Vehicle Lateral Control Based on Dynamic Boundary of Phase Plane Based on Tire Characteristics

Abstract

Lateral control is an essential safety control technology for autonomous vehicles, but the effectiveness of lateral control technology relies heavily on the precision of vehicle motion state judgements. In order to achieve accurate judgements of the vehicle motion state and to improve the control effectiveness of vehicle maneuverability and the stability controller, this paper starts with an analysis of phase plane stability. A simulation analysis is conducted to investigate the effect of the vehicle steering angle of the front wheels, the longitudinal velocity, and the tire–road adhesion coefficient on the boundary of the stability area. The stable area of the phase plane was partitioned using the proposed novel quadrilateral method, and we established a stability area regression model using machine learning methods. We analyzed the inherent connection between the lateral tire forces and the principles of vehicle maneuverability and stability control, indirectly combining the characteristics of tire forces with vehicle maneuverability and stability control. An allocation algorithm for maneuverability and stability control was designed. A co-simulation indicates that the vehicle stability controller not only accurately assesses the motion state of the vehicle but also demonstrates a considerably better performance in maneuverability and stability control compared to a controller using the traditional partitioning method of stable regions. The suggested allocation method enhances vehicle maneuverability and stability by enabling a seamless transition between the two and improving the effectiveness of stability control.

1. Introduction

As autonomous technology continues to advance, the industrial implementation of autonomous vehicles is steadily becoming a reality, and active safety technology is an indispensable control technology in the process of the industrialized application of autonomous vehicles [1]. Vehicle stability control technology is a crucial component of active safety technology; it plays a vital role in ensuring the safety of both the vehicle and its passengers [2]. Car motion control technology utilizes motion information to judge the vehicle motion state and achieves movement control by controlling the vehicle chassis system; stability control technology ensures the safety of autonomous driving vehicles.

The car’s stability control system takes into account the vehicle’s running conditions and external environmental factors to ensure stability and safety. This is achieved through the implementation of vehicle control strategies and algorithms. The car motion status judgement module determines when the operating system intervenes in car motion, and an accurate judgement of the vehicle motion state is an important prerequisite for the control system to effectively operate. Strict stability judgment conditions can result in frequent interventions from the stability control system, disrupting the vehicle’s normal driving. On the contrary, if the stability judgment conditions are too lenient, then the stability control system may intervene too late to maintain car safety. Thus, it is crucial to judge, accurately and in a timely manner, the vehicle motion state during the vehicle stability control process. Nowadays, there are several methods for judging vehicle stability, including the $\beta$ method [3], the threshold method [4], the phase plane method [5,6,7,8,9,10,11,12,13], etc. [14,15,16]. The phase plane method is commonly utilized for car stability analysis due to its ability to characterize the dynamic performance characteristics of the vehicle system. As a result, it finds extensive applications in the analysis of, and research into, vehicle maneuverability and stability.

Three primary types of phase plane method have been classified: $\beta -\dot{\beta }$, $\beta -\gamma$, and ${\nu }_y-\gamma$. Due to the limitations of the $\beta -\gamma$ method in accurately judging a car’s stable status under unstable circumstances with small fluctuations in yaw rate, $\beta -\dot{\beta }$ is more widely utilized in phase plane analyses [6]. Tang H et al. [7] conducted a study on the effects of the car steering angle on the stable area. They established a stability region library for $\beta -\dot{\beta }$ phase plane analyses. Li Q et al. [8] employed the bilinear method to partition the stability region of $\beta -\dot{\beta }$. They also established a functional relationship between the steering angle, longitudinal speed, adhesion coefficient, and stable region. Liu J et al. [9] established the $\beta -\dot{\beta }$ and $\beta -\gamma$ phase planes and combined the characteristics of both portraits to judge the vehicle stability state. The ${\nu }_y-\gamma$ phase plane method is also a commonly used method for vehicle stability analysis [10,11,12,13]. Zhu J et al. [10] analyzed the effects of the longitudinal speed, front wheel steering angle, and road adhesion coefficient on the system’s static bifurcation characteristics, redefining the stable region of the ${\nu }_y-\gamma$ phase plane and providing theoretical support for the applications of vehicle handling and stability. Sun W et al. [11] analyzed the impact of the longitudinal speed and front wheel steering angle on the stable region of the ${\nu }_y-\gamma$ phase plane; the genetic algorithm and phase space analysis methods were used to calculate the system’s equilibrium points under different running conditions. Alves J A V et al. [12] considered the effects of the front wheel steering angle, center of gravity, longitudinal speed, and road adhesion on vehicle dynamics; a method to solve the lateral stable region of the ${\nu }_y-\gamma$ phase plane was proposed, while an electronically controlled stability system, with active front steering as actuation, was implemented. Zhu Z et al. [13] used the sum-of-squares programming method to estimate the stable region of the ${\nu }_y-\gamma$ phase plane, and the effects of different driving conditions on the stable region were studied. The approximate dynamic stability boundary was established for different steering angle inputs, and a new vehicle dynamic stability controller was designed based on the Lyapunov function, improving the vehicle’s stability and dynamic performance.

According to the references, it is known that in a phase plane analysis, especially in the $\beta -\dot{\beta }$ phase plane and ${\nu }_y-\gamma$ phase plane, the front wheel steering angle, longitudinal velocity, and road adhesion coefficient are key parameters that affect the vehicle stability region. It is necessary to establish the mapping relationship between the influencing factors and the stability region and to accurately divide the stability region, which is essential for stability control of the vehicle. However, most studies have adopted the establishment of a stable region library through the look-up table method to judge the stable regions under different conditions in vehicle stability analysis using phase planes. Although the look-up table method can satisfy the accuracy requirements of the stability analysis, it cannot handle complex multi-dynamic characteristic datasets well; machine learning methods have unique advantages in dealing with such data.

Machine learning is a branch of artificial intelligence that establishes models by analyzing and learning from data, to make decisions or predictions, or improve behavior. Machine learning has been applied in various fields, such as medicine [17], finance [18], agriculture [19], industry [20,21,22,23], etc. In the field of vehicle control [24,25,26,27,28], machine learning methods have also been successfully applied. Xie F et al. [24] proposed an adaptive sliding mode controller based on an improved particle swarm optimization algorithm. Taking the desired trajectory and speed as input, the control values of the orientation angle and speed are obtained based on sliding mode control (SMC). They proposed an updated law for the inertia weight and learning rate to improve the particle swarm algorithm (PSO), avoiding the problem of local optima. Chai R et al. [25] proposed an integrated framework for trajectory planning and tracking control to address the autonomous ground vehicle parking maneuver problem. They used deep neural networks to approximate the optimal parking trajectory and explored the relationships between different vehicle states during the training process. The design of the motion controller adopted an adaptive learning tracking control algorithm, making the vehicle follow along the planned trajectory. Tian F et al. [26] presented a parallel learning-based steering control method. They constructed a neural-network-based trajectory generation model based on limited steering trajectory data. By enumerating steering actions and generating steering trajectory data, they finally learned the inverse mapping relationship between steering actions and trajectories through training the neural network. Meng W et al. [27] proposed a longitudinal speed-planning method based on fuzzy neural networks to improve comfort and to reduce the complexity of planning algorithms. They established a fuzzy planning model and used the self-learning function of neural networks to modify it, establishing a fuzzy neural network planning model. The planning method was applied to speed planning, and based on three types of driving scenes, a speed-planning model based on fuzzy neural networks was established. Wang H et al. [28] proposed a path-tracking controller based on model-free adaptive dynamic programming. They used a radial basis function neural network (RBFNN) to approximate nonlinear external disturbances, thereby compensating for the front wheel angle, and used the Lyapunov function to derive the weight update law of RBFNN, ensuring the stability and convergence of the system.

