An approximation algorithm for nonholonomic systems. (English) Zbl 0887.34063

Liu, Wensheng

SIAM J. Control Optimization 35, No. 4, 1328-1365 (1997).

Summary: In [SIAM J. Control Optimization 37 (1997; to appear)], we have studied the limiting behavior of trajectories of control affine systems \(\Sigma : \dot{x}=\sum_{k=1}^m u_k f_k(x)\) generated by a sequence \(\{u^j\}\subseteq L^1([0,T],\mathbb{R}^m)\), where the \(f_k\) are smooth vector fields on a smooth manifold \(M\). We have shown that under very general conditions the trajectories of \(\Sigma\) generated by the \(u^j\) converge to trajectories of an {extended system} of \(\Sigma\) of the form \(\Sigma_{ext}\): \(\dot{x}=\sum_{k=1}^r v_kf_k(x)\), where \(f_k,k=1,\ldots,m\), are the same as in \(\Sigma\) and \(f_{m+1},\ldots,f_r\) are Lie brackets of \(f_1,\ldots,f_m\). In this paper, we will apply these convergence results to solve the inverse problem; i.e., given any trajectory \(\gamma\) of an extended system \(\Sigma_{ext}\), find trajectories of \(\Sigma\) that converge to \(\gamma\) uniformly. This is done by means of a universal construction that only involves the knowledge of the \(v_k, k=1,\ldots,r\), and the structure of the Lie brackets in \(\Sigma_{ext}\) but does not depend on the manifold \(M\) and the vector fields \(f_1,\ldots,f_m\). These results can be applied to approximately track an arbitrary smooth path in \(M\) for controllable systems \(\Sigma\), which in particular gives an alternative approach to the motion planning problem for nonholonomic systems.

MSC:

34H05	Control problems involving ordinary differential equations
93C15	Control/observation systems governed by ordinary differential equations
93B27	Geometric methods
34E10	Perturbations, asymptotics of solutions to ordinary differential equations
93B05	Controllability
34A26	Geometric methods in ordinary differential equations

Keywords:

control affine systems; extended inputs; free associative algebras; free Lie algebras; Chen-Fliess series; nonholonomic motion planning