In this paper, we study the motion planning problem for generic sub-riemannian metrics of co-rank one. We give explicit expressions for the metric complexity (in the sense of Jean [10, 11]), in terms of the elementary invariants of the problem. We construct asymptotic optimal syntheses. It turns out that among the results we show, the most complicated case is the 3-dimensional. Besides the generic case, we study some non-generic generalizations in the analytic case.
Keywords: motion planning problem, metric complexity, normal forms, asymptotic optimal synthesis
@article{COCV_2004__10_4_634_0, author = {Romero-Mel\'endez, Cutberto and Gauthier, Jean Paul and Monroy-P\'erez, Felipe}, title = {On complexity and motion planning for co-rank one sub-riemannian metrics}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {634--655}, publisher = {EDP-Sciences}, volume = {10}, number = {4}, year = {2004}, doi = {10.1051/cocv:2004024}, mrnumber = {2111085}, zbl = {1101.93030}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2004024/} }
TY - JOUR AU - Romero-Meléndez, Cutberto AU - Gauthier, Jean Paul AU - Monroy-Pérez, Felipe TI - On complexity and motion planning for co-rank one sub-riemannian metrics JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2004 SP - 634 EP - 655 VL - 10 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2004024/ DO - 10.1051/cocv:2004024 LA - en ID - COCV_2004__10_4_634_0 ER -
%0 Journal Article %A Romero-Meléndez, Cutberto %A Gauthier, Jean Paul %A Monroy-Pérez, Felipe %T On complexity and motion planning for co-rank one sub-riemannian metrics %J ESAIM: Control, Optimisation and Calculus of Variations %D 2004 %P 634-655 %V 10 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2004024/ %R 10.1051/cocv:2004024 %G en %F COCV_2004__10_4_634_0
Romero-Meléndez, Cutberto; Gauthier, Jean Paul; Monroy-Pérez, Felipe. On complexity and motion planning for co-rank one sub-riemannian metrics. ESAIM: Control, Optimisation and Calculus of Variations, Volume 10 (2004) no. 4, pp. 634-655. doi : 10.1051/cocv:2004024. http://www.numdam.org/articles/10.1051/cocv:2004024/
[1] Transversal mappings and flows. W.A. Benjamin, Inc. (1967). | MR | Zbl
and ,[2] Chakir1996) 29-76, Canad. Math. Soc. Conf. Proc. 25, Amer. Math. Soc., Providence, RI (1998). | MR | Zbl
,[3] Sub-Riemannian Metrics and Isoperimetric Problems in the Contact case, L.S. Pontriaguine, 90th Birthday Commemoration, Contemporary Mathematics 64 (1999) 5-48 (Russian). English version: J. Math. Sci. 103, 639-663. | MR | Zbl
and ,[4] Differential Topology. Springer-Verlag (1976). | MR | Zbl
,[5] Chakir 2 (1996) 359-421. | MR | Zbl
[6] Quasi-Contact sub-Riemannian Metrics 74 (2002) 217-263. | MR | Zbl
,[7] Algorithmic foundations of robotics. AK Peters, Wellesley, Mass. (1995). | MR | Zbl
, , and ,[8] Mc Pherson Goreski, Stratified Morse Theory. Springer-Verlag, New York (1988). | MR | Zbl
[9] Carnot-Caratheodory spaces seen from within, in Sub-Riemannian geometry. A. Bellaiche, J.J. Risler Eds., Birkhauser (1996) 79-323. | MR | Zbl
,[10] Complexity of nonholonomic motion planning. Internat. J. Control 74 (2001) 776-782. | MR | Zbl
,[11] Entropy and Complexity of a Path in Sub-Riemannian Geometry. ESAIM: COCV 9 (2003) 485-508. | Numdam | MR | Zbl
,[12] Measures and transverse paths in Sub-Riemannian Geometry. J. Anal. Math. 91 (2003) 231-246. | MR | Zbl
and ,[13] Perturbation theory for linear operators. Springer-Verlag (1966) 120-122. | MR | Zbl
,[14] Géometrie sous-Riemannienne1995-96) 1-30. | Numdam
,[15] Motion Planning for controllable systems without drift, in Proc. of the 1991 IEEE Int. Conf. on Robotics and Automation (1991).
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