Probabilistic Collision Risk Estimation through Gauss-Legendre Cubature and Non-Homogeneous Poisson Processes

TL;DR

This paper presents the Gauss-Legendre Rectangle (GLR) algorithm, a fast and accurate method for real-time collision risk estimation in high-speed autonomous racing, outperforming existing approaches.

Contribution

The paper introduces the GLR algorithm, combining Gauss-Legendre integration with non-homogeneous Poisson processes for precise collision risk estimation in autonomous racing scenarios.

Findings

GLR reduces average error by 77% compared to baselines

GLR operates at 1000 Hz for real-time application

GLR outperforms five state-of-the-art methods in experiments

Abstract

Overtaking in high-speed autonomous racing demands precise, real-time estimation of collision risk; particularly in wheel-to-wheel scenarios where safety margins are minimal. Existing methods for collision risk estimation either rely on simplified geometric approximations, like bounding circles, or perform Monte Carlo sampling which leads to overly conservative motion planning behavior at racing speeds. We introduce the Gauss-Legendre Rectangle (GLR) algorithm, a principled two-stage integration method that estimates collision risk by combining Gauss-Legendre with a non-homogeneous Poisson process over time. GLR produces accurate risk estimates that account for vehicle geometry and trajectory uncertainty. In experiments across 446 overtaking scenarios in a high-fidelity Formula One racing simulation, GLR outperforms five state-of-the-art baselines achieving an average error reduction of 77% and surpassing the next-best method by 52%, all while running at 1000 Hz. The framework is general and applicable to broader motion planning contexts beyond autonomous racing.