Research on Short-Term Driver Following Habits Based on GA-BP Neural Network

Abstract

The current commercial intelligent driving systems still take the optimal strategy judged by the machine to be the only goal. Therefore, in order to improve the driving experience of the intelligent driving following scene, based on the assumption that environmental factors remain unchanged for a short time, five important parameters affecting the following scene are selected through correlation analysis, and vehicle-following research is carried out. This paper adopts a driver-following model based on a Genetic Algorithm (GA)-optimized Back Propagation (BP) neural network. Based on the data of next-generation simulation (ngsim), this paper selects vehicle 32 (32 represents the ID of the vehicle in the ngsim project) as the main vehicle in order to study short-term driving habits. A BP neural network is built using MATLAB; 60% of the data of vehicles 32 and 29 is used for the training set, 20% is used for the verification set, and 20% for the test set. Because short-term prediction requires high timeliness, the genetic algorithm is used to optimize the initial weights of the neural network, which not only accelerates the convergence speed but also plays a role in avoiding the local optimal solution. The experimental results show that compared with the traditional stimulus-response vehicle-following model, this model has a following ability that is more in line with the driver’s driving habits in terms of ensuring following safety.

1. Introduction

The statistical data from road traffic accidents show that accidents for which drivers are directly responsible account for a large proportion of the total number of accidents [1]. With the advancement of global intellectualization, the intellectualization of vehicles will also effectively reduce the accident rate [2]. Modern intelligent driving systems obtain environmental information through advanced radar, cameras, and other sensors, and realize assisted driving or replace human driving through artificial intelligence carried on a high-power vehicle specification intelligent driving chip. Advanced driver assistance systems are developing rapidly. From AEB in the early days to autopilots today, people are no longer satisfied with passive safety technology [3] but continue to climb to the level of active safety technology. The concept of vehicle road collaboration [4] has long been proposed, but it has not been realized until now due to various restrictions. Now, the birth of the 5G network makes it more possible. Nowadays, the mass-produced advanced driver assistance system (ADAS) [5] generally does not meet personalized needs. To meet the personalized needs of drivers, it is necessary to study their driving habits and establish driving habits model analysis rules, so as to further improve the function of ADAS.

As one of the most common scenarios in the process of vehicle driving, domestic and foreign scholars have conducted much research on the topic. A car-following model has always been a research hotspot of traffic flow theory. With the development of assisted driving and autonomous vehicles, the car-following model, as one of many driver models, has been widely used in the field of intelligent vehicles. The concept of the car-following model was first proposed by Reuschel. He believed that the expected distance between cars is linear to the speed of the cars behind [6,7]. Elsewhere in the literature, the authors of [8] divide car-following models into five categories: data-driven models, safe distance models, stimulus-response models, physiological and psychological models, and cellular automata models (as shown in Figure 1).

A BP (backpropagation) neural network is a concept proposed by numerous scientists, led by Rumelhart and McClelland in 1986. The concept is widely used in data-driven models. It is a multilayer feedforward neural network, trained according to the error backpropagation algorithm, and it is one of the most widely used neural network models. It can be used as a mapping equation [9] without needing to confirm the relationship between the input and output in advance. By learning the law of a given training set, the output is close to the target value when a given input value is achieved. Its basic principle includes two parts: the forward propagation of the signal, where the input signal is finally output through each node of each hidden layer, and the backward propagation of error. This is gradually adjusted from the back end to the front end by comparing the error between the output value and the expected value, and the weight of each layer is adjusted through the error of each layer, which is reciprocating, so that the error decreases in a gradient way, which is most in line with the requirements [10].

As early as 1953, the pipes model provided a structure by which to maintain a dynamic safe following distance from the vehicle in front. With the proposal of the GM model and the Herman model, the car-following model theory has been further improved. The GM model takes into account the influence of relative speed on the acceleration of the rear vehicle, while the Herman model takes into account the common influence of the multiple vehicles in front.

