Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-28T05:57:46.728Z Has data issue: false hasContentIssue false
  • English
  • Français

A continuation method for motion-planning problems

Published online by Cambridge University Press:  15 December 2005

Yacine Chitour
Yacine Chitour*
Affiliation:
Laboratoire des signaux et systèmes, Université de Paris-Sud, CNRS, Supélec, 91192, Gif-Sur-Yvette, France; yacine.chitour@lss.supelec.fr
Get access

Abstract

We apply the well-known homotopy continuation method to address the motion planning problem (MPP) for smooth driftless control-affine systems. The homotopy continuation method is a Newton-type procedure to effectively determine functions only defined implicitly. That approach requires first to characterize the singularities of a surjective map and next to prove global existence for the solution of an ordinary differential equation, the Wazewski equation. In the context of the MPP, the aforementioned singularities are the abnormal extremals associated to the dynamics of the control system and the Wazewski equation is an o.d.e. on the control space called the Path Lifting Equation (PLE). We first show elementary facts relative to the maximal solution of the PLE such as local existence and uniqueness. Then we prove two general results, a finite-dimensional reduction for the PLE on compact time intervals and a regularity preserving theorem. In a second part, if the Strong Bracket Generating Condition holds, we show, for several control spaces, the global existence of the solution of the PLE, extending a previous result of H.J. Sussmann.

Keywords

Homotopy continuation methodpath followingWazewski equationsub-Riemannian geometrynonholonomic control systemsmotion planning problem.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 1 , January 2006 , pp. 139 - 168
Copyright
© EDP Sciences, SMAI, 2006

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.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
E.L. Allgower and K. Georg, Continuation and Path Following. Acta Numerica (1992).
J.M. Bismuth, Large Deviations and the Malliavin Calculus. Birkhäuser (1984).
Cesari, L., Functional analysis and Galerkin's method. Mich. Math. J. 11 (1964) 385–418.
Chelouah, A. and Chitour, Y., On the controllability and trajectories generation of rolling surfaces. Forum Math. 15 (2003) 727–758. CrossRef
Y. Chitour, Applied and theoretical aspects of the controllability of nonholonomic systems. Ph.D. thesis, Rutgers University (1996).
Chitour, Y., Path planning on compact Lie groups using a continuation method. Syst. Control Lett. 47 (2002) 383–391. CrossRef
Chitour, Y. and Sussmann, H.J., Line-integral estimates and motion planning using a continuation method. Essays on Math. Robotics, J. Baillieul, S.S. Sastry and H.J. Sussmann Eds., IMA. Math. Appl. 104 (1998) 91–125.
S.N. Chow and J.K. Hale, Methods of Bifurcation Theory. Springer, New York 251 (1982).
Divelbiss, A. and Wen, J.T., Path Space Approach, A to Nonholonomic Motion Planning in the Presence of Obstacles. IEEE Trans. Robotics Automation 13 (1997) 443–451. CrossRef
Ge Zhong, Horizontal Path Spaces and Carnot-Carathéodory, Metrics. Pacific J. Math. 161 (1993) 255–286.
Grasse, K.A. and Sussmann, H.J., Global controllability by nice controls, Nonlinear controllability and optimal control. Dekker, NY. Mono. Text. Pure Appl. Math. 133 (1990) 33–79.
J.K. Hale, Applications of alternative problems. Lectures notes, Brown University (1971).
M.W. Hirsch, Differential Topology. Springer, New York (1976).
V. Jurdjevic, Geometric control theory. Cambridge Studies in Adv. Math., Cam. Univ. Press (1997).
G. Lafferriere, and H.J. Sussmann, Motion planning for controllable systems without drift, in Proc. Int. Conf. Robot. Auto. Sacramento, CA (1991) 1148–1153.
E.B. Lee and L. Markus, Foundations of Optimal Control Theory. Wiley, New York (1967).
Leray, J. and Schauder, J., Topologie et équations fonctionelles. Ann. Sci. Ecole Norm. Sup. 51 (1934) 45–78. CrossRef
T.Y. Li, Numerical solution of multivariate polynomial systems by homotopy continuation methods. Acta Numerica (1997) 399–436.
Liu, W., An approximation algorithm for nonholonomic systems. SIAM J. Control Optim. 35 (1997) 1328–1365. CrossRef
W. Liu and H.J. Sussmann, Shortest paths for sub-Riemannian metrics on rank 2 distributions. Memoirs of the AMS, $\#$ 564 118 (1995).
P. Martin, Contribution à l'étude des systèmes différentiellement plats. Ph.D. thesis, École des Mines de Paris, Paris, France (1992).
R. Montgomery, Abnormal Optimal Controls and Open Problems in Nonholonomic Steering. J. Dyn. Cont. Sys. 1 Plenum Pub. Corp. (1995) 49–90.
R.M. Murray and S.S. Sastry, Steering nonholonomic systems using sinusoids, in Proc. IEEE Conference on Decision and Control (1990).
Cz. Olech, On the Wazewski equation, in Proc. of the conference, Topological methods in Differential Equations and Dynamical systems, Krakow (1996). Univ. Iagel. Acta Math. 36 (1998) 55–64.
Pansu, P., Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. Math. 129 (1989) 1–60. CrossRef
S.L. Richter and R.A. Decarlo, Continuation methods: Theory and Application. IEEE Trans. Circuits Syst. 30 (1983).
E.D. Sontag, Mathematical Control Theory. Texts Appl. Math. 6, Springer-Verlag, New York, 2nd edition (1998).
Souères, P. and Laumond, J.P., Shortest paths synthesis for a car-like robot. IEEE Trans. Aut. Cont. 41 (1996) 672–688. CrossRef
Strichartz, R., Sub-Riemannian Geometry. J. Diff. Geom. 24 (1983) 221–263. CrossRef
H.J. Sussmann, A Continuation Method for Nonholonomic Path-finding Problems, in Proceedings of the 32nd IEEE CDC, San Antonio, TX (Dec. 1993).
H.J. Sussmann, New Differential Geometric Methods in Nonholonomic Path Finding, in Systems, Models, and Feedback, A. Isidori and T.J. Tarn Eds. Birkh $\ddot a$ user, Boston (1992).
T. Wazewski, Sur l'évaluation du domaine d'existence des fonctions implicites réelles ou complexes. Ann. Soc. Polon. Math. 20 (1947).