A Surface Integral Approach to the Motion Planning of Nonholonomic Systems

Nonholonomic mechanical systems are governed by constraints of motion that are nonintegrable differential expressions. Unlike holonomic constraints, these differential 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, we first converge the independent variables to their desired values and then use closed trajectories of the independent variables to converge the dependent variables. We use Stokes´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. The salient features of our algorithm is apparent in the example of a planar space robot.