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Optimal Trajectories of Curvature Constrained Motion in the Hamilton–Jacobi Formulation

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Abstract

We propose a PDE approach for computing time-optimal trajectories of a vehicle which travels under certain curvature constraints. We derive a class of Hamilton–Jacobi equations which models such motions; it unifies two well-known vehicular models, the Dubins’ and Reeds–Shepp’s cars, and gives further generalizations. Numerical methods (finite difference for the Reeds–Shepp’s car and semi-Lagrangian for the Dubins’ car) are investigated for two-dimensional domains and surfaces.

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Notes

  1. We note that a similar problem laying railroad tracks that connect already existing tracks was considered by Markov [35].

  2. Later, we demonstrate that, in practice, the Lipschitz continuity of \(f\) in with respect to space can be dropped.

  3. We note that are several notions for discontinuous viscosity solutions [13, 24, 26, 51].

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Acknowledgments

The authors thank Professors S. Osher, P. Souganidis and A. Vladimirsky for helpful discussions, and Dr. Y. Landa and Mr. H. Shen for taking part in the early stages of this project. Takei was supported by ONR Grants N00014-07-1-0810 and N00014-08-1-1119, and the CHASE MURI Grant 556016. Tsai was supported by a Sloan Fellowship, National Science Foundation Grants DMS-0914465, DMS-0914840, and a MURI sub-contract from Univ. of S. Carolina Grant No. W911NF-07-1-0185. Takei thanks Prof. Claire Tomlin and the Hybrid Systems Lab at UC Berkeley for their hospitality during the completion of this work.

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  1. Department of Electrical Engineering and Computer Sciences, UC Berkeley, Berkeley, CA, USA

    Ryo Takei

  2. Department of Mathematics, University of Texas, Austin, TX, USA

    Richard Tsai

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Takei, R., Tsai, R. Optimal Trajectories of Curvature Constrained Motion in the Hamilton–Jacobi Formulation. J Sci Comput 54, 622–644 (2013). https://doi.org/10.1007/s10915-012-9671-y

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