An Approximation Algorithm for Nonholonomic Systems

Abstract

In [SIAM J. Control Optim.}, 37 (1997), to appear], [Limiting process of control-affine systems with Holder continuous inputs, submitted], 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 $\{u^j\}\subseteq L^1([0,T],\rr^m)$, 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.