Research on V2V Time Headway Strategy Considering Multi-Front Vehicles Based on Artificial Potential Field

Abstract

Within the Internet of Vehicles, the driver’s conduct is more affected by the information about the vehicles in front of them. To meet the driver’s demand for the design of the safety vehicle distance of the adaptive cruise control system in the Vehicle-to-Vehicle (V2V) environment, this paper employs the artificial potential field (APF) theory to enhance the vehicle time headway strategy and constructs a mixed force function in the global force field with three front vehicles as the center of the three potential fields. Taking into account the inadequate control effect of conventional APF in a dynamic environment, this paper describes the relative kinematic characteristics of the vehicles in front and the host vehicle as parameters and obtains the improved APF. The improved strategy based on APF enhances the forward-looking and anti-interference ability of the distance strategy. Also, the strategy presented in this paper can achieve time headway control under all operating conditions and prevent acceleration fluctuations due to switching between various control strategies. For collaborative simulation, this research uses MATLAB/Simulink and Carsim software. The simulation is conducted under three common traffic scenarios, and it is determined that the improved control method results in smoother changes in vehicle speed, acceleration, and car-following distance, hence improving driving and riding comfort.

1. Introduction

The Internet of Vehicles (IOV) system consists of roadside units, vehicle onboard units, control centers, and other basic information interaction modes, including Vehicle-to-Infrastructure (V2I), Vehicle-to-Vehicle (V2V), and Vehicle-to-Pedestrian (V2P), to enable information communication among pedestrians, vehicles, and roads [1,2,3]. This technology is currently highly esteemed since it enables the collection of more complete and specific traffic data.

Cruise control systems [4,5], as one of the typical applications of advanced driving assistance systems (ADAS), can be classified into adaptive cruise control (ACC) systems and cooperative adaptive cruise control (CACC) systems. The ACC system senses the driving environment and self-states using sensors such as onboard radar and camera. The control unit supports the driver in controlling the vehicle by determining the input of the anti-lock braking system and engine control system, thus enhancing driving safety, economy, and comfort. The CACC system is based on the conventional adaptive cruise control system. Through the Internet of Vehicles’ wireless communication technology, it is possible to communicate information between numerous vehicles in a platoon, increase the range of vehicle awareness, and help the vehicle estimate the control time more precisely [6,7,8].

The safety distance strategy is a crucial component of the CACC system, which estimates the expected following distance when the vehicle is in motion [9,10]. It is the distance input value of the CACC system control algorithm. The choice of the projected following distance has a direct effect on the driving stability and safety of the vehicle, the comfort of the passengers, and the road capacity [11]. If the safety distance strategy selected by the CACC system is excessively conservative, the road traffic utilization rate will be reduced, resulting in traffic congestion, and it will often be overtaken by vehicles in other lanes. If the safety distance strategy selected is too radical, it will degrade vehicle safety and make rear-end collisions more likely. Designing an appropriate time headway method is, therefore, the most important aspect of the CACC system’s control algorithm design.

The two primary safety distance strategies utilized by vehicles are based on the braking process and the time headway strategy, respectively. The constant time headway (CTH) strategy [12], constant distance strategy [13], constant safety factor headway strategy, and second-order headway strategy [14] are sequentially presented depending on the characteristic values of headway, such as constant time interval or distance interval. With the development of Internet of Vehicles technologies, automobile product design has increasingly prioritized the driving experience in recent years. The constant parameter time headway control strategy is difficult to adjust to the needs of various driver control behaviors, and it also hinders the psychological acceptance of drivers and passengers to the adaptive cruise system. Because of this, a great deal of study has been undertaken on the time headway distance method in the Internet of Vehicles environment. Bian et al. establish the headway control strategy based on the information of multiple vehicles in front of the host vehicle, and demonstrate that the lower limit of the time headway can be reduced and the platoon stability can be enhanced under the V2V communication environment by building the two-predictor following (TPF) platoon topology [15]. Lu et al. propose a freeway car-following model that incorporates standstill distance and time headway distributions and applies it to estimate freeway travel time reliability. By integrating these distributions into the car-following model, more accurate estimates of travel time reliability can be obtained. This research is important for improving the efficiency and reliability of the freeway transportation system [16]. Alsuhaim et al. propose adapting the time headway parameter of a Cooperative Adaptive Cruise Control platoon to network reliability to maintain a string-stable platoon while maximizing traffic flow efficiency [17]. Abolfarzli investigates how communication delays can compromise the internal and string stability of platoons under the multiple-predecessor following (MPF) topology and proposes a lower bound for the time headway to ensure stability [18]. Wang and other researchers propose a new control strategy called Integral Sliding Mode Control (ISMC) combined with a Disturbance Observer (DO) for vehicle platoons. The goal is to achieve string stability in a finite time while considering unknown acceleration uncertainties in the following vehicles [19]. In [20,21], the researchers give an LMI method to ensure the stability of a single vehicle and platoon. By modifying the driving time headway and adjusting/optimizing the vehicle distance in a timely manner, these strategies can enhance road utilization, conserve fuel, and assure the stability, safety, and speed responsiveness of the platoon. However, the relevant literature lacks a general analysis method of the variable time headway strategy, as well as a compatibility study of converting the variable time headway strategy into the constant distance and constant time headway strategy under different working conditions. Consequently, the application scope is restricted and the versatility is low.

