Discretely Actuated Manipulator Workspace Generation by Closed-Form Convolution

We determine workspaces of discretely actuated manipulators using convolution of real-valued functions on the Special Euclidean Group. Each workspace is described in terms of a density function that provides the number of reachable frames inside any unit volume of the workspace. A manipulator consisting of n discrete actuators each with K states can reach Kn frames in space. Given this exponential growth, brute force representation of discrete manipulator workspaces is not feasible in the highly actuated case. However, if the manipulator is of macroscopically-serial architecture, the workspace can be generated by the following procedure: (1) partition the manipulator into B kinematically independent segments; (2) approximate the workspace of each segment as a continuous density function on a compact subset of the Special Euclidean Group; (3) approximate the whole workspace as a B-fold convolution of these densities. We represent density functions as finite Hermite-Fourier Series and show for the planar case how the B-fold convolution can be performed in closed form. If all segments are identical only O(log B) convolutions are necessary.