Lateral-Stability-Oriented Path-Tracking Control Design for Four-Wheel Independent Drive Autonomous Vehicles with Tire Dynamic Characteristics under Extreme Conditions

Abstract

This paper proposes a lateral-stability-oriented path-tracking controller for four-wheel independent drive (4WID) autonomous vehicles. The proposed controller aims to maintain vehicle stability under extreme conditions while minimizing lateral deviation. Firstly, a tiered control framework comprising upper-level and lower-level controllers is introduced. The upper-level controller is a lateral stability path-tracking controller that incorporates tire dynamic characteristics, developed using model predictive control (MPC) theory. This controller dynamically updates the tire lateral force constraints in real time to account for variations in tire dynamics under extreme conditions. Additionally, it enhances lateral stability and reduces path-tracking errors by applying additional yaw torque based on minimum tire utilization. The lower-level controllers execute the required steering angles and yaw moments through the appropriate component equipment and torque distribution. The joint simulation results from CarSim and MATLAB/Simulink show that, compared to the traditional MPC controller with unstable sideslip, this controller can maintain vehicle lateral stability under extreme conditions. Compared to the MPC controller, which only considers lateral force constraints, this controller can significantly reduce lateral tracking errors, with an average yaw rate reduction of 31.62% and an average sideslip angle reduction of 40.21%.

1. Introduction

As the global automotive industry rapidly expands, the number of vehicles traveling on various roads has sharply increased, leading to a more complex transportation environment. As a result, traffic congestion and vehicle safety have become increasingly critical issues [1,2]. To address these challenges and meet current demands for traffic safety, intelligence, and efficiency, autonomous driving has emerged as a major research focus for leading automotive companies and academic institutions [3,4,5,6]. In addition, with the development of sensor technology, advanced assisted driving technology, and corresponding fault detection technology, the progress of autonomous driving technology has been greatly promoted [7]. Simultaneously, electric vehicles have gained importance in the automotive industry due to their benefits in environmental protection, energy efficiency, and low noise levels [8]. Among these, 4WID electric vehicles, representing the high-end market, have attracted significant attention for their superior driving performance and handling stability [9,10]. However, lateral stability is crucial for vehicle safety under extreme conditions like emergency obstacle avoidance, high-speed turning, and large curvature turning. At present, the control technology of the motor is relatively stable, including its corresponding motor diagnostic technology, which has demonstrated reliability in various operational scenarios [11,12,13]. Thus, researching and optimizing lateral stability control strategies for 4WD electric vehicles in these situations holds significant theoretical and practical importance for advancing autonomous vehicle development [14,15,16].

The path-tracking control of vehicles has a significant impact on the autonomous driving of intelligent vehicles. It involves controlling the longitudinal and lateral movement positions of autonomous vehicles to ensure they follow the target path, thereby minimizing the lateral distance and directional deviation between the vehicle and the desired path while maintaining driving stability [17,18]. Common methods for path-tracking control include PID control [19,20], fuzzy control, dynamic speed planning, sliding mode control [4,21], preview control, MPC, and linear quadratic regulator (LQR) control [16]. These methods can generally maintain vehicle stability and provide satisfactory path-tracking performance under most conditions. However, many of these algorithms primarily consider the vehicle’s non-holonomic constraints. In scenarios of utmost challenges, such as rapid directional changes at high velocities, swift turns, and urgent evasion of obstacles, significant changes in tire dynamic load amplify the tires’ nonlinear behavior, greatly impacting vehicle stability and tracking accuracy. In addition, relying solely on the front wheel steering angle as the lateral control variable makes it difficult for the path-tracking controller to balance accuracy and dynamic stability in this situation, but its robustness is poor when encountering external disturbances [22]. Fernando Viadero Monasterio et al. proposed a multi-input, multi-output control method, where the control inputs are front wheel steering and anti-roll torque. This method, combined with Lyapunov Krasovskii functionals, enhances the robustness of vehicle path tracking with multiple control inputs [23,24]. Therefore, existing control methods make it difficult to balance the accuracy and stability of path tracking and have poor robustness in the face of external disturbances, making it difficult to meet the requirements of extreme driving scenarios [25].

MPC is characterized by its predictive capabilities, rolling optimization, and feedback correction, which allow it to incorporate both the structural and dynamic constraints of a vehicle into its control strategy [18,26]. These features provide MPC with significant advantages in path tracking under conditions such as rapid directional changes at high velocities, swift turns, and the urgent evasion of obstacles [27]. However, using MPC to achieve path tracking requires a relatively complete knowledge of the vehicle’s state, but it is generally difficult to measure directly through expensive sensors, such as vehicle sideslip angle and roll angle [28]. Some researchers assume that these state variables can be directly obtained. Others have designed estimators for it, such as Fernando Viadero Monasterio et al.’s event-triggered IoT architecture that simultaneously estimates vehicle sideslip and roll [29], providing a method for MPC without strong assumptions. Yuan Tengfei et al. proposed an LQR controller to determine the front wheel steering angle and reduce lateral tracking errors, developed an MPC controller to calculate longitudinal acceleration to track target velocity, and used state estimation methods to estimate state variables that are difficult to directly measure [30]. However, many similar papers have not considered the dynamic constraints of lateral forces within the controller.

In a controller design that considers the dynamic characteristics of vehicle tires and constraints of tire lateral forces, Xiaojian Wu et al. applied a sliding mode control (SMC) method to establish an active front steering control system that takes into account the stability domain boundary envelope of the lateral slip angle and road adhesion constraints to enhance vehicle stability [14]. Liu Yingjie et al. created a tire force estimator using an RBF neural network to calculate tire lateral forces, incorporating it into the vehicle envelope constraint’s boundary conditions to minimize tracking deviation [15]. Although both approaches consider the constraint of tire lateral forces, they do not address the crucial impact of dynamic tire load variations on these forces. Jing Changchun et al. proposed an upper- and lower-level control system for the lateral stability and vehicle energy efficiency control of 4WID electric vehicles. The study utilized a sliding mode control approach combined with adhesion ellipse and braking constraints to apply additional yaw moment control, thereby improving vehicle stability [16]. However, this method does not utilize the MPC controller’s capability to handle and update dynamic constraints. Wang Zizheng et al. proposed a coordinated control method for the four-wheel steering of vehicles. This approach employs a seven-degrees-of-freedom (7-dof) vehicle model and the tire magic formula, combined with the recursive least squares method with an adaptive forgetting factor, to estimate and update steering stiffness in real time. This enhances path-tracking performance, especially when the vehicle is making sharp turns at high speeds and large angles [17]. Nevertheless, it did not consider the effects of dynamic tire load changes and road curvature on controller design. In extreme conditions caused by dynamic tire load changes, the tires are more likely to enter a saturated range and are unable to generate enough force to ensure lateral stability. This can lead to lane deviation and loss of steering control. In such cases, the motion control for autonomous vehicles should prioritize lateral stability over path-tracking performance. Based on the analysis of the above research methods on vehicle control, it is crucial to consider the dynamic characteristics of tires and update the boundary values of lateral force constraints in real time to ensure that the tires operate within their linear range and provide sufficient lateral force to maintain the lateral stability of the vehicle under extreme operating conditions.

