Molecular Dynamics-Based Car-Following Safety Characteristics and Modeling for Connected Autonomous Vehicles

Abstract

To characterize the dynamic interaction properties of heterogeneous traffic flow in the complex human–vehicle–road environment and to enhance the safety and efficiency of connected autonomous vehicles (CAVs), this study analyzes the self-driven particle characteristics and safety interaction behavior of CAVs based on molecular interaction potential. The molecular dynamics of potential interaction functions are employed to establish a dynamic quantization model for car-following (CF) safety potential, referred to as the molecular force field quantization model. To calibrate the model parameters, the Artificial Bee Colony Algorithm and the highD dataset are utilized, subsequently validating the reasonableness and effectiveness of the molecular dynamics model for vehicle tracking. The simulation results demonstrate that the proposed model can more accurately fit actual CF data, significantly improving vehicle travel safety and efficiency. Moreover, the profile of vehicle acceleration shows a lower mean absolute error and root mean square error compared to actual data, indicating that the model provides superior anti-interference fluctuation resistance and stability in CF scenarios. Overall, the proposed model effectively captures the microscopic CF behavior and vehicle–vehicle safety interactions, offering a theoretical foundation for further research into vehicle-following dynamics.

1. Introduction

The intelligent connected transportation system deeply integrates multi-dimensional elements, such as the human–vehicle–road–environment, resulting in heterogeneous vehicle group perceptions and interactions that present complex dynamic characteristics [1,2,3]. Therefore, systematically characterizing the intrinsic “self-driven particle” and the operating state evolution law of CAVS is crucial for ensuring the safety and stable operation of these vehicle groups [4].

The application of the artificial potential field theory in vehicular path planning was pioneered in the works of [5,6]. Guldner et al. [7] introduced a sliding mode control strategy for tracking the gradient of an artificial potential field. Li et al. [8] extended this theory by establishing a risk field for CF behavior, incorporating vehicle acceleration and steering angle to address the gap in existing studies on vehicle acceleration. Ge et al. [9] explored different information flow topologies in heterogeneous platoons composed of manually driven and connected vehicles, demonstrating that the insertion of connected vehicles with optimally designed gains and delays enhances traffic flow stability. Treiber et al. [10] proposed the Intelligent Driver Model (IDM) to provide comparative insights into micro and macro traffic models. Yang et al. [11] developed and refined the modified molecular dynamics (M-MD) model for CF behaviors based on potential molecular interaction and wall functions. Wang et al. [12] devised an expected safety margin model grounded in drivers’ risk perception levels and driving characteristics to investigate the impact of diverse behavioral traits on CF dynamics. Tian et al. [13] identified that traffic oscillations arise from the interplay between speed adaptation and stochastic effects, concluding that speed differences significantly influence CF behavior more than spacing. Lee et al. [14] employed a stochastic process to simulate following behavior on multi-lane highways, resulting in the development of a multi-lane stochastic CF model. Makridis et al. [15] introduced a CF model that integrated driver and vehicle characteristics as primary inputs. Qin et al. [16] developed a cooperative adaptive cruise control (CACC) CF model and assessed the capacity of mixed traffic flow under varying CACC ratios. These CF models are pivotal in elucidating the connections between microscopic vehicle–vehicle interactions and macroscopic traffic phenomena. Traditional CF models primarily focus on the relative speed and distance between trailing and leading vehicles, often overlooking the dynamic elements within car-following scenarios. With advancements in autonomous driving technologies, modeling the following behavior of CAVs has emerged as a significant research focus. Chen et al. [17] established an integrated vehicle safety field-following model, accounting for both vehicle dynamics and driving environment fields, and proposed a risk–margin of safety evaluation index.

