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

2025-07-24

AI Summary

To address the challenge of real-time collision risk assessment under minimal wheel-to-wheel safety distances in high-speed autonomous racing overtaking scenarios, this paper proposes GLR—a high-accuracy, low-conservatism probabilistic risk estimation algorithm. GLR innovatively integrates Gaussian–Legendre quadrature with a non-homogeneous Poisson process to explicitly model both vehicle geometry and temporal uncertainty in trajectories, thereby avoiding the accuracy loss and computational overhead associated with conventional circular bounding approximations or Monte Carlo sampling. Evaluated on 446 high-fidelity F1 simulation overtaking scenarios, GLR achieves a 77% average error reduction over five state-of-the-art methods and a 52% improvement over the best-performing baseline. Moreover, it operates at a stable 1000 Hz, satisfying real-time motion planning requirements.

Technology Category

• Planning, Routing, and Scheduling: Replanning and Plan Repair
• Intelligent Robots: State Estimation
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Accurate real-time collision risk estimation in high-speed autonomous racing
• Overcoming limitations of simplified geometric and Monte Carlo methods
• General framework applicable beyond racing for motion planning

Innovation

• Gauss-Legendre Rectangle algorithm for collision risk
• Non-homogeneous Poisson process integration
• Real-time 1000 Hz risk estimation