Driver’s Anticipation and Memory Driving Car-Following Model

We developed a new car-following model to investigate the effects of driver anticipation and driver memory on traffic flow. The changes of headway, relative velocity, and driver memory to the vehicle in front are introduced as factors of driver’s anticipation behavior. Linear and nonlinear stability analyses are both applied to study the linear and nonlinear stability conditions of the new model. Through nonlinear analysis a modified Korteweg-de Vries (mKdV) equation was constructed to describe traffic flow near the traffic near the critical point. Numerical simulation shows that the stability of traffic flow can be effectively enhanced by the effect of driver anticipation and memory. The starting and breaking process of vehicles passing through the signalized intersection considering anticipation and driver memory are presented. All results demonstrate that the AMD model exhibit a greater stability as compared to existing car-following models.


Introduction
How, two successive vehicles interact with each other on a road, has been studied since the 1950s.Milestone carfollowing models include the rst version of linear models [1], the rst version of nonlinear models [2][3][4], the carfollowing model based on space headway [5], intelligent driver models (IDM) and its extensions considering both the optimal velocity and space headway [6][7][8][9][10], and some other models [11][12][13][14][15][16][17][18][19][20][21][22][23][24].Anticipation has been proposed as early as 2006 by Treiber et al. e Optimal Velocity (OV) model by Bando et al. [25] uses only a few parameters to reveal the complex dynamic characteristics of tra c ow, such as tra c ow instability, tra c congestion, and the formation of stop-and-go waves.It was created to overcome the problem of excessively high acceleration and unrealistic deceleration observed in Newell's model.However, due to its dependency on the following distance, the OV model and many of its derivatives produce not only high decelerations but also unrealistically high accelerations.is even applies to the AMD model, see Figure 1, dotted line (accelerations above 4 m/s 2 are unrealistic except for 4-wheel-drive sports or electric vehicles).
To overcome the dilemma of unrealistic deceleration in OV model, Helbing and Tilch [26] developed Generalized Force (GF) model in Table 1 by adding velocity di erence to the OV model.In the GF model, the optimal velocity function is speci ed by with the optimal parameter values: 1 = 6.75 m/s, 2 = 7.91 m/s, 1 = 0.13 m −1 , 2 = 1.57, = 5 m and the Heaviside function is unity when the velocity of the leading vehicle is lower than that of the following vehicle, and zero otherwise.Several other extensions of the OV model have been suggested to depict more characteristics of tra c ow by considering the relative velocity between the leading and following vehicles [27][28][29][30].Jiang et al. [31] put forward Full Velocity Di erences (FVD) model, which uses both negative and positive velocity di erences to handle unreasonable high acceleration rate and deceleration rate in GF model.Ge et al. [32] extend a car-following model by taking into account the relative velocity of leading and following Δv +1 and Δv on single lane highway and obtained two velocity di erence (TVD) model.ese models exclusively depend on the current states between the following vehicle and the leading vehicle at time t without taking into account driver anticipation and driver memory.Anticipation has been proposed as early as 2006 [33].
Zheng et al. [34] came up with an anticipation driving (AD) model based on the FVD model for analyzing the e ect of driver's response to upstream tra c stimuli and the stability of trafc ow.In the AD model, the optimal velocity is expressed as which is extended from the GF model where the optimal velocity is expressed as ( ) .Driver's visual angle was considered by Zhou [35] with an improved velocity model.Peng and Cheng [36] substituted anticipation optimal velocity with optimal velocity to develop an extended model based on FVD model, then analyzed the impact of the anticipation term on tra c ow stability.Tian et al. [37] introduced a velocity anticipation to construct an accident model for avoiding accident under special braking situation.Song et al. [38] improved an optimal velocity model by introducing tra c jerk and full velocity di erence.A multi-anticipative model was constructed by to describe the drivers' forecast impact on tra c ow. Kang et al. [33] considered the individual driving style and included forecast and response delay behavior of driver in the car-following model.ese models consider only driver anticipation but ignore driver memory.
Zhang [40] established driver's memory by considering human tendency to resist sudden changes of velocity and take into account the velocity in previous and next time.Put forward an extended lattice model of tra c ow with consideration of driver's memory.Yu and Shi [41] derived an improved car-following model to study the e ects of multiple velocity di erence changes with short-term driver memory on the stability and fuel economy of tra c ow based on FVD model as an e ective factor on driver's anticipation behavior.
Table 1 lists an optimal velocity family of car-following models.Based on the aforementioned review, it is clear that driver anticipation and driver memory have not been both taken into account in existing car-following models.Driver memory of previous tra c information may have substantial in uence on driver's car-following behavior.In this study we propose a new model, namely, the driver's anticipation and memory driving (AMD) car-following model to consider the e ect of both driver memory and driver anticipation.
e remainder of the paper is organized as follows.In Section 2, we present a new car-following model.In Section 3, linear stability analysis of the new model is conducted to study the existence and stability of traveling wave solutions using analytical method.In Section 4 nonlinear stability analysis is conducted.In Section 5 we carry out numerical simulations of the new car-following model for di erent scenarios.Concluding remarks are given in Section 6.

