Safe online navigation of convex potentials in spaces with convex obstacles

Consider a convex set of which we remove an arbitrary number of disjoints convex sets - the obstacles - and a convex function whose minimum is the agent's goal. A solution to this problem under some geometric conditions [1] is to construct a Rimon-Koditschek artificial potential. Then, an agent that navigates along the negative gradient of that potential is ensured to reach the desired goal while avoiding the obstacles for almost every initial configuration. To build such potential, the agent needs exact information about the obstacles and the function he is trying to minimize. In this work, we generalize the results in [1] to stochastic settings, where the information available is not deterministic but comes from a probability distribution. In particular, we show that if the agent follows a stochastic approximation of the negative gradient of the Rimon-Koditschek artificial potential, convergence to the desired destination and obstacle avoidance is guaranteed with probability one for all initial positions under the same geometrical conditions than in the deterministic case. Numerical examples explore the practical value of these theoretical results.