Liu et al. [18] introduced an integrated predictive control method for autonomous vehicles operating in dynamic traffic environments. The simulation results demonstrated that this approach accommodates the diverse control requirements of self-driving vehicles across various road traffic conditions. Li et al. [19,20,21] developed a CF model grounded in potential driving risk fields, integrating model predictive control with artificial potential fields to formulate a control method for CAV platoons. They further constructed a platoon formation and optimization model by combining the graph theory with the safety potential field theory and validated the efficacy of this control strategy. Jia et al. [22] proposed a decision-making model for CAV CF behaviors, which utilizes the virtual potential field method. Xue et al. [23] devised a two-lane CF model that thoroughly accounted for the influence of adjacent lanes on vehicles traveling in the current lane. They analyzed the sensitivity of various factors affecting vehicle stability. Xu et al. [24] introduced a CF model considering the impact of neighboring multi-vehicles, with results indicating high stability in the proposed model. Jiang et al. [25] enhanced the full-speed differential model and proposed an extended visual angle-following model, demonstrating that the type of vehicles in adjacent lanes significantly influences traffic flow stability.

With the rapid advancement of intelligent connected transportation technologies [26,27,28], ensuring the safety of CAV CF behaviors has become paramount. It is also essential to balance rationality and efficiency in these systems [29]. Existing studies on CF behavior typically quantify the safety relationship between following vehicles based on molecular interaction forces [30,31,32,33]. This paper builds on this foundation by considering the system’s similarities and comparing microscopic vehicles to self-driven particles within a molecular force field. It analyzes the interaction and relationship between vehicles and the molecular force field’s following characteristics from a force field perspective. Furthermore, it constructs a CF model for CAVs based on the potential function of the molecular force field. This model incorporates the influence of the speed of front and rear vehicles on CF by integrating a speed synergy term, aiming to achieve the safe interaction and efficient operation of CAVs and to promote the sustainable development of future urban transport.

2. Similarity Analysis of the Relationship between CF and Molecular Force Field Action

From a physical perspective, a field represents the interaction of an object with other objects within a specific spatial range around it, with the magnitude of this interaction varying according to its relative position. The molecular force field potential function analytically expresses energy as a function of the atomic nucleus coordinates. The interaction force and the spacing between particles are depicted in Figure 1, illustrating the relationship between these variables.

Accurately assessing the traffic environment around a vehicle in complex traffic scenarios is crucial for enhancing road traffic safety. In a manner analogous to the use of fields in physics to describe interactions between objects, inter-vehicle interactions can be analytically represented through vehicle coordinates. In a road traffic system, vehicles maintain a distance neither too close nor too far from the vehicle ahead, influenced by the surrounding vehicles. This behavior is akin to the rear vehicle adjusting its acceleration under the influence of the front vehicle to achieve an equilibrium following distance. The molecular force field manifests itself as an interaction force between particles under the action of a combined attractive force and repulsive force. The effects of inter-particle interactions can be distinguished by the relationship between the actual spacing $r$ and the size of the equilibrium distance $r_e$ of the particles, which can be manifested in the following three states: (1) when $r<r_e$, the particles exhibit mutual repulsion; (2) when $r=r_e$, the interaction force between the particles is zero; and (3) when $r>r_e$, the particles exhibit mutual attraction. Under the influence of these forces, the inter-particle distance oscillates around the equilibrium distance, tending towards a state of equilibrium. Similarly, a vehicle endeavors to maintain a dynamically safe distance from the preceding vehicle. When the rear vehicle is too close to the preceding vehicle, it increases its distance to avoid a collision, enhancing safety. This behavior can be explained by the repulsive force exerted by the preceding vehicle. Conversely, when the rear vehicle is too far from the preceding vehicle, it decreases the distance to improve traffic efficiency while maintaining safety. This can be likened to the attractive force exerted by the preceding vehicle. The rear vehicle, thus, achieves its driving tasks safely and efficiently by maintaining a balance near the equilibrium distance, either moving away from the preceding vehicle to avoid hazards or moving closer to improve traffic flow efficiency.

3. Molecular Force Field Theory and Modeling

3.1. Molecular Force Field Analysis

Objects exert influence over other objects within a specific range, and the magnitude of this influence varies with their relative positions. The source of the field induces force effects on other objects within the potential field, thereby altering their state of motion, as depicted in Figure 2.

