥ C* ⊥ ||C* and R ⊥ P ⊥ ⊥ P (a), R ⊥ P ⊥ ||Pa (b)

4.1 Overconstrained solutions

269

Fig. 4.17. Overconstrained PPMs with uncoupled motions of types PR*C*C*PassC*C*-RPaP (a) and PR*C*C*-PassC*C*-RPaPa (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3 (a), NF=6 (b), limb topology P||R* ⊥ C* ⊥ ⊥ C*, Pass ⊥ C* ⊥ ||C* and R||Pa ⊥ P (a), R||Pa||Pa (b)

270

4 Planar parallel robots with uncoupled motions

Fig. 4.18. Overconstrained PPMs with uncoupled motions of types PR*C*C*PassC*C*-RPaPa (a) and PR*C*C*-PassC*C*-RPaPat (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=6, limb topology P||R* ⊥ C* ⊥ ⊥ C*, Pass ⊥ C* ⊥ ||C* and R||Pa||Pa (a), R||Pa||Pat (b)

4.2 Non overconstrained solutions

271

4.2 Non overconstrained solutions Equation (1.15) indicates that non overconstrained solutions of planar parallel robots with uncoupled motions and q independent loops meet the p condition ∑ 1 f i = 3 + 6q along with MF=SF=3 and (RF)=( v1 ,v2 ,ωδ ).

The non overconstrained solutions of planar parallel robots with uncoupled motions presented in this section (Figs. 4.19-4.24) are derived from overconstrained counterparts presented in Figs. 4.1-4.6 by introducing the required idle mobilities. For example, the non overconstrained solution in Fig. 4.19a is derived from the overconstrained solution in Fig. 4.1a by combining three idle mobilities in each parallelogram loop and three idle mobilities outside each parallelogram loop in limbs G1 and G2. Two idle mobilities are introduced in each limb in two cylindrical joints denoted by C*. The rotational motion is the idle mobility in the first cylindrical joint and the translational motion in the second cylindrical joint of the limb. We recall that one translational and two rotational idle mobilities are introduced in one cylindrical and one spherical joint of the the parallelogram and telescopic parallelogram loops denoted by Pacs and Patcs (Figs. 4.21 and 4.24). In the cylindrical joints of these loops, the translational motion is an idle mobility. In the parallelogram loop Pass-type, three idle mobilities are introduced in the loop and one outside the loop. If the link adjacent to the two spherical joints is a binary link than the idle mobility introduced outside the loop becomes an internal rotational mobility of this binary link around the axis passing by the centre of the two spherical joints. This internal mobility gives one degree of structural redundancy. If the link adjacent to the two spherical joints is connected in the limb by three or more joints (polinary link) than the rotational motion around the axis passing by the centre of the two spherical joints is an idle (potential) mobility of the limb. For example in Fig. 4.19b, this rotational motion is internal mobility of binary link 5C, and idle mobility for the ternary links 4A and 4B. Table 4.11. Bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 4.19-4.24 No. Parallel mechanism 1 Figs. 4.19-4.21, 4.22a, 4.23a, 4.24 2 Figs. 4.22b, 4.23b

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

α,

β,

( v1 , v 2 , v 3 ,

β

δ

,

)

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

α,

β,

(RG3) ) δ ( v1 , v 2 ,

δ

)

α

β

δ

α

,

,

,

) ( v1 , v 2 ,

δ

)

272

4 Planar parallel robots with uncoupled motions

The bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 4.19-4.24 are given in Table 4.11. The limb topology and connecting conditions of these solutions are systematized in Table 4.12, as are their structural parameters in Tables 4.13-4.15. Table 4.12. Limb topology of the non overconstrained PPMs presented in Figs. 4.19-4.24 No. Basic PPM Type 1 2PaPR-1RPP (Fig. 4.1a) 2 2PaPR-1RPPa (Fig. 4.1b) 3 2PaPR-1RPaP (Fig. 4.2a) 4 2PaPR-1RPaPa (Fig. 4.2b) 5 2PaPR-1RPaPa (Fig. 4.3a) 6 2PaPR-1RPaPat (Fig. 4.3b) 7 PPR-PaPR-RPP (Fig. 4.4a)

NF 12 15 15 18 18 18 9

8

PPR-PaPR-RPPa (Fig. 4.4b)

12

9

PPR-PaPR-RPaP (Fig. 4.5a)

12

10

PPR-PaPR-RPaPa 15 (Fig. 4.5b)

11

PPR-PaPR-RPaPa 15 (Fig. 4.6a)

12

PPR-PaPR-RPaPat 15 (Fig. 4.6b)

PPM with NF=0 Type 2PassC*C*-1RPP (Fig. 4.19a) 2PassC*C*-1RPPass (Fig. 4.19b) 2PassC*C*-1RPassP (Fig. 4.20a) 2PassC*C*-1RPassPass (Fig. 4.20b) 2PassC*C*-1RPacsPacs (Fig. 4.21a) 2PassC*C*-1RPacsPatcs (Fig. 4.21b) PR*C*C*-PassC*C*-RPP (Fig. 4.22a)

Limb topology Pass ⊥ C* ⊥ ||C* R ⊥ P ⊥⊥ P Pass ⊥ C* ⊥ ||C* R ⊥ P ⊥ ||Pass Pass ⊥ C* ⊥ ||C* R||Pass ⊥ P Pass ⊥ C* ⊥ ||C* R||Pass||Pass Pass ⊥ C* ⊥ ||C* R||Pacs||Pacs Pass ⊥ C* ⊥ ||C* R||Pacs||Patcs P||R* ⊥ C* ⊥ ⊥ C* Pass ⊥ C* ⊥ ||C* R ⊥ P ⊥⊥ P ss ss PC*C*-Pa C*C*-RPPa P ⊥ C* ⊥ ⊥ C* (Fig. 4.22b) Pass ⊥ C* ⊥ ||C* R ⊥ P ⊥ ||Pass ss ss PR*C*C*- Pa C*C*-RPa P P||R* ⊥ C* ⊥ ⊥ C* (Fig. 4.23a) Pass ⊥ C* ⊥ ||C* R||Pass ⊥ P ss ss ss PC*C*-Pa C*C*-RPa Pa P ⊥ C* ⊥ ⊥ C* (Fig. 4.23b) Pass ⊥ C* ⊥ ||C* R||Pass||Pass ss PR*C*C*- Pa C*C*P||R* ⊥ C* ⊥ ⊥ C* cs cs RPa Pa Pass ⊥ C* ⊥ ||C* (Fig. 4.24a) R||Pacs||Pacs ss PR*C*C*-Pa C*C*P||R* ⊥ C* ⊥ ⊥ C* cs tcs RPa Pa Pass ⊥ C* ⊥ ||C* (Fig. 4.24b) R||Pacs||Patcs

4.2 Non overconstrained solutions

273

Table 4.13. Structural parametersa of non overconstrained PPMs in Figs. 4.19 and 4.20 No. Structural Solution parameter 2PassC*C*-1RPP (Fig. 4.19a)

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

12 6 6 3 15 4 1 2 3 see Table 4.11

2PassC*C*-1RPPass (Fig. 4.19b) 2PassC*C*-1RPassP (Fig. 4.20a) 14 6 6 6 18 5 0 3 3 see Table 4.11

16 6 6 9 21 6 0 3 3 see Table 4.11

6 6 3 6 6 0 6 6 3 ( v1 , v 2 , 3 12 24 3 0 0 12

6 6 3 6 6 6 6 6 4 ( v1 , v 2 , 3 18 30 4 0 1 12

6 6 3 6 6 12 6 6 5 ( v1 , v 2 , 3 24 36 5 0 2 12

δ

)

δ

)

2PassC*C*-1RPassPass (Fig. 4.20b)

fj

12

12

12

fj

3

10

17

fj

27

34

41

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

274

4 Planar parallel robots with uncoupled motions

Table 4.14. Structural parametersa of non overconstrained PPMs in Figs. 4.21 and 4.22 No. Structural Solution parameter 2PassC*C*-1RPacsPacs (Fig. 4.21a) 2PassC*C*-1RPacsPatcs (Fig. 4.21b) 1 m 16 2 p1 6 3 p2 6 4 p3 9 5 p 21 6 q 6 7 k1 0 8 k2 3 9 k 3 10 (RGi) see Table 4.11 (i=1,2,3) 11 SG1 6 12 SG2 6 13 SG3 3 14 rG1 6 15 rG2 6 16 rG3 12 17 MG1 6 18 MG2 6 19 MG3 3 20 (RF) ( v1 , v 2 , δ ) 21 SF 3 22 rl 24 23 rF 36 24 MF 3 25 NF 0 26 TF 0 p1 27 12 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

PR*C*C*PassC*C*-RPP (Fig. 4.22a)

PC*C*PassC*C*-RPPass (Fig. 4.22b)

11 4 6 3 13 3 2 1 3 see Table 4.11

12 3 6 6 15 4 1 2 3 see Table 4.11

6 6 3 0 6 0 6 6 3 ( v1 , v 2 , 3 6 18 3 0 0 6

5 6 4 0 6 6 5 6 4 ( v1 , v 2 , 3 12 24 3 0 0 5

δ

)

δ

j

fj

12

12

12

fj

15

3

10

fj

39

21

27

See footnote of Table 2.4 for the nomenclature of structural parameters

)

4.2 Non overconstrained solutions

275

Table 4.15. Structural parametersa of non overconstrained PPMs in Figs. 4.23 and 4.24 No. Structural Solution parameter PR*C*C*PassC*C*RPassP (Fig. 4.23a) 1 m 13 2 p1 4 3 p2 6 4 p3 6 5 p 16 6 q 4 7 k1 1 8 k2 2 9 k 3 10 (RGi) see Table 4.11 (i=1,2,3) 11 SG1 6 12 SG2 6 13 SG3 3 14 rG1 0 15 rG2 6 16 rG3 6 17 MG1 6 18 MG2 6 19 MG3 4 20 (RF) ( v1 , v 2 , δ ) 21 SF 3 22 rl 12 23 rF 24 24 MF 4 25 NF 0 26 TF 1 p1 27 6 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

PC*C*PassC*C*RPassPass (Fig. 4.23b) 14 3 6 9 18 5 1 2 3 see Table 4.11

PR*C*C*-PassC*C*-RPacsPacs (Fig. 4.24a) PR*C*C*-PassC*C*-RPacsPatcs (Fig. 4.24b) 15 4 6 9 19 5 1 2 3 see Table 4.11

5 6 4 0 6 12 5 6 5 ( v1 , v 2 , 3 18 30 4 0 1 5

6 6 3 0 6 12 6 6 3 ( v1 , v 2 , 3 18 30 3 0 0 6

δ

)

δ

)

j

fj

12

12

12

fj

10

17

15

fj

28

34

33

See footnote of Table 2.4 for the nomenclature of structural parameters

276

4 Planar parallel robots with uncoupled motions

Fig. 4.19. Non overconstrained PPMs with uncoupled motions of types 2PassC*C*-1RPP (a) and 2PassC*C*-1RPPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ) and MF=3, NF=0, TF=0 (a), MF=4, NF=0, TF=1 (b), limb topology Pass ⊥ C* ⊥ ||C* and R ⊥ P ⊥ ⊥ P (a), R ⊥ P ⊥ ||Pass (b)

4.2 Non overconstrained solutions

277

Fig. 4.20. Non overconstrained PPMs with uncoupled motions of types 2PassC*C*-1RPassP (a) and 2PassC*C*-1RPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ) and MF=4, NF=0, TF=1 (a), MF=5, NF=0, TF=2 (b), limb topology Pass ⊥ C* ⊥ ||C* and R||Pacs ⊥ P (a), R||Pass||Pass (b)

278

4 Planar parallel robots with uncoupled motions

Fig. 4.21. Non overconstrained PPMs with uncoupled motions of types 2PassC*C*-1RPacsPacs (a) and 2PassC*C*-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology Pass ⊥ C* ⊥ ||C* and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

4.2 Non overconstrained solutions

279

Fig. 4.22. Non overconstrained PPMs with uncoupled motions of types PR*C*C*PassC*C*-RPP (a) and PC*C*-PassC*C*-RPPass (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology P||R* ⊥ C* ⊥ ⊥ C*, Pass ⊥ C* ⊥ ||C* and R ⊥ P ⊥ ⊥ P (a), P ⊥ C* ⊥ ⊥ C*, Pass ⊥ C* ⊥ ||C* and R ⊥ P ⊥ ||Pass (b)

280

4 Planar parallel robots with uncoupled motions

Fig. 4.23. Non overconstrained PPMs with uncoupled motions of types PR*C*C*PassC*C*-RPassP (a) and PC*C*-PassC*C*-RPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, NF=0, TF=1, limb topology P||R* ⊥ C* ⊥ ⊥ C*,

Pass ⊥ C* ⊥ ||C* and R||Pass ⊥ P (a), P ⊥ C* ⊥ ⊥ C*, Pass ⊥ C* ⊥ ||C* and R||Pass||Pass (b)

4.2 Non overconstrained solutions

281

Fig. 4.24. Non overconstrained PPMs with uncoupled motions of types PR*C*C*PassC*C*-RPacsPacs (a) and PR*C*C*-PassC*C*-RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology P||R* ⊥ C* ⊥ ⊥ C*, Pass ⊥ C* ⊥ ||C* and R||Pacs||Pacs (a), R||Pa cs||Pa tcs (b)

5 Maximally regular planar parallel robots

Maximally regular planar parallel robots are actuated by one rotating and two linear actuators 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 ωδ = &q3 . The Jacobian matrix in Eq. (1.18) is the identity matrix. We call planar Isoglide3-T2R1 the parallel mechanisms of this family.

5.1 Overconstrained solutions Equation (1.16) indicates that overconstrained solutions of maximally regular planar parallel robots with q independent loops meet the condition p ∑ 1 fi < 3 + 6q . Various solutions fulfil this condition along with MF=SF=3 and (RF)=( v1 ,v2 ,ωδ ). They have two identical limbs for positioning and a different limb for rotating the moving platform. 5.1.1 Basic solutions In the basic solutions of overconstrained maximally regular planar parallel robots, F ← G1-G2-G3, the moving platform n nGi (i=1, 2, 3) is connected to the reference platform 1 1Gi 0 by three limbs with three degrees of connectivity (Figs. 5.1-5.3). Two identical simple limbs (Fig. 2.1f) are used to position the moving platform and one simple (Fig. 2.1g) or complex (Figs. 2.2f-h and Fig. 2.3) limb to rotate it upon an axis of fixed or variable position. The solution in Fig 5.3a can provide an unlimited angle of rotation of the moving platform upon an axis of variable position. No idle mobilities exist in these basic solutions. The limb topologies and connecting conditions in the overconstrained maximally regular PPMs presented in Figs. 5.1-3.3 are systematized in Table 5.1 as are their structural parameters in Table 5.2.

G. Gogu, Structural Synthesis of Parallel Robots, Solid Mechanics and Its Applications 173, DOI 10.1007/978-90-481-9831-3_5, © Springer Science + Business Media B.V. 2010

283

284

5 Maximally regular planar parallel robots

Table 5.1. Limb topology and connecting conditions of the overconstrained maximally regular PPMs with no idle mobilities presented in Figs. 5.1-5.3 No. PPM type 1

2PPR-1RPP (Fig. 5.1a)

2

2PPR-1RPPa (Fig. 5.1b)

3

2PPR-1RPaP (Fig. 5.2a)

4

2PPR-1RPaPa (Fig. 5.2b)

5

2PPR-1RPaPa (Fig. 5.3a)

6

2PPR-1RPaPat (Fig. 5.3b)

Limb topology P ⊥ P ⊥⊥ R (Fig. 2.1f) R ⊥ P ⊥⊥ P (Fig. 2.1g) P ⊥ P ⊥⊥ R (Fig. 2.1f) R ⊥ P ⊥ ||Pa (Fig. 2.2f) P ⊥ P ⊥⊥ R (Fig. 2.1f) R||Pa ⊥ P (Fig. 2.2g) P ⊥ P ⊥⊥ R (Fig. 2.1f) R||Pa||Pa (Fig. 2.2h) P ⊥ P ⊥⊥ R (Fig. 2.1f) R||Pa||Pa (Fig. 2.3a) P ⊥ P ⊥⊥ R (Fig. 2.1f) R||Pa||Pat (Fig. 2.3b)

Connecting conditions The directions of the revolute joints of the three limbs are parallel. The last revolute joint of limbs G1 and G2 have the same axis. Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

5.1 Overconstrained solutions

285

Table 5.2. Structural parametersa of maximally regular planar parallel mechanisms in Figs. 5.1-5.3 No. Structural Solution parameter 2PPR-1RPP (Fig. 5.1a)

1 2 3 4 5 6 7 8 9 10

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

8 3 3 3 9 2 3 0 3 ( v1 , v 2 ,

11

(RG2)

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 , ( v1 , v2 , 3 3 3 0 0 0 3 3 3 ( v1 , v2 , 3 0 6 3 6 0 3

30 31 32 a

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

)

2PPR-1RPPa (Fig. 5.1b) 2PPR-1RPaP (Fig. 5.2a) 10 3 3 6 12 3 2 1 3 ( v1 , v2 , δ )

)

( v1 , v2 ,

δ

)

δ

)

( v1 , v2 , 3 3 3 0 0 3 3 3 3 ( v1 , v2 , 3 3 9 3 9 0 3

δ δ

δ

)

δ

)

δ

)

2PPR-1RPaPa (Fig. 5.2b) 2PPR-1RPaPa (Fig. 5.3a) 2PPR-1RPaPat (Fig. 5.3b) 12 3 3 9 15 4 2 1 3 ( v1 , v 2 , ( v1 , v 2 , ( v1 , v 2 , 3 3 3 0 0 6 3 3 3 ( v1 , v 2 , 3 6 12 3 12 0 3

fj

3

3

3

fj

3

6

9

fj

9

12

15

δ δ

) )

δ

)

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

286

5 Maximally regular planar parallel robots

Fig. 5.1. Overconstrained maximally regular PPMs of types 2PPR-1RPP (a) and 2PPR-1RPPa (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=6 (a), NF=9 (b), limb topology P ⊥ P ⊥ ⊥ R and R ⊥ P ⊥ ⊥ P (a), R ⊥ P ⊥ ||Pa (b)

5.1 Overconstrained solutions

287

Fig. 5.2. Overconstrained maximally regular PPMs of types 2PPR-1RPaP (a) and 2PPR-1RPaPa (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=9 (a),

NF=12 (b), limb topology P ⊥ P ⊥ ⊥ R and R||Pa ⊥ P (a), R||Pa||Pa (b)

288

5 Maximally regular planar parallel robots

Fig. 5.3. Overconstrained maximally regular PPMs of types 2PPR-1RPaPa (a) and 2PPR-1RPaPat (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=12, limb topology P ⊥ P ⊥ ⊥ R and R||Pa||Pa (a), R||Pa||Pat (b)

5.1 Overconstrained solutions

289

5.1.2 Derived solutions Solutions with lower degrees of overconstraint can be derived from the basic solutions in Figs. 5.1-5.3 by using joints with idle mobilities. A large set of solutions can be obtained by introducing one or two idle mobilities outside the planar loops and up to three idle mobilities in each planar loop combined in the limbs (Figs. 5.4-5.9). We recall that the joints combining idle mobilities are denoted by an asterisk. The idle mobilities which can be combined in a parallelogram loop are systematized in Fig. 1.2 and Table 1.1. The rotational mobility of the revolute joint denoted by R* is an idle mobility. One idle mobility is combined in each cylindrical joint C* and two idle mobilities in each spherical joint S*. In the cylindrical joint denoted by C* in Figs 5.4 and 5.5, the rotational motion is an idle mobility. In the limbs with two cylindrical joints C* in Figs. 5.7-5.9, the idle mobility is the rotational motion in the first cylindrical joint and the translational motion in the second one. The notation Pacs is associated with a parallelogram loop with three idle mobilities combined in a cylindrical and a spherical joint, and Pass with four idle mobilities combined in two spherical joints adjacent to the same coupler link. In the parallelogram loop Pass-type, three idle mobilities are introduced in the loop and one outside the loop. The idle mobility introduced outside the parallelogram loop is the internal mobility of the coupler link adjacent to the two spherical joints. Examples of solutions with 2 to 8 overconstraints derived from the basic solutions in Figs. 4.1-4.6 are illustrated in Figs. 5.4-5.9. The basis of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 5.4-5.9 are given in Table 5.3. The limb topology and connecting conditions of these solutions are systematized in Table 5.4 and the structural parameters of these solutions are presented in Tables 5.5-5.7.

290

5 Maximally regular planar parallel robots

Table 5.3. Bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 5.4-5.9 No. Parallel Basis mechanism (RG1) 1 Figs. 5.4, 5.5 ( v1 , v2 ,

δ

)

(RG2) ( v1 , v2 ,

δ

)

( v1 , v 2 ,

2

Fig. 5.6

3

Figs. 5.7, 5.8 ( v1 , v2 , v3 ,

β

,

δ

)

4

Fig. 5.9

α

,

β

,

( v1 , v 2 ,

( v1 , v 2 , v 3 ,

δ

δ

)

(RG3) ( v1 , v 2 ,

α

,

δ

)

( v1 , v 2 ,

δ

)

( v1 , v 2 , v 3 ,

α

,

δ

)

) ( v1 , v 2 , v 3 ,

α

,

β

,

δ

( v1 , v 2 ,

δ

)

) ( v1 , v 2 ,

δ

)

β

,

δ

)

Table 5.4. Limb topology and the number of overconstraints NF of the derived maximally regular PPMs with idle mobilities presented in Figs. 5.4-5.9 No. Basic PPM Type 1 2PPR-1RPP (Fig. 5.1a) 2 3

2PPR-1RPPa (Fig. 5.1b)

4 5

2PPR-1RPaP (Fig. 5.2a)

6 7

2PPR-1RPaPa (Fig. 5.2b)

8 9

2PPR-1RPaPa (Fig. 5.3a)

10 11 2PPR-1RPaPat (Fig. 5.3b) 12

Derived PPM NF Type 6 2PPR-1RC*C* (Fig. 5.4a) 2PC*C*-1RPP (Fig. 5.7a) 9 2PPR-1RC*Pass (Fig. 5.4b) 2PC*C*-1RPPass (Fig. 5.7b) 9 2PPR-1RPassC* (Fig. 5.5a) 2PC*C*-1RPaP (Fig. 5.8a) 12 2PPR-1RPassPass (Fig. 5.5b) 2PC*C*-1RPaPa (Fig. 5.8b) 12 2PPR-1RPacsPacs (Fig. 5.6a) 2PR*C*C*-1RPaPa (Fig. 5.9a) 12 2PPR-1RPacsPatcs (Fig. 5.6a) 2PR*C*C*-1RPaPat (Fig. 5.9a)

NF Limb topology 4 P ⊥ P ⊥⊥ R R ⊥ C* ⊥ ⊥ C* 2 P ⊥ C* ⊥ ⊥ C* R ⊥ P ⊥⊥ P 4 P ⊥ P ⊥⊥ R R ⊥ C* ⊥ ||Pass 1 P ⊥ C* ⊥ ⊥ C* R ⊥ P ⊥ ||Pass 4 P ⊥ P ⊥⊥ R R||Pass ⊥ C* 5 P ⊥ C* ⊥ ⊥ C* R||Pa ⊥ P 4 P ⊥ P ⊥⊥ R R||Pass||Pass 8 P ⊥ C* ⊥ ⊥ C* R||Pa||Pa 6 P ⊥ P ⊥⊥ R R||Pacs||Pacs 6 P||R ⊥ C* ⊥ ⊥ C* R||Pa||Pa 6 P ⊥ P ⊥⊥ R R||Pacs||Patcs 6 P||R ⊥ C* ⊥ ⊥ C* R||Pa||Pat

5.1 Overconstrained solutions Table 5.5. Structural parametersa of overconstrained PPMs in Figs. 5.4 and 5.5 No. Structural Solution parameter 2PPR-1RC*C* (Fig. 5.4a)

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

8 3 3 3 9 2 3 0 3 see Table 5.3

2PPR-1RC*Pass (Fig. 5.4b) 2PPR-1RPassC* (Fig. 5.5a) 10 3 3 6 12 3 2 1 3 see Table 5.3

12 3 3 9 15 4 2 1 3 see Table 5.3

3 3 5 0 0 0 3 3 5 ( v1 , v 2 , 3 0 8 3 4 0 3

3 3 5 0 0 6 3 3 5 ( v1 , v 2 , 3 6 14 3 4 0 3

3 3 5 0 0 12 3 3 5 ( v1 , v 2 , 3 12 20 3 4 0 3

δ

)

δ

)

2PPR-1RPassPass (Fig. 5.5b)

fj

3

3

3

fj

5

11

17

fj

11

17

23

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

291

292

5 Maximally regular planar parallel robots

Table 5.6. Structural parametersa of overconstrained PPMs in Figs. 5.6 and 5.7 No. Structural Solution parameter 2PPR-1RPacsPacs (Fig. 5.6a) 2PPR-1RPacsPatcs (Fig. 5.6b) 1 m 12 2 p1 3 3 p2 3 4 p3 9 5 p 15 6 q 4 7 k1 2 8 k2 1 9 k 3 10 (RGi) see Table 5.3 (i=1,2,3) 11 SG1 3 12 SG2 3 13 SG3 3 14 rG1 0 15 rG2 0 16 rG3 12 17 MG1 3 18 MG2 3 19 MG3 3 20 (RF) ( v1 , v 2 , δ ) 21 SF 3 22 rl 12 23 rF 18 24 MF 3 25 NF 6 26 TF 0 p1 27 3 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

2PC*C*-1RPP (Fig. 5.7a)

2PC*C*-1RPPass (Fig. 5.7b)

8 3 3 3 9 2 3 0 3 see Table 5.3

10 3 3 6 12 3 2 1 3 see Table 5.3

5 5 3 0 0 0 5 5 3 ( v1 , v 2 , 3 0 10 3 2 0 5

5 5 4 0 0 6 5 5 4 ( v1 , v 2 , 3 6 17 3 1 0 5

δ

)

fj

3

5

5

fj

15

3

10

fj

21

13

20

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

5.1 Overconstrained solutions

293

Table 5.7. Structural parametersa of overconstrained PPMs in Figs. 5.8 and 5.9 No. Structural Solution parameter 2PC*C*-1RPaP (Fig. 5.8a)

2PC*C*-1RPaPa (Fig. 5.8b)

1 2 3 4 5 6 7 8 9 10

10 3 3 6 12 3 2 1 3 see Table 5.3

12 3 3 9 15 4 2 1 3 see Table 5.3

2PR*C*C*-1RPaPa (Fig. 5.9a) 2PR*C*C*-1RPaPat (Fig. 5.9b) 14 4 4 9 17 4 2 1 3 see Table 5.3

5 5 3 0 0 3 5 5 3 ( v1 , v 2 , 3 3 13 3 5 0 5

5 5 3 0 0 6 5 5 3 ( v1 , v 2 , 3 6 16 3 8 0 5

6 6 3 0 0 6 6 6 3 ( v1 , v 2 , 3 6 18 3 6 0 6

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

δ

)

fj

5

5

6

fj

6

9

9

fj

16

19

21

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

294

5 Maximally regular planar parallel robots

Fig. 5.4. Overconstrained maximally regular PPMs of types 2PPR-1RC*C* (a) and 2PPR-1RC*Pass (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=4, limb topology P ⊥ P ⊥ ⊥ R and R ⊥ C* ⊥ ⊥ C* (a), R ⊥ C* ⊥ ||Pass (b)

5.1 Overconstrained solutions

295

Fig. 5.5. Overconstrained maximally regular PPMs of types 2PPR-1RPassC* (a) and 2PPR-1RPassPass (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=4, limb topology P ⊥ P ⊥ ⊥ R and R||Pass ⊥ C* (a), R||Pass||Pass (b)

296

5 Maximally regular planar parallel robots

Fig. 5.6. Overconstrained maximally regular PPMs of types 2PPR-1RPacsPacs (a) and 2PPR-1RPacsPatcs (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=6,

limb topology P ⊥ P ⊥ ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

5.1 Overconstrained solutions

297

Fig. 5.7. Overconstrained maximally regular PPMs of types 2PC*C*-1RPP (a) and 2PC*C*-1RPPass (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=2 (a), NF=1 (b), limb topology P ⊥ C* ⊥ ⊥ C* and R ⊥ P ⊥ ⊥ P (a), R ⊥ P ⊥ ||Pass (b)

298

5 Maximally regular planar parallel robots

Fig. 5.8. Overconstrained maximally regular PPMs of types 2PC*C*-1RPaP (a) and 2PC*C*-1RPaPa (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=5

(a), NF=8 (b), limb topology P ⊥ C* ⊥ ⊥ C* and R||Pa ⊥ P (a), R||Pa||Pa (b)

5.1 Overconstrained solutions

299

Fig. 5.9. Overconstrained maximally regular PPMs of types 2PR*C*C*-1RPaPa (a) and 2PR*C*C*-1RPaPat (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0,

NF=6, limb topology P||R ⊥ C* ⊥ ⊥ C* and R||Pa||Pa (a), R||Pa||Pat (b)

300

5 Maximally regular planar parallel robots

5.2 Non overconstrained solutions Equation (1.15) indicates that non overconstrained solutions of maximally regular planar parallel robots with q independent loops meet the condition p ∑ 1 fi = 3 + 6q along with MF=SF=3 and (RF)=( v1 ,v2 ,ωδ ).

The non overconstrained solutions of maximally regular planar parallel robots presented in this section (Figs. 5.10-5.12) are derived from overconstrained counterparts presented in Figs. 5.1-5.3 by introducing the required idle mobilities. For example, the non overconstrained solution in Fig. 5.10a is derived from the overconstrained solution in Fig. 5.1a by combining three idle mobilities in each limb G1 and G2. They are introduced in one revolute and two cylindrical joints denoted by R* and C*. The rotational motion is the idle mobility in the first cylindrical joint and the translational motion in the second cylindrical joint of the limb. We recall that one translational and two rotational idle mobilities are introduced in one cylindrical and one spherical joint of the the parallelogram and telescopic parallelogram loops denoted by Pacs and Patcs (Figs. 5.3, 5.6, 5.9 and 5.12). In the cylindrical joints of these loops, the translational motion is an idle mobility. In the parallelogram loop Passtype four idle mobilities are combined in two spherical joints adjacent to the same link. Three idle mobilities are introduced in the loop and one outside the loop. If the link adjacent to the two spherical joints is a binary link than the idle mobility introduced outside the loop becomes an internal rotational mobility of the binary link around the axis passing by the centre of the two spherical joints. This is the case of binary links 4C (Fig. 5.11a) and 3C, 6C (Fig. 5.11b). Each internal mobility gives one degree of structural redundancy. If the link adjacent to the two spherical joints is connected in the limb by three or more joints (polinary link) than the rotational motion around the axis passing by the centre of the two spherical joints is an idle (potential) mobility of the limb. This is the case of link 6 in Fig. 5.10b. The basis of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 5.10-5.12 are given in Table 5.8. The limb topology and connecting conditions of these solutions are systematized in Table 5.9 and the structural parameters of these solutions are presented in Tables 5.10 and 5.11.

5.2 Non overconstrained solutions

301

Table 5.8. Bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 5.10-5.12 No. Parallel mechanism 1 Figs. 5.10a 5.11, 5.12 2 Figs. 5.10b

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

α,

β,

( v1 , v 2 , v 3 ,

β

δ

,

)

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

α,

β,

(RG3) ) δ ( v1 , v 2 ,

δ

)

α

β

δ

α

,

,

,

) ( v1 , v 2 ,

δ

)

Table 5.9. Limb topology of the non overconstrained maximally regular PPMs presented in Figs. 5.10-5.12 No. Basic PPM Type 1 2PPR-1RPP (Fig. 5.1a) 2 2PPR-1RPPa (Fig. 5.1b) 3 4 5 6

2PPR-1RPaP (Fig. 5.2a) 2PPR-1RPaPa (Fig. 5.2b) 2PPR-1RPaPa (Fig. 5.3a) 2PPR-1RPaPat (Fig. 5.3b)

NF 6 9

9 12 12 12

PPM with NF =0 Type 2PR*C*C*-1RPP (Fig. 5.10a) PC*C*-PR*C*C*-RPPass (Fig. 5.10b) 2PR*C*C*-1RPacsP (Fig. 5.11a) 2PR*C*C*-1RPacsPacs (Fig. 5.11b) 2PR*C*C*-1RPacsPacs (Fig. 5.12a) 2PR*C*C*-1RPacsPatcs (Fig. 5.12b)

Limb topology P||R* ⊥ C* ⊥ ⊥ C* R ⊥ P ⊥⊥ P P ⊥ C* ⊥ ⊥ C* P||R* ⊥ C* ⊥ ⊥ C* R ⊥ P ⊥ ||Pass P||R* ⊥ C* ⊥ ⊥ C* R||Pacs ⊥ P P||R* ⊥ C* ⊥ ⊥ C* R||Pacs||Pacs P||R* ⊥ C* ⊥ ⊥ C* R||Pacs||Pacs P||R* ⊥ C* ⊥ ⊥ C* R||Pacs||Patcs

302

5 Maximally regular planar parallel robots

Table 5.10. Structural parametersa of non overconstrained PPMs in Fig. 5.10 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 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

Solution 2PR*C*C*-1RPP (Fig. 5.10a) 10 4 4 3 11 2 3 0 3 see Table 5.8

PC*C*-PR*C*C*-RPPass (Fig. 5.10b) 11 3 4 6 13 3 2 1 3 see Table 5.8

6 6 3 0 0 0 6 6 3 ( v1 , v 2 , 3 0 12 3 0 0 6

5 6 4 0 0 6 5 6 3 ( v1 , v 2 , 3 6 18 3 0 0 5

δ

)

fj

6

6

fj

3

10

fj

15

21

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

5.2 Non overconstrained solutions

303

Table 5.11. Structural parametersa of non overconstrained PPMs in Figs. 5.11 and 5.12 No. Structural Solution parameter 2PR*C*C*1RPassP (Fig. 5.11a) 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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

2PR*C*C*-1RPassPass 2PR*C*C*-1RPacsPacs (Fig. 5.11b) (Fig. 5.12a) 2PPR-1RPacsPatcs (Fig. 5.12b) 12 14 14 4 4 4 4 4 4 6 9 9 14 17 17 3 4 4 2 2 2 1 1 1 3 3 3 see Table 5.8 see Table 5.8 see Table 5.8 6 6 3 0 0 6 6 6 4 ( v1 , v 2 , 3 6 18 4 0 1 6

δ

)

6 6 3 0 0 12 6 6 5 ( v1 , v 2 , 3 6 24 5 0 2 6

δ

)

6 6 3 0 0 12 6 6 3 ( v1 , v 2 , 3 6 24 3 0 0 6

fj

6

6

6

fj

10

17

15

fj

22

29

27

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

304

5 Maximally regular planar parallel robots

Fig. 5.10. Non overconstrained maximally regular PPMs of types 2PR*C*C*1RPP (a) and PC*C*-PR*C*C*-RPPass (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology P||R* ⊥ C* ⊥ ⊥ C*, R ⊥ P ⊥ ⊥ P (a)

and P ⊥ C* ⊥ ⊥ C*, P||R* ⊥ C* ⊥ ⊥ C*, R ⊥ P ⊥ ||Pass (b)

5.2 Non overconstrained solutions

305

Fig. 5.11. Non overconstrained maximally regular PPMs of types 2PR*C*C*1RPassP (a) and 2PR*C*C*-1RPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ),

NF=0, and MF=4, TF=1 (a), MF=5, TF=2 (b), limb topology P||R* ⊥ C* ⊥ ⊥ C* and R||Pass ⊥ P (a), R||Pass||Pass (b)

306

5 Maximally regular planar parallel robots

Fig. 5.12. Non overconstrained maximally regular PPMs of types 2PR*C*C*1RPacsPacs (a) and 2PR*C*C*-1RPacsPatcs (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology P||R* ⊥ C* ⊥ ⊥ C* and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

6 Spatial PMs with coupled planar motion of the moving platform

The solutions of spatial parallel mechanism with planar motion of the moving platform have in their structure at least one spatial limb. In the general case, in a spatial parallel robotic manipulator (SPM) with coupled planar motions of the moving platform each operational velocity depends in the general case on three actuated joint velocities. In this section we focus on the solutions with decoupled rotation of the moving platform & 2 ) , v2 = v2 ( &q1 ,q & 2 ) and ωδ = ωδ ( &q3 ) . In these solutions, with v1 = v1 ( &q1 ,q the Jacobian matrix in Eq. (6.1) is not triangular and the parallel robot is considered with coupled motions. They have just a few partially decoupled motions.

6.1 Overconstrained solutions The overconstrained solutions of SPMs with coupled planar motions of the moving platform and q independent loops meet the condition p ∑ 1 fi < 3 + 6q . The limbs can be simple or complex kinematic chains and

the actuators can be mounted on the fixed base or on a moving link. Basic and derived fully-parallel solutions are presented in this section. 6.1.1 Basic solutions

In the basic fully-parallel solutions of SPMs with coupled motions F ← G1G2-G3 presented in this section, the moving platform nznGi (i=1, 2, 3) is connected to the reference platform 1z1Giz0 by three limbs. One actuator is combined in a revolute or prismatic pair of each limb. Two identical planar limbs G1 and G2 are used for positioning the moving platform and a spatial limb G3 for orienting it. There are no idle mobilities in these basic solutions.

G. Gogu, Structural Synthesis of Parallel Robots, Solid Mechanics and Its Applications 173, DOI 10.1007/978-90-481-9831-3_6, © Springer Science + Business Media B.V. 2010

307

308 6 Spatial PMs with coupled planar motion of the moving platform

The various types of simple and complex planar limbs with three degrees of connectivity used in the basic solutions illustrated in this section are presented in Figs. 2.1a-c and 2.2a-c. The simple limbs combine only revolute and prismatic joints. One (Fig. 2.2a,c) or two (Figs. 2.2b) planar parallelogram loops are combined in these planar complex limbs. The various types of simple and complex spatial limbs used in the basic solutions illustrated in this section are presented in Fig. 6.1. The simple limb in Fig. 6.1a has six degrees of connectivity and combines just revolute and prismatic joints in a double universal joint with telescopic intermediary shaft. Three planar parallelogram loops are combined in the spatial complex limbs with four degrees of connectivity in Fig. 6.1band c. Various solutions of SPMs with coupled planar motion of the moving platform and no idle mobilities can be obtained by using G1- and G2-limbs with identical or different topology presented in Figs. 2.1a-c and 2.2a-c and a spatial G3-limb in Fig. 6.1. Only solutions with identical G1- and G2-limbs are illustrated in Figs. 6.2-6.16. The revolute joints in G1- and G2-limbs have parallel axes and the directions of the prismatic joints are parallel to a plane perpendicular to the rotation axes of the revolute joints. The actuated revolute joint in G3-limb is perpendicular to the translation plane of the moving platform. The actuators are mounted on the fixed base excepting the solutions in Figs. 6.3, 6.10 and 6.11 in which the linear actuators are mounted on a moving link. The limb topology and connecting conditions of the solutions in Figs. 6.2-6.16 are systematized in Table 6.1, as are their structural parameters in Tables 6.2 and 6.3. The solutions in Figs. 6.2-6.16 give a decoupled and unlimited rotational motion of the moving platform.

6.1 Overconstrained solutions

309

Fig. 6.1. Simple (a) and complex limbs (b-c) with MG=SG=6 (a) and MG=SG=4 (b) for SPMs with planar motion of the moving platform

310 6 Spatial PMs with coupled planar motion of the moving platform Table 6.1. Limb topology and connecting conditions of the overconstrained SPM with no idle mobilities presented in Figs. 6.2-6.16 No. SPM type 1

2PRR-1RRRPRR (Fig. 6.2a)

2

2RRR-1RRRPRR (Fig. 6.2b)

3

2RPR-1RRRPRR (Fig. 6.3)

4

2PPaR-1RRRPRR (Fig. 6.4)

5

2PaRR-1RRRPRR (Fig. 6.5a)

6

2PaPaR-1RRRPRR (Fig. 6.5b)

7

2PRR-1RPaPaPa (Fig. 6.6a)

8

2PRR-1RPaPaPa (Fig. 6.6b)

9

2PRR-1RPaPaPa (Fig. 6.7)

Limb topology P ⊥ R||R (Fig. 2.1c) R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R (Fig. 6.1a)

R||R||R (Fig. 2.1a) R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R (Fig. 6.1a) R ⊥ P ⊥ ||R (Fig. 2.1b) R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R (Fig. 6.1a) P ⊥ Pa||R (Fig. 2.2c) R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R (Fig. 6.1a) Pa||R||R (Fig. 2.2a) R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R (Fig. 6.1a) Pa||Pa||R (Fig. 2.2b) R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R (Fig. 6.1a) P ⊥ R||R (Fig. 2.1c) R ⊥ Pa ⊥ ⊥ Pa ⊥ ||Pa (Fig. 6.1b) P ⊥ R||R (Fig. 2.1c) R ⊥ Pa ⊥ ⊥ Pa||Pa (Fig. 6.1c) P ⊥ R||R (Fig. 2.1c) R ⊥ Pa||Pa ⊥ ⊥ Pa (Fig. 6.1d)

Connecting conditions The directions of the revolute joints of limbs G1 and G2 are parallel. The last revolute joints of limbs G1 and G2 have superposed axes. Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

6.1 Overconstrained solutions Table 6.1. (cont.) 10

2RRR-1RPaPaPa (Fig. 6.8)

11

2RRR-1RPaPaPa (Fig. 6.9a)

12

2RRR-1RPaPaPa (Fig. 6.9b)

13

2RPR-1RPaPaPa (Fig. 6.10a)

14

2RPR-1RPaPaPa (Fig. 6.10b)

15

2RPR-1RPaPaPa (Fig. 6.11)

16

2PPaR-1RPaPaPa (Fig. 6.12)

17

2PPaR-1RPaPaPa (Fig. 6.13a)

18

2PPaR-1RPaPaPa (Fig. 6.13b)

19

2PaRR-1RPaPaPa (Fig. 6.14a)

R||R||R (Fig. 2.1a) R ⊥ Pa ⊥ ⊥ Pa ⊥ ||Pa (Fig. 6.1b) R||R||R (Fig. 2.1a) R ⊥ Pa ⊥ ⊥ Pa||Pa (Fig. 6.1c) R||R||R (Fig. 2.1a) R ⊥ Pa||Pa ⊥ ⊥ Pa (Fig. 6.1d) R ⊥ P ⊥ ||R (Fig. 2.1b) R ⊥ Pa ⊥ ⊥ Pa ⊥ ||Pa (Fig. 6.1b) R ⊥ P ⊥ ||R (Fig. 2.1b) R ⊥ Pa ⊥ ⊥ Pa||Pa (Fig. 6.1c) R ⊥ P ⊥ ||R (Fig. 2.1b) R ⊥ Pa||Pa ⊥ ⊥ Pa (Fig. 6.1d) P ⊥ Pa||R (Fig. 2.2c) R ⊥ Pa ⊥ ⊥ Pa ⊥ ||Pa (Fig. 6.1b) P ⊥ Pa||R (Fig. 2.2c) R ⊥ Pa ⊥ ⊥ Pa||Pa (Fig. 6.1c) P ⊥ Pa||R (Fig. 2.2c) R ⊥ Pa||Pa ⊥ ⊥ Pa (Fig. 6.1d) Pa||R||R (Fig. 2.2a) R ⊥ Pa ⊥ ⊥ Pa ⊥ ||Pa (Fig. 6.1b)

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

311

312 6 Spatial PMs with coupled planar motion of the moving platform Table 6.1. (cont.) 20

2PaRR-1RPaPaPa (Fig. 6.14b)

21

2PaRR-1RPaPaPa (Fig. 6.15a)

22

2PaPaR-1RPaPaPa (Fig. 6.15b)

23

2PaPaR-1RPaPaPa (Fig. 6.16a)

24

2PaPaR-1RPaPaPa (Fig. 6.16b)

Pa||R||R (Fig. 2.2a) R ⊥ Pa ⊥ ⊥ Pa ||Pa (Fig. 6.1c) Pa||R||R (Fig. 2.2a) R ⊥ Pa||Pa ⊥ ⊥ Pa (Fig. 6.1d) Pa||Pa||R (Fig. 2.2b) R ⊥ Pa ⊥ ⊥ Pa ⊥ ||Pa (Fig. 6.1b) Pa||Pa||R (Fig. 2.2a) R ⊥ Pa ⊥ ⊥ Pa||Pa (Fig. 6.1c) Pa||Pa||R (Fig. 2.2a) R ⊥ Pa||Pa ⊥ ⊥ Pa (Fig. 6.1d)

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

6.1 Overconstrained solutions

313

Table 6.2. Structural parametersa of spatial parallel mechanisms in Figs. 6.2-6.5 No. Structural Solution parameter 2PRR-1RRRPRR 2RRR-1RRRPRR (Fig. 6.2a,b) 2RPR-1RRRPRR (Fig. 6.3) 1 m 11 2 p1 3 3 p2 3 4 p3 6 5 p 12 6 q 2 7 k1 3 8 k2 0 9 k 3 10 (RG1) ( v1 , v 2 , δ ) 11 (RG2) ( v1 , v 2 , δ ) 12

(RG3)

( v1 , v 2 , v 2 ,

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

3 3 6 0 0 0 3 3 6 ( v1 , v 2 , 3 0 9 3 3 0 3

30 31 32 a

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

α,

)

β,

δ

2PPaR-1RRRPRR (Fig. 6.4) 2PaRR-1RRRPRR (Fig. 6.5a)

2PaPaR-1RRRPRR (Fig. 6.5b)

15 6 6 6 18 4 1 2 3 ( v1 , v2 , ( v1 , v2 ,

19 9 9 6 24 6 1 2 3 ( v1 , v2 , ( v1 , v2 ,

δ δ

) ) α,

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

δ

)

β,

δ

δ δ

) ) α

)( v1 , v2 , v2 , 3 3 6 6 6 0 3 3 6 ( v1 , v2 , 3 12 21 3 15 0 9

fj

3

6

9

fj

6

6

6

fj

12

18

24

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

,

β

,

δ

)

314 6 Spatial PMs with coupled planar motion of the moving platform Table 6.3. Structural parametersa of spatial parallel mechanisms in Figs. 6.6-6.16 No. Structural Solution parameter 2PRR-1RPaPaPa 2RRR-1RPaPaPa 2RPR-1RPaPaPa (Figs. 6.6-6.11) 1 2 3 4 5 6 7 8 9 10

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

11

(RG2)

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 a

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

15 3 3 13 19 5 2 1 3 ( v1 , v 2 , ( v1 , v 2 ,

δ δ

) )

2PPaR-1RPaPaPa 2PaRR-1RPaPaPa (Figs. 6.12-6.14, 6.15a)

