Efficient Probabilistic Planning with Maximum-Coverage Distributionally Robust Backward Reachable Trees

This paper presents a new multi-query motion planning algorithm for linear Gaussian systems with the goal of reaching a Euclidean ball with high probability. We develop a new formulation for ball-shaped ambiguity sets of Gaussian distributions and leverage it to develop a distributionally robust belief roadmap construction algorithm. This algorithm synthe- sizes robust controllers which are certified to be safe for maximal size ball-shaped ambiguity sets of Gaussian distributions. Our algorithm achieves better coverage than the maximal coverage algorithm for planning over Gaussian distributions [1], and we identify mild conditions under which our algorithm achieves strictly better coverage. For the special case of no process noise or state constraints, we formally prove that our algorithm achieves maximal coverage. In addition, we present a second multi-query motion planning algorithm for linear Gaussian systems with the goal of reaching a region parameterized by the Minkowski sum of an ellipsoid and a Euclidean ball with high probability. This algorithm plans over ellipsoidal sets of maximal size ball-shaped ambiguity sets of Gaussian distributions, and provably achieves equal or better coverage than the best-known algorithm for planning over ellipsoidal ambiguity sets of Gaussian distributions [2]. We demonstrate the efficacy of both methods in a wide range of conditions via extensive simulation experiments.