Vehicle stability control is another significant research content in active safety control. $\gamma$ and $\beta$ are crucial arguments that characterize the car motion state in the study of stability control. The ability of the actual $\gamma$ to closely follow ${\gamma }_d$ reflects the maneuverability of the vehicle. On the other hand, the variation $\beta$ primarily indicates the lateral stability competence of the car [29]. Direct yaw moment control (DYC) methods are mainly used for vehicle stability control under routine conditions. Under extreme conditions, it is challenging to effectively track both the ${\gamma }_d$ and ${\beta }_d$ solely through the use of a DYC system [30]. Therefore, the vehicle stability control system must establish different control objectives according to the prevailing operating conditions.

In the process of vehicle motion, the tires are the only part of the vehicle to make contact with the ground, and the tires’ characteristics directly affect the car’s motion performance. To enhance the competence of car control, a scheme was proposed using the dynamic characteristics of tires in this paper. The objective is to prioritize vehicle maneuverability when the lateral force of tires is within the linear area. In situations where the lateral force of tires enters the nonlinear region, the control requirements for vehicle stability are not as stringent. However, to maintain the desired vehicle trajectory as expected by the driver, both maneuverability and stability should be simultaneously controlled. In cases where the lateral force of the tires reaches the saturation area, the car’s lateral stability margin diminishes significantly, as the tire can only generate a lateral force close to its limit. If the control objective continues to focus on vehicle maneuverability, then it is likely to lead to the vehicle being unstable. Therefore, the control objective in this case should be stability. Numerous studies have demonstrated that DYC can greatly enhance the maneuverability and stability of vehicles [31,32]; however, few studies have achieved both maneuverability and stability control using DYC. In this paper, considering the dynamic characteristics of tire lateral force, a proportional coefficient is introduced in the $\beta -\dot{\beta }$ method to partition region into the stable region, understable region, and unstable region. The stable region corresponds to the lateral force of the tires’ linear range, while the understable region corresponds to the nonlinear range of the tires, namely, lateral force. The unstable region corresponds to the saturation range of the lateral force of the tire.

This paper aims to utilize machine learning to construct the dynamic boundary of the $\beta -\dot{\beta }$ phase plane stability region for various vehicle operating conditions. Considering the inherent connection between lateral tire forces and vehicle stability control, a novel car control weight allocation algorithm for maneuverability and stability was proposed. Utilizing this approach, a stability controller for cars was developed to advance the efficacy of stability control. The main contributions are outlined below:

(1)

An improved quadrilateral method is proposed in this study to partition the $\beta -\dot{\beta }$ phase plane stability region, considering the effect of the car steering angle, longitudinal speed, and adhesion coefficient. A stability region dynamic boundary regression model is established using particle swarm optimization and the random forest method.

(2)

A mapping relationship is formed between the $\beta -\dot{\beta }$ phase plane and the dynamic characteristics of lateral tire forces by introducing a proportional coefficient, thereby improving the performance of the vehicle stability controller.

(3)

Different vehicle motion control objectives are proposed under different tire characteristics to achieve precise trade-offs between vehicle maneuverability and lateral stability while ensuring smooth transitions between different control objectives.

This article is divided into the following parts. Section 2 presents the partitioning method for the phase plane. Section 3 presents the regression model for the dynamic stable area boundary. Section 4 presents the DYC controller using the dynamic stable area boundary. Section 5 provides the simulation results of the stability control algorithm. Finally, Section 6 concludes the paper.

2. Partition of Phase Planes

2.1. Two-Degrees-of-Freedom Nonlinear Vehicle Model

The phase plane method is a graphical technique commonly used to analyze the stability of nonlinear systems, specifically those with a second-order or lower complexity. In vehicle dynamics research, the linear two-degrees-of-freedom (2-DOF) model is frequently employed. This model assumes that the tire lateral stiffness is constant and within the linear range, neglecting nonlinear tire factors. The simplified vehicle system only takes into account the yaw motion and lateral motion. However, when analyzing the phase plane of vehicle motion states, it is essential to consider the nonlinear characteristics of the actual vehicle, primarily originating from the tires. Therefore, this paper incorporates a nonlinear tire lateral force model, utilizing the magic formula tire model. Consequently, a nonlinear 2-DOF car model is established, as depicted in Figure 1.

The motion equation for the car is represented by the equations shown in Figure 1:

$$\begin{array}{l}\dot{\beta }=\frac{\cos \beta }{mu}(F_f\cos \delta +F_r)+\frac{\sin \beta }{mu}{F}_f\sin \delta -\gamma \\ \dot{\gamma }=\frac{F_{f}a\cos (\delta -\beta )-F_{r}b\cos \beta }{I_Z}\end{array}$$

(1)

where $m$ represents the car mass; $u$ is the car longitudinal speed in the vehicle coordinate system; $F_f$ represents the front wheel lateral force; $F_r$ represents the rear wheel lateral force; $I_z$ is the car inertia moment about the $z$-axis; $a$ and $b$ are the distances from the mass center to the front and rear axles, respectively; $\beta$ represents the sideslip angle; $\gamma$ is the car yaw angle rate; and $\delta$ is the car steering angle.

