Coordinated Control Strategy for Stability Control and Trajectory Tracking with Wheel-Driven Autonomous Vehicles Under Harsh Situations

Abstract

A coordinated strategy is proposed to prevent interference between trajectory tracking control and stability control in wheel-driven autonomous vehicles. A tire cornering stiffness estimate model is developed using the recursive least squares approach with a forgetting factor (FFRLS), resulting in precise estimation of tire cornering stiffness. An adaptive trajectory tracking control is developed, utilizing real-time updates of tire cornering stiffness; for the direct yaw moment required for stability control, an integral sliding-mode control is adopted, and the chatter problem of the integral sliding-mode controller is optimized by a fuzzy controller. The coordinated control of trajectory tracking and vehicle stability is ultimately attained through the application of the normalized stability index. The method’s practicality is validated by the hardware-in-the-loop simulation platform.

1. Introduction

To mitigate traffic safety concerns, automobiles are advancing towards greater intelligence. Intelligent driving technology is acknowledged as an essential element of future intelligent transportation systems, since it might significantly diminish 90% of road traffic accidents, enhance road utilization, and lower transportation expenses [1,2]. Intelligent driving technology primarily encompasses three modules: environmental perception, decision-making and planning, and motion control. These three modules cooperate to collectively achieve intelligent vehicle operation. Trajectory tracking falls within the category of motion control for intelligent vehicles, with the primary objective of accurately following the vehicle’s target speed and reference path. This requires sufficiently precise control algorithms and smooth, comfortable control processes to ensure vehicle stability and driving safety. However, when vehicles operate under extreme conditions such as low-adhesion road surfaces, they are prone to instability, significantly impacting trajectory tracking accuracy. Therefore, exploring trajectory tracking accuracy and stability under extreme conditions is of significant importance for advancing the development of intelligent driving technology [3].

Electric vehicles (EVs) represent not only an effective means to address energy consumption and atmospheric pollution but also an ideal platform for intelligent driving technology [4]. Compared to centrally driven EVs, in-wheel motor-driven EVs eliminate components such as the drive shaft and differential, installing the motors directly within the wheel hubs to drive the wheels. This configuration enables rapid response to torque demands for each individual in-wheel motor, with independently controllable torque for all four wheels, providing greater flexibility in driving modes. In-wheel motor-driven EVs are considered the ultimate form of new energy vehicles. By redistributing torque among the wheels through additional yaw moments, they can enhance vehicle driving stability.

Taking hub–motor-driven electric vehicles as the research subject and leveraging their chassis advantages, this study focuses on trajectory tracking and stability control of vehicles under extreme conditions. It investigates the integration of trajectory tracking with direct yaw moment control (DYC) to ensure that vehicles maintain good trajectory tracking accuracy while also achieving driving stability [5].

A vehicle represents a complex control system characterized by information source interaction and controller variable coupling. Under conditions of high lateral acceleration, reliance solely on trajectory tracking control can lead to deviation from the reference trajectory due to saturation of lateral forces causing ineffective adjustment of the vehicle’s front wheel angle. Consequently, numerous scholars have initiated research into methods for coordinated control of trajectory tracking and vehicle stability, aiming to achieve trajectory tracking control while simultaneously maintaining vehicle stability. Hub–motor-driven electric vehicles (HMEVs), with their advantages of four-wheel independence and rapid torque response, are conducive to enhancing overall vehicle performance under complex conditions, making them the preferred research subject for coordinated control of trajectory tracking and stability. Currently, there are two architectures applied to coordinated control of trajectory tracking and stability in HMEVs: hierarchical control and integrated control [6].

The hierarchical control architecture comprises three core modules: a lateral path tracking controller, a longitudinal speed tracking controller, and a stability controller. The lateral path tracking controller calculates the desired front wheel angle to achieve tracking of the reference trajectory. The longitudinal speed tracking controller determines the total longitudinal force required to track the target speed, while the stability controller computes the yaw moment based on the side-slip angle and yaw rate of the vehicle’s centroid. Vehicle trajectory tracking and stability control are achieved through the coordination of front wheel angle and four-wheel torque. Xie et al. [7] utilized the Model Predictive Control (MPC) algorithm to design an active steering path tracking controller and employed Sliding Mode Control (SMC) to design a stability controller for the desired yaw rate, with torque distribution based on rules. Deng et al. [8] used the MPC algorithm to compute the front wheel angle, while a PID longitudinal driver model was used to output the desired torque for speed tracking. They also designed a lower-level Direct Yaw Moment Control (DYC) controller using a Linear Quadratic Regulator (LQR) and optimized the distribution of the resulting additional yaw moment based on vertical load. Guo [9] designed a lateral controller using the MPC algorithm and optimized the parameters of the MPC controller using the Particle Swarm Optimization (PSO) algorithm. The stability controller was designed using SMC and incorporated variable universe fuzzy logic to improve the chattering issue in the control system, effectively enhancing the tracking accuracy and stability of intelligent vehicles under extreme conditions.

The integrated controller architecture achieves coordinated control for trajectory tracking and stability by utilizing a centralized controller to harmonize the steering angle of the front wheels and the torque of all four wheels of the vehicle. The centralized controller integrates lateral and longitudinal motion control, as well as stability control, to realize multi-objective control. Liang et al. [10] designed a coordinated controller that integrates both trajectory tracking and stability, utilizing a hyperbolic tangent function to dynamically adjust the target weights for lateral and longitudinal motion control under various operating conditions, thereby enabling the coupling of lateral and longitudinal motion control of the vehicle. This coordinated controller fully considers the lateral–longitudinal coupling characteristics of the vehicle under extreme operating conditions. Zhang et al. [11] took a dual-tracked vehicle model as the research subject and designed an MPC (Model Predictive Control) centralized integrated controller. This controller incorporates actuator constraints and safety constraints on vehicle states while also considering the influence of active camber on the vehicle, ensuring tracking accuracy and stability of the vehicle.

To sum up, the integrated control architecture fully takes into account the vehicle’s longitudinal and lateral coupling characteristics. This lets it find the best solution for steering and constraint conditions to make sure the system has the best control performance. However, for complex vehicle systems, the integrated control architecture is computationally intensive and lacks real-time performance. On the other hand, the hierarchical independent control architecture boasts a simple structure and easily adjustable parameters but lacks coordination, with controllers operating independently from each other, leading to chattering issues during the control process.