To investigate the interaction dynamics between vehicles, several quantitative models based on the potential field theory have been proposed. Qu et al. [34] examined the molecular dynamic characteristics of vehicle–vehicle interaction coupling, proposing a molecular dynamics CF model and validating its accuracy and effectiveness through simulations. Li et al. [35] extended a CF model grounded in the safety potential field theory. Typically, potential field theory is applied to analyze CF behavior under the assumption of homogeneous vehicle types. There is a relative scarcity of academic research analyzing the following behavior within connected heterogeneous traffic flows. Moreover, in the application of potential field theory, the molecular force field potential function presents a clear and reasonable structure with simulations closely reflecting real-world scenarios, thereby demonstrating its effectiveness.

Scholars have constructed mathematical models to describe the interaction and relationships between microscopic physical particles, which serve as sources of interacting forces on one another. Unlike some potential field models that independently express attractive and repulsive forces, the molecular force field interaction potential function unifies these terms, treating inter-particle interactions as a combination of both repulsive and attractive forces. These forces are inversely related to multiple powers of particle spacing. The potential molecular force field interaction function is thus defined as follows:

$$\phi (r)=\frac{A}{r^{\alpha }}-\frac{B}{r^{\beta }}$$

(1)

where: $r$ is the distance between the two particles; $\alpha$ and $\beta$ are powers of the repulsive and attractive force action terms, respectively; and $A$ and $B$ are intermediate variables of the potential function regarding $r$. For the molecular force field’s interaction with the potential field, two particles mutually become the source of a field subject to an interaction force. When the particle is at the center of the potential well, the first-order derivative of the molecular force field potential function with respect to $r$ has the value 0, i.e.:

$${\left[\frac{\mathrm{d}\phi (r)}{\mathrm{d}r}\right]}_{r=r_e}=0$$

(2)

$$\phi \left(r_e\right)=-\epsilon$$

(3)

where $\phi (r)$ is the potential energy, $\epsilon$ is the depth of the potential well of the molecular force field, and $r_e$ is the particle spacing when at the center of the potential well. Substituting Equation (1) into Equation (2) gives the following:

$$\frac{d\phi (r_e)}{d{r}_e}=\frac{\beta B}{{r_e}^{\beta -1}}-\frac{\alpha A}{{r_e}^{\alpha -1}}=0$$

(4)

$${r_e}^{\alpha -\beta }=\frac{\alpha A}{\beta B}$$

(5)

The molecular force field’s potential function can be expressed as follows:

$$\phi (r)=\frac{\epsilon }{\alpha -\beta }\left[\beta {\left(\frac{r_{\mathrm{e}}}{r}\right)}^{\alpha }-\alpha {\left(\frac{r_{\mathrm{e}}}{r}\right)}^{\beta }\right]$$

(6)

The force on the particle is solved by potential energy differentiation, and the resulting expression for the force on the particle is as follows:

$$F(r)=-\frac{\mathrm{d}\phi (r)}{\mathrm{d}r}=\frac{\alpha \beta \epsilon }{\alpha -\beta }\left(\frac{{r_{\mathrm{e}}}^{\beta }}{r^{\beta +1}}-\frac{{r_{\mathrm{e}}}^{\alpha }}{r^{\alpha +1}}\right)$$

(7)

As shown in Figure 3, the inter-particle interaction force of the molecular force field potential function is calculated and plotted against the spacing according to Equation (7). Here, the equilibrium distance between the particles is the distance between the field source and the position of the potential well. The stress condition of the particles in the molecular force field matches the actual situation, which proves that the molecular force field’s potential function has a better simulation effect on the inter-particle interactions.

