  We propose a method to efficiently compute the forward stochastic reach (FSR)
set and its probability measure for nonlinear systems with an affine
disturbance input, that is stochastic and bounded. This method is applicable to
systems with an a priori known controller, or to uncontrolled systems, and
often arises in problems in obstacle avoidance in mobile robotics. When used as
a constraint in finite horizon controller synthesis, the FSR set, and its
probability measure facilitates probabilistic collision avoidance, in contrast
to methods which presume the obstacles act in a worst-case fashion and generate
hard constraints that cannot be violated. We tailor our approach to accommodate
rigid body constraints, and show convexity is assured so long as the rigid body
shape of each obstacle is also convex. We extend methods for multi-obstacle
avoidance through mixed integer linear programming (with linear robot and
obstacle dynamics) to accommodate chance constraints that represent the FSR set
probability measure. We demonstrate our method on a rigid-body obstacle
avoidance scenario, in which a receding horizon controller is designed to avoid
several stochastically moving obstacles while reaching the desired goal. Our
approach can provide solutions when approaches that presume a worst-case action
from the obstacle fail.