In the research on independently distributing four-wheel torque and generating additional yaw moment to reform the lateral dynamic stability of 4WD electric vehicles under conditions such as high-speed turning, some scholars have studied the use of torque difference to generate additional yaw moment under extreme conditions, thereby improving lateral stability and reducing path-tracking errors. Ebrahim Muhammad et al. proposed a sliding-mode-based control method that converts yaw torque into the required longitudinal slip and generates the necessary distributed driving and braking torque to improve lateral stability [31]. Liu Gang et al. studied the optimal timing for applying direct yaw torque control and engine torque regulation. They developed a direct yaw torque control algorithm based on sliding mode control theory to improve the lateral stability of the controlled vehicle during extreme driving [32]. Geng Guoqing et al. combined the sliding mode control (SMC) algorithm with an adaptive fuzzy algorithm to design an adaptive fuzzy sliding mode controller (FSMC), which accurately calculates the additional yaw motion needed to improve vehicle lateral stability [33]. Liang Li et al. considered nonlinear tire combined slip constraints and generated yaw torque on the wheels through differential braking to ensure vehicle stability [34]. These studies demonstrated that generating additional yaw moments through the distribution of four-wheel drive torque significantly enhances the lateral stability of distributed-drive electric vehicles. Therefore, integrating an additional yaw moment into the MPC controller design is essential. By calculating the additional yaw moment needed to maintain dynamic stability and distributing it through four-wheel drive torque, the lateral stability of 4WD autonomous vehicles in extreme conditions can be enhanced, thereby minimizing lateral errors.

The research presented above demonstrates the impact of tire dynamic characteristics on the maximum lateral force that tires can provide under extreme operating conditions. Constraining the lateral force within the linear range of tire operation effectively enhances path tracking, lateral stability, and accuracy. However, the literature on this control method remains immature. Furthermore, given the ability of 4WID electric vehicles to independently distribute torque across all wheels, incorporating additional yaw torque into the design of the MPC controller can significantly enhance lateral stability under extreme conditions and reduce lateral errors. This is achieved by calculating the additional yaw torque required to maintain the vehicle’s dynamic stability and distributing it through four-wheel drive torque allocation.

Building on the previous discussion, this paper introduces a path-tracking controller for 4WID autonomous vehicles, emphasizing lateral stability in extreme conditions. The key innovations and contributions of this study are summarized as follows:

• A lateral-stability-oriented path-tracking controller is proposed for 4WID autonomous vehicles. The controller accounts for tire dynamic characteristics under extreme conditions by analyzing tire lateral force, introducing an adhesion ellipse constraint, and updating the lateral force constraint boundary in real time. This ensures that the vehicle maintains lateral stability while optimally following the desired path under extreme conditions.
• A novel additional yaw torque distribution system is introduced, which minimizes tire utilization and improves vehicle lateral stability under extreme operating conditions.
• An MPC controller that considers tire dynamics and road curvature under extreme conditions is proposed, and its effectiveness is demonstrated through collaborative simulations using CarSim and MATLAB/Simulink.

Compared with similar existing research papers, the main advantages of a lateral-stability-oriented path-tracking control design for 4WID autonomous vehicles proposed in this paper are as follows: Compared with References [22,35], This paper focuses on the factors of tire dynamic characteristics under extreme working conditions and designs an MPC controller for processing and updating tire lateral force constraint boundaries in real time. Compared with References [36,37], this paper considers the advantage of 4WD electric vehicles being able to distribute four-wheel torque independently. This paper introduces additional yaw torque in the design of the MPC controller. The output additional yaw torque meets the vehicle’s dynamic stability through four-wheel drive torque distribution. In addition, compared with the above paper, this paper considers the impact of road curvature on path-tracking accuracy and lateral stability under extreme working conditions.

The remaining sections and research content of this paper are as follows: Section 2 introduces the framework of the controller proposed in this paper and the symbols and their explanations used in the following text. Section 3 develops mathematical models for controller design, including a 7-dof vehicle dynamics model, a tire magic formula model, and a simplified two-degrees-of-freedom path-tracking control model. Section 4 provides a detailed introduction to the design of a lateral-stability-oriented path-tracking controller, covering the design objectives and both upper- and lower-layer controller designs. In Section 5, a fifth-degree polynomial function is used to generate the expected reference path for path-tracking control. Simulations are conducted under several extreme operating conditions to evaluate the controller’s effectiveness, followed by an analysis of the joint simulation results. Finally, Section 6 summarizes the research content of the entire paper, compares it with other similar control methods, and proposes future directions for work based on the shortcomings of the assumptions made in this paper.

2. Framework of Path-Tracking Control System

This paper introduces an upper and lower control framework for the lateral-stability-oriented path-tracking controller, as shown in Figure 1. The main goal of this system is to compute the optimal front wheel steering angle and additional yaw moment while maintaining lateral stability. To achieve this, various system elements work together to ensure precise tracking of the intended path. The upper-level controller determines the necessary steering angle and yaw moment to minimize lateral deviation. A 7-dof vehicle dynamics model is used to estimate the time-varying vertical and longitudinal forces on the wheels, which serve as constraints for tire–road interaction (adhesion ellipse) and yaw moment generation. The lower-level controller implements the upper-level commands by controlling steering and torque distribution to accurately follow the desired path.

Table 1. The representation of symbols along with their corresponding explanations.

Symbol	Description	Symbol	Description
$m\left(kg\right)$	vehicular heft	$a_x\left(m\cdot s^{-2}\right)$	vehicle’s longitudinal acceleration
$v_x\left(m\cdot s^{-1}\right)$	velocity components of V along X axis	$a_y\left(m\cdot s^{-2}\right)$	vehicle’s lateral acceleration
$v_y\left(m\cdot s^{-1}\right)$	velocity components of V along Y axis	$h_g\left(m\right)$	vehicle center of mass height
$\phi \left(rad\cdot s^{-1}\right)$	vehicle’s yaw angle	$a_{ij}\left(\mathrm{deg}\right)$	wheel side deflection angle
$F_{yij}\left(N\right)$	lateral forces on individual tires	$s_{ij}$	wheel slip ratio
$F_{xij}\left(N\right)$	longitudinal forces on individual tires	$v_{ij}\left(m\cdot s^{-1}\right)$	longitudinal speed of the wheels
$F_{zij}\left(N\right)$	dropping forces on individual tires	$I_t(kg\cdot m^2)$	inertia of wheel rotation
$B\left(m\right)$	track width	$k_f\left(N\cdot ra{d}^{-1}\right)$	sidewise rigidity of the foremost axle
$\delta \left(\mathrm{deg}\right)$	front wheel angle	$k_r\left(N\cdot ra{d}^{-1}\right)$	sidewise rigidity of the rear axle
$I_z\left(kg\cdot m^2\right)$	rotational inertia of the vehicle	$e_y\left(m\right)$	lateral tracking error
$l_f\left(m\right)$	distance extending from the mass center to the foremost axle	$e_{\phi }\left(\mathrm{deg}\right)$	directional angle tracking error
$l_r\left(m\right)$	distance extending from the mass center to the rear axle	$D_p\left(m\right)$	pre-targeting distance
${\omega }_{ij}\left(rad\cdot s^{-1}\right)$	angular velocity of the wheel	$\rho \left(m^{-1}\right)$	pre-target curvature
$T_{\mathrm{ij}}\left(N\cdot m\right)$	wheel torque	$\mu$	road surface friction coefficient
$l\left(m\right)$	wheelbase	$g\left(m\cdot s^{-2}\right)$	acceleration of gravity

provides a summary of the key symbols used in this paper, along with their respective descriptions, for the reader’s convenience.

3. Path-Tracking Model

3.1. Seven-Degree Vehicle Dynamics Model

This paper primarily focuses on four-wheel drive electric vehicles. To determine the time-varying vertical and longitudinal forces acting on the four wheels and to constrain the tire lateral forces using an attachment ellipse, a 7-dof dynamic model was developed. This model accounts for the longitudinal, lateral, and rotational dynamics of the four wheels, as illustrated in Figure 2.

