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

Trent Weiss, Madhur Behl

TL;DR

This work addresses the challenge of real-time probabilistic collision risk estimation during high-speed autonomous racing overtakes. It proposes GLR, a two-stage framework that combines spatial Gauss–Legendre cubature for instantaneous collision probability with a time-integrating non-homogeneous Poisson process to yield the total risk, achieving high accuracy at 1000 Hz. The approach avoids conservative Boole bounds and Monte Carlo sampling, and is validated on a 446-scenario dataset from a Formula One racing simulator, where GLR reports a substantial reduction in mean absolute error compared to baselines. The method preserves the full rectangular vehicle footprint and is applicable to broader motion-planning contexts requiring precise risk assessment under uncertainty, with potential extension to multi-agent scenarios.

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.

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

TL;DR

This work addresses the challenge of real-time probabilistic collision risk estimation during high-speed autonomous racing overtakes. It proposes GLR, a two-stage framework that combines spatial Gauss–Legendre cubature for instantaneous collision probability with a time-integrating non-homogeneous Poisson process to yield the total risk, achieving high accuracy at 1000 Hz. The approach avoids conservative Boole bounds and Monte Carlo sampling, and is validated on a 446-scenario dataset from a Formula One racing simulator, where GLR reports a substantial reduction in mean absolute error compared to baselines. The method preserves the full rectangular vehicle footprint and is applicable to broader motion-planning contexts requiring precise risk assessment under uncertainty, with potential extension to multi-agent scenarios.

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.

Paper Structure

This paper contains 15 sections, 2 theorems, 23 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. Preliminaries
  5. One-dimensional Gauss--Legendre Quadrature
  6. Non-homogeneous Poisson Processes
  7. Stage 1: Instantaneous Collision Probability with GLC
  8. Stage 2: Hazard Function & Total Collision Probability
  9. Experimental Results
  10. Dataset Collection and Description
  11. Experimental Setup
  12. Evaluation Metrics
  13. Comparisons to Other Methods
  14. Empirical Results
  15. Conclusion

Key Result

Lemma 1

Let $N(t)$ be an NHPP with hazard function $\lambda(t)$ on $[0,T_F]$. Then the probability that $N(t) = 0$ for all $t\in[0,\,T_F]$ is Hence, the probability of at least one event by time $T_F$ is

Figures (4)

  • Figure 1: Problem setting: the ego vehicle’s deterministic trajectory $\mathcal{T}_{ego}$ (orange) is evaluated for collisions against the target vehicle’s uncertain motion $p(\mathcal{T})$ (blue).
  • Figure 2: Two stages of the GLR algorithm. Left panel: Instantaneous collision probability $P_{col}(t)$ using GLC. Right panel: Corresponding hazard function $\lambda(t)$; $Pr[N(T_F)=0]$ readily follows from the grey area under the $\lambda(t)$ curve via equation \ref{['eqn:nhpp_PrNeq0']}. This example has $T_F=6$ seconds, but the model is extensible to other prediction horizons.
  • Figure 3: Flowchart of our method. Stage 1 (GLC) computes $P_{\mathrm{col}}(t)$ and $\lambda(t)$ at discrete times; Stage 2 (GLQ) integrates $\lambda(t)$ to yield overall collision probability.
  • Figure 4: An example overtaking scenario from our F1 dataset. $\mathcal{T}_{ego}$ is shown with some samples from $p(\mathcal{T})$

Theorems & Definitions (3)

  • Definition 1: NHPP
  • Lemma 1: Zero-event probability in an NHPP
  • Proposition 1: Boundary Conditions of $\lambda(t)$