3.2. Safety Characteristics of Car-Following in Molecular Force Fields

With the advancement of vehicle–road cooperative technologies, autonomous decision-making capabilities are enhanced so that each vehicle effectively becomes a source of the molecular force field that influences driving behavior. Assuming that driving behavior originates from a risk field, employing the potential function to construct an interaction model proves widely applicable to vehicle–vehicle interactions, as depicted in Figure 4. This approach enables a comprehensive analysis of the dynamic interactions between vehicles within the intelligent transportation ecosystem.

The CAV is traveling at a low-risk location, and the situation of the risk field where it is located is analyzed using the potential function, as shown in Figure 5.

The interaction between vehicles traveling on the road resembles the molecular force field, where the motion of one vehicle induces corresponding changes in others, thus creating a dynamic, interactive behavior. This dynamic correlation can be characterized as a coupling relationship. Vehicles in motion maintain a coupling relationship that prevents them from being either too close or too far from each other, akin to the interactions between particles in a molecular force field. The rear vehicle maintains a position at the trailing edge of the required safety distance from the vehicle in front, while the front vehicle maintains a position at the leading edge of the demanded safety distance from the rear vehicle. Essentially, vehicles continuously strive to maintain a dynamically safe distance from the vehicle ahead while traveling.

Since the safety dynamic characteristics of vehicle–vehicle interaction behavior are similar to the mechanical relationship between particles in a molecular force field, it can reflect the influence of traffic factors on the vehicle–vehicle interactions. Therefore, we analyze the safety characteristics of autonomous vehicles from the perspective of the molecular force field’s potential function to improve the stability of traffic flow.

3.3. Molecular Force Field Model of CF

3.3.1. Human Vehicle CF Model

According to the characteristics of dynamics, the displacement $x_n$ produced by the vehicle during emergency braking is as follows:

$$x_n=\frac{{v_n}^2}{2a_n}$$

(8)

where $v_n$ is the initial velocity of the rear vehicle; $a_n$ is the maximum acceleration of the rear vehicle during braking. Under the actual road traffic scenario, the rear vehicle still maintains its initial speed for a period of time during the initial phase of deceleration, which is defined as the response delay of the human-driven vehicle ${\tau }_{HV}$ (generally taken as 0.7 s), so the actual displacement distance $x_n$ of the rear vehicle is the following:

$$x_{n-HV}=v_n{\tau }_{HV}+\frac{{v_n}^2}{2a_n}$$

(9)

When the front car suddenly brakes with maximum acceleration, the displacement difference $\Delta x_{\mathrm{retardation}}$ between the two cars produced by the driver of the rear car from the time he discovers that the front car has started braking to the time when the rear car finishes braking is as follows:

$$\Delta x_{\mathrm{retardation}}=v_n{\tau }_{HV}+\frac{{v_n}^2}{2a_n}-\frac{{v_{n-1}}^2}{2a_{\mathrm{n}-1}}$$

(10)

where: $v_{n-1}$ is the initial speed of the car in front, and $a_{n-1}$ is the maximum acceleration conducted during the braking of the front car. Assuming that all vehicles have the same emergency braking capacity and the same initial velocity (i.e., $a_{n-1}=a_n=a_1$ and $v_{n-1}=v_n=v_0$), the displacement difference $\Delta x_{HV}$ between the front and rear vehicles during braking is the following:

$$\Delta x_{HV}=v_0{\tau }_{HV}$$

(11)

Under ideal conditions, the rear vehicle is traveling at $\Delta x_{HV}$ distance from the vehicle in front of it and happens not to collide with the vehicle in front after emergency braking. However, in actual traffic scenarios, it is difficult to avoid unforeseen situations, such as vehicles or pedestrians suddenly intruding from the blind spot of the field of vision. Therefore, the following safety distance should be appropriately increased by the stopping safety distance $s_{0-HV}$ (generally taking the value of 2 m); then, the following safety distance $s_{n-HV}$ is as follows:

$$s_{n-HV}=s_{0-HV}+v_0{\tau }_{\mathrm{HV}}+\frac{{v_0}^2}{2a_1}$$

(12)