$F_f$ and $F_r$ are represented by the magic tire formula as

$$\begin{array}{l}{F}_f=D_f\sin \left\{C_f\arctan \left[B_f{\alpha }_f-E_f\left(B_f{\alpha }_f-\arctan \left(B_f{\alpha }_f\right)\right)\right]\right\}\\ F_r=D_r\sin \left\{C_r\arctan \left[B_r{\alpha }_r-E_r\left(B_r{\alpha }_r-\arctan \left(B_r{\alpha }_r\right)\right)\right]\right\}\end{array}$$

(2)

Rewrite Equation (2) to the form of Equation (3) [33]:

$$\begin{array}{l}{F}_f=\mu F_{zf}\sin \left\{C_f\arctan \left[B_f{\alpha }_f-E_f\left(B_f{\alpha }_f-\arctan \left(B_f{\alpha }_f\right)\right)\right]\right\}\\ F_r=\mu F_{zr}\sin \left\{C_r\arctan \left[B_r{\alpha }_r-E_r\left(B_r{\alpha }_r-\arctan \left(B_r{\alpha }_r\right)\right)\right]\right\}\end{array}$$

(3)

where $\mu$ is the adhesion coefficient; $C_f$, $C_r$, $B_f$, $B_r$, $E_f$, and $E_r$ are the magic tire model fitting coefficients, which are related to the tire vertical force; ${\alpha }_f$ and ${\alpha }_r$ represent the front and rear wheel tire side slip angles, respectively; and their calculation formulas are shown in Equation (4):

$$\begin{array}{l}{\alpha }_f=\beta +\frac{a\gamma }{u}-\delta \\ {\alpha }_r=\beta -\frac{b\gamma }{u}\end{array}$$

(4)

$F_{zf}$ and $F_{zr}$ are the front and rear wheels’ vertical loads, respectively, and the calculation formulas are shown in Equation (5):

$$\begin{array}{l}{F}_{zf}=\frac{mgb}{l}\\ F_{zr}=\frac{mga}{l}\end{array}$$

(5)

2.2. Partition of Stability Area of Phase Plane

This paper uses a nonlinear 2-DOF vehicle model, and simulation results are obtained to generate a $\beta -\dot{\beta }$ phase plane portrait under various steering angles, longitudinal velocities, and adhesion coefficients. Zhu S et al. [34] employed the conventional bilinear method to determine the stability area of the phase plane. Nonetheless, this method has limitations, as it may mistakenly include unstable points in the first and third quadrants within the stable region, particularly at low vehicle speeds, as shown in Figure 2. Zhang H et al. [5] developed an offline database of stable regions in the $\beta -\dot{\beta }$ phase plane using the bilinear method under various operating conditions. The purpose was to improve the adaptability of car control systems under various conditions. However, this method still fails to overcome the drawback of the bilinear method in accurately partitioning the stable region at low vehicle speeds. Liu Z et al. [35] introduced the quadrilateral approach to partitioning the stability area of the $\beta -\dot{\beta }$ phase plane. This method involves establishing a coordinate axis centered on the equilibrium point and identifying the car stability area boundary as the crossover node of two parallel lines tangential to the trajectory on the phase plane and the horizontal axis. The intersection points mentioned above serve as the quadrilateral stability area’s left vertex and right vertex. Meanwhile, the top vertex and bottom vertex values are determined with the intersection points of the two parallel lines and the vertical axis passing through the equilibrium point. However, it is important to note that this method has the drawback of potentially including unstable regions within the stability area at higher speeds, as we can see in Figure 3.

It has been found that existing methods for judging the car motion state by using the $\beta -\dot{\beta }$ phase plane have disadvantages through references, such as inaccurate partition of the stable region and poor adaptability to different running conditions [5,34]. To restrain these shortcomings, we introduce a novel quadrilateral approach for determining the phase plane stability area. The approach involves partitioning the $\beta -\dot{\beta }$ phase plane into quadrilateral regions and constructing an inscribed circle within the quadrilateral region to determine the radius of the inscribed circle and the boundary of the quadrilateral. To determine the quadrilateral, phase plane data are collected through simulation experiments using a nonlinear 2-DOF vehicle model. These experiments involve varying front wheel angles, longitudinal velocities, and adhesion coefficients; a database is established to train a random forest regression model that predicts the slopes and intercepts of the quadrilateral boundaries under different running conditions. The phase plane stability area is determined with the predicted slopes and intercepts of the quadrilateral boundaries, which will be elaborated in detail in the subsequent section.

2.3. Analysis of Phase Plane Stability Area

To accurately predict the phase plane stability area, this paper conducts a survey and analysis to identify the key factors that impact it. These factors include the steering angle, longitudinal speed, and coefficient of adhesion [36,37]. Through a simulation, the effect of these three factors on the phase plane stability area is analyzed. The bilinear approach is employed to partition the stability region, with the goal of discussing the effects of these factors on it.

Figure 4 shows the trend in the phase plane stability area as the longitudinal velocity changes. It reveals that the stable region becomes narrower and decreases in area while the velocity rises from 40 km/h to 100 km/h. The slope of the stable boundary remains almost constant. This simulation suggests that the longitudinal velocity primarily affects the width of the phase plane boundary.

Figure 5 illustrates the trend in phase plane stability area as the adhesion coefficient changes. It reveals that increasing the coefficient of adhesion leads to a rise in the absolute value of the slope of the phase plane stability area boundary, as well as a rise in the width of the stable area boundary. This simulation indicates that the variation in the coefficient adhesion affects both the gradient of the phase plane border and the width of the stable region.

Figure 6 illustrates the trend in the phase plane stability area as the car steering angle changes. It reveals that increasing the front wheel angle leads to an asymmetric phase plane stability area, with the stable point moving away from the origin. The bilinear method is no longer effective in accurately partitioning the stable region. This simulation suggests that the variation in the car steering angle has a significant effect on the car’s initial stability status, causing it to lose stability.

3. Dynamic Stable Area Boundary

Through the previous study, it has been discovered that a nonlinear mapping relationship exists between the steering angle, longitudinal speed, adhesion coefficient, and phase plane stability area. To accurately describe this nonlinear mapping relationship and to provide a foundation for the stability control system to judge the car motion state, we utilize particle swarm optimization and random forest (PSO-RF) [38,39]. The simulation data for the steering angle, longitudinal speed, adhesion coefficient, and phase plane stability area are used to train a regression model, which establishes the dynamic boundary of the stable region.

The random forest regression algorithm is friendly to small datasets and has a high prediction accuracy. It is also less affected by noise, insensitive to outliers, resistant to overfitting, and capable of dealing with high-dimensional data [40].

To obtain the simulation dataset for the phase plane, the novel quadrilateral method is employed, which records the partition information of the stability area. The trajectory of the phase plane is primarily influenced by the front wheel angle, longitudinal velocity, and adhesion coefficient. In this study, the front wheel angle interval is set at 2°, ranging from 0° to 6°; the longitudinal speed interval is set at 20 km/h, ranging from 20 km/h to 80 km/h; and the coefficient of adhesion interval is set at 0.2, ranging from 0.2 to 0.8. Utilizing the sampled data, a MISO random forest regression model is established to output the slope and intercept of the quadrilateral boundary, as depicted in Figure 7.

3.1. Random Forest Model

Random forest is a machine learning algorithm that consists of multiple decision trees. There are various algorithms for decision tree construction, such as ID3, CD4, and CART. Among them, the CART algorithm is commonly used in the random forest regression model. The CART decision tree is generated by recursively constructing a binary decision tree and using the mean squared error (MSE) criterion for feature selection.