Through the analysis of Newton’s second law and the equation of torque balance during vehicle operation, the dynamic differential equations for the longitudinal, lateral, yaw, and four-wheel rotations in the 7-dof vehicle dynamics model are derived as follows [37]:

$$\left\{\begin{array}{c}m\left({\dot{v}}_x-v_y\dot{\phi }\right)=\left(F_{xfl}+F_{xfr}\right)\cos \delta -\left(F_{yfl}+F_{yfr}\right)\sin \delta +F_{xrl}+F_{xrr}\\ m\left({\dot{v}}_y+v_x\dot{\phi }\right)=\left(F_{xfl}+F_{xfr}\right)\sin \delta +\left(F_{yfl}+F_{yfr}\right)\cos \delta +F_{yrl}+F_{yrr}\\ \begin{array}{cc}\hfill I_z\ddot{\phi }=& \left[\left(F_{xfl}+F_{xfr}\right)\sin \delta +\left(F_{yfl}+F_{yfr}\right)\cos \delta \right]l_f+\left(F_{xrr}-F_{xrl}\right){\displaystyle \frac{B}{2}}\hfill \\ & \hfill +\left[\left(F_{xfr}-F_{xfl}\right)\cos \delta +\left(F_{yfl}-F_{yfr}\right)\sin \delta \right]{\displaystyle \frac{B}{2}}-\left(F_{yrl}+F_{yrr}\right)l_r\end{array}\\ I_t{\dot{\omega }}_{fl}=-F_{xfl}R+T_{fl}\hfill \\ I_t{\dot{\omega }}_{fr}=-F_{xfr}R+T_{fr}\hfill \\ I_t{\dot{\omega }}_{rl}=-F_{xrl}R+T_{rl}\hfill \\ I_t{\dot{\omega }}_{rr}=-F_{rr}R+T_{rr}\hfill \end{array}\right.$$

(1)

The variation in vertical force for all four wheels can be expressed as follows [38]:

$$\left\{\begin{array}{l}{F}_{zfl}={\displaystyle \frac{m}{l}}\left({\displaystyle \frac{1}{2}}g{l}_r-{\displaystyle \frac{1}{2}}{a}_x{h}_g-{\displaystyle \frac{l_r}{l}}{a}_y{h}_g\right)\\ F_{zfr}={\displaystyle \frac{m}{l}}\left({\displaystyle \frac{1}{2}}g{l}_r-{\displaystyle \frac{1}{2}}{a}_x{h}_g+{\displaystyle \frac{l_r}{l}}{a}_y{h}_g\right)\\ F_{zrl}={\displaystyle \frac{m}{l}}\left({\displaystyle \frac{1}{2}}g{l}_f+{\displaystyle \frac{1}{2}}{a}_x{h}_g-{\displaystyle \frac{l_f}{l}}{a}_y{h}_g\right)\\ F_{zrr}={\displaystyle \frac{m}{l}}\left({\displaystyle \frac{1}{2}}g{l}_f+{\displaystyle \frac{1}{2}}{a}_x{h}_g+{\displaystyle \frac{l_f}{l}}{a}_y{h}_g\right)\end{array}\right.$$

(2)

The four-wheel lateral deviation angle is represented as follows:

$$\left\{\begin{array}{l}{\alpha }_{fl}=\arctan \left({\displaystyle \frac{v_y+l_f\dot{\phi }}{v_x-\frac{B}{2}\dot{\phi }}}\right)-\delta ,{\alpha }_{fr}=\arctan \left({\displaystyle \frac{v_y+l_f\dot{\phi }}{v_x+\frac{B}{2}\dot{\phi }}}\right)-\delta \\ {\alpha }_{rl}=\arctan \left({\displaystyle \frac{v_y-l_r\dot{\phi }}{v_x-\frac{B}{2}\dot{\phi }}}\right),{\alpha }_{rr}=\arctan \left({\displaystyle \frac{v_y-l_r\dot{\phi }}{v_x+\frac{B}{2}\dot{\phi }}}\right)\end{array}\right.$$

(3)

The four-wheel slide rate can be expressed as follows:

$$\left\{\begin{array}{l}{s}_{fl}=\left({\displaystyle \frac{{\omega }_{fl}R-v_{fl}}{v_{fl}}}\right),s_{fr}=\left({\displaystyle \frac{{\omega }_{fr}R-v_{fr}}{v_{fr}}}\right)\\ s_{fl}=\left({\displaystyle \frac{{\omega }_{rl}R-v_{rl}}{v_{rl}}}\right),s_{fl}=\left({\displaystyle \frac{{\omega }_{rr}R-v_{rr}}{v_{rr}}}\right)\end{array}\right.$$

(4)

3.2. Magic Formula Tire Model

This paper uses the magic formula tire model, which represents the dynamic relationship between tire force and roll angle, camber angle, and slip rate through trigonometric functions. The model has the advantages of simplicity and high fitting accuracy.

The forces in the longitudinal and lateral directions can be characterized as below: [35]:

$$\left\{\begin{array}{l}{F}_x=D_x\left(C_x\arctan \left(B_x{x}_s-E_x\left(B_x{x}_s-\arctan \left(B_x{x}_s\right)\right)\right)\right)\\ F_y=D_y\left(C_y\arctan \left(B_y{x}_{\alpha }-E_x\left(B_y{x}_{\alpha }-\arctan \left(B_y{x}_{\alpha }\right)\right)\right)\right)\end{array}\right.$$

(5)

In the formula, $F_x$ and $F_y$ are the tires’ vertical and lateral forces, respectively. $B_x$ and $B_y$ are rigidity factors, $C_x$ and $C_y$ are shape factors, $D_x$ and $D_y$ are hardness factors, $E_x$ and $E_y$ are curvature factors, $x_s$ are sliding ratio, and $x_a$ are tire side sliding angles. Based on the above theory, a 7-dof vehicle dynamics model was established in MATLAB/Simulink.

3.3. Path-Tracking Control System Model

An overly complex vehicle model can consume excessive computational resources, complicating control problem solutions and preventing real-time control algorithms from meeting the requirements of practical vehicle applications. In this section, we account for the effects of lateral motion, yaw motion, and road curvature and present a simplified automotive dynamics model that captures key automotive motion characteristics through constraint simplification and approximation, as illustrated in Figure 3.

The following represents the differential equation that governs the dynamics [39]:

$$\left\{\begin{array}{l}m\left({\dot{v}}_y+v_x\dot{\phi }\right)=F_{yf}\cos \delta +F_{xf}\sin \delta +F_{yr}\\ I_{\mathrm{z}}\ddot{\phi }=l_f{F}_{yf}\cos \delta +l_f{F}_{xf}\sin \delta -l_r{F}_{yr}\end{array}\right.$$

(6)

As shown in Figure 3, based on the planned path and the relative position of the vehicle during operation, the lateral error between the vehicle and the planned path can be calculated. Here, $e_y$ represents the lateral error, which is the distance from the current operating position of the vehicle to the planned expected path; and $e_{\phi }$ indicates the relative angle error between the heading of the running vehicle at the preview distance and the tangent of the current planned path. The formulation for the differentiation of the lateral error is presented in the manner detailed below:

$${\dot{e}}_y=\sin \left(e_{\phi }\right)v_x-v_y-\phi D_p$$

(7)

