  Motion planning under sensing uncertainty is critical for robots in
unstructured environments to guarantee safety for both the robot and any nearby
humans. Most work on planning under uncertainty does not scale to
high-dimensional robots such as manipulators, assumes simplified geometry of
the robot or environment, or requires per-object knowledge of noise. Instead,
we propose a method that directly models sensor-specific aleatoric uncertainty
to find safe motions for high-dimensional systems in complex environments,
without exact knowledge of environment geometry. We combine a novel implicit
neural model of stochastic signed distance functions with a hierarchical
optimization-based motion planner to plan low-risk motions without sacrificing
path quality. Our method also explicitly bounds the risk of the path, offering
trustworthiness. We empirically validate that our method produces safe motions
and accurate risk bounds and is safer than baseline approaches.