2PaPaR-1RPaPaPa (Figs. 6.15b, 6.16)

19 6 6 13 25 7 0 3 3 ( v1 , v2 , ( v1 , v2 ,

23 9 9 13 31 9 0 3 3 ( v1 , v2 , ( v1 , v2 ,

δ δ

) )

δ δ

) )

( v1 , v 2 , v 2 , δ ) 3 3 4 0 0 9 3 3 4 ( v1 , v 2 , δ ) 3 9 16 3 14 0 3

( v1 , v 2 , v 2 , δ ) 3 3 4 3 3 9 3 3 4 ( v1 , v2 , δ ) 3 15 22 3 20 0 6

( v1 ,v2 , v2 , δ ) 3 3 4 6 6 9 3 3 4 ( v1 , v2 , δ ) 3 21 28 3 26 0 9

fj

3

6

9

fj

13

13

13

fj

19

25

31

fj

See footnote of Table 2.4 for the nomenclature of structural parameters

6.1 Overconstrained solutions

315

Fig. 6.2. Overconstrained SPMs with planar motion of the moving platform of types 2PRR-1RRRPRR (a) and 2RRR-1RRRPRR (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3, limb topology R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R and P ⊥ R||R (a), R||R||R (b)

316 6 Spatial PMs with coupled planar motion of the moving platform

Fig. 6.3. 2RPR-1RRRPRR-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3, limb topology R ⊥ P ⊥ ||R and R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R

Fig. 6.4. 2PPaR-1RRRPRR-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=9, limb topology P ⊥ Pa||R and R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R

6.1 Overconstrained solutions

317

Fig. 6.5. Overconstrained SPMs with planar motion of the moving platform of types 2PaRR-1RRRPRR (a) and 2PaPaR-1RRRPRR (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=9 (a), NF=15 (b), limb topology R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R and Pa||R||R (a), Pa||Pa||R (b)

318 6 Spatial PMs with coupled planar motion of the moving platform

Fig. 6.6. 2PRR-1RPaPaPa-type overconstrained SPMs with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14, limb

topology P ⊥ R||R and R ⊥ Pa ⊥ ⊥ Pa ⊥ ||Pa (a), R ⊥ Pa ⊥ ⊥ Pa||Pa (b)

6.1 Overconstrained solutions

319

Fig. 6.7. 2PRR-1RPaPaPa-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14, limb

topology P ⊥ R||R and R ⊥ Pa||Pa ⊥ ⊥ Pa

Fig. 6.8. 2RRR-1RPaPaPa-type overconstrained SPMs with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14, limb

topology R||R||R and R ⊥ Pa ⊥ ⊥ Pa ⊥ ||Pa

320 6 Spatial PMs with coupled planar motion of the moving platform

Fig. 6.9. 2RRR-1RPaPaPa-type overconstrained SPMs with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14, limb

topology R||R||R and R ⊥ Pa ⊥ ⊥ Pa||Pa (a), R ⊥ Pa||Pa ⊥ ⊥ Pa (b)

6.1 Overconstrained solutions

321

Fig. 6.10. 2RPR-1RPaPaPa-type overconstrained SPMs with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14, limb

topology R ⊥ P ⊥ ||R and R ⊥ Pa ⊥ ⊥ Pa ⊥ ||Pa (a), R ⊥ Pa ⊥ ⊥ Pa||Pa (b)

322 6 Spatial PMs with coupled planar motion of the moving platform

Fig. 6.11. 2RPR-1RPaPaPa-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14, limb

topology R ⊥ P ⊥ ||R and R ⊥ Pa||Pa ⊥ ⊥ Pa

Fig. 6.12. 2PPaR-1RPaPaPa-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=20, limb

topology P ⊥ Pa||R and R ⊥ Pa ⊥ ⊥ Pa ⊥ ||Pa

6.1 Overconstrained solutions

323

Fig. 6.13. 2PPaR-1RPaPaPa-type overconstrained SPMs with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=20, limb topology P ⊥ Pa||R and R ⊥ Pa ⊥ ⊥ Pa||Pa (a), R ⊥ Pa||Pa ⊥ ⊥ Pa (b)

324 6 Spatial PMs with coupled planar motion of the moving platform

Fig. 6.14. 2PaRR-1RPaPaPa-type overconstrained SPMs with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=20, limb topology Pa||R||R and R ⊥ Pa ⊥ ⊥ Pa ⊥ ||Pa (a), R ⊥ Pa ⊥ ⊥ Pa||Pa (b)

6.1 Overconstrained solutions

325

Fig. 6.15. Overconstrained SPMs with planar motion of the moving platform pf types 2PaRR-1RPaPaPa (a) and 2PaPaR-1RPaPaPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=20 (a), NF=26 (b), limb topology Pa||R||R and R ⊥ Pa||Pa ⊥ ⊥ Pa (a), Pa||Pa||R and R ⊥ Pa ⊥ ⊥ Pa ⊥ ||Pa (b)

326 6 Spatial PMs with coupled planar motion of the moving platform

Fig. 6.16. 2PaPaR-1RPaPaPa-type overconstrained SPMs with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=26, limb topology Pa||Pa||R and R ⊥ Pa ⊥ ⊥ Pa||Pa (a), R ⊥ Pa||Pa ⊥ ⊥ Pa (b)

6.1 Overconstrained solutions

327

6.1.2 Derived solutions Solutions with lower degrees of overconstraint can be derived from the basic solutions in Figs. 6.2-6.16 by using joints with idle mobilities. One or two idle mobilities can be introduced in the solutions in Figs. 6.2 and 6.3a. A large set of solutions can be obtained by introducing up to three idle mobilities in each planar parallelogram loop and up to three idle mobilities outside the planar loops combined in the limbs in Figs. 6.4b and 6.5. Up to five idle mobilities can be introduced outside the planar loops combined in the limbs in Figs. 6.6-6.16. Examples of solutions with 1-7 overconstraints derived from the basic solutions in Figs. 6.2-6.16 are illustrated in Figs. 6.17-6.31. We recall that the joints combining idle mobilities are denoted by an asterisk. The idle mobilities which can be combined in a parallelogram loop are systematized in Fig. 1.2 and Table 1.1. The rotational mobility of the revolute joint denoted by R* is an idle mobility. One idle mobility is combined in each cylindrical joint C* and two idle mobilities in each spherical joint S*. For example, in the cylindrical joint denoted by C* in Figs. 6.216.31, the idle mobility is the rotational motion. In the parallelogram loop Pass-type, three idle mobilities are introduced in the loop and one outside the loop. The idle mobility introduced outside the parallelogram loop becomes an internal mobility of the links adjacent to the two spherical joints – see for example links 3A and 3B in Figs. 6.20a, 6.29-6.31. In these cases, the parallel robot has two degrees of structural redundancy. Table 6.4. Bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 6.17-6.31 No. Parallel mechanism 1 Figs. 6.17, 6.18 2 Figs. 6.19, 6.20b 3 Fig. 6.20a 4 5

6

Basis (RG1) ( v1 , v 2 ,

α,

β ,

( v1 , v 2 ,

β

,

δ

( v1 , v 2 ,

δ

Figs. ( v1 , v 2 , 6.21-6.26 Figs. ( v1 , v 2 , 6.27, 6.28 6.30b, 6.31 Figs. 6.29, ( v1 , v2 , 6.30a

(RG2) ) ( v1 , v2 , δ

δ

)

( v1 , v 2 ,

α

,

)

( v1 , v 2 ,

δ

)

δ

)

( v1 , v 2 , v 3 ,

α

,

β

,

β

,

( v1 , v 2 , v 3 ,

α

,

δ

)

δ

)

( v1 , v 2 , v 3 ,

δ

)

δ

)

)

δ

)

δ

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

α

,

β

,

δ

)

( v1 , v 2 , v 3 ,

α

,

β

,

δ

)

( v1 , v 2 , v 3 ,

α

,

β

,

δ

)

) ( v1 , v 2 , v 3 ,

α

,

β

,

δ

)

( v1 , v 2 , v 3 ,

α

,

β

,

δ

)

( v1 , v 2 , v 3 ,

α

,

β

,

δ

)

328 6 Spatial PMs with coupled planar motion of the moving platform Table 6.5. Limb topology and the number of overconstraints NF of the derived SPMs with idle mobilities presented in Figs. 6.17-6.31 No. Basic SPM Type 1 2PRR-1RRRPRR (Fig. 6.2a)

Derived SPM NF Type 3 PS*R-PRR-RRRPRR (Fig. 6.17a)

2

2RRR-1RRRPRR (Fig. 6.2b)

3

RS*R-RRR-RRRPRR (Fig. 6.17b)

3

2RPR-1RRRPRR (Fig. 6.3)

3

S*PR-RPR-RRRPRR (Fig. 6.18)

4

2PPaR-1RRRPRR (Fig. 6.4) 2PaRR-1RRRPRR (Fig. 6.5a) 2PaPaR-1RRRPRR (Fig. 6.5b) 2PRR-1RPaPaPa (Fig. 6.6a)

9

2PPassR-1RRRPRR (Fig. 6.19) 2PassRR-1RRRPRR (Fig. 6.20a) 2PaPassR-1RRRPRR (Fig. 6.20b) PRR-PS*C*RPaPassPass (Fig. 6.21a) PRR-PS*C*RPassPaPass (Fig. 6.21b) PRR-PS*C*RPaPassPass (Fig. 6.22) RRR-RS*C*RPaPassPass (Fig. 6.23) RRR-RS*C*RPassPaPass (Fig. 6.24a) RRR-RS*C*RPaPassPass (Fig. 6.24b) RPR-S*PC*RPaPassPass (Fig. 6.25a) RPR-S*PC*RPassPaPass (Fig. 6.25b)

5 6 7

9 15 14

8

2PRR-1RPaPaPa (Fig. 6.6b)

14

9

2PRR-1RPaPaPa (Fig. 6.7)

14

10 2RRR-1RPaPaPa (Fig. 6.8)

14

11 2RRR-1RPaPaPa (Fig. 6.9a)

14

12 2RRR-1RPaPaPa (Fig. 6.9b)

14

13 2RPR-1RPaPaPa (Fig. 6.10a)

14

14 2RPR-1RPaPaPa (Fig. 6.10b)

14

NF Limb topology 1 PS*R P ⊥ R||R R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R 1 RS*R R||R||R R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R 1 SP ⊥ R R ⊥ P ⊥ ||R R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R 1 P ⊥ Pass||R R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R 3 Pass||R||R R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R 7 Pa||Pass||R R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R 3 P ⊥ R||R PS*C* R ⊥ Pa ⊥ ⊥ Pass ⊥ ||Pass 3 P ⊥ R||R PS*C* R ⊥ Pass ⊥ ⊥ Pa||Pass 3 P ⊥ R||R PS*C* R ⊥ Pa||Pass ⊥ ⊥ Pass 3 R||R||R RS*C* R ⊥ Pa ⊥ ⊥ Pass ⊥ ||Pass 3 R||R||R RS*C* R ⊥ Pass ⊥ ⊥ Pa||Pass 3 R||R||R RS*C* R ⊥ Pa||Pass ⊥ ⊥ Pass 3 R ⊥ P ⊥ ||R S*P ⊥ C* R ⊥ Pa ⊥ ⊥ Pass ⊥ ||Pass 3 R ⊥ P ⊥ ||R S*P ⊥ C* R ⊥ Pass ⊥ ⊥ Pa||Pass

6.1 Overconstrained solutions Table 6.5. (cont.) 14 RPR-S*PC*RPaPassPass (Fig. 6.26) 2PPaR-1RPaPaPa 20 PPassR-PPassC*(Fig. 6.12) RPaPassPass (Fig. 6.27) 2PPaR-1RPaPaPa 20 PPassR-PPassC*(Fig. 6.13a) RPassPaPass (Fig. 6.28a) 2PPaR-1RPaPaPa 20 PPassR-PPassC*(Fig. 6.13b) RPaPassPass (Fig. 6.28b) 2PaRR-1RPaPaPa 20 PassRR-PassRC*(Fig. 6.14a) RPaPassPass (Fig. 6.29a) 2PaRR-1RPaPaPa 20 PassRR-PassRC*(Fig. 6.14b) RPassPaPass (Fig. 6.29b) 2PaRR-1RPaPaPa 20 PassRR-PassRC*(Fig. 6.15a) RPaPassPass (Fig. 6.30a) 2PaPaR-1RPaPaPa 26 PassPassR-PassPassC*(Fig. 6.15b) RPaPassPass (Fig. 6.30b) 2PaPaR-1RPaPaPa 26 PassPassR-PassPassC*(Fig. 6.16a) RPassPaPass (Fig. 6.31a) 2PaPaR-1RPaPaPa 26 PassPassR-PassPassC*(Fig. 6.16b) RPaPassPass (Fig. 6.31b)

15 2RPR-1RPaPaPa (Fig. 6.11)

3

16

3

17

18

19

20

21

22

23

24

3

3

5

5

5

3

3

3

329

R ⊥ P ⊥ ||R SP ⊥ C* R ⊥ Pa||Pass ⊥ ⊥ Pass P ⊥ Pass||R P ⊥ Pass||C* R ⊥ Pa ⊥ ⊥ Pass ⊥ ||Pass P ⊥ Pass||R P ⊥ Pass||C* R ⊥ Pass ⊥ ⊥ Pa||Pass P ⊥ Pass||R P ⊥ Pass||C* R ⊥ Pa||Pass ⊥ ⊥ Pass Pass||R||R Pass||R||C* R ⊥ Pa ⊥ ⊥ Pass ⊥ ||Pass Pass||R||R Pass||R||C* R ⊥ Pass ⊥ ⊥ Pa||Pass Pass||R||R Pass||R||C* R ⊥ Pa||Pass ⊥ ⊥ Pass Pass||Pass||R Pass||Pass||C* R ⊥ Pa ⊥ ⊥ Pass ⊥ ||Pass Pass||Pass||R Pass||Pass||C* R ⊥ Pass ⊥ ⊥ Pa||Pass Pass||Pass||R Pass||Pass||C* R ⊥ Pa||Pass ⊥ ⊥ Pass

The basis of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 6.17-6.31 are given in Table 6.4. The limb topology and the number of overconstraints of these solutions are systematized in Table 6.5, as are their structural parameters in Tables 6.6-6.9.

330 6 Spatial PMs with coupled planar motion of the moving platform Table 6.6. Structural parametersa of spatial parallel mechanisms in Figs. 6.17-6.19 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

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

Solution PS*R-PRR-RRRPRR (Fig. 6.17a) RS*R-RRR-RRRPRR (Fig. 6.17b) S*PR-RPR-RRRPRR (Fig. 6.18) 11 3 3 6 12 2 3 0 3 see Table 6.4

15 6 6 6 18 4 1 2 3 see Table 6.4

5 3 6 0 0 0 5 3 6 ( v1 , v 2 , 3 0 11 3 1 0 5

4 4 6 6 6 0 4 4 6 ( v1 , v 2 , 3 12 23 3 1 0 10

δ

)

2PPassR-1RRRPRR (Fig. 6.19)

fj

3

10

fj

6

6

fj

14

26

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

6.1 Overconstrained solutions

331

Table 6.7. Structural parametersa of spatial parallel mechanisms in Figs. 6.20-6.25 No. Structural Solution parameter 2PassRR1RRRPRR (Fig. 6.20a)

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

m p1 p2 p3 p q k1 k2 k (RGi) SG1 SG2 SG3 rG1 rG2 rG3 MG1 MG2 MG3 (RF)

21 22 23 24 25 26 27

SF rl rF MF NF TF

28 29 30 a

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

2PaPassR- PRR-PS*C*-RPaPassPass (Fig. 6.21a) 1RRRPRR PRR-PS*C*-RPassPaPass (Fig. 6.21b) (Fig. 6.20b) PRR-PS*C*-RPaPassPass (Fig. 6.22) RRR-RS*C*-RPaPassPass (Fig. 6.23) RRR-RS*C*-RPassPaPass (Fig. 6.24a) RRR-RS*C*-RPaPassPass (Fig. 6.24b) RPR-S*PC*-RPaPassPass (Fig. 6.25a) RPR-S*PC*-RPassPaPass (Fig. 6.25b) 15 19 15 6 9 3 6 9 3 6 6 13 18 24 19 4 6 5 1 1 2 2 2 1 3 3 3 see Table 6.4 see Table 6.4 see Table 6.4 3 4 3 3 4 6 6 6 6 6 9 0 6 9 0 0 0 15 4 4 3 4 4 6 6 6 6 ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) 3 3 3 12 18 15 21 29 27 5 3 3 3 7 3 2 0 0 10 13 3

fj

10

13

6

fj

6

6

21

fj

26

32

30

See footnote of Table 2.4 for the nomenclature of structural parameters

332 6 Spatial PMs with coupled planar motion of the moving platform Table 6.8. Structural parametersa of spatial parallel mechanisms in Figs. 6.26-6.28 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 RPR-S*PC*-RPaPassPass (Fig. 6.26)

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

15 3 3 13 19 5 2 1 3 see Table 6.4

PPassR-PPassC*-RPaPassPass (Fig. 6.27) PPassR-PPassC*-RPassPaPass (Fig. 6.28a) PPassR-PPassC*-RPaPassPass (Fig. 6.28b) 19 6 6 13 25 7 0 3 3 see Table 6.4

3 6 6 0 0 15 3 6 6 ( v1 , v 2 , 3 15 27 3 3 0 3

4 5 6 6 6 15 4 5 6 ( v1 , v 2 , 3 27 39 3 3 0 10

SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

fj

6

11

fj

21

21

fj

30

42

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

6.1 Overconstrained solutions

333

Table 6.9. Structural parametersa of spatial parallel mechanisms in Figs. 6.29-6.31 No. Structural Solution parameter PassRR-PassRC*-RPaPassPass (Fig. 6.29a) PassRR-PassRC*-RPassPaPass (Fig. 6.29b) PassRR-PassRC*-RPaPassPass (Fig. 6.30a) 1 m 19 2 p1 6 3 p2 6 4 p3 13 5 p 25 6 q 7 7 k1 0 8 k2 3 9 k 3 10 (RGi) see Table 6.4 (i=1,2,3) 11 SG1 3 12 SG2 4 13 SG3 6 14 rG1 6 15 rG2 6 16 rG3 15 17 MG1 4 18 MG2 5 19 MG3 6 20 (RF) ( v1 , v 2 , δ ) 21 SF 3 22 rl 27 23 rF 37 24 MF 5 25 NF 5 26 TF 2 p1 27 10 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

PassPassR-PassPassC*-RPaPassPass (Fig. 6.30b) PassPassR-PassPassC*-RPassPaPass (Fig. 6.31a) PassPassR-PassPassC*-RPaPassPass (Fig. 6.31b) 23 9 9 13 31 9 0 3 3 see Table 6.4 4 5 6 12 12 15 5 6 6 ( v1 , v 2 , 3 39 51 5 3 2 17

δ

)

j

fj

11

18

fj

21

21

fj

42

56

See footnote of Table 2.4 for the nomenclature of structural parameters

334 6 Spatial PMs with coupled planar motion of the moving platform

Fig. 6.17. Overconstrained SPMs with planar motion of the moving platform of types PS*R-PRR-RRRPRR (a) and RS*R-RRR-RRRPRR (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=1, limb topology R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R and PS*R, P ⊥ R||R (a), RS*R, R||R||R (b)

6.1 Overconstrained solutions

335

Fig. 6.18. S*PR-RPR-RRRPRR-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=1, limb topology SP ⊥ R, R ⊥ P ⊥ ||R and R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R

Fig. 6.19. 2PPassR-1RRRPRR-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=1, limb topology P ⊥ Pass||R and R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R

336 6 Spatial PMs with coupled planar motion of the moving platform

Fig. 6.20. Overconstrained SPMs with planar motion of the moving platform of types 2PassRR-1RRRPRR (a) and 2PaPassR-1RRRPRR (b) defined by MF=5, SF=3, (RF)=( v1 , v2 , δ ), TF=2, NF=3 (a), MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=7 (b), limb topology R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R and Pass||R||R (a), Pa||Pass||R (b)

6.1 Overconstrained solutions

337

Fig. 6.21. Overconstrained SPMs with planar motion of the moving platform of types PRR-PS*C*-RPaPassPass (a) and PRR-PS*C*-RPassPaPass (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3, limb topology P ⊥ R||R, PS*C* and R ⊥ Pa ⊥ ⊥ Pass ⊥ ||Pass (a), R ⊥ Pass ⊥ ⊥ Pa||Pass (b)

338 6 Spatial PMs with coupled planar motion of the moving platform

Fig. 6.22. PRR-PS*C*-RPaPassPass-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0,

NF=3, limb topology P ⊥ R||R, PS*C* and R ⊥ Pa||Pass ⊥ ⊥ Pass

Fig. 6.23. RRR-RS*C*-RPaPassPass-type overconstrained SPMs with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0,

NF=3, limb topology R||R||R, RS*C* and R ⊥ Pa ⊥ ⊥ Pass ⊥ ||Pass

6.1 Overconstrained solutions

339

Fig. 6.24. Overconstrained SPMs with planar motion of the moving platform of types RRR-RS*C*-RPassPaPass (a) and RRR-RS*C*-RPaPassPass (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3, limb topology R||R||R, RS*C* and R ⊥ Pass ⊥ ⊥ Pa||Pass (a), R ⊥ Pa||Pass ⊥ ⊥ Pass (b)

340 6 Spatial PMs with coupled planar motion of the moving platform

Fig. 6.25. Overconstrained SPMs with planar motion of the moving platform of types RPR-S*PC*-RPaPassPass (a) and RPR-S*PC*-RPassPaPass (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3, limb topology R ⊥ P ⊥ ||R, S*P ⊥ C* and R ⊥ Pa ⊥ ⊥ Pass ⊥ ||Pass (a), R ⊥ Pass ⊥ ⊥ Pa||Pass (b)

6.1 Overconstrained solutions

341

Fig. 6.26. RPR-S*PC*-RPaPassPass-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0,

NF=3, limb topology R ⊥ P ⊥ ||R, SP ⊥ C* and R ⊥ Pa||Pass ⊥ ⊥ Pass

Fig. 6.27. PPassR-PPassC*-RPaPassPass-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0,

NF=3, limb topology P ⊥ Pass||R, P ⊥ Pass||C* and R ⊥ Pa ⊥ ⊥ Pass ⊥ ||Pass

342 6 Spatial PMs with coupled planar motion of the moving platform

Fig. 6.28. Overconstrained SPMs with planar motion of the moving platform of types PPassR-PPassC*-RPassPaPass (a) and PPassR-PPassC*-RPaPassPass (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3, limb topology P ⊥ Pass||R,

P ⊥ Pass||C* and R ⊥ Pass ⊥ ⊥ Pa||Pass (a), R ⊥ Pa||Pass ⊥ ⊥ Pass (b)

6.1 Overconstrained solutions

343

Fig. 6.29. Overconstrained SPMs with planar motion of the moving platform of types PassRR-PassRC*-RPaPassPass (a) and PassRR-PassRC*-RPassPaPass (b) defined by MF=5, SF=3, (RF)=( v1 , v2 , δ ), TF=2, NF=5, limb topology Pass||R||R,

Pass||R||C* and R ⊥ Pa ⊥ ⊥ Pass ⊥ ||Pass (a), R ⊥ Pass ⊥ ⊥ Pa ||Pass (b)

344 6 Spatial PMs with coupled planar motion of the moving platform

Fig. 6.30. Overconstrained SPMs with planar motion of the moving platform pf types PassRR-PassRC*-RPaPassPass (a), PassPassR-PassPassC*-RPaPassPass (b) defined by MF=5, SF=3, (RF)=( v1 , v2 , δ ), TF=2, NF=5 (a), NF=3 (b), limb

topology Pass||R||R, Pass||R||C*, R ⊥ Pa||Pass ⊥ ⊥ Pass (a) and Pass||Pass||R, Pass||Pass||C*, R ⊥ Pa ⊥ ⊥ Pass ⊥ ||Pass (b)

6.1 Overconstrained solutions

345

Fig. 6.31. Overconstrained SPMs with planar motion of the moving platform of types PassPassR-PassPassC*-RPassPaPass (a), PassPassR-PassPassC*-RPaPassPass (b) defined by MF=5, SF=3, (RF)=( v1 , v2 , δ ), TF=2, NF=3, limb topology Pass||Pass||R, Pass||Pass||C* and R ⊥ Pass ⊥ ⊥ Pa||Pass (a), R ⊥ Pa||Pass ⊥ ⊥ Pass (b)

346 6 Spatial PMs with coupled planar motion of the moving platform

6.2 Non overconstrained solutions Equation (1.15) indicates that non overconstrained solutions of spatial 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 ,ωδ ) and NF=0 (Figs. 6.32-6.46). These solutions are derived from the overconstrained counterparts presented in Figs. 6.2-6.16 by introducing the required idle mobilities. They can have the actuators mounted on the base or on a moving link. We recall that the idle mobilities can be introduced outside or inside the closed loops combined in the limbs. The notation Pass is associated with the parallelogram loops which combine four idle mobilities in two spherical joints adjacent to the same coupler link. In these cases, three idle mobilities are introduced in the loop and one outside the loop. If the link adjacent to the two spherical joints is a binary link than the idle mobility introduced outside the loop becomes an internal rotational mobility of this binary link around the axis passing by the centre of the two spherical joints. Each internal mobility gives one degree of structural redundancy (see Table 6.10). If the link adjacent to the two spherical joints is connected in the limb by three or more joints (polinary link) than the rotational motion around the axis passing by the centre of the two spherical joints is an idle (potential) mobility of the limb. This motion is restricted by the constraints of the parallel mechanism and remains just a potential mobility. For example (Fig. 6.36a) this rotational motion is internal mobility of binary link 3C and idle mobility of polinary links 8C and 11. The bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 6.32-6.46 are given in Table 6.11. The limb topology and connecting conditions of these solutions are systematized in Table 6.12, as are their structural parameters of these solutions are presented in Tables 6.13-6.16.

6.2 Non overconstrained solutions

347

Table 6.10. Links with internal mobilities and the degree of structural redundancy TF of non overconstrained SPMs with uncoupled planar motion of the moving platform No. Parallel mechanism Figure 1 2 3

4 5 6

TF

Fig. 6.35a 2 Fig. 6.35b 2 Figs. 6.36a, 6.37, 6.38, 1 6.39b, 6.40a, 6.41, 6.42, 6.43b Figs. 6.36b, 6.39a, 1 6.40b, 6.43a Figs. 6.44a, 6.45, 3 6.46b Fig. 6.44b, 6.46a 3

Link with internal rotational mobility in limb G1 G2 G3 3A 3A, 5 A -

3B -

3C

-

-

7C

3A

3B

3C

3A

3B

7C

Table 6.11. Bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 6.32-6.46 No. Parallel Basis mechanism (RG1) 1 Figs. 6.32, ( v1 , v2 , v3 , α , β , 6.33 2 Figs. 6.34, ( v1 , v2 , β , δ ) 6.35a, 6.45b, 6.46 3 Figs. 6.35b, ( v1 , v2 , δ ) 6.44, 6.45a 4 Figs. ( v1 , v 2 , δ ) 6.36-6.41 5 Figs. 6.42, ( v1 , v2 , β , δ ) 6.43

δ

(RG2) ) ( v1 , v 2 ,

δ

)

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

α

,

β

,

δ

)

( v1 , v 2 , v 3 ,

α

,

β

,

δ

)

( v1 , v 2 , v 3 ,

α

,

δ

)

( v1 , v 2 , v 3 ,

α

,

β

,

δ

) ( v1 , v 2 , v 3 ,

α

,

β

,

δ

)

( v1 , v 2 , v 3 ,

α

,

β

,

δ

) ( v1 , v 2 , v 3 ,

α

,

β

,

δ

)

( v1 , v 2 , v 3 ,

α

,

δ

)

( v1 , v 2 , v 3 ,

α

,

β

,

δ

)

348 6 Spatial PMs with coupled planar motion of the moving platform Table 6.12. Limb topology of the non overconstrained SPMs presented in Figs. 6.32-6.46 No. Basic SPM Type 1 2PRR-1RRRPRR (Fig. 6.2a)

SPM with NF =0 NF Type 3 PS*C*-PRR-RRRPRR (Fig. 6.32a)

2

2RRR-1RRRPRR (Fig. 6.2b)

3

3

2RPR-1RRRPRR (Fig. 6.3)

3

4

2PPaR-1RRRPRR 9 (Fig. 6.4)

5

2PaRR-1RRRPRR 9 (Fig. 6.5a)

6

2PaPaR-1RRRPRR 15 (Fig. 6.5b)

7

2PRR-1RPaPaPa (Fig. 6.6a)

14

8

2PRR-1RPaPaPa (Fig. 6.6b)

14

9

2PRR-1RPaPaPa (Fig. 6.7)

14

10 2RRR-1RPaPaPa (Fig. 6.8)

14

11 2RRR-1RPaPaPa (Fig. 6.9a)

14

12 2RRR-1RPaPaPa (Fig. 6.9b)

14

13 2RPR-1RPaPaPa (Fig. 6.10a)

14

Limb topology PS*C* P ⊥ R||R R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R RS*C*-RRR-RRRPRR RS*C* (Fig. 6.32b) R||R||R R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R S*PC*-RPR-RRRPRR SP ⊥ C* (Fig. 6.33) R ⊥ P ⊥ ||R R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R ss ss PPa R-PPa C*-RRRPRR P ⊥ Pass||R (Fig. 6.34) P ⊥ Pass||C* R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R ss ss Pa RR*R-Pa RRC*Pass||R ⊥ R* ⊥ ||R RRRPRR Pass||R ⊥ R* ⊥ ||C* (Fig. 6.35a) R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R ss ss ss ss Pa Pa R-Pa Pa C*Pass||Pass||R RRRPRR Pass||Pass||C* (Fig. 6.35b) R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R ss ss ss PRR-PS*C*-RPa Pa Pa P ⊥ R||R PS*C* (Fig. 6.36a) R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass ss ss ss PRR-PS*C*-RPa Pa Pa P ⊥ R||R PS*C* (Fig. 6.36b) R ⊥ Pass ⊥ ⊥ Pass||Pass ss ss ss PRR-PS*C*-RPa Pa Pa P ⊥ R||R PS*C* (Fig. 6.37) R ⊥ Pass||Pass ⊥ ⊥ Pass ss ss ss RRR-RS*C*-RPa Pa Pa R||R||R RS*C* (Fig. 6.38) R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass ss ss ss RRR-RS*C*-RPa Pa Pa R||R||R (Fig. 6.39a) RS*C* R ⊥ Pass ⊥ ⊥ Pa||Pass ss ss ss RRR-RS*C*-RPa Pa Pa R||R||R (Fig. 6.39b) RS*C* R ⊥ Pass||Pass ⊥ ⊥ Pass ss ss ss RPR-S*PC*-RPa Pa Pa R ⊥ P ⊥ ||R S*P ⊥ C* (Fig. 6.40a) R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass

6.2 Non overconstrained solutions Table 6.12. (cont.) 14 RPR-S*PC*RPassPassPass (Fig. 6.40b) 2RPR-1RPaPaPa 14 RPR-S*PC*(Fig. 6.11) RPassPassPass (Fig. 6.41) 2PPaR-1RPaPaPa 20 PPassR-PPassC*(Fig. 6.12) RPassPassPass (Fig. 6.42) 2PPaR-1RPaPaPa 20 PPassR-PPassC*(Fig. 6.13a) RPassPassPass (Fig. 6.43a) 2PPaR-1RPaPaPa 20 PPassR-PPassC*(Fig. 6.13b) RPassPassPass (Fig. 6.43b) 2PaRR-1RPaPaPa 20 PassRR-PassRR*R*C*(Fig. 6.14a) RPassPassPass (Fig. 6.44a) 2PaRR-1RPaPaPa 20 PassRR-PassRR*R*C*(Fig. 6.14b) RPassPassPass (Fig. 6.44b) 2PaRR-1RPaPaPa 20 PassRR-PassRR*R*C*(Fig. 6.15a) RPassPassPass (Fig. 6.45a) 2PaPaR-1RPaPaPa 26 PassPassR-PassPassC*(Fig. 6.15b) RPassPassPass (Fig. 6.45b) 2PaPaR-1RPaPaPa 26 PassPassR-PassPassC*(Fig. 6.16a) RPassPassPass (Fig. 6.46a) 2PaPaR-1RPaPaPa 26 PassPassR-PassPassC*(Fig. 6.16b) RPassPassPass (Fig. 6.46b)

14 2RPR-1RPaPaPa (Fig. 6.10b) 15

16

17

18

19

20

21

22

23

24

349

R ⊥ P ⊥ ||R S*P ⊥ C* R ⊥ Pass ⊥ ⊥ Pass||Pass R ⊥ P ⊥ ||R S*P ⊥ C* R ⊥ Pass||Pass ⊥ ⊥ Pass P ⊥ Pass||R P ⊥ Pass||C* R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass P ⊥ Pass||R P ⊥ Pass||C* R ⊥ Pass ⊥ ⊥ Pass||Pass P ⊥ Pass||R P ⊥ Pass||C* R ⊥ Pass||Pass ⊥ ⊥ Pass Pass||R||R Pass||R ⊥ R* ⊥ ⊥ R* ⊥ ⊥ C* R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass Pass||R||R Pass||R ⊥ R* ⊥ ⊥ R* ⊥ ⊥ C* R ⊥ Pass ⊥ ⊥ Pass||Pass Pass||R||R Pass||R ⊥ R* ⊥ ⊥ R* ⊥ ⊥ C* R ⊥ Pass||Pass ⊥ ⊥ Pass Pass||Pass||R Pass||Pass||C* R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass Pass||Pass||R Pass||Pass||C* R ⊥ Pass ⊥ ⊥ Pass||Pass Pass||Pass||R Pass||Pass||C* R ⊥ Pass||Pass ⊥ ⊥ Pass

350 6 Spatial PMs with coupled planar motion of the moving platform Table 6.13. Structural parametersa of spatial parallel mechanisms in Figs. 6.326.34 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

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

Solution PS*C*-PRR-RRRPRR (Fig. 6.32a) RS*C*-RRR-RRRPRR (Fig. 6.32b) S*PC*-RPR-RRRPRR (Fig. 6.33) 11 3 3 6 12 2 3 0 3 See Table 6.11

15 6 6 6 18 4 1 2 3 See Table 6.11

6 3 6 0 0 0 6 3 6 ( v1 , v 2 , 3 0 12 3 0 0 6

4 5 6 6 6 0 4 5 6 ( v1 , v 2 , 3 12 24 3 0 0 10

δ

)

PPassR-PPassC*-RRRPRR (Fig. 6.34)

fj

3

11

fj

6

6

fj

15

27

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

6.2 Non overconstrained solutions

351

Table 6.14. Structural parametersa of spatial parallel mechanisms in Figs. 6.356.40 No. Structural Solution parameter PassRR*RPassRRC*RRRPRR (Fig. 6.35a) 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)

p1

j =1

p2

j =1

p3

j =1

p

j =1

PRR-PS*C*-RPassPassPass (Fig. 6.36) PRR-PS*C*-RPassPassPass (Fig. 6.37) RRR-RS*C*-RPassPassPass (Fig. 6.38) RRR-RS*C*-RPassPassPass (Fig. 6.39) RPR-S*PC*-RPassPassPass (Fig. 6.40) 15 3 3 13 19 5 2 1 3 See Table 6.11

17 7 7 6 20 4 1 2 3 See Table 6.11 4 5 6 6 6 0 5 6 6 ( v1 , v 2 , δ ) 3 12 24 5 0 2 11

19 9 9 6 24 6 1 2 3 See Table 6.11 3 6 6 12 12 0 5 6 6 ( v1 , v 2 , δ ) 3 24 36 5 0 2 17

3 6 6 0 0 18 3 6 7 ( v1 , v 2 , 3 18 30 4 0 1 3

fj

12

18

6

fj

6

6

25

fj

29

41

34

SF rl rF MF NF TF

∑ ∑ ∑ ∑

PassPassRPassPassC*RRRPRR (Fig. 6.35b)

fj

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

352 6 Spatial PMs with coupled planar motion of the moving platform Table 6.15. Structural parametersa of spatial parallel mechanisms in Figs. 6.416.43 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 RPR-S*PC*-RPassPassPass (Fig. 6.41)

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

15 3 3 13 19 5 2 1 3 See Table 6.11

PPassR-PPassC*-RPassPassPass (Fig. 6.42) PPassR-PPassC*-RPassPassPass (Fig. 6.43) 19 6 6 13 25 7 0 3 3 See Table 6.11

3 6 6 0 0 18 3 6 7 ( v1 , v 2 , 3 18 30 4 0 1 3

4 5 6 6 6 18 4 5 7 ( v1 , v 2 , 3 30 42 4 0 1 10

SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

fj

6

11

fj

25

25

fj

34

46

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

6.2 Non overconstrained solutions

353

Table 6.16. Structural parametersa of spatial parallel mechanisms in Figs. 6.446.46 No. Structural Solution parameter PassRR-PassRR*R*C*RPassPassPass (Fig. 6.44) PassRR-PassRR*R*C*RPassPassPass (Fig. 6.45a) 1 m 21 2 p1 6 3 p2 8 4 p3 13 5 p 27 6 q 7 7 k1 0 8 k2 3 9 k 3 10 (RGi) See Table 6.11 (i=1,2,3) 11 SG1 3 12 SG2 6 13 SG3 6 14 rG1 6 15 rG2 6 16 rG3 18 17 MG1 4 18 MG2 7 19 MG3 7 20 (RF) ( v1 , v 2 , δ ) 21 SF 3 22 rl 30 23 rF 42 24 MF 6 25 NF 0 26 TF 3 p1 27 10 f

4 5 6 12 12 18 5 6 7 ( v1 , v 2 , 3 42 54 6 0 3 17

28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

PassPassR-PassPassC*RPassPassPass (Figs. 6.45b, 6.46)

23 9 9 13 31 9 0 3 3 See Table 6.11

fj

13

18

fj

25

25

fj

48

60

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

354 6 Spatial PMs with coupled planar motion of the moving platform

Fig. 6.32. Non overconstrained SPMs with planar motion of the moving platform of types PS*C*-PRR-RRRPRR (a) and RS*C*-RRR-RRRPRR (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R and PS*C*, P ⊥ R||R (a), RS*C*, R||R||R (b)

6.2 Non overconstrained solutions

355

Fig. 6.33. S*PC*-RPR-RRRPRR-type non overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology SP ⊥ C*, R ⊥ P ⊥ ||R and R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R

Fig. 6.34. PPassR-PPassC*-RRRPRR-type non overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology P ⊥ Pass||R, P ⊥ Pass||C* and R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R

356 6 Spatial PMs with coupled planar motion of the moving platform

Fig. 6.35. Non overconstrained SPMs with planar motion of the moving platform of types PassRR*R-PassRRC*-RRRPRR (a) and PassPassR-PassPassC*-RRRPRR (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, NF=0, TF=2, limb topology R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R and Pass||R ⊥ R* ⊥ ||R, Pass||R ⊥ R* ⊥ ||C* (a), Pass||Pass||R, Pass||Pass||C* (b)

6.2 Non overconstrained solutions

357

Fig. 6.36. PRR-PS*C*-RPassPassPass-type non overconstrained SPMs with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, NF=0,

TF=1, limb topology P ⊥ R||R, PS*C* and R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass (a), R ⊥ Pass ⊥ ⊥ Pass||Pass (b)

358 6 Spatial PMs with coupled planar motion of the moving platform

Fig. 6.37. PRR-PS*C*-RPassPassPass-type non overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, NF=0,

TF=1, limb topology P ⊥ R||R, PS*C* and R ⊥ Pass||Pass ⊥ ⊥ Pass

Fig. 6.38. RRR-RS*C*-RPassPassPass-type non overconstrained SPMs with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, NF=0,

TF=1, limb topology R||R||R, RS*C* and R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass

6.2 Non overconstrained solutions

359

Fig. 6.39. RRR-RS*C*-RPassPassPass-type non overconstrained SPMs with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, NF=0, TF=1, limb topology R ⊥ Pass||Pass ⊥ ⊥ Pass (b)

R||R||R,

RS*C*

and

R ⊥ Pass ⊥ ⊥ Pass||Pass

(a),

360 6 Spatial PMs with coupled planar motion of the moving platform

Fig. 6.40. RPR-S*PC*-RPassPassPass-type non overconstrained SPMs with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, NF=0,

TF=1, limb topology R ⊥ P ⊥ ||R, S*P ⊥ C* and R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass (a), R ⊥ Pass ⊥ ⊥ Pass||Pass (b)

6.2 Non overconstrained solutions

361

Fig. 6.41. RPR-S*PC*-RPassPassPass-type non overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, NF=0,

TF=1, limb topology R ⊥ P ⊥ ||R, SP ⊥ C* and R ⊥ Pass||Pass ⊥ ⊥ Pass

Fig. 6.42. PPassR-PPassC*-RPassPassPass-type non overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4,

NF=0, TF=1, limb topology P ⊥ Pass||R, P ⊥ Pass||C* and R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass

362 6 Spatial PMs with coupled planar motion of the moving platform

Fig. 6.43. PPassR-PPassC*-RPassPassPass-type non overconstrained SPMs with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4,

NF=0, TF=1, limb topology P ⊥ Pass||R, P ⊥ Pass||C* and R ⊥ Pass ⊥ ⊥ Pass||Pass (a), R ⊥ Pass||Pass ⊥ ⊥ Pass (b)

6.2 Non overconstrained solutions

363

Fig. 6.44. PassRR-PassRR*R*C*-RPassPassPass-type non overconstrained SPMs with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ),

MF=6, NF=0, TF=3, limb topology Pass||R||R, Pass||R ⊥ R* ⊥ ⊥ R* ⊥ ⊥ C* and R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass (a), R ⊥ Pass ⊥ ⊥ Pass||Pass (b)

364 6 Spatial PMs with coupled planar motion of the moving platform

Fig. 6.45. Non overconstrained SPMs with planar motion of the moving platform of types PassRR-PassRR*R*C*-RPassPassPass (a), PassPassR-PassPassC*RPassPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=6, NF=0, TF=3, limb

topology Pass||R||R, Pass||R ⊥ R* ⊥ ⊥ R* ⊥ ⊥ C*, R ⊥ Pass||Pass ⊥ ⊥ Pass (a) and Pass||Pass||R, Pass||Pass||C*, R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass (b)

6.2 Non overconstrained solutions

365

Fig. 4.46. PassPassR-PassPassC*-RPassPassPass-type non overconstrained SPMs with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=6, NF=0, TF=3, limb topology Pass||Pass||R, Pass||Pass||C* and R ⊥ Pass ⊥ ⊥ Pass||Pass (a), R ⊥ Pass||Pass ⊥ ⊥ Pass (b)

7 Spatial PMs with uncoupled planar motion of the moving platform

In the general case, in a spatial parallel robotic manipulator (SPM) with uncoupled planar motions of the moving platform at least one spatial limb exists and each operational velocity depends on just one actuated joint velocity: v1 = v1 ( &q1 ) , v2 = v2 ( &q2 ) and ωδ = ωδ ( &q3 ) . The Jacobian matrix in Eq. (1.18) is a diagonal matrix. The overconstrained and non overconstrained solutions presented in this section use rotating actuators mounted on the fixed base.

7.1 Overconstrained solutions The overconstrained solutions of SPMs with uncoupled planar motions of the moving platform and q independent loops meet the condition p ∑ 1 fi < 3 + 6q . The limbs can be simple or complex kinematic chain. Basic

and derived fully-parallel solutions are presented in this section. 7.1.1 Basic solutions

In the basic fully-parallel solutions of SPMs with uncoupled motions F ← G1-G2-G3 presented in this section, the moving platform n nGi (i=1, 2, 3) is connected to the reference platform 1 1Gi 0 by three limbs. One actuator is combined in a revolute joint of each limb. Two planar or spatial limbs G1 and G2 are used for positioning the moving platform and a spatial or planar limb G3 for orienting it. There are no idle mobilities in these basic solutions. Various solutions of SPMs with uncoupled planar motion of the moving platform and no idle mobilities can be obtained by using G1- and G2-limbs with identical or different topology presented in Figs. 2.2e, 7.1 and 7.2, and a planar or spatial G3-limb in Figs. 2.2f-h, 2.3 and 6.1. Only solutions with identical G1- and G2-limbs are illustrated in Figs. 7.3-7.34. G. Gogu, Structural Synthesis of Parallel Robots, Solid Mechanics and Its Applications 173, DOI 10.1007/978-90-481-9831-3_7, © Springer Science + Business Media B.V. 2010

367

368 7 Spatial PMs with uncoupled planar motion of the moving platform

The basic solutions illustrated in Figs. 7.3 and 7.4 combine two planar limbs with three degrees of connectivity in Fig. 2.2e with one spatial limb with four or six degrees of connectivity in Fig. 6.1. The basic solutions illustrated in Figs. 7.5-7.34 combine two spatial limbs with four or five degrees of connectivity in Figs. 7.1 and 7.2 with one planar limb in Figs. 2.1g, 2.2f-h and 3a,b. Planar parallelogram and rhombus loops with on degrees of mobility and planar loops with two or three degrees of mobility are combined in the limbs presented in Figs. 7.1 and 7.2. The limb topology and connecting conditions of the solutions in Figs. 7.3-7.34 are systematized in Tables 7.1-7.6. The bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms in Figs. 7.3-7.34 are given in Table 7.7, and the structural parameters of these solutions are systematized in Tables 7.8-7.18.

