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Chitour, Yacine

A continuation method for motion-planning problems. (English) Zbl 1105.93030

ESAIM, Control Optim. Calc. Var. 12, 139-168 (2006).
Summary: 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.

Cited in 12 Documents

MSC:

93B40 Computational methods in systems theory (MSC2010)
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
70F25 Nonholonomic systems related to the dynamics of a system of particles
70Q05 Control of mechanical systems
93C85 Automated systems (robots, etc.) in control theory
93B29 Differential-geometric methods in systems theory (MSC2000)

Keywords:

homotopy continuation method; path following; Wazewski equation; sub-Riemannian geometry; nonholonomic control systems; motion planning problem
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References:

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