Therefore, this paper takes hub–motor-driven autonomous vehicles as the research object and proposes a coordination control strategy based on a hierarchical architecture, with the following contributions: (1) An adaptive Model Predictive Control (MPC) controller based on time-varying linearity is proposed. In this method, the real-time tire cornering stiffness is obtained using the Forgetting Factor Recursive Least Squares (FFRLS) algorithm. By introducing the real-time updated tire cornering stiffness into the trajectory tracking control, the time-varying characteristics of the vehicle during trajectory tracking and the trajectory tracking accuracy are effectively improved. (2) Integral sliding mode control is utilized to obtain the direct yaw moment in stability control, and a fuzzy control method is employed to mitigate the chattering of the integral sliding-mode controller. (3) A normalized stability index is employed to evaluate the degree of vehicle instability. The intervention level of stability control is adjusted according to this normalized stability index to ensure the coordinated control of trajectory tracking and vehicle stability.

The other parts of this paper are described as follows: In Section 2, the vehicle dynamics model and trajectory tracking model of autonomous vehicles are constructed. In Section 3, trajectory tracking control, stability control, and coordinated control methods are proposed. In Section 4, the verification of the control strategy using a hardware-in-the-loop test bench and the analysis of data are described. Finally, Section 5 summarizes the conclusions of this paper.

2. Vehicle Model

2.1. Vehicle Dynamics Model with 2 Degrees of Freedom

The two-degree-of-freedom vehicle model contains the lateral motion and the transverse pendulum motion of the vehicle, and the model is shown in Figure 1. $v_x$ and $v_y$ in the figure denote the longitudinal and lateral velocities of the vehicle; $\beta$ represents the side-slip angle; $\gamma$ represents the yaw rate; and ${\delta }_f$ represents the front wheel angle of the vehicle. The two-degree-of-freedom vehicle model can be represented as follows:

$$\left\{\begin{array}{l}m{v}_x(\dot{\beta }+\gamma )=F_{yf}+F_{yr}\\ I_z\dot{\gamma }=l_f{F}_{yf}-l_r{F}_{yr}\end{array}\right.,$$

(1)

where $l_f$ denotes the distance from the front axle to the center of mass; $l_r$ denotes the distance from the rear axle to the center of mass; $I_z$ denotes the moment of inertia; $m$ denotes the mass of the car; $F_{yf}$ denotes the lateral force of the front tire; $F_{yr}$ denotes the lateral force of the rear tire; and $F_{yf}$ and $F_{yr}$ can be considered as linearly related to the tire slip angle, with the relationship between the two being expressed as follows:

$$\left\{\begin{array}{l}{F}_{yf}=C_f{\alpha }_f=C_f({\delta }_f-{\displaystyle \frac{l_f\gamma }{v_x}}-\beta )\\ F_{yr}=C_r{\alpha }_r=C_r({\displaystyle \frac{l_r\gamma }{v_x}}-\beta )\end{array}\right.,$$

(2)

where $C_f$ denotes the cornering stiffness of the front tires; $C_r$ denotes the cornering stiffness of the rear tires; ${\alpha }_f$ denotes the slip angles of the front tires; and ${\alpha }_r$ denotes the slip angles of the rear tires. Equations (1) and (2) are combined as follows:

$$\left\{\begin{array}{l}\dot{\beta }=-{\displaystyle \frac{C_f+C_r}{m{v}_x^2}}\beta -({\displaystyle \frac{l_f{C}_f-l_r{C}_r}{m{v}_x^2}}+1)\gamma +{\displaystyle \frac{C_f}{m{v}_x}}{\delta }_f\\ \dot{\gamma }=-{\displaystyle \frac{l_f{C}_f-l_r{C}_r}{I_z}}\beta -{\displaystyle \frac{l_f^2{C}_f-l_r^2{C}_r}{I_z{v}_x}}\gamma +{\displaystyle \frac{l_f{C}_f}{I_z}}{\delta }_f\end{array}\right.,$$

(3)

2.2. Vehicle Trajectory Tracking Model

In this paper, a trajectory tracking algorithm is designed using a single-point preview driver model. This model takes into account the driver’s preview deviation, and the model is shown in Figure 2. The vehicle trajectory tracking model is shown in Figure 2. In the figure, $d$ denotes the nearest distance from the vehicle center of mass to the reference trajectory; $e_y$ represents the lateral distance from the vehicle center of mass to the preview point $d_p$ on the reference trajectory; $e_{yo}$ represents the distance from the vehicle center of mass to point $d$; $L$ denotes the distance between $e_y$ and $e_{yo}$; $\Psi$ denotes the actual heading angle of the vehicle; and ${\Psi }_d$ denotes the desired heading angle.

By analyzing the positional relationship between the vehicle and the reference trajectory, the trajectory tracking model based on single-point preview can be represented as follows:

$$\left\{\begin{array}{l}{\dot{e}}_y=v_y+v_x{e}_{\Psi }+L\dot{\Psi }\\ {\dot{e}}_{\Psi }=\dot{\Psi }-{\dot{\Psi }}_d=\gamma -k_d{v}_x\end{array}\right.,$$

(4)

where $e_y$ denotes the lateral displacement error; $e_{\Psi }$ denotes the heading angle error; and $k_d$ is the curvature of the reference track.

When the vehicle is moving at elevated velocity, $L$ can be expressed as follows:

$$L=v_x\Delta t, L_{\min }\le L\le L_{\max },$$

(5)

where $\Delta t$ denotes the preview time, which can be expressed as $\Delta t=N_{p}T$; $L_{\min }$ is the lower limit of the preview distance, whereas $L_{\max }$ denotes the upper limit of the preview distance; and $N_p$ denotes the prediction range in the model prediction algorithm.

Considering the complex coupling relationship between longitudinal and lateral motions in the trajectory tracking model, this paper simplifies the longitudinal velocity tracking model in the trajectory tracking model as follows [12]:

$$\left\{\begin{array}{l}{e}_x=v_x-v_{xd}\\ {\dot{e}}_x={\displaystyle \frac{F_x}{m}}-a_d\end{array}\right.,$$

(6)

where $e_x$ denotes the vehicle longitudinal velocity error; $F_x$ denotes the longitudinal force; $v_{xd}$ denotes the longitudinal desired velocity; and $a_d$ denotes the desired acceleration of the vehicle.