According to the molecular force field’s potential function, it is analogous to establish the following model of the molecular force field’s potential function:

$${\phi }_{HV}\left(l\right)=\frac{\epsilon }{\alpha -\beta }\left[\beta {\left(\frac{s_{n-HV}}{l}\right)}^{\alpha }-\alpha {\left(\frac{s_{n-HV}}{l}\right)}^{\beta }\right]$$

(13)

where $l$ is the actual distance between the front and rear vehicles, which corresponds to the actual distance $r$ between particles. The acceleration $a_{n-HV}(l)$ of the rear vehicle resulting from the force of the front vehicle is the following:

$$a_{n-HV}\left(l\right)=\frac{\alpha \beta \epsilon }{m(\alpha -\beta )}\left(\frac{{s^{\beta }}_{n-HV}}{l^{\beta +1}}-\frac{{s^{\alpha }}_{n-HV}}{l^{\alpha +1}}\right)$$

(14)

where $m$ is the mass of the rear vehicle. To simplify the model, let $\lambda =\frac{\alpha \beta \epsilon }{m(\alpha -\beta )}$, assuming that all vehicles have the same mass; then, $\lambda$ is only linearly related to $\epsilon$, and the model of the longitudinal acceleration subjected to the action of the front vehicle is expressed as follows:

$$a_{n-HV}=\lambda \left(\frac{{s^{\beta }}_{n-HV}}{l^{\beta +1}}-\frac{{s^{\alpha }}_{n-HV}}{l^{\alpha +1}}\right)$$

(15)

The driver of the rear vehicle dynamically adjusts the distance to the target according to the speed relationship between the front and rear vehicles, therefore introducing the velocity covariance term $a_{k-HV}$, i.e.,

$$a_{k-HV}=\mu \left(1-\frac{v_n}{v_{n-1}}\right)\cdot \left|\Delta v\right|$$

(16)

where $\mu$ is the coefficient to be determined for the velocity synergy term. The human vehicle (HV) longitudinal acceleration of molecular force fields with the introduction of a velocity synergy term can be modeled, i.e.,

$$a_{HV}=\lambda \left(\frac{{s^{\beta }}_{HV}}{l^{\beta +1}}-\frac{{s^{\alpha }}_{HV}}{l^{\alpha +1}}\right)+\mu \left(1-\frac{v_n}{v_{n-1}}\right)\cdot \left|\Delta v\right|$$

(17)

where $a_{HV}$ is the HV longitudinal acceleration.

3.3.2. CAV CF Model

Equipped with connected communication technology, CAVs utilize real-time data on relative positions, speed, and other operational status information to facilitate interactions between vehicles. This real-time transmission of collected operational status data enhances the safety and efficiency of connected heterogeneous traffic flow. As depicted in Figure 6, the operational status of each vehicle is denoted as Ci for the i-th vehicle.

To investigate traffic flow operations involving a mix of CAVs and HVs, adjusting the acceleration of CAVs can mitigate the propagation of traffic fluctuations. This adjustment prevents the instability from the leading vehicles from extending backward, thereby achieving the overall stable operation of the connected heterogeneous vehicle groups, as illustrated in Figure 7.

During braking, under the influence of the vehicle in front, the shortest braking distance for the CAV to avoid a collision is the following:

$$x_{n-CAV}=v_n{\tau }_{\mathrm{CAV}}+\frac{{v_n}^2}{2a_n}$$

(18)

where ${\tau }_{\mathrm{CAV}}$ is the CAV delayed response. For an appropriate increase in the stopping distance, $s_{0-CAV}$, the CAV following safety distance is the following:

$$s_{0-CAV}+v_0{\tau }_{CAV}<s_{n-CAV}⩽s_{0-CAV}+v_0{\tau }_{CAV}+\frac{{v_0}^2}{2a_1}$$

(19)

For the following safety distance factor, $\delta$, the range of values is (0, 1]; then, the above equation can be simplified as follows:

$$s_{n-CAV}=s_{0-CAV}+v_0{\tau }_{CAV}+\delta \frac{{v_0}^2}{2a_1}$$

(20)