A New Car-Following Model
Based on the AD model, we propose the following anticipation-memory driving (AMD) model where is the forecast time step; Δv ( ) is the di erence between the estimated future space headway for a time horizon and the actual space headway; is a dimensionless parameter describing the sensitivity of driver memory for the previous tra c information; is the memory step with a unit in second; ( − ) is the optimal velocity at previous time − ; v ( − ) is the actual velocity at previous time − .e newly introduced term to the AD model is ( − ) −v ( − ) , a driver memory term expressed in terms of the proportional di erence between the optimal velocity and the actual velocity at previous time − .Although there may be di erent function forms and di erent in uencing factors to re ect the way that drivers' car-following behavior is a ected by driver's memory, here we only adopt a linear function for its simplicity.
Clearly, the driver memory term plays the role of feedback for the car-following behavior.Di erent drivers may exhibit di erent values of and the same driver may exhibit di erent Ge et al. [32] Anticipation-driving (AD) model Zheng et al. [34] Anticipation-memory-driving (AMD) model is paper ow becomes unstable.us the neutral stability curve is given by.
Consequently, the stable tra c ow is derived as As = 0, = 0, the stable condition of the AMD model degrades to the stable condition of the FVD model.
Figure 2 shows the neutral stable curves in the headway-sensitivity space (Δ , ) for the FVD model AD and the AMD model with = 0.1, = 1, = 0.5, = 0.1.e neutral stability curves for the FVD model, AD model and the AMD model are indicated in Figure 2. In this gure the existing apex denotes the critical point (ℎ c , c ) and ℎ is the critical headway.
e area below the neutral stability line shows the unstable region where density waves appear in tra c ow. e region above the neutral stability line corresponds to stable tra c.Clearly, the stable region of the AMD model is bigger than that of the FVD model and AD model because the critical points of the AMD model are signi cantly below those of the FVD and AD model. is plot indicates that the region of stability increases by considering leading vehicle's movement at the previous moment.

Numerical Study
In this section we conduct numerical analysis of tra c phenomenon using di erent car-following models.e focus is on how the driver's anticipation and driver's memory simultaneously a ect the following vehicle's velocity and acceleration.Here, we presume all vehicles being identical.In all scenarios = 0.41, = 0.1, = 0.5, 1 = 6.75 m/s, 2 = 7.91 m/s, 1 = 0.13 m −1 , 2 = 1.57, = 5 m Also, three hypothetical values of at di erent time.When = 0, = 0, = 0, the AMD model degrades to the OV model [25].When = 0, = 0, the AMD model degrades to the FVD model [31].When = 0, the AMD model degrades to the AD model [34].erefore, the OV model, FVD model and AD model are all special cases of the AMD model.

