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Second-Order Time to Collision With Non-Static Acceleration

Hossein Nick Zinat Matin, Yuneil Yeo, Amelie Ju-Kang Ngo, Antonio R. Paiva, Jean Utke, Maria Laura Delle Monache

TL;DR

This work advances collision risk assessment by formulating a second-order time-to-collision metric that accommodates non-static acceleration and turning, addressing the limitations of traditional first-order TTC in curved trajectories. It introduces STAR, a region-based search algorithm that leverages circle and line trajectory intersections and RK45 integration to efficiently compute the earliest collision time, with theoretical guarantees that the search covers all viable collision regions. Through numerical simulations and real-world data (Argoverse 2), the authors show that 2D-TTC more accurately reflects collision timing in turning scenarios and yields fewer false alarms compared to 1D-TTC, while STAR provides substantial speedups over step-by-step simulations. The results suggest significant practical impact for ADAS and autonomous planning, particularly at intersections and during complex maneuvers, and point to future work in relaxing movement assumptions, incorporating vehicle geometry, and expanding real-world validation.

Abstract

We propose a second-order time to collision (TTC) considering non-static acceleration and turning with realistic assumptions. This is equivalent to considering that the steering wheel is held at a fixed angle with constant pressure on the gas or brake pedal and matches the well-known bicycle model. Past works that use acceleration to compute TTC consider only longitudinally aligned acceleration. We additionally develop and present the Second-Order Time-to-Collision Algorithm using Region-based search (STAR) to efficiently compute the proposed second-order TTC and overcome the current limitations of the existing built-in functions. The evaluation of the algorithm in terms of error and computation time is conducted through statistical analysis. Through numerical simulations and publicly accessible real-world trajectory datasets, we show that the proposed second-order TTC with non-static acceleration is superior at reflecting accurate collision times, especially when turning is involved.

Second-Order Time to Collision With Non-Static Acceleration

TL;DR

This work advances collision risk assessment by formulating a second-order time-to-collision metric that accommodates non-static acceleration and turning, addressing the limitations of traditional first-order TTC in curved trajectories. It introduces STAR, a region-based search algorithm that leverages circle and line trajectory intersections and RK45 integration to efficiently compute the earliest collision time, with theoretical guarantees that the search covers all viable collision regions. Through numerical simulations and real-world data (Argoverse 2), the authors show that 2D-TTC more accurately reflects collision timing in turning scenarios and yields fewer false alarms compared to 1D-TTC, while STAR provides substantial speedups over step-by-step simulations. The results suggest significant practical impact for ADAS and autonomous planning, particularly at intersections and during complex maneuvers, and point to future work in relaxing movement assumptions, incorporating vehicle geometry, and expanding real-world validation.

Abstract

We propose a second-order time to collision (TTC) considering non-static acceleration and turning with realistic assumptions. This is equivalent to considering that the steering wheel is held at a fixed angle with constant pressure on the gas or brake pedal and matches the well-known bicycle model. Past works that use acceleration to compute TTC consider only longitudinally aligned acceleration. We additionally develop and present the Second-Order Time-to-Collision Algorithm using Region-based search (STAR) to efficiently compute the proposed second-order TTC and overcome the current limitations of the existing built-in functions. The evaluation of the algorithm in terms of error and computation time is conducted through statistical analysis. Through numerical simulations and publicly accessible real-world trajectory datasets, we show that the proposed second-order TTC with non-static acceleration is superior at reflecting accurate collision times, especially when turning is involved.

Paper Structure

This paper contains 34 sections, 1 theorem, 49 equations, 41 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Second-Order TTC
  3. Notation
  4. Dynamical model
  5. First-Order TTC
  6. Second-Order TTC
  7. $\bullet$ Straight line.
  8. $\bullet$ Circular.
  9. Second-Order Time-to-Collision Algorithm using Region-based search (STAR)
  10. Algorithmic Structure
  11. Intersection Points Computation
  12. A. Circle-Circle Trajectories.
  13. B. Line-Line Trajectories.
  14. C. Circle-Line Trajectories.
  15. Search Region and Boundary Hitting Times
  16. ...and 19 more sections

Key Result

Theorem 3.1

Considering vehicles driving according to the dynamics E:car_dynamic. Let us fix the time $t_\circ$ and assume the trajectories of the vehicles on $t \ge t_\circ$ follow the dynamics E:2dttc_traj_linear or E:2dttc_traj_circle. If the collision time $\textcolor{black}{\mathcal{T}_S} < \infty$, define

Figures (41)

  • Figure 1: Representation of position and velocity of a vehicle. Vehicles' dimension is approximated by circles with dimension $\phi$.
  • Figure 2: A vehicle with diameter $\phi$ is illustrated. The two assumptions behind the Second-Order TTC computation are shown in blue.
  • Figure 3: In the case of linear trajectories, if the collision happens the distance between the vehicles will be decreasing along a linear trajectory.
  • Figure 4: Possible Collision Cases when one vehicle's estimated trajectory is linear while the other vehicle's estimated trajectory is circular.
  • Figure 5: Possible Collision Cases when both vehicles' estimated trajectories are circular.
  • ...and 36 more figures

Theorems & Definitions (5)

  • Theorem 3.1
  • proof
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5