Table 7.1. Limb topology and connecting conditions of the overconstrained SPM with no idle mobilities presented in Figs. 7.3 and 7.34 No. SPM type 1

2PaPR-1RRRPRR (Fig. 7.3a)

2

2PaPR-1RPaPaPa (Fig. 7.3b)

3

2PaPR-1RPaPaPa (Fig. 7.4a)

4

2PaPR-1RPaPaPa (Fig. 7.4b)

Limb topology Pa ⊥ P ⊥ ||R (Fig. 2.2e) R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R (Fig. 6.1a) Pa ⊥ P ⊥ ||R (Fig. 2.2e) R ⊥ Pa ⊥ ⊥ Pa ⊥ ||Pa (Fig. 6.1b) Pa ⊥ P ⊥ ||R (Fig. 2.2e) R ⊥ Pa ⊥ ⊥ Pa||Pa (Fig. 6.1c) Pa ⊥ P ⊥ ||R (Fig. 2.2e) R ⊥ Pa|| Pa ⊥ ⊥ Pa (Fig. 6.1d)

Connecting conditions The directions of the revolute joints of limbs G1 and G2 are parallel. The last revolute joints of limbs G1 and G2 have superposed axes. Idem No. 1

Idem No. 1

Idem No. 1

7.1 Overconstrained solutions

369

Fig. 7.1. Complex limbs G1 and G2 with MG=SG=4 (a, d) and MG=SG=5 (b, c) for SPMs with planar motion of the moving platform

370 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.2. Complex limbs G1 and G2 with MG=SG=5 for SPMs with planar motion of the moving platform

7.1 Overconstrained solutions

371

Table 7.2. Limb topology and connecting conditions of the overconstrained SPM with no idle mobilities presented in Figs. 7.5-7.9 No. SPM type 1

2PaRRRR-1RPP (Fig. 7.5a)

2

2PaPaRRR-1RPP (Fig. 7.5b)

3

2PaRRPaR-1RPP (Fig. 7.6a)

4

2PaPaPaR-1RPP (Fig. 7.6b)

5

2PaRRbRR-1RPP (Fig. 7.7)

6

2PaRRbRbRR-1RPP (Fig. 7.8)

7

2PaPn2RR-1RPP (Fig. 7.9a)

8

2PaPn3R-1RPP (Fig. 7.9b)

Limb topology Pa ⊥ R||R||R ⊥ R (Fig. 7.1a) R ⊥ P ⊥⊥ P (Fig. 2.1g)

Pa ⊥ Pa||R||R ⊥ R (Fig. 7.1b) R ⊥ P ⊥⊥ P (Fig. 2.1g) Pa ⊥ R||R||Pa ⊥ R (Fig. 7.1c) R ⊥ P ⊥⊥ P (Fig. 2.1g) Pa ⊥ Pa||Pa ⊥ R (Fig. 7.1d) R ⊥ P ⊥⊥ P (Fig. 2.1g) Pa ⊥ R||Rb||R ⊥ R (Fig. 7.2a) R ⊥ P ⊥⊥ P (Fig. 2.1g) Pa ⊥ R||Rb||Rb||R ⊥ R (Fig. 7.2b) R ⊥ P ⊥⊥ P (Fig. 2.1g) Pa ⊥ Pn2||R ⊥ R (Fig. 7.2c) R ⊥ P ⊥⊥ P (Fig. 2.1g) Pa ⊥ Pn3 ⊥ R (Fig. 7.2d) R ⊥ P ⊥⊥ P (Fig. 2.1g)

Connecting conditions The last revolute joints of limbs G1 and G2 have superposed axes and parallel to the axis of the first revolute joint of G3-limb. Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

372 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.3. Limb topology and connecting conditions of the overconstrained SPM with no idle mobilities presented in Figs. 7.10-7.14 No. SPM type 1

2PaRRRR-1RPPa (Fig. 7.10a)

2

2PaPaRRR-1RPPa (Fig. 7.10b)

3

4

5

6

7

8

Limb topology Pa ⊥ R||R||R ⊥ R (Fig. 7.1a) R ⊥ P ⊥ ||Pa (Fig. 2.2f)

Pa ⊥ Pa||R||R ⊥ R (Fig. 7.1b) R ⊥ P ⊥ ||Pa (Fig. 2.2f) 2PaRRPaR-1RPPa Pa ⊥ R||R||Pa ⊥ R (Fig. 7.11a) (Fig. 7.1c) R ⊥ P ⊥ ||Pa (Fig. 2.2f) 2PaPaPaR-1RPPa Pa ⊥ Pa||Pa ⊥ R (Fig. 7.11b) (Fig. 7.1d) R ⊥ P ⊥ ||Pa (Fig. 2.2f) 2PaRRbRR-1RPPa Pa ⊥ R||Rb||R ⊥ R (Fig. 7.12) (Fig. 7.2a) R ⊥ P ⊥ ||Pa (Fig. 2.2f) 2PaRRbRbRR-1RPPa Pa ⊥ R||Rb||Rb||R ⊥ R (Fig. 7.13) (Fig. 7.2b) R ⊥ P ⊥ ||Pa (Fig. 2.2f) Pa ⊥ Pn2||R ⊥ R 2PaPn2RR-1RPPa (Fig. 7.14a) (Fig. 7.2c) R ⊥ P ⊥ ||Pa (Fig. 2.2f) 2PaPn3R-1RPPa Pa ⊥ Pn3 ⊥ R (Fig. 7.14b) (Fig. 7.2d) R ⊥ P ⊥ ||Pa (Fig. 2.2f)

Connecting conditions The last revolute joints of limbs G1 and G2 have superposed axes and parallel to the axis of the first revolute joint of G3-limb. Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

7.1 Overconstrained solutions

373

Table 7.4. Limb topology and connecting conditions of the overconstrained SPM with no idle mobilities presented in Figs. 7.15-7.19 No. SPM type 1

2PaRRRR-1RPaP (Fig. 7.15a)

2

2PaPaRRR-1RPaP (Fig. 7.15b)

3

4

5

6

7

8

Limb topology Pa ⊥ R||R||R ⊥ R (Fig. 7.1a) R||Pa ⊥ P (Fig. 2.2g)

Pa ⊥ Pa||R||R ⊥ R (Fig. 7.1b) R||Pa ⊥ P (Fig. 2.2g) 2PaRRPaR-1RPaP Pa ⊥ R||R||Pa ⊥ R (Fig. 7.16a) (Fig. 7.1c) R||Pa ⊥ P (Fig. 2.2g) 2PaPaPaR-1RPaP Pa ⊥ Pa||Pa ⊥ R (Fig. 7.16b) (Fig. 7.1d) R||Pa ⊥ P (Fig. 2.2g) 2PaRRbRR-1RPaP Pa ⊥ R||Rb||R ⊥ R (Fig. 7.17) (Fig. 7.2a) R||Pa ⊥ P (Fig. 2.2g) 2PaRRbRbRR-1RPaP Pa ⊥ R||Rb||Rb||R ⊥ R (Fig. 7.18) (Fig. 7.2b) R||Pa ⊥ P (Fig. 2.2g) Pa ⊥ Pn2||R ⊥ R 2PaPn2RR-1RPaP (Fig. 7.19a) (Fig. 7.2c) R||Pa ⊥ P (Fig. 2.2g) 2PaPn3R-1RPaP Pa ⊥ Pn3 ⊥ R (Fig. 7.19b) (Fig. 7.2d) R||Pa ⊥ P (Fig. 2.2g)

Connecting conditions The last revolute joints of limbs G1 and G2 have superposed axes and parallel to the axis of the first revolute joint of G3-limb. Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

374 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.5. Limb topology and connecting conditions of the overconstrained SPM with no idle mobilities presented in Figs. 7.20-7.24 No. SPM type 1

2

3

4

5

6

7

8

2PaRRRR-1RPaPa (Fig. 7.20a)

Limb topology Pa ⊥ R||R||R ⊥ R (Fig. 7.1a) R||Pa||Pa (Fig. 2.2h)

Pa ⊥ Pa||R||R ⊥ R (Fig. 7.1b) R||Pa||Pa (Fig. 2.2h) 2PaRRPaR-1RPaPa Pa ⊥ R||R||Pa ⊥ R (Fig. 7.21a) (Fig. 7.1c) R||Pa||Pa (Fig. 2.2h) 2PaPaPaR-1RPaPa Pa ⊥ Pa||Pa ⊥ R (Fig. 7.21b) (Fig. 7.1d) R||Pa||Pa (Fig. 2.2h) 2PaRRbRR-1RPaPa Pa ⊥ R||Rb||R ⊥ R (Fig. 7.22) (Fig. 7.2a) R||Pa||Pa (Fig. 2.2h) 2PaRRbRbRR-1RPaPa Pa ⊥ R||Rb||Rb||R ⊥ R (Fig. 7.23) (Fig. 7.2b) R||Pa||Pa (Fig. 2.2h) 2PaPn2RR-1RPaPa Pa ⊥ Pn2||R ⊥ R (Fig. 7.24a) (Fig. 7.2c) R||Pa||Pa (Fig. 2.2h) 2PaPn3R-1RPaPa Pa ⊥ Pn3 ⊥ R (Fig. 7.24b) (Fig. 7.2d) R||Pa||Pa (Fig. 2.2h) 2PaPaRRR-1RPaPa (Fig. 7.20b)

Connecting conditions The last revolute joints of limbs G1 and G2 have superposed axes and parallel to the axis of the first revolute joint of G3limb. Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

7.1 Overconstrained solutions

375

Table 7.6. Limb topology and connecting conditions of the overconstrained SPM with no idle mobilities presented in Figs. 7.25-7.34 No. SPM type 1

2PaRRRR-1RPaPa (Fig. 7.25a)

2

2PaRRRR-1RPaPat (Fig. 7.25b)

3

2PaPaRRR-1RPaPa (Fig. 7.26a)

4

2PaPaRRR-1RPaPat (Fig. 7.26b)

5

2PaRRPaR-1RPaPa (Fig. 7.27a)

6

2PaRRPaR-1RPaPat (Fig. 7.27b)

7

2PaPaPaR-1RPaPa (Fig. 7.28a)

8

2PaPaPaR-1RPaPat (Fig. 7.28b)

9

2PaRRbRR-1RPaPa (Fig. 7.29)

10

2PaRRbRR-1RPaPat (Fig. 7.30)

Limb topology Pa ⊥ R||R||R ⊥ R (Fig. 7.1a) R||Pa||Pa (Fig. 2.3a) Pa ⊥ R||R||R ⊥ R (Fig. 7.1a) R||Pa||Pat (Fig. 2.3b) Pa ⊥ Pa||R||R ⊥ R (Fig. 7.1b) R||Pa||Pa (Fig. 2.3a) Pa ⊥ Pa||R||R ⊥ R (Fig. 7.1b) R||Pa||Pat (Fig. 2.3b) Pa ⊥ R||R||Pa ⊥ R (Fig. 7.1c) R||Pa||Pa (Fig. 2.3a) Pa ⊥ R||R||Pa ⊥ R (Fig. 7.1c) R||Pa||Pat (Fig. 2.3b) Pa ⊥ Pa||Pa ⊥ R (Fig. 7.1d) R||Pa||Pa (Fig. 2.3a) Pa ⊥ Pa||Pa ⊥ R (Fig. 7.1d) R||Pa||Pat (Fig. 2.3b) Pa ⊥ R||Rb||R ⊥ R (Fig. 7.2a) R||Pa||Pa (Fig. 2.3a) Pa ⊥ R||Rb||R ⊥ R (Fig. 7.2a) R||Pa||Pat (Fig. 2.3b)

Connecting conditions The last revolute joints of limbs G1 and G2 have superposed axes and parallel to the revolute axes of G3limb. 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

376 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.6. (cont.) 11

2PaRRbRbRR-1RPaPa (Fig. 7.31)

12

2PaRRbRbRR-1RPaPat (Fig. 7.32)

13

2PaPn2RR-1RPaPa (Fig. 7.33a)

14

2PaPn2RR-1RPaPat (Fig. 7.33b)

15

2PaPn3R-1RPaPa (Fig. 7.34a)

16

2PaPn3R-1RPaPat (Fig. 7.34b)

Pa ⊥ R||Rb||Rb||R ⊥ R (Fig. 7.2b) R||Pa||Pa (Fig. 2.3a) Pa ⊥ R||Rb||Rb||R ⊥ R (Fig. 7.2b) R||Pa||Pat (Fig. 2.3b) Pa ⊥ Pn2||R ⊥ R (Fig. 7.2c) R||Pa||Pa (Fig. 2.3a) Pa ⊥ Pn2||R ⊥ R (Fig. 7.2c) R||Pa||Pat (Fig. 2.3b) Pa ⊥ Pn3 ⊥ R (Fig. 7.2d) R||Pa||Pa (Fig. 2.3a) Pa ⊥ Pn3 ⊥ R (Fig. 7.2d) R||Pa||Pat (Fig. 2.3b)

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Table 7.7. Bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 7.3-7.34 No. Parallel mechanism 1 Fig. 7.3a

Basis (RG1) ( v1 , v 2 ,

2

Figs. 7.3b, 7.4

( v1 , v 2 ,

3

Figs. 7.5, 7.6a, ( v1 , v2 , v2 , 7.7-7.10, 7.11a 7.12-7.15, 7.16a, 7.17-7.20, 7.21a, 7.22-7.27, 7.297.34 Figs. 7.6b, 7.11b ( v1 ,v2 , v2 , 7.16b, 7.21b, 7.28

4

δ

)

(RG2) ( v1 , v 2 ,

δ

)

( v1 , v 2 , α

,

δ

)

δ

δ

)

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

α

,

δ

)

( v1 , v 2 , v 2 ,

δ

)

) ( v1 , v 2 , v 2 ,

β

,

( v1 , v 2 , v 2 ,

δ

)

δ

) ( v1 , v 2 ,

δ

)

( v1 , v 2 ,

δ

)

β

,

δ

)

7.1 Overconstrained solutions

377

Table 7.8. Structural parametersa of spatial parallel mechanisms in Figs. 7.3 and 7.4 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 2PaPR-1RRRPRR (Fig. 7.3a)

2PaPR-1RPaPaPa (Figs. 7.3b, 7.4)

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

15 6 6 6 18 4 1 2 3 See Table 7.7

19 6 6 13 25 7 0 3 3 See Table 7.7

3 3 6 3 3 0 3 3 6 ( v1 , v 2 , 3 6 15 3 9 0 6

3 3 4 3 3 9 3 3 4 ( v1 , v 2 , 3 15 22 3 20 0 6

SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

fj

6

6

fj

6

13

fj

18

25

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

378 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.9. Structural parametersa of spatial parallel mechanisms in Figs. 7.5 and 7.6 No. Structural Solution parameter 2PaRRRR-1RPP (Fig. 7.5a)

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

2PaPaPaR-1RPP (Fig. 7.6b)

16 8 8 3 19 4 1 2 3 See Table 7.7

2PaPaRRR-1RPP (Fig. 7.5b) 2PaRRPaR-1RPP (Fig. 7.6a) 20 11 11 3 25 6 1 2 3 See Table 7.7

22 13 13 3 29 8 1 2 3 See Table 7.7

5 5 3 3 3 0 5 5 3 ( v1 , v 2 , 3 6 16 3 8 0 8

5 5 3 6 6 0 5 5 3 ( v1 , v 2 , 3 12 22 3 14 0 11

4 4 3 9 9 0 4 4 3 ( v1 , v 2 , 3 18 26 3 22 0 13

δ

)

δ

)

fj

8

11

13

fj

3

3

3

fj

19

25

29

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

7.1 Overconstrained solutions

379

Table 7.10. Structural parametersa of spatial parallel mechanisms in Figs. 7.7-7.9 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 2PaRRbRR-1RPP (Fig. 7.7) 2PaPn2RR-1RPP (Fig. 7.9a) 2PaPn3R-1RPP (Fig. 7.9b) m 20 p1 11 p2 11 p3 3 p 25 q 6 k1 1 k2 2 k 3 (RGi) See Table 7.7 i=1,2,3 SG1 5 SG2 5 SG3 3 rG1 6 rG2 6 rG3 0 MG1 5 MG2 5 MG3 3 (RF) ( v1 , v 2 , δ ) SF 3 rl 12 rF 22 MF 3 NF 14 TF 0 p1 11 f

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

2PaRRbRbRR-1RPP (Fig. 7.8)

24 14 14 3 31 8 1 2 3 See Table 7.7 5 5 3 9 9 0 5 5 3 ( v1 , v 2 , 3 18 28 3 20 0 14

δ

)

j

fj

11

14

fj

3

3

fj

25

31

See footnote of Table 2.4 for the nomenclature of structural parameters

380 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.11. Structural parametersa of spatial parallel mechanisms in Figs. 7.10 and 7.11 No. Structural Solution parameter 2PaRRRR-1RPPa (Fig. 7.10a)

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

2PaPaPaR-1RPPa (Fig. 7.11b)

18 8 8 6 22 5 0 3 3 See Table 7.7

2PaPaRRR-1RPPa (Fig. 7.10b) 2PaRRPaR-1RPPa (Fig. 7.11a) 22 11 11 6 28 7 0 3 3 See Table 7.7

24 13 13 6 32 9 0 3 3 See Table 7.7

5 5 3 3 3 3 5 5 3 ( v1 , v 2 , 3 9 19 3 11 0 8

5 5 3 6 6 3 5 5 3 ( v1 , v 2 , 3 15 25 3 17 0 11

4 4 3 9 9 3 4 4 3 ( v1 , v 2 , 3 21 29 3 25 0 13

δ

)

δ

)

fj

8

11

13

fj

6

6

6

fj

22

28

32

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

7.1 Overconstrained solutions

381

Table 7.12. Structural parametersa of spatial parallel mechanisms in Figs. 7.127.14 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 2PaRRbRR-1RPPa (Fig. 7.12) 2PaPn2RR-1RPPa (Fig. 7.14a) 2PaPn3R-1RPPa (Fig. 7.14b) m 22 p1 11 p2 11 p3 6 p 28 q 7 k1 0 k2 3 k 3 (RGi) See Table 7.7 i=1,2,3 SG1 5 SG2 5 SG3 3 rG1 6 rG2 6 rG3 3 MG1 5 MG2 5 MG3 3 (RF) ( v1 , v 2 , δ ) SF 3 rl 15 rF 25 MF 3 NF 17 TF 0 p1 11 f

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

2PaRRbRbRR-1RPPa (Fig. 7.13) 26 14 14 6 34 9 0 3 3 See Table 7.7 5 5 3 9 9 3 5 5 3 ( v1 , v 2 , 3 21 31 3 23 0 14

fj

11

14

fj

6

6

fj

28

34

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

382 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.13. Structural parametersa of spatial parallel mechanisms in Figs. 7.15 and 7.16 No. Structural Solution parameter 2PaRRRR-1RPaP (Fig. 7.15a)

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

2PaPaPaR-1RPaP (Fig. 7.16b)

18 8 8 6 22 5 0 3 3 See Table 7.7

2PaPaRRR-1RPaP (Fig. 7.15b) 2PaRRPaR-1RPaP (Fig. 7.16a) 22 11 11 6 28 7 0 3 3 See Table 7.7

24 13 13 6 32 9 0 3 3 See Table 7.7

5 5 3 3 3 3 5 5 3 ( v1 , v 2 , 3 9 19 3 11 0 8

5 5 3 6 6 3 5 5 3 ( v1 , v 2 , 3 15 25 3 17 0 11

4 4 3 9 9 3 4 4 3 ( v1 , v 2 , 3 21 29 3 25 0 13

δ

)

δ

)

fj

8

11

13

fj

6

6

6

fj

22

28

32

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

7.1 Overconstrained solutions

383

Table 7.14. Structural parametersa of spatial parallel mechanisms in Figs. 7.177.19 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 2PaRRbRR-1RPaP (Fig. 7.17) 2PaPn2RR-1RPaP (Fig. 7.19a) 2PaPn3R-1RPaP (Fig. 7.19b) m 22 p1 11 p2 11 p3 6 p 28 q 7 k1 0 k2 3 k 3 (RGi) See Table 7.7 i=1,2,3 SG1 5 SG2 5 SG3 3 rG1 6 rG2 6 rG3 3 MG1 5 MG2 5 MG3 3 (RF) ( v1 , v 2 , δ ) SF 3 rl 15 rF 25 MF 3 NF 17 TF 0 p1 11 f

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

2PaRRbRbRR-1RPaP (Fig. 7.18) 26 14 14 6 34 9 0 3 3 See Table 7.7 5 5 3 9 9 3 5 5 3 ( v1 , v 2 , 3 21 31 3 23 0 14

fj

11

14

fj

6

6

fj

28

34

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

384 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.15. Structural parametersa of spatial parallel mechanisms in Figs. 7.20 and 7.21 No. Structural Solution parameter 2PaRRRR-1RPaPa (Fig. 7.20a)

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

2PaPaPaR-1RPaPa (Fig. 7.21b)

20 8 8 9 25 6 0 3 3 See Table 7.7

2PaPaRRR-1RPaPa (Fig. 7.20b) 2PaRRPaR-1RPaPa (Fig. 7.21a) 24 11 11 9 31 8 0 3 3 See Table 7.7

26 13 13 9 35 10 0 3 3 See Table 7.7

5 5 3 3 3 6 5 5 3 ( v1 , v 2 , 3 12 22 3 14 0 8

5 5 3 6 6 6 5 5 3 ( v1 , v 2 , 3 18 28 3 20 0 11

4 4 3 9 9 6 4 4 3 ( v1 , v 2 , 3 24 32 3 28 0 13

δ

)

δ

)

fj

8

11

13

fj

9

9

9

fj

25

31

35

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

7.1 Overconstrained solutions

385

Table 7.16. Structural parametersa of spatial parallel mechanisms in Figs. 7.227.24 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 2PaRRbRR-1RPaPa (Fig. 7.22) 2PaPn2RR-1RPaPa (Fig. 7.24a) 2PaPn3R-1RPaPa (Fig. 7.24b) m 24 p1 11 p2 11 p3 9 p 31 q 8 k1 0 k2 3 k 3 (RGi) See Table 7.7 i=1,2,3 SG1 5 SG2 5 SG3 3 rG1 6 rG2 6 rG3 6 MG1 5 MG2 5 MG3 3 (RF) ( v1 , v 2 , δ ) SF 3 rl 18 rF 28 MF 3 NF 20 TF 0 p1 11 f

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

2PaRRbRbRR-1RPaPa (Fig. 7.23) 28 14 14 9 37 10 0 3 3 See Table 7.7 5 5 3 9 9 6 5 5 3 ( v1 , v 2 , 3 24 34 3 26 0 14

fj

11

14

fj

9

9

fj

31

37

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

386 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.17. Structural parametersa of spatial parallel mechanisms in Figs. 7.257.28 No. Structural Solution parameter 2PaRRRR-1RPaPa 2PaPaRRR-1RPaPa 2PaPaPaR-1RPaPa 2PaRRRR-1RPaPat 2PaPaRRR-1RPaPat 2PaPaPaR-1RPaPat (Fig. 7.25a,b) (Fig. 7.28a,b) (Fig. 7.26a,b) 2PaRRPaR-1RPaPa 2PaRRPaR-1RPaPat (Fig. 7.27a,b) 1 m 20 24 26 2 p1 8 11 13 3 p2 8 11 13 4 p3 9 9 9 5 p 25 31 35 6 q 6 8 10 7 k1 0 0 0 8 k2 3 3 3 9 k 3 3 3 10 (RGi) See Table 7.7 See Table 7.7 See Table 7.7 i=1,2,3 11 SG1 5 5 4 12 SG2 5 5 4 13 SG3 3 3 3 14 rG1 3 6 9 15 rG2 3 6 9 16 rG3 6 6 6 17 MG1 5 5 4 18 MG2 5 5 4 19 MG3 3 3 3 20 (RF) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) 21 SF 3 3 3 22 rl 12 18 24 23 rF 22 28 32 24 MF 3 3 3 25 NF 14 20 28 26 TF 0 0 0 p1 27 8 11 13 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

fj

8

11

13

fj

9

9

9

fj

25

31

35

See footnote of Table 2.4 for the nomenclature of structural parameters

7.1 Overconstrained solutions

387

Table 7.18. Structural parametersa of spatial parallel mechanisms in Figs. 7.297.34 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 Solution parameter 2PaRRbRR-1RPaPa (Fig. 7.29) 2PaRRbRR-1RPaPat (Fig. 7.30) 2PaPn2RR-1RPaPa (Fig. 7.33a) 2PaPn2RR-1RPaPat (Fig. 7.33b) 2PaPn3R-1RPaPa (Fig. 7.34a) 2PaPn3R-1RPaPat (Fig. 7.34b) m 24 p1 11 p2 11 p3 9 p 31 q 8 k1 0 k2 3 k 3 (RGi) See Table 7.7 i=1,2,3 SG1 5 SG2 5 SG3 3 rG1 6 rG2 6 rG3 6 MG1 5 MG2 5 MG3 3 (RF) ( v1 , v 2 , δ ) SF 3 rl 18 rF 28 MF 3 NF 20 TF 0 p1 11 f

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

2PaRRbRbRR-1RPaPa (Fig. 7.31) 2PaRRbRbRR-1RPaPat (Fig. 7.32)

28 14 14 9 37 10 0 3 3 See Table 7.7 5 5 3 9 9 6 5 5 3 ( v1 , v 2 , 3 24 34 3 26 0 14

fj

11

14

fj

9

9

fj

31

37

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

388 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.3. Overconstrained SPMs with planar motion of the moving platform of types 2PaPR-1RRRPRR (a) and 2PaPR-1RPaPaPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=9 (a), NF=20 (b), limb topology Pa ⊥ P ⊥ ||R and

R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R (a), R ⊥ Pa ⊥ ⊥ Pa ⊥ ||Pa (b)

7.1 Overconstrained solutions

389

Fig. 7.4. 2PaPR-1RPaPaPa-type overconstrained SPMs with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=20, limb topology Pa ⊥ P ⊥ ||R and R ⊥ Pa ⊥ ⊥ Pa||Pa (a), R ⊥ Pa|| Pa ⊥ ⊥ Pa (b)

390 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.5. Overconstrained SPMs with planar motion of the moving platform of types 2PaRRRR-1RPP (a) and 2PaPaRRR-1RPP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=8 (a), NF=14 (b), limb topology R ⊥ P ⊥ ⊥ P and Pa ⊥ R||R||R ⊥ R (a), Pa ⊥ Pa||R||R ⊥ R (b)

7.1 Overconstrained solutions

391

Fig. 7.6. Overconstrained SPMs with planar motion of the moving platform of types 2PaRRPaR-1RPP (a) and 2PaPaPaR-1RPP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14 (a), NF=22 (b), limb topology R ⊥ P ⊥ ⊥ P and Pa ⊥ R||R||Pa ⊥ R (a), Pa ⊥ Pa||Pa ⊥ R (b)

392 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.7. 2PaRRbRR-1RPP-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14, limb

topology R ⊥ P ⊥ ⊥ P and Pa ⊥ R||Rb||R ⊥ R

7.1 Overconstrained solutions

393

Fig. 7.8. 2PaRRbRbRR-1RPP-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=20, limb

topology R ⊥ P ⊥ ⊥ P and Pa ⊥ R||Rb||Rb||R ⊥ R

394 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.9. Overconstrained SPMs with planar motion of the moving platform of types 2PaPn2RR-1RPP (a) and 2PaPn3R-1RPP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14, limb topology R ⊥ P ⊥ ⊥ P and Pa ⊥ Pn2||R ⊥ R (a), Pa ⊥ Pn3 ⊥ R (b)

7.1 Overconstrained solutions

395

Fig. 7.10. Overconstrained SPMs with planar motion of the moving platform of types 2PaRRRR-1RPPa (a) and 2PaPaRRR-1RPPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=11 (a), NF=17 (b), limb topology R ⊥ P ⊥ ||Pa and Pa ⊥ R||R||R ⊥ R (a), Pa ⊥ Pa||R||R ⊥ R (b)

396 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.11. Overconstrained SPMs with planar motion of the moving platform of types 2PaRRPaR-1RPPa (a) and 2PaPaPaR-1RPPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=17 (a), NF=25 (b), limb topology R ⊥ P ⊥ ||Pa and Pa ⊥ R||R||Pa ⊥ R (a), Pa ⊥ Pa||Pa ⊥ R (b)

7.1 Overconstrained solutions

397

Fig. 7.12. 2PaRRbRR-1RPPa-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=17, limb topology Pa ⊥ R||Rb||R ⊥ R and R ⊥ P ⊥ ||Pa

398 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.13. 2PaRRbRbRR-1RPPa-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=23, limb topology Pa ⊥ R||Rb||Rb||R ⊥ R and R ⊥ P ⊥ ||Pa and (a)

7.1 Overconstrained solutions

399

Fig. 7.14. Overconstrained SPMs with planar motion of the moving platform of types 2PaPn2RR-1RPPa (a) and 2PaPn3R-1RPPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=17, limb topology R ⊥ P ⊥ ||Pa and Pa ⊥ Pn2||R ⊥ R (a), Pa ⊥ Pn3 ⊥ R (b)

400 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.15. Overconstrained SPMs with planar motion of the moving platform of types 2PaRRRR-1RPaP (a) and 2PaPaRRR-1RPaP (b), defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=11 (a), NF=17 (b) limb topology R||Pa ⊥ P and Pa ⊥ R||R||R ⊥ R (a), Pa ⊥ Pa||R||R ⊥ R (b)

7.1 Overconstrained solutions

401

Fig. 7.16. Overconstrained SPMs with planar motion of the moving platform of types 2PaRRPaR-1RPaP (a) and 2PaPaPaR-1RPaP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=17 (a), NF=25 (b), limb topology R||Pa ⊥ P and Pa ⊥ R||R||Pa ⊥ R (a), Pa ⊥ Pa||Pa ⊥ R (b)

402 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.17. 2PaRRbRR-1RPaP-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=17, limb topology Pa ⊥ R||Rb||R ⊥ R and R||Pa ⊥ P

7.1 Overconstrained solutions

403

Fig. 7.18. 2PaRRbRbRR-1RPaP-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=23, limb topology Pa ⊥ R||Rb||Rb||R ⊥ R and R||Pa ⊥ P

404 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.19. Overconstrained SPMs with planar motion of the moving platform of types 2PaPn2RR-1RPaP (a) and 2PaPn3R-1RPaP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=17, limb topology R||Pa ⊥ P and Pa ⊥ Pn2||R ⊥ R (a), Pa ⊥ Pn3|| ⊥ R (b)

7.1 Overconstrained solutions

405

Fig. 7.20. Overconstrained SPMs with planar motion of the moving platform of types 2PaRRRR-1RPaPa (a) and 2PaPaRRR-1RPaPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14 (a), NF=20 (b), limb topology R||Pa||Pa and Pa ⊥ R||R||R ⊥ R (a), Pa ⊥ Pa||R||R ⊥ R (b)

406 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.21. Overconstrained SPMs with planar motion of the moving platform of types 2PaRRPaR-1RPaPa (a) and 2PaPaPaR-1RPaPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=20 (a), NF=28 (b), limb topology R||Pa||Pa and Pa ⊥ R||R||Pa ⊥ R (a), Pa ⊥ Pa||Pa ⊥ R (b)

7.1 Overconstrained solutions

407

Fig. 7.22. 2PaRRbRR-1RPaPa-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=20, limb topology Pa ⊥ R||Rb||R ⊥ R and R||Pa||Pa

408 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.23. 2PaRRbRbRR-1RPaPa-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=26, limb topology Pa ⊥ R||Rb||Rb||R ⊥ R and R||Pa||Pa

7.1 Overconstrained solutions

409

Fig. 7.24. Overconstrained SPMs with planar motion of the moving platform of types 2PaPn2RR-1RPaPa (a) and 2PaPn3R-1RPaPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=20, limb topology R||Pa||Pa and Pa ⊥ Pn2||R ⊥ R (a), Pa ⊥ Pn3|| ⊥ R (b)

410 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.25. Overconstrained SPMs with planar motion of the moving platform of types 2PaRRRR-1RPaPa (a) and 2PaRRRR-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14, limb topology Pa ⊥ R||R||R ⊥ R and R||Pa||Pa (a), R||Pa||Pat (b)

7.1 Overconstrained solutions

411

Fig. 7.26. Overconstrained SPMs with planar motion of the moving platform of types 2PaPaRRR-1RPaPa (a) and 2PaPaRRR-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=20, limb topology Pa ⊥ Pa||R||R ⊥ R and R||Pa||Pa (a), R||Pa||Pat (b)

412 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.27. Overconstrained SPMs with planar motion of the moving platform of types 2PaRRPaR-1RPaPa (a) and 2PaRRPaR-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=20, limb topology Pa ⊥ R||R||Pa ⊥ R and R||Pa||Pa (a), R||Pa||Pat (b)

7.1 Overconstrained solutions

413

Fig. 7.28. Overconstrained SPMs with planar motion of the moving platform of types 2PaPaPaR-1RPaPa (a) and 2PaPaPaR-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=28, limb topology Pa ⊥ Pa||Pa ⊥ R and R||Pa||Pa (a), R||Pa||Pat (b)

414 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.29. 2PaRRbRR-1RPaPa-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=20, limb topology Pa ⊥ R||Rb||R ⊥ R and R||Pa||Pa

7.1 Overconstrained solutions

415

Fig. 7.30. 2PaRRbRR-1RPaPat-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=20, limb topology Pa ⊥ R||Rb||R ⊥ R and R||Pa||Pat

416 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.31. 2PaRRbRbRR-1RPaPa-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=26, limb topology Pa ⊥ R||Rb||Rb||R ⊥ R and R||Pa||Pa

7.1 Overconstrained solutions

417

Fig. 7.32. 2PaRRbRbRR-1RPaPat-type overconstrained SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=26, limb topology Pa ⊥ R||Rb||Rb||R ⊥ R and R||Pa||Pat

418 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.33. Overconstrained SPMs with planar motion of the moving platform of types 2PaPn2RR-1RPaPa (a) and 2PaPn2RR-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=20, limb topology Pa ⊥ Pn2||R ⊥ R and R||Pa||Pa (a), R||Pa||Pat (b)

7.1 Overconstrained solutions

419

Fig. 7.34. Overconstrained SPMs with planar motion of the moving platform of types 2PaPn3R-1RPaPa (a) and 2PaPn3R-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=20, limb topology Pa ⊥ Pn3 ⊥ R and R||Pa||Pa (a), R||Pa||Pat (b)

420 7 Spatial PMs with uncoupled planar motion of the moving platform

7.1.2 Derived solutions Solutions with lower degrees of overconstraint can be derived from the basic solutions in Figs. 7.3-7.34 by using joints with idle mobilities. A large set of solutions can be obtained by introducing up to three idle mobilities in each planar closed loop and up to three idle mobilities outside the planar loops combined in the limbs. Examples of solutions with 1 to 8 overconstraints derived from the basic solutions in Figs. 7.3-7.34 are illustrated in Figs. 7.35-7.66. We recall that the idle mobilities can be introduced outside or inside the planar loops combined in the limbs. The idle mobilities which can be combined in a parallelogram loop are systematized in Fig. 1.2 and Table 1.1. In the cylindrical joints of the rhombus loops denoted by Rbcs (Figs. 7.39, 7.40, 7.44, 7.45, 7.49, 7.50, 7.54, 7.55 and 7.61-7.66) and the parallelogram loops denoted by Pacs (Figs. 7.57, 7.58 and 7.61-7.66), the translational motion is an idle mobility. The notations Pacs and Rbcs are associated with the parallelogram and rhombus loops with three idle mobilities combined in a cylindrical and a spherical joint. The notations Pass, Pn2ss and Pn3ss are associated with parallelogram loops and planar loops with 2 and 3 degrees of freedom which combine four idle mobilities in two spherical joints adjacent to the same coupler link. In these cases, three idle mobilities are introduced in the loop and one outside the loop. If the link adjacent to the two spherical joints is a binary link than the idle mobility introduced outside the loop becomes an internal rotational mobility of this binary link around the axis passing by the centre of the two spherical joints. Each internal mobility gives one degree of structural redundancy (see Table 7.19). If the link adjacent to the two spherical joints is connected in the limb by three or more joints (polinary link) than the rotational motion around the axis passing by the centre of the two spherical joints can be either active or idle (potential) mobility of the limb. The active motion is compatible with the non constrained motions of the limb, while the idle mobility is restricted by the constraints of the parallel mechanism and remains just a potential mobility. For example in Fig. 7.43a, this rotational motion is an active mobility of the ternary links 4A and 4B and an idle mobility for the ternary links 8A and 8B. The bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms in Figs. 7.35-7.66 are given in Table 7.20. The limb topology and connecting conditions of these solutions are systematized in Tables 7.21-7.23, as are their structural parameters in Tables 7.24-7.38.

7.1 Overconstrained solutions

421

Table 7.19. Links with internal mobilities and the degree of structural redundancy TF of overconstrained SPMs with uncoupled planar motion of the moving platform No. Parallel mechanism Figure

TF

Link with internal rotational mobility in limb G1 G2 G3

1

1

-

-

3C

1 4

3A, 6A

3B, 6B

7C -

2 4 2

7A 6A, 9A 5A

7B 6B, 9B 5B

-

4

3A, 5A

3B, 5B

-

2 5 2

6A 3A, 6A 6A

6B 3B, 6B 6B

3C -

3 5

5A 3A, 5A

5B 3B, 5B

3C 3C

2 3 4 5 6

7 8 9 10 11 12

Figs. 7.35b, 7.36b, 7.47a Fig. 7.36a Figs. 7.37b, 7.42b, 7.52b, 7.58 Fig. 7.38a Fig. 7.38b Figs. 7.39, 7.40, 7.44, 7.45, 7.54, 7.55, 7.61-7.64 Figs. 7.41, 7.46, 7.56, 7.65, 7.66 Figs. 7.43b Fig. 7.47b Figs. 7.48b, 7.53b, 7.60 Figs. 7.49, 7.50 Fig. 7.51

Table 7.20. Bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 7.35-7.66 No. Parallel mechanism 1 Figs. 7.35, 7.36 2 Figs. 7.37-7.41, 7.47, 7.497.52, 7.547.58, 7.617.66 3 Figs. 7.42, 7.44-7.46, 4 Figs. 7.43, 7.48, 7.53, 7.59, 7.60

Basis (RG1) ( v1 , v 2 ,

α

δ

,

(RG2) ( v1 , v 2 ,

)

β

δ

,

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

)

α

( v1 , v 2 , v 2 ,

α

,

δ

)

( v1 , v 2 , v 2 ,

β

,

δ

)

( v1 , v 2 ,

δ

)

( v1 , v 2 , v 2 ,

α

,

δ

)

( v1 , v 2 , v 2 ,

β

,

δ

)

( v1 , v 2 ,

α

,

) ( v1 , v 2 ,

δ

)

( v1 ,v2 , v3 ,

α

,

β

,

δ

) ( v1 ,v2 , v3 ,

α

,

β

,

δ

β

,

δ

)

,

δ

)

422 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.21. Limb topology and the number of overconstraints NF of the derived SPMs with idle mobilities presented in Figs. 7.35-7.45 No. Basic SPM Type 1 2PaPR-1RRRPRR (Fig. 7.3a) 2 2PaPR-1RPaPaPa (Fig. 7.3b) 3 2PaPR-1RPaPaPa (Fig. 7.4a) 4 2PaPR-1RPaPaPa (Fig. 7.4b) 5 2PaRRRR-1RPP (Fig. 7.5a) 6 2PaPaRRR-1RPP (Fig. 7.5b) 7 2PaRRPaR-1RPP (Fig. 7.6a) 8 2PaPaPaR-1RPP (Fig. 7.6b) 9 2PaRRbRR-1RPP (Fig. 7.7) 10 2PaRRbRbRR1RPP (Fig. 7.8) 11 2PaPn2RR-1RPP (Fig. 7.9a) 12 2PaPn3R-1RPP (Fig. 7.9b) 13 2PaRRRR-1RPPa (Fig. 7.10a) 14 2PaPaRRR-1RPPa (Fig. 7.10b) 15 2PaRRPaR-1RPPa (Fig. 7.11a) 16 2PaPaPaR-1RPPa (Fig. 7.11b) 17 2PaRRbRR-1RPPa (Fig. 7.12) 18 2PaRRbRbRR1RPPa (Fig. 7.13)

Derived SPM NF Type NF 9 2PassPR-1RRRPRR 1 (Fig. 7.35a) 20 2PassPR-1RPassPassPass 1 (Fig. 7.35b) 20 2PassPR-1RPassPassPass 1 (Fig. 7.36a) 20 2PassPR-1RPassPassPass 1 (Fig. 7.36b) 8 2PassRRR-1RPP 2 (Fig. 7.37a) 14 2PassPassRRR-1RPP 2 (Fig. 7.37b) 2 14 2PassRPassR-1RPP (Fig. 7.38a) 22 2PassPassPassR-1RPP 2 (Fig. 7.38b) 14 2PassRbssRR-1RPP 2 (Fig. 7.39) 20 2PassRbssRbcsRR-1RPP 2 (Fig. 7.40)

14 2PassPn2ssRR-1RPP (Fig. 7.41a) 14 2PassPn3ssR-1RPP (Fig. 7.41b) 11 2PassRRR-1RPPass (Fig. 7.42a) 17 2PassPassRRR-1RPPass (Fig. 7.42b) 17 2PassRPassR-1RPPa (Fig. 7.43a) 25 2PassPassPassR-1RPPa (Fig. 7.43b) 17 2PassRbssRR-1RPPass (Fig. 7.44) 23 2PassRbssRbcsRR1RPPass (Fig. 7.45)

2 2 1 1 3 3 1 1

Limb topology Pass ⊥ P ⊥ ||R R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R Pass ⊥ P ⊥ ||R R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass Pass ⊥ P ⊥ ||R R ⊥ Pass ⊥ ⊥ Pass||Pass Pass ⊥ P ⊥ ||R R ⊥ Pass||Pass ⊥ ⊥ Pass Pass ⊥ R||R ⊥ R R ⊥ P ⊥⊥ P Pass ⊥ Pass||R||R ⊥ R R ⊥ P ⊥⊥ P Pass ⊥ R||Pass ⊥ R R ⊥ P ⊥⊥ P Pass ⊥ Pass||Pass ⊥ R R ⊥ P ⊥⊥ P Pass ⊥ Rbss||R ⊥ R R ⊥ P ⊥⊥ P Pass ⊥ Rbss||Rbcs||R ⊥ R R ⊥ P ⊥⊥ P

Pass ⊥ Pn2ss||R ⊥ R R ⊥ P ⊥⊥ P Pass ⊥ Pn3ss ⊥ R R ⊥ P ⊥⊥ P Pass ⊥ R||R ⊥ R R ⊥ P ⊥ ||Pass Pass ⊥ Pass||R||R ⊥ R R ⊥ P ⊥ ||Pass Pass ⊥ R||Pass ⊥ R R ⊥ P ⊥ ||Pa Pass ⊥ Pass||Pass ⊥ R R ⊥ P ⊥ ||Pa Pass ⊥ Rbss||R ⊥ R R ⊥ P ⊥ ||Pass Pass ⊥ Rbss||Rbcs||R ⊥ R R ⊥ P ⊥ ||Pass

7.1 Overconstrained solutions

423

Table 7.22. Limb topology and the number of overconstraints NF of the derived SPMs with idle mobilities presented in Figs. 7.46-7.56 No. Basic SPM Type NF 1 2PaPn2RR-1RPPa 17 (Fig. 7.14a) 2 2PaPn3R-1RPPa 17 (Fig. 7.14b) 3 2PaRRRR-1RPaP 11 (Fig. 7.15a) 4 2PaPaRRR-1RPaP 17 (Fig. 7.15b) 5 2PaRRPaR-1RPaP 17 (Fig. 7.16a) 6 2PaPaPaR-1RPaP 25 (Fig. 7.16b) 7 2PaRRbRR-1RPaP 17 (Fig. 7.17) 8 2PaRRbRbRR23 1RPaP (Fig. 7.18) 9 2PaPn2RR-1RPaP 17 (Fig. 7.19a) 10 2PaPn3R-1RPaP 17 (Fig. 7.19b) 11 2PaRRRR-1RPaPa 14 (Fig. 7.20a) 12 2PaPaRRR-1RPaPa 20 (Fig. 7.20b) 13 2PaRRPaR-1RPaPa 20 (Fig. 7.21a) 14 2PaPaPaR-1RPaPa 28 (Fig. 7.21b) 15 2PaRRbRR-1RPaPa20 (Fig. 7.22) 16 2PaRRbRbRR26 1RPaPa (Fig. 7.23) 17 2PaPn2RR-1RPaPa 20 (Fig. 7.24a) 18 2PaPn3R-1RPaPa 20 (Fig. 7.24b)

Derived SPM Type NF 2PassPn2ssRR-1RPPass 1 (Fig. 7.46a) 2PassPn3ssR-1RPPass 1 (Fig. 7.46b) 2 2PassRRR-1RPassP (Fig. 7.47a) 2PassPassRRR-1RPassP 2 (Fig. 7.47b) 3 2PassRPassR-1RPaP (Fig. 7.48a) 2PassPassPassR-1RPaP 3 (Fig. 7.48b) 2PassRbssRR-1RPassP 2 (Fig. 7.49) 2 2PassRbssRbcsRR1RPassP (Fig. 7.50) 2PassPn2ssRR-1RPassP 2 (Fig. 7.51a) 2PassPn3ssR-1RPassP 2 (Fig. 7.51b) 8 2PassRRR-1RPaPa (Fig. 7.52a) 2PassPassRRR-1RPaPa 8 (Fig. 7.52b) 2PassRPassR-1RPaPa 6 (Fig. 7.53a) 2PassPassPassR-1RPaPa 6 (Fig. 7.53b) 2PassRbssRR-1RPaPa 8 (Fig. 7.54) 8 2PassRbssRbcsRR1RPaPa (Fig. 7.55) 2PassPn2ssRR-1RPaPa 8 (Fig. 7.56a) 2PassPn3ssR-1RPaPa 8 (Fig. 7.56b)

Limb topology Pass ⊥ Pn2ss||R ⊥ R R ⊥ P ⊥ ||Pass Pass ⊥ Pn3ss ⊥ R R ⊥ P ⊥ ||Pass Pass ⊥ R||R ⊥ R R||Pass ⊥ P Pass ⊥ Pass||R||R ⊥ R R||Pass ⊥ P Pass ⊥ R||Pass ⊥ R R||Pa ⊥ P Pass ⊥ Pass||Pass ⊥ R R||Pa ⊥ P Pass ⊥ Rbss||R ⊥ R R||Pass ⊥ P Pass ⊥ Rbss||Rbcs||R ⊥ R R||Pass ⊥ P

Pass ⊥ Pn2ss||R ⊥ R R||Pass ⊥ P Pass ⊥ Pn3ss ⊥ R R||Pass ⊥ P Pass ⊥ R||R ⊥ R R||Pa||Pa Pass ⊥ Pass||R||R ⊥ R R||Pa||Pa Pass ⊥ R||Pass ⊥ R R||Pa||Pa Pass ⊥ Pass||Pass ⊥ R R||Pa||Pa Pass ⊥ Rbss||R ⊥ R R||Pa||Pa Pass ⊥ Rbss||Rbcs||R ⊥ R R||Pa||Pa Pass ⊥ Pn2ss||R ⊥ R R||Pa||Pa Pass ⊥ Pn3ss ⊥ R R||Pa||Pa

424 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.23. Limb topology and the number of overconstraints NF of the derived SPMs with idle mobilities presented in Figs. 7.57-7.66 No. Basic SPM Type NF 1 2PaRRRR-1RPaPa 14 (Fig. 7.25a) 2 2PaRRRR-1RPaPat 14 (Fig. 7.25b) 3 2PaPaRRR-1RPaPa 20 (Fig. 7.26a) 4

