Nonholonomic mechanical systems are governed by constraints of motion that are nonintegrable differential expressions. Unlike holonomic constraints, these constraints do not reduce the number of dimensions of the configuration space of a system. Therefore a nonholonomic system can access a configuration space of dimension higher than the number of the degrees of freedom of the system. In this paper, we develop an algorithm for planning admissible trajectories for nonholonomic systems that will take the system from one point in its configuration space to another. In our algorithm the independent variables are first converged to their desired values. Subsequently, closed trajectories of the independent variables are used to converge the dependent variables. We use Green’s theorem in our algorithm to convert the problem of finding a closed path into that of finding a surface area in the space of the independent variables such that the dependent variables converge to their desired values as the independent variables traverse along the boundary of this surface area. Using this approach, we specifically address issues related to the reachability of the system, motion planning amidst additional constraints, and repeatable motion of nonholonomic systems. The salient features of our algorithm are quite apparent in the two examples we discuss: a planar space robot and a disk rolling without slipping on a flat surface.
Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning