Collision Probabilities for Continuous-Time Systems Without Sampling [with Appendices]

Demand for high-performance, robust, and safe autonomous systems has grown substantially in recent years. Fulfillment of these objectives requires accurate and efficient risk estimation that can be embedded in core decision-making tasks such as motion planning. On one hand, Monte-Carlo (MC) and other sampling-based techniques can provide accurate solutions for a wide variety of motion models but are cumbersome to apply in the context of continuous optimization. On the other hand, "direct" approximations aim to compute (or upper-bound) the failure probability as a smooth function of the decision variables, and thus are widely applicable. However, existing approaches fundamentally assume discrete-time dynamics and can perform unpredictably when applied to continuous-time systems operating 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 improves as the 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, partially-observable systems.