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On complexity and motion planning for co-rank one sub-Riemannian metrics

Published online by Cambridge University Press:  15 October 2004

Cutberto Romero-Meléndez ,
Jean Paul Gauthier  and
Felipe Monroy-Pérez
Cutberto Romero-Meléndez
Affiliation:
Laboratoire d'Analyse Appliquée et Optimisation, Département de Mathématiques, Université de Bourgogne, 21078 Dijon, France.
Jean Paul Gauthier
Affiliation:
Departement Maths, Lab. LE2I, UMR CNRS 5158, Université de Bourgogne, BP 47870, 21078 Dijon, France.
Felipe Monroy-Pérez
Affiliation:
Basic Sciences Department, UAM-Azcapotzalco, 02200, México D.F., Mexico; fmp@correo.azc.uam.mx.
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Abstract

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 [CITE]), 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 C∞ case, we study some non-generic generalizations in the analytic case.

Keywords

Motion planning problemmetric complexitynormal formsasymptotic optimal synthesis.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 4 , October 2004 , pp. 634 - 655
Copyright
© EDP Sciences, SMAI, 2004

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References

R. Abraham and J. Robbin, Transversal mappings and flows. W.A. Benjamin, Inc. (1967).
A. Agrachev, El-A. Chakir, El-H. and J.P. Gauthier, Sub-Riemannian metrics on R 3, in Geometric Control and non-holonomic mechanics, Mexico City (1996) 29-76, Canad. Math. Soc. Conf. Proc. 25, Amer. Math. Soc., Providence, RI (1998).
A. Agrachev and J.P. Gauthier, 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.
M.W. Hirsch, Differential Topology. Springer-Verlag (1976).
El-A. Chakir, El-H., J.P. Gauthier, I. Kupka, Small Sub-Riemannian balls on R 3. J. Dynam. Control Syst. 2 (1996) 359-421.
Charlot, G., Quasi-Contact sub-Riemannian Metrics, Normal Form in R2n , Wave front and Caustic in R 4. Acta Appl. Math. 74 (2002) 217-263. CrossRef
K. Goldberg, D. Halperin, J.C. Latombe and R. Wilson, Algorithmic foundations of robotics. AK Peters, Wellesley, Mass. (1995).
Mc Pherson Goreski, Stratified Morse Theory. Springer-Verlag, New York (1988).
M. Gromov, Carnot-Caratheodory spaces seen from within, in Sub-Riemannian geometry. A. Bellaiche, J.J. Risler Eds., Birkhauser (1996) 79-323.
Jean, F., Complexity of nonholonomic motion planning. Internat. J. Control 74 (2001) 776-782. CrossRef
Jean, F., Entropy and Complexity of a Path in Sub-Riemannian Geometry. ESAIM: COCV 9 (2003) 485-508. CrossRef
Jean, F. and Falbel, E., Measures and transverse paths in Sub-Riemannian Geometry. J. Anal. Math. 91 (2003) 231-246.
T. Kato, Perturbation theory for linear operators. Springer-Verlag (1966) 120-122.
I. Kupka, Géometrie sous-Riemannienne, in Séminaire Bourbaki, 48e année, No. 817 (1995-96) 1-30.
G. Lafferiere and H. Sussmann, Motion Planning for controllable systems without drift, in Proc. of the 1991 IEEE Int. Conf. on Robotics and Automation (1991).