The longitudinal acceleration function of the CAV following model based on the molecular force field is expressed as follows:

$${a_{n-}}_{CAV}\left(l\right)=\lambda \left(\frac{{s^{\beta }}_{n-CAV}}{l^{\beta +1}}-\frac{{s^{\alpha }}_{n-CAV}}{l^{\alpha +1}}\right)$$

(21)

4. Calibration of Model Parameters

The Artificial Bee Colony (ABC) Algorithm is an optimization technique inspired by the foraging behavior of bees. As a global optimization algorithm rooted in swarm intelligence, it exhibits a rapid convergence rate. In this study, the ABC algorithm was employed for calibration purposes. The algorithm’s workflow is depicted in Figure 8.

To filter the highD dataset of German highways, the following criteria were applied:

(1)

Vertical spacing less than 150 m to exclude vehicles in free-flow conditions.

(2)

A duration greater than 20 s to ensure that each following segment possesses enough CF data.

Based on these criteria, 2100 segments of CF data were extracted, with 2060 segments manually verified. From this dataset, 1560 and 500 data segments were randomly selected as the calibration and validation sets, respectively.

The number of iterations is set to 350, and the calibration results of the parameters in the molecular force field following the model are shown in

Table 1. Parameters calibration results.

Parameters	Parameter Value
$\alpha$	0.7123
$\beta$	1.6753
${\lambda }_1$	28.7231
${\lambda }_2$	42.1964

.

Due to the lack of sufficient conditions and data for real-world measurements of CAV traffic flow, the differences between traditional traffic flow and CAV traffic flow are compared and analyzed, which results in the calibration of the parameters of the CAV CF model based on molecular force fields using measured data from HV CF behavior. Three sets of vehicle acceleration, $a$, safety distance $s_{CAV}$, and actual car-following distance $l$, are used to solve the equation; that is,

$$\mu \left(\frac{{s^{\beta }}_{CAV-i}}{{l_i}^{\beta +1}}-\frac{{s^{\alpha }}_{CAV-i}}{{l_i}^{\alpha +1}}\right)=a_i$$

(22)

For the prescribed function ${\psi }_i\left(\alpha ,\beta \right)$, let ${\psi }_1=a_2{a}_3\left(\frac{{s^{\beta }}_{CAV-1}}{{l_1}^{\beta +1}}-\frac{{s^{\alpha }}_{CAV-1}}{{l_1}^{\alpha +1}}\right)$, ${\psi }_2=a_1{a}_3\left(\frac{{s^{\beta }}_{CAV-2}}{{l_2}^{\beta +1}}-\frac{{s^{\alpha }}_{CAV-2}}{{l_2}^{\alpha +1}}\right)$, and ${\psi }_3=a_1{a}_2\left(\frac{{s^{\beta }}_{CAV-3}}{{l_3}^{\beta +1}}-\frac{{s^{\alpha }}_{CAV-3}}{{l_3}^{\alpha +1}}\right)$ function ${\psi }_i({\alpha }_1,{\beta }_1)$ in the ${\alpha }_1$, ${\beta }_1$, ${\mu }_1$ direction of the three-dimensional space surface composed of the intersection of ${\alpha }_1$, ${\beta }_1$ coordinates for the value that is the solution of ${\alpha }_1$, ${\beta }_1$, and then the value of ${\alpha }_1$, ${\beta }_1$ into Equation (22) to obtain the value of ${\mu }_1$; that is,

$$\mu =\frac{a_i}{\left(\frac{{s^{\beta }}_{CAV-i}}{{l_i}^{\beta +1}}-\frac{{s^{\alpha }}_{CAV-\mathrm{i}}}{{l_i}^{\alpha +1}}\right)}$$

(23)

