ENOC 2011, 24-29 July 2011, Rome, Italy Dynamics of delayed car-following models: human vs. robotic drivers G´ abor Orosz * , Jeff Moehlis ** , Francesco Bullo ** and G´ abor St´ ep´ an † * Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109 USA ** Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106 USA † Department of Applied Mechanics, Budapest University of Technology, Budapest 1521, Hungary Summary . A general class of car-following models is studied with different driver reaction time configurations. In particular the stability of traffic flow is investigated in case of human-driven vehicles and computer-controlled (robotic) cars. It is shown that in both cases, time delays can change the frequency of arising oscillations and, consequently, the wavelength of the emerging traveling waves, leading to high-frequency / short-wavelength oscillations. Furthermore, interacting the with the nonlinearities in the system, time delays can make the dynamics excitable, such that waves may be triggered when the uniform flow is linearly stable. Finally, low-frequency and high-frequency oscillations can interact at the nonlinear level leading to very complex dynamics. Modeling car following Car following has been studied for six decades and there exist about a hundred different models which can reproduce traffic behavior for chosen sets of driver parameters. Still, no fundamental modeling principles has been established to guide scientists and engineers how to construct car-following models and what kind of qualitative requirements these models must satisfy. In order to change this tendency, a new direction has been identified in transportation science: instead of studying the dynamics of individual models quantitatively, the qualitative dynamics of classes of models shall be studied [1, 3]. Such investigations are useful for classifying the existing models and can also provide guidance for future models. Here, we investigate a class of models where each driver monitors the kinematic properties of the driver ahead and reacts to such stimuli with reaction time delays. These models can describe the behavior of human drivers fairly accurately, and may also be applied when programming Autonomous Cruise Control (ACC) devices (that consist of a radar sensor and an on board computer and can actuate the vehicle based on information collected about the environment). To obtain tractable models, usually identical vehicles are considered, so the acceleration of the ith vehicle is given by ˙ v i (t)= f ( h i (t - τ ), ˙ h i (t - σ),v i (t - κ) ) , (1) where the dot stands for differentiation with respect to time t, v i is the velocity of the ith vehicle, h i is the distance between the ith and the i +1st vehicles, that is, ˙ h i (t)= v i+1 (t) - v i (t). The delays τ, σ, κ represent driver reaction times to different stimuli. Depending on whether human-driven or computer-controlled vehicles are considered, the following approximations can be made. 1. Human driver setup: τ = σ> 0,κ =0. This setup represents that drivers react to the distance and to the velocity difference with (the same) delay but they are aware of their own velocity immediately [3]. 2. Robotic driver setup: τ = σ = κ> 0. Here the delay accounts for the time needed for sensing, computation and actuation in a computer controlled vehicle [2]. Note that these time delays differ from the desired time gap (a time distance the driver wishes to keeps from the leader) and from the relaxation time (which is related to the inertia of the vehicle); such parameters are incorporated in the nonlinear function f . In order to obtain a feasible model one must assume that the traffic system (1) possesses a one-parameter set of uniform flow equilibria: h i (t) ≡ h * , ˙ h i (t) ≡ 0 , v i (t) ≡ v * , (2) and that there exists a functional relationship between the equilibrium headway h * and the equilibrium velocity v * : 0= f (h * , 0,v * ) ⇒ v * = V (h * ) . (3) Here the monotonically increasing non-negative function V expresses that the more sparse traffic is, the faster drivers want to go and the quantity 1/V  (h * ) corresponds to the desired time gap. For simplicity, we assume periodic boundary conditions, that is, N vehicles are placed on a circular road of length L, which yields the algebraic equation ∑ N i=1 h i (t)= L. This determines the equilibrium headway h * = L/N and the equilibrium velocity v * through (3). Analytical results can be obtained for arbitrarily large N , but here we present the diagrams for N = 33 vehicles, which is small enough to keep the illustrations readable but is large enough to represent the limit N →∞ for realistic traffic scenarios.