Compared with alternative methodologies, the artificial potential field approach is favored for its inherent qualities, including simplicity, real-time responsiveness, wide applicability, computational efficiency, and adaptability [22,23,24]. This method finds frequent application in the domain of intelligent vehicle and robotics path planning. In the study by Li et al. [25], they utilize this approach to extend its utilization to the design of time headway and lane-changing functions, yielding tailored safety trajectory planning across diverse driving styles. Furthermore, in the research conducted by Li and colleagues [26], they enhance the artificial potential field paradigm to address cooperative control challenges pertinent to local obstacle avoidance and path tracking in autonomous vehicles. Collectively, these advancements underscore the substantial potential of the artificial potential field method within the realm of intelligent transportation systems.

In this paper, through the use of an intelligent transportation system, vehicles can obtain the acceleration, speed, and location information of non-adjacent vehicles. Based on the artificial potential field theory, each front vehicle is regarded as the center of the potential field so that when the host vehicle is following at any time, the acceleration, speed, and position of multiple vehicles in front of the host vehicle are taken into account, and the optimal following time headway is calculated, and the vehicle safety distance strategy based on the artificial potential field is established, which improves the forward-looking of the CACC system, and solves the problem of reaction delay of CACC system. The improved strategy can implement the full condition control of longitudinal car-following, reduce the abrupt acceleration change caused by the switching of different operating conditions, and improve the ride comfort of the vehicle. Using MATLAB/Simulink and Carsim in tandem, the simulated verification of the vehicle safety distance strategy proposed in this paper is achieved. The results show that the safety distance strategy established in this paper is capable of calculating a more accurate safety distance. The main contributions of this paper include:

• The traditional artificial potential field is improved by adding two parameters of vehicle speed and acceleration, and taking into account the information of multiple vehicles in front, the artificial potential field function in the dynamic scenario of car-following platoons is proposed;
• Based on the theory of artificial potential field, the expression of variable time headway strategy is given, the model of vehicle distance error is established, and the comprehensive performance of the improved strategy, VTH strategy, and CTH strategy under various working conditions is determined;
• Using MATLAB/Simulink and Carsim in tandem, the simulated verification of the vehicle safety distance strategy proposed in this paper is achieved.

2. Research and Analysis of the Safety Distance Algorithm

Safety distance algorithms can be divided into constant safety distance and variable safety distance. Constant safety distance signifies that the safety distance of the vehicle remains consistent during driving, regardless of the host vehicle’s speed and road conditions. This paper will not examine this algorithm because it is simple and straightforward to construct, but it is difficult to suit practical needs. The variable safety distance is based on the time headway, which can be subdivided into two types: constant time headway and variable time headway (VTH). The following is an analysis of the existing time headway algorithm.

2.1. Constant Time Headway Algorithm

The time headway in the CTH algorithm is constant, but this time headway can be set manually. The expression of the algorithm is as follows:

$$d_{safe}=t_h{v}_{host}+d_{min}$$

(1)

In the above equation, d_safe stands for the safe distance between two vehicles, t_h stands for the V2V time headway, the value is 1.4–3 s, v_host stands for the actual speed of the host vehicles, d_min stands for the minimum safety distance between vehicles in the static process, the value is 2–5 m. Based on this equation, it can be concluded that when t_h is constant, the speed of the host vehicle is proportional to the safe distance, which corresponds to the actual driving habits.

2.2. Time Headway Strategy Considering the Relative Speed of Two Vehicles

The time headway changes in the VTH algorithm, and the algorithm has better adaptability to the change of driving environment and operating conditions. Broqua mainly uses the VTH algorithm in his CACC system, which defines the time headway as proportional to the vehicle speed [27].

$$d_{safe}=t_{h1}{v}_{host}+t_{h2}{(v_{host})}^2+d_{min}$$

(2)

$$t_h=t_{h1}+t_{h2}{v}_{host}$$

(3)

In the above equation, t_h1, t_h2 are constants greater than 0. The time headway is proportional to the vehicle speed. As the vehicle speed needs to be lower than the maximum speed v_max, the corresponding time headway equation is as follows:

$$t_h=\left\{\begin{array}{cc}{t}_{h1}+t_{h2}·v_{host}& v_{host}<v_{\max }\\ t_{h1}+t_{h2}·v_{\max }& Otherwise\end{array}\right.$$

(4)

Yanakiev et al. further studied this and introduced the effect of relative velocity on time headway [28]. They used a saturation function with specific upper and lower limits of 1 and 0 to define the headway as follows:

$$\begin{array}{l}{t}_h=sat(t_{h0}-c_h{v}_r)\\ =\left\{\begin{array}{cc}1& t_{h0}-c_h{v}_r>1\\ t_{h0}-c_h{v}_r& 0<t_{h0}-c_h{v}_r<1\\ 0& Otherwise\end{array}\right.\end{array}$$

(5)

In the above equation, sat(·) represents the saturation function, t_h0 and c_h are constants greater than 0, and v_r represents the relative speed of the front and rear vehicles.