3.1.1. Generation of Regression Tree

We suppose that we have input variable $x$ and output variable $y$, where $y$ is a variable. Let $D$ be the given training dataset.

$$D=\left\{(x_1,y_1),(x_2,y_2),\cdots (x_n,y_n)\right\}$$

(6)

A regression tree is a partition of the input space (feature space) and assigns output values to each partition unit. Assume that the input space is divided into $M$ units, denoted as $R_1$, $R_2$,$\cdot \cdot \cdot$, and $R_m$, and there is a fixed output value $c_m$ for each unit. Therefore, the regression tree model can be represented as follows:

$$f(x)={\displaystyle \sum _{m=1}^M{c}_{m}I(x\in R_m)}$$

(7)

Once the input space is partitioned, the regression tree can be evaluated using ${\displaystyle \sum _{x_i\in R_m}{(y_i-f(x_i))}^2}$, to measure the prediction error on the training data. The optimal output value for each unit is determined by minimizing the MSE. The optimal value ${\widehat{c}}_m$ for unit $R_m$ is calculated as the mean of the output $y_i$ for all input instances $x_i$ in unit $R_m$. Mathematically, this can be represented as follows:

$$\widehat{c_m}=ave(y_i|x_i\in R_m)$$

(8)

To partition the input space, a heuristic method is employed, where the jth variable and its value $s$ are chosen as the splitting variable and splitting point, respectively. This results in the creation of two regions:

$$\begin{array}{l}{R}_1(j,s)=\left\{x|x^{(j)}\le s\right\}\\ R_2(j,s)=\left\{x|x^{(j)}>s\right\}\end{array}$$

(9)

To determine the optimal splitting variable $j$ and splitting point $s$, the following approach can be taken:

$$\underset{j,s}{\min }\left[\underset{c_1}{\min }{\displaystyle \sum _{x_i\in R_1(j,s)}{(y_i-c_1)}^2+\underset{c_2}{\min }{\displaystyle \sum _{x_i\in R_1(j,s)}{(y_i-c_2)}^2}}\right]$$

(10)

For a given input variable $j$, the optimal split point $s$ can be found using the following:

$$\begin{array}{l}\widehat{c_1}=ave(y_i|x_i\in R_1(j,s))\\ \widehat{c_2}=ave(y_i|x_i\in R_2(j,s))\end{array}$$

(11)

To find the optimal splitting variable and splitting point $\left(j,s\right)$, iterate through all input variables. Partition the input space into two regions based on the selected variable and point. Repeat this partitioning process for each region until the stop condition is satisfied. This iterative process generates a least squares regression tree.

3.1.2. Random Forest

Next, we will describe the generation of random forest. This algorithm includes two parts of randomness. Firstly, assuming there are $M$ samples, select $m$ samples using the method of putting them back (each time one sample is selected and then put back), which is the first part of randomness. Secondly, each node in the decision-making process needs to be split. At this point, a random selection of $n$ features is made from a total of $N$ features for splitting. Here, $n$ is chosen to satisfy $n\ll N$, which represents a subset of $n$ features. Subsequently, an optimal attribute is selected from this subset for partitioning, introducing a second level of randomness. The parameter $n$ plays a role in controlling the extent of randomness introduced: in general, the recommended value $n={\log }_{2}N$ [40]. By following the aforementioned steps, it is possible to construct a decision tree using $M$ samples and $n$ features. By repeating the above steps, a large number of decision trees can be constructed and integrated to form a random forest. For regression problems, use the method of mean value to calculate the results of multiple decision trees.

3.2. Hyperparameter Optimization

Random forest regression also has some limitations, such as the need to select appropriate hyperparameters to enhance the efficiency and algorithm property. The key parameters that impact the efficiency and performance of the random forest algorithm include the size of the forest (nTree), the minimum number of samples in a leaf node (MinLeaf), and the size of the feature subset (n).

The out-of-bag estimate can be utilized to assess the generalization capability of the model. As each decision tree employs only around 63.2% of the samples from the original training set, the remaining approximately 36.8% of samples can be utilized as a validation set to estimate the generalization performance [41]. The out-of-bag (OOB) estimate error can be used to evaluate the reasonableness of the parameters.

Due to the unique advantages of intelligent algorithms in parameter optimization, many researchers have conducted relevant studies and achieved good results [42,43]. The PSO algorithm imitates birds searching for food. Each particle represents a candidate solution and has three characteristics: position, velocity, and fitness value. The particle with the best fitness value is selected from both individual and global extreme values. In this paper, the PSO algorithm is used for parameter search, with the goal of selecting parameters that minimize the out-of-bag estimate error.

4. Design of Vehicle Stability Controller

The DYC used in this paper adopted a hierarchical control structure, and its control scheme is shown in Figure 8.

The driver model generates the front wheel steering angle signal $\delta$, while the desired vehicle model generates the signals that the driver expected, including the desired yaw rate ${\gamma }_d$ and sideslip angle ${\beta }_d$. The control algorithm block includes the control diagnosis block, control mode block, and DYC block. The control diagnosis block includes the intervention condition block and phase-plane stability judgment block, which, respectively, determine the intervention moment of the control algorithm and the results of the phase-plane stability judgment block. The control mode block includes the without control mode, handling-oriented control mode, stability-oriented control mode, and transitional control mode, corresponding to the different motion states of the vehicle. The control algorithm block receives signals from the front wheel angle $\delta$, expected yaw rate ${\gamma }_d$, and expected sideslip angle ${\beta }_d$, as well as vehicle status information, including vehicle speed $u$, road adhesion coefficient $\mu$, actual yaw rate $\gamma$, actual sideslip angle $\beta$, and actual sideslip angle rate $\dot{\beta }$. The phase plane method was used to judge the vehicle motion state and to determine the switching moment for the maneuverability control mode, stability control mode, and transition control mode. Additionally, The weight coefficients $\lambda$ and $\beta$ were computed by using the phase plane approach and applied to the DYC system. The DYC system has a hierarchical control structure, including a global fast terminal sliding mode control (GFT-SMC) block and optimal torque distribution block, the GFT-SMC block using the GFT-SMC method to track the $\gamma$ and $\beta$ deviation and calculate additive yaw moment $\Delta M_z$. The optimal torque distribution block aims to achieve optimal tire slip ratio control and to distribute the four-wheel braking force $T_f{}_l,T_{fr},T_{rl},T_{rr}$ to achieve effective vehicle motion control. The vehicle model block receives the front wheel angle signal $\delta$ of the driver model and the longitudinal tire force signal $T_f{}_l,T_{fr},T_{rl},T_{rr}$ of the four wheels of the control algorithm block, simulating vehicle motion in running conditions.

