  Consider a convex set of which we remove an arbitrarily number of disjoints
convex sets -- the obstacles -- and a convex function whose minimum is the
agent's goal. We consider a local and stochastic approximation of the gradient
of a Rimon-Koditschek navigation function where the attractive potential is the
convex function that the agent is minimizing. In particular we show that if the
estimate available to the agent is unbiased convergence to the desired
destination while obstacle avoidance is guaranteed with probability one under
the same geometrical conditions than in the deterministic case. Qualitatively
these conditions are that the ratio of the maximum over the minimum eigenvalue
of the Hessian of the objective function is not too large and that the
obstacles are not too flat or too close to the desired destination. Moreover,
we show that for biased estimates a similar result holds under some assumptions
on the bias. These assumptions are motivated by the study of the estimate of
the gradient of a Rimon-Koditschek navigation function for sensor models that
fit circles or ellipses around the obstacles. Numerical examples explore the
practical value of these theoretical results.
