Probability of success in stochastic robot navigation with state feedback

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.