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


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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  1. 1.
    1. A. A. Agrachev, H. E. A. Chakir, and J. P. Gauthier, Sub-Riemannian metrics on ℝ3. In: Geometric Control and Nonholonomic Mechanics, Mexico City (1996), pp. 29–76. Proc. Can. Math. Soc. 25 (1998).zbMATHGoogle Scholar
  2. 2.
    2. A. A. Agrachev and J. P. Gauthier, Sub-Riemannian metrics and isoperimetric problems in the contact case. J. Math. Sci. 103 (2001), No. 6, 639–663.zbMATHCrossRefGoogle Scholar
  3. 3.
    3. G. Charlot, Quasicontact SR metrics: normal form in ℝ2n, wave front and caustic in ℝ4. Acta Appl. Math. 74 (2002), No. 3, 217–263.zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    4. H. E. A. Chakir, J. P. Gauthier, and I. A. K. Kupka, Small sub-Riemannian balls on ℝ3. J. Dynam. Control Systems 2 (1996), No. 3, 359–421.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    5. F. H. Clarke, Optimization and nonsmooth analysis. John Wiley & Sons (1983).zbMATHGoogle Scholar
  6. 6.
    6. J. P. Gauthier, F. Monroy-Perez, and C. Romero-Melendez, On complexity and motion planning for corank one SR metrics. ESAIM Control Optim. Calc. Var. 10 (2004), 634–655.zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    7. J. P. Gauthier and V. Zakalyukin, On the codimension one motion planning problem. J. Dynam. Control Systems 11 (2005), No. 1, 73–89.zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    8. ———, On the one-step bracket-generating motion planning problem. J. Dynam. Control Systems 11 (2005), No. 2, 215–235.zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    9. ———, Robot motion planning, a wild case. To appear in Proc 2004 Suszdal Conf. on Dynam. Systems (2006).Google Scholar
  10. 10.
    10. M. Gromov, Carnot-Caratheodory spaces seen from within. Prog. Math. 144 (1996), 79–323.zbMATHMathSciNetGoogle Scholar
  11. 11.
    11. F. Jean, Complexity of nonholonomic motion planning. Int. J. Control 74 (2001), No. 8, 776–782.zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    12. ———, Entropy and complexity of a path in SR geometry. ESAIM Control Optim. Calc. Var. 9 (2003), 485–506.zbMATHMathSciNetGoogle Scholar
  13. 13.
    13. F. Jean and E. Falbel, Measures and transverse paths in SR geometry. J. Anal. Math. 91 (2003), 231–246.zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    14. A. E. H. Love, A treatise on the mathematical theory of elasticity. Dover, New York (1944).zbMATHGoogle Scholar
  15. 15.
    15. Mathematica®, Reference manual. France (1997).Google Scholar
  16. 16.
    16. L. Pontryagin, V. Boltyanski, R. Gamkelidze, and E. Michenko, The mathematical theory of optimal processes. Wiley, New York (1962).zbMATHGoogle Scholar

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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