2.3. Safety Distance Strategy Based on Acceleration of Front Vehicle

By using vehicular communication technology to obtain the acceleration signal of the preceding vehicle and incorporating the preceding vehicle’s acceleration to represent its velocity change trend, it is possible to obtain the following:

$$\begin{array}{l}{t}_h=sat(t_0-k_a{v}_r-k_b{a}_f)\\ =\left\{\begin{array}{cc}{t}_{h\_\max }& if t_0-k_a{v}_r-k_b{a}_f>t_{h\_\max }\\ t_0-k_a{v}_r-k_b{a}_f& if t_{h\_\min }<t_0-k_a{v}_r-k_b{a}_f<t_{h\_\max }\\ t_{h\_\min }& if t_0-k_a{v}_r-k_b{a}_f<t_{h\_\min }\end{array}\right.\end{array}$$

(6)

In the equation, the parameters t₀, k_a, and k_b are greater than 0, t__min is the lower limit of time headway, and t__max is the upper limit of time headway.

Under the Internet of Vehicles environment, vehicles can acquire data from several vehicles in front of them. The above time headway strategy solely takes into account the effect of the preceding vehicle on the following attributes of the host vehicle. However, in the environment of vehicle communication, it is possible to obtain the state characteristics of several vehicles ahead. Therefore, based on the theory of artificial potential field, this paper establishes a V2V time headway optimization strategy considering multiple vehicles ahead.

3. V2V Time Headway Strategy Based on Artificial Potential Field

3.1. Methodological Overview

To illustrate the proposed approach, a methodological overview is presented in Figure 1. The host vehicle utilizes V2V communication within the vehicular network and sensor information to acquire status data of both its own vehicle and multiple preceding vehicles. Subsequently, employing the APF function, the host vehicle computes potential field forces exerted by each preceding vehicle, including gravitational and repulsive forces. The superposition of these forces yields the total potential field force experienced by the host vehicle, allowing for the determination of the time headway. Consequently, the safe distance between the host vehicle and the preceding one is calculated.

3.2. Artificial Potential Field Theory

Khatib first proposes the artificial potential field to solve the problem of robot obstacle avoidance [29]. The method involves defining the mobile robot’s environment with a potential field, i.e., constructing an artificial potential field consisting of the target position gravitational field and the obstacle repulsion field. The gravitational field will produce an attraction to the moving object, causing it to move toward the target position, while the repulsion field will prevent the moving object from moving in the obstacle direction.

The following potential field function is introduced by the classical artificial potential field method:

As shown in Figure 2 and Figure 3, the gravitational potential field function is:

$$U_a(X)=\frac{1}{2}{K}_a{\left|X-X_g\right|}^2$$

(7)

The gravitation of a moving object is:

$$F_a=-grad\left|U_a(X)\right|=-K_a\left|X-X_g\right|$$

(8)

where K_a is the position gain coefficient, X is the current position of the individual, and X_g is the position of the target point.

As shown in Figure 4 and Figure 5, the repulsion potential field function is:

$$U_r(X)=\left\{\begin{array}{cc}\frac{1}{2}{K}_r{\left[\frac{1}{\rho (X,X_g)}-\frac{1}{{\rho }_0}\right]}^2& \rho \le {\rho }_0\\ 0& \rho >{\rho }_0\end{array}\right.$$

(9)

The repulsion of moving objects is:

$$\begin{array}{l}{F}_r=-grad\left|U_r(X)\right|\\ =\left\{\begin{array}{cc}{K}_r\left[\frac{1}{\rho (X,X_0)}-\frac{1}{{\rho }_0}\right]\frac{\Delta \rho (X,X_g)}{\rho {(X,X_g)}^2}& \rho \le {\rho }_0\\ 0& \rho >{\rho }_0\end{array}\right.\end{array}$$

(10)

where K_r is the constant of the repulsion potential field, ρ(X,X_g) is the distance between a moving object and an obstacle, ρ₀ is the range of influence for the repulsive potential field.

The total potential field is the superposition of the repulsive field and gravitational field, that is:

$$U={\displaystyle \sum U_a}+{\displaystyle \sum U_r}$$

(11)

3.3. Potential Field Modeling of Front Vehicles

Several researchers have investigated the application of artificial potential field theory to obstacle avoidance control, lane maintenance, and lane change in automobiles. The conventional potential field technique serves as the theoretical foundation for this paper. On the basis of the aforementioned equations, the effect of the speed and acceleration change trend of the vehicles ahead on the host vehicle is evaluated. The three vehicles ahead are regarded as three potential fields. The mixed force function in the global potential field is produced using the new artificial potential field technique. Use the gravitational and repulsive forces of the three potential fields to balance and secure the position of the host vehicle. The adjustment effect is shown in Figure 6.