The safe distance car-following model simulates the driving characteristics of the driver to a certain extent, offering strong logic and explanations, but it does not consider the structured mathematical model, which may cause large errors in describing the driving behavior and makes it difficult to simulate the nonlinearity and uncertainty of the driving process [11]. A model based on data-driven methods can simulate different driving habits and predict different drivers’ car-following behavior in different environments by fitting the input characteristic parameters, but it is easy to ignore the state relationship between the internal characteristic parameters [12,13], and the key feature mining ability is not perfect. At the same time, an a priori theory related to driver characteristics is required, which is prone to overfitting problems. Therefore, this paper proposes a short-term assumption that environmental factors are invariant, which is used to solve the problem that environmental factors are not exhaustive. At the same time, the physical model and the data-driven model are integrated; that is, based on the stimulus-response model, the BP neural network is the core, and the GA algorithm is used to further optimize the network. Finally, the model is validated using ngsim data.

Based on the stimulus-response model, a BP neural network is used to learn driving habits in order to optimize the traditional stimulus-response car-following model and increase the personalized driving needs of drivers. As a data-driven model, a BP neural network needs a large amount of data as support. In this paper, ngsim data are used for research to analyze drivers’ driving habits and their influencing factors. Based on a BP neural network optimized by a genetic algorithm, a car-following model for drivers’ driving habits is established. Compared with the traditional safe-distance model, the novel model has better habit-learning and prediction ability through experiments, and it can greatly enhance the human–computer interaction experience of the automatic vehicle-following function.

2. Data Acquisition and Analysis

Data Sources

The data in this paper mainly come from next-generation simulation (ngsim), which is widely used in intelligent driving research [14]. The data sets were collected from four different regions of the United States, namely, the southbound US 101 highway in California, the Lankershim Boulevard map in Los Angeles, California, the eastbound 1-80 highway in Emeryville, California, and the Peacetree in Atlanta, Georgia. The data contained in this data set are shown in

Table 1. Ngsim field data description.

Column Name	Description	Unit
Vehicle_ID	Vehicle identification number.	-
Frame_ID	The number of frames of the data at a certain time.	0.1 s
Total_Frame	The total number of frames of this vehicle in this dataset.	0.1 s
Global Time	Time stamp.	ms
Local_X	Abscissa of the center of the front of the vehicle.	Feet
Local_Y	Ordinate of the center of the front of the vehicle.	Feet
Global_X	Abscissa of the vehicle in the global coordinate system.	Feet
Global_Y	Ordinate of the vehicle in the global coordinate system.	Feet
v_length	Vehicle length.	Feet
v_width	Vehicle width.	Feet
v_Class	Vehicle class: 1—Motorcycle; 2—Light-Duty Vehicle; 3—Large Vehicle.	-
v_Vel	Instantaneous speed of the vehicle.	Feet/s
v_Acc	Instantaneous acceleration of the vehicle.	Feet/s2
Lane_ID	Current lane position of the vehicle.	-
Preceding	Vehicle ID of the vehicle in front of the same lane.	-
Following	Vehicle ID of the vehicle following the vehicle in the same lane.	-
Space_Headway	Distance from the front center of the vehicle to the front center of the front vehicle.	Feet
Time_Headway	Time required for two vehicles to pass the same position	s

. Since the units in the table are not convenient for calculation and observation, all the data in this paper have been converted into international system units. The time unit is seconds, the distance unit is in meters, the speed unit is m/s, and the acceleration unit is m/s². However, it should be clarified here that the data unit in the source table is the unit shown in the table. All the data in this paper have been converted before calculation.

The actual ngsim data set contains many parameters, but for the purposes of this experiment, it is necessary to select those parameters with greater influence. Therefore, the author conducted preliminary experiments on many parameters before beginning the process. Because of the large number of experiments and because the steps are too repetitive, the experimental process is not shown here; six parameters with greater influence are selected.

Based on the initial test results, this paper preliminarily selected six parameters of v_Vel, v_Acc, Time_Headway, Space_Headway, Preceding_Vel, and Preceding_Acc for subsequent analysis.

