  Demand for high-performance, robust, and safe autonomous systems has grown
substantially in recent years. These objectives motivate the desire for
efficient safety-theoretic reasoning that can be embedded in core
decision-making tasks such as motion planning, particularly in constrained
environments. On one hand, Monte-Carlo (MC) and other sampling-based techniques
provide accurate collision probability estimates for a wide variety of motion
models but are cumbersome in the context of continuous optimization. On the
other, "direct" approximations aim to compute (or upper-bound) the failure
probability as a smooth function of the decision variables, and thus are
convenient for optimization. However, existing direct approaches fundamentally
assume discrete-time dynamics and can perform unpredictably when applied to
continuous-time systems ubiquitous in the real world, often manifesting as
severe conservatism. State-of-the-art attempts to address this within a
conventional discrete-time framework require additional Gaussianity
approximations that ultimately produce inconsistency of their own. In this
paper we take a fundamentally different approach, deriving a risk approximation
framework directly in continuous time and producing a lightweight estimate that
actually converges as the underlying discretization is refined. Our
approximation is shown to significantly outperform state-of-the-art techniques
in replicating the MC estimate while maintaining the functional and
computational benefits of a direct method. This enables robust, risk-aware,
continuous motion-planning for a broad class of nonlinear and/or
partially-observable systems.
