An approximation algorithm for nonholonomic systems

In [SIAM J. Control Optim., 37 (1997), to appear], [Limiting process of control-affine systems with Holder continuous inputs: submitted], rye have studied the limiting behavior of trajectories of control affine systems Sigma : (x) over dot = Sigma(k=1)(m) u(k)f(k)(x) generated by a sequence {u(j)} subset of or equal to L-1([0,T],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) : (x) over dot = Sigma(k=1)(r)v(k)f(k)(x), where f(k), k = 1,...,m, are the same as in Sigma and f(m+1),...,f(r) are Lie brackets of f(1),...,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,...,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),...,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.