3. Adaptive Processing

3.1. Data Preprocessing

The ngsim data set is used as the observation data. Due to the influence of equipment, environment, and human factors, there are certain errors and noises in the data set. If this data were directly used for analysis, there would be data abnormalities. Therefore, it is necessary to filter the data. In this paper, Kalman filtering is used to reduce the noise of the data [15].

$${\mathrm{P}}_{\overline{\mathrm{k}}}={\mathrm{AP}}_{\mathrm{k}-1}{\mathrm{A}}^{\mathrm{T}}+\mathrm{Q}$$

(1)

$${\mathrm{K}}_{\mathrm{k}}=\frac{{\mathrm{P}}_{\overline{\mathrm{k}}}{\mathrm{H}}^{\mathrm{T}}}{{\mathrm{HP}}_{\overline{\mathrm{k}}}{\mathrm{H}}^{\mathrm{T}}+\mathrm{R}}$$

(2)

$${\hat{\mathrm{x}}}_{\mathrm{k}}={\hat{\mathrm{x}}}_{\hat{\mathrm{k}}}+{\mathrm{K}}_{\mathrm{k}}\left({\mathrm{z}}_{\mathrm{k}}-{\mathrm{H}\hat{\mathrm{x}}}_{\overline{\mathrm{k}}}\right)$$

(3)

$${\mathrm{P}}_{\mathrm{k}}=\left(\mathrm{I}-{\mathrm{K}}_{\mathrm{k}}\mathrm{H}\right){\mathrm{P}}_{\overline{\mathrm{k}}}$$

(4)

Here, ${\hat{\mathrm{x}}}_{\mathrm{k}-1}$ and ${\hat{\mathrm{x}}}_{\mathrm{k}}$ represent the posterior state estimates at time k−1 and time k, respectively, which calculation is also called the optimal estimation.

${\hat{\mathrm{x}}}_{\overline{\mathrm{k}}}$ represents a priori state estimation value at time k, which is the result of optimal estimation prediction based on the previous time (time k−1).

${\mathrm{P}}_{\mathrm{k}}$ and ${\mathrm{P}}_{\mathrm{k}-1}$ represent the posterior estimated covariance at time k−1 and time k, indicates the uncertainty of the state.

${\mathrm{P}}_{\overline{\mathrm{k}}}$ represents the a priori estimated covariance at time k.

H represents the transformation matrix from the state variable to the measurement (observation) and represents the relationship between the state and the observation. In the Kalman filter, it is a linear relationship, which is responsible for converting the m-dimensional measurement value to the n-dimensional measurement value, to make it conform to the mathematical form of the state variable.

${\mathrm{z}}_{\mathrm{k}}$ represents the measured value.

${\mathrm{K}}_{\mathrm{k}}$ represents the filter gain matrix, also known as the Kalman gain or Kalman coefficient.

A represents the state transition matrix.

Q represents the process excitation noise covariance (system process covariance).

R is the measurement noise covariance.

B represents a matrix in which the input quantity is converted to the state quantity.

${\mathrm{z}}_{\mathrm{k}}-{\mathrm{H}\hat{\mathrm{x}}}_{\overline{\mathrm{k}}}$ represents the residual error between the actual observation and the predicted observation and corrects the prior (prediction), together with the Kalman gain, to obtain the posterior value.

Through the above Kalman filtering process, the original data set in ngsim is processed for subsequent research. The filtering results are shown in Figure 2, below.

3.2. Correlation Analysis

In this paper, the Pearson correlation coefficient, Kendall correlation coefficient, and Spearman correlation coefficient are mainly used for correlation analysis [16,17]. The Pearson correlation coefficient, as a coefficient reflecting the linear relationship between two variables, is mainly used in this paper to analyze whether there is a linear relationship between the six variables screened through empirical analysis. If there is a strong correlation between them, this will reduce the training effect and efficiency of the subsequent neural networks. Therefore, it is necessary to compare the Pearson correlation coefficient to reveal those variables with strong correlations. Since the data operation in this paper is based on the MATLAB platform, the Pearson coefficient correlation function provided by MATLAB is also used for analysis, by which method the results shown in Figure 3 below are obtained.

By analyzing the information in Figure 3, when the value of the Pearson coefficient is greater than 0.05, it can be considered that there is a certain linear relationship between the two variables. It can be seen that there is a certain linear relationship between Time_Headway and v_Acc and Preceding_Acc, while there is an obvious linear relationship between the other five variables. Therefore, Time_Headway should be excluded from the subsequent study and the other five variables should be retained. Similarly, the Kendall correlation coefficient and Spearman correlation coefficient are still analyzed with MATLAB’s own functions; the results are shown in Figure 4 and Figure 5, below.

