Delayed Car-following Dynamics for Human and Robotic Drivers

A general class of car-following models is analyzed where the longitudinal acceleration of a vehicle is determined by a nonlinear function of the distance to the vehicle in front, their velocity difference, and the vehicle’s own velocity. The driver’s response to these stimuli includes the driver reaction time that appears as a time delay in governing differential equations. The linear stability of the uniform flow is analyzed for human-driven and computer-controlled (robotic) vehicles. It is shown that the stability conditions are equivalent when considering ring-road and platoon configurations. It is proven that time delays result in novel high-frequency oscillations that manifest themselves as short-wavelength traveling waves. The theoretical results are illustrated using an optimal velocity model where the nonlinear behavior is also revealed by numerical simulations. The results may lead to better understanding of multi-vehicle dynamics and allow one to design cooperative autonomous cruise control algorithms. INTRODUCTION Vehicular traffic is one of the most complex interconnected dynamical systems created by mankind. Each vehicle is controlled by a human operator (sometimes assisted by an on-board computer) who senses the environment (i.e., the motion of other vehicles, traffic signals and road conditions), makes decisions ∗Address all correspondence to this author. based on the collected information and actuates the car accordingly. This process takes a finite amount time, known as the driver reaction time. The emergent dynamics of a traffic system, i.e., the time evolution of traffic patterns over large time and length scales, is determined by these delayed, nonlinear driverto-driver and driver-to-infrastructure interactions. In this paper we focus our attention on the former one and study the corresponding car-following dynamics. By now, a vast number of different car-following models have been constructed [1–3], but still no first principles have been established to guide the modeling procedure (if such principles exist at all). In many cases, authors have claimed that the developed model described traffic better than models prior to that point, and such claims were often justified by fitting the models to empirical data. This approach may easily lead to models capturing, but also missing, some essential characteristics and a model fit to one set of data may no longer be predictive when extrapolated to new sets of data. We believe that another way to conduct research in traffic can be by studying general classes of models and classifying their qualitative dynamical features when varying model parameters. Of particular interest is the stability of the uniform traffic flow in which vehicles follow each other with the same velocity because this state is beneficial for traffic safety and throughput. The approaches taken to analyze this state are very different in the physics, applied mathematics and control engineering communities. To bridge the gap between these approaches, here we 1 Copyright c © 2011 by ASME n n + 1 n + 2 vn vn+1 vn+2 xn xn+1 xn+2 l l l hn hn+1 FIGURE 1. SEQUENCE OF CARS ON A SINGLE LANE SHOWING VEHICLES’ POSITIONS, VELOCITIES, AND HEADWAYS. calculate the flow stability by two different approaches and show that they lead to the same result for the considered general class of delayed car-following models when the number of vehicles is sufficiently large. (Such proof was presented for non-delayed models in the appendix of [3].) Both methods provide valuable insights into the dynamics underlying jam formation. In particular, apart from the location of the stability boundaries in parameter space, the frequencies of the arising oscillations and the wavelength of the developing traveling waves can be determined. Although, this paper focuses on the linear stability of the uniform flow, we emphasize that car-following models are inherently nonlinear due to a fundamental speed-headway (or an equivalent flux-density) relation built into them. Since the detailed bifurcation analysis of car-following models goes beyond the scope of this paper we demonstrate the implications of the linear stability analysis on the nonlinear dynamics by numerical simulations. MODELING CAR-FOLLOWING In car-following models each driver-vehicle system is modelled by a set of differential equations that are coupled to other driver-vehicle systems based on the driver’s responses to external stimuli. Fig. 1 shows a queue of vehicles on a single lane where vehicles have equal length l. At time t, the position of the front