The established potential field model of the front vehicles should meet the following conditions:

• When the acceleration a of the front car and the relative speed v_r with the host vehicle are 0, the gravitation and repulsion of the front vehicle to the host vehicle are both 0;
• When the relative speed between the front vehicle and the host vehicle is v_r > 0, and the speed of the host vehicle is less than the speed of the front vehicle, then the potential field of the front vehicle should be a gravitational field; otherwise, the potential field should be a repulsive field;
• When the relative speed of the front vehicle and the host vehicle is v_r = 0, if the front vehicle is accelerating, the potential field of the front vehicle should be the gravitational field; otherwise, the potential field should be the repulsive field;
• In contrast to the conventional artificial potential field theory, the effect of the gravitational and repulsive fields in the car-following model is inversely proportional to the distance between the following vehicle and the vehicle in front;
• When the relative speed between the host vehicle and the vehicle in front is constant, it can include two states of acceleration and deceleration. While the vehicle in front is decelerating, there should be a greater time headway to prevent the possibility of rear-end collision caused by braking ahead.

The traditional artificial potential field method is primarily used to solve path-planning problems in static environments. However, in the case of a vehicle following a leading vehicle in a time-varying dynamic environment, the traditional potential field theory cannot account for dynamically changing obstacles. Based on the aforementioned analysis and using the method proposed by Rasekhipour et al. [30], an improved artificial potential field function that considers velocity and acceleration has been derived.

The gravitational potential field function expression is:

$$U_a=-\frac{1}{2\Delta {x_i}^2}\left[k_{av}{\left(v_i-v\right)}^2+k_{aa}{a}_i{}^2\right]$$

(12)

The gravitational force generated by the i-th front vehicle is:

$$\begin{array}{l}{F}_{ai}=-grad\left|U_a\right|\\ =\left\{\begin{array}{cc}\frac{k_{av}\left|v_i-v\right|+k_{aa}\left|a_i\right|}{\Delta x_i{}^2}& v_r>0 or a_i>0\\ 0& Otherwise\end{array}\right.\end{array}$$

(13)

where k_av and k_aa are parameters greater than 0; v_i and a_i are the speed and acceleration of the i-th front vehicle; v is the speed of the host vehicle; and Δx_i is the distance between the i-th front vehicle and the host vehicle.

The repulsive force field function expression is:

$$U_r=\left\{\begin{array}{cc}\frac{1}{2\Delta x_i{}^2}\left[k_r+k_{rv}{\left(v_i-v\right)}^2+k_{ra}{a}_i{}^2\right]& \begin{array}{l}{v}_r<0 \\ or a_i<0\end{array}\\ 0& Otherwise\end{array}\right.$$

(14)

The repulsive force generated by the i-th front vehicle is:

$$\begin{array}{l}{F}_{ri}=-grad\left|U_r\right|\\ =\left\{\begin{array}{cc}-\frac{1}{\Delta x_i{}^2}\left[k_r+k_{rv}\left|v_i-v\right|+k_{ra}\left|a_i\right|\right]& \begin{array}{l}{v}_r>0 \\ or a_i>0\end{array}\\ 0& Otherwise\end{array}\right.\end{array}$$

(15)

In the above equation, k_r, k_rv, and k_ra are parameters greater than 0.

3.4. Safety Distance Strategy Based on Artificial Potential Field

Based on Yanakiev’s VTH strategy and the leading vehicle potential field force theory proposed in this paper, the time headway expression is obtained as:

$$t_h=sat\left[t_0-k_a{\displaystyle \sum _{i=1,2,3}{F}_{ai}-k_b{\displaystyle \sum _{i=1,2,3}{F}_{ri}}}\right]$$

(16)

In the equation, t₀, k_a, and k_b are a parameter greater than 0. F_ai and F_ri are the gravitational and repulsive force of the i-th front vehicle on the host vehicle.

Assuming that the signal delay during signal acquisition is ignored because the time headway cannot be negative, and a greater time headway will induce vehicles in the adjacent lane to change lanes and insert, the limit value of the time headway is set to make the strategy more practical:

$$\begin{array}{l}{t}_h=sat\left[t_0-k_a{\displaystyle \sum _{i=1,2,3}{F}_{ai}-k_b{\displaystyle \sum _{i=1,2,3}{F}_{ri}}}\right]\\ =\left\{\begin{array}{cc}{t}_{h\_\max }& \begin{array}{l}if t_0-k_a{\displaystyle \sum _{i=1,2,3}{F}_{ai}-k_b{\displaystyle \sum _{i=1,2,3}{F}_{ri}}}\\ >t_{h\_\max }\end{array}\\ t_0-k_a{\displaystyle \sum _{i=1,2,3}{F}_{ai}-k_b{\displaystyle \sum _{i=1,2,3}{F}_{ri}}}& \begin{array}{l}if t_{h\_\max }>\\ t_0-k_a{\displaystyle \sum _{i=1,2,3}{F}_{ai}-k_b{\displaystyle \sum _{i=1,2,3}{F}_{ri}}}\\ >t_{h\_\min }\end{array}\\ t_{h\_\min }& Otherwise\end{array}\right.\end{array}$$

(17)

In the above equation, t_{h_min}—lower limit of time headway; t_{h_max}—upper limit of time headway.

