Precise and Efficient Collision Prediction under Uncertainty in Autonomous Driving

TL;DR

This work addresses the challenge of estimating collision risk for planned autonomous-driving trajectories under uncertainty. It introduces two semi-analytic formulations—Probability of Spatial Overlap (PSO) and Boundary Crossing Probability (BCP)—that compute the collision probability $P_c$ against convex obstacles while accounting for pose, orientation, and velocity uncertainties. The methods rely on relative pose modeling, Minkowski-sum collision volumes, Gaussian whitening, and edge-based 1D integrations, with orientation integration to handle uncertain headings; BCP additionally uses a rate-based boundary-crossing formulation. Validation against Monte Carlo shows strong agreement, while runtimes are significantly faster, enabling real-time, risk-aware trajectory planning; the approaches are implemented in open-source software for practical adoption.

Abstract

This research introduces two efficient methods to estimate the collision risk of planned trajectories in autonomous driving under uncertain driving conditions. Deterministic collision checks of planned trajectories are often inaccurate or overly conservative, as noisy perception, localization errors, and uncertain predictions of other traffic participants introduce significant uncertainty into the planning process. This paper presents two semi-analytic methods to compute the collision probability of planned trajectories with arbitrary convex obstacles. The first approach evaluates the probability of spatial overlap between an autonomous vehicle and surrounding obstacles, while the second estimates the collision probability based on stochastic boundary crossings. Both formulations incorporate full state uncertainties, including position, orientation, and velocity, and achieve high accuracy at computational costs suitable for real-time planning. Simulation studies verify that the proposed methods closely match Monte Carlo results while providing significant runtime advantages, enabling their use in risk-aware trajectory planning. The collision estimation methods are available as open-source software: https://github.com/TUM-AVS/Collision-Probability-Estimation

Figures (9)

• Figure 1: Exemplary scenario with large state uncertainties for the obstacle vehicle (blue) and small ones for the av, illustrating the necessity of a probabilistic collision risk measure in motion planning.
• Figure 2: Combined collision volume of the ego av (orange) and an obstacle vehicle (blue). Vehicles are approximated as rectangles. The ellipses over the blue vehicle illustrate the combined positional covariance $\mathbf{\Sigma_x}$
• Figure 3: The orientation-dependent collision polyhedron. It is the same scenario as in \ref{['fig:colloc']} but with an orientation variance of $\sigma_\theta^2 = 0.01\square rad$. The third dimension illustrates the change in the shape of the collision volume. The distribution on the right indicates the pdf of the relative orientation $\theta$.
• Figure 4: The collision volume from the scenario illustrated in \ref{['fig:colloc']} after whitening transformation.
• Figure 5: Illustration of the normal velocity for the boundary crossing rate along the segment $1-2$ of the collision volume.