bumper of the n-th car is denoted by xn(t), its velocity is vn(t) = ẋn(t) and the bumper-to-bumper distance to the vehicle in front (called the headway) is hn(t). It can be read from the figure that hn(t) = xn+1(t)− xn(t)− l, which results in ḣn(t) = vn+1(t)− vn(t) , (1) when differentiated with respect to time t. To complete the model, this equation has to be supplemented with a car-following rule, that is, the velocity or the acceleration has to be given as the function of stimuli that are usually the distance hn, the velocity difference ḣn and the vehicle’s own velocity vn. To represent the fact that the longitudinal dynamics of automobiles are controlled by varying the engine torque we choose a class of models where the acceleration of vehicles is prescribed: v̇n(t) = f ( hn(t − τ), ḣn(t −σ),vn(t −κ) ) . (2) For simplicity, drivers with identical characteristics are considered. The delays τ,σ ,κ represent driver reaction times to different stimuli (dead times required to process information and initiate action). To make the models more tractable, simple relations may be assumed between the different delays. There are three simplifications commonly used in the literature: 1. Zero reaction times: τ = σ = κ = 0. This is usually justified by saying that dynamic models (2) may reproduce uniform flow as well as traveling waves for zero reaction time by varying some other characteristic times [4]. 2. ‘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 [5, 6]. 3. ‘Robotic driver setup’: τ = σ = κ > 0. This setup is mainly used in the adaptive/automatic/autonomous cruise control (ACC) literature. The delay accounts for the time needed for sensing, computation and actuation in computer controlled vehicles [7, 8]. Many other setups are also possible, for example, one may account for human memory effects by using distributed delays as in [9]. We remark that in the first case the system (1,2) consists of ordinary differential equations (ODEs) where the initial conditions are given by hn(0),vn(0). In the latter cases, systems of delay differential equations (DDEs) are obtained where hn(t),vn(t), t ∈ [−τ,0] must be specified as initial conditions. Determining the general properties of the multi-variable nonlinear function f in (2) is a difficult task. However, the model must be able to reproduce the uniform flow where both the velocities and the headways are time independent: hn(t) ≡ h∗ , ḣn(t) ≡ 0 , vn(t) ≡ v∗ . (3) 2 Copyright c © 2011 by ASME We also assume a functional relationship between the equilibrium headway h∗ and the equilibrium velocity v∗, that is, 0 = f (h∗,0,v∗) ⇒ v∗ = V (h∗) ⇔ h∗ = V−1(v∗) , (4) where V is assumed to have the following properties: 1. V is continuous and monotonically increasing (the more sparse traffic is, the faster drivers want to travel). 2. V (h) ≡ 0 for h ≤ hstop (in very dense traffic, drivers intend to stop). 3. V (h) ≡ vmax for large h (in very sparse traffic, drivers intend to drive with maximum speed – often called free flow). This function is often called the range policy in the control literature [10]. Two examples are shown on the top panels in Fig. 2. The function on the left represents that between stopping and free-flow conditions, drivers intend to keep a constant time gap Tgap (also called time-headway), while the function on top right shows a scenario when the intended time gap changes with the distance/velocity. One may define the equilibrium density and the flux as ρ∗ = 1 h∗ + l , q∗ = ρ∗ v∗ = ρ∗V (1/ρ∗− l) := Q(ρ∗) . (5) This way the equilibrium speed-headway diagrams can be transformed into the equilibrium flux-density (fundamental) diagrams displayed at the bottom of Fig. 2. The rising part of the fundamental diagrams (that represents free flow) can be observed in empirical traffic data, collected by loop detectors, while usually a cloud appears instead of the decaying part (indicating unstable equilibria) [3]. Nevertheless the triangular fundamental diagram is often used for designing flow control strategies for ramp metering and variable speed limit control [11]. The equilibrium speed-density function (range policy) may be explicitly built into car-following models. The corresponding so-called optimal velocity (OV) model [4, 6, 10, 12] can be formulated as f (h, ḣ,v) = 1 T ( V (h)− v ) +bḣ . (6) The first term corresponds to relaxation to a density dependent optimal velocity given by the increasing OV function V with a relaxation time T , while in the second, relative-velocity term we have b ≥ 0. Despite its simplicity, the model (6) can reproduce qualitatively almost all kinds of traffic behavior. 0 hstop h ∗ v∗ vmax