The limited speed of highways is set to 33.3 m·s⁻¹ (i.e., 120 km·h⁻¹) and the critical speed for congested conditions is set to 21.67 m·s⁻¹ (i.e., 78 km·h⁻¹), and the latter is used as the benchmark speed for parameter calibration. The CAV response time is usually less than 0.2 s, and the maximum acceleration during CAV braking is taken to be 8 m·s⁻² with a 5 m vehicle length. The following distance of the CACC vehicle in a steady traffic flow at a free stream speed (i.e., 33.3 m·s⁻¹) is 26.98 m. The following safety distance coefficient = 0.2643 in the CACC car-following model, and the following safety distance in the CACC car-following model at a speed of 21.67 m·s⁻¹ is 14.09 m. Bringing the three sets of values into Equation (22) gives the values of ${\alpha }_1$, ${\beta }_1$, and ${\mu }_1$ as 2.241, 0.573, and 183.912, respectively.

5. Model Stability Analysis and Effectiveness Evaluation

To objectively evaluate the effectiveness of the molecular force field following the model developed in this paper, the M-MD model and the IDM model were selected for comparison. The M-MD model equation is expressed as follows:

$$a_{M-MD}={\omega }_1\left(2\frac{{s_n}^6}{l^7}-\frac{1}{l}\right){\left(\frac{s_n}{l}\right)}^6+{\omega }_2\left(1-\frac{v_n}{v_{n-1}}\right)$$

(24)

where $a_{M-MD}$ is the acceleration of the rear vehicle; ${\omega }_1$, ${\omega }_2$ is a parameter to be determined.

Based on the same dataset, the parameters of the M-MD model were calibrated according to the methodology used in the literature [11] and the results are shown in

Table 2. Calibration results of M-MD model parameters.

Motion State	${\mathit{\omega }}_1$	${\mathit{\omega }}_2$
regular driving	1.4792	9.4095
getting up to speed	0.9605	19.2781
decelerating and stopping	67.4112	0.0201

.

The IDM model [10] equation can be expressed as follows:

$$a_{IDM}=a_{max}\left[1-{\left(\frac{v_n}{v_d}\right)}^4-{\left(\frac{s_0+v_n{t}_d-\frac{v_n\left(v_n-v_0\right)}{2\sqrt{a_{max}\cdot a_c}}}{l}\right)}^2\right]$$

(25)

where $a_{IDM}$ is the rear vehicle acceleration; $v_d$ is the desired vehicle speed; $t_d$ is the safe following time distance; $a_{max}$ is the braking maximum acceleration; and $a_c$ is the braking comfortable acceleration. The parameters of the IDM model were calibrated using the same dataset, and the results are shown in

Table 3. Parameter value of IDM model.

Parameter	Parameter Meaning	Parameter Value
$a_{max}$	braking maximum acceleration	1.24 m·s−2
$a_c$	braking comfortable acceleration	1.89 m·s−2
$v_d$	desired vehicle speed	33.33 m·s−1
$s_0$	minimum vehicle distance	2.11 m
$t_d$	safe following time distance	1.44 s

.

The vehicle motion modeling is constructed in the simulation environments built on MATLAB R2019b. The front vehicle’s position, velocity, and acceleration data were set based on measured values, while the model simulated the rear vehicle by initializing its velocity. The simulation control step was set to 0.1 s. The results of the simulation experiments, including the measured data [36] and the acceleration profile of the rear vehicle, were plotted for comparison, as illustrated in Figure 9.

The experimental results indicate that both the M-MD model and the molecular force field CF model effectively capture the motion state changes in the front vehicle, producing a CF response closely resembling that of an actual vehicle. In contrast, the acceleration profile of the IDM model is relatively smooth but exhibits a large response variance, reflecting small acceleration fluctuations and the smooth driving characteristics of the self-driving vehicle.