5 6

7 8

9 10

11

12

13

14

15 16

Derived SPM Type NF 2PassRRR-1RPacsPacs 2 (Fig. 7.57a) 2PassRRR-1RPacsPatcs 2 (Fig. 7.57b) 2PassPassRRR2 1RPacsPacs (Fig. 7.58a) 2PaPaRRR2 20 2PassPassRRR1RPaPat 1RPacsPatcs (Fig. 7.26b) (Fig. 7.58b) 2PaRRPaR-1RPaPa 20 2PassRPassR-1RPaPa 6 (Fig. 7.27a) (Fig. 7.59a) 20 2PassRPassR-1RPaPat 6 2PaRRPaR(Fig. 7.59b) 1RPaPat (Fig. 7.27b) 2PaPaPaR-1RPaPa 28 2PassPassPassR-1RPaPa 6 (Fig. 7.28a) (Fig. 7.60a) 2PaPaPaR28 2PassPassPassR-1RPaPat 6 (Fig. 7.60b) 1RPaPat (Fig. 7.28b) 2PaRRbRR-1RPaPa20 2PassRbssRR-1RPacsPacs 2 (Fig. 7.29) (Fig. 7.61) 2PaRRbRR20 2PassRbssRR-1RPacsPatcs2 (Fig. 7.62) 1RPaPat (Fig. 7.30) 2PaRRbRbRR2 26 2PassRbssRbcsRR1RPaPa 1RPacsPacs (Fig. 7.31) (Fig. 7.63) 2PaRRbRbRR2 26 2PassRbssRbcsRR1RPaPat 1RPacsPatcs (Fig. 7.32) (Fig. 7.64) 2PaPn2RR-1RPaPa 20 2PassPn2ssRR2 (Fig. 7.33a) 1RPacsPacs (Fig. 7.65a) 2PaPn2RR2 20 2PassPn2ssRR1RPaPat 1RPacsPatcs (Fig. 7.33b) (Fig. 7.65b) 2PaPn3R-1RPaPa 20 2PassPn3ssR-1RPacsPacs 2 (Fig. 7.34a) (Fig. 7.66a) 2PaPn3R-1RPaPat 20 2PassPn3ssR-1RPacsPatcs 2 (Fig. 7.34b) (Fig. 7.66b)

Limb topology Pass ⊥ R||R ⊥ R R||Pacs||Pacs Pass ⊥ R||R ⊥ R R||Pacs||Patcs Pass ⊥ Pass||R||R ⊥ R R||Pacs||Pacs

Pass ⊥ Pass||R||R ⊥ R R||Pacs||Patcs Pass ⊥ R||Pass ⊥ R R||Pa||Pa Pass ⊥ R||Pass ⊥ R R||Pa||Pat Pass ⊥ Pass||Pass ⊥ R R||Pa||Pa Pass ⊥ Pass||Pass ⊥ R R||Pa||Pat Pass ⊥ Rbss||R ⊥ R R||Pacs||Pacs Pass ⊥ Rbss||R ⊥ R R||Pacs||Patcs Pass ⊥ Rbss||Rbcs||R ⊥ R R||Pacs||Pacs Pass ⊥ Rbss||Rbcs||R ⊥ R R||Pacs||Patcs Pass ⊥ Pn2ss||R ⊥ R R||Pacs||Pacs Pass ⊥ Pn2ss||R ⊥ R R||Pacs||Patcs Pass ⊥ Pn3ss ⊥ R R||Pacs||Pacs Pass ⊥ Pn3ss ⊥ R R||Pacs||Patcs

7.1 Overconstrained solutions

425

Table 7.24. Structural parametersa of spatial parallel mechanisms in Figs. 7.35 and 7.36 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 2PassPR-1RRRPRR (Fig. 7.35a)

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

15 6 6 6 18 4 1 2 3 See Table 7.20

2PassPR-1RPassPassPass (Fig. 7.35b) 2PassPR-1RPassPassPass (Fig. 7.36) 19 6 6 13 25 7 0 3 3 See Table 7.20

4 4 6 6 6 0 4 4 6 ( v1 , v 2 , 3 12 23 3 1 0 10

4 4 6 6 6 18 4 4 7 ( v1 , v 2 , 3 30 41 4 1 1 10

SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

fj

10

10

fj

6

25

fj

26

45

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

426 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.25. Structural parametersa of spatial parallel mechanisms in Fig. 7.37 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

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

Solution 2PassRRR-1RPP (Fig. 7.37a) 14 7 7 3 17 4 1 2 3 See Table 7.20

2PassPassRRR-1RPP (Fig. 7.37b) 20 11 11 3 25 6 1 2 3 See Table 7.20

5 5 3 6 6 0 5 5 3 ( v1 , v 2 , 3 12 22 3 2 0 11

5 5 3 12 12 0 7 7 3 ( v1 , v 2 , 3 24 34 7 2 4 19

δ

)

fj

11

19

fj

3

3

fj

25

41

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

7.1 Overconstrained solutions

427

Table 7.26. Structural parametersa of spatial parallel mechanisms in Figs. 7.387.40 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 2PassRPassR-1RPP (Fig. 7.38a) 2PassRbssRR-1RPP (Fig. 7.39) m 18 p1 10 p2 10 p3 3 p 23 q 6 k1 1 k2 2 k 3 (RGi) See Table 7.20 (i=1,2,3) SG1 5 SG2 5 SG3 3 rG1 12 rG2 12 rG3 0 MG1 6 MG2 6 MG3 3 (RF) ( v1 , v 2 , δ ) SF 3 rl 24 rF 34 MF 5 NF 2 TF 2 p1 18 f

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

2PassPassPassR-1RPP 2PassRbssRbcsRR(Fig. 7.38b) 1RPP (Fig. 7.40) 22 13 13 3 29 8 1 2 3 See Table 7.20

22 13 13 3 29 8 1 2 3 See Table 7.20

5 5 3 18 18 0 7 7 3 ( v1 , v 2 , 3 36 46 7 2 4 25

5 5 3 18 18 0 6 6 3 ( v1 , v 2 , 3 36 46 5 2 2 24

δ

)

j

fj

18

25

24

fj

3

3

3

fj

39

53

51

See footnote of Table 2.4 for the nomenclature of structural parameters

δ

)

428 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.27. Structural parametersa of spatial parallel mechanisms in Figs. 7.41 and 7.42 No. Structural Solution parameter 2PassPn2ssRR-1RPP (Fig. 7.41a) 2PassPn3ssR-1RPP (Fig. 7.41b) 1 m 20 2 p1 11 3 p2 11 4 p3 3 5 p 25 6 q 6 7 k1 1 8 k2 2 9 k 3 10 (RGi) See Table 7.20 (i=1,2,3) 11 SG1 5 12 SG2 5 13 SG3 3 14 rG1 12 15 rG2 12 16 rG3 0 17 MG1 7 18 MG2 7 19 MG3 3 20 (RF) ( v1 , v 2 , δ ) 21 SF 3 22 rl 24 23 rF 34 24 MF 7 25 NF 2 26 TF 4 p1 27 19 f

16 7 7 6 20 5 0 3 3 See Table 7.20

22 11 11 6 28 7 0 3 3 See Table 7.20

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

5 5 4 12 12 6 7 7 4 ( v1 , v 2 , 3 30 41 7 1 4 19

28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

2PassRRR-1RPPass 2PassPassRRR-1RPPass (Fig. 7.42a) (Fig. 7.42b)

δ

)

δ

)

j

fj

19

11

19

fj

3

10

10

fj

41

32

48

See footnote of Table 2.4 for the nomenclature of structural parameters

7.1 Overconstrained solutions

429

Table 7.28. Structural parametersa of spatial parallel mechanisms in Figs. 7.43 and 7.44 No. Structural Solution parameter 2PassRPassR-1RPPa 2PassPassPassR-1RPPa 2PassRbssRR(Fig. 7.43a) (Fig. 7.43b) 1RPPass (Fig. 7.44) 1 m 20 24 20 2 p1 10 13 10 3 p2 10 13 10 4 p3 6 6 6 5 p 26 32 26 6 q 7 9 7 7 k1 0 0 0 8 k2 3 3 3 9 k 3 3 3 10 (RGi) See Table 7.20 See Table 7.20 See Table 7.20 (i=1,2,3) 11 SG1 6 6 5 12 SG2 6 6 5 13 SG3 3 3 4 14 rG1 12 18 12 15 rG2 12 18 12 16 rG3 3 3 6 17 MG1 6 7 6 18 MG2 6 7 6 19 MG3 3 3 4 20 (RF) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) 21 SF 3 3 3 22 rl 27 39 30 23 rF 39 51 41 24 MF 3 5 5 25 NF 3 3 1 26 TF 0 2 2 p1 27 18 25 18 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

fj

18

25

18

fj

6

6

10

fj

42

56

46

See footnote of Table 2.4 for the nomenclature of structural parameters

430 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.29. Structural parametersa of spatial parallel mechanisms in Figs. 7.45 and 7.46 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 2PassRbssRbcsRR-1RPPass (Fig. 7.45)

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

24 13 13 6 32 9 0 3 3 See Table 7.20

2PassPn2ssRR-1RPPass (Fig. 7.46a) 2PassPn3ssR-1RPPass (Fig. 7.46b) 22 11 11 6 28 7 0 3 3 See Table 7.20

5 5 4 18 18 6 6 6 4 ( v1 , v 2 , 3 42 53 5 1 2 24

5 5 4 12 12 6 7 7 4 ( v1 , v 2 , 3 30 41 7 1 4 19

SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

fj

24

19

fj

10

10

fj

58

48

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

7.1 Overconstrained solutions Table 7.30. Structural parametersa of spatial parallel mechanisms in Fig. 7.47 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

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

Solution 2PassRRR-1RPassP (Fig. 7.47a) 16 7 7 6 20 5 0 3 3 See Table 7.20

2PassPassRRR-1RPassP (Fig. 7.47b) 22 11 11 6 28 7 0 3 3 See Table 7.20

5 5 3 6 6 6 5 5 4 ( v1 , v 2 , 3 18 28 4 2 1 11

5 5 3 12 12 6 7 7 4 ( v1 , v 2 , 3 30 40 8 2 5 19

δ

)

fj

11

19

fj

10

10

fj

32

48

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

431

432 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.31. Structural parametersa of spatial parallel mechanisms in Fig. 7.48 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

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

Solution 2PassRPassR-1RPaP (Fig. 7.48a) 20 10 10 6 26 7 0 3 3 See Table 7.20

2PassPassPassR-1RPaP (Fig. 7.48b) 24 13 13 6 32 9 0 3 3 See Table 7.20

6 6 3 12 12 3 6 6 3 ( v1 , v 2 , 3 27 39 3 3 0 18

6 6 3 18 18 3 7 7 3 ( v1 , v 2 , 3 39 51 5 3 2 25

δ

)

fj

18

25

fj

6

6

fj

42

56

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

7.1 Overconstrained solutions

433

Table 7.32. Structural parametersa of spatial parallel mechanisms in Figs. 7.497.51 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 2PassRbssRR1RPassP (Fig. 7.49)

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

20 10 10 6 26 7 0 3 3 See Table 7.20

24 13 13 6 32 9 0 3 3 See Table 7.20

2PassPn2ssRR-1RPassP (Fig. 7.51a) 2PassPn3ssR-1RPassP (Fig. 7.51b) 22 11 11 6 28 7 0 3 3 See Table 7.20

5 5 3 12 12 6 6 6 4 ( v1 , v 2 , 3 30 40 6 2 3 18

5 5 3 18 18 6 6 6 4 ( v1 , v 2 , 3 42 52 6 2 3 24

5 5 3 12 12 6 7 7 4 ( v1 , v 2 , 3 30 40 8 2 5 19

δ

)

2PassRbssRbcsRR1RPassP (Fig. 7.50)

δ

)

fj

18

24

19

fj

10

10

10

fj

46

58

48

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

434 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.33. Structural parametersa of spatial parallel mechanisms in Fig. 7.52 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

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

Solution 2PassRRR-1RPaPa (Fig. 7.52a) 18 7 7 9 23 6 0 3 3 See Table 7.20

2PassPassRRR-1RPaPa (Fig. 7.52b) 24 11 11 9 31 8 0 3 3 See Table 7.20

5 5 3 6 6 6 5 5 3 ( v1 , v 2 , 3 18 28 3 8 0 11

5 5 3 12 12 6 7 7 3 ( v1 , v 2 , 3 30 40 7 8 4 19

δ

)

fj

11

19

fj

9

9

fj

31

47

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

7.1 Overconstrained solutions Table 7.34. Structural parametersa of spatial parallel mechanisms in Fig. 7.53 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

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

Solution 2PassRPassR-1RPaPa (Fig. 7.53a) 22 10 10 9 29 8 0 3 3 See Table 7.20

2PassPassPassR-1RPaPa (Fig. 7.53b) 26 13 13 9 35 10 0 3 3 See Table 7.20

6 6 3 12 12 6 6 6 3 ( v1 , v 2 , 3 30 42 3 6 0 18

6 6 3 18 18 6 7 7 3 ( v1 , v 2 , 3 42 54 5 6 2 25

δ

)

fj

18

25

fj

9

9

fj

45

59

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

435

436 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.35. Structural parametersa of spatial parallel mechanisms in Figs. 7.547.56 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 2PassRbssRR1RPaPa (Fig. 7.54)

2PassRbssRbcsRR1RPaPa (Fig. 7.55)

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

22 10 10 9 29 8 0 3 3 See Table 7.20

26 13 13 9 35 10 0 3 3 See Table 7.20

2PassPn2ssRR1RPaPa (Fig. 7.56a) 2PassPn3ssR1RPaPa (Fig. 7.56b) 24 11 11 9 31 8 0 3 3 See Table 7.20

5 5 3 12 12 6 6 6 3 ( v1 , v 2 , 3 30 40 5 8 2 18

5 5 3 18 18 6 6 6 3 ( v1 , v 2 , 3 42 52 5 8 2 24

5 5 3 12 12 6 7 7 3 ( v1 , v 2 , 3 30 40 7 8 4 19

SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

δ

)

fj

18

24

19

fj

9

9

9

fj

45

57

47

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

7.1 Overconstrained solutions

437

Table 7.36. Structural parametersa of spatial parallel mechanisms in Figs. 7.577.59 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

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

Solution 2PassRRR1RPacsPacs (Fig. 7.57a) 2PassRRR1RPacsPatcs (Fig. 7.57b) 18 7 7 9 23 6 0 3 3 See Table 7.20

2PassPassRRR1RPacsPacs (Fig. 7.58a) 2PassPassRRR1RPacsPatcs (Fig. 7.58b) 24 11 11 9 31 8 0 3 3 See Table 7.20

2PassRPassR1RPaPa (Fig. 7.59a) 2PassRPassR1RPaPat (Fig. 7.59b) 22 10 10 9 29 8 0 3 3 See Table 7.20

5 5 3 6 6 12 5 5 3 ( v1 , v 2 , 3 24 34 3 2 0 11

5 5 3 12 12 12 7 7 3 ( v1 , v 2 , 3 36 46 7 2 5 19

6 6 3 12 12 6 6 6 3 ( v1 , v 2 , 3 30 42 3 6 0 18

δ

)

δ

)

fj

11

19

18

fj

15

15

9

fj

37

53

45

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

438 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.37. Structural parametersa of spatial parallel mechanisms in Figs. 7.607.62 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

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

Solution 2PassPassPassR-1RPaPa (Fig. 7.60a) 2PassPassPassR-1RPaPat (Fig. 7.60b) 26 13 13 9 35 10 0 3 3 See Table 7.20

2PassRbssRR-1RPacsPacs (Fig. 7.61) 2PassRbssRR-1RPacsPatcs (Fig. 7.62) 22 10 10 9 29 8 0 3 3 See Table 7.20

6 6 3 18 18 6 7 7 3 ( v1 , v 2 , 3 42 54 5 6 2 25

5 5 3 12 12 12 6 6 3 ( v1 , v 2 , 3 36 46 5 2 2 18

δ

)

fj

25

18

fj

9

15

fj

59

51

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

7.1 Overconstrained solutions

439

Table 7.38. Structural parametersa of spatial parallel mechanisms in Figs. 7.637.66 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

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

Solution 2PassRbssRbcsRR1RPacsPacs (Fig. 7.63) 2PassRbssRbcsRR1RPacsPatcs (Fig. 7.64) 26 13 13 9 35 10 0 3 3 See Table 7.20 5 5 3 18 18 12 6 6 3 ( v1 , v 2 , 3 48 58 5 2 2 24

δ

)

2PassPn2ssRR-1RPacsPacs (Fig. 7.65a) 2PassPn2ssRR-1RPacsPatcs (Fig. 7.65b) 2PassPn3ssR-1RPacsPacs (Fig. 7.66a) 2PassPn3ssR-1RPacsPatcs (Fig. 7.66b)

24 11 11 9 31 8 0 3 3 See Table 7.20 5 5 3 12 12 12 7 7 3 ( v1 , v 2 , 3 36 46 7 2 4 19

fj

24

19

fj

15

15

fj

63

53

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

440 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.35. Overconstrained SPMs with planar motion of the moving platform of types 2PassPR-1RRRPRR (a) and 2PassPR-1RPassPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=1 and MF=3, TF=0, (a), MF=4, TF=1, (b), limb topology

Pass ⊥ P ⊥ ||R and R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R (a), R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass (b)

7.1 Overconstrained solutions

441

Fig. 7.36. 2PassPR-1RPassPassPass-type overconstrained SPMs with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, TF=1, NF=1, limb topology Pass ⊥ P ⊥ ||R and R ⊥ Pass ⊥ ⊥ Pass||Pass (a), R ⊥ Pass||Pass ⊥ ⊥ Pass (b)

442 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.37. Overconstrained SPMs with planar motion of the moving platform of types 2PassRRR-1RPP (a) and 2PassPassRRR-1RPP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=2 and MF=3, TF=0, (a), MF=7, TF=4 (b), limb topology

R ⊥ P ⊥ ⊥ P and Pass ⊥ R||R ⊥ R (a), Pass ⊥ Pass||R||R ⊥ R (b)

7.1 Overconstrained solutions

443

Fig. 7.38. Overconstrained SPMs with planar motion of the moving platform of types 2PassRPassR-1RPP (a) and 2PassPassPassR-1RPP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=2 and MF=5, TF=2, (a), MF=7, TF=4 (b), limb topology

R ⊥ P ⊥ ⊥ P and Pass ⊥ R||Pass ⊥ R (a), Pass ⊥ Pass||Pass ⊥ R (b)

444 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.39. 2PassRbssRR-1RPP-type overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, TF=2, NF=2, limb

topology R ⊥ P ⊥ ⊥ P and Pass ⊥ Rbss||R ⊥ R

7.1 Overconstrained solutions

445

Fig. 7.40. 2PassRbssRbcsRR-1RPP-type overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, TF=2, NF=2, limb topology R ⊥ P ⊥ ⊥ P and Pass ⊥ Rbss||Rbcs||R ⊥ R

446 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.41. Overconstrained SPMs with planar motion of the moving platform of types 2PassPn2ssRR-1RPP (a) and 2PassPn3ssR-1RPP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=7, TF=4, NF=2, limb topology R ⊥ P ⊥ ⊥ P and Pass ⊥ Pn2ss||R ⊥ R (a), Pass ⊥ Pn3ss ⊥ R (b)

7.1 Overconstrained solutions

447

Fig. 7.42. Overconstrained SPMs with planar motion of the moving platform of types 2PassRRR-1RPPass (a) and 2PassPassRRR-1RPPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=1 and MF=3, TF=0, (a), MF=7, TF=4 (b), limb topology R ⊥ P ⊥ ||Pass and Pass ⊥ R||R ⊥ R (a), Pass ⊥ Pass||R||R ⊥ R (b)

448 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.43. Overconstrained SPMs with planar motion of the moving platform of types 2PassRPassR-1RPPa (a) and 2PassPassPassR-1RPPa (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=3 and MF=3, TF=0, (a), MF=5, TF=2 (b), limb topology R ⊥ P ⊥ ||Pa and Pass ⊥ R||Pass ⊥ R (a), Pass ⊥ Pass||Pass ⊥ R (b)

7.1 Overconstrained solutions

449

Fig. 7.44. 2PassRbssRR-1RPPass-type overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, TF=2, NF=1, limb topology R ⊥ P ⊥ ||Pass and Pass ⊥ Rbss||R ⊥ R

450 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.45. 2PassRbssRbcsRR-1RPPass-type overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, TF=2, NF=1, limb topology R ⊥ P ⊥ ||Pass and Pass ⊥ Rbss||Rbcs||R ⊥ R

7.1 Overconstrained solutions

451

Fig. 7.46. Overconstrained SPMs with planar motion of the moving platform of types 2PassPn2ssRR-1RPPass (a) and 2PassPn3ssR-1RPPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=7, TF=4, NF=1, limb topology R ⊥ P ⊥ ||Pass and Pass ⊥ Pn2ss||R ⊥ R (a), Pass ⊥ Pn3ss ⊥ R (b)

452 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.47. Overconstrained SPMs with planar motion of the moving platform of types 2PassRRR-1RPassP (a) and 2PassPassRRR-1RPassP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=2 and MF=4, TF=1, (a), MF=8, TF=5 (b), limb topology R||Pass ⊥ P and Pass ⊥ R||R ⊥ R (a), Pass ⊥ Pass||R||R ⊥ R (b)

7.1 Overconstrained solutions

453

Fig. 7.48. Overconstrained SPMs with planar motion of the moving platform of types 2PassRPassR-1RPaP (a) and 2PassPassPassR-1RPaP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=3 and MF=3, TF=0, (a), MF=5, TF=2 (b), limb topology R||Pa ⊥ P and Pass ⊥ R||Pass ⊥ R (a), Pass ⊥ Pass||Pass ⊥ R (b)

454 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.49. 2PassRbssRR-1RPassP-type overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=6, TF=3, NF=2, limb topology Pass ⊥ Rbss||R ⊥ R and R||Pass ⊥ P

7.1 Overconstrained solutions

455

Fig. 7.50. 2PassRbssRbcsRR-1RPassP-type overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=6, TF=3, NF=2, limb topology Pass ⊥ Rbss||Rbcs||R ⊥ R and R||Pass ⊥ P

456 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.51. Overconstrained SPMs with planar motion of the moving platform of types 2PassPn2ssRR-1RPassP (a) and 2PassPn3ssR-1RPassP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=8, TF=5, NF=2, limb topology R||Pass ⊥ P and Pass ⊥ Pn2ss||R ⊥ R (a), Pass ⊥ Pn3ss ⊥ R (b)

7.1 Overconstrained solutions

457

Fig. 7.52. Overconstrained SPMs with planar motion of the moving platform of types 2PassRRR-1RPaPa (a) and 2PassPassRRR-1RPaPa (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=8 and MF=3, TF=0, (a), MF=7, TF=4 (b), limb topology R||Pa||Pa and Pass ⊥ R||R ⊥ R (a), Pass ⊥ Pass||R||R ⊥ R (b)

458 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.53. Overconstrained SPMs with planar motion of the moving platform of types 2PassRPassR-1RPaPa (a) and 2PassPassPassR-1RPaPa (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=6 and MF=3, TF=0, (a), MF=5, TF=2 (b), limb topology R||Pa||Pa and Pass ⊥ R||Pass ⊥ R (a), Pass ⊥ Pass||Pass ⊥ R (b)

7.1 Overconstrained solutions

459

Fig. 7.54. 2PassRbssRR-1RPaPa-type overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, TF=2, NF=8, limb topology Pass ⊥ Rbss||R ⊥ R and R||Pa||Pa

460 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.55. 2PassRbssRbcsRR-1RPaPa-type overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, TF=2, NF=8, limb topology Pass ⊥ Rbss||Rbcs||R ⊥ R and R||Pa||Pa

7.1 Overconstrained solutions

461

Fig. 7.56. Overconstrained SPMs with planar motion of the moving platform of types 2PassPn2ssRR-1RPaPa (a) and 2PassPn3ssR-1RPaPa (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=7, TF=4, NF=8, limb topology R||Pa||Pa and Pass ⊥ Pn2ss||R ⊥ R (a), Pass ⊥ Pn3ss ⊥ R (b)

462 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.57. Overconstrained SPMs with planar motion of the moving platform of types 2PassRRR-1RPacsPacs (a) and 2PassRRR-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=2, limb topology Pass ⊥ R||R ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

7.1 Overconstrained solutions

463

Fig. 7.58. Overconstrained SPMs with planar motion of the moving platform of types 2PassPassRRR-1RPacsPacs (a) and 2PassPassRRR-1RPacsPatcs (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=7, TF=4, NF=2, limb topology Pass ⊥ Pass||R||R ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

464 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.59. Overconstrained SPMs with planar motion of the moving platform of types 2PassRPassR-1RPaPa (a) and 2PassRPassR-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=6, limb topology Pass ⊥ R||Pass ⊥ R and R||Pa||Pa (a), R||Pa||Pat (b)

7.1 Overconstrained solutions

465

Fig. 7.60. Overconstrained SPMs with planar motion of the moving platform of types 2PassPassPassR-1RPaPa (a) and 2PassPassPassR-1RPaPat (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, TF=2, NF=6, limb topology Pass ⊥ Pass||Pass ⊥ R and R||Pa||Pa (a), R||Pa||Pat (b)

466 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.61. 2PassRbssRR-1RPacsPacs-type overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, TF=2, NF=2, limb topology Pass ⊥ Rbss||R ⊥ R and R||Pacs||Pacs

7.1 Overconstrained solutions

467

Fig. 7.62. 2PassRbssRR-1RPacsPatcs-type overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, TF=2, NF=2, limb topology Pass ⊥ Rbss||R ⊥ R and R||Pacs||Patcs

468 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.63. 2PassRbssRbcsRR-1RPacsPacs-type overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, TF=2, NF=2, limb topology Pass ⊥ Rbss||Rbcs||R ⊥ R and R||Pacs||Pacs

7.1 Overconstrained solutions

469

Fig. 7.64. 2PassRbssRbcsRR-1RPacsPatcs-type overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, TF=2, NF=2, limb topology Pass ⊥ Rbss||Rbcs||R ⊥ R and R||Pacs||Patcs

470 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.65. Overconstrained SPMs with planar motion of the moving platform of types 2PassPn2ssRR-1RPacsPacs (a) and 2PassPn2ssRR-1RPacsPatcs (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=7, TF=4, NF=2, limb topology Pass ⊥ Pn2ss||R ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

7.1 Overconstrained solutions

471

Fig. 7.66. Overconstrained SPMs with planar motion of the moving platform of types 2PassPn3ssR-1RPacsPacs (a) and 2PassPn3ssR-1RPacsPatcs (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=7, TF=4, NF=2, limb topology Pass ⊥ Pn3ss ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

472 7 Spatial PMs with uncoupled planar motion of the moving platform

7.2 Non overconstrained solutions Equation (1.15) indicates that non overconstrained solutions of spatial parallel robots with uncoupled 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 ,ωδ ) and NF=0 (Figs. 7.67-7.100). These solutions are derived from the overconstrained counterparts presented in Figs. 7.3-7.34 by introducing the required idle mobilities. We recall that the joints combining idle mobilities are denoted by an asterisk. These idle mobilities can be introduced outside or inside the loops combined in the limbs. The rotational mobility of the revolute joint denoted by R* is an idle mobility. One idle mobility is combined in each cylindrical joint C* and two idle mobilities in each spherical joint. For example, in the cylindrical joint denoted by C* in Figs. 7.67 and 7.68, the idle mobility is the rotational motion. The translational motion is an idle mobility in the cylindrical joint of the rhombus loops denoted by Rbcs (Figs. 7.72, 7.77, 7.82 and 7.87) and in the parallelogram loops denoted by Pacs (Figs. 7.89-7.100). The notations Pass, Pn2ss and Pn3ss are associated with parallelogram loops and planar loops with 2 and 3 degrees of freedom which combine four idle mobilities in two spherical joints adjacent to the same coupler link. In these cases, three idle mobilities are introduced in the loop and one outside the loop. If the link adjacent to the two spherical joints is a binary link than the idle mobility introduced outside the loop becomes an internal rotational mobility of the binary link around the axis passing by the centre of the two spherical joints. Each internal mobility gives one degree of structural redundancy (see Table 7.39). If the link adjacent to the two spherical joints is connected in the limb by three or more joints (polinary link) than the rotational motion around the axis passing by the centre of the two spherical joints can be either active or idle (potential) mobility of the limb. The active motion is compatible with the non constrained motions of the limb, while the idle mobility is restricted by the constraints of the parallel mechanism and remains just a potential mobility. For example in Fig. 7.75a, this rotational motion is an active mobility of ternary links 4A and 4B, an idle mobility for ternary links 8A and 8B and an internal mobility of link 5C. The bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms in Figs. 7.67-7.100 are given in Table 7.40. The limb topology and connecting conditions of these solutions are systematized in Tables 7.41-7.44, as are their structural parameters in Tables 7.45-7.60.

7.2 Non overconstrained solutions

473

Table 7.39. Links with internal mobilities and the degree of structural redundancy TF of overconstrained SPMs with uncoupled planar motion of the moving platform No. Parallel mechanism Figure

TF

Link with internal rotational mobility in limb G1 G2 G3

1

1

-

-

3C

1 4 2 2

3A, 6A 6A 5A

3B, 6B 6B 5B

7C -

4 1 5 3 3 5 5 3 3 5 2 6 4 6 4 2

3A, 5A 3A, 6A 6A 5A 3A, 5A 3A, 6A 6A 5A 3A, 5A 3A, 6A 5A 3A, 5A 3A, 6A 6A

3B, 5B 3B, 6B 6B 5B 3B, 5B 3B, 6B 6B 5B 3B, 5B 3B, 6B 5B 3B, 5B 3B, 6B 6B

5C 5C 5C 5C 5C 3C 3C 3C 3C 3 C, 6 C 3 C, 6 C 3 C, 6 C 3 C, 6 C -

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

Figs. 7.67b, 7.68b, 7.79a, 7.80a Fig. 7.68a Fig. 7.69b, Fig. 7.70b Figs. 7.71, 7.72, 7.93-7.96 Figs. 7.73, 7.97-7.100 Figs. 7.74a, 7.75a, Fig. 7.74b, Fig. 7.75b Figs. 7.76, 7.77 Fig. 7.78 Fig. 7.79b, Figs. 7.80b Figs. 7.81, 7.82 Fig. 7.83 Figs. 7.84a, 7.85a, Fig. 7.84b, 7.85b Fig. 7.86, 7.87 Fig. 7.88, Fig. 7.90 Fig. 7.92

Table 7.40. Bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 7.67-7.100 No. Parallel mechanism 1 Figs. 7.67, 7.68 2 Figs. 7.697.100

Basis (RG1) ( v1 , v 2 ,

α,

( v1 , v 2 , v 3 ,

δ α

,

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

) β

,

δ

) ( v1 , v 2 , v 3 ,

β,

δ )

α

β

,

,

(RG3) ( v1 , v 2 , v 3 , δ

) ( v1 , v 2 ,

δ

α

)

,

β

,

δ

)

474 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.41. Limb topology of the non overconstrained SPMs presented in Figs. 7.67-7.75 No. Basic SPM SPM with NF=0 Type NF Type 1 2PaPR-1RRRPRR 9 PassPR- PassPC*(Fig. 7.3a) RRRPRR (Fig. 7.67a) 2 2PaPR-1RPaPaPa 20 PassPR-PassPC*(Fig. 7.3b) RPassPassPass (Fig. 7.67b) 3 2PaPR-1RPaPaPa 20 PassPR-PassPC*(Fig. 7.4a) RPassPassPass (Fig. 7.68a) 4 2PaPR-1RPaPaPa 20 PassPR-PassPC*(Fig. 7.4b) RPassPassPass (Fig. 7.68b) 8 2PassRRR*R-1RPP 5 2PaRRRR-1RPP (Fig. 7.5a) (Fig. 7.69a) 6 2PaPaRRR-1RPP 14 2PassPassRRR*R-1RPP (Fig. 7.5b) (Fig. 7.69b) 7 2PaRRPaR-1RPP 14 2PassRPassR-1RPP (Fig. 7.6a) (Fig. 7.70a) 8 2PaPaPaR-1RPP 22 2PassPassPassR-1RPP (Fig. 7.6b) (Fig. 7.70b) 9 2PaRRbRR-1RPP 14 2PassRbssRR*R-1RPP (Fig. 7.7) (Fig. 7.71) 10 2PaRRbRbRR20 2PassRbssRbcsRR*R1RPP 1RPP (Fig. 7.8) (Fig. 7.72) 11 2PaPn2RR-1RPP (Fig. 7.9a) 12 2PaPn3R-1RPP (Fig. 7.9b) 13 2PaRRRR-1RPPa (Fig. 7.10a) 14 2PaPaRRR-1RPPa (Fig. 7.10b) 15 2PaRRPaR-1RPPa (Fig. 7.11a) 16 2PaPaPaR-1RPPa (Fig. 7.11b)

14 14 11 17 17 25

Limb topology Pass ⊥ P ⊥ ||R Pass ⊥ P ⊥ ||C* R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R Pass ⊥ P ⊥ ||R Pass ⊥ P ⊥ ||C* R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass Pass ⊥ P ⊥ ||R Pass ⊥ P ⊥ ||C* R ⊥ Pass ⊥ ⊥ Pass||Pass Pass ⊥ P ⊥ ||R Pass ⊥ P ⊥ ||C* R ⊥ Pass||Pass ⊥ ⊥ Pass Pass ⊥ R||R ⊥ ⊥ R ⊥ ⊥ R R ⊥ P ⊥⊥ P Pass ⊥ Pass||R||R ⊥ ⊥ R ⊥ ⊥ R R ⊥ P ⊥⊥ P Pass ⊥ R||Pass ⊥ ||R R ⊥ P ⊥⊥ P Pass ⊥ Pass||Pass ⊥ ||R R ⊥ P ⊥⊥ P Pass ⊥ Rbss||R ⊥ ⊥ R ⊥ ⊥ R R ⊥ P ⊥⊥ P Pass ⊥ Rbss||Rbcs||R ⊥ ⊥ R ⊥⊥ R R ⊥ P ⊥⊥ P ss ss 2Pa Pn2 RR*R-1RPP Pass ⊥ Pn2ss||R ⊥ ⊥ R ⊥ ⊥ R (Fig. 7.73a) R ⊥ P ⊥⊥ P ss ss 2Pa Pn3 R*R-1RPP Pass ⊥ Pn3ss ⊥ ⊥ R ⊥ ⊥ R (Fig. 7.73b) R ⊥ P ⊥⊥ P ss ss 2Pa RRR*R-1RPPa Pass ⊥ R||R ⊥ ⊥ R ⊥ ⊥ R (Fig. 7.74a) R ⊥ P ⊥ ||Pass ss ss ss 2Pa Pa RRR*R-1RPPa Pass ⊥ Pass||R||R ⊥ ⊥ R ⊥ ⊥ R (Fig. 7.74b) R ⊥ P ⊥ ||Pass 2PassRPassR-1RPPass Pass ⊥ R||Pass ⊥ ||R (Fig. 7.75a) R ⊥ P ⊥ ||Pass ss ss ss ss 2Pa Pa Pa R-1RPPa Pass ⊥ Pass||Pass ⊥ ||R R ⊥ P ⊥ ||Pass (Fig. 7.75b)

7.2 Non overconstrained solutions

475

Table 7.42. Limb topology of the non overconstrained SPMs presented in Figs. 7.76-7.85 No. Basic SPM SPM with NF=0 Type NF Type 1 2PaRRbRR-1RPPa 17 2PassRbssRR*R-1RPPass (Fig. 7.12) (Fig. 7.76) 2 2PaRRbRbRR23 2PassRbssRbcsRR*R1RPPa 1RPPass (Fig. 7.13) (Fig. 7.77) 3 4 5 6

7 8 9 10

11

12 13 14

15 16

17 2PassPn2ssRR*R-1RPPass (Fig. 7.78a) 17 2PassPn3ssR*R-1RPPass (Fig. 7.78b) 11 2PassRRR*R-1RPassP (Fig. 7.79a) 17 2PassPassRRR*R1RPassP (Fig. 7.79b) 2PaRRPaR-1RPaP 17 2PassRPassR-1RPassP (Fig. 7.16a) (Fig. 7.80a) 2PaPaPaR-1RPaP 25 2PassPassPassR-1RPassP (Fig. 7.16b) (Fig. 7.80b) 2PaRRbRR-1RPaP 17 2PassRbssRR*R-1RPassP (Fig. 7.17) (Fig. 7.81) 2PaRRbRbRR23 2PassRbssRbcsRR*R1RPaP 1RPassP (Fig. 7.18) (Fig. 7.82) 2PaPn2RR-1RPaP 17 2PassPn2ssRR*R(Fig. 7.19a) 1RPassP (Fig. 7.83a) 2PaPn3R-1RPaP 17 2PassPn3ssR*R-1RPassP (Fig. 7.19b) (Fig. 7.83b) 2PaRRRR-1RPaPa 14 2PassRRR*R-1RPassPass (Fig. 7.20a) (Fig. 7.84a) 2PaPaRRR-1RPaPa 20 2PassPassRRR*R(Fig. 7.20b) 1RPassPass (Fig. 7.84b) 2PaRRPaR-1RPaPa 20 2PassRPassR-1RPassPass (Fig. 7.21a) (Fig. 7.85a) 2PaPaPaR-1RPaPa 28 2PassPassPassR(Fig. 7.21b) 1RPassPass (Fig. 7.85b) 2PaPn2RR-1RPPa (Fig. 7.14a) 2PaPn3R-1RPPa (Fig. 7.14b) 2PaRRRR-1RPaP (Fig. 7.15a) 2PaPaRRR-1RPaP (Fig. 7.15b)

Limb topology Pass ⊥ Rbss||R ⊥ ⊥ R ⊥ ⊥ R R ⊥ P ⊥ ||Pass Pass ⊥ Rbss||Rbcs||R ⊥ ⊥ R ⊥⊥ R R ⊥ P ⊥ ||Pass Pass ⊥ Pn2ss||R ⊥ ⊥ R ⊥ ⊥ R R ⊥ P ⊥ ||Pass Pass ⊥ Pn3ss ⊥ ⊥ R ⊥ ⊥ R R ⊥ P ⊥ ||Pass Pass ⊥ R||R ⊥ ⊥ R ⊥ ⊥ R R||Pass ⊥ P Pass ⊥ Pass||R||R ⊥ ⊥ R ⊥ ⊥ R R||Pass ⊥ P Pass ⊥ R||Pass ⊥ ||R R||Pass ⊥ P Pass ⊥ Pass||Pass ⊥ ||R R||Pass ⊥ P Pass ⊥ Rbss||R ⊥ ⊥ R ⊥ ⊥ R R||Pass ⊥ P Pass ⊥ Rbss||Rbcs||R ⊥ ⊥ R ⊥⊥ R R||Pass ⊥ P Pass ⊥ Pn2ss||R ⊥ ⊥ R ⊥ ⊥ R R||Pass ⊥ P

Pass ⊥ Pn3ss ⊥ ⊥ R ⊥ ⊥ R R||Pass ⊥ P Pass ⊥ R||R ⊥ ⊥ R ⊥ ⊥ R R||Pass||Pass Pass ⊥ Pass||R||R ⊥ ⊥ R ⊥ ⊥ R R||Pass||Pass Pass ⊥ R||Pass ⊥ ||R R||Pass||Pass Pass ⊥ Pass||Pass ⊥ ||R R||Pass||Pass

476 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.43. Limb topology of the non overconstrained SPMs presented in Figs. 7.86-7.92 No. Basic SPM Type 1 2PaRRbRR1RPaPa (Fig. 7.22) 2 2PaRRbRbRR1RPaPa (Fig. 7.23)

SPM with NF=0 NF Type 20 2PassRbssRR*R1RPassPass (Fig. 7.86) 26 2PassRbssRbcsRR*R1RPassPass (Fig. 7.87)

2PaPn2RR1RPaPa (Fig. 7.24a) 4 2PaPn3R1RPaPa (Fig. 7.24b) 5 2PaRRRR1RPaPa (Fig. 7.25a) 6 2PaRRRR1RPaPat (Fig. 7.25b) 7 2PaPaRRR1RPaPa (Fig. 7.26a) 8 2PaPaRRR1RPaPat (Fig. 7.26b) 9 2PaRRPaR1RPaPa (Fig. 7.27a) 10 2PaRRPaR1RPaPat (Fig. 7.27b) 11 2PaPaPaR1RPaPa (Fig. 7.28a) 12 2PaPaPaR1RPaPat (Fig. 7.28b)

20 2PassPn2ssRR*R1RPassPass (Fig. 7.88a) 20 2PassPn3ssR*R1RPassPass (Fig. 7.88b) 14 2PassRRR*R1RPacsPacs (Fig. 7.89a) 14 2PassRRR*R1RPacsPatcs (Fig. 7.89b) 20 2PassPassRRR*R1RPacsPacs (Fig. 7.90a) 20 2PassPassRR*R1RPacsPatcs (Fig. 7.90b) 20 2PassRPassR1RPacsPacs (Fig. 7.91a) 20 2PaRRPaR1RPacsPatcs (Fig. 7.91b) 28 2PassPassPassR1RPacsPacs (Fig. 7.92a) 28 2PassPassPassR1RPacsPatcs (Fig. 7.92b)

3

Limb topology Pass ⊥ Rbss||R ⊥ ⊥ R ⊥ ⊥ R R||Pass||Pass

Pass ⊥ Rbss||Rbcs||R ⊥ ⊥ R ⊥⊥ R R||Pass||Pass Pass ⊥ Pn2ss||R ⊥ ⊥ R ⊥ ⊥ R R||Pass||Pass Pass ⊥ Pn3ss ⊥ ⊥ R ⊥ ⊥ R R||Pass||Pass Pass ⊥ R||R ⊥ R* ⊥ ⊥ R R||Pacs||Pacs Pass ⊥ R||R ⊥ R* ⊥ ⊥ R R||Pacs||Patcs Pass ⊥ Pass||R||R ⊥ R* ⊥ ⊥ R R||Pacs||Pacs Pass ⊥ Pass||R||R ⊥ R* ⊥ ⊥ R R||Pacs||Patcs Pass ⊥ R||Pass ⊥ R R||Pacs||Pacs Pass ⊥ R||Pass ⊥ R R||Pacs||Patcs Pass ⊥ Pass||Pass ⊥ R R||Pacs||Pacs Pass ⊥ Pass||Pass ⊥ R R||Pacs||Patcs

7.2 Non overconstrained solutions

477

Table 7.44. Limb topology of the non overconstrained SPMs presented in Figs. 7.93-7.100 No. Basic SPM Type 13 2PaRRbRR1RPaPa (Fig. 7.29) 14 2PaRRbRR1RPaPat (Fig. 7.30) 15 2PaRRbRbRR1RPaPa (Fig. 7.31) 16 2PaRRbRbRR1RPaPat (Fig. 7.32) 2PaPn2RR1RPaPa (Fig. 7.33a) 2PaPn2RR1RPaPat (Fig. 7.33b) 2PaPn3R1RPaPa (Fig. 7.34a) 2PaPn3R1RPaPat (Fig. 7.34b)

SPM with NF=0 NF Type 20 2PassRbssRR*R1RPacsPacs (Fig. 7.93) 20 2PassRbssRR*R1RPacsPatcs (Fig. 7.94) 26 2PassRbssRbcsRR*R1RPacsPacs (Fig. 7.95) 26 2PassRbssRbcsRR*R1RPacsPatcs (Fig. 7.96) 20 2PassPn2ssRR*R1RPacsPacs (Fig. 7.97) 20 2PassPn2ssRR*R1RPacsPatcs (Fig. 7.98) 20 2PassPn3ssR*R1RPacsPacs (Fig. 7.99) 20 2PassPn3ssR*R1RPacsPatcs (Fig. 7.100)

Limb topology Pass ⊥ Rbss||R ⊥ R* ⊥ ⊥ R R||Pacs||Pacs

Pass ⊥ Rbss||R ⊥ R* ⊥ ⊥ R R||Pacs||Patcs

Pass ⊥ Rbss||Rbcs||R ⊥ R* ⊥⊥ R R||Pacs||Pacs Pass ⊥ Rbss||Rbcs||R ⊥ R* ⊥⊥ R R||Pacs||Patcs Pass ⊥ Pn2ss||R ⊥ R* ⊥ ⊥ R R||Pacs||Pacs Pass ⊥ Pn2ss||R ⊥ R* ⊥ ⊥ R R||Pacs||Patcs Pass ⊥ Pn3ss ⊥ R* ⊥ ⊥ R R||Pacs||Pacs Pass ⊥ Pn3ss ⊥ R* ⊥ ⊥ R R||Pacs||Patcs

478 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.45. Structural parametersa of spatial parallel mechanisms in Figs. 7.67 and 7.68 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 PassPR-PassPC*-RRRPRR (Fig. 7.67a)

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

15 6 6 6 18 4 1 2 3 See Table 7.40

PassPR-PassPC*-RPassPassPass (Fig. 7.67b) PassPR-PassPC*-RPassPassPass (Fig. 7.68a,b) 19 6 6 13 25 7 0 3 3 See Table 7.40

4 5 6 6 6 0 4 5 6 ( v1 , v 2 , 3 12 24 3 0 0 10

4 5 6 6 6 18 4 5 7 ( v1 , v 2 , 3 30 42 4 0 1 10

SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

fj

11

11

fj

6

25

fj

27

46

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

7.2 Non overconstrained solutions Table 7.46. Structural parametersa of spatial parallel mechanisms in Fig. 7.69 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

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

Solution 2PassRRR*R-1RPP (Fig. 7.69a) 16 8 8 3 19 4 1 2 3 See Table 7.40

2PassPassRRR*R-1RPP (Fig. 7.69b) 22 12 12 3 27 6 1 2 3 See Table 7.40

6 6 3 6 6 0 6 6 3 ( v1 , v 2 , 3 12 24 3 0 0 12

6 6 3 12 12 0 8 8 3 ( v1 , v 2 , 3 24 36 7 0 4 20

δ

)

fj

12

20

fj

3

3

fj

27

43

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

479

480 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.47. Structural parametersa of spatial parallel mechanisms in Figs. 7.70 and 7.71 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 2PassRPassR-1RPP 2PassPassPassR-1RPP 2PassRbssRR*R(Fig. 7.70a) (Fig. 7.70b) 1RPP (Fig. 7.71) m 18 22 20 p1 10 13 11 p2 10 13 11 p3 3 3 3 p 23 29 25 q 6 8 6 k1 1 1 1 k2 2 2 2 k 3 3 3 (RGi) See Table 7.40 See Table 7.40 See Table 7.40 (i=1,2,3) SG1 6 6 6 SG2 6 6 6 SG3 3 3 3 rG1 12 18 12 rG2 12 18 12 rG3 0 0 0 MG1 6 7 7 MG2 6 7 7 MG3 3 3 3 (RF) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) SF 3 3 3 rl 24 36 24 rF 36 48 36 MF 3 5 5 NF 0 0 0 TF 0 2 2 p1 18 25 19 f