Since the relative angle error is typically considered small in the horizontal control of autonomous automobiles, we define it as $\sin \left(e_{\phi }\right)\approx e_{\phi }$. The horizontal driver model used for path tracking can be described as follows:

$$\begin{array}{l}{\dot{e}}_y=\sin \left(e_{\phi }\right)v_x-v_y-\phi D_p\\ {\dot{e}}_{\phi }=-\phi +v_x\rho \end{array}$$

(8)

Since the angle of the front wheel is relatively small, it is possible to approximate $\sin \delta =\delta ,\cos \delta =1$, bringing the (3) forms into the (6) forms can be sorted as in the following differential equation:

$$\left\{\begin{array}{l}{\dot{v}}_y={\displaystyle \frac{2\left(C_r+C_f\right)}{m{v}_x}}{v}_y+\left({\displaystyle \frac{2\left(l_f{C}_f-l_r{C}_r\right)}{m{v}_x}}-v_x\right)\dot{\phi }-{\displaystyle \frac{2C_f}{m}}\delta \\ \ddot{\phi }={\displaystyle \frac{2\left(l_f{C}_f-l_r{C}_r\right)}{I_z{v}_x}}{v}_y+{\displaystyle \frac{2\left({l_f}^2{C}_f+{l_r}^2{C}_r\right)}{I_{\mathrm{z}}{v}_x}}\dot{\phi }-{\displaystyle \frac{2l_f{C}_f}{I_z}}\delta +{\displaystyle \frac{\Delta M_{\mathrm{z}}}{I_z}}\\ {\dot{e}}_y=v_x\sin \left(e_{\phi }\right)-v_y-\dot{\phi }{D}_p\\ {\dot{e}}_{\phi }=v_x\rho -\dot{\phi }\end{array}\right.$$

(9)

In the formula, $\Delta M_{\mathrm{z}}$ is the additional yaw moment required for vehicle stability as defined by us.

The front wheel angle and additional yaw moment of the controlled vehicle are selected as control variables to control the lateral tracking error, steering angle tracking error, lateral velocity, and yaw rate of the vehicle, as well as the system state variables $x={\left[\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}{e}_y& e_{\phi }\end{array}& v_y\end{array}& \dot{\phi }\end{array}\right]}^T$. The control input is selected as $u={\left[\begin{array}{cc}{\delta }_f& \Delta M_Z\end{array}\right]}^T$. The state-space representation can be derived in the following manner:

$$\left\{\begin{array}{l}\dot{x}=A_{x}x+B_{u}u+D_{\rho }\rho \\ y=C_{x}x\end{array}\right.$$

(10)

where:

$$\begin{array}{l}{A}_x=\left[\begin{array}{cccc}0& v_x& -1& -D_{\rho }\\ 0& 0& 0& -1\\ 0& 0& {\displaystyle \frac{2\left(k_r+k_f\right)}{m{v}_x}}& {\displaystyle \frac{2\left(l_f{k}_f-l_r{k}_r\right)}{m{v}_x}}-v_x\\ 0& 0& {\displaystyle \frac{2\left(l_f{k}_f-l_r{k}_r\right)}{I_z{v}_x}}& {\displaystyle \frac{2\left({l_f}^2{k}_f+{l_r}^2{k}_r\right)}{I_{\mathrm{z}}{v}_x}}\end{array}\right]\\ \end{array}$$

$$B_u=\left[\begin{array}{cc}\begin{array}{c}0\\ 0\\ -{\displaystyle \frac{2k_f}{m}}\\ -{\displaystyle \frac{2l_f{k}_f}{I_z}}\end{array}& \begin{array}{c}0\\ 0\\ 0\\ {\displaystyle \frac{1}{I_z}}\end{array}\end{array}\right],C_x=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],D_{\rho }=\left[\begin{array}{c}0\\ v_x\\ 0\\ 0\end{array}\right]$$

4. Control System Design

4.1. Control Objectives

The controller design aims to develop a path-tracking controller that ensures lateral stability by considering the relationship between vehicle lateral stability and path-tracking control. To achieve accurate tracking of the time-varying planned path, it is essential to control both the vehicle’s longitudinal dynamics and lateral position. The specific objectives of the controller are as follows:

$$Objective(A):e_y\to 0$$

(11)

$$Objective(B):e_{\phi }\to 0$$

(12)

Among them, $e_y$ and $e_{\phi }$ represent the lateral position deviation and heading error, respectively, and they should try to approach 0 as much as possible while keeping the vehicle stable.

Additionally, the controller design must account for variations in tire dynamic characteristics under extreme operating conditions, ensuring that the maximum lateral force of the tire remains within the permissible range. The tire lateral force constraint boundary should be updated in real time. This can be formulated as outlined below:

$$F_y\le F_{y\max }$$

(13)

4.2. Upper-Level Controller Design

By discretizing Equation (10) using the forward Euler method in mathematics, the discrete-time linear state–space formulation can be derived as described below:

$$\left\{\begin{array}{l}x\left(\tilde{k}+1\right)=Ax\left(\tilde{k}\right)+Bu\left(\tilde{k}\right)+D\rho \left(\tilde{k}\right)\\ y\left(\tilde{k}\right)=Cx\left(\tilde{k}\right)\end{array}\right.$$

(14)

In the formula, $A=A_x{T}_e+I, B=B_x{T}_e, C=C_x{T}_e, D=D_{\rho }{T}_e, I$ is the unit matrix, and $T_e$ is the controller sampling time, where $\tilde{k}=t,t+1,\dots ,t+N_p-1$.

The state quantity and control quantity are integrated to construct new state quantities $\tilde{x}\left(\tilde{k}\right)={\left[\begin{array}{cc}x\left(\tilde{k}\right)& u\left(\tilde{k}-1\right)\end{array}\right]}^T$; thus, rewriting Equation (14) can be achieved.

$$\left\{\begin{array}{l}\tilde{x}\left(\tilde{k}+1\right)=\tilde{A}\tilde{x}\left(k\right)+\tilde{B}\Delta u\left(\tilde{k}\right)+\tilde{D}\rho \left(\tilde{k}\right)\\ \tilde{y}\left(\tilde{k}\right)=C\tilde{x}\left(\tilde{k}\right)\end{array}\right.$$

(15)

where: $\tilde{A}=\left[\begin{array}{cc}A& B\\ 0_{m\times n}& {\mathrm{I}}_m\end{array}\right],\tilde{B}=\left[\begin{array}{c}B\\ {\mathrm{I}}_m\end{array}\right],\tilde{D}=\left[\begin{array}{c}D\\ 0_m\end{array}\right],\tilde{C}=\left[\begin{array}{cc}C& 0_{4\times 2}\end{array}\right]$.

Among them, m and n are the control quantity dimension and state quantity dimension, respectively. The control input changes to $\Delta u\left(\tilde{k}\right)=u\left(\tilde{k}\right)-u\left(\tilde{k}-1\right)$.

