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On the Line-Symmetry of Self-motions of Linear Pentapods

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Advances in Robot Kinematics 2016

Part of the book series: Springer Proceedings in Advanced Robotics ((SPAR,volume 4))

Abstract

We show that all self-motions of pentapods with linear platform of Type 1 and Type 2 can be generated by line-symmetric motions . Thus this paper closes a gap between the more than 100 year old works of Duporcq and Borel and the extensive study of line-symmetric motions done by Krames in the 1930s. As a consequence we also get a new solution set for the Borel Bricard problem. Moreover we discuss the reality of self-motions and give a sufficient condition for the design of linear pentapods of Type 1 and Type 2, which have a self-motion free workspace.

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Notes

  1. 1.

    These are the only non-trivial motions where every point of the moving space has a spherical trajectory (cf. [3, Chap. VI]).

  2. 2.

    The corresponding sphere centers of lines belonging to the other regulus are again located on lines (cf. [9, p. 24]), which imply linear pentapods with an architecturally singular design.

  3. 3.

    With respect to the notation introduced in Sect. 2 these five parameters are \(C,a_r,a_c,a_4\) and \(p_5\) or \(R_1\) (cf. Eq. (7)) by canceling the factor of similarity by setting \(A=1\).

  4. 4.

    These are the parameters a, c, g, k used in [9, Sect. 2.3].

  5. 5.

    This angle condition can be seen as the limit of the sphere condition (cf. [12, Sect. 4.1]).

  6. 6.

    A self-motion is dangerous as it is uncontrollable and thus a hazard to man and machine.

  7. 7.

    Note that all basic surfaces and trajectories can be parametrized due to Remark 2.

References

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  2. Borel, E.: Mémoire sur les déplacements à trajectoires sphériques. Mémoire présenteés par divers savants à l’Académie des Sciences de l’Institut National de France 33(1), 1–128 (1908)

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  3. Bricard, R.: Mémoire sur les déplacements à trajectoires sphériques. Journal de École Polytechnique(2) 11, 1–96 (1906)

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Acknowledgements

This research is funded by Grant No. P 24927-N25 of the Austrian Science Fund FWF within the project “Stewart Gough platforms with self-motions”.

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Authors and Affiliations

  1. Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Vienna, Austria

    Georg Nawratil

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  1. Georg Nawratil
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Correspondence to Georg Nawratil .

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

    Jadran Lenarčič

  2. INRIA Sophia Antipolis , Sophia Antipolis Cedex, France

    Jean-Pierre Merlet

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Nawratil, G. (2018). On the Line-Symmetry of Self-motions of Linear Pentapods. In: Lenarčič, J., Merlet, JP. (eds) Advances in Robot Kinematics 2016. Springer Proceedings in Advanced Robotics, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-56802-7_16

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  • DOI: https://doi.org/10.1007/978-3-319-56802-7_16

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