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

fj

18

25

19

fj

3

3

3

fj

39

53

41

See footnote of Table 2.4 for the nomenclature of structural parameters

7.2 Non overconstrained solutions

481

Table 7.48. Structural parametersa of spatial parallel mechanisms in Figs. 7.72 and 7.73 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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

Solution 2PassRbssRbcsRR*R-1RPP (Fig. 7.72)

24 14 14 3 31 8 1 2 3 See Table 7.40

2PassPn2ssRR*R-1RPP (Fig. 7.73a) 2PassPn3ssR*R-1RPP (Fig. 7.73b) 22 12 12 3 27 6 1 2 3 See Table 7.40

6 6 3 18 18 0 7 7 3 ( v1 , v 2 , 3 36 48 5 0 2 25

6 6 3 12 12 0 8 8 3 ( v1 , v 2 , 3 24 36 7 0 4 20

δ

)

fj

25

20

fj

3

3

fj

53

43

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

482 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.49. Structural parametersa of spatial parallel mechanisms in Fig. 7.74 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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

Solution 2PassRRR*R-1RPPass (Fig. 7.74a) 18 8 8 6 22 5 0 3 3 See Table 7.40

2PassPassRRR*R-1RPPass (Fig. 7.74b) 24 12 12 6 30 7 0 3 3 See Table 7.40

6 6 3 6 6 6 6 6 4 ( v1 , v 2 , 3 18 30 4 0 1 12

6 6 3 12 12 6 8 8 4 ( v1 , v 2 , 3 30 42 8 0 5 20

δ

)

fj

12

20

fj

10

10

fj

34

50

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

7.2 Non overconstrained solutions

483

Table 7.50. Structural parametersa of spatial parallel mechanisms in Figs. 7.75 and 7.76 No. Structural Solution parameter 2PassRPassR-1RPPass 2PassPassPassR-1RPPass 2PassRbssRR*R(Fig. 7.75a) (Fig. 7.75b) 1RPPass (Fig. 7.76) 1 m 20 24 22 2 p1 10 13 11 3 p2 10 13 11 4 p3 6 6 6 5 p 26 32 28 6 q 7 9 7 7 k1 0 0 0 8 k2 3 3 3 9 k 3 3 3 10 (RGi) See Table 7.40 See Table 7.40 See Table 7.40 (i=1,2,3) 11 SG1 6 6 6 12 SG2 6 6 6 13 SG3 3 3 3 14 rG1 12 18 12 15 rG2 12 18 12 16 rG3 6 6 6 17 MG1 6 7 7 18 MG2 6 7 7 19 MG3 4 4 4 20 (RF) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) 21 SF 3 3 3 22 rl 30 42 30 23 rF 42 54 42 24 MF 4 6 6 25 NF 0 0 0 26 TF 1 3 3 p1 27 18 25 19 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

fj

18

25

19

fj

10

10

10

fj

46

60

48

See footnote of Table 2.4 for the nomenclature of structural parameters

484 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.51. Structural parametersa of spatial parallel mechanisms in Figs. 7.77 and 7.78 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 2PassRbssRbcsRR*R-1RPPass (Fig. 7.77)

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 14 14 6 34 9 0 3 3 See Table 7.40

2PassPn2ssRR*R-1RPPass (Fig. 7.78a) 2PassPn3ssR*R-1RPPass (Fig. 7.78b) 24 12 12 6 30 7 0 3 3 See Table 7.40

6 6 3 18 18 6 7 7 4 ( v1 , v 2 , 3 42 54 6 0 3 25

6 6 3 12 12 6 8 8 4 ( v1 , v 2 , 3 30 42 8 0 5 20

SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

fj

25

20

fj

10

10

fj

60

50

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

7.2 Non overconstrained solutions

485

Table 7.52. Structural parametersa of spatial parallel mechanisms in Fig. 7.79 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

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

Solution 2PassRRR*R-1RPassP (Fig. 7.79a) 18 8 8 6 22 5 0 3 3 See Table 7.40

2PassPassRRR*R-1RPassP (Fig. 7.79b) 24 12 12 6 30 7 0 3 3 See Table 7.40

6 6 3 6 6 6 6 6 4 ( v1 , v 2 , 3 18 30 4 0 1 12

6 6 3 12 12 6 8 8 4 ( v1 , v 2 , 3 30 42 8 0 5 20

δ

)

fj

12

20

fj

10

10

fj

34

50

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

486 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.53. Structural parametersa of spatial parallel mechanisms in Figs. 7.80 and 7.81 No. Structural Solution parameter 2PassRPassR-1RPassP 2PassPassPassR-1RPassP 2PassRbssRR*R(Fig. 7.80a) (Fig. 7.80b) 1RPassP (Fig. 7.81) 1 m 20 24 22 2 p1 10 13 11 3 p2 10 13 11 4 p3 6 6 6 5 p 26 32 28 6 q 7 9 7 7 k1 0 0 0 8 k2 3 3 3 9 k 3 3 3 10 (RGi) See Table 7.40 See Table 7.40 See Table 7.40 (i=1,2,3) 11 SG1 6 6 6 12 SG2 6 6 6 13 SG3 3 3 3 14 rG1 12 18 12 15 rG2 12 18 12 16 rG3 6 6 6 17 MG1 6 7 7 18 MG2 6 7 7 19 MG3 4 4 4 20 (RF) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) 21 SF 3 3 3 22 rl 30 42 30 23 rF 42 54 42 24 MF 4 6 6 25 NF 0 0 0 26 TF 1 3 3 p1 27 18 25 19 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

fj

18

25

19

fj

10

10

10

fj

46

60

48

See footnote of Table 2.4 for the nomenclature of structural parameters

7.2 Non overconstrained solutions

487

Table 7.54. Structural parametersa of spatial parallel mechanisms in Figs. 7.82 and 7.83 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 2PassRbssRbcsRR*R-1RPassP (Fig. 7.82)

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 14 14 6 34 9 0 3 3 See Table 7.40

2PassPn2ssRR*R-1RPassP (Fig. 7.83a) 2PassPn3ssR*R-1RPassP (Fig. 7.83b) 24 12 12 6 30 7 0 3 3 See Table 7.40

6 6 3 18 18 6 7 7 4 ( v1 , v 2 , 3 42 54 6 0 3 25

6 6 3 12 12 6 8 8 4 ( v1 , v 2 , 3 30 42 8 0 5 20

SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

fj

25

20

fj

10

10

fj

60

50

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

488 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.55. Structural parametersa of spatial parallel mechanisms in Fig. 7.84 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

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

Solution 2PassRRR*R-1RPassPass (Fig. 7.84a) 20 8 8 9 25 6 0 3 3 See Table 7.40

2PassPassRRR*R-1RPassPass (Fig. 7.84b) 26 12 12 9 33 8 0 3 3 See Table 7.40

6 6 3 6 6 12 6 6 5 ( v1 , v 2 , 3 24 36 5 0 2 12

6 6 3 12 12 12 8 8 5 ( v1 , v 2 , 3 36 48 9 0 6 20

δ

)

fj

12

20

fj

17

17

fj

41

57

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

7.2 Non overconstrained solutions Table 7.56. Structural parametersa of spatial parallel mechanisms in Fig. 7.85 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

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

Solution 2PassRPassR-1RPassPass (Fig. 7.85a) 22 10 10 9 29 8 0 3 3 See Table 7.40

2PassPassPassR-1RPassPass (Fig. 7.85b) 26 13 13 9 35 10 0 3 3 See Table 7.40

6 6 3 12 12 12 6 6 5 ( v1 , v 2 , 3 36 48 5 0 2 18

6 6 3 18 18 12 7 7 5 ( v1 , v 2 , 3 48 60 7 0 4 25

δ

)

fj

18

25

fj

17

17

fj

53

67

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

489

490 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.57. Structural parametersa of spatial parallel mechanisms in Figs. 7.867.88 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 parameter

Solution 2PassRbssRR*R1RPassPass (Fig. 7.86)

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

24 11 11 9 31 8 0 3 3 See Table 7.40

2PassRbssRbcsRR*R- 2PassPn2ssRR*R1RPassPass 1RPassPass (Fig. 7.87) (Fig. 7.88a) 2PassPn3ssR*R1RPassPass (Fig. 7.88b) 28 26 14 12 14 12 9 9 37 33 10 8 0 0 3 3 3 3 See Table 7.40 See Table 7.40

6 6 3 12 12 12 7 7 5 ( v1 , v 2 , 3 36 48 7 0 4 19

6 6 3 18 18 12 7 7 5 ( v1 , v 2 , 3 48 60 7 0 4 25

SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

δ

)

6 6 3 12 12 12 8 8 5 ( v1 , v 2 , 3 36 48 9 0 6 20

fj

19

25

20

fj

17

17

17

fj

55

67

57

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

7.2 Non overconstrained solutions

491

Table 7.58. Structural parametersa of spatial parallel mechanisms in Figs. 7.897.91 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

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

Solution 2PassRRR*R1RPacsPacs (Fig. 7.89a) 2PassRRR*R1RPacsPatcs (Fig. 7.89b) 20 8 8 9 25 6 0 3 3 See Table 7.40

2PassPassRRR*R1RPacsPacs (Fig. 7.90a) 2PassPassRR*R1RPacsPatcs (Fig. 7.90b) 26 12 12 9 33 8 0 3 3 See Table 7.40

2PassRPassR1RPacsPacs (Fig. 7.91a) 2PaRRPaR1RPacsPatcs (Fig. 7.91b) 22 10 10 9 29 8 0 3 3 See Table 7.40

6 6 3 6 6 12 6 6 3 ( v1 , v 2 , 3 24 36 3 0 0 12

6 6 3 12 12 12 8 8 3 ( v1 , v 2 , 3 36 48 7 0 4 20

6 6 3 12 12 12 6 6 3 ( v1 , v 2 , 3 36 48 3 0 0 18

δ

)

δ

)

fj

12

20

18

fj

15

15

15

fj

39

55

51

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

492 7 Spatial PMs with uncoupled planar motion of the moving platform Table 7.59. Structural parametersa of spatial parallel mechanisms in Figs. 7.927.94 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

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

Solution 2PassPassPassR-1RPacsPacs (Fig. 7.92a) 2PassPassPassR-1RPacsPatcs (Fig. 7.92b) 26 13 13 9 35 10 0 3 3 See Table 7.40

2PassRbssRR*R-1RPacsPacs (Fig. 7.93) 2PassRbssRR*R-1RPacsPatcs (Fig. 7.94) 24 11 11 9 31 8 0 3 3 See Table 7.40

6 6 3 18 18 12 7 7 3 ( v1 , v 2 , 3 48 60 5 0 2 25

6 6 3 12 12 12 7 7 3 ( v1 , v 2 , 3 36 48 5 0 2 19

δ

)

fj

25

19

fj

15

15

fj

65

53

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

7.2 Non overconstrained solutions

493

Table 7.60. Structural parametersa of spatial parallel mechanisms in Figs. 7.957.100 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 2PassRbssRbcsRR*R1RPacsPacs (Fig. 7.95) 2PassRbssRbcsRR*R1RPacsPatcs (Fig. 7.96) m 28 p1 14 p2 14 p3 9 p 37 q 10 k1 0 k2 3 k 3 (RGi) See Table 7.40 (i=1,2,3) SG1 6 SG2 6 SG3 3 rG1 18 rG2 18 rG3 12 MG1 7 MG2 7 MG3 3 (RF) ( v1 , v 2 , δ ) SF 3 rl 48 rF 60 MF 5 NF 0 TF 2 p1 25 f

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

2PassPn2ssRR*R-1RPacsPacs (Fig. 7.97) 2PassPn2ssRR*R-1RPacsPatcs (Fig. 7.98) 2PassPn3ssR*R-1RPacsPacs (Fig. 7.99) 2PassPn3ssR*R-1RPacsPatcs (Fig. 7.100)

26 12 12 9 33 8 0 3 3 See Table 7.40 6 6 3 12 12 12 8 8 3 ( v1 , v 2 , 3 36 48 7 0 4 20

fj

25

20

fj

15

15

fj

65

55

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

494 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.67. Non overconstrained SPMs with planar motion of the moving platform of types PassPR-PassPC*-RRRPRR (a) and PassPR-PassPC*-RPassPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=0 and MF=3, TF=0, (a), MF=4, TF=1 (b), limb topology Pass ⊥ P ⊥ ||R, Pass ⊥ P ⊥ ||C* and R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R (a), R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass (b)

7.2 Non overconstrained solutions

495

Fig. 7.68. PassPR-PassPC*-RPassPassPass-type non overconstrained SPMs with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, TF=1, limb topology Pass ⊥ P ⊥ ||R, Pass ⊥ P ⊥ ||C* and NF=0, ss ss ss ⊥ ss ss ⊥ ss R ⊥ Pa ⊥ Pa ||Pa (a), R ⊥ Pa ||Pa ⊥ Pa (b)

496 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.69. Non overconstrained SPMs with planar motion of the moving platform of types 2PassRRR*R-1RPP (a) and 2PassPassRRR*R-1RPP defined by SF=3, (RF)=( v1 , v2 , δ ), NF=0 and MF=3, TF=0, (a), MF=7, TF=4 (b), limb topology

R ⊥ P ⊥ ⊥ P and Pass ⊥ R||R ⊥ ⊥ R ⊥ ⊥ R (a), Pass ⊥ Pass||R||R ⊥ ⊥ R ⊥ ⊥ R (b)

7.2 Non overconstrained solutions

497

Fig. 7.70. Non overconstrained SPMs with planar motion of the moving platform of types 2PassRPassR-1RPP (a) and 2PassPassPassR-1RPP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=0 and MF=3, TF=0, (a), MF=5, TF=2 (b), limb topology

R ⊥ P ⊥ ⊥ P and Pass ⊥ R||Pass ⊥ ||R (a), Pass ⊥ Pass||Pass ⊥ ||R (b)

498 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.71. 2PassRbssRR*R-1RPP-type non overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, NF=0,

TF=2, limb topology Pass ⊥ Rbss||R ⊥ ⊥ R ⊥ ⊥ R and R ⊥ P ⊥ ⊥ P

7.2 Non overconstrained solutions

499

Fig. 7.72. 2PassRbssRbcsRR*R-1RPP-type non overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, NF=0,

TF=2, limb topology Pass ⊥ Rbss||Rbcs||R ⊥ ⊥ R ⊥ ⊥ R and R ⊥ P ⊥ ⊥ P

500 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.73. Non overconstrained SPMs with planar motion of the moving platform of types 2PassPn2ssRR*R-1RPP (a) and 2PassPn3ssR*R-1RPP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=7, NF=0, TF=4 limb topology R ⊥ P ⊥ ⊥ P and

Pass ⊥ Pn2ss||R ⊥ ⊥ R ⊥ ⊥ R (a), Pass ⊥ Pn3ss ⊥ ⊥ R ⊥ ⊥ R (b)

7.2 Non overconstrained solutions

501

Fig. 7.74. Non overconstrained SPMs with planar motion of the moving platform of types 2PassRRR*R-1RPPass (a) and 2PassPassRRR*R-1RPPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=0 and MF=4, TF=1, (a), MF=8, TF=5 (b), limb

topology R ⊥ P ⊥ ||Pass and Pass ⊥ R||R ⊥ ⊥ R ⊥ ⊥ R (a), Pass ⊥ Pass||R||R ⊥ ⊥ R ⊥ ⊥ R (b)

502 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.75. Non overconstrained SPMs with planar motion of the moving platform of types 2PassRPassR-1RPPass (a) and 2PassPassPassR-1RPPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=0 and MF=4, TF=1, (a), MF=6, TF=3 (b), limb topology R ⊥ P ⊥ ||Pass and Pass ⊥ R||Pass ⊥ ||R (a), Pass ⊥ Pass||Pass ⊥ ||R (b)

7.2 Non overconstrained solutions

503

Fig. 7.76. 2PassRbssRR*R-1RPPass-type non overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=6, NF=0,

TF=3, limb topology Pass ⊥ Rbss||R ⊥ ⊥ R ⊥ ⊥ R and R ⊥ P ⊥ ||Pass

504 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.77. 2PassRbssRbcsRR*R-1RPPass-type non overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=6, NF=0,

TF=3, limb topology Pass ⊥ Rbss||Rbcs||R ⊥ ⊥ R ⊥ ⊥ R and R ⊥ P ⊥ ||Pass

7.2 Non overconstrained solutions

505

Fig. 7.78. Non overconstrained SPMs with planar motion of the moving platform of types 2PassPn2ssRR*R-1RPPass (a) and 2PassPn3ssR*R-1RPPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=8, NF=0, TF=5, limb topology R ⊥ P ⊥ ||Pass and

Pass ⊥ Pn2ss||R ⊥ ⊥ R ⊥ ⊥ R (a), Pass ⊥ Pn3ss ⊥ ⊥ R ⊥ ⊥ R (b)

506 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.79. Non overconstrained SPMs with planar motion of the moving platform of types 2PassRRR*R-1RPassP (a) and 2PassPassRRR*R-1RPassP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=0 and MF=4, TF=1, (a), MF=8, TF=5 (b), limb topology R||Pass ⊥ P and Pass ⊥ R||R ⊥ ⊥ R ⊥ ⊥ R (a), Pass ⊥ Pass||R||R ⊥ ⊥ R ⊥ ⊥ R (b)

7.2 Non overconstrained solutions

507

Fig. 7.80. Non overconstrained SPMs with planar motion of the moving platform of types 2PassRPassR-1RPassP (a) and 2PassPassPassR-1RPassP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=0 and MF=4, TF=1, (a), MF=6, TF=3 (b), limb topology R||Pass ⊥ P and Pass ⊥ R||Pass ⊥ ||R (a), Pass ⊥ Pass||Pass ⊥ ||R (b)

508 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.81. 2PassRbssRR*R-1RPassP-type non overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=6, NF=0,

TF=3, limb topology Pass ⊥ Rbss||R ⊥ ⊥ R ⊥ ⊥ R and R||Pass ⊥ P

7.2 Non overconstrained solutions

509

Fig. 7.82. 2PassRbssRbcsRR*R-1RPassP-type non overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=6, NF=0,

TF=3, limb topology Pass ⊥ Rbss||Rbcs||R ⊥ ⊥ R ⊥ ⊥ R and R||Pass ⊥ P

510 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.83. Non overconstrained SPMs with planar motion of the moving platform of types 2PassPn2ssRR*R-1RPassP (a) and 2PassPn3ssR*R-1RPassP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=8, NF=0, TF=5, limb topology R||Pass ⊥ P and

Pass ⊥ Pn2ss||R ⊥ ⊥ R ⊥ ⊥ R (a), Pass ⊥ Pn3ss ⊥ ⊥ R ⊥ ⊥ R (b)

7.2 Non overconstrained solutions

511

Fig. 7.84. Non overconstrained SPMs with planar motion of the moving platform of types 2PassRRR*R-1RPassPass (a) and 2PassPassRRR*R-1RPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=0 and MF=5, TF=2, (a), MF=9, TF=6 (b), limb topology R||Pass||Pass and Pass ⊥ R||R ⊥ ⊥ R ⊥ ⊥ R (a), Pass ⊥ Pass||R||R ⊥ ⊥ R ⊥ ⊥ R (b)

512 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.85. Non overconstrained SPMs with planar motion of the moving platform of types 2PassRPassR-1RPassPass (a) and 2PassPassPassR-1RPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=0 and MF=5, TF=2, (a), MF=7, TF=4 (b), limb topology R||Pass||Pass and Pass ⊥ R||Pass ⊥ ||R (a), Pass ⊥ Pass||Pass ⊥ ||R (b)

7.2 Non overconstrained solutions

513

Fig. 7.86. 2PassRbssRR*R-1RPassPass-type non overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=7, NF=0,

TF=4, limb topology Pass ⊥ Rbss||R ⊥ ⊥ R ⊥ ⊥ R and R||Pass||Pass

514 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.87. 2PassRbssRbcsRR*R-1RPassPass-type non overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=7,

NF=0, TF=4, limb topology Pass ⊥ Rbss||Rbcs||R ⊥ ⊥ R ⊥ ⊥ R and R||Pass||Pass

7.2 Non overconstrained solutions

515

Fig. 7.88. Non overconstrained SPMs with planar motion of the moving platform of types 2PassPn2ssRR*R-1RPassPass (a) and 2PassPn3ssR*R-1RPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=9, NF=0, TF=6, limb topology

R||Pass||Pass and Pass ⊥ Pn2ss||R ⊥ ⊥ R ⊥ ⊥ R (a), Pass ⊥ Pn3ss ⊥ ⊥ R ⊥ ⊥ R (b)

516 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.89. Non overconstrained SPMs with planar motion of the moving platform of types 2PassRRR*R-1RPacsPacs (a) and 2PassRRR*R-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology Pass ⊥ R||R ⊥ R* ⊥ ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

7.2 Non overconstrained solutions

517

Fig. 7.90. Non overconstrained SPMs with planar motion of the moving platform of types 2PassPassRRR*R-1RPacsPacs (a) and 2PassPassRR*R-1RPacsPatcs (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=7, NF=0, TF=4, limb topology

Pass ⊥ Pass||R||R ⊥ R* ⊥ ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

518 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.91. Non overconstrained SPMs with planar motion of the moving platform of types 2PassRPassR-1RPacsPacs (a) and 2PaRRPaR-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=0, limb topology Pass ⊥ R||Pass ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

7.2 Non overconstrained solutions

519

Fig. 7.92. Non overconstrained SPMs with planar motion of the moving platform of types 2PassPassPassR-1RPacsPacs (a) and 2PassPassPassR-1RPacsPatcs (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, NF=0, TF=2, limb topology Pass ⊥ Pass||Pass ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

520 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.93. 2PassRbssRR*R-1RPacsPacs-type non overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, NF=0,

TF=2, limb topology Pass ⊥ Rbss||R ⊥ R* ⊥ ⊥ R and R||Pacs||Pacs

7.2 Non overconstrained solutions

521

Fig. 7.94. 2PassRbssRR*R-1RPacsPatcs-type non overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, NF=0,

TF=2, limb topology Pass ⊥ Rbss||R ⊥ R* ⊥ ⊥ R and R||Pacs||Patcs

522 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.95. 2PassRbssRbcsRR*R-1RPacsPacs-type non overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5,

NF=0, TF=2, limb topology Pass ⊥ Rbss||Rbcs||R ⊥ R* ⊥ ⊥ R and R||Pacs||Pacs

7.2 Non overconstrained solutions

523

Fig. 7.96. 2PassRbssRbcsRR*R-1RPacsPatcs-type non overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5,

NF=0, TF=2, limb topology Pass ⊥ Rbss||Rbcs||R ⊥ R* ⊥ ⊥ R and R||Pacs||Patcs

524 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.97. 2PassPn2ssRR*R-1RPacsPacs-type non overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=7, NF=0,

TF=4, limb topology Pass ⊥ Pn2ss||R ⊥ R* ⊥ ⊥ R and R||Pacs||Pacs

7.2 Non overconstrained solutions

525

Fig. 7.98. 2PassPn2ssRR*R-1RPacsPatcs-type non overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=7, NF=0,

TF=4, limb topology Pass ⊥ Pn2ss||R ⊥ R* ⊥ ⊥ R and R||Pacs||Patcs

526 7 Spatial PMs with uncoupled planar motion of the moving platform

Fig. 7.99. 2PassPn3ssR*R-1RPacsPacs-type non overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=7, NF=0,

TF=4, limb topology Pass ⊥ Pn3ss ⊥ R* ⊥ ⊥ R and R||Pacs||Pacs

7.2 Non overconstrained solutions

527

Fig. 7.100. 2PassPn3ssR*R-1RPacsPatcs-type non overconstrained SPM with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=7, NF=0,

TF=4, limb topology Pass ⊥ Pn3ss ⊥ R* ⊥ ⊥ R and R||Pacs||Patcs

8 Maximally regular SPMs with planar motion of the moving platform

Maximally regular spatial parallel robotic manipulators (SPMs) are actuated by one rotating and two linear actuators 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 ωδ = &q3 . The Jacobian matrix in Eq. (1.18) is the identity matrix. We call spatial Isoglide3-T2R1 with planar motion of the moving platform the parallel mechanisms of this family in which at least one limb is a spatial kinematic chain.

8.1 Overconstrained solutions Equation (1.16) indicates that overconstrained solutions of maximally regular spatial parallel robots with q independent loops meet the condition p ∑ 1 fi < 3 + 6q . Various solutions fulfil this condition along with MF=SF=3 and (RF)=( v1 ,v2 ,ωδ ). They have two identical limbs for positioning and a different limb for rotating the moving platform. 8.1.1 Basic solutions In the basic solutions of overconstrained maximally regular spatial parallel robots, F ← G1-G2-G3, the moving platform nznGi (i=1, 2, 3) is connected to the reference platform 1z1Giz0 by three limbs. Two planar or spatial limbs G1 and G2 are used for positioning the moving platform and a spatial or planar limb G3 for orienting it. Thera are no idle mobilities in the basic solutions. Various solutions of maximally regular SPMs with planar motion of the moving platform and no idle mobilities can be obtained by using G1- and G2-limbs with identical or different topology presented in Figs. 2.1f, 8.1 and 8.2, and a planar or spatial G3-limb in Figs. 2.1g, 2.2f-h, 2.3 and 6.1. G. Gogu, Structural Synthesis of Parallel Robots, Solid Mechanics and Its Applications 173, DOI 10.1007/978-90-481-9831-3_8, © Springer Science + Business Media B.V. 2010

529

530

8 Maximally regular SPMs with planar motion of the moving platform

Only solutions with identical G1- and G2-limbs are illustrated in Figs. 8.38.29. The bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms in Figs. 8.3-8.29 are given in Table 8.1. The limb topology and connecting conditions of these solutions are systematized in Tables 8.2-8.8, as are their structural parameters in Tables 8.9-8.19.

Table 8.1. Bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 8.3-8.29 No. Parallel mechanism 1 Fig. 8.3a 2 3

4

Basis (RG1) ( v1 , v 2 ,

δ

)

Figs. 8.3b, 8.4 ( v1 , v2 , δ ) Figs. 8.5, 8.6a, ( v1 , v2 , v2 , α , 8.7-8.9, 8.10a, 8.11-8.13, 8.14a 8.15-8.17, 8.18a, 8.19-8.23, 8.258.29 Figs. 8.6b, 8.10b, ( v1 ,v2 , v2 , δ ) 7.11b, 8.14b, 8.18b, 8.24

δ

(RG2) ( v1 , v 2 ,

δ

)

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

α

,

( v1 , v 2 ,

δ

)

( v1 , v 2 , v 2 ,

δ

)

) ( v1 , v 2 , v 2 ,

β

,

( v1 , v 2 , v 2 ,

δ

)

δ

) ( v1 , v 2 ,

δ

)

( v1 , v 2 ,

δ

)

β

,

δ

)

8.1 Overconstrained solutions

531

Table 8.2. Limb topology and connecting conditions of the overconstrained maximally regular SPMs with planar motion of the moving platform and no idle mobilities presented in Figs. 8.3-8.5 No. SPM type 1

2PPR-1RRRPRR (Fig. 8.3a)

2

2PPR-1RPaPaPa (Fig. 8.3b)

3

2PPR-1RPaPaPa (Fig. 8.4a)

4

2PPR-1RPaPaPa (Fig. 8.4b)

5

2PRRRR-1RPP (Fig. 8.5a)

6

2PPaRRR-1RPP (Fig. 8.5b)

Limb topology P ⊥ P ⊥⊥ R (Fig. 2.1f) R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R (Fig. 6.1a)

P ⊥ P ⊥⊥ R (Fig. 2.1f) R ⊥ Pa ⊥ ⊥ Pa ⊥ ||Pa (Fig. 6.1b) P ⊥ P ⊥⊥ R (Fig. 2.1f) R ⊥ Pa ⊥ ⊥ Pa||Pa (Fig. 6.1c) P ⊥ P ⊥⊥ R (Fig. 2.1f) R ⊥ Pa||Pa ⊥ ⊥ Pa (Fig. 6.1d) P||R||R||R ⊥ R (Fig. 8.1a) R ⊥ P ⊥⊥ P (Fig. 2.1g) P||Pa||R||R ⊥ R (Fig. 8.1b) R ⊥ P ⊥⊥ P (Fig. 2.1g)

Connecting conditions The directions of the translational actuated joints are orthogonal. The last revolute joints of limbs G1 and G2 have the same axis which is parallel to the axis of the actuated revolute joint of limb G3. Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

532

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.3. Limb topology and connecting conditions of the overconstrained maximally regular SPMs with planar motion of the moving platform and no idle mobilities presented in Figs. 8.6-8.9 No. SPM type 1

2PRRPaR-1RPP (Fig. 8.6a)

2

2PPaPaR-1RPP (Fig. 8.6b)

3

2PRRbRR-1RPP (Fig. 8.7a)

4

2PRRbRbRR-1RPP (Fig. 8.7b)

5

2PPn2RR-1RPP (Fig. 8.8a)

6

2PPn3R-1RPP (Fig. 8.8b)

7

2PRRRR-1RPPa (Fig. 8.9a)

8

2PPaRRR-1RPPa (Fig. 8.9b)

Limb topology P||R||R||Pa ⊥ R (Fig. 8.1c) R ⊥ P ⊥⊥ P (Fig. 2.1g)

P||Pa||Pa ⊥ R (Fig. 8.1d) R ⊥ P ⊥⊥ P (Fig. 2.1g) P||R||Rb||R ⊥ R (Fig. 8.2a) R ⊥ P ⊥⊥ P (Fig. 2.1g) P||R||Rb||Rb||R ⊥ R (Fig. 8.2b) R ⊥ P ⊥⊥ P (Fig. 2.1g) P||Pn2||R ⊥ R (Fig. 8.2c) R ⊥ P ⊥⊥ P (Fig. 2.1g) P||Pn3 ⊥ R (Fig. 8.2d) R ⊥ P ⊥⊥ P (Fig. 2.1g) P||R||R||R ⊥ R (Fig. 8.1a) R ⊥ P ⊥ ||Pa (Fig. 2.2f) P||Pa||R||R ⊥ R (Fig. 8.1b) R ⊥ P ⊥ ||Pa (Fig. 2.2f)

Connecting conditions The directions of the translational actuated joints are orthogonal. The last revolute joints of limbs G1 and G2 have the same axis which is parallel to the axis of the actuated revolute joint of limb G3. Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

8.1 Overconstrained solutions

533

Table 8.4. Limb topology and connecting conditions of the overconstrained maximally regular SPMs with planar motion of the moving platform and no idle mobilities presented in Figs. 8.10-8.13 No. SPM type 1

2PRRPaR-1RPPa (Fig. 8.10a)

2

2PPaPaR-1RPPa (Fig. 8.10b)

3

2PRRbRR-1RPPa (Fig. 8.11a)

4

2PRRbRbRR-1RPPa (Fig. 8.11b)

5

2PPn2RR-1RPPa (Fig. 8.12a)

6

2PPn3R-1RPPa (Fig. 8.12b)

7

2PRRRR-1RPaP (Fig. 8.13a)

8

2PPaRRR-1RPaP (Fig. 8.13b)

Limb topology P||R||R||Pa ⊥ R (Fig. 8.1c) R ⊥ P ⊥ ||Pa (Fig. 2.2f)

P||Pa||Pa ⊥ R (Fig. 8.1d) R ⊥ P ⊥ ||Pa (Fig. 2.2f) P||R||Rb||R ⊥ R (Fig. 8.2a) R ⊥ P ⊥ ||Pa (Fig. 2.2f) P||R||Rb||Rb||R ⊥ R (Fig. 8.2b) R ⊥ P ⊥ ||Pa (Fig. 2.2f) P||Pn2||R ⊥ R (Fig. 8.2c) R ⊥ P ⊥ ||Pa (Fig. 2.2f) P||Pn3 ⊥ R (Fig. 8.2c) R ⊥ P ⊥ ||Pa (Fig. 2.2f) P||R||R||R ⊥ R (Fig. 8.1a) R||Pa ⊥ P (Fig. 2.2g) P||Pa||R||R ⊥ R (Fig. 8.1b) R||Pa ⊥ P (Fig. 2.2g)

Connecting conditions The directions of the translational actuated joints are orthogonal. The last revolute joints of limbs G1 and G2 have the same axis which is parallel to the axis of the actuated revolute joint of limb G3. Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

534

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.5. Limb topology and connecting conditions of the overconstrained maximally regular SPMs with planar motion of the moving platform and no idle mobilities presented in Figs. 8.14-8.17 No. SPM type 1

2PRRPaR-1RPaP (Fig. 8.14a)

2

2PPaPaR-1RPaP (Fig. 8.14b)

3

2PRRbRR-1RPaP (Fig. 8.15a)

4

2PRRbRbRR-1RPaP (Fig. 8.15b)

5

2PPn2RR-1RPaP (Fig. 8.16a)

6

2PPn3R-1RPaP (Fig. 8.16b)

7

2PRRRR-1RPaPa (Fig. 8.17a)

8

2PPaRRR-1RPaPa (Fig. 8.17b)

Limb topology P||R||R||Pa ⊥ R (Fig. 8.1c) R||Pa ⊥ P (Fig. 2.2g)

P||Pa||Pa ⊥ R (Fig. 8.1d) R||Pa ⊥ P (Fig. 2.2g) P||R||Rb||R ⊥ R (Fig. 8.2a) R||Pa ⊥ P (Fig. 2.2g) P||R||Rb||Rb||R ⊥ R (Fig. 8.2b) R||Pa ⊥ P (Fig. 2.2g) P||Pn2||R ⊥ R (Fig. 8.2c) R||Pa ⊥ P (Fig. 2.2g) P||Pn3 ⊥ R (Fig. 8.2d) R||Pa ⊥ P (Fig. 2.2g) P||R||R||R ⊥ R (Fig. 8.1a) R||Pa||Pa (Fig. 2.2h) P||Pa||R||R ⊥ R (Fig. 8.1b) R||Pa||Pa (Fig. 2.2h)

Connecting conditions The directions of the translational actuated joints are orthogonal. The last revolute joints of limbs G1 and G2 have the same axis which is parallel to the axis of the actuated revolute joint of limb G3. Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

8.1 Overconstrained solutions

535

Table 8.6. Limb topology and connecting conditions of the overconstrained maximally regular SPMs with planar motion of the moving platform and no idle mobilities presented in Figs. 8.18-8.21 No. SPM type 1

2PRRPaR-1RPaPa (Fig. 8.18a)

2

2PPaPaR-1RPaPa (Fig. 8.18b)

3

4

5

6

7

8

Limb topology P||R||R||Pa ⊥ R (Fig. 8.1c) R||Pa||Pa (Fig. 2.2h)

P||Pa||Pa ⊥ R (Fig. 8.1d) R||Pa||Pa (Fig. 2.2h) 2PRRbRR-1RPaPa P||R||Rb||R ⊥ R (Fig. 8.19a) (Fig. 8.2a) R||Pa||Pa (Fig. 2.2h) 2PRRbRbRR-1RPaPa P||R||Rb||Rb||R ⊥ R (Fig. 8.19b) (Fig. 8.2b) R||Pa||Pa (Fig. 2.2h) 2PPn2RR-1RPaPa P||Pn2||R ⊥ R (Fig. 8.20a) (Fig. 8.2c) R||Pa||Pa (Fig. 2.2h) P||Pn3 ⊥ R 2PPn3R-1RPaPa (Fig. 8.20b) (Fig. 8.2d) R||Pa||Pa (Fig. 2.2h) 2PRRRR-1RPaPa P||R||R||R ⊥ R (Fig. 8.21a) (Fig. 8.1a) R||Pa||Pa (Fig. 2.3a) 2PRRRR-1RPaPat P||R||R||R ⊥ R (Fig. 8.21b) (Fig. 8.1a) R||Pa||Pat (Fig. 2.3b)

Connecting conditions The directions of the translational actuated joints are orthogonal. The last revolute joints of limbs G1 and G2 have the same axis which is parallel to the axis of the actuated revolute joint of limb G3. Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

536

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.7. Limb topology and connecting conditions of the overconstrained maximally regular SPMs with planar motion of the moving platform and no idle mobilities presented in Figs. 8.22-8.25 No. SPM type 1

2PPaRRR-1RPaPa (Fig. 8.22a)

2

2PPaRRR-1RPaPat (Fig. 8.22b)

3

2PRRPaR-1RPaPa (Fig. 8.23a)

4

2PRRPaR-1RPaPat (Fig. 8.23b)

5

2PPaPaR-1RPaPa (Fig. 8.24a)

6

2PPaPaR-1RPaPat (Fig. 8.24b)

7

2PRRbRR-1RPaPa (Fig. 8.25a)

8

2PRRbRR-1RPaPat (Fig. 8.25b)

Limb topology P||Pa||R||R ⊥ R (Fig. 8.1b) R||Pa||Pa (Fig. 2.3a)

P||Pa||R||R ⊥ R (Fig. 8.1b) R||Pa||Pat (Fig. 2.3b) P||R||R||Pa ⊥ R (Fig. 8.1c) R||Pa||Pa (Fig. 2.3a) P||R||R||Pa ⊥ R (Fig. 8.1c) R||Pa||Pat (Fig. 2.3b) P||Pa||Pa ⊥ R (Fig. 8.1d) R||Pa||Pa (Fig. 2.3a) P||Pa||Pa ⊥ R (Fig. 8.1d) R||Pa||Pat (Fig. 2.3b) P||R||Rb||R ⊥ R (Fig. 8.2a) R||Pa||Pa (Fig. 2.3a) P||R||Rb||R ⊥ R (Fig. 8.2a) R||Pa||Pat (Fig. 2.3b)

Connecting conditions The directions of the translational actuated joints are orthogonal. The last revolute joints of limbs G1 and G2 have the same axis which is parallel to the axis of the actuated revolute joint of limb G3. Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

8.1 Overconstrained solutions

537

Table 8.8. Limb topology and connecting conditions of the overconstrained maximally regular SPMs with planar motion of the moving platform and no idle mobilities presented in Figs. 8.26-8.29 No. SPM type 1

2

3

4

5

6

Limb topology

2PRRbRbRR-1RPaPa P||R||Rb||Rb||R ⊥ R (Fig. 8.26) (Fig. 8.2b) R||Pa||Pa (Fig. 2.3a)

2PRRbRbRR-1RPaPat P||R||Rb||Rb||R ⊥ R (Fig. 8.27) (Fig. 8.2b) R||Pa||Pat (Fig. 2.3b) 2PPn2RR-1RPaPa P||Pn2||R ⊥ R (Fig. 8.28a) (Fig. 8.2c) R||Pa||Pa (Fig. 2.3a) 2PPn2RR-1RPaPat P||Pn2||R ⊥ R (Fig. 8.28b) (Fig. 8.2c) R||Pa||Pat (Fig. 2.3a) 2PPn3R-1RPaPa P||Pn3 ⊥ R (Fig. 8.29a) (Fig. 8.2d) R||Pa||Pa (Fig. 2.3a) P||Pn3 ⊥ R 2PPn3R-1RPaPat (Fig. 8.29b) (Fig. 8.2d) R||Pa||Pat (Fig. 2.3b)

Connecting conditions The directions of the translational actuated joints are orthogonal. The last revolute joints of limbs G1 and G2 have the same axis which is parallel to the axis of the actuated revolute joint of limb G3. Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

Idem No. 1

538

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.9. Structural parametersa of spatial parallel mechanisms in Figs. 8.3 and 8.4 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 2PPR-1RRRPRR (Fig. 8.3a)

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 3 3 6 12 2 3 0 3 See Table 8.1

2PPR-1RPaPaPa (Fig. 8.3b) 2PPR-1RPaPaPa (Fig. 8.4a,b) 15 3 3 13 19 5 2 1 3 See Table 8.1

3 3 6 0 0 0 3 3 6 ( v1 , v 2 , 3 0 9 3 3 0 3

3 3 4 0 0 9 3 3 4 ( v1 , v 2 , 3 9 16 3 14 0 3

SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

fj

3

3

fj

6

13

fj

12

19

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

8.1 Overconstrained solutions

539

Table 8.10. Structural parametersa of spatial parallel mechanisms in Figs. 8.5 and 8.6 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 2PRRRR-1RPP (Fig. 8.5a)

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

2PPaPaR-1RPP (Fig. 8.6b)

12 5 5 3 13 2 3 0 3 See Table 8.1

2PPaRRR-1RPP (Fig. 8.5b) 2PRRPaR-1RPP (Fig. 8.6a) 16 8 8 3 19 4 1 2 3 See Table 8.1

5 5 3 0 0 0 5 5 3 ( v1 , v 2 , 3 0 10 3 2 0 5

5 5 3 3 3 0 5 5 3 ( v1 , v 2 , 3 6 16 3 8 0 8

4 4 3 6 6 0 4 4 3 ( v1 , v 2 , 3 12 20 3 16 0 10

δ

)

δ

)

18 10 10 3 23 6 1 2 3 See Table 8.1

fj

5

8

10

fj

3

3

3

fj

13

19

23

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

540

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.11. Structural parametersa of spatial parallel mechanisms in Figs. 8.7 and 8.8 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

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

Solution 2PRRbRR-1RPP (Fig. 8.7a) 2PPn2RR-1RPP (Fig. 8.8a) 2PPn3R-1RPP (Fig. 8.8b) 16 8 8 3 19 4 1 2 3 See Table 8.1 5 5 3 3 3 0 5 5 3 ( v1 , v 2 , 3 6 16 3 8 0 8

δ

)

2PRRbRbRR-1RPP (Fig. 8.7b)

20 11 11 3 25 6 1 2 3 See Table 8.1 5 5 3 6 6 0 5 5 3 ( v1 , v 2 , 3 12 22 3 14 0 11

fj

8

11

fj

3

3

fj

19

25

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

8.1 Overconstrained solutions

541

Table 8.12. Structural parametersa of spatial parallel mechanisms in Figs. 8.9 and 8.10 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 2PRRRR-1RPPa (Fig. 8.9a)

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

2PPaPaR-1RPPa (Fig. 8.10b)

14 5 5 6 16 3 2 1 3 See Table 8.1

2PPaRRR-1RPPa (Fig. 8.9b) 2PRRPaR-1RPPa (Fig. 8.10a) 18 8 8 6 22 5 0 3 3 See Table 8.1

20 10 10 6 26 7 0 3 3 See Table 8.1

5 5 3 0 0 3 5 5 3 ( v1 , v 2 , 3 3 13 3 5 0 5

5 5 3 3 3 3 5 5 3 ( v1 , v 2 , 3 9 19 3 11 0 8

4 4 3 6 6 3 4 4 3 ( v1 , v 2 , 3 15 23 3 19 0 10

δ

)

δ

)

fj

5

8

10

fj

6

6

6

fj

16

22

26

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

542

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.13. Structural parametersa of spatial parallel mechanisms in Figs. 8.11 and 8.12 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

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

Solution 2PRRbRR-1RPPa (Fig. 8.11a) 2PPn2RR-1RPPa (Fig. 8.12a) 2PPn3R-1RPPa (Fig. 8.12b) 18 8 8 6 22 5 0 3 3 See Table 8.1

22 11 11 6 28 7 0 3 3 See Table 8.1

5 5 3 3 3 3 5 5 3 ( v1 , v 2 , 3 9 19 3 11 0 8

5 5 3 6 6 3 5 5 3 ( v1 , v 2 , 3 15 25 3 17 0 11

δ

)

2PRRbRbRR-1RPPa (Fig. 8.11b)

fj

8

11

fj

6

6

fj

22

28

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

8.1 Overconstrained solutions

543

Table 8.14. Structural parametersa of spatial parallel mechanisms in Figs. 8.13 and 8.14 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 2PRRRR-1RPaP (Fig. 8.13a)

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

2PPaPaR-1RPaP (Fig. 8.14b)

14 5 5 6 16 3 2 1 3 See Table 8.1

2PPaRRR-1RPaP (Fig. 8.13b) 2PRRPaR-1RPaP (Fig. 8.14a) 18 8 8 6 22 5 0 3 3 See Table 8.1

20 10 10 6 26 7 0 3 3 See Table 8.1

5 5 3 0 0 3 5 5 3 ( v1 , v 2 , 3 3 13 3 5 0 5

5 5 3 3 3 3 5 5 3 ( v1 , v 2 , 3 9 19 3 11 0 8

4 4 3 6 6 3 4 4 3 ( v1 , v 2 , 3 15 23 3 19 0 10

δ

)

δ

)

fj

5

8

10

fj

6

6

6

fj

16

22

26

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

544

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.15. Structural parametersa of spatial parallel mechanisms in Figs. 8.15 and 8.16 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

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

Solution 2PRRbRR-1RPaP (Fig. 8.15a) 2PPn2RR-1RPaP (Fig. 8.16a) 2PPn3R-1RPaP (Fig. 8.16b) 18 8 8 6 22 5 0 3 3 See Table 8.1

22 11 11 6 28 7 0 3 3 See Table 8.1

5 5 3 3 3 3 5 5 3 ( v1 , v 2 , 3 9 19 3 11 0 8

5 5 3 6 6 3 5 5 3 ( v1 , v 2 , 3 15 25 3 17 0 11

δ

)

2PRRbRbRR-1RPaP (Fig. 8.15b)

fj

8

11

fj

6

6

fj

22

28

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

8.1 Overconstrained solutions

545

Table 8.16. Structural parametersa of spatial parallel mechanisms in Figs. 8.17 and 8.18 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 2PRRRR-1RPaPa (Fig. 8.17a)

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

2PPaPaR-1RPaPa (Fig. 8.18b)

16 5 5 9 19 4 2 1 3 See Table 8.1

2PPaRRR-1RPaPa (Fig. 8.17b) 2PRRPaR-1RPaPa (Fig. 8.18a) 20 8 8 9 25 6 0 3 3 See Table 8.1

22 10 10 9 29 8 0 3 3 See Table 8.1

5 5 3 0 0 6 5 5 3 ( v1 , v 2 , 3 6 16 3 8 0 5

5 5 3 3 3 6 5 5 3 ( v1 , v 2 , 3 12 22 3 14 0 8

4 4 3 6 6 6 4 4 3 ( v1 , v 2 , 3 18 26 3 22 0 10

δ

)

δ

)

fj

5

8

10

fj

9

9

9

fj

19

25

29

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

546

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.17 Structural parametersa of spatial parallel mechanisms in Figs. 8.19 and 8.20 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

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

Solution 2PRRbRR-1RPaPa (Fig. 8.19a) 2PPn2RR-1RPaPa (Fig. 8.20a) 2PPn3R-1RPaPa (Fig. 8.20b) 20 8 8 9 25 6 0 3 3 See Table 8.1 5 5 3 3 3 6 5 5 3 ( v1 , v 2 , 3 12 22 3 14 0 8

δ

)

2PRRbRbRR-1RPaPa (Fig. 8.19b)