Based on the above derivation, the system control time domain is set as $N_c$ and the system prediction time domain as $N_p$, and iterative calculations are performed to predict the system state variables and output variables in the time domain:

$$\left\{\begin{array}{l}\tilde{x}\left(\tilde{k}+1\right)=\tilde{A}\tilde{x}\left(\tilde{k}\right)+\tilde{B}\Delta u\left(\tilde{k}\right)+\tilde{D}\rho \left(\tilde{k}\right)\\ \tilde{x}\left(\tilde{k}+2\right)={\tilde{A}}^2\tilde{x}\left(\tilde{k}\right)+\tilde{A}\tilde{B}\Delta u\left(\tilde{k}\right)+\tilde{B}\Delta u\left(\tilde{k}\right)+\tilde{A}\tilde{D}\rho \left(\tilde{k}\right)+\tilde{D}\rho \left(\tilde{k}+1\right)\\ \tilde{x}\left(\tilde{k}+3\right)={\tilde{A}}^3\tilde{x}\left(\tilde{k}\right)+{\tilde{A}}^2\tilde{B}\Delta u\left(\tilde{k}\right)+\tilde{A}\tilde{B}\Delta u\left(\tilde{k}+1\right)+\tilde{B}\Delta u\left(\tilde{k}+2\right)+\cdots \\ +{\tilde{A}}^2\tilde{D}\rho \left(\tilde{k}\right)+\tilde{A}\tilde{D}\rho \left(\tilde{k}+1\right)+\tilde{D}\rho \left(\tilde{k}+2\right)\\ ⋮\\ \tilde{x}\left(\tilde{k}+N_c\right)={\tilde{A}}^{N_c}\tilde{x}\left(\tilde{k}\right)+{\tilde{A}}^{N_c-1}\tilde{B}\Delta u\left(\tilde{k}\right)+\cdots +\tilde{B}\Delta u\left(\tilde{k}+N_c-1\right)\\ +{\tilde{A}}^{N_c-1}\tilde{D}\rho \left(\tilde{k}\right)+\cdots +\tilde{D}\rho \left(\tilde{k}+N_c-1\right)\\ ⋮\\ \tilde{x}\left(\tilde{k}+N_p\right)={\tilde{A}}^{N_p}\tilde{x}\left(\tilde{k}\right)+{\tilde{A}}^{N_p-1}\tilde{B}\Delta u\left(\tilde{k}\right)+\cdots +\tilde{B}\Delta u\left(\tilde{k}+N_p-1\right)\\ +{\tilde{A}}^{N_p-1}\tilde{D}\rho \left(\tilde{k}\right)+\cdots +\tilde{D}\rho \left(\tilde{k}+N_p-1\right)\end{array}\right.$$

(16)

The predicted output vector is expressed in the form of a matrix:

$$Y\left(\tilde{k}\right)={\psi }_t\tilde{x}\left(\tilde{k}\right)+\Theta \Delta U\left(\tilde{k}\right)+\Pi \Omega \left(\tilde{k}\right)$$

(17)

where:

$$Y(\tilde{k})={\left[\begin{array}{cccc}y(\left.\tilde{k}+1\right|\tilde{k})& y(\left.\tilde{k}+2\right|\tilde{k})& \cdots & y(\left.\tilde{k}+N_p\right|\tilde{k})\end{array}\right]}^T$$

$$\Delta U(\tilde{k})={\left[\begin{array}{cccc}\Delta u\left(\tilde{k}\right)& \Delta u(\tilde{k}+1)& \cdots & \Delta u(\tilde{k}+N_p)\end{array}\right]}^T$$

$$\Omega \left(\tilde{k}\right)={\left[\begin{array}{cccc}\rho \left(\tilde{k}\right)& \rho \left(\tilde{k}+1\right)& \cdots & \rho \left(\tilde{k}+N_p-1\right)\end{array}\right]}^T$$

$${\psi }_t=\left[\begin{array}{c}\tilde{C}\tilde{A}\\ \tilde{C}{\tilde{A}}^2\\ ⋮\\ \tilde{C}{\tilde{A}}^{N_c-1}\\ ⋮\\ \tilde{C}{\tilde{A}}^{N_P-1}\end{array}\right] \Theta =\left[\begin{array}{cccc}C\tilde{B}& 0& \cdots & 0\\ C\tilde{A}\tilde{B}& C\tilde{B}& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮\\ C{\tilde{A}}^{N_c-1}\tilde{B}& C{\tilde{A}}^{N_c-2}\tilde{B}& \cdots & C\tilde{B}\\ C{\tilde{A}}^{N_c}\tilde{B}& C{\tilde{A}}^{N_c-1}\tilde{B}& \cdots & C\tilde{A}\tilde{B}\\ ⋮& ⋮& \ddots & ⋮\\ C{\tilde{A}}^{N_p-1}\tilde{B}& C{\tilde{A}}^{N_p-2}\tilde{B}& \cdots & C{\tilde{A}}^{N_p-N_c}\tilde{B}\end{array}\right]$$

$$\Pi =\left[\begin{array}{cccc}C\tilde{D}& 0& \cdots & 0\\ C\tilde{A}\tilde{D}& C\tilde{D}& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮\\ C{\tilde{A}}^{N_c-1}\tilde{D}& C{\tilde{A}}^{N_c-2}\tilde{D}& \cdots & C\tilde{D}\\ C{\tilde{A}}^{N_c}\tilde{D}& C{\tilde{A}}^{N_c-1}\tilde{D}& \cdots & C\tilde{A}\tilde{D}\\ ⋮& ⋮& \ddots & ⋮\\ C{\tilde{A}}^{N_p-1}\tilde{D}& C{\tilde{A}}^{N_p-2}\tilde{D}& \cdots & C{\tilde{A}}^{N_p-N_c}\tilde{D}\end{array}\right]$$

In order to obtain the expected control increase, the path-tracking problem can be converted to the following binary optimum problem:

$$J={\displaystyle \sum _{\tilde{k}=1}^{N_p}{‖Y\left(\tilde{k}\right)-Y_r\left(\tilde{k}\right)‖}_Q^2}+{\displaystyle \sum _{\tilde{k}=0}^{N_c}{‖\Delta U(|\tilde{k})‖}_R^2}$$

(18)

where $Q$ and $R$ are the matrix of weights.

When designing multi-objective optimization functions, the friction ellipse constraint is considered. The friction ellipse constraint is as follows [37]:

$$F_{xij}^2+F_{yij}^2\le {\left(\mu F_{zij}\right)}^2$$

(19)

Within the linear range, it can be organized as:

$${\left|F_{yij}\right|}_{\max }\le \sqrt{{\left(\mu F_{zij}\right)}^2-F_{xij}^2}=k{\alpha }_{\max }$$

(20)

Alternatively, the linear tire model achieves high fitting accuracy solely when the tire sideslip angle is small. The small angle assumption (3) can be summarized as:

$$\left\{\begin{array}{l}{\alpha }_f=\arctan \left({\displaystyle \frac{v_y+l_f\phi }{v_x}}\right)-\delta \approx {\displaystyle \frac{v_y+l_f\dot{\phi }}{v_x}}-\delta \\ {\alpha }_r=\arctan \left({\displaystyle \frac{v_y-l_r\phi }{v_x}}\right)\approx {\displaystyle \frac{v_y-l_r\dot{\phi }}{v_x}}\end{array}\right.$$

(21)

The constraint on the lateral deviation angle is converted into the constraint on the control and state variables, and Equation (21) can be written in the following matrix form:

$$\left[\begin{array}{c}{\alpha }_f\\ {\alpha }_r\end{array}\right]=\left[\begin{array}{cccc}0& 0& -{\displaystyle \frac{1}{v_x}}& -{\displaystyle \frac{l_f}{v_x}}\\ 0& 0& -{\displaystyle \frac{1}{v_x}}& {\displaystyle \frac{l_f}{v_x}}\end{array}\right]\left[\begin{array}{c}{e}_x\\ e_{\phi }\\ v_y\\ \dot{\phi }\end{array}\right]+\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]u\left(k\right)$$

(22)

