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21st Century Kinematics pp 85–123Cite as

Kinematics and Algebraic Geometry

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Abstract

This overview paper is a collection of several papers (Husty and Schröcker, Nonlinear Computational Geometry, The IMA Volumes in Mathematics and Its Applications, vol. 151, pp. 85–107, Springer, Berlin, 2010; Husty et al., Mech. Mach. Theory 42:66–81, 2007; Robotica 25:661–675, 2007; Schröcker et al., J. Mech. Des. 129:924–929, 2007; Walter et al., Contemporary Mathematics, vol. 496, pp. 331–346, American Mathematical Society, Providence, 2009; Walter and Husty, Mach. Des. Res. 26:218–226, 2010) that were published by the authors and their collaborators within the last ten years. As basic paradigm we show how problems in computational kinematics can be translated into the language of algebraic geometry and subsequently solved using techniques developed in this field mostly with help of an algebraic manipulation system.

The idea to transform kinematic features into the language of algebraic geometry is old and goes back to E. Study. Recent advances in algebraic geometry and symbolic computation gave the motivation to resume these ideas and make them successful in the solution of kinematic problems. It is not the aim of the paper to provide detailed solutions, but basic accounts to the used tools and examples where these techniques were applied within the last years. We start with Study’s kinematic mapping and show how kinematic entities can be transformed into algebraic varieties. The transformations in the image space that preserve the kinematic features are introduced. The main topics are the definition of constraint varieties and their application to the solution of direct and inverse kinematics of serial and parallel robots, to the analysis of workspaces and especially to the decomposition of these kinematic entities. We provide a new definition of the degree of freedom of a mechanical system and discuss singularities and global pathological behavior of selected mechanisms.

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Notes

  1. 1.

    An English translation of Blaschke’s work is in preparation: W. Blaschke, Kinematics and Quaternions, translated by M. Husty and P. Zsombor-Murray, Springer 2013.

  2. 2.

    Note that homogeneous coordinates in this chapter are written in the European notation, with homogenizing coordinate on first place.

  3. 3.

    For more detailed explanations see [4] or [27].

  4. 4.

    Note that in the examples of this section often only the input of maple is displayed. To obtain the results one has to type these commands into Maple.

  5. 5.

    In Sect. 4.2 we have defined affine varieties. When all defining polynomials of the variety are homogeneous then the zero set of these polynomials is called projective variety. Because in mechanism analysis affine as well as projective varieties occur we use the generic term algebraic variety in this section.

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Acknowledgement

The authors thank M. Pfurner and D. Walter for their agreement to use common results for this paper.

References [12] and [20] were used in this chapter reprinted with kind permission © Springer 2012.

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Authors and Affiliations

  1. Institute of Basic Sciences in Engineering, Unit Geometry and CAD, University Innsbruck, Technikerstraße 13, 6020, Innsbruck, Austria

    Manfred L. Husty & Hans-Peter Schröcker

Authors
  1. Manfred L. Husty
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  2. Hans-Peter Schröcker
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Correspondence to Manfred L. Husty .

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Editors and Affiliations

  1. Dept. of Mech. & Aerospace Engineering, Robotics and Automation Laboratory, University of California, Engineering Gateway S4218, Irvine, 92697-3975, USA

    J. Michael McCarthy

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Husty, M.L., Schröcker, HP. (2013). Kinematics and Algebraic Geometry. In: McCarthy, J. (eds) 21st Century Kinematics. Springer, London. https://doi.org/10.1007/978-1-4471-4510-3_4

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  • DOI: https://doi.org/10.1007/978-1-4471-4510-3_4

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