The final expected distance:

$$\Delta x_{\exp }=t_h{v}_b+\Delta x_{\min }$$

(18)

In the equation: Δx_exp is the expected distance between the two vehicles; Δx_min is the minimum safety distance; and v_b is the speed of the vehicle behind.

The improved time headway strategy can detect the driving changes of the vehicle ahead earlier, make judgments in advance, reduce the host car’s acceleration limit, and improve driving stability. The reduction in time headway improves the utilization rate of road traffic.

4. Stability Analysis of the Time Headway Strategy

When the vehicles in the front of the platoon are traveling at a constant speed, the distance error of the distance strategy must be convergent; otherwise, the control strategy of the CACC system is unstable. Hence, when the speed between two vehicles tends to be consistent, the relative speed v_r and the distance error δ should both tend to zero.

Relative speed v_r:

$$v_r=v_f-v_b$$

(19)

where vf is the speed of the front vehicle.

Vehicle distance error δ is the actual distance between vehicles minus the expected distance between vehicles.

$$\delta =\Delta x-\Delta x_{\exp }$$

(20)

In the above equation, δ represents the error of the distance, and Δx represents the actual distance.

Bringing the Equation (18) into (20) can get:

$$\delta =\Delta x-\Delta x_{\min }-t_{h}v$$

(21)

where v is the speed of the host vehicle.

By differentiating Equation (21), we can get:

$$\stackrel{·}{\delta }=v_r-{\stackrel{·}{t}}_{h}v-t_h\stackrel{·}{v}$$

(22)

No matter what control strategy the CACC system adopts, it is to ensure that the actual distance is consistent with the expected safe distance obtained from the distance strategy. If Δx > Δx_exp, the host vehicle speed should be increased to reduce the vehicle distance; if Δx < Δx_exp, the host vehicle speed should be decreased and lower than the front vehicle speed to increase the vehicle distance. The above description can be expressed as an equation:

$$v_r+k\delta =0$$

(23)

where k is a coefficient greater than 0.

Bringing (23) into (22) gives:

$$\stackrel{·}{\delta }=-k\delta -{\stackrel{·}{t}}_{h}v-t_h\stackrel{·}{v}$$

(24)

By differentiating the variable time headway strategy Equation (17), we can get:

$$\begin{array}{l}{\stackrel{·}{t}}_h=-k_a{\displaystyle \sum _{i=1,2,3}{\stackrel{·}{F}}_{ai}}-k_b{\displaystyle \sum _{i=1,2,3}{\stackrel{·}{F}}_{ri}}\\ ={\displaystyle \sum _{i=1,2,3}(-1)(k_a{\stackrel{·}{F}}_{ai}+k_b{\stackrel{·}{F}}_{ri})}\end{array}$$

(25)

In this equation

$${\stackrel{·}{F}}_{ai}=\frac{k_{av}{\stackrel{·}{v}}_{r\_i}+k_{aa}{\stackrel{·}{a}}_i}{\Delta x_i{}^2}-\frac{2v_{r\_i}(k_{av}{v}_{r\_i}+k_{aa}{a}_i)}{\Delta x_i{}^3}$$

(26)

And

$${\stackrel{·}{F}}_{ri}=\frac{-k_{rv}{\stackrel{·}{v}}_{r\_i}-k_{ra}{\stackrel{·}{a}}_i}{\Delta x_i{}^2}-\frac{2v_{r\_i}(-k_r-k_{rv}{v}_{r\_i}-k_{ra}{a}_i)}{\Delta x_i{}^3}$$

(27)

where v_ri is the relative speed of the i-th front vehicle and the host vehicle, and,${\stackrel{·}{v}}_{r\_i}={\stackrel{·}{v}}_i-\stackrel{·}{v}$.

Take (25) into (24), and you get:

$$\stackrel{·}{\delta }=-k\delta -t_h\stackrel{·}{v}-\left[{\displaystyle \sum _{i=1,2,3}(-1)(k_a{\stackrel{·}{F}}_{ai}+k_b{\stackrel{·}{F}}_{ri})}\right]v$$

(28)

When the vehicles in front are driving at a constant speed, $\stackrel{·}{v}$ = 0, and you can get:

$${\stackrel{·}{v}}_{r\_i}=-\stackrel{·}{v}$$

(29)

Take Equations (23) and (29) into Equation (28) to get:

$$\begin{array}{l}\stackrel{·}{\delta }=-k\delta -t_h\stackrel{·}{v}-\left[{\displaystyle \sum _{i=1,2,3}(-1)(k_a{\stackrel{·}{F}}_{ai}+k_b{\stackrel{·}{F}}_{ri})}\right]v\\ =-k\delta -t_{h}k\stackrel{·}{\delta }-\left[{\displaystyle \sum _{i=1,2,3}(-1)(k_a{\stackrel{·}{F}}_{ai}+k_b{\stackrel{·}{F}}_{ri})}\right]v\end{array}$$