It can be expressed as:

$$\alpha ={\zeta }_{1}y\left(k\right)+{\zeta }_{2}u\left(k\right)$$

where:

$$\begin{array}{l}\alpha =\left[\begin{array}{c}{\alpha }_f\\ {\alpha }_r\end{array}\right],{\zeta }_1=\left[\begin{array}{cccc}0& 0& -{\displaystyle \frac{1}{v_x}}& -{\displaystyle \frac{l_f}{v_x}}\\ 0& 0& -{\displaystyle \frac{1}{v_x}}& {\displaystyle \frac{l_f}{v_x}}\end{array}\right],\\ y\left(k\right)=\left[\begin{array}{c}{e}_x\\ e_{\phi }\\ v_y\\ \dot{\phi }\end{array}\right],{\zeta }_2=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\end{array}$$

$\alpha \le {\alpha }_{\max }$ i.e., $\left({\zeta }_{1}y\left(\tilde{k}\right)+{\zeta }_{2}u\left(\tilde{k}\right)\right)\le {\alpha }_{\max }$, simplified:

$$u\left(\tilde{k}\right)\le {\displaystyle \frac{1}{{\zeta }_2}}\left({\alpha }_{\max }-{\zeta }_{1}y\left(\tilde{k}\right)\right)=u_{\max }$$

(23)

During the controller design process, in addition to ensuring tracking accuracy, it is also necessary to provide stability and safety performance for the vehicle’s driving. Therefore, it is essential to set constraints on control variables and control increments. The constraints can be expressed in the following manner:

$$\left\{\begin{array}{l}{u}_{\min }\le u(\tilde{k}+i\left|\tilde{k}\right.)\le u_{\max }\begin{array}{cc}& i=0,1,\cdots ,N_c-1\end{array}\\ \Delta u_{\min }\le \Delta u(\tilde{k}+i\left|\tilde{k}\right.)\le \Delta u_{\max }\begin{array}{cc}& i=0,1,\cdots ,N_c-1\end{array}\end{array}\right.$$

(24)

The mathematical relationship between control quantity and control increment can be expressed as follows:

$$\left\{\begin{array}{c}u\left(\tilde{k}\right)=u\left(\tilde{k}-1\right)+\Delta u\left(\tilde{k}\right)\hfill \\ u\left(\tilde{k}+1\right)=u\left(\tilde{k}\right)+\Delta u\left(\tilde{k}+1\right)=u\left(\tilde{k}-1\right)+\Delta u\left(\tilde{k}\right)+\Delta u\left(\tilde{k}+1\right)\hfill \\ ⋮\hfill \\ \begin{array}{cc}\hfill u\left(\tilde{k}+N_c-1\right)& =u\left(\tilde{k}+N_c-2\right)+\Delta u\left(\tilde{k}+N_c-1\right)=u\left(\tilde{k}-1\right)+\Delta u\left(\tilde{k}\right)\hfill \\ & +\Delta u\left(\tilde{k}+1\right)+\cdots +\Delta u\left(\tilde{k}+N_c-1\right)\hfill \end{array}\hfill \end{array}\right.$$

(25)

The formula can be sorted as:

$$\begin{array}{l}U=\left[\begin{array}{c}u\left(\tilde{k}\right)\\ u\left(\tilde{k}+1\right)\\ ⋮\\ u\left(\tilde{k}+N_c-1\right)\end{array}\right]=\left[\begin{array}{c}u\left(\tilde{k}-1\right)\\ u\left(\tilde{k}-1\right)\\ ⋮\\ u\left(\tilde{k}-1\right)\end{array}\right]+\left[\begin{array}{cccc}1& 0& \cdots & 0\\ 1& 1& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 1& 1& 1& 1\end{array}\right]\left[\begin{array}{c}\Delta u\left(\tilde{k}\right)\\ \Delta u\left(\tilde{k}+1\right)\\ ⋮\\ \Delta u\left(\tilde{k}+N_c-1\right)\end{array}\right]\\ =U_t+A_I\Delta U\left(\tilde{k}\right)\end{array}$$

(26)

Thus, the constraints can be translated into:

$$\left\{\begin{array}{c}{U}_{\min }\le A_I\Delta U_t+U_t\le U_{\max }\\ \Delta U_{\min }\le \Delta U_t\le \Delta U_{\max }\end{array}\right.$$

(27)

4.3. Lower-Layer Controller Design

To facilitate the conversion of the output of the proposed lateral-stability-oriented path-tracking controller into the input of the simulated vehicle actuator, we have established a lower-level controller. The control input of the actuator is the optimal front wheel steering angle and additional yaw moment calculated by the controller based on real-time vehicle status, where the expected additional yaw moment is obtained by distributing torque between each wheel. The proposed lateral-stability-oriented path-tracking controller allocates four-wheel torque algorithm as follows.

The ratio of real-time tire force to the maximum adhesion between the wheel and the road surface is called the adhesion utilization coefficient, which is the reciprocal of the adhesion margin. Generally, higher adhesion utilization values indicate lower available tire potential. When the adhesion utilization value approaches 100%, the tire–road adhesion nears its limit, placing the vehicle in an unstable critical state. Therefore, the optimization goal of the proposed lateral-stability-oriented path-tracking controller is to reduce the lateral error of the controlled vehicle while minimizing the utilization rate of each tire’s grip. By selecting tire adhesion utilization and total power as optimization objectives, the tire utilization and total power can be expressed as [37]:

$$\begin{array}{l}{\Psi }_1={\displaystyle \frac{F_{xfl}^2}{{\left(\mu F_{zfl}\right)}^2}}+{\displaystyle \frac{F_{xfr}^2}{{\left(\mu F_{zfr}\right)}^2}}+{\displaystyle \frac{F_{xrl}^2}{{\left(\mu F_{zl}\right)}^2}}+{\displaystyle \frac{F_{xrr}^2}{{\left(\mu F_{zrr}\right)}^2}}\\ {\Psi }_2={\left(F_{xfl}R{\omega }_{fl}\right)}^2+{\left(F_{xfr}R{\omega }_{fr}\right)}^2+{\left(F_{xrl}R{\omega }_{rl}\right)}^2+{\left(F_{xrr}R{\omega }_{rr}\right)}^2\end{array}$$

(28)

Firstly, the control allocation error is minimized to ensure that the sum of the torques generated by the four wheels equals the total required torque and that the yaw moment produced by these wheels matches the additional yaw moment. The driving and braking forces of each wheel must satisfy the following relationship:

$$\begin{array}{l}{F}_x=F_{xfl}+F_{xfr}+F_{xrl}+F_{xrr}\\ \Delta M={\displaystyle \frac{d}{2}}\left(-F_{xfl}+F_{xfr}-F_{xrl}+F_{xrr}\right)\end{array}$$

(29)

The first cost function can be set to:

$$J_1={\displaystyle \frac{1}{2}}{\left(B_{b}u-v_d\right)}^T{W}_u\left(B_{b}u-v_d\right)$$

(30)

where $v_d=B_{b}u=\left[\begin{array}{c}{F}_x\\ \Delta M_z\end{array}\right],u={\left[\begin{array}{cccc}{T}_{fl}& T_{fr}& T_{rl}& T_{rr}\end{array}\right]}^T$

$B_b={\displaystyle \frac{1}{R}}\left[\begin{array}{cccc}1& 1& 1& 1\\ -d_f& d_f& -d_r& d_r\end{array}\right]$.

Here, $W_u$ is the weight matrix of the distribution error, $T_{ij}$ is the four-wheel torque, and $d_f$, $d_r$ is half of the wheel range.

