# Geometric Design of Linkages

Springer Science & Business Media, Nov 11, 2010 - Science - 448 pages

This book is an introduction to the mathematical theory of design for articulated mechanical systems known as linkages. The focus is on sizing mechanical constraints that guide the movement of a work piece, or end-effector, of the system. The function of the device is prescribed as a set of positions to be reachable by the end-effector; and the mechanical constraints are formed by joints that limit relative movement. The goal is to find all the devices that can achieve a specific task. Formulated in this way the design problem is purely geometric in character. Robot manipulators, walking machines, and mechanical hands are examples of articulated mechanical systems that rely on simple mechanical constraints to provide a complex workspace for the end- effector. The principles presented in this book form the foundation for a design theory for these devices. The emphasis, however, is on articulated systems with fewer degrees of freedom than that of the typical robotic system, and therefore, less complexity. This book will be useful to mathematics, engineering and computer science departments teaching courses on mathematical modeling of robotics and other articulated mechanical systems.

This new edition includes research results of the past decade on the synthesis of multi loop planar and spherical linkages, and the use of homotopy methods and Clifford algebras in the synthesis of spatial serial chains. One new chapter on the synthesis of spatial serial chains introduces numerical homotopy and the linear product decomposition of polynomial systems. The second new chapter introduces the Clifford algebra formulation of the kinematics equations of serial chain robots. Examples are use throughout to demonstrate the theory.

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### Contents

 Chapter 1 Introduction 1 Chapter 2 Analysis of Planar Linkages 15 Chapter 3 Graphical Synthesis in the Plane 55 Chapter 4 Planar Kinematics 75 Chapter 5 Algebraic Synthesis of Planar Chains 93 Chapter 6 Multiloop Planar Linkages 125 Chapter 7 Analysis of Spherical Linkages 155 Chapter 8 Spherical Kinematics 179
 Chapter 13 Algebraic Synthesis of Spatial Chains 307 Chapter 14 Synthesis of Spatial Chains with Reachable Surfaces 335 Chapter 15 Clifford Algebra Synthesis of Serial Chains 357 Chapter 16 Platform Manipulators 393 Appendix A Solving Constraint Equations 411 Appendix B Graphical Constructions 415 Appendix C Spherical Trigonometry 419 Appendix D Operations with Dual Numbers 425

 Chapter 9 Algebraic Synthesis of Spherical Chains 203 Chapter 10 Multiloop Spherical Linkages 231 Chapter 11 Analysis of Spatial Chains 253 Chapter 12 Spatial Kinematics 281
 Appendix E Kinematics Equations 427 References 433 Index 441 Copyright

### About the author (2010)

J. Michael McCarthy is a Professor in the Department of Mechanical Engineering at University of California, Irvine.