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

Cutberto Romero-Meléndez; Jean Paul Gauthier; Felipe Monroy-Pérez

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 10, Issue: 4, page 634-655
- ISSN: 1292-8119

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topRomero-Meléndez, Cutberto, Gauthier, Jean Paul, and Monroy-Pérez, Felipe. "On complexity and motion planning for co-rank one sub-Riemannian metrics." ESAIM: Control, Optimisation and Calculus of Variations 10.4 (2010): 634-655. <http://eudml.org/doc/90748>.

@article{Romero2010,

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.
},

author = {Romero-Meléndez, Cutberto, Gauthier, Jean Paul, Monroy-Pérez, Felipe},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Motion planning problem; metric complexity;
normal forms; asymptotic optimal synthesis.; motion planning problem; normal forms; asymptotic optimal synthesis},

language = {eng},

month = {3},

number = {4},

pages = {634-655},

publisher = {EDP Sciences},

title = {On complexity and motion planning for co-rank one sub-Riemannian metrics},

url = {http://eudml.org/doc/90748},

volume = {10},

year = {2010},

}

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

DA - 2010/3//

PB - EDP Sciences

VL - 10

IS - 4

SP - 634

EP - 655

AB -
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.

LA - eng

KW - Motion planning problem; metric complexity;
normal forms; asymptotic optimal synthesis.; motion planning problem; normal forms; asymptotic optimal synthesis

UR - http://eudml.org/doc/90748

ER -

## References

top- 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 R3, 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 Mathematics64 (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 and I. Kupka, Small Sub-Riemannian balls on R3. J. Dynam. Control Syst.2 (1996) 359-421.
- G. Charlot, Quasi-Contact sub-Riemannian Metrics, Normal Form in R2n, Wave front and Caustic in R4. Acta Appl. Math.74 (2002) 217-263.
- 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.
- F. Jean, Complexity of nonholonomic motion planning. Internat. J. Control74 (2001) 776-782.
- F. Jean, Entropy and Complexity of a Path in Sub-Riemannian Geometry. ESAIM: COCV9 (2003) 485-508.
- F. Jean and E. Falbel, 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).