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

TL;DR

This paper introduces a novel continuous-time risk approximation method for collision probabilities that outperforms existing discrete-time approaches, enabling more accurate and efficient safety analysis in autonomous system motion planning.

Contribution

It develops a continuous-time risk approximation framework that converges with discretization refinement, addressing limitations of prior discrete-time methods.

Findings

Outperforms state-of-the-art techniques in replicating Monte Carlo estimates

Provides a lightweight, convergent risk estimate for continuous-time systems

Enables robust, risk-aware motion planning for nonlinear and partially-observable systems

Abstract

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.