For a more intuitive comparison of the fit of the model to the actual traffic scenario, the model error was calculated using the mean absolute error $E_{MAE}$ (mean absolute error, MAE) and the root mean square error $E_{RMSE}$ (root mean square error, RMSE), the formula of which is given as follows:

$$E_{MAE}=\frac{1}{N}{\displaystyle \sum _{i=1}^N\left|y_i-{\widehat{y}}_i\right|}$$

(26)

$$E_{RMSE}=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(y_i-{\widehat{y}}_i\right)}^2}}$$

(27)

where $N$ is the total amount of sample data; $y_i$ is the true value; and ${\widehat{y}}_i$ is the theoretical value. Both can reflect the model error scale more objectively; the smaller the error, the better the fit. The model output values were evaluated for errors using data with a sample size of 2060, and the results are shown in

Table 4. Model fitting error.

Following Model	${\mathit{E}}_{\mathit{M}\mathit{A}\mathit{E}}$	${\mathit{E}}_{\mathit{R}\mathit{M}\mathit{S}\mathit{E}}$
M-MD model	1.5760	0.7337
IDM model	0.7611	0.5905
molecular force field model	0.4012	0.5129

.

The analysis based on the error evaluation results indicated that, in terms of model fit, the molecular force field model outperforms the IDM model, which, in turn, outperforms the M-MD model. This suggests that the molecular force field of the CF model more accurately simulates the CF interaction behavior in road traffic scenarios.

Simulation experiments were designed to introduce perturbations to observe and record the evolution of traffic flow. A steady-state traffic flow of 40 vehicles was initialized at a speed of 21.67 m/s. The applied perturbation destabilized the traffic, allowing for the observation of changes in the motion state of CAVs. The simulation step size was set to 0.01 s, and the simulation duration was set to 120 s. The results are illustrated in Figure 10.

When the CAVs in the platoon were perturbed, the velocity fluctuations propagated backward, and the degree of fluctuation gradually decreased, mirroring real-world scenarios. The numerical simulation results indicate that after perturbation, the CAV traffic flow in both models can recover to a stable driving state relatively quickly, demonstrating local stability in each model. Within approximately 30 s, the speed in both models’ CAV traffic flows tended to stabilize, and speed fluctuations gradually diminished, indicating superior asymptotic stability. Compared to the IDM model, the CAV traffic flow under the molecular force field CF model recovered its previous driving state in a shorter time, exhibiting lower overall velocity volatility and operating more stably and efficiently. The simulated CF scenarios closely matched actual traffic flow behavior. The proposed steady-state control strategy, based on the speed and distance between vehicles, aligned well with real traffic environments. This strategy significantly reduces the speed fluctuations caused by drivers’ delayed information perception and processing, thereby enhancing driving safety.

6. Conclusions

The main conclusions drawn from the study of CAVs in complex traffic scenarios are as follows:

By treating CAVs as self-driven particles constrained by the multi-dimensional elements of human–vehicle–road–environment interactions in complex traffic scenarios, the characteristics of CF behaviors based on molecular force field relations can be analyzed. This approach simplifies dynamic CF behaviors into the process of maintaining a dynamically demanded safe distance to the rear vehicle. The molecular force field’s potential function is employed to address the limitations of existing molecular dynamics-based CF models, smoothing the transition between repulsive and attractive forces and introducing a velocity synergy term. This leads to the establishment of a molecular force field CF model.

A comparative analysis of the M-MD and IDM models shows that the vehicle acceleration results from the molecular force field model are significantly lower than those derived from actual data, with reduced overall traffic flow fluctuations when the speed of the leading vehicle oscillates. The molecular force field model demonstrates superior accuracy and stability, effectively reflecting the actual motion conditions of CF.

The theory of CF based on molecular force fields provides a robust model and theoretical reference for large-scale CACC operations. Given the complexity and variability of the road traffic environment, CF is influenced by numerous factors. Future research will focus on improving the existing model by fully considering the factors that constrain the stability of CF behavior, such as the impact of communication delays in connected autonomous driving environments. A multi-dimensional and multi-state vehicle interaction behavior model will be developed, and the molecular force field CF model will be extended to address complex scenarios involving connected heterogeneous vehicle groups with multi-factor constraints. This will facilitate intelligent collaboration and the safe control of connected heterogeneous traffic flows.