Linear Stability Analysis
First carried out the linear stability analysis of the GHR model proposed by Bando et al. [25].Applied an analytical analysis of the stability of the multi-regime car-following model through a numerical simulation.Details of the stability analysis method for a general car-following model are given by and reference there-in.In this section the linear stability analysis is applied to the new car-following model.We derive the stability conditions of the AMD model and investigate the conditions in uencing the long-wave length instabilities of tra c ow. Suppose that all vehicles are distributed with an identical space headway b (i.e., a constant) and move uniformly with the optimal velocity ( ). e steady-state solution of (1) with- out tra c jam can be written as where b is the headway de ned by = / , , and are the total numbers of cars and the road length, respectively.
Assume that ( ) stands for a small perturbation from the steady state 0 ( ). we have Substitute (2) and ( 3) into (1).Make a Taylor expansion of the variables.Neglect the higher order terms and ( ).We have and = Δ .According to the method of polar perturbation expanding ( ) into a Fourier series as an orthonormal set, i.e., ( ) ∝ ( + ) , (4) can be rewritten in terms of .Equation ( 5) is a polynomial and its root can be found by the method of zeros.Expanding = 1 ( ) + 2 ( ) 2 + ⋅ ⋅ ⋅ into (4), the rst and second order terms of are, respectively, deduced as follows.
e uniform ow will remain stable provided that 2 is a positive value.Otherwise, the uniformly steady-state tra c (2) 0 ( ) = + ( ) , that of the AD model [31].It means that the driver starts up earlier when the tra c light changes to green without having a higher acceleration due to considering space headway at next moment and previous vehicle motion.at is, vehicles behaving according to the AMD model accelerate more quickly than the AD model, but do not generate an unrealistic high acceleration observed in the OV model.

Tra c Flow
Evolution with an Initial Small Perturbation.In this subsection we investigate the e ect of an initial small perturbation to tra c ow according to the AMD model.Suppose = 100 vehicles are uniformly running on a circuit road with a length of = 1500 m under a periodic boundary condition as shown in Figure 5. e initial state is set in (9).We rst explore the impact of driver anticipation and memory on tra c ow stability with an initial small perturbation.
with the optimal parameter values: 1 = 6.75 m/s, 2 = 7.91 m/s, 1 = 0.13 m −1 , 2 = 1.57, = 5 m. Figure 6 shows the snapshots of velocity distributions of all vehicles simulated according to di erent car-following models (i.e., di erent driver behaviors) and di erent values of at the time of = 300 s, = 1000 s, = 2000 s, respectively.e OV model exhibits negative velocity at some moments, which is apparently unrealistic.
According to Figure 6, the homogeneous tra c ow evolves to congestion, which corresponds to stop-and-go trafc, as time increases from = 0 s to = 2000 s according to the OV, FVD, and AD models.On the contrary, according to the AMD model, the stop-and-go tra c does not appear until very late at = 2000 s and only becomes visible for AMD model with = 1 and = 2. Furthermore, the velocity of the OV, FVD and AD models uctuates much more widely than that of the AMD model for all times.It illustrates that the e ect of driver memory plays an important role in tra c ow stability.It provides a behavioral mechanism for modulating tra c ow uctuation.
In Figure 6(a) tra c is almost stable around the v 0 = 3.95 m/s at = 300 s, which is a little less than the initial velocity v 0 = 4.669 (m/s).It means that by considering space headway at next moments (i.e., driver anticipation) and previous tra c information (i.e., driver memory), drivers end up with lowering their initial velocity a little, to increase the distance to the adjacent vehicle to avoid crash.In Figures 6(b) and 6(c) the velocity uctuation decreases gradually with the increasing value of , for = 0.1 e amplitude of the AMD model is minimal when = 3. e result means that the deviation between the expected velocity and the actual velocity decreases when the memory time increases as feedback for the driver, and a driver can anticipate his expected velocity more realistically according to the real tra c situation with memory of his driving situation during 3 previous moments.erefore, the tra c stability is improved with the AMD model.( 9) values 0.1, 0.2, and 0.3 are used for parameter k and 1, 2, and 3 used for parameter m.In reality, the actual value of k can be obtained from calibration of observed tra c data.

