## Geometric Design of Linkages (Google eBook)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 |

433 | |

441 | |

### Common terms and phrases

6R loop angular velocity assembly CC chain center-point Clifford algebra common normal components compute configuration constraint equations constructed coordinate rotations coordinates coupler crank angle define denote design equations determine dual angle dual quaternion dyad triangle eight-bar linkage end-link fixed pivot G floating link formula geometric given input angle input crank interior angle intersect inverse kinematics Jacobian joint angles kinematics equations linear loop equations monomials moving axis moving body moving frame moving pivot obtain open chain output crank perpendicular bisector plane platform Plücker coordinates pole triangle polynomial prismatic joint relative displacement relative inverse relative rotation result revolute joint Rodrigues’s rotation angle rotation axis rotation matrix RR constraints screw axis serial chain six-bar linkage slide slider solution solve spatial displacement specified spherical linkage spherical RR chain Springer Science+Business Media synthesis equations theorem transformation unit vector x-axis yields z-axis zero