AN APPROXIMATION ALGORITHM FOR NONHOLONOMIC SYSTEMS

In [SIAM J. Control Optim., 37 (1997), to appear], [Limiting process o f control-affine systems with Holder continuous inputs: submitted], ry e 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 t he 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 br ackets of f(1),...,f(m). In this paper, we will apply these convergenc e results to solve the inverse problem; i.e., given any trajectory gam ma of an extended system Sigma(ext), find trajectories of Sigma that c onverge to gamma uniformly. This is done by: means of a universal cons truction 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 o n the manifold M and the vector fields f(1),...,f(m). These results ca n be applied to approximately track an arbitrary smooth path in M for controllable systems Sigma, which in particular gives an alternative a pproach to the motion planning problem for nonholonomic systems.