4.1. Vehicle Dynamics Model

A 3-DOF vehicle dynamics model was created, considering longitudinal, lateral, and yaw motions. The model state variables are the yaw rate and sideslip angle, represented by Equation (12).

$$\begin{array}{l}{a}_x=\frac{1}{m}((F_{x1}+F_{x2})\cos \delta -(F_{y1}+F_{y2})\sin \delta +F_{x3}+F_{x4})\\ a_y=\frac{1}{m}((F_{x1}+F_{x2})\sin \delta +(F_{y1}+F_{y2})\cos \delta +F_{y3}+F_{y4})\\ \dot{\gamma }=\frac{1}{I_z}(a(F_{x1}+F_{x2})\sin \delta +a(F_{y1}+F_{y2})\cos \delta -b(F_{y3}+F_{y4})-\frac{t_f}{2}(F_{x1}-F_{x2})\cos \delta +\frac{t_f}{2}(F_{y1}-F_{y2})\sin \delta -\frac{t_r}{2}(F_{x3}-F_{x4}))\end{array}$$

(12)

where $a_x$ and $a_y$ represent car longitudinal and lateral acceleration, $F_{xi}$ and $F_{yi}$ represent the forces exerted on the tires in the longitudinal and lateral directions, and $t_f$ and $t_r$ represent the front and rear axle track widths.

Simplify Equation (12) to obtain Equation (13):

$$\begin{array}{l}\dot{\beta }=-\gamma +\frac{F_y}{mu}\\ \dot{\gamma }=\frac{M_z}{I_z}+\frac{M_u}{I_z}\end{array}$$

(13)

where $F_y$ and $M_z$ represent the lateral forces and yaw moment exerted on the vehicle, respectively:

$$\begin{array}{l}{F}_y={\displaystyle \sum _{i=1}^2(F_{xi}\sin \delta )}+2F_{yf}\cos \delta +2F_{yr}\\ M_z=a{\displaystyle \sum _{i=1}^2(F_{xi}\sin \delta )}+2a{F}_{yf}\cos \delta -2b{F}_{yr}+\frac{t_f}{2}{\displaystyle \sum _{i=1}^2{(-1)}^i(F_{xi}\cos \delta )}+\frac{t_r}{2}{\displaystyle \sum _{i=3}^4{(-1)}^i(F_{xi})}\end{array}$$

(14)

To align with the expectations of human drivers for vehicle trajectory, a linear 2-DOF car model was developed in this paper. This model is capable of calculating ${\gamma }_d$ and ${\beta }_d$, as depicted in Equation (15).

$$\begin{array}{l}\dot{\beta }=\frac{k_f+k_r}{mu}\beta +\left(\frac{a{k}_f-b{k}_r}{m{u}^2}-1\right)\gamma -\frac{k_f}{mu}\delta \\ \dot{\gamma }=\frac{a{k}_f-b{k}_r}{I_z}\beta +\frac{a^2{k}_f+b^2{k}_r}{I_{z}u}\gamma -\frac{a{k}_f}{I_z}\delta \end{array}$$

(15)

when the vehicle is stable, and $\dot{\beta }=0$ and $\dot{\gamma }=0$ are satisfied, we obtain ${\beta }_d$ and ${\gamma }_d$.

$${\gamma }_d=\frac{u/l}{1+K{u}^2}\delta$$

(16)

$${\beta }_d=\frac{\frac{b}{l}+\frac{ma{u}^2}{l^2{k}_r}}{1+K{u}^2}\delta$$

(17)

where $K=\frac{m}{l^2}\left(\frac{a}{k_r}-\frac{b}{k_f}\right)$ is the stability factor, and $k_f$ and $k_r$ represent the front and rear wheels’ lateral stiffnesses, respectively.

However, the tires are limited by their adhesion coefficient, meaning that they cannot provide the required forces for high yaw rates. This renders the ideal yaw rate obtained from Equation (16) unsafe. Therefore, it is essential to consider the tire adhesion coefficient to determine the maximum allowable values for $\gamma$ and $\beta$.

The lateral acceleration at the centroid can be expressed by Equation (18):

$$a_y=u\gamma +\tan (\beta )a_y+\frac{u\beta }{\sqrt{1+{\tan }^2(\beta )}}$$

(18)

To account for the tire–road adhesion, Equation (19) must be satisfied by the lateral acceleration.

$$a_y\le \mu g$$

(19)

when $\beta$ is small, the last two terms in Equation (18) become negligible. By combining Equations (18) and (19), the maximum allowable value of ${\gamma }_d$ can be shown as Equation (20), with $\xi$ set to 0.8.

$$\left|{\gamma }_d\right|=\left|\xi \frac{\mu g}{u}\right|$$

(20)

Therefore, Equation (21) can be utilized to express ${\gamma }_d$.

$${\gamma }_d=\min \left\{\left|\frac{u/l}{1+K{u}^2}\delta \right|,\left|\xi \frac{\mu g}{u}\right|\right\}\times \mathrm{sgn}(\delta )$$

(21)

By substituting Equations (16) and (20) into Equation (15), the maximum allowable value of ${\beta }_d$ can be obtained, as shown in Equation (22).

$${\beta }_d=\min \left\{\left|\frac{b/l+ma{u}^2/(l^2{k}_r)}{1+K{u}^2}\delta \right|,\left|\xi \mu g(\frac{b}{u^2}+\frac{ma}{k_{f}l})\right|\right\}\times \mathrm{sgn}\left(\frac{b/l+ma{u}^2/(l^2{k}_r)}{1+k{u}^2}\delta \right)$$

(22)

4.2. Design of the Top-Level Controller

The time-varying characteristic of the vehicle motion state requires that the vehicle stability controller has good real-time performance. According to the references of vehicle dynamics and control [44], the yaw rate $\gamma$ is a parameter that characterizes the speed of the vehicle’s direction change, and the ideal yaw rate ${\gamma }_d$ can be seen as the expected speed ${\nu }_d$ through the expected path $R_d$. Therefore, the error between the actual yaw rate $\gamma$ and the ideal yaw rate ${\gamma }_d$ mainly affects the maneuverability of the vehicle. The sideslip angle $\beta$ is the angle between the velocity ${\nu }_c$ of the vehicle’s centroid and the vehicle’s X-axis; there is a fixed geometric relationship between the steady-state side slip angle ${\alpha }_f$, ${\alpha }_r$ of wheels, and the sideslip angle $\beta$, so there is a mapping relationship between the sideslip angle $\beta$ and the tire force. However the most important reason for the lateral instability state of the vehicle is that the tires cannot provide enough lateral force $F_y$; thus, the error between the actual sideslip angle $\beta$ and the ideal sideslip angle ${\beta }_d$ mainly affects the stability of the vehicle. In addition, in the phase plane stability area, the tire forces operate within the linear region, resulting in a large tire adhesion margin. Thus, the control objective should prioritize vehicle maneuverability by focusing on controlling the deviation of the yaw rate. In the understable region, where tire forces operate in the nonlinear region, the need for vehicle stability control is less significant. Instead, the emphasis should be on simultaneously controlling the deviations of both $\gamma$ and $\beta$, ensuring the desired vehicle trajectory and stability. In the unstable region, where lateral tire forces approach extreme values, the control objective shifts to vehicle stability, with a focus on controlling the deviation of $\beta$. Therefore, we designed a sliding surface based on the error ${\gamma }_e$ in yaw rate and the error ${\beta }_e$ in sideslip angle, and based on that, a GFT-SMC controller was designed. At the same time, to improve the control performance, we considered tire characteristics, by dividing the lateral force into linear, nonlinear, and saturation regions, dividing the phase plane into stable, transitional, and unstable regions, indirectly combining the vehicle’s maneuverability and stability with tire force, thereby achieving coupling and decoupling of the vehicle’s maneuverability and stability control.

