  As drones and autonomous cars become more widespread it is becoming
increasingly important that robots can operate safely under realistic
conditions. The noisy information fed into real systems means that robots must
use estimates of the environment to plan navigation. Efficiently guaranteeing
that the resulting motion plans are safe under these circumstances has proved
difficult. We examine how to guarantee that a trajectory or policy is safe with
only imperfect observations of the environment. We examine the implications of
various mathematical formalisms of safety and arrive at a mathematical notion
of safety of a long-term execution, even when conditioned on observational
information. We present efficient algorithms that can prove that trajectories
or policies are safe with much tighter bounds than in previous work. Notably,
the complexity of the environment does not affect our methods ability to
evaluate if a trajectory or policy is safe. We then use these safety checking
methods to design a safe variant of the RRT planning algorithm.