The vehicle’s trajectory tracking model is structured into state space equations as per Equations (3)–(6).

$$\left\{\begin{array}{l}\dot{x}=Ax+Bu+\omega \\ y=Cx\end{array}\right.,$$

(7)

where $x={[e_x,e_y,\beta ,e_{\Psi },\gamma ]}^T$, $u={[F_x,{\delta }_f]}^T$, $\omega ={[-a_d,0,0,-k_d{v}_x,0]}^T$, $y={[e_x,e_y,e_{\Psi }]}^T$, $A=\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 1& v_x& 0\\ 0& 0& -\frac{C_f+C_r}{m{v}_x}& 0& \frac{l_r{C}_r-l_f{C}_f}{m{v}_x}-v_x\\ 0& 0& 0& 0& 1\\ 0& 0& \frac{l_r{C}_r-l_f{C}_f}{I_z{v}_x}& 0& -\frac{l_f^2{C}_f+l_r^2{C}_r}{I_z{v}_x}\end{array}\right]$, $B=\left[\begin{array}{cc}{\displaystyle \frac{1}{m}}& 0\\ 0& 0\\ 0& {\displaystyle \frac{C_f}{m}}\\ 0& 0\\ 0& {\displaystyle \frac{l_f{C}_f}{I_z}}\end{array}\right]$, $C=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0\end{array}\right]$.

3. Design of Coordinated Control Strategies

The coordinated control system architecture for stability and trajectory tracking is shown in Figure 3. The planning layer relies on the on-board sensor data and real-time vehicle state parameters to calculate the reference trajectory and reference speed of the vehicle. The MPC-based trajectory tracking controller calculates the optimal front wheel angle and longitudinal force required to maintain the current vehicle trajectory. The optimal front wheel angle is substituted into the desired state calculation module to obtain the desired value of the yaw rate and the sideslip angle to maintain the stability of the vehicle. The SMC-based stability control module calculates the active yaw moment required to maintain vehicle stability. In this paper, the weight coefficients are set to adaptively adjust the yaw rate required for trajectory tracking and the active yaw moment required for maintaining vehicle stability so as to achieve the coordinated control of vehicle stability and trajectory tracking control.

3.1. Trajectory Tracking Controller

The discretization of Equation (7) with the forward Euler method is expressed as follows [13]:

$$\left\{\begin{array}{l}x(k+1)=A_{d}x(k)+B_{d}u(k)+T\omega (k)\\ y(k)=C_{d}x(k)\end{array}\right.,$$

(8)

where $A_d=I+AT$, $B_d=BT$, and $C_d=C$.

The expanded state vector consists of a discrete state matrix and a control matrix and can be expressed as follows:

$$\xi (k)=\left[\begin{array}{l}x(k)\\ u(k-1)\end{array}\right],$$

(9)

By combining Equations (8) and (9), the discrete system state equation can be obtained as follows:

$$\left\{\begin{array}{l}\xi (k+1)={\tilde{A}}_d\xi (k)+{\tilde{B}}_d\Delta u(k)+T\tilde{\omega }(k)\\ y(k)={\tilde{C}}_d\xi (k)\end{array}\right.,$$

(10)

where ${\tilde{A}}_d=\left[\begin{array}{cc}{A}_d& B_d\\ 0& I\end{array}\right]$, ${\tilde{B}}_d=\left[\begin{array}{c}{B}_d\\ I\end{array}\right]$, ${\tilde{C}}_d=\left[\begin{array}{cc}{C}_d& 0\end{array}\right]$, $\tilde{\omega }={\left[\begin{array}{cc}\omega & 0\end{array}\right]}^T$.

State prediction based on Equation (10) can be obtained as follows:

$$\begin{array}{l}\xi (k+1)={\tilde{A}}_d\xi (k)+{\tilde{B}}_d\Delta u(k)+T\tilde{\omega }(k)\\ \xi (k+2)={\tilde{A}}_d^2\xi (k)+{\tilde{A}}_d{\tilde{B}}_d\Delta u(k)+{\tilde{A}}_{d}T\tilde{\omega }(k)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\tilde{B}}_d\Delta u(k+1)+T\tilde{\omega }(k+1)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}⋮\\ \xi (k+N_c)={\tilde{A}}_d^{N_c}\xi (k)+{\tilde{A}}_d^{N_c-1}{\tilde{B}}_d\Delta u(k)+{\tilde{A}}_d^{N_c-1}T\tilde{\omega }(k)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\cdots +{\tilde{B}}_d\Delta u(k+N_c-1)+T\tilde{\omega }(k+N_c-1)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}⋮\\ \xi (k+N_p)={\tilde{A}}_d^{N_p}\xi (k)+{\tilde{A}}_d^{N_p-1}{\tilde{B}}_d\Delta u(k)+{\tilde{A}}_d^{N_p-1}T\tilde{\omega }(k)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\cdots +{\tilde{A}}_d^{N_p-N_c}{\tilde{B}}_d\Delta u(k+N_c-1)+{\tilde{A}}_d^{N_p-N_c}T\tilde{\omega }(k+N_c-1)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\cdots +T\tilde{\omega }(k+N_c-1)\end{array},$$

(11)

where $\Delta u(k)=u(k+1)-u(k)$ represents the control increment.

