Robot Motion Planning: A~Wild Case

J.-P. Gauthier ; V. M. Zakalyukin

Informatics and Automation, Differential equations and dynamical systems, Tome 250 (2005), pp. 64-78.

Résumé

A basic problem in robotics is a constructive motion planning problem: given an arbitrary (nonadmissible) trajectory $\Gamma$ of a robot, find an admissible $\varepsilon$-approximation (in the sub-Riemannian (SR) sense) $\gamma(\varepsilon)$ of $\Gamma$ that has the minimal sub-Riemannian length. Then, the (asymptotic behavior of the) sub-Riemannian length $L(\gamma (\varepsilon))$ is called the metric complexity of $\Gamma$ (in the sense of Jean). We have solved this problem in the case of an SR metric of corank 3 at most. For coranks greater than 3, the problem becomes much more complicated. The first really critical case is the 4–10 case (a four-dimensional distribution in $\mathbb {R}^{10}$. Here, we address this critical case. We give partial but constructive results that generalize, in a sense, the results of our previous papers.

Bibliographie

[1] Agrachev A. A., El-Alaoui El-H.Ch., Gauthier J. P., “Sub-Riemannian metrics on $\mathbb{R}^3$”, Geometric control and non-holonomic mechanics, Proc. Conf. Mexico City, 1996, Can. Math. Soc. Conf. Proc., 25, Amer. Math. Soc., Providence, RI, 1998, 29–76 | MR

[2] Agrachev A. A., Gote Zh. P. A., “Subrimanovy metriki i izoperimetricheskie zadachi v kontaktnom sluchae”, Tr. Mezhdunar. konf., posv. 90-letiyu L. S. Pontryagina. T. 3: Geometricheskaya teoriya upravleniya, Itogi nauki i tekhniki. Sovr. matematika i ee pril. Tematich. obzory, 6, VINITI, M., 1999, 5–48 ; J. Math. Sci., 103:6 (2001), 639–663 | MR | Zbl | DOI | Zbl

[3] Bierstone E., Milman P., “Semianalytic and subanalytic sets”, Publ. Math. IHES, 1988, no. 67, 5–42 | MR | Zbl

[4] Charlot G., “Quasi-contact S-R metrics: Normal form in $\mathbb{R}^{2n}$, wave front and caustic in $\mathbb{R}^4$”, Acta Appl. Math., 74:3 (2002), 217–263 | DOI | MR | Zbl

[5] El-Alaoui El-H. Ch., Gauthier J.-P., Kupka I., “Small sub-Riemannian balls on $\mathbb{R}^3$”, J. Dyn. and Control Syst., 2:3 (1996), 359–421 | DOI | MR | Zbl

[6] Clarke F. H., Optimization and nonsmooth analysis, J. Wiley Sons, New York etc., 1983 | MR

[7] Romero-Meléndez C., Gauthier J. P., Monroy-Pérez F., “On complexity and motion planning for co-rank one sub-Riemannian metrics”, ESAIM Control Optim. and Calc. Var., 10 (2004), 634–655 | DOI | MR | Zbl

[8] Gauthier J.-P., Zakalyukin V., “On the codimension one motion planning problem”, J. Dyn. and Control Syst., 11:1 (2005), 73–89 | DOI | MR | Zbl

[9] Gauthier J.-P., Zakalyukin V., “On the one-step-bracket-generating motion planning problem”, J. Dyn. and Control Syst., 11:2 (2005), 215–235 | DOI | MR | Zbl

[10] Goresky M., MacPherson R., Stratified Morse theory, Springer, Berlin, 1988 | MR | Zbl

[11] Gromov M., “Carnot–Carathéodory spaces seen from within”, Sub-Riemannian geometry, Progr. Math., 144, Birkhäuser, Basel, 1996, 79–323 | MR | Zbl

[12] Hardt R., “Stratification of real analytic mappings and images”, Invent. Math., 28 (1975), 193–208 | DOI | MR | Zbl

[13] Jean F., “Complexity of nonholonomic motion planning”, Intern. J. Control., 74:8 (2001), 776–782 | DOI | MR | Zbl

[14] Jean F., “Entropy and complexity of a path in sub-Riemannian geometry”, ESAIM Control Optim. and Calc. Var., 9 (2003), 485–506 | MR

[15] Falbel E., Jean F., “Measures and transverse paths in sub-Riemannian geometry”, J. Anal. Math., 91 (2003), 231–246 | DOI | MR | Zbl

[16] Pontryagin L. S., Boltyanskij V. G., Gamkrelidze R. V., Mishchenko E. F., The mathematical theory of optimal processes, Wiley, New York, 1962 | MR | Zbl

[17] Ekeland I., Temam R., Analyse convexe et problèmes variationnels, Gauthier-Villars, Paris, 1974 ; Ekland I., Temam R., Vypuklyi analiz i variatsionnye problemy, Mir, M., 1979 | MR | Zbl | MR