(30)

When the front vehicle acceleration a_i → 0, $\stackrel{·}{v}$ simplifying the above equation can be obtained:

$$[1+(t_{h}k+2k_{a}v)k]\stackrel{·}{\delta }+k\delta =0$$

(31)

Because t_h ≥ 0, k > 0, k_a > 0, v ≥ 0,

$$1+(t_{h}k+2k_{a}v)k\ge 1$$

(32)

Let K = 1 + (t_hk + 2k_av) k, Equation (31) can be transformed into:

$$\stackrel{·}{\delta }=-\frac{k}{K}\delta$$

(33)

Equation (33) shows that when δ > 0, $\stackrel{·}{\delta }$ < 0, the value of δ decreases, and when δ < 0, $\stackrel{·}{\delta }$ > 0, the value of δ increases. In short, the value of δ is eventually going to zero. Again v_r + kδ = 0. And k > 0, so v_r is also close to zero. So, when v_r → 0, δ → 0 is established. It can be seen that the distance error and relative speed are stable and convergent. In summary, under this strategy, the platoon is stable when driving.

5. Simulation Verification

5.1. Simulation Platform Setup

In order to demonstrate the advantages of the V2V time headway strategy proposed in this paper, each simulation-required vehicle is built based on the established vehicle dynamics model. In this paper, a basic and mature PID control algorithm is selected as the control algorithm for computer simulation, and the road environment and initial driving information of the car are calculated. Then, the V2V time headway simulation system based on artificial potential fields is constructed by combining MATLAB/Simulink and Carsim software. The experimental setup utilized for this study involved a computer system featuring an NVIDIA GeForce RTX 3080 graphics card, complemented by an Intel Core i7 processor, 32 GB of RAM, and a 1 TB SSD. The simulation platform is built as shown in Figure 7. The proposed strategy is compared with the CTH strategy and VTH strategy.

5.2. Simulation Comparison Analysis

This paper will focus on the primary traffic circumstances that can arise when following a vehicle: approaching the front vehicle, the leading vehicle accelerating, and the leading vehicle performing an emergency stop. These three kinds are also present in additional, more complex traffic scenarios. The simulated setting has four vehicles ahead; the first vehicle of the platoon is the leading vehicle, and the remaining three are the front vehicles. This study only considers the influence of the host vehicle’s first three vehicles’ potential field forces. The leading vehicle and other front vehicles are all equipped with an adaptive cruise system. The constant time headway strategy is used to control the vehicle distance, and the mature PID control algorithm is used for the acceleration controller.

The initial conditions set for the smooth car-following situation are: the initial speed of each vehicle is 15 m/s, and the distance is 50 m. During the computer simulation, the speed change of the first leading vehicle is the same as shown in

Table 1. State change of the leading car.

Leading Car	Time	Working Conditions of the Leading Car
Approach the car ahead	0–40 s	15 m/s
Speed up	40–45 s	1 m/s2
Constant speed	45–65 s	20 m/s
Slow down	65–70 s	−4 m/s2
Stop	of the leadi 70–100 s	Stop

, when t = 40 s, driving at a constant acceleration of 1 m/s² for 5 s; accelerate to 20 m/s and drive at a constant speed; when t = 65 s, driving at a constant deceleration of −4 m/s² for 5 s and then keep it stopped. VTH strategy, CTH strategy, and the improvement strategies proposed in this paper are selected for comparison.

5.2.1. Approach the Front Vehicle

The initial simulation conditions are 50 m between vehicles moving at 15 m/s. The leading vehicle in a computer simulation always keeps up its constant speed. Figure 8 depicts the response curves of predicted distance, speed, acceleration, and potential field force obtained from computer simulation.

According to Figure 8a, the acceleration change curve reveals that initially, the distance between the host vehicle and the first vehicle in front is greater than the expected distance, but as the following distance decreases, it is required to decelerate to maintain the safe following distance. VTH and CTH strategies start to decelerate around t = 20 s, and the improved APF strategy starts to decelerate at t = 17 s. As shown in Figure 8d, further analysis is made in combination with the potential field force change diagram of the front vehicles and the host vehicle. When t = 17 s, the comprehensive potential field force reaches the influence threshold, the expected distance starts to increase, and the host vehicle slows down. Because the APF strategy adopts the deceleration time earlier, the required deceleration limit value is smaller, and the deceleration limit value of the APF strategy is reduced by 32.3% compared with the VTH strategy.

The improved APF strategy can perceive the driving situation of multiple vehicles ahead in advance, allowing it to adjust the expected distance earlier (Figure 8c), reduce the deceleration, and adopt a gentler deceleration behavior so that the speed of the host vehicle approaches the speed of the first front vehicle and a steady car-following state is achieved (Figure 8b). However, the CTH strategy and VTH strategy can only perceive the first front vehicle. Owing to the information lag, when the vehicle in front decelerates sharply, the vehicle behind must adopt a larger deceleration to adjust the following distance. Under the control of the two strategies, the following distance, speed, and acceleration vary substantially over the entire operation. In contrast, the improved APF strategy allows for a more gradual shift in distance, velocity, and acceleration.

