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

Abstract

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. Stateof-the-art attempts to address this within a conventional discretetime 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, riskaware, continuous motion-planning for a broad class of nonlinear, partially-observable systems.

Scenario-Based Trajectory Optimization in Uncertain Dynamic Environments

“…The gap between the risk guarantee and the obtained risk can be reduced, for example, by assuming some knowledge of the distribution or by running multiple scenario programs in parallel. Alternatively, the risk could be analyzed in continuous time (see for example [43]) to reduce discretization errors. The proposed method is widely applicable.…”

Tight Collision Probability for UAV Motion Planning in Uncertain Environment

“…The collision probability for each step is taken as the maximum one among all obstacles. As mentioned in [24], summing the collision probabilities at discrete time steps will double count the probability. Thus the risk here (i.e.…”