[Paper Review] Resolving Collisions for the Gipps Car-Following Model

Leonhard Lücken|arXiv (Cornell University)|Feb 13, 2019

Traffic control and management10 references4 citations

TL;DR

This paper resolves collisions in the Gipps car-following model by extending its safety principle to account for heterogeneous braking capabilities, introducing a tangential calculus approach that ensures collision-free behavior when followers can brake harder than leaders. The key contribution is a modified speed-headway function that prevents trajectory intersections, with numerical results showing enhanced stability in heterogeneous traffic flows.

ABSTRACT

The Gipps car-following model is a widely used tool for studying and simulation traffic dynamics. Despite its popularity an often disregarded property is that under heterogeneous parametrization on the individual vehicles in the traffic flow, the model may produce collisions. This stands in crude contrast to the principle, from which the model was derived: drive as fast as possible while guaranteeing a safe headway in case that the leading vehicle starts braking hard. Indeed, Gipps proof for the model being collision-free only holds for ensembles of identical vehicles. In this work we examine the circumstances leading to collisions in heterogeneous ensembles and propose a natural model extension, which conveys the original models principles to situations, where collisions occur. For these cases we present analytical and numerical results on the stability of the equilibrium flow.

Motivation & Objective

• To identify the root cause of collisions in the Gipps car-following model under heterogeneous vehicle parameters, particularly when followers have higher braking capacity than leaders.
• To address the failure of the original Gipps safety principle—based on non-intersecting hypothetical trajectories—when such intersections occur due to differing deceleration rates.
• To propose a natural extension of the Gipps model that maintains its core principle of driving at the maximum safe speed while preventing collisions in heterogeneous ensembles.
• To analyze the stability of equilibrium flow under the extended model, especially in cases where the tangential regime becomes active due to parameter heterogeneity.
• To evaluate whether heterogeneity in braking rates stabilizes traffic flow, contrary to the instability typically associated with such models.

Proposed method

• Reformulate the Gipps safety principle to require non-negative distance and zero relative speed at the point of tangency between hypothetical trajectories of leader and follower, rather than just at final stop.
• Derive a new safe acceleration function based on the condition that the distance function $ g(t) $ between leader and follower reaches zero with zero derivative at some future time $ t_{\parallel} $, ensuring no trajectory crossing.
• Introduce a three-phase trajectory model (acceleration, constant speed, deceleration) as in the original Gipps, but apply the tangency condition to determine the maximum safe speed.
• Use analytical and numerical methods to study equilibrium flow stability under the extended model, comparing identical and heterogeneous vehicle ensembles.
• Conduct numerical experiments with varying braking rate heterogeneity $ \Delta B $, measuring speed deviation $ \delta $ from equilibrium to assess stability.
• Compare the proposed extension with existing heuristic fixes (e.g., SUMO’s max-braking rule), highlighting its principled derivation and avoidance of arbitrary parameter constraints.

Experimental results

Research questions

• RQ1 Why does the original Gipps model produce collisions in heterogeneous traffic, despite being derived from a safety principle?
• RQ2 What specific condition in the trajectory geometry leads to collisions when follower braking capacity exceeds that of the leader?
• RQ3 How can the Gipps safety principle be extended to prevent such collisions while preserving its original intent of maximizing safe speed?
• RQ4 What is the stability behavior of equilibrium flow under the extended model, particularly in the tangential regime?
• RQ5 Does increasing heterogeneity in braking rates improve or worsen the stability of equilibrium traffic flow?

Key findings

• Collisions in the original Gipps model arise when the follower’s hypothetical trajectory intersects the leader’s due to higher braking capacity, violating the assumption of decoupled trajectories.
• The proposed extension resolves collisions by enforcing a tangency condition between trajectories, ensuring $ g(t_{\parallel}) = g'(t_{\parallel}) = 0 $, which prevents intersection and guarantees safe stopping.
• The extended model introduces a discontinuous speed-headway relation due to the tangential regime, which is a known limitation but reflects a physically meaningful boundary condition.
• Analytical analysis shows that the tangential regime leads to unstable equilibrium flow, as it requires a large underestimation $ \Delta\hat{B} $ of the leader’s braking rate to activate.
• Numerical experiments indicate that for realistic heterogeneous ensembles without misestimation, increased braking rate variability $ \Delta B $ enhances flow stability, suggesting a stabilizing role of heterogeneity.
• The model extension preserves the original Gipps principle of maximizing safe speed while avoiding arbitrary parameter restrictions, offering a more principled alternative to heuristic fixes like using the maximum of $ B_i $ and $ \hat{B}_{i+1} $.