By observing the Kendall correlation coefficient analysis results and Spearman correlation coefficient analysis results, it can be seen that the v_Vel variable has a large correlation with Space_Headway and Preceding_Vel, and it can be preliminarily concluded that:

(1) v_Vel, as expected by the driver, is greatly affected by the Space_Headway and the Preceding_Vel, and it is found from the data changes that the greater the Space_Headway, the greater the expected v_Vel, and the greater the Preceding_Vel, the greater the expected v_Vel.

(2) The expected v_Acc of the driver is greatly affected by the Preceding_Acc, and it can be seen from the data change that the greater the Preceding_Acc, the greater the expected acceleration.

(3) The driver’s desired Space_Headway is greatly affected by the Preceding_Vel and the v_Vel. From the data changes, it is found that the larger the Preceding_Vel and the smaller the v_Vel, the greater the desired Space_Headway. The above three conclusions are fully consistent with the stimulus-response model. The driver’s response behavior (v_Acc) is mainly affected by the Preceding_Vel (excitation). The v_Vel is mainly balanced with the Preceding_Vel, and the Space_Headway is related to the speed of the two cars.

Therefore, the data show that the driver’s behavior is not in good agreement with the safety distance model [18]. It is obvious that the driver’s style will greatly affect the most appropriate car-following model-matching [19]. This paper is based on the stimulus-response model.

4. Car-following Model

4.1. Stimulus-Response Model

The basic idea of the stimulus-response model is to define the behavior of the vehicle ahead as a stimulus s; the sensitivity of the following car to the stimulus is defined by the coefficient C, and the following car’s behavior is expressed by r:

r = Cs

(5)

There is a reaction time for the car following it to stimulate the vehicle in front. Here, t_r is used to define the reaction time. The changes to the acceleration, relative speed, and headway of the car in front can be regarded as the stimulation of the car in front to the car following behind. At the same time, the acceleration change of the car following behind (that is, the driver’s control of the accelerator pedal and brake pedal) is defined as the behavior of the car following it, under the excitation of the car ahead, from which we can establish:

$${\mathrm{a}}_{\mathrm{r}}\left(\mathrm{t}+{\mathrm{t}}_{\mathrm{r}}\right)=\frac{{\mathrm{Cv}}_{\mathrm{r}}^{{\mathrm{m}}_1}\left(\mathrm{t}\right)}{{\mathrm{s}}^{{\mathrm{m}}_4}\left(\mathrm{t}\right)}{[{\mathrm{a}}_{\mathrm{f}}\left(\mathrm{t}\right)-{\mathrm{a}}_{\mathrm{f}}\left(\mathrm{t}-{\mathrm{t}}_{\mathrm{r}}\right)]}^{{\mathrm{m}}_2}{\left. [{\mathrm{v}}_{\mathrm{f}}\left(\mathrm{t}\right)-{\mathrm{v}}_{\mathrm{r}}\left(\mathrm{t}\right)\right]}^{{\mathrm{m}}_3}.$$

(6)

As the key parameters in Equation (6), m₁, m₂, m₃, and m₄ are very difficult to obtain more accurate values.

4.2. Driver Habit Learning Model Based on GA-BP Neural Network

4.2.1. BP Neural Network

Since the parameters given above are difficult to obtain accurately, the idea is changed and the Equation (6) is fitted by BP neural network to achieve an approximate effect. The BP neural network used in this paper is a common feedforward neural network. It transfers data through the activation function and weight of neurons and trains the neural network by using the algorithm of error forward transfer and gradient descent to achieve the purpose of fitting the objective function. Scholars have also proven that a well-trained three-layer neural network can realize the fitting of any function. The structure diagram of BP neural network is shown in Figure 6.

