Discretely actuated manipulator workspace generation using numerical convolution on the Euclidean group

The concept of a convolution product of real-valued functions on the special Euclidean group is applied to the determination of workspaces of discretely actuated manipulators. If a discretely actuated manipulator has n actuators each with K states, then it can reach Kⁿ frames in space. Given this exponential growth in the number of reachable frames, brute force representation of discretely actuated manipulator workspaces is not feasible in the highly actuated case. However, by partitioning a discretely actuated manipulator into segments, and approximating the workspace of each segment as a density function on a compact subset of the special Euclidean group, the whole workspace can be approximated as an n-fold convolution of these densities. A numerical approximation of this convolution is presented in this paper that requires O(logn) computation time as compared to the O(K ⁿ) computations required by brute force workspace generation