Collision-free nonuniform dynamics within continuous optimal velocity models

Optimal velocity (OV) car-following models give with few parameters stable stop-and -go waves propagating like in empirical data. Unfortunately, classical OV models locally oscillate with vehicles colliding and moving backward. In order to solve this problem, the models have to be completed with additional parameters. This leads to an increase of the complexity. In this paper, a new OV model with no additional parameters is defined. For any value of the inputs, the model is intrinsically asymmetric and collision-free. This is achieved by using a first-order ordinary model with two predecessors in interaction, instead of the usual inertial delayed first-order or second-order models coupled with the predecessor. The model has stable uniform solutions as well as various stable stop-and -go patterns with bimodal distribution of the speed. As observable in real data, the modal speed values in congested states are not restricted to the free flow speed and zero. They depend on the form of the OV function. Properties of linear, concave, convex, or sigmoid speed functions are explored with no limitation due to collisions.

Figure 1

Spacing variables: Vehicle length, $\ell$; position, $x$; and distance spacing, $\Delta x$. The vehicle $n$ collides with the vehicle $n+1$ if the distance gap $\Delta x_n\left(t\right)-\ell$ is negative.

Figure 2

Trajectories of 22 vehicles on a ring with the collision-free model and bounded linear OV function.

Figure 3

Vehicle speeds according to the mean spacing in the stationary state for the bounded linear speed function (16) according to the size of the ring ($L=505$, 1005, and 2005 m). The middle curve is the mean value, while upper and lower curves give the modal values. The vertical dotted lines correspond to the linear stability conditions.

Figure 4

Vehicle speeds according to the mean spacing in the stationary state for the nonlinear speed functions (17), (18), and (19). The middle curve is the mean value, while the upper and lower curves give the modal values. The vertical dotted lines correspond to the linear stability conditions. $L=2005$ m.

Figure 5

Normalized histograms of the vehicles' speeds in the stationary state with varying $d$. From top to bottom: convex, concave, and sigmoid optimal speed functions.