The calculation method of data passing through a single neuron in the network is shown in Figure 7. The data transmitted from the first n neurons are weighted and summed, and then enter the next n neurons through the activation function. The neuron used in this paper is also a three-layer structure, in which the number of neurons in the input layer is 5. The input data included can be established from the theory of the stimulus-response model: following vehicle speed, following vehicle acceleration, Preceding_Vel, front vehicle acceleration, and headway. The output is the acceleration of the following car at the next moment. To determine the number of neurons in the hidden layer, we first determine the range through the empirical Equations (7) and (8).

$$\mathrm{n}=2\mathrm{m}+1$$

(7)

where n is the number of hidden layer nodes and m is the number of inputs.

$${\mathrm{n}}_1=\sqrt{\mathrm{n}+\mathrm{m}}+\mathsf{\alpha }$$

(8)

where n₁ is the number of hidden layer nodes, n is the number of input quantities, m is the number of output quantities, and α is a number between 1 and 10.

From the above two empirical formulas, it can be preliminarily determined that the number of hidden layer nodes is between 3 and 13. After repeated tests, it was confirmed that the number was 12, which showed a good effect.

4.2.2. Genetic Algorithm (GA)

The genetic algorithm (GA), as a randomized search method that evolved from natural evolution, is widely used in various fields. In this paper, it is used to optimize the weights and thresholds of BP neural networks. The basic flow is shown in Figure 8.

(1)

Population initialization:

The individual coding method is real-number coding, and each individual is a real-number string, which consists of four parts: the connection weight between the input layer and the hidden layer, the threshold value of the hidden layer, the connection weight between the hidden layer and the output layer, and the threshold value of the output layer. The individual coding method contains all the weights and values of the neural network. When the network structure is known, it can form a definite neural network.

(2)

Fitness function:

We take the absolute value of the difference between the output ${\mathrm{o}}_{\mathrm{i}}$ predicted by the BP neural network and the expected output ${\mathrm{y}}_{\mathrm{i}}$ as the fitness value of the current individual, as shown in Equation (9).

$$\mathrm{F}={\mathrm{k}}_1\left({\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}\mathrm{abs}\left({\mathrm{y}}_{\mathrm{i}}-{\mathrm{o}}_{\mathrm{i}}\right)\right)$$

(9)

Here, ${\mathrm{k}}_1$ is the coefficient.

(3)

Selection operation:

The strategy of selecting individuals based on the proportion of fitness is selected here. The probability i of individuals ${\mathrm{p}}_{\mathrm{i}}$ being selected is as shown in Equation (10).

$${\mathrm{p}}_{\mathrm{i}}=\frac{\mathrm{k}/{\mathrm{F}}_{\mathrm{i}}}{{{\displaystyle \sum }}_{\mathrm{j}=1}^{\mathrm{N}}{\mathrm{f}}_{\mathrm{j}}}$$

(10)

F_i denotes the fitness of an individual i, and N denotes the number of populations.

(4)

Crossover operation:

The individual coding adopted real numbers, so the real number crossover method can be directly used, and the chromosome ${\mathrm{a}}_{\mathrm{k}}$ and the position ${\mathrm{a}}_{\mathrm{l}}$ are crossed by Equation (11).

$$\left\{\begin{array}{c}{\mathrm{a}}_{\mathrm{kj}}={\mathrm{a}}_{\mathrm{kj}}\left(1-\mathrm{b}\right)+{\mathrm{a}}_{\mathrm{lj}}\mathrm{b}\\ {\mathrm{a}}_{\mathrm{lj}}={\mathrm{a}}_{\mathrm{lj}}\left(1-\mathrm{b}\right)+{\mathrm{a}}_{\mathrm{kj}}\mathrm{b}\end{array}\right.$$

(11)

where b is a random number between 0 and 1.

(5)

Variation operation:

Select the genes $\mathrm{j}$ in the individual $\mathrm{i}$ for variation operation. The operation mode is shown in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8.

$${\mathrm{A}}_{\mathrm{ij}}=\left\{\begin{array}{c}{\mathrm{a}}_{\mathrm{ij}}+({\mathrm{a}}_{\mathrm{ij}}-{\mathrm{a}}_{\max })\times \mathrm{f}\left(\mathrm{g}\right) \mathrm{r}>0.5\\ {\mathrm{a}}_{\mathrm{ij}}+\left({\mathrm{a}}_{\min }-{\mathrm{a}}_{\mathrm{ij}}\right)\times \mathrm{f}\left(\mathrm{g}\right) \mathrm{r}\le 0.5\end{array}\right.$$

(12)

$$\mathrm{f}\left(\mathrm{g}\right)={\mathrm{r}}_2{\left(1-\mathrm{g}/{\mathrm{G}}_{\max }\right)}^2$$

(13)

In the formula, ${\mathrm{a}}_{\max }$ and ${\mathrm{a}}_{\min }$ represent the maximum and minimum values in the gene ${\mathrm{a}}_{\mathrm{i}}$, $\mathrm{r}$ is a random number between 0 and 1, ${\mathrm{r}}_2$ is a random number, $\mathrm{g}$ is the current iteration number, and ${\mathrm{G}}_{\max }$ is the current maximum evolution number.

