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Provably safe and human-like car-following behaviors: Part 2. A parsimonious multi-phase model with projected braking

Wen-Long Jin

TL;DR

Provable safety and human-likeness in car-following under real-world uncertainties is addressed by introducing a parsimonious multi-phase projection-based framework. The approach merges an extended Newell nominal-driving component with a projection-based braking control law to enforce bounded acceleration and deceleration and to anticipate leader braking via projected trajectories. The authors formalize projected braking for both leader and follower, define projection-based safety phases, and prove collision-free operation along with a bounded $a(t)$ and nonnegative $v(t)$ under reasonable initial conditions, supported by analysis in the speed-spacing phase plane. Numerical validation on the stationary lead-vehicle problem demonstrates human-like braking profiles and safe stopping distances, and the work discusses extensions to moving leaders, empirical calibration, and jerk constraints for practical deployment.

Abstract

Ensuring safe and human-like trajectory planning for automated vehicles amidst real-world uncertainties remains a critical challenge. While existing car-following models often struggle to consistently provide rigorous safety proofs alongside human-like acceleration and deceleration patterns, we introduce a novel multi-phase projection-based car-following model. This model is designed to balance safety and performance by incorporating bounded acceleration and deceleration rates while emulating key human driving principles. Building upon a foundation of fundamental driving principles and a multi-phase dynamical systems analysis (detailed in Part 1 of this study \citep{jin2025WA20-02_Part1}), we first highlight the limitations of extending standard models like Newell's with simple bounded deceleration. Inspired by human drivers' anticipatory behavior, we mathematically define and analyze projected braking profiles for both leader and follower vehicles, establishing safety criteria and new phase definitions based on the projected braking lead-vehicle problem. The proposed parsimonious model combines an extended Newell's model for nominal driving with a new control law for scenarios requiring projected braking. Using speed-spacing phase plane analysis, we provide rigorous mathematical proofs of the model's adherence to defined safe and human-like driving principles, including collision-free operation, bounded deceleration, and acceptable safe stopping distance, under reasonable initial conditions. Numerical simulations validate the model's superior performance in achieving both safety and human-like braking profiles for the stationary lead-vehicle problem. Finally, we discuss the model's implications and future research directions.

Provably safe and human-like car-following behaviors: Part 2. A parsimonious multi-phase model with projected braking

TL;DR

Provable safety and human-likeness in car-following under real-world uncertainties is addressed by introducing a parsimonious multi-phase projection-based framework. The approach merges an extended Newell nominal-driving component with a projection-based braking control law to enforce bounded acceleration and deceleration and to anticipate leader braking via projected trajectories. The authors formalize projected braking for both leader and follower, define projection-based safety phases, and prove collision-free operation along with a bounded and nonnegative under reasonable initial conditions, supported by analysis in the speed-spacing phase plane. Numerical validation on the stationary lead-vehicle problem demonstrates human-like braking profiles and safe stopping distances, and the work discusses extensions to moving leaders, empirical calibration, and jerk constraints for practical deployment.

Abstract

Ensuring safe and human-like trajectory planning for automated vehicles amidst real-world uncertainties remains a critical challenge. While existing car-following models often struggle to consistently provide rigorous safety proofs alongside human-like acceleration and deceleration patterns, we introduce a novel multi-phase projection-based car-following model. This model is designed to balance safety and performance by incorporating bounded acceleration and deceleration rates while emulating key human driving principles. Building upon a foundation of fundamental driving principles and a multi-phase dynamical systems analysis (detailed in Part 1 of this study \citep{jin2025WA20-02_Part1}), we first highlight the limitations of extending standard models like Newell's with simple bounded deceleration. Inspired by human drivers' anticipatory behavior, we mathematically define and analyze projected braking profiles for both leader and follower vehicles, establishing safety criteria and new phase definitions based on the projected braking lead-vehicle problem. The proposed parsimonious model combines an extended Newell's model for nominal driving with a new control law for scenarios requiring projected braking. Using speed-spacing phase plane analysis, we provide rigorous mathematical proofs of the model's adherence to defined safe and human-like driving principles, including collision-free operation, bounded deceleration, and acceptable safe stopping distance, under reasonable initial conditions. Numerical simulations validate the model's superior performance in achieving both safety and human-like braking profiles for the stationary lead-vehicle problem. Finally, we discuss the model's implications and future research directions.
Paper Structure (14 sections, 7 theorems, 38 equations, 4 figures, 1 table)

This paper contains 14 sections, 7 theorems, 38 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Brief review of variables, behavioral principles, and extended Newell's car-following model
  3. Recap of core variables and behavioral principles
  4. The extended Newell's car-following model as a nominal driving component
  5. Foundation of projection in car-following: problems, concepts, and definitions
  6. Projected braking lead-vehicle problem
  7. Projected safety and projection-based phases
  8. A multi-phase projection-based car-following model with assured safety and bounded deceleration
  9. A multi-phase projection-based car-following model
  10. Behavioral analysis
  11. Application to stationary lead-vehicle problem
  12. Analytical solutions of the stationary lead-vehicle problem
  13. A numerical example
  14. Conclusion

Key Result

Lemma 3.1

The projected spacing at time $t\geq t'$ attains its minimum either at time $t$ or when both vehicles come to a complete stop under the following situations. In other words, if the conditions defined in (projected-spacing-lemma-conditions) are satisfied, we have (for $t'\geq t$) In contrast, if the follower decelerates more rapidly, is not slower at $t$, and stops earlier than the leader; i.e.,

Figures (4)

  • Figure 1: Projected braking lead-vehicle problem
  • Figure 2: Projection-based phases in the $(v(t),z(t))$ plane
  • Figure 3: Extended triangular fundamental diagram for the multi-phase projection-based car-following model (\ref{['MPP-CFM']})
  • Figure 4: Replication of Figure 2 in treiber2000congested with the multi-phase projection-based car-following model

Theorems & Definitions (7)

  • Lemma 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Theorem 5.1