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Journal of Dynamical and Control Systems

, Volume 12, Issue 3, pp 371–404 | Cite as

On the Motion Planning Problem, Complexity, Entropy, and Nonholonomic Interpolation

  • Jean-Paul GauthierEmail author
  • Vladimir ZakalyukinEmail author
Article

Abstract.

We consider the sub-Riemannian motion planning problem defined by a sub-Riemannian metric (the robot and the cost to minimize) and a non-admissible curve to be ε-approximated in the sub-Riemannian sense by a trajectory of the robot. Several notions characterize the ε-optimality of the approximation: the “metric complexity” MC and the “entropy” E (Kolmogorov-Jean). In this paper, we extend our previous results. 1. For generic one-step bracketgenerating problems, when the corank is at most 3, the entropy is related to the complexity by E = 2πMC. 2. We compute the entropy in the special 2-step bracket-generating case, modelling the car plus a single trailer. The ε-minimizing trajectories (solutions of the “ε-nonholonomic interpolation problem”), in certain normal coordinates, are given by Euler's periodic inflexional elastica. 3. Finally, we show that the formula for entropy which is valid up to corank 3 changes in a wild case of corank 6: it has to be multiplied by a factor which is at most 3/2.

Key words and phrases.

Robotics Subriemannian geometry 

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.LE2I, UMR CNRS 5158Universit'e de BourgogneDijon CEDEXFrance
  2. 2.Moscow State UniversityMoscowRussia

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