The output matrix in the prediction horizon can be expressed as follows:

$$Y(k)={\tilde{C}}_p\xi (k)+{\tilde{D}}_p\Delta U(k)+{\tilde{E}}_p\tilde{W}(k),$$

(12)

where

• $Y(k)={\left[\begin{array}{cccc}y(k+1),& y(k+2),& \dots ,& y(k+N_p)\end{array}\right]}^T$,
• $\Delta U(k)={\left[\begin{array}{cccc}\Delta u(k),& \Delta u(k+1),& \dots ,& \Delta u(k+N_c-1)\end{array}\right]}^T$,
• $\tilde{W}(k)={\left[\begin{array}{cccc}T\tilde{\omega }(k),& T\tilde{\omega }(k+1)& \dots ,& T\tilde{\omega }(k+N_c-1)\end{array}\right]}^T$,
• ${\tilde{C}}_p={\left[\begin{array}{cccc}{\tilde{C}}_p{\tilde{A}}_p,& {\tilde{C}}_p{\tilde{A}}_d^2& \dots ,& {\tilde{C}}_p{\tilde{A}}_d^{N_p}\end{array}\right]}^T$,
• ${\tilde{D}}_p=\left[\begin{array}{ccccc}{\tilde{C}}_d{\tilde{B}}_d& 0& \dots & 0& 0\\ & & ⋮& & \\ {\tilde{C}}_d{\tilde{A}}_d^{N_c}{\tilde{B}}_d& {\tilde{C}}_d{\tilde{A}}_d^{N_c-1}{\tilde{B}}_d& \dots & {\tilde{C}}_d{\tilde{A}}_d{\tilde{B}}_d& {\tilde{C}}_d{\tilde{B}}_d\\ & & ⋮& & \\ {\tilde{C}}_d{\tilde{A}}_d^{N_p-1}{\tilde{B}}_d& {\tilde{C}}_d{\tilde{A}}_d^{N_p-2}{\tilde{B}}_d& \dots & {\tilde{C}}_d{\tilde{A}}_d^{N_p-N_c+1}{\tilde{B}}_d& {\tilde{C}}_d{\tilde{A}}_d^{N_p-N_c}{\tilde{B}}_d\end{array}\right]$,
• ${\tilde{E}}_p=\left[\begin{array}{ccccc}{\tilde{C}}_d& 0& \dots & 0& 0\\ {\tilde{C}}_d& {\tilde{C}}_d{\tilde{A}}_d& \dots & 0& 0\\ & & ⋮& & \\ {\tilde{C}}_d{\tilde{A}}_d^{N_p-1}& {\tilde{C}}_d{\tilde{A}}_d^{N_p-2}& \dots & {\tilde{C}}_d{\tilde{A}}_d& {\tilde{C}}_d\end{array}\right]$.

The objective function of the MPC-based trajectory tracking controller can be expressed as follows:

$$J=Y^{T}QY+\Delta U^{T}R\Delta U+\rho {\epsilon }^2,$$

(13)

where $Q$ is the weight matrix of the system output, $R$ is the weight matrix of the system input, $\epsilon$ is the relaxation factor, and $\rho$ is the weight coefficient.

The system constraints can be expressed as follows:

$$\begin{gathered} u_{\min }\le u\le u_{\max } \\ \Delta u_{\min }\le \Delta u\le \Delta u_{\max }, \end{gathered}$$

(14)

The optimization problem is converted into a standard quadratic programming problem to be solved:

$$\min J={[\Delta U^T,\epsilon ]}^{T}H[\Delta U^T,\epsilon ]+G[\Delta U^T,\epsilon ]+P,$$

(15)

where $H=[\begin{array}{cc}{\tilde{D}}_p^{T}Q{\tilde{D}}_p+R& 0\\ 0& \rho \end{array}]$, $G=[2{({\tilde{C}}_p\xi (k))}^{T}Q{\tilde{D}}_p 0]$, and $P={({\tilde{C}}_p\xi (k))}^{T}Q({\tilde{C}}_p\xi (k))$.

3.2. Estimation of Tire Cornering Stiffness

When the road surface adhesion coefficient and vertical load are constant, the tire force calculated by linear stiffness can be approximated to the real value. However, the vehicle driving conditions are complex and variable, and the road surface adhesion coefficient and vertical load will change with the driving conditions, which leads to a large error between the tire force calculated by linear stiffness and the real value. Therefore, the cornering stiffness of the front tire and rear tire in the MPC algorithm needs to be accurately estimated so as to adapt to the changes in vehicle driving conditions.

From the kinetic model shown in Figure 1, the following expression can be obtained:

$$\left\{\begin{array}{l}m{a}_y=F_{xf}\delta +C_f{\alpha }_f+C_r{\alpha }_r\\ I_z\dot{\gamma }=l_r{F}_{xf}\delta +l_r{C}_f{\alpha }_f-l_f{C}_r{\alpha }_r\end{array}\right.,$$

(16)

Eliminating the side-slip angle β from Equation (16) yields its matrix expression as follows:

$$bm{a}_y+I_z\dot{\gamma }-l{F}_{xf}\delta =[\begin{array}{cc}l(m{a}_y-F_{xf}\delta )& l({\displaystyle \frac{l\gamma }{v_x}}-\delta )\end{array}]\left[\begin{array}{c}{X}_{e1}\\ X_{e2}\end{array}\right],$$

(17)

where $l=l_f+l_r$, $X_{e1}={\displaystyle \frac{C_f}{C_f+C_r}}$, and $X_{e2}={\displaystyle \frac{C_f{C}_r}{C_f+C_r}}$.

Thus, the cornering stiffness of the tire can be expressed as follows:

$$\left\{\begin{array}{l}{C}_f={\displaystyle \frac{X_{e1}{C}_r}{1-X_{e1}}}\\ C_r={\displaystyle \frac{X_{e2}}{X_{e1}}}\end{array}\right.,$$

(18)

Based on the known input and output data of the model, FFRLS can conveniently find the unknown system parameters and make the sum of squares of estimation errors minimum [14]. Therefore, the FFRLS algorithm is used for online estimation of the cornering stiffness.

In order to avoid the extra noise caused by the derivation of the angular velocity of the pendulum, Equation (17) is discretized to obtain the expressions for the inputs and outputs of the estimator and the parameters to be estimated.

