Computation of Robot Configuration and Workspaces via the Fourier Transform on the Discrete-Motion Group

We apply the Fourier transform on the discrete-motion group to the problem of computing the configuration-space obstacles of mobile robots which move among static obstacles, the problem of finding the workspace density of binary manipulators with many actuators, and the problem of determining workspace boundaries of manipulators with continuous-motion actuators. We develop and implement Fourier transforms for the discrete-motion group of the plane. These transforms allow us to apply fast Fourier transform methods to the computation of convolution-like integrals that arise in robot kinematics and motion planning. The results of the implementation are discussed for particular examples.