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Line Geometry in Terms of the Null Geometric Algebra over ℝ3,3, and Application to the Inverse Singularity Analysis of Generalized Stewart Platforms

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Guide to Geometric Algebra in Practice

Abstract

In this chapter, the classical line geometry is modeled in ℝ3,3, where lines are represented by null vectors, and points and planes by null 3-blades. The group of 3D special projective transformations SL(4) when acting upon points in space induces a Lie group isomorphism, with SO(3,3) acting upon lines.

As an application of the use of the ℝ3,3 model of line geometry, this chapter analyzes the inverse singularity analysis of generalized Stewart platforms, using vectors of ℝ3,3 to encode the force and torque wrenches to classify their singular configurations.

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Notes

  1. 1.

    Editorial note: See also Chap. 14.

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Author information

Authors and Affiliations

  1. Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, P.R. China

    Hongbo Li & Lixian Zhang

Authors
  1. Hongbo Li
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  2. Lixian Zhang
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Corresponding author

Correspondence to Hongbo Li .

Editor information

Editors and Affiliations

  1. Informatics Institute, University of Amsterdam, Science Park 107, Amsterdam, 1098 XG, Netherlands

    Leo Dorst

  2. Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, United Kingdom

    Joan Lasenby

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Li, H., Zhang, L. (2011). Line Geometry in Terms of the Null Geometric Algebra over ℝ3,3, and Application to the Inverse Singularity Analysis of Generalized Stewart Platforms. In: Dorst, L., Lasenby, J. (eds) Guide to Geometric Algebra in Practice. Springer, London. https://doi.org/10.1007/978-0-85729-811-9_13

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  • DOI: https://doi.org/10.1007/978-0-85729-811-9_13

  • Publisher Name: Springer, London

  • Print ISBN: 978-0-85729-810-2

  • Online ISBN: 978-0-85729-811-9

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