The input expression of the system is as follows:

$${\varphi }^T(t)=[\begin{array}{cc}l(m{a}_y(t)-F_{xf}(t)\delta (t))& l({\displaystyle \frac{l\gamma (t)}{v_x(t)}}-\delta (t))]T,\end{array}$$

(19)

The output expression of the system is as follows:

$$Y(t)=bm{a}_y(t)T+I_z(\gamma (t)-\gamma (t-1))-l{F}_{xf}\delta (t)\delta (t)T,$$

(20)

The formula for the parameter to be estimated is expressed as follows:

$$\theta (t)={[\begin{array}{cc}{X}_{c1}(t)& X_{c2}(t)\end{array}]}^T,$$

(21)

The expression for the FFRLS algorithm is as follows:

$$\left\{\begin{array}{l}\chi (t)=P(t-1)\varphi (t){[\lambda I_{2\times 2}+{\varphi }^T(t)P(t-1)\varphi (t)]}^{-1}\\ \widehat{\theta }(t)=\widehat{\theta }(t-1)+\chi (t)[Y(t)-{\varphi }^T(t)\widehat{\theta }(t-1)]\\ P(t)={\displaystyle \frac{1}{\lambda }}[P(t-1)-\chi (t){\varphi }^T(t)P(t-1)]\end{array}\right.,$$

(22)

In the formula, $\chi (t)$ is the recursive gain matrix; $P(t)$ is the covariance matrix; $\lambda$ is the forgetting factor, whose value ranges from 0 to 1, and the larger the value is, the higher the recognition accuracy is. The larger the value, the higher the recognition accuracy is, but the convergence speed becomes slower; on the contrary, the recognition accuracy is reduced, but the convergence speed becomes faster [15]. Therefore, the value of $\lambda$ needs to be considered comprehensively.

The intermediate variable $\theta (t)$ can be obtained by substituting Equations (19)–(21) into the FFRLS expression Equation (22) for iterative operation. The cornering stiffness is then converted by Equation (18).

3.3. Design of Stability Controller-Based SMC

According to the literature [16], when the vehicle enters the steady state, $\dot{\beta }=0$, $\dot{\gamma }=0$. Therefore, substituting the stability conditions into the linear two-degree-of-freedom vehicle dynamics equations, the desired values of yaw rate and side-slip angle can be expressed as follows:

$${\gamma }_1^{\ast }={\displaystyle \frac{v_x}{L(1+K_w{v}_x^2)}}{\delta }_f,$$

(23)

$${\beta }^{\ast }=({\displaystyle \frac{b}{v_x^2}}+{\displaystyle \frac{ma}{C_{r}L}}){\displaystyle \frac{v_x}{L(1+K_w{v}_x^2)}}{\delta }_f,$$

(24)

where $K_w$ is the stability factor.

Since the side-slip angle is very small, the desired value of the side-slip angle is set to 0, which can be obtained as follows:

$$K_w={\displaystyle \frac{m}{L^2}}({\displaystyle \frac{l_r}{C_r}}-{\displaystyle \frac{l_f}{C_f}}),$$

(25)

The vehicle is limited by the road surface attachment conditions, and the tire adhesion cannot exceed the tire attachment limit, i.e.,

$$\left|a_y\right|\le \mu g,$$

(26)

where $a_y$ is the lateral acceleration; $\mu$ is the road adhesion coefficient; and $g$ is the gravitational acceleration.

Considering the limitation of the road adhesion coefficient, the desired value of the yaw rate can be obtained as follows:

$${\gamma }^{\ast }=\min \{{\displaystyle \frac{v_x}{L(1+K_w{v}_x^2)}}{\delta }_f,{\displaystyle \frac{\mu g}{v_x}}\}\mathrm{sgn}{\delta }_f,$$

(27)

The sliding mode control has the characteristics of fast response, insensitivity to parameter changes and perturbations, strong anti-interference, and good adaptability to complex and less deterministic nonlinear time-varying systems.

The integral sliding mode control is designed to solve the disturbance term brought by external perturbation. Define the deviation as follows:

$$e=(\gamma -{\gamma }^{\ast })+c_1(\beta -{\beta }^{\ast }),$$

(28)

where C is the proportion of the side-slip angle and the yaw rate in the control system, $c_1>0$.

The integral sliding mode function is expressed as follows:

$$s=e+c_2{\displaystyle {\int }_0^{t}edt},$$

(29)

where $c_2>0$ and t is time.

When the sliding mode surface tends to be stable, the first-order derivative of the sliding mode switching function gives the following equation:

$$\dot{s}=(\dot{\gamma }-{\dot{\gamma }}^{\ast })+c_1(\dot{\beta }-{\dot{\beta }}^{\ast })+c_{2}e,$$

(30)

Considering the longitudinal wheel forces, the dynamic equations of the two-degree-of-freedom vehicle model are as follows:

$$I_z\dot{\gamma }=a{F}_{y1}-b{F}_{y2}+M_z,$$

(31)

$$m{v}_x(\dot{\beta }+\gamma )\dot{\gamma }=F_{y1}+F_{y2},$$

(32)

The lateral force of the front and rear wheels can be expressed as follows:

$$F_{y1}=C_f(\beta +{\displaystyle \frac{a\gamma }{v_x}}-{\delta }_f),$$

(33)

$$F_{y2}=C_r(\beta -{\displaystyle \frac{b\gamma }{v_x}}),$$

(34)

Substituting Equations (31)–(34) into Equation (30), the following is obtained:

$$\begin{array}{l}\dot{s}={\displaystyle \frac{(a{C}_f-b{C}_r)\beta -a{C}_f{\delta }_f+M_z}{I_z}}+{\displaystyle \frac{a^2{C}_f+b^2{C}_r}{I_z{v}_d}}\gamma -{\dot{\gamma }}_d+\\ c_1[{\displaystyle \frac{C_f+C_r}{m{v}_d}}\beta +({\displaystyle \frac{a{C}_f-b{C}_r}{m{v}_d^2}})\gamma ]-c_1{\displaystyle \frac{C_f}{m{v}_d}}{\delta }_f-c_1{\dot{\beta }}_d+c_{2}e\end{array},$$

(35)

where $v_d$ is the desired vehicle velocity; ${\dot{\gamma }}_d$ is the derivative of the desired yaw rate; and ${\dot{\beta }}_d$ is the derivative of the desired sideslip angle.

By choosing the exponential reaching law, the system state can approach the sliding mode at a faster speed, and the chattering of the system can be reduced.

$$\dot{s}=-K\mathrm{sgn}s-k_{1}s,$$

(36)

where k is the reaching law coefficient, $K>0$; $k_1>0$.

