  This paper presents a tool for addressing a key component in many algorithms
for planning robot trajectories under uncertainty: evaluation of the safety of
a robot whose actions are governed by a closed-loop feedback policy near a
nominal planned trajectory. We describe an adaptive importance sampling Monte
Carlo framework that enables the evaluation of a given control policy for
satisfaction of a probabilistic collision avoidance constraint which also
provides an associated certificate of accuracy (in the form of a confidence
interval). In particular this adaptive technique is well-suited to addressing
the complexities of rigid-body collision checking applied to non-linear robot
dynamics. As a Monte Carlo method it is amenable to parallelization for
computational tractability, and is generally applicable to a wide gamut of
simulatable systems, including alternative noise models. Numerical experiments
demonstrating the effectiveness of the adaptive importance sampling procedure
are presented and discussed.
