Regularised Homotopy Continuation Method for the Motion Planning Problem

We address the motion planning problem for smooth control-nonlinear systems by presenting a regularised version of the well-established homotopy continuation method. The proposed approach (inspired by Tikhonov regularisation in the Moore-Penrose pseudo-inverse theory) deals with the two issues of the classical method: the singularities of the endpoint map of the nonlinear control system and the global existence of solutions of a differential equation posed on the control space, the path lifting equation (PLE). We analyse the properties of our method and show that it is always well-posed and that the regularised PLE always possesses a global solution. We additionally examine the effect of the regularisation and we give conditions under which the regularized solution converges to the initial solution. Finally, we illustrate the potential of our method through several numerical examples.