The direct yaw moment can be expressed as follows:

$$\begin{array}{l}{M}_z=I_z\left\{-K\mathrm{sgn}s-k_{1}s-\left\{\begin{array}{l}{\displaystyle \frac{a^2{C}_f+b^2{C}_r}{I_z{v}_d}}\gamma -{\dot{\gamma }}_d+c_1[{\displaystyle \frac{C_f+C_r}{m{v}_d}}\beta +\\ ({\displaystyle \frac{a{C}_f-b{C}_r}{m{v}_d^2}}-1)\gamma ]-c_1{\displaystyle \frac{C_f}{m{v}_d}}{\delta }_f-c_1{\dot{\beta }}_d+c_{2}e\end{array}\right\}\right\}-\\ (a{C}_f-b{C}_r)\beta -a{C}_f{\delta }_f\end{array},$$

(37)

Define Lyapunov function as follows:

$$V={\displaystyle \frac{1}{2}}{s}^2,$$

(38)

The proof of the stability of sliding mode control system can be expressed as follows:

$$V=s\dot{s}=s(-K\mathrm{sgn}s-k_{1}s)=-K\left|s\right|-k_1{s}^2\le -K\left|s\right|\le 0$$

(39)

3.4. Chattering Reduction Based on Fuzzy Controller

In the modeling process, the system is subject to external interference, which requires a larger reaching law coefficient, and the larger reaching law coefficient will produce chattering when the system state is close to the sliding mode surface. The excessive chattering of the sliding mode control system will greatly affect the dynamic quality of the system. In this paper, fuzzy control rules are used to adaptively adjust the reaching law coefficient. Fuzzy control is used to effectively adjust the reaching law coefficient to achieve adaptive approximation of model uncertainty to reduce chattering. The structure of fuzzy sliding mode control is shown in Figure 4, in which the function of $K_f$ is to reduce the chattering of the sliding mode control system. When the system state is close to the sliding mode surface, the value of $K_f$ should be appropriately reduced; otherwise, when it is far from the sliding mode surface, the value of $K_f$ should be appropriately increased, but the chattering will be more severe if the value of $K_f$ is too large.

The fuzzy controller is designed in the form of double input and single output. The inputs of the fuzzy controller are $S$ and $\dot{s}$, and $K_f$ is the output of the fuzzy controller. The input and output fuzzy sets are defined as follows:

$$\begin{gathered} s=\{NB,NM,NS,ZE,PS,PM,PB\}, \\ \dot{s}=\{NB,NM,NS,ZE,PS,PM,PB\}, \\ K=\{VS,S,M,B,VB\}. \end{gathered}$$

where NB is negative; NM is negative middle; NS is negative small; ZE is 0; PS is positive small; PM is the center; PB is positive; VS is very small; S is small; M is medium; B is large; and VB is very large.

According to the vehicle state parameters and experience, it is determined that the input universe $\dot{s}$ in the fuzzy controller is [−1.5, 1.5], the input universe $S$ is [−0.3, 0.3], and the output universe $K_f$ is [−2, 2]. The membership functions of the input and output are shown in Figure 5, Figure 6 and Figure 7, where f ($K_f$), f ($S$), and f ($\dot{s}$) are the membership functions of $K_f$, s, and $\dot{s}$, respectively.

In order to reduce the chattering and the existence of the sliding mode, a fuzzy rule is established. When $\dot{s}$ is NB and $s\dot{s}>0$, the state trajectory is far away from the sliding mode surface. At this time, $K_f$ should be increased to make it move towards the sliding mode surface, and the greater the $\left|s\right|$ is, the greater the increase of $K_f$ is. When $s$ is NB and $s\dot{s}<0$, the state trajectory is close to the sliding surface. Currently, $K_f$ should be diminished to mitigate chattering, and the smaller the s, the more significant the reduction of $K_f$.

To further reduce chattering, the symbolic function $\mathrm{sgn}s$ in sliding mode control is substituted with the saturation function $sat s$ as follows:

$$sat s=\left\{\begin{array}{l}1, s>\epsilon \\ qs , |s|\le \epsilon ,q={\displaystyle \frac{1}{\epsilon }} \\ -1, s<-\epsilon \end{array}\right.,$$

(40)

3.5. Coordinated Strategy for Path Tracking and Vehicle Stability

Generally, MPC-based trajectory tracking control can ensure that autonomous vehicles have better path tracking effects in most working conditions. However, when the vehicle runs at high speed and encounters extreme conditions such as emergency obstacle avoidance and sudden deterioration of road friction coefficient, the extreme nonlinear characteristics of vehicle tire force will seriously affect the tracking accuracy of vehicle trajectory based on the active steering system. In addition, because there is only one front wheel angle as the lateral control variable of the vehicle, the dynamic stability of the vehicle under extreme conditions cannot be ensured. Most scholars regard the active yaw moment as the control variable [17,18] in trajectory tracking control. This control strategy can improve the effect of trajectory tracking control to a certain extent, but there are still two problems: (1) When active yaw moment assist and active steering coordinated control realize path tracking control, the yaw rate may exceed the vehicle stability boundary during trajectory tracking control, resulting in vehicle instability [19]. (2) If the active yaw moment is used as the only control variable of the MPC controller, it will produce unnecessary longitudinal force, which will cause an increase in energy consumption and excessive wear of the actuator. In this case, once the lateral error or heading error exceeds the threshold, the active yaw moment control will intervene. Obviously, this control strategy cannot meet the control demand.

To address the aforementioned issues, we propose a trajectory tracking and stability coordination control strategy based on distributed architecture. In this paper, adaptive weight coefficients are utilized to adjust the desired yaw rate in trajectory tracking control and the participation degree in active yaw moment control. A normalized stability index is employed to effectively evaluate the degree of vehicle instability:

$$\lambda =\sqrt{p{({\displaystyle \frac{\gamma }{{\gamma }_0}})}^2+{({\displaystyle \frac{\beta }{{\beta }_0}})}^2},$$

(41)

where p represents the coefficient. Under extreme conditions, the weight of yaw rate should be greater than that of the side slip angle [20]. This article uses p = 1.8.

In this paper, a coefficient q is defined to ensure that the active yaw moment exhibits a substantial degree of participation solely in instances of severe vehicle instability.

$$q=\left\{\begin{array}{l}{\displaystyle \frac{e^{c(\lambda -1)}}{e^{c({\lambda }_0-1)}}} \lambda \le {\lambda }_0\\ 1 \lambda >{\lambda }_0 \end{array}\right.,$$

(42)

where ${\lambda }_0$ represents the limit value, the value is set to 1.5; c, primarily a parameter reflecting the adjustment speed, is set to 5 based on simulation results; and when $\lambda$ exceeds the parameter ${\lambda }_0$, the weight coefficient q is set to 1.

