Collision Probability Estimation for Optimization-based Vehicular Motion Planning

TL;DR

This paper introduces a deterministic, efficient method for estimating collision probability in vehicular motion planning by over-approximating vehicle shapes and modeling uncertainties, enabling safer and more reliable autonomous driving.

Contribution

It proposes an analytical POC estimation method using multi-circular shape approximation and Gaussian uncertainty modeling, suitable for optimization-based motion planning.

Findings

The method provides a guaranteed over-approximation of collision probability.

It enables reproducible trajectories in stochastic model predictive control.

The approach effectively handles varying levels of uncertainty in vehicle motion.

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. For the particular case of Gaussian uncertainty in the position and heading, we present a computationally efficient algorithm for computing the POC estimate. 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.