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Fast Collision Probability Estimation for Automated Driving using Multi-circular Shape Approximations

Leon Tolksdorf, Christian Birkner, Arturo Tejada, Nathan van de Wouw

TL;DR

This work tackles the problem of estimating the probability of collision (POC) for automated driving under Gaussian position uncertainty. It replaces costly Monte Carlo sampling with fast, multi-circle shape approximations, deriving four circle-to-circle POC methods (local, global, polar, and Monte Carlo) and identifying the local-coordinate approach as the most computationally efficient. The authors extend the framework to multi-circle ego models, introduce lens-overlap corrections, and provide upper and lower bounds on the approximation error, demonstrated in simulation. The results show substantial real-time benefits for motion-planning use cases while maintaining guaranteed bounds on the POC estimate, enabling safer and more scalable planning in autonomous systems.

Abstract

Many state-of-the-art methods for safety assessment and motion planning for automated driving require estimation of the probability of collision (POC). To estimate the POC, a shape approximation of the colliding actors and probability density functions of the associated uncertain kinematic variables are required. Even with such information available, the derivation of the POC is in general, i.e., for any shape and density, only possible with Monte Carlo sampling (MCS). Random sampling of the POC, however, is challenging as computational resources are limited in real-world applications. We present expressions for the POC in the presence of Gaussian uncertainties, based on multi-circular shape approximations. In addition, we show that the proposed approach is computationally more efficient than MCS. Lastly, we provide a method for upper and lower bounding the estimation error for the POC induced by the used shape approximations.

Fast Collision Probability Estimation for Automated Driving using Multi-circular Shape Approximations

TL;DR

This work tackles the problem of estimating the probability of collision (POC) for automated driving under Gaussian position uncertainty. It replaces costly Monte Carlo sampling with fast, multi-circle shape approximations, deriving four circle-to-circle POC methods (local, global, polar, and Monte Carlo) and identifying the local-coordinate approach as the most computationally efficient. The authors extend the framework to multi-circle ego models, introduce lens-overlap corrections, and provide upper and lower bounds on the approximation error, demonstrated in simulation. The results show substantial real-time benefits for motion-planning use cases while maintaining guaranteed bounds on the POC estimate, enabling safer and more scalable planning in autonomous systems.

Abstract

Many state-of-the-art methods for safety assessment and motion planning for automated driving require estimation of the probability of collision (POC). To estimate the POC, a shape approximation of the colliding actors and probability density functions of the associated uncertain kinematic variables are required. Even with such information available, the derivation of the POC is in general, i.e., for any shape and density, only possible with Monte Carlo sampling (MCS). Random sampling of the POC, however, is challenging as computational resources are limited in real-world applications. We present expressions for the POC in the presence of Gaussian uncertainties, based on multi-circular shape approximations. In addition, we show that the proposed approach is computationally more efficient than MCS. Lastly, we provide a method for upper and lower bounding the estimation error for the POC induced by the used shape approximations.
Paper Structure (18 sections, 1 theorem, 25 equations, 6 figures, 1 table)

This paper contains 18 sections, 1 theorem, 25 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Collision Probability for Arbitrary Shapes
  4. Multi-Circle Shape Approximations
  5. Circle-to-Circle Collisions
  6. Baseline, Approach 1: Monte Carlo Sampling
  7. Proposed Approach 2: Local Coordinates
  8. Approach 3: Global Coordinates
  9. Approach 4: Polar Coordinates
  10. Comparison of Approaches
  11. Multi-Circle-to-Circle Collision
  12. Single-Axis Circle Placement
  13. Multi-Axis Placement
  14. POC of the Lens
  15. Error Bounding
  16. ...and 3 more sections

Key Result

Lemma 1

The POC of the two-circle-to-circle case with $d_c$ and $r_e$ identified by (eq_placing) is a probability given by where $\{ \boldsymbol{q}_o \in \tilde{\mathcal{A}}_{1}^{B} \}$ represents the event of the front ego circle colliding with the object's circle, $\{ \boldsymbol{q}_o \in \tilde{\mathcal{A}}_{2}^{B} \}$ represents the event of the ego's rear circle colliding with the object, and $\{ \b

Figures (6)

  • Figure 1: Depiction of the problem statement. The global coordinate axis are represented by $\hat{c}_{1},\hat{c}_{2}$, the vehicle coordinate system by $c_1, c_2$.
  • Figure 2: Determination of the integral bounds in local coordinates, where $c_2 \in [-R, R]$ and $c_1 \in \left[-\sqrt{R^2-c_2^2}, \sqrt{R^2-c_2^2}\right]$.
  • Figure 3: Geometry of the lens, the red line denotes all object circle positions leading to collisions with both ego circles.
  • Figure 4: Exemplary coverage (left) and quadruple collision lens overlap, green area in right depiction. For better visibility, the blue circles are shaded in gray in the right depiction.
  • Figure 5: Left: Scenario A, right: Scenario B. The green vehicle represents the ego vehicle and the red vehicle the object. The actors are displayed at three different time steps: 1: $t=$0s, 2: $t=$4s, 3: $t=$8s.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof