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Derivatives of Screw Systems in Body-Fixed Representation

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Advances in Robot Kinematics

Abstract

The configuration of a kinematic chain can be uniquely expressed in terms of the joint screws via the product of exponentials. Twists on the other hand can be represented in various forms. The particular representation is determined by the reference frame in which the velocity is measured and the reference frame in which this velocity is expressed. For kinematic analyses the spatial twists are commonly used. Analytical mechanism dynamics, on the other hand, uses body-fixed twists. The body-fixed twist of a moving body is the velocity of a body-attached frame relative to the spatial frame expressed in the body-attached moving frame. Accordingly the spatial and body-fixed twists are expressed in terms of spatial and body-fixed instantaneous joint screw coordinates, respectively. Crucial for analytical kinematics and dynamics are the derivatives of twists, and thus of the mechanism’s screw system. Whereas higher-order derivatives of screw systems in spatial representation have been a subject of intensive research, the body-fixed representation has not yet been addressed systematically. In this chapter a closed form expression for higher-order partial derivatives of the screw system of a kinematic chain w.r.t. the joint variables is reported. The final expression is a nested Lie bracket of the body-fixed instantaneous joint screws. It resembles the previously presented results for the spatial representation.

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Authors and Affiliations

  1. University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai, People’s Republic of China

    Andreas Müller

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  1. Andreas Müller
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Correspondence to Andreas Müller .

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Editors and Affiliations

  1. J. Stefan Institute, Ljubljana-Vic-Rudnik, Slovenia

    Jadran Lenarčič

  2. Department of Computer Science Artificial Intelligence Laboratory, Stanford University, Stanford, California, USA

    Oussama Khatib

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Müller, A. (2014). Derivatives of Screw Systems in Body-Fixed Representation. In: Lenarčič, J., Khatib, O. (eds) Advances in Robot Kinematics. Springer, Cham. https://doi.org/10.1007/978-3-319-06698-1_14

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  • DOI: https://doi.org/10.1007/978-3-319-06698-1_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-06697-4

  • Online ISBN: 978-3-319-06698-1

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