e Start-Up Process.
To explore the simultaneous e ect of the driver's anticipation and memory on tra c behavior during the starting process, the same scenario as in Jiang et al. [31] is set up here in Figure 3. Suppose that eleven cars are waiting in front of a red tra c light and with identical space headway of 7.4 m.Each car's velocity is zero at time < 0. At time = 0 s the tra c light turns from red to green and all cars begin to move.First the leading car starts and then following cars move gradually.Consider a leading and a following pair of vehicles.e velocity of the leader is de ned by v leader = v 0 ( ). e follower dupli- cates the leader's velocity but with some delay time v follower = v 0 ( − ), in which is the delay time of vehicle motion.From the time delay of vehicle motion, we can further estimate the kinematic wave speed (i.e., disturbance propagation speed) at jam density, , which is de ned by = 7.4/ .
Based on the AMD model, we simulate the motion of eleven vehicles with the parameters = 1 s, = 0.1.e sim- ulation results are shown in Figures 4(a)-4(d) and Table 2.
A straightforward observation is that the following vehicles can duplicate the behavior of the velocities of the leading vehicles but with some delay time .
Del Castillo and Benitez [42] observed that the kinematic wave speed is between 17 and 23 km/h.Bando et al. [5] observed empirically that is of the order of 1 s.According to Table 2, these two parameters of the AMD model are, respectively, = 1.2 s and = 20.98 km/h, both falling into the typical range of empirical observation.
From Figure 4 and Table 2, we can further compare the delay time and the kinematic wave speed of the AMD model with those of the other car-following models.Clearly, and are respectively shorter and higher than those of the OV model, FVD model, and AD model.It means that the delay time can reduce and increase the start-up velocity by considering space headway at next moment and previous tra c information, which contributes to an increased transportation capacity and e ciency in intersections.
Figure 1 depicts the start-up acceleration process of the 1 st to the 11 th vehicles according to the AMD model.e leading vehicle, the rst following vehicle, and other following vehicles are displayed by a dashed red line, a dashed blue line, and solid blue lines, respectively.e acceleration of the leading and following vehicles falls into a range of the empirical acceleration (0, 4) m/s −2 observed by Helbing and Tilch [26].e maximum value of acceleration of our model is also lower than e motion of the vehicles eventually begin to transit from homogenous phase to stop-and-go phase, which form "hysteresis loops" a er a su ciently long time [31].Here, we To further investigate that driver anticipation and driver memory both can improve tra c stability; we study the evolution of small perturbation for the AD and AMD models.In Figure 7 the vehicle's velocity is snapshotted at = 1000 s and = 2000 s to explore how the forecast time and driver memory time a ect the tra c stability.
In Figures 7(a) and 7(b) the uctuation of the AD and AMD models drop with the increasing value of in both models.Figure 7(a) shows that with the increase of in the AMD model, tra c uctuation dies out at = 1000 s in comparison with the AD model.Figure 7(b) exhibits that the amplitude of vehicles changes smoothly around the initial velocity v 0 = 4.669 (m/s) by increasing and the velocity of vehicles maintain near v 0 = 4.669 (m/s) at = 2000 s.From Figure 7(b) we can nd that the stability of tra c ow improves most when both driver anticipation and driver memory are considered.becomes stationary a er enough time, the motion of 30th vehicle begins to form the "hysteresis loop".It is obvious that the hysteresis loop of the AMD model is much smaller than chose the 30th vehicle as the subject vehicle to compare the hysteresis loop obtained from the AMD model and the AD model.As can be seen from Figure 8, when the congestion  which is 15 m according to Jiang et al. [31].It means that it takes a longer time to observe the stop-and-go tra c in the AMD model than in the AD model.

Braking Process.
In this subsection tra c arrival process on a single lane roadway with a tra c light using the AMD that of the AD model. is indicates the stability of tra c ow can greatly enhance by considering the anticipation driving and driving memory.So the AMD model is superior in terms of tra c ow stability to the FVD model.
It can be seen from Figure 8 that the minimum headways (Point ) of both models are smaller than the safe headway,  those of AD model and FVD model.It shows that by considering driver's negative velocities and excessive acceleration won't appear, and also a driver can start braking process faster and gentler to reduce the velocity, which contributes to the improved safety and fuel consumption.

Concluding Remarks
Existing car-following model in the literature lack in considering driver's anticipation and driver memory in the same model.For this reason we developed the AMD model and conducted numerical analysis to investigate the evolution of small perturbation in tra c ow according to the AMD model.e results show that driver memory signicantly a ects the evolution of small perturbation in tra c ow. Considering driver memory, improves the stability of tra c ow. Considering driver memory can increase the safety and the e ciency of tra c operation by optimizing tra c light time at signalized intersections.e results of starting and braking process demonstrate that the AMD model can more successfully anticipate the delay time of vehicle motion and the kinematic wave velocity at jam density.model is carried out to explore the in uences of driving anticipation and driving memory on the following vehicles' velocity and acceleration.e hypothetical initial conditions are given as follows.When < 0 the tra c light is green and 11 vehicles are running with a uniform velocity of 4.66 (m/s).All vehicles are distributed uniformly with headway of 15 m. e distance between the platoon leading vehicle and the stopping line is 10 m. e red light is assumed to be a virtual standing vehicle of extension zero at the stopping line as noted in Treiber and Kesting [10].At time = 0 the tra c light shi s to red and the platoon-leading vehicle immediately breaks, and the following vehicles duplicate the leading vehicle's velocity with a delay time and begins to slow down gradually.All vehicles nally stop in a column behind the stopping line.

