This paper presents a receding horizon control design for a robot subject to stochastic uncertainty, moving in a constrained environment. Instead of minimizing the expectation of a cost functional while ensuring satisfaction of probabilistic state constraints, we propose a two-stage solution where the path that minimizes the cost functional is planned deterministically, and a local stochastic optimal controller with exit constraints ensures satisfaction of probabilistic state constraints while following the planned path. This control design strategy ensures boundedness of errors around the reference path and collision-free convergence to the goal with probability one under the assumption of unbounded inputs. We show that explicit expressions for the control law are possible for certain cases. We provide simulation results for a point robot moving in a constrained two-dimensional environment under Brownian noise. The method can be extended to systems with bounded inputs, if a small nonzero probability of failure can be accepted.

The analysis in this paper applies to robots with dynamics described by a stochastic differential equation, which need to navigate in constrained environments. The approach offers a method to calculate the probability that a feedback control policy designed for the drift component of the dynamics, will succeed in allowing the robot to avoid collisions and converge to its navigation goal in the presence of stochastic (white) noise. The problem is formulated as an exit problem and known techniques in the field of stochastic processes are brought to bear to determine the probabilities that the stochastic process describing the motion of the robot will ¿exit¿ the workspace through a particular part of the boundary. We motivate the use of this analysis using a controller constructed using negative gradient of a navigation function and give the analytic solution for the case of a constrained but obstacle-free workspace.