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
Mots-clés : 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 = {https://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 - https://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 https://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, Tome 10 (2004) no. 4, pp. 634-655. doi : 10.1051/cocv:2004024. https://www.numdam.org/articles/10.1051/cocv:2004024/
[1] and, Transversal mappings and flows. W.A. Benjamin, Inc. (1967). | MR | Zbl
[2] , Chakir1996) 29-76, Canad. Math. Soc. Conf. Proc. 25, Amer. Math. Soc., Providence, RI (1998). | MR | Zbl
[3] and, 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
[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] ,, and, Algorithmic foundations of robotics. AK Peters, Wellesley, Mass. (1995). | MR | Zbl
[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] and, Measures and transverse paths in Sub-Riemannian Geometry. J. Anal. Math. 91 (2003) 231-246. | MR | Zbl
[13] , Perturbation theory for linear operators. Springer-Verlag (1966) 120-122. | MR | Zbl
[14] , Géometrie sous-Riemannienne1995-96) 1-30. | Numdam
[15] and, Motion Planning for controllable systems without drift, in Proc. of the 1991 IEEE Int. Conf. on Robotics and Automation (1991).
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