Secondly, the tire loading rate can be used to characterize the adhesion effect of the tire, which is the reciprocal of the tire and the adhesion margin. When the tire loading rate decreases, the adhesion margin of the tire increases, and the adhesion ability of the tire improves. Therefore, tire load rate is one of the indicators that characterize vehicle stability. $J_2$ is chosen as the second cost function [40].

$$J_2={\displaystyle \frac{1}{2}}{u}^T{W}_{v}u$$

(31)

where $W_v=diag\left({\displaystyle \frac{1}{{\left(\mu R{F}_{zij}\right)}^2}}\right)$.

The total cost function can be expressed as:

$$\min J=J_1+J_2={\displaystyle \frac{1}{2}}{\left(B_{b}u-v_d\right)}^T{W}_u\left(B_{b}u-v_d\right)+{\displaystyle \frac{1}{2}}{u}^T{W}_{v}u$$

(32)

5. Simulation Verification

To verify the comprehensive performance of the controller, several extreme simulation conditions were selected for comparative analysis. On the joint simulation platform of CarSim and MATLAB/Simulink, we compared controller A, which only performs path-tracking control, and controller B, which considers tire vertical load transfer and lateral force constraints, with controller C, proposed in this paper.

Table 2. Controller types and their corresponding descriptions.

Controller Type	Corresponding Description
Controller A	Traditional MPC controller that only considers path tracking
Controller B	MPC controller that considers changes in tire dynamic characteristics and real-time updating of lateral force constraints
Controller C (Proposed Controller)	MPC controller that simultaneously considers changes in tire dynamic characteristics and performs real-time updates of lateral force constraints and torque distribution (proposed controller)

summarizes the three controllers mentioned in this paper and their corresponding descriptions for readers’ convenience. The following sections will provide detailed information on the simulation.

5.1. Desired Path Setting

The fifth-degree polynomial function is commonly used in lane-changing scenarios due to its ability to plan trajectories with continuous third-order derivatives and smooth curvatures based on the initial state at the beginning of vehicle operation and the final state at the end of vehicle operation. This paper uses a fifth-degree polynomial function to generate the expected reference path for path-tracking control and validates the proposed lateral-stability-oriented path-tracking controller through several different extreme lane-changing scenarios. A fifth-degree polynomial function can be expressed as:

$$\left\{\begin{array}{c}x(t)=c_0+c_{1}t+c_2{t}^2+c_3{t}^3+c_4{t}^4+c_5{t}^5\\ y(x)=d_0+d_{1}x+d_2{x}^2+d_3{x}^3+d_4{x}^4+d_5{x}^5\end{array}\right.$$

(33)

In the formula, $c_0,c_1,\cdots c_5,d_0,d_1\cdots d_5$ is a design parameter that can be determined by the initial state at the beginning of vehicle operation and the final state at the end of vehicle operation.

$${\theta }_r(t)=\arctan \left\{y^{\prime }\left[x\left(t\right)\right]\right\}$$

(34)

$$k_r\left(t\right)={\displaystyle \frac{y^{″}\left[x\left(t\right)\right]}{{\left(1+{\left\{y^{\prime }\left[x\left(t\right)\right]\right\}}^2\right)}^{\frac{3}{2}}}}$$

(35)

5.2. Simulation Scenario Setting for Lateral-Stability-Oriented Path-Tracking Control

A collaborative simulation environment was established in CarSim and MATLAB/Simulink to validate the proposed lateral-stability-oriented path-tracking controller. Figure 4 displays the block diagram of the simulation, while

Table 3. Parameters for co-simulation.

Definition	Symbol	Value
Vehicle mass	$m\left(kg\right)$	1723
Gravitational acceleration	$g\left(m\cdot s^{-2}\right)$	9.81
Center of mass to front axle distance	$l_f\left(m\right)$	1.015
Center of mass to rear axle distance	$l_r\left(m\right)$	1.895
Front axle lateral stiffness	$k_f\left(N\cdot ra{d}^{-1}\right)$	−46,241
Rear axle lateral stiffness	$k_r\left(N\cdot ra{d}^{-1}\right)$	−66,660
Wheel track	$B\left(m\right)$	1.675
The rotational inertia of the vehicle	$I_z\left(kg\cdot m^2\right)$	1537
Pre-targeting distance	$D_p\left(m\right)$	10
Joint simulation running time	$T\left(s\right)$	10

lists the key parameters. The four simulation scenarios are outlined in

Table 4. Simulation scenarios.

Simulation Scenario	Track Type	Tire-Road Friction Coefficient	Simulated Vehicle Speed
Scenario 1	Large-curvature double shift line	0.95	70 km/h
Scenario 2	Large-curvature double shift line	0.75	70 km/h
Scenario 3	General double shift line	0.95	90 km/h
Scenario 4	General double shift line	0.75	90 km/h

.

5.3. Analysis and Discussion of Simulation Results

5.3.1. Scenario 1

The results show that, at a simulation speed of 70 km/h on asphalt with a road adhesion coefficient of 0.95, the designed controller offers better control accuracy and stability than controllers A and B. As shown in Figure 5a, during lane changes on a reference path with high curvature, controller A gradually becomes unstable and loses its path-tracking capability. Controller B maintains vehicle stability but incurs larger path-tracking errors, while controller C effectively preserves stability.

Figure 5b demonstrates that the front wheel angle of the MPC controller, which only considers path tracking, becomes significantly abnormal during lane changes, with a maximum steering wheel angle of 24.6 degrees. Meanwhile, controller B accounts for vertical load transfer and lateral force constraints, substantially reducing the steering angle amplitude during turning. Building upon controller B, controller C incorporates active yaw moment as a control variable in the path-tracking system, achieving smoother turns and further enhancing stability during high-speed lane changes.

Figure 6 compares the stability parameters of the three controllers. As is evident from the figure, controller A’s instability in the sideslip results in diverging lateral errors, yaw rates, and sideslip angles. Figure 6a shows that controller C controls the vehicle’s lateral error within a smaller range compared to controllers A and B. Figure 6b reveals that controller C achieves a smoother yaw rate change than controller B, enhancing vehicle stability, with a maximum yaw rate reduced by 2.33% and an average yaw rate reduced by 31.62%. Figure 6c illustrates that controller C maintains the vehicle’s sideslip angle within a smaller range compared to controller B, reducing the maximum sideslip angle by 28.45% and the average sideslip angle by 40.21%.

Figure 7a illustrates that the control strategy proposed by Controller C requires active yaw moment only briefly and only when vehicle stability is compromised, thereby minimizing the interference with the path-tracking control. The distribution of the four-wheel torque, calculated based on the additional lateral moment, is presented in Figure 7b.

To better understand the improvement in the stability of controller C relative to controller B under extreme operating conditions, we have comprehensively compared the stability parameters of controller B and controller C in the following table.

Table 5. The maximum and average values of various parameters of controller B and controller C.

	Controller B	Controller C	Improvement (Controller C Relative to Controller B)
Maximum yaw rate	0.4245 rad/s	0.4146 rad/s	2.33%
Maximum sideslip angle	1.1101 deg	0.7943 deg	28.45%
Average yaw rate	0.1151 rad/s	0.0787 rad/s	31.62%
Average sideslip angle	0.1798 deg	0.1075 deg	40.21%

presents the maximum yaw rate, maximum sideslip angle, average yaw rate, and average sideslip angle.

5.3.2. Scenario 2

When setting the vehicle simulation speed to 70 km/h on an asphalt pavement with a road adhesion coefficient of 0.75, the designed controller demonstrates superior control accuracy and stability compared to controllers B and C. As illustrated in Figure 8a, all controllers are capable of tracking the reference path during the initial lane change. However, the designed path-tracking controller and controller B take longer to converge to the target path compared to controller A, which focuses solely on path tracking. This delay is due to the consideration of vertical load transfer and the constraints on lateral forces. Nevertheless, during subsequent lane changes, controller A progressively loses stability and path-tracking capability, whereas controllers B and C maintain stability effectively.

