  As robots are being increasingly used in close proximity to humans and
objects, it is imperative that robots operate safely and efficiently under
real-world conditions. Yet, the environment is seldom known perfectly. Noisy
sensors and actuation errors compound to the errors introduced while estimating
features of the environment. We present a novel approach (1) to incorporate
these uncertainties for robot state estimation and (2) to compute the
probability of collision pertaining to the estimated robot configurations. The
expression for collision probability is obtained as an infinite series and we
prove its convergence. An upper bound for the truncation error is also derived
and the number of terms required is demonstrated by analyzing the convergence
for different robot and obstacle configurations. We evaluate our approach using
two simulation domains which use a roadmap-based strategy to synthesize
trajectories that satisfy collision probability bounds.