5.2.2. Speed-Up Conditions of the Leading Vehicle

When t = 40 s, the leading car starts accelerating with an acceleration of 1 m/s². As can be seen from Figure 9d, the three front vehicles start accelerating at t = 40 s, t = 41 s, and t = 42.5 s, respectively. At 41 s, the potential field function begins to affect the expected distance of the host vehicle. It can be seen from Figure 9a that under the APF strategy, the acceleration starts at t = 41 s, and the CTH strategy and the VTH strategy begin to accelerate at the same time at t = 42.3 s. As shown in Figure 9b, the speed change of the improved APF strategy is smoother compared to the other two strategies. The improved strategy adopts a more gradual acceleration, with a maximum acceleration of 0.4 m/s², whereas the other two strategies adopt an acceleration behavior with a greater rate of change when they detect that the vehicle in front of them is accelerating, causing passengers to experience abrupt speed changes and reducing ride comfort.

It can be seen from Figure 9c that under the improved distance strategy, when the relative speed of the leading vehicle and the host vehicle is greatly different, the host vehicle adopts a smaller expected distance to reduce the distance from the first front vehicle, that is, accelerate in advance. The other two strategies will accelerate simultaneously with the first vehicle in front; therefore, there is no procedure for decreasing the expected space between vehicles.

5.2.3. Emergency Deceleration of the Leading Vehicle

Emergency braking occurs when two vehicles are in close proximity, and the one in front abruptly stops. All vehicles are initially going at a consistent pace once the simulation begins. When t = 65 s, the leading vehicle brakes suddenly with an acceleration of −4 m/s² and finally stops. Figure 10 depicts the response curves of expected distance, actual distance, velocity, and acceleration determined from computer simulation.

Figure 10 illustrates how, upon receiving the brake signal from the leading vehicle in the event of emergency braking, the improved distance strategy takes immediate braking steps (Figure 10a) and adjusts the following distance from 32 m to 48 m to ensure the safety of the following distance (Figure 10c). It effectively avoids possible rear-end collisions during emergency braking, and the minimum space between the host vehicle and the car in front when the host stops, i.e., 5 m, is maintained during the whole braking process (Figure 10d). The other two strategies start to brake when t = 68.5 s. Because of the reaction lag, the host vehicle speed becomes negative. The maximum negative speed of the CTH strategy is −1.9 m/s, and the maximum negative speed of the VTH strategy is −1.2 m/s (Figure 10b). It signals the possibility of a rear-end collision with the car ahead. Not only does the distance strategy based on the APF theory prevent rear-end collisions, but it also maintains an actual following distance with the vehicle ahead that exceeds the maximum safe distance. Until the end of the following distance, there is a progressive decrease. It not only increases comfort but also ensures safety.

6. Discussion

6.1. Analysis of Simulation Results

We will discuss the above results:

• With the exception of the emergency braking condition, the safety distance calculated by the APF strategy established in this paper is always less than the safety distance calculated using the CTH and VTH, thereby resolving the issue that the traditional safety distance strategy is overly conservative. Figure 8 demonstrates that the safety distance determined by the APF strategy differs by 5 m from the safety distance calculated by the VTH strategy when the leading vehicle’s speed is constant. The safety distance calculated by the APF strategy established in this paper is generally equivalent to 85% of the safety distance calculated by the VTH strategy; the APF strategy requires less expected distance than VTH and CTH strategies, thus improving road traffic utilization;
• Figure 9 demonstrates that at this time, the potential field force of the leading vehicle on the host vehicle is positive, and the risk is considerably diminished. It is possible to minimize the following distance so that the host vehicle can match the speed of the car in front. The safety distance calculated by the APF strategy is less than that calculated by the CTH and VTH strategies, and the change is more gradual, indicating that the vehicle’s speed is relatively stable during driving and that speed changes are also relatively smooth, resulting in good driving safety and comfort;
• It can be seen from Figure 10 that the safety distance calculated by the APF strategy is greater than the safety distance calculated by the VTH strategy. This is because the acceleration of the front vehicles is negative at this stage, and the risk is increased. It is necessary to appropriately increase the following distance to improve driving safety. At the same time, when other conditions are the same, the greater the deceleration of the front vehicles, the greater the safety distance obtained from the APF strategy, which is consistent with the actual driving situation;
• It can be seen from (c) and (d) in Figure 8 and Figure 9 that the change in the expected vehicle distance is based on the influence of the comprehensive potential field force of the vehicles ahead. When the speed of the vehicles in front of the host vehicle changes, the host vehicle will alter the expected distance based on the relative location of the first front vehicle and the host vehicle;
• It can be seen from the working conditions of approaching the front vehicle and the emergency brake that the reaction of VTH and CTH is delayed, and the negative speed value appears due to the conservative time headway strategy. Therefore, VTH and CTH are only suitable for stable car-following working conditions and cannot deal with the sudden dangerous working conditions quickly and effectively. The traditional CACC system needs to switch the control algorithm according to different driving conditions. During the switching process of the control algorithm, the acceleration will be discontinuous, resulting in a large acceleration jump, which will affect the driving comfort. After preliminary simulation analysis, the improved APF strategy is able to obtain the information of multiple vehicles ahead in real-time and adjust the expected time headway more flexibly, allowing it to adapt to a variety of driving conditions without adopting a multi-control algorithm strategy, thereby enhancing driving stability;
• According to the acceleration change curve of each working condition, when the vehicles in front of the host vehicle accelerate or decelerate, the acceleration curve of the host vehicle derived from the APF strategy will fluctuate but without a major effect on driving stability. This variation is due to the fact that the time headway method employed by each front vehicle is the CTH strategy. When the speed of the leading vehicle changes, the vehicles behind will accelerate or decelerate after a certain delay time. After the speed change of each vehicle in front reaches the threshold Q, the potential field force generated on the host vehicle will be input into the expected time headway model, thus causing the acceleration fluctuation of the vehicle. Due to the conservative CTH strategy of the front vehicles, when the leading vehicle is driving at a constant speed, the driving speed of the front vehicles is slightly lower than that of the leading vehicle, resulting in a slightly lower speed of the host vehicle.

