Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-06T01:00:35.611Z Has data issue: false hasContentIssue false
  • English
  • Français

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; [email protected].
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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).