Probabilistic Occupancy via Forward Stochastic Reachability for Markov Jump Affine Systems

Probabilistic occupancy, the likelihood that the state at a known future time lies in a given set, is important in a variety of stochastic motion planning problems. We provide efficient computational techniques, based in Fourier transforms, to characterize the stochasticity of the future state for Markov jump affine systems. This class of systems captures a variety of important dynamics in planning problems, including the Dubins' vehicle. We employ convex optimization to compute outer approximations of the superlevel sets of the probabilistic occupancy function, which is a key for preserving the safety guarantees sought in collision-avoidance problems. In contrast to traditional approaches, our approach does not rely on gridding, recursion, or sampling, accommodates non-Gaussian perturbed dynamics, and affords outer-approximation guarantees. We demonstrate our methods on the target pursuit problem with multiple robots pursuing a nonadversarial target with stochastic dynamics, and on the problem of computing keep-out regions for stochastic collision avoidance of a Dubins' vehicle.