To ensure car system maneuverability within the linear region of the tires, this paper adopted the yaw rate tolerance band method to control the intervention timing of the car maneuverability control system. The $\gamma$ tolerance band method is shown in Equation (23).

$$|\gamma -{\gamma }_d|\le k\left|{\gamma }_d\right|$$

(23)

where $k=0.05$ is the yaw rate control coefficient. According to reference [45], when the error between the control target value and the expected value is less than 5% of the target value, the control accuracy is considered to meet the requirements, so $k=0.05$.

To enhance the vehicle stability control, a regression model was employed in this study to calculate the dynamic weight coefficient $\rho$, utilizing the dynamic boundary of the stable region. It serves as both the judgment condition for the intervention of the sideslip angle deviation control algorithm and determines the proportion of vehicle stability control in the algorithm. This enables a seamless transition between vehicle maneuverability and stability. The specific calculation method of $\rho$ is given in the following text.

For vehicle stability control, this paper adopted the SMC method, and the designed tracking error sliding surface is shown in Equation (24):

$$s_0=(1-\rho )(\gamma -{\gamma }_d)+\rho ({\beta }_d-\beta )$$

(24)

The global fast terminal sliding surface is defined as shown in Equation (25):

$$s_1=s_0+{\displaystyle \int c{s}_0+d{s}_0^{q/p}}d\tau$$

(25)

where $c$, $d>0$, $q$, and $p$ are odd numbers and $0<q<p$.

The designed GFT-SMC law is shown in Equation (26):

$$M_u=\lambda (-M_z+I_z{\dot{\gamma }}_d+\frac{I_z}{(1-\rho )}(-\rho ({\dot{\beta }}_d+\gamma -\frac{F_y}{mu})-c{s}_0-d{s}_0{}^{q/p}-k_1|s_1{|}^{\eta }\mathrm{sgn}(s_1)-k_2\mathrm{sgn}(s_1)-k_3{s}_1))$$

(26)

where $\lambda$ is the vehicle maneuverability control coefficient. When Equation (23) is satisfied, $\lambda =1$; otherwise $\lambda =0$. $\rho$ is the dynamic weight coefficient, and its value is determined by the degree to which the system state point deviates from the stable area. In a stable state, no control system intervention is required, $\lambda =0$, and $\rho =0$; when the vehicle is controlled for maneuverability, $\lambda =1$ and $\rho =0$; when the vehicle is controlled for stability, $\lambda =1$ and $\rho =1$; and when the vehicle is controlled for both maneuverability and stability, $\lambda =1$. The value of $\rho$ is calculated based on the proposed novel quadrilateral phase plane method in this paper.

In Equation (26), $\mathrm{sgn}(\cdot )$ represents the sign function. The closed-loop system formed by the GFT-SMC law is finite-time stable if the control law parameters are selected to meet the condition $0<\eta <1$, $k_1>0$, $k_2>0$, and $k_3>0$. This means that there exists a settling time $T_{total}$ such that for any $t\ge T_{total}$, the system state variable $s_0=0$.

Select the Lyapunov candidate function as shown in Equation (27):

$$V=\frac{1}{2}{s}_1{}^2$$

(27)

The derivative of both sides of the above equation is taken:

$$\begin{array}{l}\dot{V}=s_1{\dot{s}}_1\\ =s_1({\dot{s}}_0+c{s}_0+d{s}_0{}^{q/p})\\ =s_1(\lambda (1-\rho )((\frac{M_z}{I_z}+\frac{M_u}{I_z}-{\dot{\gamma }}_d)+\rho ({\dot{\beta }}_d+\gamma -\frac{F_y}{mu})+\\ c{s}_0+d{s}_0{}^{q/p}-k_1|s_1{|}^{\eta }\mathrm{sgn}(s_1)-k_2\mathrm{sgn}(s_1)-k_3{s}_1)\\ =s_1(-k_1|s_1{|}^{\eta }\mathrm{sgn}(s_1)-k_2\mathrm{sgn}(s_1)-k_3{s}_1)\\ =-k_1\left|s_1{|}^{\eta }\right|s_1|-k_2|s_1|-k_3{s}_1{}^2\\ =-k_1|s_1{|}^{\eta +1}-k_2|s_1|-k_3{s}_1{}^2\end{array}$$

(28)

Therefore,

$${\dot{V}}_2\le 0$$

(29)

So, the Lyapunov candidate function is negative-definite.

If $s_1=0$, it can be obtained from Equation (25):

$$s_0{}^{-q/p}\frac{d{s}_0}{dt}+c{s}_0{}^{1-q/p}=-d$$

(30)

If $y=s_0^{1-q/p}$ and $\frac{dy}{dt}=\frac{p-q}{p}{s}_0^{-\frac{q}{p}}\frac{d{s}_0}{dt}$, Equation (30) becomes

$$\frac{dy}{dt}+\frac{p-q}{p}cy=-\frac{p-q}{p}d$$

(31)

Due to the general solution of the first-order linear differential equation $\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$ being

$$y=e^{-{\displaystyle \int P(x)dx}}({\displaystyle \int Q(x)e^{{\displaystyle \int P(x)dx}}dx+C})$$

the solution of Equation (31) is

$$\begin{array}{cc}\hfill y& =e^{-{\displaystyle {\int }_0^t\frac{p-q}{p}c}dt}({\displaystyle {\int }_0^t-\frac{p-q}{p}d{e}^{{\displaystyle {\int }_0^t\frac{p-q}{p}cdt}}}dt+C)\hfill \\ & =e^{-{\displaystyle {\int }_0^t\frac{p-q}{p}cdt}}({\displaystyle {\int }_0^t-\frac{p-q}{p}d{e}^{\frac{p-q}{p}ct}dt+C})\hfill \end{array}$$

(32)

When $t=0$ and $C=y\left(0\right)$, Equation (32) becomes

$$\begin{array}{cc}\hfill y& =e^{-\frac{p-q}{p}ct}({-\frac{p-q}{p}d\frac{p}{(p-q)c}{e}^{\frac{p-q}{p}ct}|}_0^t+y(0))\hfill \\ & =-\frac{d}{c}+\frac{d}{c}{e}^{-\frac{p-q}{p}ct}+y(0)e^{-\frac{p-q}{p}ct}\hfill \end{array}$$

(33)

When $s_0=0$ and $y=0$, $t=t_s$, Equation (33) becomes

$$\frac{d}{c}{e}^{-\frac{p-q}{p}c{t}_s}+y(0)e^{-\frac{p-q}{p}c{t}_s}=\frac{d}{c}$$

(34)

Therefore,

$$(\frac{d}{c}+y(0))e^{-\frac{p-q}{p}c{t}_s}=\frac{d}{c}$$

(35)

$$\frac{d+cy(0)}{d}=e^{\frac{p-q}{p}c{t}_s}$$

(36)

where $y\left(0\right)=s_0{\left(0\right)}^{\left(p-q\right)/p}$.

