Abstract

We propose a general strategy for solving the motion planning problem for real analytic, controllable systems without drift. The procedure starts by computing a control that steers the given initial point to the desired target point for an extended system, in which a number of Lie brackets of the system vector fields are added to the right-hand side. The main point then is to use formal calculations based on the product expansion relative to a P. Hall basis, to produce another control that achieves the desired result on the formal level. It then turns out that this control provides an exact solution of the original problem if the given system is nilpotent. When the system is not nilpotent, one can still produce an iterative algorithm that converges very fast to a solution. Using the theory of feedback nilpotentization, one can find classes of non-nilpotent systems for which the algorithm, in cascade with a precompensator, produces an exact solution in a finite number of steps. We also ...

Citations