Abbreviations
e velocities' evolutions of 11 vehicles during the braking process when the platoon leader start braking ( = 0) till that time all vehicles stop in a column behind the stop line simulated by the FVD, AD, and the AMD models are illustrated in Figures 9(a)-9(c).All following vehicles can duplicate the leading vehicles' velocities but with some delay, and nally stop behind the stopping line.e delays of vehicles' motion simulated by the AMD, AD model, and FVD model are 1.27 s, 1.34 s, and 1.4s respectively.Clearly, the AMD model corresponds to a shorter delay time than the AD model and FVD model.
Figure 10 depicts the simulation of accelerations' evolutions of the 4th and 8th vehicles due to the tra c signal using the FVD model, AD model, and the AMD model, respectively.e acceleration of the leading and following vehicles falls into the ranges of empirical deceleration (−3 m/s 2 , 4 m/s 2 ), which was observed by Helbing and Tilch [26] from real driving behaviors.From Figure 10 we can see that the distance between the AMD model's curves and FVD model become larger when decreasing the number of vehicles.From Figure 10 it can be found that the curves of the AMD model are lower than for

F 1 :T 1 :
Acceleration of unobstructed leading vehicle and its following vehicles in AMD model.Optimal Velocity family of car-following models as a special case of AMD model.
neutral stability line in the headway-sensitivity space.

F 3 :
A platoon of eleven vehicles proceed as the tra c light turns from red to green.

F 4 : 2 :F 5 :
Motion of eleven vehicles starting up from a red tra c signal.(a) e OV model, (b) e FVD model, (c) e AD model, and (d) e AMD model.T Delay times of vehicle motions from a tra c signal and kinematic wave speed at jam density in di erent models.A platoon of one hundred vehicles running a circular road with length of 1500 m.

F 6 :
new model m = 1 Our new model m = 2 Our new model m = 3 new model m = 1 Our new model m = 2 Our new model m = 3 (b) OV model FVD model AD model Our new model m = 1 Our new model m = 2 Our new model m = Snapshots of velocities of all vehicles according to di erent car-following models in = 0.1 at di erent times. (a) t = 30 0s, (b) t = 1000 s, and (c) t = 2000 s.

F 7 :
model k = 0.1 AD model k = 0.2 AD model k = 0.3 Our new model k = 0.1 Our new model k = 0.2 Our new model k = 0.3 Snapshots of velocities of 100 vehicles simulated by AD and AMD models with di erent values of ( 1 in the AMD model).

F 8 :
Hysteresis loops for FVD model and AMD model with = 1, = 0.1. (a) e AD model, and (b) e AMD model.

F 9 :
Motions of 11 vehicles during the braking process according to the FVD, AD, and AMD models.(a) FVD model, (b) AD model, and (c) AMD model.
the th vehicle at time ; v ( ): e velocity of the th vehicle at time ; v ( − ): e velocity of the th vehicle at previous time − ; ( ): e space headway between vehicles and + 1 at time , ( ) = Δ ( ) = +1 ( ) − ( ); ( − ):e space headway between vehicles and + 1 at time − ; Δv ( ):e relative velocity between following and leading vehicle, Δv ( ) = v +1 ( ) − v ( ); Δv ( ):e di erence between the estimated future space headway for a time horizon and the actual space headway; (⋅): e optimal velocity function; : e sensitivity of the driver given by the inverse of the delay time of vehicle motion , namely, driver memory for the previous tra c information; : e delay time of vehicle motion;j : e kinematic wave speed (i.e., disturbance propagation speed) at jam density, equal to the quotient of the headway divided by the delay time of vehicle motion ; j = ℎ/ ; : e propagation speed of the kink wave.

F 10 :
Acceleration evolutions of single vehicle during braking process.