  We present an efficient algorithm to compute tight upper bounds of collision
probability between two objects with positional uncertainties, whose error
distributions are represented with non-Gaussian forms. Our approach can handle
noisy datasets from depth sensors, whose distributions may correspond to
Truncated Gaussian, Weighted Samples, or Truncated Gaussian Mixture Model. We
derive tight probability bounds for convex shapes and extend them to non-convex
shapes using hierarchical representations. We highlight the benefits of our
approach over prior probabilistic collision detection algorithms in terms of
tighter bounds ($10$x) and improved running time ($3$x). Moreover, we use our
tight bounds to design an efficient and accurate motion planning algorithm for
a 7-DOF robot arm operating in tight scenarios with sensor and motion
uncertainties.