24 11 11 9 31 8 0 3 3 See Table 8.1 5 5 3 6 6 6 5 5 3 ( v1 , v 2 , 3 18 28 3 20 0 11

fj

8

11

fj

9

9

fj

25

31

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

8.1 Overconstrained solutions

547

Table 8.18. Structural parametersa of spatial parallel mechanisms in Figs. 8.218.24 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 2PRRRR-1RPaPa (Fig. 8.21a) 2PRRRR-1RPaPat (Fig. 8.21b)

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

16 5 5 9 19 4 2 1 3 See Table 8.1

2PPaRRR-1RPaPa 2PPaRRR-1RPaPat (Fig. 8.22a,b) 2PRRPaR-1RPaPa 2PRRPaR-1RPaPat (Fig. 8.23a,b) 20 8 8 9 25 6 0 3 3 See Table 8.1

22 10 10 9 29 8 0 3 3 See Table 8.1

5 5 3 0 0 6 5 5 3 ( v1 , v 2 , 3 6 16 3 8 0 5

5 5 3 3 3 6 5 5 3 ( v1 , v 2 , 3 12 22 3 14 0 8

4 4 3 6 6 6 4 4 3 ( v1 , v 2 , 3 18 26 3 22 0 10

δ

)

δ

)

2PPaPaR-1RPaPa (Fig. 8.24a) 2PPaPaR-1RPaPat (Fig. 8.24b)

fj

5

8

10

fj

9

9

9

fj

19

25

29

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

548

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.19. Structural parametersa of spatial parallel mechanisms in Figs. 8.258.29 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

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

Solution 2PRRbRR-1RPaPa (Fig. 8.25a) 2PRRbRR-1RPaPat (Fig. 8.25b) 2PPn2RR-1RPaPa (Fig. 8.28a) 2PPn2RR-1RPaPat (Fig. 8.28b) 2PPn3R-1RPaPa (Fig. 8.29a) 2PPn3R-1RPaPat (Fig. 8.29b) 20 8 8 9 25 6 0 3 3 See Table 8.1

24 11 11 9 31 8 0 3 3 See Table 8.1

5 5 3 3 3 6 5 5 3 ( v1 , v 2 , 3 12 22 3 14 0 8

5 5 3 6 6 6 5 5 3 ( v1 , v 2 , 3 18 28 3 20 0 11

δ

)

2PRRbRbRR-1RPaPa (Fig. 8.26) 2PRRbRbRR-1RPaPat (Fig. 8.27)

fj

8

11

fj

9

9

fj

25

31

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

8.1 Overconstrained solutions

549

Fig. 8.1. Simple (a) and complex (b-d) limbs G1 and G2 with MG=SG=5 (a-c) and MG=SG=4 (d) for SPMs with planar motion of the moving platform

550

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.2. Complex limbs G1 and G2 with MG=SG=5 for SPMs with planar motion of the moving platform

8.1 Overconstrained solutions

551

Fig. 8.3. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPR-1RRRPRR (a) and 2PPR-1RPaPaPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=3 (a), NF=14 (b), limb topology

P ⊥ P ⊥ ⊥ R and R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R (a), R ⊥ Pa ⊥ ⊥ Pa ⊥ ||Pa (b)

552

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.4. 2PPR-1RPaPaPa-type overconstrained maximally regular SPMs with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14, limb R ⊥ Pa||Pa ⊥ ⊥ Pa (b)

topology

P ⊥ P ⊥⊥ R

and

R ⊥ Pa ⊥ ⊥ Pa||Pa

(a),

8.1 Overconstrained solutions

553

Fig. 8.5. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRRR-1RPP (a) and 2PPaRRR-1RPP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=2 (a), NF=8 (b), limb topology R ⊥ P ⊥ ⊥ P and P||R||R||R ⊥ R (a), P||Pa||R||R ⊥ R (b)

554

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.6. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRPaR-1RPP (a) and 2PPaPaR-1RPP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=8 (a), NF=16 (b), limb topology R ⊥ P ⊥ ⊥ P and P||R||R||Pa ⊥ R (a), P||Pa||Pa ⊥ R (b)

8.1 Overconstrained solutions

555

Fig. 8.7. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRbRR-1RPP (a) and 2PRRbRbRR-1RPP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=8 (a), NF=14 (b), limb topology R ⊥ P ⊥ ⊥ P and P||R||Rb||R ⊥ R (a), P||R||Rb||Rb||R ⊥ R (b)

556

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.8. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPn2RR-1RPP (a) and 2PPn3R-1RPP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=8, limb topology R ⊥ P ⊥ ⊥ P and P||Pn2||R ⊥ R (a), P||Pn3 ⊥ R (b)

8.1 Overconstrained solutions

557

Fig. 8.9. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRRR-1RPPa (a) and 2PPaRRR-1RPPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=5 (a), NF=11 (b), limb topology R ⊥ P ⊥ ||Pa and P||R||R||R ⊥ R (a), P||Pa||R||R ⊥ R (b)

558

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.10. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRPaR-1RPPa (a) and 2PPaPaR-1RPPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=11 (a), NF=19 (b), limb topology R ⊥ P ⊥ ||Pa and P||R||R||Pa ⊥ R (a), P||Pa||Pa ⊥ R (b)

8.1 Overconstrained solutions

559

Fig. 8.11. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRbRR-1RPPa (a) and 2PRRbRbRR-1RPPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=11 (a), NF=17 (b), limb topology R ⊥ P ⊥ ||Pa and P||R||Rb||R ⊥ R (a), P||R||Rb||Rb||R ⊥ R (b)

560

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.12. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPn2RR-1RPPa (a) and 2PPn3R-1RPPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=11, limb topology R ⊥ P ⊥ ||Pa and P||Pn2||R ⊥ R (a), P||Pn3 ⊥ R (b)

8.1 Overconstrained solutions

561

Fig. 8.13. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRRR-1RPaP (a) and 2PPaRRR-1RPaP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=5 (a), NF=11 (b), limb topology R||Pa ⊥ P and P||R||R||R ⊥ R (a), P||Pa||R||R ⊥ R (b)

562

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.14. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRPaR-1RPaP (a) and 2PPaPaR-1RPaP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=11 (a), NF=19 (b), limb topology R||Pa ⊥ P and P||R||R||Pa ⊥ R (a), P||Pa||Pa ⊥ R (b)

8.1 Overconstrained solutions

563

Fig. 8.15. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRbRR-1RPaP (a) and 2PRRbRbRR-1RPaP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=11 (a), NF=17 (b), limb topology R||Pa ⊥ P and P||R||Rb||R ⊥ R (a), P||R||Rb||Rb||R ⊥ R (b)

564

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.16. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPn2RR-1RPaP (a) and 2PPn3R-1RPaP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=11, limb topology R||Pa ⊥ P and P||Pn2||R ⊥ R (a), P||Pn3 ⊥ R (b)

8.1 Overconstrained solutions

565

Fig. 8.17. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRRR-1RPaPa (a) and 2PPaRRR-1RPaPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=8 (a), NF=14 (b), limb topology R||Pa||Pa and P||R||R||R ⊥ R (a), P||Pa||R||R ⊥ R (b)

566

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.18. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRPaR-1RPaPa (a) and 2PPaPaR-1RPaPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14 (a), NF=22 (b), limb topology R||Pa||Pa and P||R||R||Pa ⊥ R (a), P||Pa||Pa ⊥ R (b)

8.1 Overconstrained solutions

567

Fig. 8.19. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRbRR-1RPaPa (a) and 2PRRbRbRR-1RPaPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14 (a), NF=20 (b), limb topology R||Pa||Pa and P||R||Rb||R ⊥ R (a), P||R||Rb||Rb||R ⊥ R (b)

568

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.20. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPn2RR-1RPaPa (a) and 2PPn3R-1RPaPa (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14, limb topology R||Pa||Pa and P||Pn2||R ⊥ R (a), P||Pn3 ⊥ R (b)

8.1 Overconstrained solutions

569

Fig. 8.21. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRRR-1RPaPa (a) and 2PRRRR-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=8, limb topology P||R||R||R ⊥ R and R||Pa||Pa (a), R||Pa||Pat (b)

570

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.22. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPaRRR-1RPaPa (a) and 2PPaRRR-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14, limb topology P||Pa||R||R ⊥ R and R||Pa||Pa (a), R||Pa||Pat (b)

8.1 Overconstrained solutions

571

Fig. 8.23. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRPaR-1RPaPa (a) and 2PRRPaR-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14, limb topology P||R||R||Pa ⊥ R and R||Pa||Pa (a), R||Pa||Pat (b)

572

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.24. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPaPaR-1RPaPa (a) and 2PPaPaR-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=22, limb topology P||Pa||Pa ⊥ R and R||Pa||Pa (a), R||Pa||Pat (b)

8.1 Overconstrained solutions

573

Fig. 8.25. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRbRR-1RPaPa (a) and 2PRRbRR-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14, limb topology P||R||Rb||R ⊥ R and R||Pa||Pa (a), R||Pa||Pat (b)

574

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.26. 2PRRbRbRR-1RPaPa-type overconstrained maximally regular SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=20, limb topology P||R||Rb||Rb||R ⊥ R and R||Pa||Pa

8.1 Overconstrained solutions

575

Fig. 8.27. 2PRRbRbRR-1RPaPat-type overconstrained maximally regular SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=20, limb topology P||R||Rb||Rb||R ⊥ R and R||Pa||Pat

576

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.28. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPn2RR-1RPaPa (a) and 2PPn2RR-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14, limb topology P||Pn2||R ⊥ R and R||Pa||Pa (a), R||Pa||Pat (b)

8.1 Overconstrained solutions

577

Fig. 8.29. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPn3R-1RPaPa (a) and 2PPn3R-1RPaPat (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=14, limb topology P||Pn3 ⊥ R and R||Pa||Pa (a), R||Pa||Pat (b)

578

8 Maximally regular SPMs with planar motion of the moving platform

8.1.2 Derived solutions Solutions with lower degrees of overconstraint can be derived from the basic solutions in Figs. 8.3-8.29 by using joints with idle mobilities. A large set of solutions can be obtained by introducing idle mobilities outside and/or in the planar loops combined in the limbs (Figs. 8.30-8.56). We recall that the idle mobilities which can be combined in a parallelogram loop are systematized in Fig. 1.2 and Table 1.1. In the cylindrical joint denoted by C*, the rotational motion is an idle mobility in Figs 8.30, 8.31 8.32a. In the cylindrical joints of the rhombus loops denoted by Rbcs (Figs. 8.34, 8.38, 8.42, 8.46) and the parallelogram loops denoted by Pacs (Figs. 8.52-8.56), the translational motion is an idle mobility. The notations Pacs and Rbcs are associated with the parallelogram and rhombus loops with three idle mobilities combined in a cylindrical and a spherical joint. The notations Pass, Pn2ss and Pn3ss are associated with parallelogram loops and planar loops with 2 and 3 degrees of freedom which combine four idle mobilities in two spherical joints adjacent to the same link. In these cases, three idle mobilities are introduced in the loop and one outside the loop. If the link adjacent to the two spherical joints is a binary link than the idle mobility introduced outside the loop becomes an internal rotational mobility of this binary link around the axis passing by the centre of the two spherical joints. Each internal mobility gives one degree of structural redundancy (see Table 8.20). If the link adjacent to the two spherical joints is connected in the limb by three or more joints (polinary link) than the rotational motion around the axis passing by the centre of the two spherical joints becomes an idle (potential) mobility of the limb. This idle mobility is restricted by the constraints of the parallel mechanism and remains just a potential mobility. For example in Fig. 8.68b, this rotational motion is internal mobility for binary links 3A, 3B and 4C, and idle mobility for ternary links 8A and 8B. Examples of solutions with 1 to 8 overconstraints derived from the basic solutions in Figs. 8.3-8.29 are illustrated in Figs. 8.30-8.56. The bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 8.30-8.56 are given in Table 8.21. The limb topology and connecting conditions of these solutions are systematized in Tables 8.22-8.24, as are their structural parameters in Tables 8.25-8.35.

8.1 Overconstrained solutions

579

Table 8.20. Links with internal mobilities and the degree of structural redundancy TF of overconstrained maximally regular SPMs with planar motion of the moving platform No. Parallel mechanism Figure 1 2 3 4 5 6 7 8 9 10

Link with internal rotational mobility in limb G1 G2 G3

TF

Figs. 8.30b, 8.31b, 1 8.40a, 8.42 Fig. 8.31a 1 Figs. 8.32b, 8.33b, 8.35, 2 8.36b, 8.51, 8.55, 8.56 Figs. 8.33a, 8.37a 2 Figs. 8.37b, 8.39 3 Fig. 8.38 1 Figs. 8.41b, 8.43, 3 Figs. 8.44a, 8.46 2 Figs. 8.44b, 8.45b, 8.47 4 Fig. 8.45a 4

-

-

3C

3A

3B

7C -

5A 3A 3A 3A 5A

5B 3B 3B 3B 5B

4C 4C 3C 3 C, 6C 3 C, 6C 3 C, 6C

Table 8.21. Bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 8.30-8.56 No. Parallel mechanism 1 Figs. 8.30, 8.31, 2 Fig. 8.32a 3

4

5 6

Basis (RG1) ( v1 , v 2 ,

β

δ

,

(RG2) ( v1 , v 2 ,

)

α

δ

,

(RG3) ( v1 ,v2 , v3 ,

)

α

( v1 , v 2 , v 3 ,

α

,

δ

)

( v1 , v 2 , v 3 ,

β

,

δ

)

( v1 , v 2 ,

β

,

Figs. 8.32b, ( v1 , v2 , v3 , 8.33a, 8.34, 8.35, 8.38, 8.39, 8.40a, 8.42-8.44, 8.45a, 8.46-8.50, 8.52-8.56 Figs. 8.33b, ( v1 , v2 , v3 , 8.37b, 8.41b, 8.45b, 8.51 Figs. 8.36, ( v1 , v2 , v3 , 8.37a, Fig. 8.40b, ( v1 ,v2 , v3 , 8.41a

α

,

δ

)

( v1 , v 2 , v 3 ,

β

,

δ

)

( v1 , v 2 ,

δ

)

β

,

δ

)

( v1 , v 2 , v 3 ,

α

,

δ

)

( v1 , v 2 ,

δ

)

α

,

δ

)

( v1 , v 2 , v 3 ,

β

,

δ

)

( v1 , v 2 ,

α

,

) ( v1 , v 2 ,

δ

)

α

,

β

,

δ

) ( v1 ,v2 , v3 ,

α

,

β

,

δ

β

, δ

)

δ

)

,

δ

)

580

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.22. Limb topology and the number of overconstraints NF of the derived maximally regular SPMs with idle mobilities presented in Figs. 8.30-8.38 No. Basic SPM Type 1 2PPR-1RRRPRR (Fig. 8.3a) 2 2PPR-1RPaPaPa (Fig. 8.3b) 3 2PPR-1RPaPaPa (Fig. 8.4a) 4 2PPR-1RPaPaPa (Fig. 8.4b) 5 2PRRRR-1RPP (Fig. 8.5a) 6 2PPaRRR-1RPP (Fig. 8.5b) 7 2PRRPaR-1RPP (Fig. 8.6a) 8 2PPaPaR-1RPP (Fig. 8.6b) 9 2PRRbRR-1RPP (Fig. 8.7a) 10 2PRRbRbRR-1RPP (Fig. 8.7b) 11 2PPn2RR-1RPP (Fig. 8.8a) 12 2PPn3R-1RPP (Fig. 8.8b) 13 2PRRRR-1RPPa (Fig. 8.9a) 14 2PPaRRR-1RPPa (Fig. 8.9b) 15 2PRRPaR-1RPPa (Fig. 8.10a) 16 2PPaPaR-1RPPa (Fig. 8.10b) 17 2PRRbRR-1RPPa (Fig. 8.11a) 18 2PRRbRbRR1RPPa (Fig. 8.11b)

Derived SPM NF Type 3 2PC*R-1RRRPRR (Fig. 8.30a) 14 2PC*R-1RPassPassPass (Fig. 8.30b) 14 2PC*R-1RPassPassPass (Fig. 8.31a) 14 2PC*R-1RPassPassPass (Fig. 8.31b) 2 2PRRRR-1RPC* (Fig. 8.32a) 8 2PPassRRR-1RPP (Fig. 8.32b) 8 2PRRPassR-1RPP (Fig. 8.33a) 16 2PPassPassR-1RPP (Fig. 8.33b) 8 2PRRbcsRR-1RPP (Fig. 8.34a) 14 2PRRbcsRbcsRR-1RPP (Fig. 8.34b) 8 2PPn2ssRR-1RPP (Fig. 8.35a) 8 2PPn3ssR-1RPP (Fig. 8.35b) 5 2PRRRR-1RPPass (Fig. 8.36a) 11 2PPassRRR-1RPPass (Fig. 8.36b) 11 2PRRPassR-1RPPass (Fig. 8.37a) 19 2PPassPassR-1RPPass (Fig. 8.37b) 11 2PRRbcsRR-1RPPass (Fig. 8.38a) 17 2PRRbcsRbcsRR1RPPass (Fig. 8.38b)

NF Limb topology 1 P ⊥ C* ⊥ ⊥ R R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R 1 P ⊥ C* ⊥ ⊥ R R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass 1 P ⊥ C* ⊥ ⊥ R R ⊥ Pass ⊥ ⊥ Pass||Pass 1 P ⊥ C* ⊥ ⊥ R R ⊥ Pass||Pass ⊥ ⊥ Pass 1 P||R||R||R ⊥ R R ⊥ P ⊥ ⊥ C* 2 P||Pass||R||R ⊥ R R ⊥ P ⊥⊥ P 2 P||R||R||Pass ⊥ R R ⊥ P ⊥⊥ P 2 P||Pass||Pass ⊥ R R ⊥ P ⊥⊥ P 2 P||R||Rbcs||R ⊥ R R ⊥ P ⊥⊥ P 2 P||R||Rbcs||Rbcs||R ⊥ R R ⊥ P ⊥⊥ P 2 P||Pn2ss||R ⊥ R R ⊥ P ⊥⊥ P 2 P||Pn3ss ⊥ R R ⊥ P ⊥⊥ P 1 P||R||R||R ⊥ R R ⊥ P ⊥ ||Pass 1 P||Pass||R||R ⊥ R R ⊥ P ⊥ ||Pass 1 P||R||R||Pass ⊥ R R ⊥ P ⊥ ||Pass 2 P||Pass||Pass ⊥ R R ⊥ P ⊥ ||Pass 2 P||R||Rbcs||R ⊥ R R ⊥ P ⊥ ||Pass 2 P||R||Rbcs||Rbcs||R ⊥ R R ⊥ P ⊥ ||Pass

8.1 Overconstrained solutions

581

Table 8.23. Limb topology and the number of overconstraints NF of the derived maximally regular SPMs with idle mobilities presented in Figs. 8.39-8.47 No. Basic SPM Type 1 2PPn2RR-1RPPa (Fig. 8.12a) 2 2PPn3R-1RPPa (Fig. 8.12b) 3 2PRRRR-1RPaP (Fig. 8.13a) 4 2PPaRRR-1RPaP (Fig. 8.13b) 5 2PRRPaR-1RPaP (Fig. 8.14a) 6 2PPaPaR-1RPaP (Fig. 8.14b) 7 2PRRbRR-1RPaP (Fig. 8.15a) 8 2PRRbRbRR1RPaP (Fig. 8.15b) 9 2PPn2RR-1RPaP (Fig. 8.16a) 10 2PPn3R-1RPaP (Fig. 8.16b) 11 2PRRRR-1RPaPa (Fig. 8.17a) 12 2PPaRRR-1RPaPa (Fig. 8.17b) 13 2PRRPaR-1RPaPa (Fig. 8.18a) 14 2PPaPaR-1RPaPa (Fig. 8.18b) 15 2PRRbRR-1RPaPa (Fig. 8.19a) 16 2PRRbRbRR1RPaPa (Fig. 8.19b) 17 2PPn2RR-1RPaPa (Fig. 8.20a) 18 2PPn3R-1RPaPa (Fig. 8.20b)

Derived SPM NF Type NF 11 2PPn2ssRR-1RPPass 2 (Fig. 8.39a) 11 2PPn3ssR-1RPPass 2 (Fig. 8.39b) 5 2PRRRR-1RPassP 2 (Fig. 8.40a) 11 2PPassRRR-1RPaP 3 (Fig. 8.40b) 11 2PRRPassR-1RPaP 3 (Fig. 8.41a) 19 2PPassPassR-1RPassP 2 (Fig. 8.41b) 11 2PRRbcsRR-1RPassP 2 (Fig. 8.42a) 17 2PRRbcsRbcsRR2 1RPassP (Fig. 8.42b) 11 2PPn2ssRR-1RPassP 2 (Fig. 8.43a) 11 2PPn3ssR-1RPassP 2 (Fig. 8.43b) 8 2PRRRR-1RPassPass 2 (Fig. 8.44a) 14 2PPassRRR-1RPassPass 2 (Fig. 8.44b) 14 2PRRPassR-1RPassPass 2 (Fig. 8.45a) 22 2PPassPassR-1RPassPass 2 (Fig. 8.45b) 14 2PRRbssRR-1RPassPass 2 (Fig. 8.46a) 20 2PRRbssRbssRR2 1RPassPass (Fig. 8.46b) 14 2PPn2ssRR-1RPassPass 2 (Fig. 8.47a) 14 2PPn3ssR-1RPassPass 2 (Fig. 8.47a)

Limb topology P||Pn2ss||R ⊥ R R ⊥ P ⊥ ||Pass P||Pn3ss ⊥ R R ⊥ P ⊥ ||Pass P||R||R||R ⊥ R R||Pass ⊥ P P||Pa||R||R ⊥ R R||Pa ⊥ P P||R||R||Pass ⊥ R R||Pa ⊥ P P||Pass||Pass ⊥ R R||Pass ⊥ P P||R||Rbcs||R ⊥ R R||Pass ⊥ P P||R||Rbcs||Rbcs||R ⊥ R R||Pass ⊥ P P||Pn2ss||R ⊥ R R||Pass ⊥ P P||Pn3ss ⊥ R R||Pass ⊥ P P||R||R||R ⊥ R R||Pass||Pass P||Pass||R||R ⊥ R R||Pass||Pass P||R||R||Pass ⊥ R R||Pass||Pass P||Pass||Pass ⊥ R R||Pass||Pass P||R||Rbss||R ⊥ R R||Pass||Pass P||R||Rbss||Rbss||R ⊥ R R||Pass||Pass P||Pn2ss||R ⊥ R R||Pass||Pass P||Pn3ss ⊥ R R||Pass||Pass

582

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.24. Limb topology and the number of overconstraints NF of the derived maximally regular SPMs with idle mobilities presented in Figs. 8.48-8.56 No. Basic SPM Type 1 2PRRRR-1RPaPa (Fig. 8.21a) 2 2PRRRR-1RPaPat (Fig. 8.21b) 3 2PPaRRR-1RPaPa (Fig. 8.22a) 4 2PPaRRR-1RPaPat (Fig. 8.22b) 5 2PRRPaR-1RPaPa (Fig. 8.23a) 6 2PRRPaR-1RPaPat (Fig. 8.23b) 7 2PPaPaR-1RPaPa (Fig. 8.24a) 8 2PPaPaR-1RPaPat (Fig. 8.24b) 9 2PRRbRR-1RPaPa (Fig. 8.25a) 10 2PRRbRR-1RPaPat (Fig. 8.25b) 11 2PRRbRbRR1RPaPa (Fig. 8.26) 12 2PRRbRbRR1RPaPat (Fig. 8.27) 13 2PPn2RR-1RPaPa (Fig. 8.28a) 14 2PPn2RR-1RPaPat (Fig. 8.28b) 15 2PPn3R-1RPaPa (Fig. 8.29a) 16 2PPn3R-1RPaPat (Fig. 8.29b)

Derived SPM NF Type NF 8 2PRRRR-1RPacsPacs 2 (Fig. 8.48a) 8 2PRRRR-1RPacsPatcs 2 (Fig. 8.48b) 14 2PPaRRR-1RPacsPacs 8 (Fig. 8.49a) 14 2PPaRRR-1RPacsPatcs 8 (Fig. 8.49b) 14 2PRRPaR-1RPacsPacs 8 (Fig. 8.50a) 14 2PRRPaR-1RPacsPatcs 8 (Fig. 8.50b) 22 2PPassPassR-1RPacsPacs 2 (Fig. 8.51a) 22 2PPassPassR-1RPacsPatcs2 (Fig. 8.51b) 14 2PRRbcsRR-1RPacsPacs 2 (Fig. 8.52a) 14 2PRRbcsRR-1RPacsPatcs 2 (Fig. 8.52b) 20 2PRRbcsRbcsRR2 1RPacsPacs (Fig. 8.53) 20 2PRRbcsRbcsRR2 1RPacsPatcs (Fig. 8.54) 14 2PPn2ssRR-1RPacsPacs 2 (Fig. 8.55a) 14 2PPn2ssRR-1RPacsPatcs 2 (Fig. 8.55b) 14 2PPn3ssR-1RPacsPacs 2 (Fig. 8.56a) 14 2PPn3ssR-1RPacsPatcs 2 (Fig. 8.56b)

Limb topology P||R||R||R ⊥ R R||Pacs||Pacs P||R||R||R ⊥ R R||Pacs||Patcs P||Pa||R||R ⊥ R R||Pacs||Pacs P||Pa||R||R ⊥ R R||Pacs||Patcs P||R||R||Pa ⊥ R R||Pacs||Pacs P||R||R||Pa ⊥ R R||Pacs||Patcs P||Pass||Pass ⊥ R R||Pacs||Pacs P||Pass||Pass ⊥ R R||Pacs||Patcs P||R||Rbcs||R ⊥ R R||Pacs||Pacs P||R||Rbcs||R ⊥ R R||Pacs||Patcs P||R||Rbcs||Rbcs||R ⊥ R R||Pacs||Pacs P||R||Rbcs||Rbcs||R ⊥ R R||Pacs||Patcs P||Pn2ss||R ⊥ R R||Pacs||Pacs P||Pn2ss||R ⊥ R R||Pacs||Patcs P||Pn3ss ⊥ R R||Pacs||Pacs P||Pn3ss ⊥ R R||Pacs||Patcs

8.1 Overconstrained solutions

583

Table 8.25. Structural parametersa of spatial parallel mechanisms in Figs. 8.30 and 8.31 No. Structural parameter

Solution 2PC*R-1RRRPRR (Fig. 8.30a)

1 2 3 4 5 6 7 8 9 10

11 3 3 6 12 2 3 0 3 See Table 8.21

2PC*R-1RPassPassPass (Fig. 8.30b) 2PC*R-1RPassPassPass (Fig. 8.31a,b) 15 3 3 13 19 5 2 1 3 See Table 8.21

4 4 6 0 0 0 4 4 6 ( v1 , v 2 , 3 0 11 3 1 0 4

4 4 6 0 0 18 4 4 7 ( v1 , v 2 , 3 18 29 4 1 1 4

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

fj

4

4

fj

6

25

fj

14

33

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

584

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.26. Structural parametersa of spatial parallel mechanisms in Figs. 8.32 and 8.33 No. Structural Solution parameter 2PRRRR-1RPC* (Fig. 8.32a)

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

2PPassPassR-1RPP (Fig. 8.33b)

12 5 5 3 13 2 3 0 3 See Table 8.21

2PPassRRR-1RPP (Fig. 8.32b) 2PRRPassR-1RPP (Fig. 8.33a) 16 8 8 3 19 4 1 2 3 See Table 8.21

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

5 5 3 6 6 0 6 6 3 ( v1 , v 2 , 3 12 22 5 2 2 12

5 5 3 12 12 0 6 6 3 ( v1 , v 2 , 3 24 34 5 2 2 18

δ

)

δ

)

18 10 10 3 23 6 1 2 3 See Table 8.21

fj

5

12

18

fj

4

3

3

fj

14

27

39

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

8.1 Overconstrained solutions

585

Table 8.27. Structural parametersa of spatial parallel mechanisms in Figs. 8.34 and 8.35 No. Structural Solution parameter 2PRRbcsRR-1RPP 2PRRbcsRbcsRR-1RPP 2PPn2ssRR-1RPP (Fig. 8.34a) (Fig. 8.34b) (Fig. 8.35a) 2PPn3ssR-1RPP (Fig. 8.35b) 1 m 16 20 16 2 p1 8 11 8 3 p2 8 11 8 4 p3 3 3 3 5 p 19 25 19 6 q 4 6 4 7 k1 1 1 1 8 k2 2 2 2 9 k 3 3 3 10 (RGi) See Table 8.21 See Table 8.21 See Table 8.21 i=1,2,3 11 SG1 5 5 5 12 SG2 5 5 5 13 SG3 3 3 3 14 rG1 6 12 6 15 rG2 6 12 6 16 rG3 0 0 0 17 MG1 5 5 6 18 MG2 5 5 6 19 MG3 3 3 3 20 (RF) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) ( v1 , v 2 , δ ) 21 SF 3 3 3 22 rl 12 24 12 23 rF 22 34 22 24 MF 3 3 5 25 NF 2 2 2 26 TF 0 0 2 p1 27 11 17 12 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

fj

11

17

12

fj

3

3

3

fj

25

37

27

See footnote of Table 2.4 for the nomenclature of structural parameters

586

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.28. Structural parametersa of spatial parallel mechanisms in Figs. 8.36 and 8.37 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 2PRRRR-1RPPass (Fig. 8.36a)

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

14 5 5 6 16 3 2 1 3 See Table 8.21

2PPassRRR-1RPPass (Fig. 8.36b) 2PRRPassR-1RPPass (Fig. 8.37a) 18 8 8 6 22 5 0 3 3 See Table 8.21

20 10 10 6 26 7 0 3 3 See Table 8.21

5 5 4 0 0 6 5 5 4 ( v1 , v 2 , 3 6 17 3 1 0 5

5 5 4 6 6 6 6 6 4 ( v1 , v 2 , 3 18 29 5 1 2 12

5 5 3 12 12 6 6 6 4 ( v1 , v 2 , 3 30 40 6 2 3 18

δ

)

δ

)

2PPassPassR1RPPass (Fig. 8.37b)

fj

5

12

18

fj

10

10

10

fj

20

34

46

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

8.1 Overconstrained solutions

587

Table 8.29. Structural parametersa of spatial parallel mechanisms in Figs. 8.38 and 8.39 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 2PRRbcsRR1RPPass (Fig. 8.38a)

2PRRbcsRbcsRR1RPPass (Fig. 8.38b)

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

18 8 8 6 22 5 0 3 3 See Table 8.21

22 11 11 6 28 7 0 3 3 See Table 8.21

2PPn2ssRR-1RPPass (Fig. 8.39a) 2PPn3ssR-1RPPass (Fig. 8.39b) 18 8 8 6 22 5 0 3 3 See Table 8.21

5 5 3 6 6 6 5 5 4 ( v1 , v 2 , 3 18 28 4 2 1 11

5 5 3 12 12 6 5 5 4 ( v1 , v 2 , 3 30 40 4 2 1 17

5 5 3 6 6 6 6 6 4 ( v1 , v 2 , 3 18 28 6 2 3 12

SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

δ

)

fj

11

17

12

fj

10

10

10

fj

32

44

34

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

588

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.30. Structural parametersa of spatial parallel mechanisms in Figs. 8.40 and 8.41 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 2PRRRR-1RPassP (Fig. 8.40a)

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

14 5 5 6 16 3 2 1 3 See Table 8.21

2PPassRRR-1RPaP (Fig. 8.40b) 2PRRPassR-1RPaP (Fig. 8.41a) 18 8 8 6 22 5 0 3 3 See Table 8.21

20 10 10 6 26 7 0 3 3 See Table 8.21

5 5 3 0 0 6 5 5 4 ( v1 , v 2 , 3 6 16 4 2 1 5

6 6 3 6 6 3 6 6 3 ( v1 , v 2 , 3 15 27 3 3 0 12

5 5 3 12 12 6 6 6 4 ( v1 , v 2 , 3 30 40 6 2 3 18

δ

)

δ

)

2PPassPassR1RPassP (Fig. 8.41b)

fj

5

12

18

fj

10

6

10

fj

20

30

46

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

8.1 Overconstrained solutions

589

Table 8.31. Structural parametersa of spatial parallel mechanisms in Figs. 8.42 and 8.43 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 2PRRbcsRR1RPassP (Fig. 8.42a) 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

18 8 8 6 22 5 0 3 3 See Table 8.21

22 11 11 6 28 7 0 3 3 See Table 8.21

2PPn2ssRR-1RPassP (Fig. 8.43a) 2PPn3ssR-1RPassP (Fig. 8.43b) 18 8 8 6 22 5 0 3 3 See Table 8.21

5 5 3 6 6 6 5 5 4 ( v1 , v 2 , 3 18 28 4 2 1 11

5 5 3 12 12 6 5 5 4 ( v1 , v 2 , 3 30 40 4 2 1 17

5 5 3 6 6 6 6 6 4 ( v1 , v 2 , 3 18 28 6 2 3 12

δ

)

2PRRbcsRbcsRR1RPassP (Fig. 8.42b)

δ

)

fj

11

17

12

fj

10

10

10

fj

32

44

34

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

590

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.32. Structural parametersa of spatial parallel mechanisms in Figs. 8.44 and 8.45 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 2PRRRR1RPassPass (Fig. 8.44a)

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

16 5 5 9 19 4 2 1 3 See Table 8.21

2PPassRRR1RPassPass (Fig. 8.44b) 2PRRPassR1RPassPass (Fig. 8.45a) 20 8 8 9 25 6 0 3 3 See Table 8.21

22 10 10 9 29 8 0 3 3 See Table 8.21

5 5 3 0 0 12 5 5 5 ( v1 , v 2 , 3 12 22 5 2 2 5

5 5 3 6 6 12 6 6 5 ( v1 , v 2 , 3 24 34 7 2 4 12

5 5 3 12 12 12 6 6 5 ( v1 , v 2 , 3 36 46 7 2 4 18

δ

)

δ

)

2PPassPassR1RPassPass (Fig. 8.45b)

fj

5

12

18

fj

17

17

17

fj

27

41

53

See footnote of Table 2.4 for the nomenclature of structural parameters

δ

)

8.1 Overconstrained solutions

591

Table 8.33. Structural parametersa of spatial parallel mechanisms in Figs. 8.46 and 8.47 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 2PRRbssRR1RPassPass (Fig. 8.46a)

2PRRbssRbssRR1RPassPass (Fig. 8.46b)

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 9 25 6 0 3 3 See Table 8.21

24 11 11 9 31 8 0 3 3 See Table 8.21

2PPn2ssRR1RPassPass (Fig. 8.47a) 2PPn3ssR1RPassPass (Fig. 8.47b) 20 8 8 9 25 6 0 3 3 See Table 8.21

5 5 3 6 6 12 5 5 5 ( v1 , v 2 , 3 24 34 5 2 2 11

5 5 3 12 12 12 5 5 5 ( v1 , v 2 , 3 36 46 5 2 2 17

5 5 3 6 6 12 6 6 5 ( v1 , v 2 , 3 24 34 7 2 4 12

SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

δ

)

fj

11

17

12

fj

17

17

17

fj

39

51

41

See footnote of Table 2.4 for the nomenclature of structural parameters

δ

)

592

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.34. Structural parametersa of spatial parallel mechanisms in Figs. 8.488.51 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 2PRRRR1RPacsPacs (Fig. 8.48a) 2PRRRR1RPacsPatcs (Fig. 8.48b m 16 p1 5 p2 5 p3 9 p 19 q 4 k1 2 k2 1 k 3 (RGi) See Table 8.21 i=1,2,3 SG1 5 SG2 5 SG3 3 rG1 0 rG2 0 rG3 12 MG1 5 MG2 5 MG3 3 (RF) ( v1 , v 2 , δ ) SF 3 rl 12 rF 22 MF 3 NF 2 TF 0 p1 5 f

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

2PPaRRR-1RPacsPacs 2PPaRRR-1RPacsPatcs (Fig. 8.49a,b) 2PRRPaR-1RPacsPacs 2PRRPaR-1RPacsPatcs (Fig. 8.50a,b) 20 8 8 9 25 6 0 3 3 See Table 8.21

2PPassPassR1RPacsPacs (Fig. 8.51a) 2PPassPassR1RPacsPatcs (Fig. 8.51b) 22 10 10 9 29 8 0 3 3 See Table 8.21

5 5 3 3 3 12 5 5 3 ( v1 , v 2 , 3 18 28 3 8 0 8

5 5 3 12 12 12 6 6 3 ( v1 , v 2 , 3 36 46 5 2 2 18

δ

)

fj

5

8

18

fj

15

15

15

fj

25

31

51

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

8.1 Overconstrained solutions

593

Table 8.35. Structural parametersa of spatial parallel mechanisms in Figs. 8.528.56 No. Structural Solution parameter 2PRRbcsRR1RPacsPacs (Fig. 8.52a) 2PRRbcsRR1RPacsPatcs (Fig. 8.52b) 1 m 20 2 p1 8 3 p2 8 4 p3 9 5 p 25 6 q 6 7 k1 0 8 k2 3 9 k 3 10 (RGi) See Table 8.21 i=1,2,3 11 SG1 5 12 SG2 5 13 SG3 3 14 rG1 6 15 rG2 6 16 rG3 12 17 MG1 5 18 MG2 5 19 MG3 3 20 (RF) ( v1 , v 2 , δ ) 21 SF 3 22 rl 24 23 rF 34 24 MF 3 25 NF 2 26 TF 0 p1 27 11 f

2PRRbcsRbcsRR1RPacsPacs (Fig. 8.53) 2PRRbcsRbcsRR1RPacsPatcs (Fig. 8.54) 24 11 11 9 31 8 0 3 3 See Table 8.21

2PPn2ssRR-1RPacsPacs 2PPn2ssRR-1RPacsPatcs (Fig. 8.55a,b) 2PPn3ssR-1RPacsPacs 2PPn3ssR-1RPacsPatcs (Fig. 8.56a,b) 20 8 8 9 25 6 0 3 3 See Table 8.21

5 5 3 12 12 12 5 5 3 ( v1 , v 2 , 3 36 46 3 2 0 17

5 5 3 6 6 12 6 6 3 ( v1 , v 2 , 3 24 34 5 2 2 12

28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

δ

)

fj

11

17

12

fj

15

15

15

fj

37

49

39

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

594

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.30. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PC*R-1RRRPRR (a) and 2PC*R-1RPassPassPass (b), defined by SF=3, (RF)=( v1 , v2 , δ ), NF=1 and MF=3, TF=0, (a) MF=4, TF=1, (b) limb topology P ⊥ C* ⊥ ⊥ R and R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R (a), R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass (b)

8.1 Overconstrained solutions

595

Fig. 8.31. 2PC*R-1RPassPassPass-type overconstrained maximally regular SPMs with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ),

NF=1, MF=4, TF=1, limb topology P ⊥ C* ⊥ ⊥ R and R ⊥ Pass ⊥ ⊥ Pass||Pass (a), R ⊥ Pass||Pass ⊥ ⊥ Pass (b)

596

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.32. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRRR-1RPC* (a) and 2PPassRRR-1RPP (b) defined by SF=3, (RF)=( v1 , v2 , δ ) and MF=3, TF=0, NF=1 (a), MF=5, TF=2, NF=2 (b), limb topology P||R||R||R ⊥ R and R ⊥ P ⊥ ⊥ C* (a), P||Pass||R||R ⊥ R and R ⊥ P ⊥ ⊥ P (b)

8.1 Overconstrained solutions

597

Fig. 8.33. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRPassR-1RPP (a) and 2PPassPassR-1RPP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, TF=2, NF=2, limb topology R ⊥ P ⊥ ⊥ P and P||R||R||Pass ⊥ R (a), P||Pass||Pass ⊥ R (b)

598

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.34. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRbcsRR-1RPP (a) and 2PRRbcsRbcsRR-1RPP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=2, limb topology R ⊥ P ⊥ ⊥ P and P||R||Rbcs||R ⊥ R (a), P||R||Rbcs||Rbcs||R ⊥ R (b)

8.1 Overconstrained solutions

599

Fig. 8.35. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPn2ssRR-1RPP (a) and 2PPn3ssR-1RPP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, TF=2, NF=2, limb topology R ⊥ P ⊥ ⊥ P and P||Pn2ss||R ⊥ R (a), P||Pn3ss ⊥ R (b)

600

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.36. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRRR-1RPPass (a) and 2PPassRRR-1RPPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=1 and MF=3, TF=0 (a), MF=5, TF=2 (b), limb topology R ⊥ P ⊥ ||Pass and P||R||R||R ⊥ R (a), P||Pass||R||R ⊥ R (b)

8.1 Overconstrained solutions

601

Fig. 8.37. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRPassR-1RPPass (a) and 2PPassPassR-1RPPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ) and MF=5, TF=2, NF=1 (a), MF=6, TF=3, NF=2 (b), limb topology R ⊥ P ⊥ ||Pass and P||R||R||Pass ⊥ R (a), P||Pass||Pass ⊥ R (b)

602

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.38. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRbcsRR-1RPPass (a) and 2PRRbcsRbcsRR-1RPPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, TF=1, NF=2, limb topology R ⊥ P ⊥ ||Pass and P||R||Rbcs||R ⊥ R (a), P||R||Rbcs||Rbcs||R ⊥ R (b)

8.1 Overconstrained solutions

603

Fig. 8.39. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPn2ssRR-1RPPass (a) and 2PPn3ssR-1RPPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=6, TF=3, NF=2, limb topology R ⊥ P ⊥ ||Pass and P||Pn2ss||R ⊥ R (a), P||Pn3ss ⊥ R (b)

604

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.40. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRRR-1RPassP (a) and 2PPassRRR-1RPaP (b) defined by SF=3, (RF)=( v1 , v2 , δ ) and MF=4, TF=1, NF=2 (a), MF=3, TF=0, NF=3 (b), limb topology P||R||R||R ⊥ R and R||Pass ⊥ P (a), P||Pa||R||R ⊥ R and R||Pa ⊥ P (b)

8.1 Overconstrained solutions

605

Fig. 8.41. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRPassR-1RPaP (a) and 2PPassPassR-1RPassP (b) defined by SF=3, (RF)=( v1 , v2 , δ ) and MF=3, TF=0, NF=3 (a), MF=6, TF=3, NF=2 (b), limb topology P||R||R||Pass ⊥ R and R||Pa ⊥ P (a), P||Pass||Pass ⊥ R and R||Pass ⊥ P (b)

606

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.42. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRbcsRR-1RPassP (a) and 2PRRbcsRbcsRR-1RPassP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, TF=1, NF=2, limb topology R||Pass ⊥ P and P||R||Rbcs||R ⊥ R (a), P||R||Rbcs||Rbcs||R ⊥ R (b)

8.1 Overconstrained solutions

607

Fig. 8.43. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPn2ssRR-1RPassP (a) and 2PPn3ssR-1RPassP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=6, TF=3, NF=2, limb topology R||Pass ⊥ P and P||Pn2ss||R ⊥ R (a), P||Pn3ss ⊥ R (b)

608

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.44. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRRR-1RPassPass (a) and 2PPassRRR-1RPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=2 and MF=5, TF=2 (a), MF=7, TF=4 (b), limb topology R||Pass||Pass and P||R||R||R ⊥ R (a), P||Pass||R||R ⊥ R (b)

8.1 Overconstrained solutions

609

Fig. 8.45. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRPassR-1RPassPass (a) and 2PPassPassR-1RPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=7, TF=4, NF=2, limb topology R||Pass||Pass and P||R||R||Pass ⊥ R (a), P||Pass||Pass ⊥ R (b)

610

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.46. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRbssRR-1RPassPass (a) and 2PRRbssRbssRR1RPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, TF=2, NF=2, limb topology R||Pass||Pass and P||R||Rbss||R ⊥ R (a), P||R||Rbss||Rbss||R ⊥ R (b)

8.1 Overconstrained solutions

611

Fig. 8.47. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPn2ssRR-1RPassPass (a) and 2PPn3ssR-1RPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=7, TF=4, NF=2, limb topology R||Pass||Pass and P||Pn2ss||R ⊥ R (a), P||Pn3ss ⊥ R (b)

612

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.48. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRRR-1RPacsPacs (a) and 2PRRRR-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=2, limb topology P||R||R||R ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

8.1 Overconstrained solutions

613

Fig. 8.49. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPaRRR-1RPacsPacs (a) and 2PPaRRR-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=8, limb topology P||Pa||R||R ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

614

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.50. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRPaR-1RPacsPacs (a) and 2PRRPaR-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=8, limb topology P||R||R||Pa ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

8.1 Overconstrained solutions

615

Fig. 8.51. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPassPassR-1RPacsPacs (a) and 2PPassPassR1RPacsPatcs (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, NF=2, TF=2, limb topology P||Pass||Pass ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

616

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.52. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRbcsRR-1RPacsPacs (a) and 2PRRbcsRR-1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=2, limb topology P||R||Rbcs||R ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

8.1 Overconstrained solutions

617

Fig. 8.53. 2PRRbcsRbcsRR-1RPacsPacs-type overconstrained maximally regular SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=2, limb topology P||R||Rbcs||Rbcs||R ⊥ R and R||Pacs||Pacs

618

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.54. 2PRRbcsRbcsRR-1RPacsPatcs-type overconstrained maximally regular SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), TF=0, NF=2, limb topology P||R||Rbcs||Rbcs||R ⊥ R and R||Pacs||Patcs

8.1 Overconstrained solutions

619

Fig. 8.55. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPn2ssRR-1RPacsPacs (a) and 2PPn2ssRR-1RPacsPatcs (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, TF=2, NF=2, limb topology P||Pn2ss||R ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

620

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.56. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPn3ssR-1RPacsPacs (a) and 2PPn3ssR-1RPacsPatcs (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, TF=2, NF=2, limb topology P||Pn3ss ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

8.2 Non overconstrained solutions

621

8.2 Non overconstrained solutions Equation (1.15) indicates that non overconstrained solutions of maximally regular spatial parallel robots with q independent loops meet the condition p ∑ 1 fi = 3 + 6q along with MF=SF=3 and (RF)=( v1 ,v2 ,ωδ ).