5. Simulation Verification

5.1. Simulation Scenario

The simulation scene is set as the simulation car-following scene, and the setting conditions are as follows:

(1)

Only two vehicles (the preceding vehicle and main vehicle) exist in this scenario;

(2)

The driving line is in a straight line (the same as for the ngsim data set);

(3)

The driving speed of the preceding vehicle in the simulation process uses the Preceding_Vel data in ngsim;

(4)

In the experimental group, the driving strategy of the main vehicle was decided by the trained neural network, while in the control group, the driving strategy of the main vehicle was a stimulus-response model;

(5)

An instantaneous change in vehicle acceleration is allowed, but speed is not allowed.

5.2. Car-following Simulation

5.2.1. Explaining This Assumption

The assumption is that environmental factors remain unchanged for a short time.

The definition of car-following habits in this paper is somewhat special. First of all, we recognize that there may be large differences in driving habits between different drivers (for example, radical and conservative driving). Therefore, many papers will classify drivers based on general drivers and set a default that the habits of a certain driver will remain unchanged [20]. In fact, this is not the case.

Real driving experience tells us that the driver’s behavior is not only determined by the several parameters mentioned in the text but is also influenced by numerous factors. For example, when a large vehicle is running side by side in the right-hand lane, the driver’s choice will tend to be to accelerate to overtake the vehicle in front, rather than to follow as usual. In this case, the driver’s driving habits will change with the environment, and the influencing factors cannot be enumerated one by one. Thus, this paper puts forward a hypothesis: we assume that the decision-making environment (environmental factors) faced by the driver will not change in a short time. As shown in the Figure 9, the driver’s driving state will remain basically unchanged before the first emergency occurs.

5.2.2. Application Scheme

This paper only studies a short-term state (that is, the default environmental elements do not change), so an additional scheme is proposed to make the model theoretically applicable to the real scene. The scheme is as follows:

(1)

To maintain data collection during the whole process of the driver driving the vehicle; this data should at least include the five main parameters mentioned in the paper.

(2)

Using the latest data collected, the model is constantly updated over time (blue cube in Figure 10).

(3)

The model used in the automatic car-following function is the green cube in Figure 10. This model will be updated in the following three cases:

• Automatic update with a cycle of 5 unit times.
• Driver’s active intervention (the automatic car-following function is on).
• When the automatic car-following function is not enabled, the system will also use the model (green cube) to predict the car-following status in real time and compare it with the real status of the vehicle. When the error is greater than the threshold, the model will be updated. (If the vehicle leaves the following state for 5 s, the system stops).

(4)

The automatic updating speed of the model is determined by the computing power of the vehicle VCU.

5.3. Car-following Simulation

In this paper, the simulation is based on MATLAB. Because the car-following behavior in this paper is only straight-line driving, lateral movement control is not considered. The simulation results are shown in the following figures.

Figure 11 is the simulation result diagram of the improved car-following strategy in this paper, and Figure 12 is the simulation result diagram of the traditional speed excitation stimulus-response model.

Because this paper is based on the stimulus-response model, the following habits are defined as the driving strategy (acceleration, deceleration or maintenance) that is selected by a driver when facing a specific scene (event), and the intensity of acceleration and deceleration. Therefore, acceleration and speed are selected as the main indicators, and Space_Headway is selected to observe the safety situation.

