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Constraint Equations of Inverted Kinematic Chains

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Computational Kinematics

Part of the book series: Mechanisms and Machine Science ((Mechan. Machine Science,volume 50))

Abstract

A lot of different kinematic chains have been investigated focusing on constraint equations, singularities, assembly modes and motion capabilities. The approach to obtain constraint equations via inverted chains however is rarely considered. We provide a detailed look on the constraint varieties of inverted chains, beginning with the basics of quaternion conjugation. The transformation of the Denavit- Hartenberg parameters needed for the quaternion conjugation is discussed in the paper. The quaternion conjugation is a fast way to obtain the variety corresponding to the inverted kinematic chain. Geometrically the conjugation is a reflection in the kinematic image space \(\mathbb {P}^7\) with respect to a line and a five- dimensional subspace. Some examples of constraint equations of kinematic chains and their inverted chains complete the paper.

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Notes

  1. 1.

    Due to space limitation this mapping cannot be explained in detail, but a comprehensive introduction can be found in [4] or [3].

  2. 2.

    The kinematic images of planar and spherical displacements subordinate completely to this description because both cases are obtained by three dimensional sub-spaces of \(\mathbb {P}^7\). The corresponding geometry of their image spaces and the algorithms to derive these geometries can be found in [1] p. 393ff. resp. [5] p. 60ff.

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Acknowledgements

The authors acknowledge the support of the FWF project I 1750-N26 “Kinematic Analysis of Lower-Mobility Parallel Manipulators Using Efficient Algebraic Tools”.

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Authors and Affiliations

  1. Unit Geometry and CAD, University of Innsbruck, Innsbruck, Austria

    Thomas Stigger & Manfred L. Husty

Authors
  1. Thomas Stigger
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  2. Manfred L. Husty
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Corresponding author

Correspondence to Manfred L. Husty .

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Editors and Affiliations

  1. Institut PPRIME, UPR 3346, University of Poitiers, Poitiers, France

    Saïd Zeghloul

  2. American University of Sharjah, UAE, on leave from the National School of Engineering of Sousse , Sousse Erriadh, Tunisia

    Lotfi Romdhane

  3. SP2MI, Téléport 2, University of Poitiers, Poitiers, France

    Med Amine Laribi

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Stigger, T., Husty, M.L. (2018). Constraint Equations of Inverted Kinematic Chains. In: Zeghloul, S., Romdhane, L., Laribi, M. (eds) Computational Kinematics. Mechanisms and Machine Science, vol 50. Springer, Cham. https://doi.org/10.1007/978-3-319-60867-9_56

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  • DOI: https://doi.org/10.1007/978-3-319-60867-9_56

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-60866-2

  • Online ISBN: 978-3-319-60867-9

  • eBook Packages: EngineeringEngineering (R0)

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