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Provably safe and human-like car-following behaviors: Part 1. Analysis of phases and dynamics in standard models

Wen-Long Jin

TL;DR

The paper addresses the challenge of obtaining provably safe and human-like car-following by formulating a multi-phase dynamical-systems framework and systematically applying it to well-known rules. It derives and analyzes zeroth-, first-, and second-order principles, including the safe-stopping distance $B(t)=v(t)\tau'+\frac{v(t)^2}{2\beta}$ and the steady-state FD $v=V(k)$ with $\kappa=1/\zeta$, illustrating how Newell's, IDM, and Gipps satisfy some but not all criteria. The analysis reveals critical limitations: BA-/BDA-Newell can violate spacing or forward-travel under bounded acceleration/deceleration; IDM can exhibit backward travel and overly long braking; Gipps can be ill-defined for certain spacings and yield FD inconsistencies with safe-stopping assumptions. These insights justify developing a new, parsimonious multi-phase, projection-based car-following model (Part 2) that anticipates braking and enforces safety while preserving human-like behavior for mixed-traffic and automated driving scenarios.

Abstract

Trajectory planning is essential for ensuring safe driving in the face of uncertainties related to communication, sensing, and dynamic factors such as weather, road conditions, policies, and other road users. Existing car-following models often lack rigorous safety proofs and the ability to replicate human-like driving behaviors consistently. This article applies multi-phase dynamical systems analysis to well-known car-following models to highlight the characteristics and limitations of existing approaches. We begin by formulating fundamental principles for safe and human-like car-following behaviors, which include zeroth-order principles for comfort and minimum jam spacings, first-order principles for speeds and time gaps, and second-order principles for comfort acceleration/deceleration bounds as well as braking profiles. From a set of these zeroth- and first-order principles, we derive Newell's simplified car-following model. Subsequently, we analyze phases within the speed-spacing plane for the stationary lead-vehicle problem in Newell's model and its extensions, which incorporate both bounded acceleration and deceleration. We then analyze the performance of the Intelligent Driver Model and the Gipps model. Through this analysis, we highlight the limitations of these models with respect to some of the aforementioned principles. Numerical simulations and empirical observations validate the theoretical insights. Finally, we discuss future research directions to further integrate safety, human-like behaviors, and vehicular automation in car-following models, which are addressed in Part 2 of this study \citep{jin2025WA20-02_Part2}, where we develop a novel multi-phase projection-based car-following model that addresses the limitations identified here.

Provably safe and human-like car-following behaviors: Part 1. Analysis of phases and dynamics in standard models

TL;DR

The paper addresses the challenge of obtaining provably safe and human-like car-following by formulating a multi-phase dynamical-systems framework and systematically applying it to well-known rules. It derives and analyzes zeroth-, first-, and second-order principles, including the safe-stopping distance and the steady-state FD with , illustrating how Newell's, IDM, and Gipps satisfy some but not all criteria. The analysis reveals critical limitations: BA-/BDA-Newell can violate spacing or forward-travel under bounded acceleration/deceleration; IDM can exhibit backward travel and overly long braking; Gipps can be ill-defined for certain spacings and yield FD inconsistencies with safe-stopping assumptions. These insights justify developing a new, parsimonious multi-phase, projection-based car-following model (Part 2) that anticipates braking and enforces safety while preserving human-like behavior for mixed-traffic and automated driving scenarios.

Abstract

Trajectory planning is essential for ensuring safe driving in the face of uncertainties related to communication, sensing, and dynamic factors such as weather, road conditions, policies, and other road users. Existing car-following models often lack rigorous safety proofs and the ability to replicate human-like driving behaviors consistently. This article applies multi-phase dynamical systems analysis to well-known car-following models to highlight the characteristics and limitations of existing approaches. We begin by formulating fundamental principles for safe and human-like car-following behaviors, which include zeroth-order principles for comfort and minimum jam spacings, first-order principles for speeds and time gaps, and second-order principles for comfort acceleration/deceleration bounds as well as braking profiles. From a set of these zeroth- and first-order principles, we derive Newell's simplified car-following model. Subsequently, we analyze phases within the speed-spacing plane for the stationary lead-vehicle problem in Newell's model and its extensions, which incorporate both bounded acceleration and deceleration. We then analyze the performance of the Intelligent Driver Model and the Gipps model. Through this analysis, we highlight the limitations of these models with respect to some of the aforementioned principles. Numerical simulations and empirical observations validate the theoretical insights. Finally, we discuss future research directions to further integrate safety, human-like behaviors, and vehicular automation in car-following models, which are addressed in Part 2 of this study \citep{jin2025WA20-02_Part2}, where we develop a novel multi-phase projection-based car-following model that addresses the limitations identified here.
Paper Structure (19 sections, 4 theorems, 33 equations, 7 figures, 1 table)

This paper contains 19 sections, 4 theorems, 33 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definitions of variables and behavioral principles
  3. Core variables and kinematics
  4. Behavioral principles
  5. Collision-free principles (Zeroth-order)
  6. Operational constraints (First-order)
  7. Acceleration constraints (Second-order)
  8. Driver objectives and macroscopic properties
  9. Analysis of Newell's simplified car-following model and its extensions
  10. Newell's simplified car-following model
  11. Extension with bounded acceleration
  12. Extension with bounded acceleration and deceleration
  13. Analysis of the Intelligent Driver Model
  14. Analytical results for the stationary lead-vehicle problem
  15. A numerical example
  16. ...and 4 more sections

Key Result

Lemma 3.1

Given initial conditions satisfying the comfort jam spacing and forward traveling principles for all vehicles, i.e., $z(0)\geq \zeta$ and $v(0)\geq 0$, Newell's simplified car-following model with $\epsilon\leq \tau$ adheres to the comfort jam spacing and forward traveling principles at any later ti

Figures (7)

  • Figure 1: Variables for car-following rules
  • Figure 2: Four phases in the BA-Newell model
  • Figure 3: An example of solutions of the BA-Newell model for the stationary lead-vehicle problem
  • Figure 4: Five phases in the BDA-Newell model in (\ref{['BDA-Newell']})
  • Figure 5: An example of bounded deceleration solutions of the BDA-Newell model in (\ref{['BDA-Newell']})
  • ...and 2 more figures

Theorems & Definitions (5)

  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Definition 3.5: Stationary lead-vehicle problem