In the sliding mode, the convergence time from any initial state $s_0\left(0\right)\ne 0$ to the equilibrium state $s_0=0$ is as follows:

$$t_s=\frac{p}{c(p-q)}\ln \frac{c{s}_0{(0)}^{(p-q)/p}+d}{d}$$

(37)

By setting $c$, $d$, $p$, $q$ the system can reach equilibrium within a finite time of $t_s$.

4.3. Dynamic Adjustment Strategy of Weight Coefficients

4.3.1. Proportional Coefficients

Based on the previous content, this paper establishes a phase plane portrait that relates to steering angle, longitudinal vehicle speed, and adhesion coefficient. The proposed novel quadrilateral method is utilized to determine the phase plane stability area. To enhance the effectiveness of the vehicle stability controller, this paper integrates the dynamic characteristics of the tire with the phase plane. The phase plane area is divided into stable, understable, and unstable regions based on the quadrilateral stable region. The stable region corresponds to the tire lateral force linear area, the understable region corresponds to the tire lateral force nonlinear region, and the unstable region corresponds to the saturated region of tire lateral force; refer to Figure 9 and Figure 10.

For instance, considering the quadrilateral border of the first quadrant of the phase plane, the tire slip angle corresponding to the lateral force of the tire can be calculated based on the tire model, given the tire vertical load and road adhesion information during vehicle motion. Based on the tire model, the maximum value of the linear part of the lateral force and the minimum value of the saturated part can be determined.

${\alpha }_1$ and ${\alpha }_2$ can be calculated from Equation (38):

$$\begin{array}{l}{\beta }_1=\tan ({\alpha }_1+\delta )-\frac{a\gamma }{u}\\ {\beta }_2=\tan ({\alpha }_2+\delta )-\frac{a\gamma }{u}\end{array}$$

(38)

By utilizing Equation (38) to calculate the intersection points $\left({\beta }_1,0\right)$, $\left({\beta }_2,0\right)$ of the border of the stability area and the unstable region border with the horizontal axis, it is possible to plot the boundaries of phase plane stability and instability areas. This establishes an inherent connection between the phase plane and the tire’s dynamic characteristics.

The proportion coefficients $p_1$ and $p_2$ can be calculated using Equation (39).

$$\begin{array}{l}{p}_1=\frac{{\beta }_1}{{\beta }_0}\\ p_2=\frac{{\beta }_2}{{\beta }_0}\end{array}$$

(39)

4.3.2. Weight Coefficient

Assume that the current state of the vehicle is the point $\left({\beta }_0,{\dot{\beta }}_0\right)$, as shown in Figure 11.

The description of the state point of the vehicle can be represented by Equation (40):

$$R=\sqrt{{\dot{\beta }}_0{}^2+{\beta }_0{}^2}$$

(40)

The blue quadrilateral in the figure represents the original stable region boundary obtained through machine learning methods. The green and red quadrilaterals represent the boundaries of the stable and unstable regions, respectively, which are determined based on tire lateral force dynamic characteristics. These quadrilaterals correspond to the maximum in the linear region and minimum in the saturated region of tire lateral force. The region between the green and red quadrilaterals is known as the understable region, which corresponds to the tire lateral force nonlinear area. Drawing the inscribed circles of the stable region boundary quadrilateral, original stable region boundary quadrilateral, and unstable region boundary quadrilateral, we obtained the stable region circle, original stable region circle, and unstable region circle from the innermost to the outermost, respectively.

The method for switching control modes is as follows: First, verify if the car status is within the stable area boundary quadrilateral. If it is, then examine if the car status falls within the stability area circle. If it is inside the stable region circle, no intervention from the control system is needed, and the vehicle’s maneuverability can be maintained. In this case, the weight coefficient $\rho =0$. If the car status is outside the stable area circle but falls within the unstable region boundary quadrilateral, further check if it is outside the unstable region circle. If the car status is outside the unstable area circle, give priority to stability control algorithms, and the weight coefficient $\rho =1$. If the car status falls within the unstable area circle, calculate the weight coefficient $\rho$ based on Equation (41) and control both the maneuverability and stability of the vehicle. This method allows for different control strategies based on the vehicle position relative to the stable and unstable region boundaries, ensuring appropriate intervention to maintain both maneuverability and stability.

$$\rho =\frac{R-D_S}{R_U-D_S}{, \mathrm{D}}_S\le \mathrm{R}\le R_U$$

(41)

Let us assume that the intersection point between the line connecting the state point and the origin $O$ and the stable region boundary quadrilateral in the same quadrant is point $\epsilon$, $D_s$ represents the distance between the state point and point $\epsilon$, and $R$ represents the distance between the state point and the stable point.

Since the original stable region boundary quadrilateral changes with the vehicle steering angle, longitudinal velocity, and coefficient of adhesion, a mapping relationship exists between the boundary quadrilateral of the stable region and the tire dynamic characteristics. By dynamically adjusting the original stable region boundary quadrilateral and the dynamic proportional factor $p_1$, $p_2$, we can determine the boundaries of the stable and unstable regions. This further determines the dynamic weight coefficient $\rho$ for the corresponding running conditions, improving the adaptability of the vehicle stability control system under different conditions.

4.4. Design of the Bottom-Level Controller

This paper describes a control system for a vehicle braking system, where a bottom-level controller optimizes the distribution of braking forces among the four wheels to meet the additional yaw moment requirements of a top-level controller.

In the process of tire braking force control, if the constraints of the braking actuator and the external environment are not considered, applying a braking force to the wheels will not maximize the utilization of road adhesion and cannot ensure the vehicle’s lateral stability. Therefore, to achieve optimal distribution of the additional yaw moment, this paper aims to minimize tire adhesion utilization as the control objective.

The formula for calculating tire adhesion utilization is shown as Equation (42):

$${\rho }_{\mu i}=\frac{\sqrt{F_{xi}^2+F_{yi}^2}}{\mu F_{zi}}$$

(42)

where $F_{xi}$ represents the tire’s longitudinal force; $i=1$, $2$, $3$, and $4$ represent the left front tire, right front tire, left rear tire, and right rear tire, respectively, with the same notation is used in the following text; ${\rho }_{\mu i}$ represents the tire adhesion utilization, $i=1$, $2$, $3$, and $4$; and $F_{zi}$ represents the vertical load on the tire, $i=1$, $2$, $3$, and $4$.