In order to coordinate trajectory tracking control and stability control, the desired yaw rate in trajectory tracking control and the desired active yaw moment in stability control are modified as follows:

$${\gamma }_c=(1-q){\gamma }_{\mathrm{track}}+q{\gamma }_{\mathrm{stability}},$$

(43)

$$M_c=q{M}_z,$$

(44)

where ${\gamma }_{\mathrm{track}}={\dot{\Psi }}_d$ and ${\gamma }_{\mathrm{stability}}={\gamma }_d$. In the coordinated control strategy, to avoid interference between stability control and trajectory tracking control, ${\dot{\Psi }}_d$ in Equation (4) is substituted by ${\gamma }_c$ from Equation (43).

Figure 8 illustrates the relationship between the weight coefficient and the normalized stability index. When the normalized stability index increases, it indicates that the yaw rate or the side-slip angle is approaching its limit value. At this point, the weight coefficient approaches 1, implying a primary focus on stability control. Conversely, when the normalized stability index is relatively small, the weight coefficient tends towards 0, suggesting a primary emphasis on trajectory tracking control to avoid any adverse impact of stability control on trajectory tracking control.

The longitudinal force of velocity can be expressed as follows:

$$F_{xv,i}={\displaystyle \frac{F_x}{4}},$$

(45)

where $F_{xv,i}$ represents the longitudinal force acting on each wheel when employed for velocity tracking control. $i=1,2,3,4$ represent the left front wheel, right front wheel, left rear wheel, and right rear wheel, respectively.

In order to make full use of the adhesion of each tire, the longitudinal force of the yaw moment on each wheel can be expressed as follows:

$$\left\{\begin{array}{l}{F}_{xm,1}=-w_1{M}_c/(0.5B_s)\\ F_{xm,2}=w_2{M}_c/(0.5B_s)\\ F_{xm,3}=-w_3{M}_c/(0.5B_s)\\ F_{xm,4}=w_4{M}_c/(0.5B_s)\end{array}\right.,$$

(46)

where $B_s$ denotes the track width; $w_i$ represents the load coefficient for each of the four wheels and is expressed as follows:

$$w_i={\displaystyle \frac{F_{zi}}{F_z}},$$

(47)

where $F_z=mg$ represents the total load; $F_{zi}$ denotes the vertical load borne by each wheel, which can be expressed according to the following formula:

$$\left\{\begin{array}{l}{F}_{z1,2}={\displaystyle \frac{mg{l}_r-m{a}_x{h}_c}{2(l_f+l_r)}}\mp {\displaystyle \frac{a_y}{B_f}}{\displaystyle \frac{m{h}_c{l}_r}{(l_f+l_r)}}\\ F_{z3,4}={\displaystyle \frac{mg{l}_f+m{a}_x{h}_c}{2(l_f+l_r)}}\mp {\displaystyle \frac{a_y}{B_r}}{\displaystyle \frac{m{h}_c{l}_f}{(l_f+l_r)}}\end{array}\right.,$$

(48)

Therefore, the resultant longitudinal force for each wheel can be expressed as follows:

$$F_{xi}=F_{xv,i}+F_{xm,i},$$

(49)

Due to the saturation characteristics limiting tire forces, the torque for each wheel is as follows:

$$T_{xi}={\displaystyle \frac{sign(F_{xi})}{R_e}}\min \left\{\left|F_{xi}\right|,\sqrt{{(\mu F_{zi})}^2-{(F_{yi})}^2}\right\},$$

(50)

where $R_e$ represents the wheel radius and $F_{yi}$ denotes the lateral force of each tire. The $F_{yi}$ can be expressed as follows:

$$\left\{\begin{array}{l}{F}_{y1}={\displaystyle \frac{w_1}{w_1+w_2}}{\widehat{F}}_{yf},F_{y2}={\displaystyle \frac{w_2}{w_1+w_2}}{\widehat{F}}_{yf}\\ F_{y3}={\displaystyle \frac{w_3}{w_3+w_4}}{\widehat{F}}_{yr},F_{y4}={\displaystyle \frac{w_4}{w_3+w_4}}{\widehat{F}}_{yr}\end{array}\right.,$$

(51)

4. Hardware-in-the-Loop Verification

To validate the effectiveness of the algorithm proposed in this paper, three operating conditions were selected for verification. Figure 9, Figure 10 and Figure 11 depict the typical cases utilized during the validation process. Among them, Case 1 is employed to verify the real-time performance and reliability of the algorithm in trajectory tracking. In Cases 2 and 3, a comparison is made between the coordinated control strategy presented in this paper and the control strategies from other literature. Specifically, Controller 1 incorporates only the trajectory tracking control algorithm, Controller 2 employs the stability and trajectory tracking coordinated control strategy from reference [21], and Controller 3 utilizes the coordinated control strategy proposed in this paper. The validation of control strategies was conducted based on the CarSim 2020 and MATLAB/Simulink 2021co-simulation platforms. The primary parameters of the vehicle and controllers are listed in

Table 1. Model parameters.

Symbol	Values
$m$	$1450 \mathrm{kg}$
$I_z$	$2050 {\mathrm{kgm}}^2$
$l_f$	$1.565 \mathrm{m}$
$l_r$	$1.685 \mathrm{m}$
$B$	$1.55 \mathrm{m}$
$R_e$	$0.325 \mathrm{m}$
$Q$	$\mathrm{diag}(100,50,2000)$
$R$	$\mathrm{diag}(0.32,500)$
$T$	$0.01 \mathrm{s}$

. The simulation parameters for Model Predictive Control (MPC) are identical across all three controllers, and the MPC is solved using MATLAB’s Quadratic Programming (QP) solver.

Validating autonomous and assisted driving control strategies for vehicles at high speeds is extremely hazardous. Therefore, the verification of the control strategies presented in this paper was conducted exclusively on a hardware-in-the-loop (HIL) platform [21]. The structure of the HIL platform is illustrated in Figure 12. The HIL encompasses a rapid prototyping controller, a host PC, DSpace, a display interface, a CAN card, a data acquisition card, converters, motor drivers, steering motors, and other components. DSpace is equipped with CarSim software 2020, which is used to configure the vehicle model and simulation conditions. The control strategies, programmed in Simulink, are executed on the rapid prototyping controller.