Figure 8b shows that the front-wheel steering angle of the MPC controller, which only considers path tracking, becomes significantly abnormal during the second lane change, reaching a maximum of 24.6 degrees. Under high-speed conditions, large and rapid steering wheel adjustments can lead to oversteering and vehicle instability. Controller B, which takes into account load transfer and limits lateral forces, significantly reduces the steering wheel angle’s amplitude during the maneuvering process. Building upon controller B, controller C introduces active yaw torque as a control variable, further smoothing the steering process and enhancing stability during high-speed lane changes.

Figure 9 compares the stability parameters of the three controllers. As depicted in the figure, due to the unstable sideslip in controller A, the vehicle’s lateral error, yaw rate, and sideslip angle tend to diverge. Figure 9a indicates that controller C maintains the vehicle’s lateral error within a smaller range compared to controllers A and B. Figure 9b shows that controller C achieves a smoother change in yaw rate compared to controller B, enhancing vehicle stability; the maximum yaw rate is reduced by 4.61%, and the average yaw rate is reduced by 35.11%. Figure 9c demonstrates that controller C effectively constrains the vehicle’s sideslip angle within a narrower range than controller B, reducing the maximum sideslip angle by 8.93% and the average sideslip angle by 37.33%.

Figure 10a illustrates that the control strategy proposed by controller C requires active yaw moment only briefly and only when vehicle stability is compromised, thereby minimizing interference with path-tracking control. The distribution of four-wheel torque, calculated based on the additional lateral moment, is presented in Figure 10b.

To better understand the improvement in stability of controller C relative to controller B under extreme operating conditions, we conducted a thorough comparison of the stability parameters of the two controllers.

Table 6. The maximum and average values of various parameters of controller B and controller C.

	Controller B	Controller C	Improvement (Controller C Relative to Controller B)
Maximum yaw rate	0.4297 rad/s	0.4099 rad/s	4.61%
Maximum sideslip angle	1.4809 deg	1.3487 deg	8.93%
Average yaw rate	0.1424 rad/s	0.0924 rad/s	35.11%
Average sideslip angle	0.2936 deg	0.1840 deg	37.33%

presents the maximum yaw rate, maximum sideslip angle, average yaw rate, and average sideslip angle.

5.3.3. Scenario 3 and Scenario 4

To comprehensively evaluate the performance of the controller, a double-lane reference path with a standard lane-changing curvature was selected, and the lane-changing speed was increased to 90 km/h. The simulation results at a speed of 90 km/h and a road adhesion coefficient of 0.95 are presented in Figure 11, while the results for a road adhesion coefficient of 0.75 are shown in Figure 12. As the speed increases to 90 km/h, it is evident that controllers A and B lose lateral stability and begin to slide, ultimately failing to maintain path tracking. In contrast, the proposed controller C is able to sustain lateral stability and accurately track the desired path.

5.4. Real-Time Calculation of the Designed Controller

Due to the fact that the lateral-stability-oriented path-tracking controller designed in this paper is based on MPC, its computational efficiency has a significant impact on the feasibility of real-time applications. Therefore, we have listed the real-time computation time for the joint simulation experiment in this paper, which was conducted based on the software and hardware conditions in

Table 7. Controller simulation software and hardware parameters.

Configuration Name	Configuration Version
CPU	12th Gen Intel(R) Core (TM) i7-12700 2.10 GHz (16 GB)
GPU	Intel(R) UHD Graphics 770
Computer operating system	Windows 10 (64-bit)
MATLAB version	MATLAB R2022a
CarSim version	CarSim 2020.0

. Under these software and hardware conditions, the real-time calculation time curve of the lateral-stability-oriented path-tracking controller proposed in this paper is shown in Figure 13. From the calculation of the time curve in Figure 13, it can be intuitively seen that the designed controller has a relatively low average calculation time in simulation scenarios 1 and 2, with average values of 3.2658 ms and 3.4786 ms, respectively. By using smaller prediction and control time domains or better hardware facilities for calculation, the real-time performance of the controller can be further improved, ensuring the feasibility of the controller designed in this paper.

5.5. Summary and Comparison of Simulation Results

From the joint simulation results of the four different extreme operating conditions mentioned above, we saw that in the scenario of lane changing at medium and high speeds, the traditional MPC controller A, which only considered path tracking in References [22,35], experienced instability and sideslip. The lateral error, yaw rate, and sideslip angle tended to diverge. MPC controller B, which took into account changes in tire dynamic characteristics and updated lateral force constraints in real time, was able to maintain lateral stability. However, when compared to controller C proposed in this paper, which also considered changes in tire dynamic characteristics and applied additional lateral torque, controller B had a larger lateral error. Controller C reduced the average lateral rate by 31.62% and the average sideslip angle by 40.21% compared to controller B. The designed path-tracking controllers C and B took longer to approach the target path at the beginning of tracking. This delay was due to the consideration of vertical load transfer and lateral force constraints. Additionally, compared to controllers A and B, controller C exhibited smoother changes in front wheel steering angle and yaw rate, enhancing the vehicle’s stability. As the speed further increased to high speed, controller B also lost lateral stability, whereas the designed controller C continued to maintain lateral stability and track the desired path, further proving that the designed controller could track the desired path while maintaining lateral stability under extreme working conditions. The simulation results have verified the effectiveness of the proposed controller, providing a more stable method for tracking target trajectories in four-wheel independent-drive electric vehicles under extreme conditions and offering new insights for the design of future intelligent vehicle control systems.

6. Conclusions

This paper presents a path-tracking controller focused on lateral stability for 4WID autonomous vehicles. It addresses the issue of uncrewed vehicles failing to generate adequate lateral force due to tire vertical load transfer under extreme conditions, such as high-speed lane changes, which can lead to poor tracking performance and even lateral instability. The approach is first based on a 7-dof vehicle dynamics model; the time-varying vertical load of the tires was calculated and transmitted to the upper-level path-tracking controller. In this upper-level controller, tire friction ellipse constraints were considered to limit the lateral forces, with the boundary values of these constraints being updated in real time. Additionally, an extra yaw moment was calculated to enhance the vehicle’s lateral stability. The lateral tracking error, steering angle tracking error, lateral velocity, and yaw rate were controlled by adjusting the front wheel angle and applying the additional yaw moment. In the lower-level controller, the front wheel steering angle was determined by corresponding components, while the four-wheel torque was distributed based on minimizing tire utilization to generate additional yaw moment.

Several typical operating conditions were simulated and verified using Carsim2020 and MATLAB/Simulink2020a. The results demonstrated that the proposed path-tracking controller could maintain vehicle stability under extreme conditions, such as high-speed steering, while keeping the lateral error, yaw rate, and sideslip angle within minimal limits. However, the MPC controller under the same operating conditions produced greater lateral errors or lost path-tracking ability due to unstable sideslips; MPC controllers that only consider lateral force constraints generate significant lateral errors. Compared with controller B, the proposed controller C reduced the average lateral velocity by 31.62% and the average sideslip angle by 40.21%. The proposed controller provided a more stable method for tracking target trajectories under extreme conditions, thereby improving the safety and stability of intelligent vehicles in high-speed environments and offering new insights for future intelligent vehicle control system design.

However, as the road friction coefficient might vary in real time, the real-time estimation of the road friction coefficient should be considered in future work. In addition, future work could involve further validation of the effectiveness of the algorithm in practical vehicle applications.