Equation (42) suggests that, to provide the tire with a significant lateral force margin, ${\rho }_{\mu i}$ should be minimized. Therefore, the following objective function can be obtained:

$$J={\displaystyle \sum _{i=1,2,3,4}\frac{F_{xi}^2+F_{yi}^2}{{\left(\mu F_{zi}\right)}^2}}$$

(43)

According to Equation (43), a smaller value of $J$ corresponds to a greater stability margin for the vehicle, reducing the likelihood of instability. In this paper, additional lateral force distribution was achieved by modifying the longitudinal force of the tires. Due to the challenge of directly controlling the lateral force, only optimization control of the longitudinal force was considered. Equation (43) was simplified as shown in Equation (44):

$$\min J={\displaystyle \sum _{i=1,2,3,4}\frac{F_{xi}^2}{{\left(\mu F_{zi}\right)}^2}}$$

(44)

The bottom-level controller optimizes the allocation of additive yaw moments generated by the top-level controller, requiring Equation (44) to satisfy the constraint condition for an additive yaw moment:

$$\Delta M_z=t_f(F_{x2}-F_{x1})\cos \delta +t_r(F_{x4}-F_{x3})+a(F_{x1}+F_{x2})\sin \delta$$

(45)

The tires’ longitudinal force must not exceed the limits of the ground adhesion conditions. The braking force of each wheel must satisfy Equation (46).

$$0\le F_{xi}\le \sqrt{{(\mu F_{zi})}^2-F_{yi}^2}$$

(46)

Solve by converting Equations (44)–(46) into the standard form of quadratic programming:

$$\begin{array}{cc}\min \hfill & \frac{1}{2}{x}^{T}Hx+f^{T}x\\ s.t.\hfill & Ax\le b\\ & A_{eq}x=b_{eq}\\ & LB\le x\le UB\end{array}$$

(47)

According to Equation (47), optimal allocation of the lateral yaw moment can be achieved.

5. Simulation

To test the competence of a novel quadrilateral-based phase plane method for a vehicle stability controller, this paper utilized a combination of Simulink and Carsim for simulations. The car model and driver model were built in Carsim, while the desired vehicle model and the car control model were constructed in MATLAB/Simulink. In order to validate the effectiveness of the proposed vehicle stability control algorithm, this study conducted tests on the vehicle maneuverability performance under tight double-lane-change maneuvers and the stability control performance was tested by using the sine steering maneuver and fishhook steering maneuver. The simulated results were also compared with the traditional bilinear-based sliding mode control algorithm and PSO-RF-based SMC algorithm by using the same test running condition.

5.1. Tight Double-Lane-Change Maneuver

In the tight double-lane-change simulation test, a longitudinal vehicle speed of 40 km/h and an adhesion coefficient of 0.9 were chosen. The driver model was employed to follow the tight double-lane-change path depicted in Figure 12a, and the simulation outcomes are presented in Figure 12b–f.

From Figure 12a, the results illustrate the validity of the suggested control algorithm in accurately tracking the desired path and achieving good maneuvering performance, while the trajectory tracking competence of the traditional phase plane method is poor. Figure 12b,c exhibit the vehicle’s stability during operation, affirming the validity of the suggested control algorithm in maintaining car stability. Figure 12d,e illustrate the changes in tire forces during operation, specifically for the left front wheel. These figures depict the correlation between tire lateral force and tire side slip angle, as well as tire lateral force and time. The results demonstrate the effective utilization of tire forces by the proposed control algorithm in ensuring vehicle stability. Figure 12f shows the weight allocation results of the control algorithm for maneuverability and stability during the simulation. The PSO-RF-based method achieved a good weight allocation for vehicle maneuverability and stability.

5.2. Sine Steering Maneuver

In the sine steering simulation, a longitudinal speed of 100 km/h and an adhesion coefficient of 0.9 were chosen. The car steering angle was adjusted by using the signal displayed in Figure 13a, and the simulation outcomes are presented in Figure 13b–f.

Figure 13b,c demonstrate the superior stability control performance of the PSO-RF-based control algorithm compared to the traditional phase plane method, as evident from the smaller sideslip angle. Additionally, Figure 13d,e showcase the correlation between lateral force and side slip angle of the tire for the left front wheel, as well as tire lateral force and time. These figures highlight the effective utilization of tire forces by the PSO-RF-based control algorithm in maintaining vehicle stability. Figure 13f shows the weight allocation results of the control algorithm for maneuverability and stability during the simulation, indicating that the traditional phase plane approach inaccurately judges the car stable status.

5.3. Fishhook Steering Maneuver

The fishhook steering maneuver is considered an extreme steering test. A longitudinal vehicle speed of 120 km/h and an adhesion coefficient of 0.9 were chosen, as depicted in Figure 14a. The simulation outcomes for this maneuver are presented in Figure 14b–f.

Figure 14b,c exhibit the improved stability control performance of the PSO-RF-based control algorithm compared to the traditional phase plane method, as evidenced by the smaller sideslip angle. Furthermore, Figure 14d,e illustrate the correlation between tire lateral force and tire lateral slip angle for the left front wheel, as well as tire lateral force and time. These figures highlight the effective utilization of tire forces by the PSO-RF-based control algorithm in maintaining vehicle stability. Figure 14f shows the weight allocation results of the control algorithm for maneuverability and stability during the simulation, indicating that the traditional phase plane method inaccurately judges the vehicle stability state. The PSO-RF-based control algorithm possesses the ability to accurately assess the stability condition of the vehicle and to assign appropriate weights to enhance maneuverability and stability.

6. Conclusions

To enhance vehicle stability controller validity and to accurately assess a vehicle’s motion state, a 2-DOF nonlinear model was established in this study. Through a simulation analysis, the impact of steering angle, longitudinal speed, and coefficient of adhesion on the phase plane stable area boundary were examined. A novel approach for partitioning a phase plane stability area was employed to design the vehicle stability controller. Simulations were conducted to evaluate the controller’s effectiveness, leading to the following conclusions:

(1)

By stating the limitations of existing phase plane methods for judging the motion state of the vehicle, a novel phase plane stability region partitioning method is proposed, which achieves accurate judgement of car motion status.

(2)

The relationship between phase plane stable area boundary and car steering angle, longitudinal speed, and coefficient of adhesion is unveiled. A random forest regression model based on PSO is established to improve the adaptability of the vehicle motion state judgement method under different running conditions.

(3)

An analysis is performed to examine the correlation between tire forces and vehicle stability control. By considering the tires’ characteristics and the demands of car stability control, the lateral force of the tire is indirectly integrated into the car stability control system. An allocation algorithm for vehicle maneuverability and stability is designed, improving the effectiveness of vehicle stability control while achieving a smooth transition between vehicle maneuverability and stability.

Overall, a novel approach for accurately judging the vehicle motion state was proposed in this paper, solving the limitations of existing methods and improving the adaptability and effectiveness of vehicle stability control. Additionally, an inherent connection between tire forces and vehicle stability control was established, enhancing both vehicle maneuverability and stability.