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On certain hyperelliptic signals that are natural controls for nonholonomic motion planning

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Abstract

In this paper, we address the general problem of approximating, in a certain optimal way, non-admissible motions of a kinematic system with nonholonomic constraints. Since this kind of problems falls into the general subriemannian geometric setting, it is natural to consider optimality in the sense of approximating by means of subriemannian geodesics. We consider systems modeled by a subriemannian Goursat structure, a particular case being the well-known system of a car with trailers, along with the associated parallel parking problem. Several authors approximate the successive Lie brackets using trigonometric functions. By contrast, we show that more natural optimal motions are related with closed hyperelliptic plane curves with a certain number of loops.

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Acknowledgments

This paper was prepared during the sabbatical leave of the second author at the Laboratoire des Sciences de l’Information et des Systèmes (LSIS, UMR 7296) in the Université du Sud Toulon-Var, France. The author was financially supported by the CONACYT under the program of sabbatical leaves abroad for the reinforcement of the research groups, project number 204051.

Conflict of interest

Herewith I confirm, on behalf of all authors, that the information provided is accurate, and that we have no potential conflicts of interest.

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Authors and Affiliations

  1. LSIS, UMR CNRS 7296 and Equipe INRIA GECO, Université de Toulon, UTLN, 83957, La Garde Cedex, France

    Jean-Paul Gauthier

  2. Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana-Azcapotzalco, 02200, México D.F., México

    Felipe Monroy-Pérez

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  1. Jean-Paul Gauthier
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  2. Felipe Monroy-Pérez
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Correspondence to Felipe Monroy-Pérez.

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Gauthier, JP., Monroy-Pérez, F. On certain hyperelliptic signals that are natural controls for nonholonomic motion planning. Math. Control Signals Syst. 27, 415–437 (2015). https://doi.org/10.1007/s00498-015-0145-2

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  • DOI: https://doi.org/10.1007/s00498-015-0145-2

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