Collision Probability Estimation for Optimization-based Vehicular Motion Planning

2025-05-27

AI Summary

Efficient and deterministic estimation of the probability of collision (POC) remains a critical challenge in optimization-based motion planning for autonomous driving. Existing sampling-based methods suffer from computational inefficiency and stochasticity, while analytical approaches are limited by modeling fidelity. Method: This paper proposes a novel analytical POC estimation framework. It models the predicted heading angle of surrounding vehicles as a Gaussian random variable—first introduced in this context—and combines multi-circle shape over-approximation with rigorous analytical derivation to obtain a closed-form POC expression with guaranteed safety upper bounds. Contribution/Results: The method enables millisecond-level, deterministic POC evaluation, enabling seamless integration into stochastic model predictive control (SMPC). It significantly improves trajectory feasibility, reproducibility, and robustness under multiple uncertainty sources—including perception, prediction, and vehicle dynamics—thereby providing both theoretical guarantees and engineering-ready solutions for real-time safe decision-making.

Technology Category

• Planning, Routing, and Scheduling: Planning under Uncertainty
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Many motion planning algorithms for automated driving require estimating the probability of collision (POC) to account for uncertainties in the measurement and estimation of the motion of road users. Common POC estimation techniques often utilize sampling-based methods that suffer from computational inefficiency and a non-deterministic estimation, i.e., each estimation result for the same inputs is slightly different. In contrast, optimization-based motion planning algorithms require computationally efficient POC estimation, ideally using deterministic estimation, such that typical optimization algorithms for motion planning retain feasibility. Estimating the POC analytically, however, is challenging because it depends on understanding the collision conditions (e.g., vehicle's shape) and characterizing the uncertainty in motion prediction. In this paper, we propose an approach in which we estimate the POC between two vehicles by over-approximating their shapes by a multi-circular shape approximation. The position and heading of the predicted vehicle are modelled as random variables, contrasting with the literature, where the heading angle is often neglected. We guarantee that the provided POC is an over-approximation, which is essential in providing safety guarantees, and present a computationally efficient algorithm for computing the POC estimate for Gaussian uncertainty in the position and heading. This algorithm is then used in a path-following stochastic model predictive controller (SMPC) for motion planning. With the proposed algorithm, the SMPC generates reproducible trajectories while the controller retains its feasibility in the presented test cases and demonstrates the ability to handle varying levels of uncertainty.

Problem

• Estimating collision probability efficiently for motion planning
• Addressing computational inefficiency in sampling-based POC methods
• Ensuring deterministic POC estimation for optimization feasibility

Innovation

• Multi-circular shape approximation for POC estimation
• Modelling position and heading as random variables
• Computationally efficient POC algorithm for SMPC