## 21st Century Kinematics: The 2012 NSF WorkshopJ. Michael McCarthy 21st Century Kinematics focuses on algebraic problems in the analysis and synthesis of mechanisms and robots, compliant mechanisms, cable-driven systems and protein kinematics. The specialist contributors provide the background for a series of presentations at the 2012 NSF Workshop. The text shows how the analysis and design of innovative mechanical systems yield increasingly complex systems of polynomials, characteristic of those systems. In doing so, it takes advantage of increasingly sophisticated computational tools developed for numerical algebraic geometry and demonstrates the now routine derivation of polynomial systems dwarfing the landmark problems of even the recent past. The 21st Century Kinematics workshop echoes the NSF-supported 1963 Yale Mechanisms Teachers Conference that taught a generation of university educators the fundamental principles of kinematic theory. As such these proceedings will provide admirable supporting theory for a graduate course in modern kinematics and should be of considerable interest to researchers in mechanical design, robotics or protein kinematics or who have a broader interest in algebraic geometry and its applications. |

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### Contents

1 | |

Kinematic Synthesis | 13 |

Synthesis of Spatial Mechanisms to Model Human Joints | 49 |

Kinematics and Algebraic Geometry | 85 |

Applying Numerical Algebraic Geometry to Kinematics | 125 |

The Kinematics of 3D CableTowing Systems | 161 |

Compliant Mechanisms | 189 |

Protein Molecules Evolutions Design for Kinematic Machines | 217 |

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### Common terms and phrases

algebra element algebraic geometry algorithm analysis anatomical ankle ASME axis Biomech cables Clifford algebra complex compliant mechanisms components compute constraint coordinates corresponding deﬁne defined degrees of freedom design equations dimension dimensional direct kinematics displacement dual quaternion elements energy equilibrium configurations equivalent mechanism example experimental femur fibula formulation four-bar frame given Hessian matrix homotopy hydrogen bonds ideal inverse kinematics irreducible irreducible component Jacobian Jacobian matrix joint angles kinematic mapping kinematics equations knee LEMs ligaments linear linkage loops matrix McCarthy Mech monomial order monomials numerical algebraic geometry obtained optimization parallel manipulators Parenti-Castelli passive motion patella payload planar plane platform points polynomial systems protein molecules pseudo-rigid-body model quadric result revolute joint rotation Sect segments serial chain shown in Fig singular solving Sommese space spatial spherical stiffness structures Study quadric surfaces synthesis equations task positions tensegrity tibia tion transform variables variety vector Wampler zero