6.2. Practical Implementation and Adaptation

Practical implementation of the proposed V2V APF time headway strategy encompasses various facets, including sensing requirements, computational demands, and real-world implementation challenges. In this section, we delve into these critical considerations to provide a comprehensive view of the strategy’s practical feasibility and potential hurdles.

• Sensing Requirements: The successful deployment of the V2V APF time headway strategy hinges on accurate and timely information exchange between vehicles. To achieve this, sensor-equipped vehicles must capture and process real-time data pertaining to relative speeds, distances, and acceleration trends of surrounding vehicles. These sensing capabilities enable the calculation of potential field forces and the dynamic adjustment of time headway parameters. It is imperative that the sensors employed are reliable, robust, and capable of operating in diverse environmental conditions;
• Computational Demands: Computational requirements play a pivotal role in the practical viability of the strategy. The real-time calculation of potential field forces and the adjustment of time headway settings necessitate onboard processing capabilities. This involves rapid data processing and decision-making to ensure responsive and accurate adjustments in varying traffic scenarios. Consequently, the computational load can be influenced by factors such as traffic density and communication frequency among vehicles. Addressing these computational demands is vital to ensure the strategy’s effectiveness and reliability;
• Real-World Implementation Challenges: Implementing the V2V APF time headway strategy in real-world settings presents several challenges. Integrating the strategy into existing vehicular communication systems requires the development of standardized communication protocols and algorithms. Collaborative efforts among stakeholders are crucial to ensure interoperability and seamless integration. Furthermore, adapting regulatory frameworks to accommodate dynamic time headway adjustments and establishing standardized testing and certification procedures are essential steps toward practical deployment.

In conclusion, the practical implementation of the V2V APF time headway strategy necessitates careful consideration of sensing requirements, computational demands, and real-world challenges. Addressing these aspects is paramount to ensure the strategy’s successful deployment and its contribution to enhancing traffic safety and efficiency.

6.3. Limitations of the Proposed Model

• Scenario Complexity: While we have validated the performance of the strategy across diverse simulation scenarios, the complexity of the scenarios covered might still be relatively simplified. Future research could encompass more intricate traffic behaviors and environmental variations;
• Dynamic Performance: Our model assumes constant speed for leading vehicles, while real-world driving conditions are more dynamic. Therefore, considering more complex speed variations and traffic flow behaviors could impact the strategy’s performance;
• Sensor Accuracy: Our strategy assumes accurate information from sensors about preceding vehicles. However, the actual accuracy and noise of sensors might influence the strategy’s performance. Thus, considering sensor uncertainty is a worthwhile avenue for research;
• Strategy Tuning: Despite achieving some performance enhancement in simulations, our strategy might require parameter adjustments in different traffic environments and driving conditions. Further research can focus on achieving strategy adaptability and robustness.

7. Conclusions

In this paper, an artificial potential field is built according to the longitudinal car-following characteristics, and the information on the front vehicles is obtained with the help of the Internet of Vehicles communication technology so as to expand the existing variable time headway strategy to consider multiple front vehicles, and thus build an improved V2V APF time headway strategy. On this basis, the host car is able to understand the driving conditions of the vehicles ahead in order to make corrections earlier. Also, the stability of the proposed strategy is confirmed. Finally, through simulation analysis, it is found that the minimum expected vehicle distance of the improved V2V APF time headway strategy is 15% smaller than that of the VTH strategy. The results show that the expected vehicle distance obtained by the proposed strategy has a good effect on improving vehicle safety, driving comfort, and road utilization. For the various limitations of this study, improvements will be pursued in future work.