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Differential Kinematics

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Humanoid Robotics: A Reference

Abstract

Kinematics play an essential role in motion analysis, motion generation, and control. This chapter focuses on the differential kinematic relations pertinent to open-loop and closed-loop kinematic chains. It is assumed that readers are familiar with the basic concepts concerning systems of rigid bodies, such as representations of positions and orientations, the kinematics of joints, coordinate frame assignment techniques for serial-link and parallel-link mechanisms, coordinate transformations, and the forward and inverse kinematic problems. First, the role of differential kinematics for instantaneous motion analysis is highlighted. The forward and inverse differential kinematic problems for open-loop systems are then formulated using first-order (velocity level) and second-order (acceleration level) relationships. Further on, instantaneous motion analysis and methods for motion generation at special (singular) configurations are discussed. Generic singular configurations for humanoid robots with and without kinematically redundant limbs are highlighted, and the notion of manipulability ellipsoid is introduced. Special attention is paid to solutions to the inverse instantaneous motion problem for kinematic chains with kinematic redundancy. Two basic cases are discussed for redundancy pertinent to a single limb and the whole-body chain. In the latter case, solution methods for multiple-task constraints are highlighted, covering both fixed and variable-priority tasks within a hierarchical structure. Finally, differential kinematic relations within chains with motion constraints stemming from contacts with the environment are derived.

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Acknowledgements

The figures were created with the help of Jie Chen and Keita Sagara.

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Authors and Affiliations

  1. Department of Mechanical Systems Engineering, Tokyo City University, Tokyo, Japan

    Dragomir Nenchev

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  1. Dragomir Nenchev
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Correspondence to Dragomir Nenchev .

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  1. Intuitive Surgical, Sunnyvale, CA, USA

    Ambarish Goswami

  2. Department of Electrical and Computer Engineering, National University of Singapore, Singapore, Singapore

    Prahlad Vadakkepat

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Nenchev, D. (2019). Differential Kinematics. In: Goswami, A., Vadakkepat, P. (eds) Humanoid Robotics: A Reference. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6046-2_2

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