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Computational Kinematics pp 323–330Cite as

Single Exponential Motion and Its Kinematic Generators

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Part of the book series: Mechanisms and Machine Science ((Mechan. Machine Science,volume 15))

Abstract

Both constant velocity (CV) joints and zero-torsion parallel kinematic machines (PKMs) possess special geometries in their subchains. They are studied as two different subjects in the past literature. In this paper we provide an alternative analysis method based on the symmetric product on \(SE(3)\) (the Special Euclidean group). Under this theoretical framework CV joints and zero-torsion mechanisms are unified into single exponential motion generators (SEMG). The properties of single exponential motion are studied and sufficient conditions are derived for the arrangement of joint screws of a serial chain so that the motion pattern of the resulting mechanism is indeed a single exponential motion generator.

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Notes

  1. 1.

    Sometimes motion pattern is also called motion type.

  2. 2.

    The Listing’s law about human eye movement, also called the half-angle law, states that the instantaneous velocity plane tilts exactly one half of that of the line of sight.

  3. 3.

    According to the Baker-Cambell-Hausdoff formula, we have

    $$\begin{aligned} e^{\hat{\omega }_1\theta _1}e^{\hat{\omega }_2\theta _2} = e^{\hat{\omega }_1\theta _1+\hat{\omega }_2\theta _2+\frac{1}{2}\left[ \hat{\omega }_1,\hat{\omega }_2 \right] \theta _1\theta _2 + O(\theta _1^2,\theta _2^2)}, \end{aligned}$$

    which is not a single exponential of a twist in the plane \(\{\hat{\omega }_1,\hat{\omega }_2\}\), but a twist in the three-dimensional Lie algebra \(\{\hat{\omega }_1,\hat{\omega }_2, \left[ \hat{\omega }_1,\hat{\omega }_2 \right] \}\), which is the Lie algebra \(so(3)\) of the rotation group \(SO(3)\).

  4. 4.

    \([,[,]]\) could be replaced by \([[,],]\) based on the Jacobian identity on any Lie algebra.

  5. 5.

    Its proof can be found in [5] (Theorem 7.2, Chap. 4)

References

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Acknowledgments

This research is supported by Talents Introduction Startup Funds of High Education of Guangdong Province (2050205) and supported by \(1000\) Young Investigator Plan of the Chinese Government.

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Authors and Affiliations

  1. Guangdong University of Technology, Guangdong, China

    Guanfeng Liu, Yuanqin Wu & Xin Chen

  2. Hong Kong University of Science and Technology, Hong Kong, China

    Yuanqin Wu

Authors
  1. Guanfeng Liu
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  2. Yuanqin Wu
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  3. Xin Chen
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Corresponding author

Correspondence to Guanfeng Liu .

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Editors and Affiliations

  1. Llorens Artigas, Inst Robótica e Informática Industrial, Barcelona, Spain

    Federico Thomas

  2. Mechanical Engineering, Idaho State University, Pocatello, Idaho, USA

    Alba Perez Gracia

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Liu, G., Wu, Y., Chen, X. (2014). Single Exponential Motion and Its Kinematic Generators. In: Thomas, F., Perez Gracia, A. (eds) Computational Kinematics. Mechanisms and Machine Science, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7214-4_36

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  • DOI: https://doi.org/10.1007/978-94-007-7214-4_36

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  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-7213-7

  • Online ISBN: 978-94-007-7214-4

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