For the convenience of analysis, the average error is used here for analysis:

$${\mathrm{e}}_{\mathrm{a}}=\frac{1}{\mathrm{t}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{t}}|{\mathrm{a}}_0\left(\mathrm{i}\right)-{\mathrm{a}}_{\mathrm{s}}\left(\mathrm{i}\right)|$$

(14)

$${\mathrm{e}}_{\mathrm{v}}=\frac{1}{\mathrm{t}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{t}}|{\mathrm{v}}_0\left(\mathrm{i}\right)-{\mathrm{v}}_{\mathrm{s}}\left(\mathrm{i}\right)|$$

(15)

$${\mathrm{e}}_{\mathrm{s}}=\frac{1}{\mathrm{t}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{t}}|{\mathrm{s}}_0\left(\mathrm{i}\right)-{\mathrm{s}}_{\mathrm{s}}\left(\mathrm{i}\right)|.$$

(16)

In Equation (14), ${\mathrm{e}}_{\mathrm{a}}$ represents the average error of acceleration, t indicates the number of test discretization, ${\mathrm{a}}_0\left(\mathrm{i}\right)$ represents the actual acceleration at time i, and ${\mathrm{a}}_{\mathrm{s}}\left(\mathrm{i}\right)$ represents the simulated acceleration at time i. Speed and space are the same in Equations (15) and (16).

The average error of the two models is calculated as follows.

In the test of the GA-BP model, the average error ${\mathrm{e}}_{\mathrm{a}1}$ is 0.8715 m/s², the average error ${\mathrm{e}}_{\mathrm{v}1}$ is 0.6932 m/s, and the average error ${\mathrm{e}}_{\mathrm{s}1}$ is 1.2056 m. In the other model, the average error ${\mathrm{e}}_{\mathrm{a}2}$ is 1.2198 m/s², the average error ${\mathrm{e}}_{\mathrm{v}2}$ is 1.1263 m/s, and the average error ${\mathrm{e}}_{\mathrm{s}2}$ is 7.9822 m.

(1) Obviously, this improved strategy has less deviation from the actual data in the control of Space_Headway and can better fit the driver’s driving habits.

(2) It can be seen from the acceleration change diagram that the simple speed excitation stimulus-response model can make the vehicle speed quickly maintain a small error with the vehicle in front. However, the disadvantages are also obvious and will cause large disturbances, and the acceleration change will be very frequent, which would not meet the comfort criteria.

(3) In addition, the improved model also has a certain integration capability. The three acceleration peaks near 5 in Figure 11 are redundant and are integrated into a gentle acceleration peak by this strategy, which can effectively increase the occupants’ comfort and meet the driving intention. The two acceleration peaks and one deceleration peak near 20 are integrated into a single long-lasting and gentle acceleration peak. The author believes that the deceleration peak is not the driving intention, being due to the road conditions or the driver’s unintentional behavior, so it is integrated into the acceleration peak here.

(4) In terms of safety, the vehicle distance is relatively safe in the whole simulation process, and there is no collision.

In order to test the safety performance of the model under the edge conditions, the motion state of the front vehicle is set to continuously accelerate and decelerate rapidly at an acceleration of 6 m/s, and the results are shown in Figure 13. It can be seen that although there are large fluctuations in the v_Vel and v_Acc, the Space_Headway remains good and there is no collision.

6. Conclusions

In this paper, a short-term environmental factor invariant assumption is proposed and an application scheme is designed. Based on the ngsim data, through the correlation analysis of the Pearson correlation coefficient, Kendall correlation coefficient, and Spearman correlation coefficient of the selected data segment, the parameters related to the car-following strategy were obtained, and the stimulus-response model with the highest matching degree with the driver was selected for later research. The model is trained and adjusted by ngsim data, and an improved car-following model was established by using a BP neural network based on the genetic algorithm. The results show that the improved stimulus-response model can truly reflect the driver’s intention and ensure safety.

However, this paper still has some shortcomings. The model data in this paper come from NGSIM, while the data duration of a driver is short, often only a few tens of seconds, which makes the simulation of the application scheme impossible. In the follow-up study of this topic, we may try to use real vehicles for testing. In addition, ngsim data are all from the United States, and whether there are differences between American drivers and Chinese drivers in this paper must be ignored for the time being, but should be considered in the follow-up real-vehicle experiments. Although the assumptions and schemes proposed in this paper have not been verified in terms of real vehicle verification, they can still offer other researchers new ideas. At the same time, it addresses the problem that the driver characteristics change with time in the car-following problem to a certain extent.