The non overconstrained solutions of maximally regular spatial parallel robots presented in this section (Figs. 8.57-5.83) are derived from the overconstrained counterparts presented in Figs. 8.3-8.29 by introducing the required idle mobilities. For example, the non overconstrained solution in Fig. 8.57a is derived from the overconstrained solution in Fig. 8.3a by combining one idle mobility in limb G1 and two idle mobilities in limb G2. They are introduced in the three cylindrical joints denoted by C*. The rotational motion is the idle mobility in the cylindrical joints between the links 2A-3A and 2B-3B, and the translational motion is idle mobility in the cylindrical joint between the links 3B-7. In the cylindrical joints of the rhombus loops denoted by Rbcs (Figs. 8.61, 8.65, 8.69, 8.73, 8.80 and 8.81) and the parallelogram loops denoted by Pacs (Figs. 8.75-8.83), the translational motion is an idle mobility. We recall that the notations Pacs and Rbcs are associated with the parallelogram and rhombus loops with three idle mobilities combined in a cylindrical and a spherical joint. The notations Pass, Pn2ss and Pn3ss are associated with parallelogram loops and planar loops with 2 and 3 degrees of freedom which combine four idle mobilities in two spherical joints adjacent to the same coupler link. In these cases, three idle mobilities are introduced in the loop and one outside the loop. If the link adjacent to the two spherical joints is a binary link than the idle mobility introduced outside the loop becomes an internal rotational mobility of this binary link around the axis passing by the centre of the two spherical joints. Each internal mobility gives one degree of structural redundancy (see Table 8.36). If the link adjacent to the two spherical joints is connected in the limb by three or more joints (polinary link) than the rotational motion around the axis passing by the centre of the two spherical joints becomes an idle (potential) mobility of the limb. This idle mobility is restricted by the constraints of the parallel mechanism and remains just a potential mobility. For example in Fig. 8.68b, this rotational motion is an internal mobility for binary links 4A, 4B and 3C, and an idle mobility for ternary links 9A and 9B. The bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 8.57-5.83 are given in Table 8.37. The limb topology and connecting conditions of these solutions are systematized in Tables 8.38-8.41, as are their structural parameters in Tables 8.42 and 8.52.

622

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.36. Links with internal mobilities and the degree of structural redundancy of non overconstrained maximally regular SPMs with planar motion of the moving platform No. Parallel mechanism Figure

TF

Link with internal rotational mobility in limb G1 G2 G3

1

1

-

-

3C

1 2 2

4A 3A

4B 3B

7C -

1 3 3 3 3 2 4 4

4A 3A 4 3A 4 3A

4B 3 4 3B 4 3B

4C 4C 4C 3C 3C 3C , 6 C 3 C, 6C 3 C, 6C

2 3 4 5 6 7 8 9 10 11 12

Figs. 8.57b, 8.58b, 8.67, 8.68a, 8.69 Fig. 8.58a Fig. 8.60b Figs. 8.62, 8.78, 8.82, 8.83 Figs. 8.63, 8.64a, 8.65 Fig. 8.64b Fig. 8.66 Fig. 8.68b Fig. 8.70 Figs. 8.76, 8.72a, 8.73 Fig. 8.72b Fig. 8.74

Table 8.37. Bases of the operational velocity spaces of the limbs isolated from the parallel mechanisms presented in Figs. 8.57-8.83 No. Parallel mechanism 1 Figs. 8.57, 8.58 2 Fig. 8.59-8.83

Basis (RG1) ( v1 , v 2 ,

β ,

( v1 ,v2 , v3 ,

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

δ ) α

,

β

,

δ

) ( v1 ,v2 , v3 ,

α, α

,

δ β

(RG3) ( v1 ,v2 , v3 ,

) ,

δ

) ( v1 , v 2 ,

δ

α

)

,

β

,

δ

)

8.2 Non overconstrained solutions

623

Table 8.38. Limb topology of the non overconstraint maximally regular SPMs presented in Figs. 8.57-8.64 No. Basic SPM Type 1 2PPR-1RRRPRR (Fig. 8.3a)

SPM with NF=0 NF Type 3 PC*R-PC*C*RRRPRR (Fig. 8.57a) 14 PC*R-PC*C*RPassPassPass (Fig. 8.57b)

2

2PPR-1RPaPaPa (Fig. 8.3b)

3

2PPR-1RPaPaPa (Fig. 8.4a)

14

4

2PPR-1RPaPaPa (Fig. 8.4b)

14

5

2PRRRR-1RPP (Fig. 8.5a) 6 2PPaRRR-1RPP (Fig. 8.5b) 7 2PRRPaR-1RPP (Fig. 8.6a) 8 2PPaPaR-1RPP (Fig. 8.6b) 9 2PRRbRR-1RPP (Fig. 8.7a) 10 2PRRbRbRR-1RPP (Fig. 8.7b)

11 2PPn2RR-1RPP (Fig. 8.8a) 12 2PPn3R-1RPP (Fig. 8.8b) 13 2PRRRR-1RPPa (Fig. 8.9a) 14 2PPaRRR-1RPPa (Fig. 8.9b) 15 2PRRPaR-1RPPa (Fig. 8.10a) 16 2PPaPaR-1RPPa (Fig. 8.10b)

2 8 8 16 8 14

8 8 5 11 11 19

Limb topology P ⊥ C* ⊥ ⊥ R P ⊥ C* ⊥ ⊥ C* R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R P ⊥ C* ⊥ ⊥ R P ⊥ C* ⊥ ⊥ C* R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass PC*R-PC*C*P ⊥ C* ⊥ ⊥ R ss ss ss RPa Pa Pa P ⊥ C* ⊥ ⊥ C* (Fig. 8.58a) R ⊥ Pass ⊥ ⊥ Pass||Pass PC*R-PC*C*P ⊥ C* ⊥ ⊥ R RPassPassPass P ⊥ C* ⊥ ⊥ C* (Fig. 8.58b) R ⊥ Pass||Pass ⊥ ⊥ Pass 2PRRRR*R-1RPP P||R||R||R ⊥ R* ⊥ ⊥ R (Fig. 8.59a) R ⊥ P ⊥⊥ P ss 2PPa RRR-1RPP P||Pass||R||R ⊥ R (Fig. 8.59b) R ⊥ P ⊥⊥ P ss 2PRRPa R-1RPP P||R||R||Pass ⊥ R (Fig. 8.60a) R ⊥ P ⊥⊥ P ss ss 2PPa Pa R-1RPP P||Pass||Pass ⊥ R (Fig. 8.60b) R ⊥ P ⊥⊥ P cs 2PRRb RR*R-1RPP P||R||Rbcs||R ⊥ R* ⊥ ⊥ R (Fig. 8.61a) R ⊥ P ⊥⊥ P cs cs 2PRRb Rb RR*RP||R||Rbcs||Rbcs||R ⊥ R* ⊥ ⊥ R 1RPP R ⊥ P ⊥⊥ P (Fig. 8.61b) 2PPn2ssRR*R-1RPP P||Pn2ss||R ⊥ R* ⊥ ⊥ R (Fig. 8.62a) R ⊥ P ⊥⊥ P ss 2PPn3 R*R-1RPP P||Pn3ss ⊥ R* ⊥ ⊥ R (Fig. 8.62b) R ⊥ P ⊥⊥ P ss 2PRRRR*R-1RPPa P||R||R||R ⊥ R* ⊥ ⊥ R (Fig. 8.63a) R ⊥ P ⊥ ||Pass ss ss 2PPa RRR-1RPPa P||Pass||R||R ⊥ R (Fig. 8.63b) R ⊥ P ⊥ ||Pass ss ss 2PRRPa R-1RPPa P||R||R||Pass ⊥ R (Fig. 8.64a) R ⊥ P ⊥ ||Pass ss ss 2PR*Pa Pa RP||R*||Pass||Pass ⊥ R ss 1RPPa R ⊥ P ⊥ ||Pass (Fig. 8.64b)

624

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.39. Limb topology and the number of overconstraints NF of the derived maximally regular SPMs with idle mobilities presented in Figs. 8.65-8.72 No. Basic SPM Type 1 2PRRbRR-1RPPa (Fig. 8.11a)

SPM with NF=0 Limb topology NF Type 11 2PRRbcsRR*RP||R||Rbcs||R ⊥ R* ⊥ ⊥ R 1RPPass R ⊥ P ⊥ ||Pass (Fig. 8.65a) 2 2PRRbRbRR17 2PRRbcsRbcsRR*RP||R||Rbcs||Rbcs||R ⊥ R* ⊥ ⊥ R ss 1RPPa 1RPPa R ⊥ P ⊥ ||Pass (Fig. 8.11b) (Fig. 8.65b) 3 2PPn2RR-1RPPa 11 2PPn2ssR*RR-1RPPass P||Pn2ss||R ⊥ R* ⊥ ⊥ R (Fig. 8.12a) (Fig. 8.66a) R ⊥ P ⊥ ||Pass ss ss 4 2PPn3R-1RPPa 11 2PPn3 R*R-1RPPa P||Pn3ss ⊥ R* ⊥ ⊥ R (Fig. 8.12b) (Fig. 8.66b) R ⊥ P ⊥ ||Pass ss 5 2PRRRR-1RPaP 5 2PRRR*RR-1RPa P P||R||R||R ⊥ R* ⊥ ⊥ R (Fig. 8.13a) (Fig. 8.67a) R||Pass ⊥ P ss ss 6 2PPaRRR-1RPaP 11 2PPa RRR-1RPa P P||Pa||R||R ⊥ R (Fig. 8.13b) (Fig. 8.67b) R||Pass ⊥ P ss ss 7 2PRRPaR-1RPaP 11 2PRRPa R-1RPa P P||R||R||Pass ⊥ R (Fig. 8.14a) (Fig. 8.68a) R||Pass ⊥ P ss ss 8 2PPaPaR-1RPaP 19 2PR*Pa Pa RP||R*||Pass||Pass ⊥ R ss (Fig. 8.14b) 1RPa P R||Pass ⊥ P (Fig. 8.68b) 9 2PRRbRR-1RPaP 11 2PRRbcsRR*RP||R||Rbcs||R ⊥ R* ⊥ ⊥ R ss (Fig. 8.15a) 1RPa P R||Pass ⊥ P (Fig. 8.69a) 10 2PRRbRbRR17 2PRRbcsRbcsRR*RP||R||Rbcs||Rbcs||R ⊥ R* ⊥ ⊥ R ss 1RPaP 1RPa P R||Pass ⊥ P (Fig. 8.15b) (Fig. 8.69b) 11 2PPn2RR-1RPaP 11 2PPn2ssRR*RP||Pn2ss||R ⊥ R* ⊥ ⊥ R (Fig. 8.16a) 1RPassP R||Pass ⊥ P (Fig. 8.70a) 12 2PPn3R-1RPaP 11 2PPn3ssR*R-1RPassP 2PPn3ssR*R-1RPassP (Fig. 8.16b) (Fig. 8.70b) R||Pass ⊥ P ss ss 13 2PRRRR-1RPaPa 8 2PRRRR*R-1RPa Pa P||R||R||R ⊥ R* ⊥ ⊥ R (Fig. 8.17a) (Fig. 8.71a) R||Pass||Pass ss ss ss 14 2PPaRRR-1RPaPa 14 2PPa RRR-1RPa Pa P||Pass||R||R ⊥ R (Fig. 8.17b) (Fig. 8.71b) R||Pass||Pass ss 15 2PRRPaR-1RPaPa 14 2PRRPa RP||R||R||Pass ⊥ R ss ss (Fig. 8.18a) 1RPa Pa R||Pass||Pass (Fig. 8.72a) 16 2PPaPaR-1RPaPa 22 2PR*PassPassRP||R*||Pass||Pass ⊥ R ss ss (Fig. 8.18b) 1RPa Pa R||Pass||Pass (Fig. 8.72b)

8.2 Non overconstrained solutions

625

Table 8.40. Limb topology and the number of overconstraints NF of the derived maximally regular SPMs with idle mobilities presented in Figs. 8.73-8.78 No. Basic SPM SPM with NF=0 Type NF Type 1 2PRRbRR-1RPaPa 14 2PRRbssRR*R(Fig. 8.19a) 1RPassPass (Fig. 8.73a) 2 2PRRbRbRR20 2PRRbssRbssRR*R1RPaPa 1RPassPass (Fig. 8.19b) (Fig. 8.73b) 3 2PPn2RR-1RPaPa 14 2PPn2ssRR*R(Fig. 8.20a) 1RPassPass (Fig. 8.74a) 4 2PPn3R-1RPaPa 14 2PPn3ssR*R(Fig. 8.20b) 1RPassPass (Fig. 8.74b) 5 2PRRRR-1RPaPa 8 2PRRRR*R(Fig. 8.21a) 1RPacsPacs (Fig. 8.75a) 6 2PRRRR-1RPaPat 8 2PRRRR*R(Fig. 8.21b) 1RPacsPatcs (Fig. 8.75b) 7 2PPaRRR-1RPaPa 14 2PPassRRR(Fig. 8.22a) 1RPacsPacs (Fig. 8.76a) 8 2PPaRRR-1RPaPat 14 2PPassRRR(Fig. 8.22b) 1RPacsPatcs (Fig. 8.76b) 9 2PRRPaR-1RPaPa 14 2PRRPassR(Fig. 8.23a) 1RPacsPacs (Fig. 8.77a) 10 2PRRPaR-1RPaPat 14 2PRRPassR(Fig. 8.23b) 1RPacsPatcs (Fig. 8.77b) 11 2PPaPaR-1RPaPa 22 2PPassPassR*R(Fig. 8.24a) 1RPacsPacs (Fig. 8.78a) 12 2PPaPaR-1RPaPat 22 2PPassPassR*R(Fig. 8.24b) 1RPacsPatcs (Fig. 8.78b)

Limb topology P||R||Rbss||R ⊥ R* ⊥ ⊥ R R||Pass||Pass P||R||Rbss||Rbss||R ⊥ R* ⊥ ⊥ R R||Pass||Pass P||Pn2ss||R ⊥ R* ⊥ ⊥ R R||Pass||Pass P||Pn3ss ⊥ R* ⊥ ⊥ R R||Pass||Pass P||R||R||R ⊥ R* ⊥ ⊥ R R||Pacs||Pacs P||R||R||R ⊥ R* ⊥ ⊥ R R||Pacs||Patcs P||Pass||R||R ⊥ R R||Pacs||Pacs P||Pass||R||R ⊥ R R||Pacs||Patcs P||R||R||Pass ⊥ R R||Pacs||Pacs P||R||R||Pass ⊥ R R||Pacs||Patcs P||Pass||Pass||R ⊥ R R||Pacs||Pacs P||Pass||Pass||R ⊥ R R||Pacs||Patcs

626

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.41. Limb topology and the number of overconstraints NF of the derived maximally regular SPMs with idle mobilities presented in Figs. 8.79-8.83 No. Basic SPM SPM with NF=0 Type NF Type 1 2PRRbRR-1RPaPa 14 2PRRbcsRR*R(Fig. 8.25a) 1RPacsPacs (Fig. 8.79a) 2 2PRRbRR-1RPaPat 14 2PRRbcsRR*R(Fig. 8.25b) 1RPacsPatcs (Fig. 8.79b) 3 2PRRbRbRR26 2PRRbcsRbcsRR*R1RPaPa 1RPacsPacs (Fig. 8.26) (Fig. 8.80) 4 2PRRbRbRR27 2PRRbcsRbcsRR*R1RPaPat 1RPacsPatcs (Fig. 8.27) (Fig. 8.81) 5 2PPn2RR-1RPaPa 14 2PPn2ssRR*R(Fig. 8.28a) 1RPacsPacs (Fig. 8.82a) 6 2PPn2RR-1RPaPat 14 2PPn2ssRR*R(Fig. 8.28b) 1RPacsPatcs (Fig. 8.82b) 7 2PPn3R-1RPaPa 14 2PPn3ssR*R(Fig. 8.29a) 1RPacsPacs (Fig. 8.83a) 8 2PPn3R-1RPaPat 14 2PPn3ssR*R(Fig. 8.29b) 1RPacsPatcs (Fig. 8.83b)

Limb topology P||R||Rbcs||R ⊥ R* ⊥ ⊥ R R||Pacs||Pacs P||R||Rbcs||R ⊥ R* ⊥ ⊥ R R||Pacs||Patcs P||R||Rbcs||Rbcs||R ⊥ R* ⊥ ⊥ R R||Pacs||Pacs P||R||Rbcs||Rbcs||R ⊥ R* ⊥ ⊥ R R||Pacs||Patcs P||Pn2ss||R ⊥ R* ⊥ ⊥ R R||Pacs||Pacs P||Pn2ss||R ⊥ R* ⊥ ⊥ R R||Pacs||Patcs P||Pn3ss ⊥ R* ⊥ ⊥ R R||Pacs||Pacs P||Pn3ss ⊥ R* ⊥ ⊥ R R||Pacs||Patcs

8.2 Non overconstrained solutions

627

Table 8.42. Structural parametersa of spatial parallel mechanisms in Figs. 8.57 and 8.58 No. Structural parameter

Solution PC*R-PC*C*-RRRPRR (Fig. 8.57a)

1 2 3 4 5 6 7 8 9 10

11 3 3 6 12 2 3 0 3 See Table 8.37

PC*R-PC*C*-RPassPassPass (Fig. 8.57b) PC*R-PC*C*-RPassPassPass (Fig. 8.58a,b) 15 3 3 13 19 5 2 1 3 See Table 8.37

4 5 6 0 0 0 4 5 6 ( v1 , v 2 , 3 0 12 3 0 0 4

4 5 6 0 0 18 4 5 7 ( v1 , v 2 , 3 18 30 4 0 1 4

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

fj

5

5

fj

6

25

fj

15

34

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

628

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.43. Structural parametersa of spatial parallel mechanisms in Figs. 8.59 and 8.60 No. Structural Solution parameter 2PRRRR*R-1RPP (Fig. 8.59a)

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

2PPassPassR-1RPP (Fig. 8.60b)

14 6 6 3 15 2 3 0 3 See Table 8.37

2PPassRRR-1RPP (Fig. 8.59b) 2PRRPassR-1RPP (Fig. 8.60a) 16 8 8 3 19 4 1 2 3 See Table 8.37

6 6 3 0 0 0 6 6 3 ( v1 , v 2 , 3 0 12 3 0 0 6

6 6 3 6 6 0 6 6 3 ( v1 , v 2 , 3 12 24 3 0 0 12

6 6 3 12 12 0 7 7 3 ( v1 , v 2 , 3 24 36 5 0 2 19

δ

)

δ

)

20 11 11 3 25 6 1 2 3 See Table 8.37

fj

6

12

19

fj

3

3

3

fj

15

27

41

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

8.2 Non overconstrained solutions

629

Table 8.44. Structural parametersa of spatial parallel mechanisms in Figs. 8.61 and 8.62 No. Structural Solution parameter 2PRRbcsRR*R1RPP (Fig. 8.61a)

2PRRbcsRbcsRR*R1RPP (Fig. 8.61b)

1 2 3 4 5 6 7 8 9 10

18 9 9 3 21 4 1 2 3 See Table 8.37

22 12 12 3 27 6 1 2 3 See Table 8.37

2PPn2ssRR*R-1RPP (Fig. 8.62a) 2PPn3ssR*R-1RPP (Fig. 8.62b) 18 9 9 3 21 4 1 2 3 See Table 8.37

6 6 3 6 6 0 6 6 3 ( v1 , v 2 , 3 12 24 3 0 0 12

6 6 3 12 12 0 6 6 3 ( v1 , v 2 , 3 24 36 3 0 0 18

6 6 3 6 6 0 7 7 3 ( v1 , v 2 , 3 12 24 5 0 2 13

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) SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

δ

)

fj

12

18

13

fj

3

3

3

fj

27

39

29

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

630

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.45. Structural parametersa of spatial parallel mechanisms in Figs. 8.63 and 8.64 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 2PRRRR*R1RPPass (Fig. 8.63a) 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

16 6 6 6 18 3 2 1 3 See Table 8.37

2PPassRRR-1RPPass (Fig. 8.63b) 2PRRPassR-1RPPass (Fig. 8.64a) 18 8 8 6 22 5 0 3 3 See Table 8.37

22 11 11 6 28 7 0 3 3 See Table 8.37

6 6 3 0 0 6 6 6 4 ( v1 , v 2 , 3 6 18 4 0 1 6

6 6 3 6 6 6 6 6 4 ( v1 , v 2 , 3 18 30 4 0 1 12

6 6 3 12 12 6 7 7 4 ( v1 , v 2 , 3 30 42 6 0 3 19

δ

)

δ

)

2PR*PassPassR1RPPass (Fig. 8.64b)

fj

6

12

19

fj

10

10

10

fj

22

34

48

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

8.2 Non overconstrained solutions

631

Table 8.46. Structural parametersa of spatial parallel mechanisms in Figs. 8.65 and 8.66 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 2PRRbcsRR*R1RPPass (Fig. 8.65a)

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 6 24 5 0 3 3 See Table 8.37

2PRRbcsRbcsRR*R- 2PPn2ssR*RR1RPPass 1RPPass (Fig. 8.65b) (Fig. 8.66a) 2PPn3ssR*R1RPPass (Fig. 8.66b) 24 20 12 9 12 9 6 6 30 24 7 5 0 0 3 3 3 3 See Table 8.37 See Table 8.37

6 6 3 6 6 6 6 6 4 ( v1 , v 2 , 3 18 30 4 0 1 12

6 6 3 12 12 6 6 6 4 ( v1 , v 2 , 3 30 42 4 0 1 18

SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

δ

)

6 6 3 6 6 6 7 7 4 ( v1 , v 2 , 3 18 30 6 0 3 13

fj

12

18

13

fj

10

10

10

fj

34

46

36

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

632

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.47. Structural parametersa of spatial parallel mechanisms in Figs. 8.67 and 8.68 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 2PRRR*RR1RPassP (Fig. 8.67a) 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

16 6 6 6 18 3 2 1 3 See Table 8.37

2PPassRRR-1RPassP (Fig. 8.67b) 2PRRPassR-1RPassP (Fig. 8.68a) 18 8 8 6 22 5 0 3 3 See Table 8.37

22 11 11 6 28 7 0 3 3 See Table 8.37

6 6 3 0 0 6 6 6 4 ( v1 , v 2 , 3 6 18 4 0 1 6

6 6 3 6 6 6 6 6 4 ( v1 , v 2 , 3 18 30 4 0 1 12

6 6 3 12 12 6 7 7 4 ( v1 , v 2 , 3 30 42 6 0 3 19

δ

)

δ

)

2PR*PassPassR1RPassP (Fig. 8.68b)

fj

6

12

19

fj

10

10

10

fj

22

34

48

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

8.2 Non overconstrained solutions

633

Table 8.48. Structural parametersa of spatial parallel mechanisms in Figs. 8.69 and 8.70 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 2PRRbcsRR*R1RPassP (Fig. 8.69a)

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

20 9 9 6 24 5 0 3 3 See Table 8.37

2PRRbcsRbcsRR*R- 2PPn2ssRR*R1RPassP 1RPassP (Fig. 8.69b) (Fig. 8.70a) 2PPn3ssR*R1RPassP (Fig. 8.70b) 24 20 12 9 12 9 6 6 30 24 7 5 0 0 3 3 3 3 See Table 8.37 See Table 8.37

6 6 3 6 6 6 6 6 4 ( v1 , v 2 , 3 18 30 4 0 1 12

6 6 3 12 12 6 6 6 4 ( v1 , v 2 , 3 30 42 4 0 1 18

δ

)

δ

)

6 6 3 6 6 6 7 7 4 ( v1 , v 2 , 3 18 30 6 0 3 13

fj

12

18

13

fj

10

10

10

fj

34

46

36

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

634

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.49. Structural parametersa of spatial parallel mechanisms in Figs. 8.71 and 8.72 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 2PRRRR*R1RPassPass (Fig. 8.71a)

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

18 6 6 9 21 4 2 1 3 See Table 8.37

2PPassRRR1RPassPass (Fig. 8.71b) 2PRRPassR1RPassPass (Fig. 8.72a) 20 8 8 9 25 6 0 3 3 See Table 8.37

24 11 11 9 31 8 0 3 3 See Table 8.37

6 6 3 0 0 12 6 6 5 ( v1 , v 2 , 3 12 24 5 0 2 6

6 6 3 6 6 12 6 6 5 ( v1 , v 2 , 3 24 36 5 0 2 12

6 6 3 12 12 12 7 7 5 ( v1 , v 2 , 3 36 48 7 0 4 19

δ

)

δ

)

2PR*PassPassR1RPassPass (Fig. 8.72b)

fj

6

12

19

fj

17

17

17

fj

29

41

55

See footnote of Table 2.4 for the nomenclature of structural parameters

δ

)

8.2 Non overconstrained solutions

635

Table 8.50. Structural parametersa of spatial parallel mechanisms in Figs. 8.73 and 8.74 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 2PRRbssRR*R1RPassPass (Fig. 8.73a)

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

22 9 9 9 27 6 0 3 3 See Table 8.37

2PRRbssRbssRR*R- 2PPn2ssRR*R1RPassPass 1RPassPass (Fig. 8.73b) (Fig. 8.74a) 2PPn3ssR*R1RPassPass (Fig. 8.74b) 26 22 12 9 12 9 9 9 33 27 8 6 0 0 3 3 3 3 See Table 8.37 See Table 8.37

6 6 3 6 6 12 6 6 5 ( v1 , v 2 , 3 24 36 5 0 2 12

6 6 3 12 12 12 6 6 5 ( v1 , v 2 , 3 36 48 5 0 2 18

SF rl rF MF NF TF

∑ ∑ ∑ ∑

p1

j =1

p2

j =1

p3

j =1

p

j =1

fj

δ

)

δ

)

6 6 3 6 6 12 7 7 5 ( v1 , v 2 , 3 24 36 7 0 4 13

fj

12

18

13

fj

17

17

17

fj

41

53

43

See footnote of Table 2.4 for the nomenclature of structural parameters

δ

)

636

8 Maximally regular SPMs with planar motion of the moving platform

Table 8.51. Structural parametersa of spatial parallel mechanisms in Figs. 8.758.78 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 2PRRRR*R1RPacsPacs (Fig. 8.75a) 2PRRRR*R1RPacsPatcs (Fig. 8.75b) m 18 p1 6 p2 6 p3 9 p 21 q 4 k1 2 k2 1 k 3 (RGi) See Table 8.37 i=1,2,3 SG1 6 SG2 6 SG3 3 rG1 0 rG2 0 rG3 12 MG1 6 MG2 6 MG3 3 (RF) ( v1 , v 2 , δ ) SF 3 rl 12 rF 24 MF 3 NF 0 TF 0 p1 6 f

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

2PPassRRR-1RPacsPacs 2PPassRRR-1RPacsPatcs (Fig. 8.76a,b) 2PRRPassR-1RPacsPacs 2PRRPassR-1RPacsPatcs (Fig. 8.77a,b) 20 8 8 9 25 6 0 3 3 See Table 8.37

2PPassPassR*R1RPacsPacs (Fig. 8.78a) 2PPassPassR*R1RPacsPatcs (Fig. 8.78b) 24 11 11 9 31 8 0 3 3 See Table 8.37

6 6 3 6 6 12 6 6 3 ( v1 , v 2 , 3 24 36 3 0 0 12

6 6 3 12 12 12 7 7 3 ( v1 , v 2 , 3 36 48 5 0 2 19

δ

)

fj

6

12

19

fj

15

15

15

fj

27

39

53

δ

See footnote of Table 2.4 for the nomenclature of structural parameters

)

8.2 Non overconstrained solutions

637

Table 8.52. Structural parametersa of spatial parallel mechanisms in Figs. 8.798.83 No. Structural Solution parameter 2PRRbcsRR*R1RPacsPacs (Fig. 8.79a) 2PRRbcsRR*R1RPacsPatcs (Fig. 8.79b) 1 m 22 2 p1 9 3 p2 9 4 p3 9 5 p 27 6 q 6 7 k1 0 8 k2 3 9 k 3 10 (RGi) See Table 8.37 i=1,2,3 11 SG1 6 12 SG2 6 13 SG3 3 14 rG1 6 15 rG2 6 16 rG3 12 17 MG1 6 18 MG2 6 19 MG3 3 20 (RF) ( v1 , v 2 , δ ) 21 SF 3 22 rl 24 23 rF 36 24 MF 3 25 NF 0 26 TF 0 p1 27 12 f 28 29 30 a

∑ ∑ ∑ ∑

j =1

p2

j =1

p3

j =1

p

j =1

j

2PRRbcsRbcsRR*R-2PPn2ssRR*R-1RPacsPacs 1RPacsPacs 2PPn2ssRR*R-1RPacsPatcs (Fig. 8.80) (Fig. 8.82a,b) 2PRRbcsRbcsRR*R-2PPn3ssR*R-1RPacsPacs 1RPacsPatcs 2PPn3ssR*R-1RPacsPatcs (Fig. 8.81) (Fig. 8.83a,b) 26 22 12 9 12 9 9 9 33 27 8 6 0 0 3 3 3 3 See Table 8.37 See Table 8.37 6 6 3 12 12 12 6 6 3 ( v1 , v 2 , 3 36 48 3 0 0 18

δ

)

6 6 3 6 6 12 7 7 3 ( v1 , v 2 , 3 24 36 5 0 2 13

fj

12

18

13

fj

15

15

15

fj

39

51

41

δ

)

See footnote of Table 2.4 for the nomenclature of structural parameters

638

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.57. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types PC*R-PC*C*-RRRPRR (a) and PC*R-PC*C*RPassPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=0 and MF=3, TF=0, (a)

MF=4, TF=1, (b) limb topology P ⊥ C* ⊥ ⊥ R, R ⊥ R ⊥ R ⊥ P ⊥ ||R ⊥ R (a), R ⊥ Pass ⊥ ⊥ Pass ⊥ ||Pass (b)

P ⊥ C* ⊥ ⊥ C*

and

8.2 Non overconstrained solutions

639

Fig. 8.58. PC*R-PC*C*-RPassPassPass-type non overconstrained maximally regular SPMs with planar motion of the moving platform defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, NF=0, TF=1, limb topology P ⊥ C* ⊥ ⊥ R, P ⊥ C* ⊥ ⊥ C* and R ⊥ Pass ⊥ ⊥ Pass||Pass (a), R ⊥ Pass||Pass ⊥ ⊥ Pass (b)

640

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.59. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRRR*R-1RPP (a) and 2PPassRRR-1RPP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), NF=0, TF=0, limb topology R ⊥ P ⊥ ⊥ P

and P||R||R||R ⊥ R* ⊥ ⊥ R (a), P||Pass||R||R ⊥ R (b)

8.2 Non overconstrained solutions

641

Fig. 8.60. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRPassR-1RPP (a) and 2PPassPassR-1RPP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=0 and MF=3, TF=0 (a), MF=5, TF=2 (a),

limb topology R ⊥ P ⊥ ⊥ P and P||R||R||Pass ⊥ R (a), P||Pass||Pass ⊥ R (b)

642

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.61. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRbcsRR*R-1RPP (a) and 2PRRbcsRbcsRR*R1RPP (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), NF=0, TF=0, limb topology R ⊥ P ⊥ ⊥ P and P||R||Rbcs||R ⊥ R* ⊥ ⊥ R (a), P||R||Rbcs||Rbcs||R ⊥ R* ⊥ ⊥ R (b)

8.2 Non overconstrained solutions

643

Fig. 8.62. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPn2ssRR*R-1RPP (a) and 2PPn3ssR*R-1RPP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, NF=0, TF=2, limb topology R ⊥ P ⊥ ⊥ P and P||Pn2ss||R ⊥ R* ⊥ ⊥ R (a), P||Pn3ss ⊥ R* ⊥ ⊥ R (b)

644

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.63. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRRR*R-1RPPass (a) and 2PPassRRR-1RPPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, NF=0, TF=1, limb topology R ⊥ P ⊥ ||Pass and P||R||R||R ⊥ R* ⊥ ⊥ R (a), P||Pass||R||R ⊥ R (b)

8.2 Non overconstrained solutions

645

Fig. 8.64. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRPassR-1RPPass (a) and 2PR*PassPassR1RPPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=0 and MF=4, TF=1, (a), MF=6, TF=3 (b), limb topology R ⊥ P ⊥ ||Pass and P||R||R||Pass ⊥ R (a), P||R*||Pass||Pass ⊥ R (b)

646

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.65. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRbcsRR*R-1RPPass (a) and 2PRRbcsRbcsRR*R1RPPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, NF=0, TF=1, limb

topology R ⊥ P ⊥ ||Pass P||R||Rbcs||Rbcs||R ⊥ R* ⊥ ⊥ R (b)

and

P||R||Rbcs||R ⊥ R* ⊥ ⊥ R

(a),

8.2 Non overconstrained solutions

647

Fig. 8.66. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPn2ssR*RR-1RPPass (a) and 2PPn3ssR*R-1RPPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=6, NF=0, TF=3, limb topology R ⊥ P ⊥ ||Pass and P||Pn2ss||R ⊥ R* ⊥ ⊥ R (a), P||Pn3ss ⊥ R* ⊥ ⊥ R (b)

648

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.67. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRR*RR-1RPassP (a) and 2PPassRRR-1RPassP (b) defined by SF=3, (RF)=( v1 , v2 , δ ) and MF=4, NF=0, TF=1, limb topology R||Pass ⊥ P and P||R||R||R ⊥ R* ⊥ ⊥ R (a), P||Pa||R||R ⊥ R (b)

8.2 Non overconstrained solutions

649

Fig. 8.68. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRPassR-1RPassP (a) and 2PR*PassPassR1RPassP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=0, and MF=4, TF=1 (a), MF=6, TF=3 (b), limb topology R||Pass ⊥ P and P||R||R||Pass ⊥ R (a), P||R*||Pass||Pass ⊥ R (b)

650

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.69. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRbcsRR*R-1RPassP (a) and 2PRRbcsRbcsRR*R1RPassP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=4, NF=0, TF=1, limb topology

R||Pass ⊥ P and P||R||Rbcs||R ⊥ R* ⊥ ⊥ R (a), P||R||Rbcs||Rbcs||R ⊥ R* ⊥ ⊥ R (b)

8.2 Non overconstrained solutions

651

Fig. 8.70. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPn2ssRR*R-1RPassP (a) and 2PPn3ssR*R-1RPassP (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=6, NF=0, TF=3, limb topology R||Pass ⊥ P and P||Pn2ss||R ⊥ R* ⊥ ⊥ R (a), P||Pn3ss ⊥ R* ⊥ ⊥ R (b)

652

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.71. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRRR*R-1RPassPass (a) and 2PPassRRR1RPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, NF=0, TF=2, limb

topology R||Pass||Pass and P||R||R||R ⊥ R* ⊥ ⊥ R (a), P||Pass||R||R ⊥ R (b)

8.2 Non overconstrained solutions

653

Fig. 8.72. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRPassR-1RPassPass (a) and 2PR*PassPassR1RPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), NF=0 and MF=5, TF=2 (a), MF=7, TF=4 (b), limb topology R||Pass||Pass and P||R||R||Pass ⊥ R (a), P||R*||Pass||Pass ⊥ R (b)

654

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.73. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRbssRR*R-1RPassPass (a) and 2PRRbssRbssRR*R-1RPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5,

NF=0, TF=2, limb topology R||Pass||Pass and P||R||Rbss||R ⊥ R* ⊥ ⊥ R (a), P||R||Rbss||Rbss||R ⊥ R* ⊥ ⊥ R (b)

8.2 Non overconstrained solutions

655

Fig. 8.74. Overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPn2ssRR*R-1RPassPass (a) and 2PPn3ssR*R1RPassPass (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=7, NF=0, TF=4, limb

topology R||Pass||Pass and P||Pn2ss||R ⊥ R* ⊥ ⊥ R (a), P||Pn3ss ⊥ R* ⊥ ⊥ R (b)

656

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.75. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRRR*R-1RPacsPacs (a) and 2PRRRR*R1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), NF=0, TF=0, limb topology P||R||R||R ⊥ R* ⊥ ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

8.2 Non overconstrained solutions

657

Fig. 8.76. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPassRRR-1RPacsPacs (a) and 2PPassRRR1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), NF=0, TF=0, limb topology P||Pass||R||R ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

658

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.77. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRPassR-1RPacsPacs (a) and 2PRRPassR1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), NF=0, TF=0, limb topology P||R||R||Pass ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

8.2 Non overconstrained solutions

659

Fig. 8.78. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPassPassR*R-1RPacsPacs (a) and 2PPassPassR*R1RPacsPatcs (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, NF=0, TF=2, limb topology P||Pass||Pass||R ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

660

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.79. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PRRbcsRR*R-1RPacsPacs (a) and 2PRRbcsRR*R1RPacsPatcs (b) defined by MF=SF=3, (RF)=( v1 , v2 , δ ), NF=0, TF=0, limb topology P||R||Rbcs||R ⊥ R* ⊥ ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

8.2 Non overconstrained solutions

661

Fig. 8.80. 2PRRbcsRbcsRR*R-1RPacsPacs-type non overconstrained maximally regular SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), NF=0, TF=0, limb topology P||R||Rbcs||Rbcs||R ⊥ R* ⊥ ⊥ R and R||Pacs||Pacs

662

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.81. 2PRRbcsRbcsRR*R-1RPacsPatcs-type non overconstrained maximally regular SPM with planar motion of the moving platform defined by MF=SF=3, (RF)=( v1 , v2 , δ ), NF=0, TF=0, limb topology P||R||Rbcs||Rbcs||R ⊥ R* ⊥ ⊥ R and R||Pacs||Patcs

8.2 Non overconstrained solutions

663

Fig. 8.82. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPn2ssRR*R-1RPacsPacs (a) and 2PPn2ssRR*R1RPacsPatcs (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, NF=0, TF=2, limb

topology P||Pn2ss||R ⊥ R* ⊥ ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

664

8 Maximally regular SPMs with planar motion of the moving platform

Fig. 8.83. Non overconstrained maximally regular SPMs with planar motion of the moving platform of types 2PPn3ssR*R-1RPacsPacs (a) and 2PPn3ssR*R1RPacsPatcs (b) defined by SF=3, (RF)=( v1 , v2 , δ ), MF=5, NF=0, TF=2, limb

topology P||Pn3ss ⊥ R* ⊥ ⊥ R and R||Pacs||Pacs (a), R||Pacs||Patcs (b)

References

Alba-Gomes O, Wenger P, Pamanes JA (2005) Consitent kinetostatic indices for planar 3-DOF parallel manipulators, application to the optimal kinematic inversion. Proc ASME Design Eng Tech Conf, Long Beach Alba-Gomes O, Pamanes JA, Wenger P (2007) Trajectory planning of a redundant parallel manipulator changing of working mode. In: Proc of the 12th World Congress in Mechanism and Machine Science, Besançon Alici G, Shirinzadeh B (2003) Kinematics and stiffness analyses of a flexurejointed planar micromanipulation system for a decoupled compliant motion. In: Proc IEEE Int Conf Inteligent Robots Systems, Las Vegas, pp 3282 - 3287 Alici G, Shirinzadeh B (2004a) Optimum synthesis of planar parallel manipulators based on kinematic isotropy and force balancing. Robotica 22:97-108 Alici G, Shirinzadeh B (2004b) Optimum dynamic balancing of planar parallel manipulators. In: Proc IEEE International Conference on Robotics and Automation, New Orleans, pp 4527 – 4532 Alici G, Shirinzadeh B (2006) Optimum dynamic balancing of planar parallel manipulators based on sensitivity analysis. Mech Mach Theory 41(12): 15201532 Angeles J (1997) Fundamentals of robotic mechanical systems: Theory, methods, and algorithms, Springer, New York Arakelian VH, Smith MR (2008) Design of planar 3-DOF 3-RRR reactionless parallel manipulators. Mechatronics 18(10):601-606 Arsenault M, Bourdeau R (2004a) The synthesis of three-degree-of-freedom planar parallel mechanisms with revolute joints (3-RRR) for an optimal singularity-free workspace. J Robotic Systems 21(5):259-274 Arsenault M, Bourdeau R (2004b) The synthesis of a general planar parallel manipulator with prismatic joints for optimal stiffness. In: Proc of the 11th World Congress in Mechanism and Machine Science, vol 4, China Machine Press, Beijing, pp 1633-1637 Arsenault M, Bourdeau R (2006) Synthesis of planar parallel mechanisms while considering workspace, dexterity, stiffness and singularity avoidance. ASME J Mechanical Design 128:69-79 Balan R, Maties V, Stan S, Lapusan C (2005) On the control of a 3-RRR planar parallel minirobot. J Mecatronica 4:1583-7653 Bamberger H, Shoham M, Wolf A (2006) Kinematics of micro planar parallel robot comprising large joint clearances. In: Lenarčič J, Roth B (eds) Advances in Robot Kinematics: Mechanisms and Motion, Springer, Dordrecht, pp 75-84

666

References

Binaud N, Caro S, Wenger P (2009a) Sensitivity Analysis of Degenerate and NonDegenerate Planar Parallel Manipulators. In: Ceccarelli M (ed) Proceedings of EUCOMES 08, The Second European Conference on Mechanism Science, Springer, Dordrecht, pp 5005-512 Binaud N, Caro S, Wenger P (2009b) Sensitivity and dexterity comparison of 3RRR planar parallel manipulators. In: Kecskeméthy A, Müller A (eds) Computational Kinematics, Proceedings of the 5th International Workshop on Computational Kinematics, Springer-Verlag, Berlin, pp 77-84 Bonev IA (2008) Planar parallel mechanism and method. US patent application 2008/0229860 Bonev IA, Gosselin CM (2001) Singularity loci of planar parallel manipulators with revolute joints, 2nd Workshop on Computational Kinematics, Seoul, pp 291–299 Bonev IA, Zlatanov D, Gosselin CM (2001) Singularity analysis of 3-dof planar parallel mechanisms. In: Proc CCToMM Symp Mechanisms, Machines and Mechatronics, St-Hubert Bonev IA, Zlatanov D, Gosselin MG (2003) Singularity analysis of 3-DOF planar parallel mechanisms via screw theory. Trans ASME J Mech Design 125:573581 Bonev IA, Briot S, Wenger P, Chablat D (2008) Changing Assembly modes without Passing Parallel Singularities in Non-Cuspidal 3-RPR Planar Parallel Robots. In: Proc 2nd Int Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, Montpellier Boudreau R, Gosselin CM (1999) The synthesis of planar parallel manipulators with a genetic algorithm. Trans ASME J Mech Design 121:533-537 Bouzagarrou CB, Ray P, Gogu G (2000) Manipulabilité: critère de synthèse de manipulateurs parallèles plans. In: Proc Int Conf on Integrated Design and Manufacturing in Mechanical Engineering, Montréal Briot S, Bonev IA (2008) Accuracy analysis of 3-DOF planar parallel robots. Mech Mach Theory 43(4):445–458 Briot S, Bonev IA (2009) Pantopteron a new fully decoupled 3DOF translational parallel robot for pick-and-place applications. J Mechanisms and Robotics 1(2):1-9 Briot S, Arakelian V, Bonev IA, Chablat D, Wenger P (2008) Self motions of general 3-RPR planar parallel robots. Int J Robotics Research 27(7):855–866 Briot S, Bonev IA, Gosselin, CM, Arakelian V (2009) Complete shaking force and shaking moment balancing of planar parallel manipulators with prismatic pairs. IMech J Multi-body Dynamics 223(1):43–52 Burton GL, Burton PJ (1996) X-Y-Theta positioning mechanism. US patent 5523941 Caro S, Chablat D, Wanger P, Angeles J (2003) The isoconditioning loci of planar three-dof parallel manipulators. In: Gogu G, Coutellier D, Chedmail P, Ray P (eds) Recent advances in integrated design and manufacturing in mechanical engineering. Kluwer Academic Publishers Carricato M (2005) Fully-isotropic four degrees-of-freedom parallel mechanisms for Schoenflies motion. Int. J Robotics Research 24(5):397-414

References

667

Carricato M, Parenti-Castelli V (2002) Singularity-free fully-isotropic translational parallel mechanism. Int J Robot Res 21(2):161-174 Castillo-Castaneda E, Fonseca-Reyes MJ, Lopez-Cajun CS (2007) Non-linear control to compensate low velocity friction of a planar parallel robot. In: Proc of the 12th World Congress in Mechanism and Machine Science, Besançon Cha SH, Lasky TA, Velinsky SA (2007) Singularity avoidance for the 3-RRR mechanisms using kinematic redundancy. Proc IEEE Int Conf Robotics and Automation, Rome, pp 1195-1200 Cha SH, Lasky TA, Velinsky SA (2009) Determination of the kinematically redundant active prismatic joint variable ranges of a planar parallel mechanism for singularity-free trajectories. Mech Mach Theory 44(5):1032-1044 Chablat D, Staicu S (2009) Kinematics of A 3-PRP planar parallel robot. University “Politehnica” of Bucharest Scientific Bulletin, Series D: Mechanical Engineering 71(1):3-16 Chablat D, Wenger P (2001) Séparation des solutions aux modèles géométriques direct et inverse pour les manipulateurs pleinement parallèles. Mech Mach Theory 36: 763-783 Chablat D, Wenger P (2004) The kinematic analysis of a symmetrical threedegree-of-freedom planar parallel manipulator. In: Proc CISM-IFToMM Symp Robot Design, Dynamics and Control, Montreal Chablat D, Wenger P Bonev I (2006) Self motions of a special 3-RPR planar parallel robot. Lenarčič J, Roth B (eds) Advances in Robot Kinematics: Mechanisms and Motion, Springer, Dordrecht, pp 221-228 Choi JK (2003) Kinematic analysis and optimal design of 3-PPR planar parallel manipulator. Journal of Mechanical Science and Technology 17(4):528-537 Choi JK, Mori O, Omata T (2004a) Dynamic and stable reconfiguration of selfreconfigurable planar parallel robots. Advanced Robotics 18(6):565-582 Choi JK, Omata T, Mori O (2004b) Self-reconfigurable planar parallel robot. In: Proc IEEE Int Conf Inteligent Robots Systems, Sendai, pp 2654-2660 Chung GJ, Choi KB (2004) Development of nano order manipulation system based on 3-PPR planar parallel mechanism. In: Proc IEEE Int Conference on Robotics and Biomimetics, Kunming, pp 612-616 Collins CL (2002) Forward kinematics of planar parallel manipulators in the Clifford algebra of P2. Mech Mach Theory 37:799-813 Collins CL, McCarthy JM (1998) The quartic singularity surfaces of planar platforms in the Cliford algebra of the projective plane. Mech Mach Theory 33(7):931–944 Company O, Krut S, Pierrot F (2007) Analysis of a high resolution planar PKM. In: Proc of the 12th World Congress in Mechanism and Machine Science, Besançon Constantinescu D, Chau I, DiMaio SP, Filipozzi L, Salcudean SE, Ghassemi F (2000) Haptic rendering of planar rigid-body motion using a redundant parallel mechanism. Proc IEEE Int Conf Robotics and Automation, San Francisco, pp 2440-2445 Dai JS, Jones JR (1999) Mobility in metamorphic mechanisms of foldable/ erectable kinds. ASME J Mech Design 121(3):375-382

