The planning of optimal motions of non-holonomic systems

A new method to the planning of optimal motions of the non-holonomic systems is presented. It is based on a non-classical formulation of the Pontryagin Maximum Principle given in variational form, which handles efficiently various control and/or state-dependent constraints. They arise naturally due to both physical limits of the actuators of the non-holonomic systems and potential existence of obstacles in the workspace. The method proposed here provides continuous solutions in infinite-dimensional control space. It seems to be in contrast to majority of known optimization algorithms which project infinite-dimensional control space into finite-dimensional one and then apply techniques of linear and/or nonlinear programming, thus resulting only in near-optimal trajectories. Moreover, the offered control schemes do not require computation of inverse or pseudo-inverse of the Jacobian in the case of classic non-holonomic motion planning what also results in numerical stability of our approach. The performance of the proposed control strategies is illustrated through computer simulations for a chosen class of non-holonomic structures operating in both an obstacle-free workspace and a workspace including obstacles. Numerical comparison of our control scheme with the representative algorithms known from the literature is also given.