668

References

Dasgupta B, Mruthyunjaya TS (1998) Force redundancy in parallel manipulators: theoretical and practical issues. Mech Mach Theory 33:727-742 Degani A, Wolf A (2006a) Graphical singularity analysis of planar parallel manipulators. In: Proc IEEE Int Conf Robotics and Automation, Orlando, pp 751-756 Degani A, Wolf A (2006b) Graphical singularity analysis of 3-DOF planar parallel manipulators. In: In: Lenarčič J, Roth B (eds) Advances in Robot Kinematics: Mechanisms and Motion, Springer, Dordrecht, pp 229-238 Di Gregorio R (2009) A novel method for the singularity analysis of planar mechanisms with more than one degree of freedom. Mech Mach Theory 44(1):83-102 Du Plessis LJ, Snyman JA (2002) Design and optimum operation of a reconfigurable planar Gough–Stewart machining platform. In: Neugebauer R (ed.) Proceedings of PKS 2002 Parallel Kinematics Seminar, Development Methods and Application Experience of Parallel Kinematics, Chemnitz, pp 729–749 Du Plessis LJ, Snyman JA (2006a) Determination of optimum geometries for a planar re-configurable machining platform using the LFOPC optimization algorithm. Mech Mach Theory 41:307-333 Du Plessis LJ, Snyman JA (2006b) An optimally re-configurabe planar GoughStewart machining platform. Mech Mach Theory 41:334-357 Du Z, Yu Y (2006) Dynamic modeling and analysis of flexible planar parallel robots. In: Proc. 7th Int. Conf. Frontier of Design and Manufacturing, Guangzhou, pp 463-468 Dudi F, Diaconescu DV, Gogu G (1989) Mecanisme articulate: inventica si cinematica in abordare filogenetica. Ed Tehnica, Bucureşti Dudi F, Diaconescu D, Jaliu C, Bârsan A, Neagoe M (2001a) Cuplaje mobile articulate. Ed Orientul Latin, Braşov Dudi F, Diaconescu D, Lateş M, Neagoe M (2001b) Cuplaje mobile podomorfe, Ed Trisedes Press, Braşov Duffy J (1980) Analysis of mechanisms and robot manipulators, Arnold, London Ebrahimi I, Carretero JA, Boudreau R (2007a) 3-PRRR redundant planar parallel manipulator: Inverse displacement, workspace and singularity analysis. Mech Mach Theory 42:1007-1016 Ebrahimi I, Carretero JA, Boudreau R (2007b) Actuation scheme for a 6-dof kinematically redundant planar parallel manipulator. In: Proc of the 12th World Congress in Mechanism and Machine Science, Besançon Ebrahimi I, Carretero JA, Boudreau R (2008) Kinematic analysis and path planning of a new kinematically redundant planar parallel manipulator. Robotica 26(3):405-413 Fanghella P, Galletti C, Giannotti E (2006) Parallel robots that change their group of motion. In: Lenarčič J, Roth, B. (eds) Advances in Robot Kinematics: Mechanisms and Motion. Springer, Dordrecht, pp 49-56 Fattah A, Hasan Ghasemi AM (2002) Isotropic design of spatial parallel manipulators. Int J Robot Res 21(9):811-824

References

669

Fattah A, Agrwal (2006) On the design of reactionless 3-DOF planar parallel mchanisms. Mech Mach Theory 41:70-82 Fattah A, Misra A, Angeles J (1994) Dynamics of a flexible-link planar parallel manipulator in Cartesian space. Proc ASME Design Eng Tech Conf, Minneapolis Firmani F, Podhorodeski RP (2004) Force-unconstrained poses for a redundantlyactuated planar parallel manipulator. Mech Mach Theory 39:459-476 Firmani F, Podhorodeski RP (2005) Force-unconstrained poses of the 3-PRR and 4-PRR planar parallel manipulators. Trans Canadian Society for Mechanical Engineering 29(4):617-628 Firmani F, Podhorodeski RP (2007) Singularity loci of revolute-jointed planar parallel manipulators with redundant actuated branches. Proc of the 12th World Congress in Mechanism and Machine Science, Besançon Firmani F, Podhorodeski RP (2009) Singularity analysis of planar parallel manipulators based on forward kinematic solutions. Mech Mach Theory 44(7): 1386-1399 Firmani F, Zibil A, Nokleby S, Podhorodeski RP (2007) Wrench capabilities of planar parallel manipulators. Part II: Redundancy and wrench workspace analysis. Robotica 26(06):803-815 Fischer R, Podhorodeski RP, Nokleby SB (2001) A reconfigurable planar parallel manipulator. Proc Symp Mechanisms, Machines and Mechatronics, Montreal Fischer R, Podhorodeski RP, Nokleby SB (2004) Design of a reconfigurable planar parallel manipulator. J Robotic Syst 21(12):665-675 Foucault S, Gosselin CM (2002) On the development of a planar 3-DOF reactionless parallel mechanism. In: Proc Proc ASME Design Eng Tech Conf Montreal Foucault S, Gosselin CM (2004) Synthesis, design, and prototyping of a planar three degree-of-freedom reactionless parallel mechanism. Trans ASME J Mech Design 126: 992-999 Fried G, Djouani K, Borojeni D, Iqbal S (2008) Determination of 3-RPR planar parallel robot assembly modes by Jacobian matrix factorization. WSEAS Trans on Systems 7(2):41-48 Fu K, Mills J (2005) A planar parallel manipulator-dynamics revisited and controller design. In: Proc IEEE Int Conf Inteligent Robots Systems, Edmonton, pp 331-336 Gallant M, Boudreau R (2000) An optimal singularity-free planar parallel manipulator for a prescribed workspace using a genetic algorithm. In: Proc Int Conf on Integrated Design and Manufacturing in Mechanical Engineering, Montréal Gallant M, Boudreau R (2002) The synthesis of planar parallel manipulators with prismatic joints for an optimal, singularity-free workspace. J Robotic Syst 19(1):13-24 Gallant M, Gosselin CM (2003) The effect of joint clearances on the singular configurations of planar parallel manipulators. In: Proc CCToMM Symp Mechanisms, Machines, and Mechatronics, St-Hubert

670

References

Galletti C, Fanghella P (2001) Single-loop kinematotropic mechanisms. Mech Mach Theory 36:743-761 Gao F, Zhao YS, Zhang XQ, ZH (1996) Physical model of the solution space of 3DOF parallel planar manipulators. Mech Mach Theory 31(2): 161-171 Gao F, Liu XJ, Chen X (2001) The relationships between the shapes of the workspace and the link lengths of 3-DOF symmetrical planar parallel manipulators. Mech Mach Theory 36:205-220 Geike T, McPhee J (2002) On the automatic generation of inverse dynamic solutions for parallel manipulators with full and reduced mobility. In: Gosselin CM, Ebert-Uphoff I (eds) Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, Quebec Geike T, McPhee J (2003) Inverse dynamic analysis of parallel manipulators with full mobility. Mech Mach Theory 38:549-562 Gogu G (2002) Structural synthesis of parallel robotic manipulators with decoupled motions. Report ROBEA MAX-CNRS Gogu G (2004a) Structural synthesis of fully-isotropic translational parallel robots via theory of linear transformations. Eur J Mech A-Solids 23(6):1021-1039 Gogu G (2004b). Fully-isotropic T3R1-type parallel manipulators. In: Lenarčič J, Galletti C (eds) On advances in robot kinematics, Kluwer Academic Publishers, Dordrecht, pp 265-272 Gogu G (2004c) Fully-isotropic over-constrained planar parallel manipulators. In: Proc IEEE/RSJ International Conference on Intelligent Robots and Systems, Sendai, pp 3519-3524 Gogu G (2005a) Evolutionary morphology: a structured approach to inventive engineering design. In: Bramley A, Brissaud D, Coutellier D, McMahon C (eds) Advances in integrated design and manufacturing in mechanical engineering. Springer, Dordrecht, pp 389-402 Gogu G (2005b) Mobility of mechanisms: a critical review. Mech Mach Theory 40:1068-1097 Gogu G (2005c) Chebychev-Grubler-Kutzbach’s criterion for mobility calculation of multi-loop mechanisms revisited via theory of linear transformations. Eur J Mech A–Solids 24:427-441 Gogu G (2005d) Mobility and spatiality of parallel robots revisited via theory of linear transformations. Eur J Mech A–Solids 24: 690-711 Gogu G (2005e) Mobility criterion and overconstraints of parallel manipulators. In: Proc International Workshop on Computational Kinematics Cassino, Paper 22-CK2005 Gogu G (2005f) Fully-isotropic over-constrained parallel wrists with two degrees of freedom. In: Proc IEEE International Conference on Robotics and Automation, Barcelona, pp 4025-4030 Gogu G (2005g) Singularity-free fully-isotropic parallel manipulators with Schönflies motions. In: Proc 12th International Conference on Advanced Robotics, Seattle, pp 194-201 Gogu G (2005h) Fully-isotropic parallel robots with four degrees of freedom T2R2-type. In: Proc of 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems, Edmonton, pp 1190-1195

References

671

Gogu G (2005i) Fully-isotropic T1R2-type parallel robots with three degrees of freedom. In: Proc International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Long Beach Gogu G (2006a) Fully-isotropic redundantly-actuated parallel manipulators with five degrees of freedom. In: Husty M, Schröcker HP(eds) Proc First European Conference on Mechanism Science, Obergurgl Gogu G (2006b) Fully-isotropic parallel manipulators with five degrees of freedom. In:Proc IEEE International Conference on Robotics and Automation, Orlando, pp 1141-1146 Gogu G (2006c) Fully-isotropic parallel manipulators with Schönflies motions and complex legs with rhombus loops. In: Proc IEEE International Conference on Robotics and Automation, Orlando, pp 1147-1152 Gogu G (2006d) Fully-isotropic T3R2-type parallel manipulators. Proc. IEEE International Conference on Robotics, Automation and Mecatronics, Bangkok, pp 248-253 Gogu G (2006e) Fully-isotropic hexapods. In: Lenarčič J, Roth B (eds) Advances in robot kinematics: Mechanisms and motion, Springer, Dordrecht, pp 323-330 Gogu G (2007a) Structural synthesis of fully-isotropic parallel robots with Schönflies motions via theory of linear transformations and evolutionary morphology. Eur J Mech A–Solids 26 (2):242-269 Gogu G (2007b) Fully-isotropic three-degree-of-freedom parallel wrists. In: Proc IEEE International Conference on Robotics and Automation, Rome, pp 895900 Gogu G (2007c) Isogliden-TaRb: a Family of up to five axes reconfigurable and maximally regular parallel kinematic machines. J of Manufacturing Systems 36 (5):419-426 Gogu G (2007d) Fully-isotropic T2R3-type redundantly-actuated parallel robots. In: Proc IEEE/RSJ International Conference on Intelligent Robots and Systems, San Diego, pp 3937-3942 Gogu G (2007e) Fully-isotropic redundantly-actuated parallel wrists with three degrees of freedom. In: Proc International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Las Vegas Gogu G (2007f) Reangularity: cross-coupling kinetostatic index for parallel robots. Proc of the 12th World Congress in Mechanism and Machine Science, Besançon Gogu G (2008a) Structural Synthesis of Parallel Robots: Part 1-Methodology, Springer, Dordrecht Gogu G (2008b) Constrained singularities and the structural parameters of parallel robots. In: Lenarčič J, Wenger P (eds) Advances in Robot Kinematics: Analysis and design, Springer, Dordrecht, pp 21-28 Gogu G (2008c) Bifurcation in constraint singularities and structural parameters of parallel. In: Andreff N, Company O, Gouttefarde M, Krut S, Pierrot F (eds.) Proceedings of the Second International Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, Montpellier, pp 137-143

672

References

Gogu G (2008d) Mobility, connectivity, overconstraint and redundancy of parallel robots. In Proc 4th International Conference Robotics, Brasov, pp 15-32. Gogu G (2008e) Fully-Isotropic T1R3-type redundantly-actuated parallel manipulators. In: Lee S et al. (eds) Recent Progress in Robotics: Viable Robotic Service to Human, Springer, Heildelberg, pp 79-90 Gogu G (2008f) Fully-isotropic parallel mechanisms - an innovative concept for haptic devices. In: Talaba D, Amditis A (eds) Product Engineering: Tools and Methods Based on Virtual Reality, Springer, Dordrecht, pp; 169-195 Gogu G (2008g) A new family of maximally regular T2R1-type spatial parallel manipulators with unlimited rotation of the moving platform.In Proc Australasian Conference on Robotics and Automation, Canberra Gogu G (2008h) Fully-isotropic T1R3-type parallel manipulators. In: Proc 4th International Conference Robotics, Brasov, pp 525-532 Gogu G (2008i) Kinematic criteria for structural synthesis of maximally regular parallel robots with planar motion of the moving platform. In: Kecskeméthy A (ed.) Interdisciplinary Applications of Kinematics, Lima, pp 61-79 Gogu G (2008j) Semangularity: kinetostatic index of input-output propensity in parallel robots. In: Proc 10th International Conference on Mechanisms and Mechanical Transmissions, Timisoara, pp 163-170 Gogu G (2009a) Structural Synthesis of Parallel Robots: Part 2-Translational Topologies with Two and Three Degrees of Freedom, Springer, Dordrecht Gogu G (2009b) Branching singularities in kinematotropic parallel mechanisms. In: Kecskeméthy A, Müller A (eds) Computational Kinematics, Proceedings of the 5th International Workshop on Computational Kinematics, SpringerVerlag, Berlin, pp 341-348 Gogu G (2009c) T2R1-type parallel manipulators with decoupled and bifurcated planar-spatial motion of the moving platform. In: Proc IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Singapore, pp 1022-1027 Gogu G (2009d) Structural synthesis of maximally regular T3R2-type parallel robots via theory of linear transformations and evolutionary morphology, Robotica 27:79-101 Gosselin CM, Angeles J (1988) The optimum kinematic design of a planar threedegree-of-freedom parallel manipulator. Trans ASME, J Mechanisms Trans and Automation in Design 110(1):35-41 Gosselin CM, Jean M (1996). Determination of the workspace of planar parallel manipulators with joint limits. Robotics and Autonomous Systems 17(3):129138. Gosselin CM, Kong X (2002) Cartesian parallel manipulators. International patent WO 02/096605 A1 Gosselin CM, Merlet JP (1994) On the direct kinematics of planar parallel manipulators: special architectures and number of solutions. Mech Mach Theory 29(8): 1083-1097 Gosselin CM, Sefrioui J (1992) Graphical representation of the singularity loci of planar parallel manipulators. In: Proc 4th Int Symp Robotics and Automation, Santa Fe, pp 333-338

References

673

Gosselin CM, Wang J (1995) Singularity loci of planar parallel manipulator. In: Proc of the 9th World Congress on the Theory of Machines and Mechanisms, pp1982-1986 Gosselin CM, Wang J (1997) Singularity loci of planar parallel manipulator with revolute actuators. Robotics and Autonomous Systems 21:377-398 Gosselin CM, Sefrioui J, Richard JM (1992) Solution polynomiale au problème de la cinématique directe des manipulateurs parallèles plans à 3 degrés de liberté. Mech Mach Theory 27(2):107-119 Gosselin CM, Lemieux S, Merlet JP (1996) A new architecture of planar threedegree-of freedom parallel manipulator. In: Proc of IEEE Int Conf on Robotics and Automation, Minneapolis, pp 3738-3743 Gosselin CM, Kong X, Foucault S, Bonev IA (2004). A fully-decoupled 3-dof translational parallel mechanism. In: Parallel Kinematic Machines in Research and Practice, 4th Chemnitz Parallel Kinematics Seminar, pp 595-610 Guo X, Zhu S, Yu M, Kong L (2004) Kinematic performance analysis for planar parallel mechanism 3RRR. In: Proc of the 11th World Congress in Mechanism and Machine Science, vol 3, China Machine Press, Beijing, pp 1223-1227 Hahn H, Klier W, Leimbach K (1999) Nonlinear control of planar parallel robots with redundant servopneumatic actuators. Zeitschrift für angewandte Mathematik und Mechanik 79(Suppl 1): S79-S82 Harms E, Kounias S, Vroomen LJ, Zsombor-Murray PJ (1991) A binarydecision/transputer network for the control of a planar three degree of freedom parallel robotic manipulator. Microprocessing and Microprogramming 31(1-5):137-142 Hay AM, Snyman JA (2000) The determination of nonconvex workspaces of generally constrained planar Stewart platforms. Computers and Mathematics with Applications 40(8-9):1043-1060 Hay AM, Snyman JA (2002) The chord method for the determination of nonconvex workspaces of planar parallel manipulators. Computers and Mathematics with Applications 43(8-9): 1135-1151 Hay AM, Snyman JA (2005) A multi-level optimization methodology for determining the dextrous workspaces of planar parallel manipulators. Structural and Multidisciplinary Optimization 30(6):422-427. Hay AM, Snyman JA (2006) Optimal synthesis for a continuous prescribed dexterity interval of a 3-dof parallel planar manipulator for different prescribed output workspaces. Int J Numerical Methods Eng 68(1):1-12 Hayes MJD, Husty ML (2000) Workspace characterization of planar three-legged platforms with holonomic higher pairs. Lenarčič J, Stanišić MM (eds) Advances in robot kinematics, Kluwer Academic Publishers, Dordrecht, pp 267–276 Hayes MJD, Husty ML (2003) On the kinematic constraint surfaces of general three-legged planar robot platforms. Mech Mach Theory 38:379-394 Hayes MJD, Zsombor-Murray P (1996) A planar parallel manipulator with holonomic higher pairs: inverse kinematics. In Proc. CSME Forum, Symposium on the Theory of Machines and Mechanisms, Hamilton

674

References

Hayes MJD, Zsombor-Murray P (1998) Inverse kinematics of a planar manipulator with holonomic higher pairs. In: Lenarčič J, Husty M (eds) Advances in robot kinematics: analysis and control, Kluwer Academic Publishers, Dordrecht, pp 59-68 Hayes MJD, Zsombor-Murray P, Chen C (2004) Unified kinematic analysis of general planar parallel robots. Trans ASME J Mech Design 126(5):866-874 Heerah I, Kang B, Mills J, Benhabib B (2002) Architecture selection and singularity analysis of a 3-degree-of-freedom planar parallel manipulator. Proc ASME Design Eng Tech Conf, Montreal Heerah I, Benhabib B, Kang B, Mills J (2003) Architecture selection and singularity analysis of a three-degree-of-freedom planar parallel manipulator. J Intelligent Robotic Systems 37(4):355-374 Hunt KH (1978) Kinematic Geometry of Mechanisms. Oxford University Press, Oxford Hunt KH (1982) Geometry of robotic devices. Mechanical Engineering Transactions 7(4):213-220 Hunt KH (1983) Structural kinematics of in-parallel actuated robot-arms. Trans ASME, J Mechanisms, Transmissions, and Automation in Design 105:705712 Husty M (1996). On the workspace of planar parallel three-legged platforms. In: Proc 2nd World Automation Congress–6th Int Symp Robotic and Manufacturing Systems, Montpellier, pp 28-30 Husty M (2009) Non-singular assembling mode change in 3-RPR parallel manipulators. In: Kecskeméthy A, Müller A (eds) Computational Kinematics, Proceedings of the 5th International Workshop on Computational Kinematics, Springer-Verlag, Berlin, pp 51-60 Husty M, Gosselin CM (2008) On the singularity surface of planar 3-RPR parallel mechanisms. In: Proc Int Symp Multibody Systems and Mechatronics, San Juan Ionescu TG (2003) Terminology for mechanisms and machine science. Mech Mach Theory 38:597-901 Jean M, Gosselin CM (1996) Static balancing of planar parallel manipulators. In: Proc IEEE Int Conf Robotics and Automation, Minneapolis, pp 3732-3737 Jeanneau A., Herder J, Laliberte T, Gosselin CM (2004) A compliant rolling contact joint and its application in a 3-dof planar parallel mechanism with kinematic analysis. Proc ASME Design Eng Tech Conf, Salt Lake City Ji Z (2003) Study of planar three-degree-of-freedom 2-RRR parallel manipulators. Mech Mach Theory 38(5):409-416 Ji P, Wu H (2002) An efficient approach to the forward kinematics of a planar parallel manipulator with similar platforms. IEEE Trans Robotics Automation 18(4):647-649 Jiang Q, Gosselin CM (2006) The maximal singularity-free workspace of planar 3-RPR parallel mechanisms. In: Proc IEEE Int Conf Mechatronics and Automation, Luoyang

References

675

Jiang Q, Gosselin CM (2008) Geometric Optimization of Planar 3-RPR Parallel Mechanisms, Trans Canadian Society for Mechanical Engineering 31(4):457468 Kang B, Mills JK (2002) Dynamic modelling of structurally-flexible planar parallel manipulator. Robotica 20:329-339 Kang B, Mills JK (2003) Study on piezoelectric actuators in vibration control of a planar parallel manipulator. In: Proc IEEE Int Conf Robotics and Automation, Taipei, pp 1268-1272 Kang B, Mills JK (2005) Vibration control of a planar parallel manipulator using piezoelectric actuators. J Intelligent Robotic Systems 42(1):51-70 Kang B, Chu J, Mills JK (2001) Design of high speed planar parallel manipulator and multiple simultaneous specification control. In: Proc IEEE Int Conf Robotics and Automation, Seoul, pp 2723-2728 Kang B, Yeung B, Mills JK (2002) Two-time scale controller design for a high speed planar parallel manipulator with structural flexibility. Robotica 20:519528 Khan W, Krovi V, Saha S, Angeles J (2005) Recursive kinematics and inverse dynamics for a planar 3R parallel manipulator. J Dynamic Systems, Measurement, Control 127: 529-537 Kim SH, Tsai L-W (2002) Evaluation of a cartesian parallel manipulator. In: Lenarčič J, Thomas F (eds) Advances in robot kinematics. Kluwer Academic Publishers, pp 21-28 Kim WK, Kim DG, Yi BJ (1996a) Analysis of a planar 3 degree-of-freedom adjustable compliance mechanism. J Mechanical Science and Technology 10(3): 286-295 Kim WK, Lee JY, Yi BJ (1996b) RCC characteristics of planar/spherical three degree of freedom parallel mechanisms with joint compliances. In: Proc IEEE Int Conf Inteligent Robots Systems, Osaka, pp 360-367 Kim WK, Lee JY, Yi BJ (1997) Analysis for a planar 3 degree-of-freedom parallel mechanism with actively adjustable stiffness characteristics. J Mechanical Science and Technology 11(4):408-418 Kong X (2008) Forward Kinematics and Singularity Analysis of a 3-RPR Planar Parallel Manipulator. In: Lenarčič J, Wenger P (eds) Advances in Robot Kinematics: Analysis and design, Springer, Dordrecht, pp 29-38 Kong X (2009) Forward displacement analysis of a 3-RPR planar parallel manipulator revisited. In: Kecskeméthy A, Müller A (eds) Computational Kinematics, Proceedings of the 5th International Workshop on Computational Kinematics, Springer-Verlag, Berlin, pp 69-76 Kong X, Gosselin CM (2000) Determination of the uniqueness domains of 3-RPR planar parallel manipulators with similar platforms. In: Proc ASME Design Eng Tech Conf, Baltimore Kong X Gosselin CM (2001) Forward displacement analysis of thrd-class analytic 3-RPR planar parallel manipulator. Mech Mach Theory 36:1009-1018 Kong X, Gosselin CM (2002a) Kinematics and singularity analysis of a novel type of 3-CRR 3-dof translational parallel manipulator. Int J Robot Res 21(9):791798

676

References

Kong X, Gosselin CM (2002b) Type synthesis of linear translational parallel manipulators. In: Lenarčič J, Thomas F (eds) Advances in robot kinematics. Kluwer Academic Publishers, pp 453-462 Kong X, Gosselin CM (2002c) A class of 3-DOF translational parallel manipulators with linear input-output equations. In: Proc of the Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, Québec, pp 25–32 Kong X, Gosselin CM (2002d) Generation and forward displacement analysis of RPR-PR-RPR analytic planar parallel manipulators. Trans ASME, J Mech Design 124:294-300. Kong X ,Gosselin CM (2008) Forward displacement analysis of a quadratic planar parallel manipulator: 3-RPR parallel manipulator with similar triangular platforms. In: Proc ASME Design Eng Tech Conf, New-York Kwan YBP, Wetzels SFCL, Vedhuis GP (2003) Positionning system for use in lithographic apparatus. US patent 6635887 Laliberte T, C. Gosselin CM, Jean M (1999). Static balancing of 3-DOF planar parallel mechanisms. IEEE/ASME Trans Mechatronics 4(4):363-377. Last P, Schutz D, Raatz A, Hesselbach J (2007) Singularity based calibration of 3DOF fully parallel planar manipulators. Proc of the 12th World Congress in Mechanism and Machine Science, Besançon Leblond M. Gosselin CM (1998). Static balancing of spatial and planar parallel manipulators with prismatic actuators. Proc ASME Design Eng Tech Conf, Atlanta Lee J, Duffy J, Keler M (1999) The optimum quality index for the stability of in-parallel planar platform devices. Trans ASME J Mech Design 121:15-20 Li S, Gosselin CM (2007) Stiffness analysis of 3-RRR planar parallel mechanisms based on CTT. In: Proc of the 12th World Congress in Mechanism and Machine Science, Besançon Li DS, Gosselin CM (2008a) Stiffness characteristics of 3-RPR planar parallel mechanism based on CCT stiffness matrix. In: Proc 8th Int Conf Frontiers of Design and Manufacturing, Tianjin Li DS, Gosselin CM (2008b) Determination of singularity-free zones in the workspace of planar parallel mechanisms with revolute actuators In: Proc 8th Int Conf Frontiers of Design and Manufacturing, Tianjin Li H, Gosselin GM, Richard MJ (2006) Determination of maximal singularity-free zones in the workspace of planar three-degree-of-freedom parallel mechanisms. Mech Mach Theory 41(10):1157-1167 Li W, Liu X, Liu K (2007) Tracking control of a planar parallel robot via adaptive backstepping. Proc of the 12th World Congress in Mechanism and Machine Science, Besançon Liu XJ, Wang J, Gao F (2000) Performance atlases of the workspace for planar 3DOF parallel manipulators. Robotica 18:563-568 Long CS, Snyman JA, Groenwold AA (2003) Optimal structural design of a planar parallel platform for machining. Applied Mathematical Modelling 27(8):581-609

References

677

Lösch S (1995) Parallel redundant manipulator based on open and closed normal Assur chains. In: Merlet JP, Ravani B (eds) Computational Kinematics, Kluwer Academic Publishers, Dordrecht, pp 251-260 Ma O, Angeles J (1989) Direct kinematics and dynamics of a planar 3-dof parallel manipulator. Advances in Design Automation 3:313–320. Macho E, Altuzarra O, Pinto C, Hernandez A (2007) Singularity free change of assembly mode in parallel manipulators: application to the 3-RPR planar platform. In: Proc of the 12th World Congress in Mechanism and Machine Science, Besançon Marquet F, Krut S, Company O, Pierrot F (2001a) ARCHI: a new redundant mechanism - modeling, control and first results. Proc IEEE International Conference on Intelligent Robots and Systems, pp 183-188 Marquet F, Krut S, Company O, Pierrot F (2001b) ARCHI: a redundant mechanism for machining with unlimited rotation capacities. In: Proc IEEE International Conference on Advanced Robotics, pp 683-689 Masouleh MT, Gosselin C (2007) Determination of singularity-free zones in the workspace of planar 3-PRR parallel mechanisms. Trans ASME J Mech Design 129:649-652 Matsumoto H (1992) Movable table. US patent 5163651 Merlet JP (1996a) Direct kinematics of planar parallel manipulators. In: Proc IEEE Int Conf on Robotics and Automation, pp 3744-3749 Merlet JP (1996b) On the separability of the solutions of the direct kinematics of a special class of planar 3-RPR parallel manipulator. Proc ASME Design Eng Tech Conf, Baltimore Merlet J-P (1997) Les robots parallèles, 2nd edn, Hermès, Paris Merlet JP (2000) Parallel robots, 1st edn, Kluwer, Dordrecht Merlet JP (2006) Parallel robots, 2nd edn, Springer, Dordrecht Merlet JP, Gosselin CM, Mouly N (1998) Workspaces of planar parallel manipulators. Mech Mach Theory 33(1-2):7-20 Mohamed M. Duffy J (1985) A direct determination of the instantaneous kinematics of 3 dof planar and spherical doubletriangular parallel manipulators. ASME J. Mech. Trans. Autom. Des 107:226–229 Mohamadi Daniali HR (2005) Instantaneous center of rotation and singularities of planar parallel manipulators. I Int J Mech Eng Education 33(3):251-259 Mohamadi Daniali HR, Zsombor-Murray P-J (1994). The design of isotropic planar parallel painpulators. Intelligent Automation and Soft Computing: 273280 Mohamadi Daniali HR, Zsombor-Murray P (1999) The design of isotropic planar parallel manipulators. In Jamshidi M, Yuh I, Nguyen CC, Lumia R (eds) Intelligent Automation and Soft Computing. TSI Press, Albuquerque, vol 2, pp. 273-280 Mohamadi Daniali HR, Zsombor-Murray P-J, Angeles J (1993) The kinematics of a 3 dof planar and spherical double-triangle parallel manipulator. In: Angeles J, Kovacs P, Hommel G (eds) Computational kinematics, Kluwer, pp 153-164

678

References

Mohamadi Daniali HR, Zsombor-Murray P-J, Angeles J (1995a) Singularity analysis of a general class of planar parallel manipulators. In: Proc of IEEE Int Conf on Robotics and Automation, pp 1547-1552 Mohamadi Daniali HR, Zsombor-Murray PJ, Angeles J (1995b) singularity analysis of planar parallel manipulators. Mech Mach Theory 30(5): 665-678 Mohamadi Daniali HR, Zsombor-Murray P-J, Angeles J (1995c) The isotropic design of two general classes of planar parallel manipulators. Journal of Robotic Systems 12(12):795-805 Mukhopadhyay D, Dong J, Pengwang E, Ferreira P (2008) A SOI-MEMS-based 3-DOF planar parallel-kinematics nanopositioning stage. Sensors and Actuators A: Physical 147(1):340-351 Murray A, Hanchak M (2000) Kinematic synthesis of planar platforms with RPR, PRR, and RRR chains. In: Lenarčič J, Stanišić MM (eds) Advances in robot kinematics, Kluwer Academic Publishers, Dordrecht Murray AP, Pierrot F (1998) N-position synthesis of parallel planar RPR platforms. In: Lenarčič J, Husty M (eds) Advances in robot kinematics: analysis and control, Kluwer Academic Publishers, Dordrecht, pp 69-78 Murray AP, Pierrot F, Dauchez P, McCarthy JM (1997) A planar quaternion approach to the kinematic synthesis of a parallel manipulator. Robotica 15:361-365 Müller A (2005) Internal preload control of redundantly actuated parallel manipulators-its application to backlash avoiding control. IEEE Trans Robotics 21(4):668-677 Nokleby NS, Firmani F, Zibil A, Podhorodeski RP (2007a) An analysis of the force-moment capabilities of branch-redundant planar-parallel manipulators. Proc ASME Design Eng Tech Conf, Las Vegas Nokleby NS, Firmani F, Zibil A, Podhorodeski RP (2007b) Force-moment capabilities of redundantly-actuated planar-parallel architectures. Proc of the 12th World Congress in Mechanism and Machine Science, Besançon Piras G, Cleghorn W, Mills J (2005) Dynamic finite-element analysis of a planar high-speed, high-precision parallel manipulator with flexible links. Mech Mach Theory 40(7):849-862 Rakotomanga N, Chablat D, Caro S (2008) Kinetostatic performance of a planar parallel mechanism with variable actuation. In: Lenarčič J, Wenger P (eds) Advances in Robot Kinematics: Analysis and design, Springer, Dordrecht, pp 311-320 Ren L, Mills J, Sun D (2004) Adaptive synchronization control of a planar parallel manipulator. In: Proc American Control Conference, Boston Ren L, Mills J, Sun D (2005) Controller design applied to planar parallel manipulators for trajectory tracking control. Proc IEEE Int Conf Robotics and Automation, Barcelona, pp 968-973 Ren L, Mills J, Sun D (2006) Convex synchronized control for a 3-DOF planar parallel manipulator. Proc IEEE Int Conf Robotics and Automation, Orlando, pp 968-973

References

679

Ridgeway S, Crane CD, Adist P, Harrell R (1992) The mechanical design of a parallel actuated joint for an articulated mobile robot. In: Proc ASME Design Engineering Technical Conferences, DE-Vol 45, pp 591-597 Rizk R, Fauroux JC, Munteanu M, Gogu G (2006) A comparative stiffness analysis of a reconfigurable parallel machine with three or four degrees of mobility. Machine Engineering 6(2):45-55 Ronchi S, Company O, Pierrot F, Fournier A (2004) PRP planar parallel mechanism in configurations improving displacement resolution. In: Proc 1st Int Conf Positioning Technology, Shizuoka Rolland LH (2006) Synthesis on forward kinematics problem algebraic modeling for the planar parallel manipulator: Displacement-based equation systems. Advanced Robotics 20(9):1035-1065 Rooney J, Earle CF (1983) Manipulator postures and kinematics assembly configurations. In: Proc of the 6th World Congress on the Theory of Machines and Mechanisms, pp 1014-1020 Rooney J, Tanev TK (2002) Pose, postures, formation and contortion in kinematic systems. In: Bianchi G, Guinot JC, Rzymkowski C (eds) Proc 13th CISMIFToMM Symp. Theory and Practice of Robots and Manipulators, SpringerVerlag, pp. 77-86 Sakai J, Fukasawa Y, Okamura S (1999) Movable table unit. US patent 6196138 Salsbury JK, Craig JJ (1982) Articulated hands: force and kinematic issues. Int J Robot Res 1(1):1-17 Scheidegger A, Liechti R (2002) Positionning device. US patent 6622586 Sefrioui J, Gosselin CM (1992) Singularity analysis and representation of planar parallel manipulators. Robotics Autonomous Systems 10(4): 209-224 Sefrioui J, Gosselin CM (1995) On the quadratic nature of the singularity curves of planar three-degree-of-freedom parallel manipulators. Mech Mach Theory 30(4):533-531 Shao H, Wang L, Guan L, Wu J (2009) Dynamic manipulability and optimization of a redundant three DOF planar parallel manipulator. In: Dai JS, Zoppi M, Kong X (eds) Reconfigurable Mechanisms and Robots, KC Edizioni, Genova, pp 308-314 Shirinzadeh B, Alici G (2004). Optimum dynamic balancing of planar parallel manipulators. In: Proc IEEE Int Conf Robotics and Automation, New Orleans Shvalb N, Shoham M, Liu G, Trinkle JC (2007) Motion planning for a class of planar closed-chain manipulators. Int J Robotics Research 26(5):457-473 Simaan N, Shoham M (2002) Stiffness synthesis of a variable geometry planar robot. In: Lenarčič J, Thomas F (eds) Advances in robot kinematics, Kluwer Academic Publishers, Dordrecht, pp 463-472 Slutski L (1996) Closed plane mechanisms as a basis of parallel manipulators. In: Lenarčič J, Parenti-Castelli V (eds) Recent advances in robot kinematics, Kluwer Academic Publishers, Dordrecht, pp 441-450 Snyman JA, Hay AM (2000) The chord method for the determination of nonconvex workspaces of planar parallel manipulators. In: Lenarčič J, Stanišić MM (eds) Advances in robot kinematics, Kluwer Academic Publishers, Dordrecht, pp 285-294

680

References

Snyman JA, Smit WJ (2002) The optimal design of a planar parallel platform for prescribed machining tasks. Multibody System Dynamics 8(2):103-115 Stachera K. (2005) An approach to direct kinematics of a planar parallel elastic manipulator and analysis for the proper definition of its workspace. In: Proc 11th IEEE Conference on Methods and Models in Automation and Robotics, Miedzyzdroje Staicu S (2008) Power requirement comparison in the 3-RPR planar parallel robot dynamics. Mech Mach Theory 44(5):1045-1057 Staicu S (2009) Inverse dynamics of the 3-PRR planar parallel robot. Robotics and Autonomous Systems 57(5):556-563 Staicu S, Carp-Ciocardia D, Codoban A (2007) Kinematics modelling of a planar parallel robot with prismatic actuators. In: University "Politehnica" of Bucharest Scientific Bulletin, Series D: Mechanical Engineering 69(3):3-14 Stan SD, Gogu G, Manic M, Balan R Rad C (2008) Fuzzy control of a 3 degree of freedom parallel robot. In: Proc International Joint Conferences on Computer, Information, and Systems Sciences, and Engineering, Bridgeport Sun D, Lu R, Mills JK, Wang C (2006) Synchronous tracking control of parallel manipulators using cross-coupling approach. Int J Robotics Research 25(11):1137-1147 Takeda Y (2005) Kinematic analysis of parallel mechanisms at singular points at which a connecting chain has local mobility. In: Proc International Workshop on Computational Kinematics Cassino, Paper 21-CK2005 Theingi I, Cheng IM, Li C, Angeles J (2004) Managing singularities of 3-DOF planar parallel manipulators using joint-coupling. In: Proc of the 11th World Congress in Mechanism and Machine Science, vol 4, China Machine Press, Beijing, pp 1966-1970 Tsai L-W (1999) Robot analysis: the mechanics of serial and parallel manipulators. Willey, New York Tsai KY, Huang KD (2003) The design of isotropic 6-DOF parallel manipulators using isotropy generators. Mech Mach Theory 38:1199-1214 Urizar M, Petuya V, Altuzarra O, Hernandez A (2009) Computing the configuration space for motion planning between assembly modes. In: Kecskeméthy A, Müller A (eds) Computational Kinematics, Proceedings of the 5th International Workshop on Computational Kinematics, Springer-Verlag, Berlin, pp 35-42 Wang L, Wang J, Li Y, Lu Y (2003) Kinematic and dynamic equations of a planar parallel manipulator. IMech J Mech Eng Science, Part C 217(5):525-531 Wang X, Mills J (2005a) FEM dynamic model for active vibration control of flexible linkages and its application to a planar parallel manipulator. Applied Acoustics 66(10):1151-1161 Wang X, Mills J (2005b) Active control of configuration-dependent linkage vibration with application to a planar parallel platform. In: Proc IEEE Int Conf Robotics and Automation, Barcelona, pp 4327-4332 Wang X, Mills J (2006) Dynamic modeling of a flexible-link planar parallel platform using a substructuring approach. Mech Mach Theory 41(6):671-687

References

681

Weiwei S, Shuang C (2006) Study on force transmission performance for planar parallel manipulator. In: Proc World Congress on Intelligent Control and Automation, Dalian Wenger P, Chablat D (2004) The kinematic analysis of a symmetrical threedegree-of-freedom planar parallel manipulator. In: Proc CISM-IFToMM Symp Robot Design, Dynamics and Control, Montreal Wenger P, Chablat D (2009) Kinematic analysis of a class of analytic planar 3RPR parallel manipulators. In: Kecskeméthy A, Müller A (eds) Computational Kinematics, Proceedings of the 5th International Workshop on Computational Kinematics, Springer-Verlag, Berlin, pp. 43-50 Wenger P, Chablat D, Zein M (2007) Degeneracy study of the forward kinematics of planar 3-RPR parallel manipulators. Trans ASME J Mech Design 129: 1265-1268 Williams RL, Joshi AR (1999) Planar parallel 3-RPR manipulator. In: Proc 6th Conference on Applied Mechanisms and Robotics, Cincinnati, pp 1-8 Williams RL, Shelley BH (1997) Inverse kinematics for planar parallel manipulators. Proc ASME Design Tech Conf , Sacramento Wohlhart K (1996) Kinematotropic linkages. In: Lenarčič J, Parenti-Castelli, V. (eds) Advances in Robot Kinematics, Kluwer Academic Publishers, pp.359– 368. Wu J, Wang J, Wang L (2008) Optimal kinematic design and application of a redundantly actuated 3DOF planar parallel manipulator. Trans ASME J Mech Design 130: paper 054503 Wu J, Wang J, Wang L, Li T (2009) Dynamics and control of a planar 3-DOF parallel manipulator with actuation redundancy. Mech Mach Theory 44(4):835-849 Wu Z, Rizk R, Fauroux JC, Gogu G (2007) Numerical simulation of parallel robots with decoupled motions and complex structure in a modular design approach. In: Tichkiewitch S, Tollenaere M, Ray P (eds) Advances in Integrated Design and Manufacturing in Mechanical Engineering II, Springer, Dordrecht, pp 129-144 Yang G, Chen W, Chen I (2002) A geometrical method for the singularity analysis of 3-RRR planar parallel robots with different actuation schemes. In: Proc IEEE Int Conf Inteligent Robots Systems, Lausanne, pp 2055-2060 Yang Y, O'Brien J (2007) A case study of planar 3-RPR parallel robot singularity free workspace design. In: Proc Int Conf Mechatronics and Automation, Harbin, pp 1834-1838 Yi BJ, Na HY, Lee JH, Hong YS, Oh SR, Suh IH, Kim WK (2002) Design of a parallel-type gripper mechanism; Int J Robotics Research 21(7):661-676 Yoon J, Ryu J (2004) The development of the 3-DOF planar parallel robot (RRR Type) for omni-directional locomotion interface. In: Proc 3rd IFAC Symp. Mechatronic Systems, Sydney Yu A, Bonev I, Zsombor-Murray P (2008) Geometric approach to the accuracy analysis of a class of 3-DOF planar parallel robots. Mech Mach Theory 43(3):364-375

682

References

Zanganeh KE, Angeles J (1997) Kinematic isotropy and the optimum design of parallel manipulators. Int J Robot Res 16(2):185-197 Zein M, Wenger P, Chablat D (2006a) An algorithm for computing cusp points in the joint space of 3-RPR parallel manipulators. In: Husty M, Schröcker HP(eds) Proc First European Conference on Mechanism Science, Obergurgl Zein M, Wenger P, Chablat D (2006b) Singular curves and cusp points in the joint space of 3-RPR parallel manipulators. In: Proc IEEE Int Conf Robotics and Automation, Orlando, pp 777-782 Zein M, Wenger P, Chablat D (2007a) Singular curves in the joint space and cusp points of 3-RPR parallel manipulators. Robotica 25(06):717-724. Zein M, Wenger P, Chablat D (2007b) Singularity surfaces and maximal singularityfree boxes in the joint space of planar 3-RPR parallel manipulators. In: Proc of the 12th World Congress in Mechanism and Machine Science, Besançon Zein M, Wenger P, Chablat D (2008) Non-singular assembly-mode changing motions for 3-RPR parallel manipulators. Mech Mach Theory 43(4):480-490 Zhang Y, Liu H, Wu X (2009) Kinematics analysis of a novel parallel manipulator. Mech Mach Theory 44(9):1648-1657 Zibil A, Firmani F, Nokleby SB, Podhorodeski RP (2007) An explicit method for determining the force-moment capabilities of redundantly-actuated planar parallel manipulators. Trans ASME J Mech Design 129(10):1046–1055 Zsombor-Murray PJ, Chen C, Hayes MJD (2002) Direct kinematic mapping for general planar parallel manipulators. In Proc CSME Forum, Kingston

Index

A actuator, 2 algorithm evolutionary, 16 approach systematic, 10 B base, 2 fixed, 2, 3 basic solution, 107, 283 basis, 13 of operational velocity space, 15 of vector space, 15 branch mobility, 12

dimension vector space, 14 E element 3 pairing, 3 reference, 3 end-effector, 9 equation constraint, 11 evolutionary morphology, 16 F frame, 3 fully-isotropic, 16 G

C characteristic point, 15 condition number, 17 connectivity, 12, 13, 78 connecting conditions, 28 constraint equation, 11 coupled motions, 16, 27 CPM, 18 D decoupled motions, 16 degree of freedom, 10 of overconstraint, 289 design objectives, 16

graph, 7 structural, 7 H hexapod, 2 I idle mobility, 5, 88, 183, 228, 251, 271, 289, 300, IFMA, 17 independent motion, 13 Isoglide3-T2R1

planar, 283 spatial, 529 Isoglide3-T3, 18 isotropy, 17

684

Index

J Jacobian matrix, 16 joint, 3 Cardan, 4 heterokinetic, 4 homokinetic, 4 universal, 4 K kinematic pair, 2, 4 kinematic chain, 2, 3 closed, 4 complex, 4 open, 4 serial, 2 simple, 4 L limb, 2 complex, 4 simple, 4 topology, 28, 89, 184, 228, 240, 272, 283, 289, 300, 329, link, 3 binary, 4 distal, 4 monary, 3 polinary, 4 loop parallelogram, 5 rhombus, 472 M mechanism, 2, 3 element, 3 parallel, 2 kinematotropic, 12 mobility, 10, 13 external, 11 full-cycle, 11 general, 11 idle, 5

instantaneous, 12 internal, 11 88, 184, 228, 251, model kinematic, 19 direct, 19 morphological operator, 16 motion coupling, 16 N number of overconstraints, 12, 13, 251 O operational vector space, 89 velocity space, 13 Orthogonal Tripteron, 18 overconstraint, 12 P paire, 2 cylindrical, 4 helical, 4 kinematic, 4 lower , 4 passive, 9 planar, 4 revolute, 4 spherical, 4 pairing element, 3 Pantopteron, 18 parallel mechanism, 5, 12 parallel robotic manipulator, 1, 16 fully-isotropic, 17 maximally regular, 16 non overconstrained, 15 non redundant, 14 overconstrained, 15 redundant, 14 redundantly-actuated, 2 translational, 1 R2-type, 18

Index R3-type, 18 T1R2-type, 18 T1R3-type, 18 T2R1-type, 18, 19 T2R2-type, 18 T2R3-type, 18 T3-type, 17, 18 T3R1-type, 18 T3R2-type, 18 T3R3-type, 18 with coupled motions, 17 with decoupled motions, 17 with planar motion of the moving platform, 19 with uncoupled motions, 16 3-PRR-type, 21 3-RPR-type, 22 3-RRR-type, 23 performance index, 17 platform, 2 fixed, 2 moving, 2, 4 reference, 4 PPM, 27 basic solutions, 27, 239, 283 coupled motions, 27, 183 decoupled motions, 240 derived solutions , 88, 251, 289 fully-parallel, 27, 183, 240 maximally regular, 283, 300 non fully-parallel, 27, 78, 228 non overconstrained, 183, 271, 300, overconstrained, 27, 239, 283 uncoupled motions, 239 point characteristic, 15 protoelement, 16 R rank, 11 redundancy, 13 robot, 1, fully parallel, 9 hexapod, 2

685

hybrid, 9 non fully-parallel, 10 parallel, 2, 9 serial, 9 robotics, 2 S singular configuration,12 SPM, 307 basic solutions, 307, 367, 529 coupled motions, 307 derived solutions, 327, 577 fully-parallel, 307, 367 maximally regular, 529, 620 non overconstrained, 346, 472, 620 overconstrained, 307, 367, 529 uncoupled motions, 367, 472 structural diagram, 7 graph, 7 parameters, 10, redundancy, 13 228 synthesis, 10 synthesis structural, 10 systematic approach, 10 T theory of linear transformations, 12 topology, 10 U uncoupled motions, 16 universal joint, 4 V velocity, 20 velocity vector space, 12