During the testing process, vehicle state signals and reference trajectory information are transmitted from DSpace to the controller. The controller then sends control signals to the motor driver via the CAN bus to control the steering motor in real time. Sensors transmit the angle signals of the steering motor to DSpace, which subsequently forwards them to the controller, forming a closed-loop control system.

4.1. Case 1

In Case 1, the road friction coefficient is set to 0.3, and the host vehicle is traveling at a speed of 70 km/h. A vehicle in the same lane suddenly brakes to a complete stop during its travel, while a vehicle in the left lane is traveling at approximately 70 km/h. At this point, the host vehicle executes a lane change based on the ideal trajectory path. In terms of trajectory tracking control, a comparison is made between the adaptive MPC algorithm designed in this paper and the traditional MPC algorithm. Figure 13 presents the test data results for operating condition 1.

Figure 13a,b demonstrate that, in terms of trajectory tracking performance, the control algorithm proposed in this paper outperforms the MPC. This is attributed to the real-time estimation method of tire cornering stiffness presented in this paper, which enhances the accuracy of the control model. Figure 13c displays the lateral force data for both the front and rear tires, indicating a high level of accuracy in tire force estimation. Figure 13d shows the tire cornering stiffness, revealing that the tire cornering stiffness undergoes significant changes as the tire forces gradually approach saturation. The lateral tire forces calculated based on cornering stiffness exhibit larger errors in the tire force saturation region, especially under icy, snowy, or wet conditions. Errors in cornering stiffness can lead to poor performance of MPC. Therefore, the issue of tire cornering stiffness is considered in the design of the trajectory tracking controller in this paper. Figure 13e indicates that the trajectory tracking controller based on AMPC can effectively track the target front wheel angle. Figure 13f shows that the solution time for each step of the algorithm proposed in this paper is within the range of 0–10 ms, which is less than the sampling time, suggesting that the real-time solution speed of the algorithm designed in this paper meets the design requirements.

The test results under Case 1, with the road adhesion coefficient set at 0.8 and the vehicle speed at 120 km/h while other conditions remain unchanged, are illustrated in Figure 14. From Figure 14a,b, both control methods enable the vehicle to track the desired trajectory, but the AMPC demonstrates smaller trajectory deviations and lower speed reduction compared to MPC. Figure 14c displays the lateral force data of front/rear tires, indicating high estimation accuracy of tire forces. The tire cornering stiffness shown in Figure 14d exhibits minimal variation amplitude under high-adhesion road conditions. As demonstrated in Figure 14e, the AMPC-based trajectory tracking controller effectively follows the target front wheel steering angle. Figure 14f further verifies that the proposed algorithm operates within the sampling time threshold, fulfilling the design requirements.

4.2. Case 2

Case 2 simulates an emergency obstacle avoidance scenario, where the target driving trajectory resembles the double lane change condition. This condition is used to validate the coordinated control algorithm proposed in this paper. In the current case, the road friction coefficient is 0.8, and the host vehicle is traveling at a speed of 110 km/h. During the travel, a vehicle ahead suddenly stops, while a vehicle to the left rear is traveling at 90 km/h and a vehicle to the left front is moving at 70 km/h. Therefore, the host vehicle must first change lanes to avoid the obstacle (the stopped vehicle) and then quickly change lanes back to the original lane. Figure 15 presents the verification results of three controllers. Figure 15a,b show that the vehicle, equipped only with trajectory tracking control 1, fails to follow the target trajectory (reference trajectory) and cannot maintain vehicle stability during the lane change. Vehicles equipped with Controller 2 and Controller 3, respectively, can achieve lane changes according to the desired trajectory, and both vehicles’ speeds closely follow the expected values in terms of speed control. Figure 15c displays the lateral error of the vehicles; Figure 15d shows the heading error; Figure 15e presents the yaw rate; and Figure 15f shows the sideslip angle. It can be observed that the vehicle equipped with the coordinated control strategy proposed in this paper exhibits smaller lateral error, heading error, yaw rate, and sideslip angle compared to the other two vehicles, indicating that the proposed method has better overall performance.

Figure 16a presents the front wheel angle data for the three vehicles. The vehicle equipped with Controller 1 is unable to complete the lane change in Case 2. Figure 16b,d show the torque output values for Controllers 2 and 3, respectively. It can be seen that the torque intervention of the algorithm proposed in this paper is only required for a short period, yet it still ensures that the vehicle can complete the lane change.

The validation results under Case 2 with the road adhesion coefficient set to 0.3 (other conditions unchanged) are shown in Figure 17. As shown in Figure 17a, vehicles equipped with Controller 1 fail to track the reference trajectory and exhibit instability during lane-changing maneuvers. Although both Controllers 2 and 3 achieve trajectory tracking, Controller 3 demonstrates smaller trajectory deviations. Figure 17b–d present vehicle dynamics parameters: lateral acceleration, yaw rate, and sideslip angle. The proposed coordinated control strategy (Controller 3) achieves lower magnitudes in all three parameters compared to Controller 2. The phase plane diagrams in Figure 17e,f reveal stability characteristics. The convergence of Controller 3’s phase trajectories in both sideslip angle–yaw rate and sideslip angle–sideslip angular velocity planes surpasses those of other controllers, indicating enhanced stability.

Table 2. Peak parameters of Case 2 under low adhesion.

Controller	Sideslip Angle	Yaw Rate
1	-	-
2	9.81 deg	0.73 rad/s
3	5.01 deg	0.67 rad/s

and

Table 3. Peak parameters of Case 2 under high adhesion.

Controller	Sideslip Angle	Yaw Rate
1	-	-
2	5.31 deg	0.69 rad/s
3	4.63 deg	0.51 rad/s

present the peak parameters of Case 2 under low- and high-adhesion road conditions, respectively. In both scenarios, Controller 1 failed to stabilize the vehicle, resulting in unmeasurable peaks for yaw rate and sideslip angle. Compared to Controller 2, Controller 3 demonstrated the following superior performance: under low adhesion, the yaw rate peak was reduced by 15.1%, and the sideslip angle was reduced by 48.0%.

4.3. Case 3

Case 3 is primarily designed to validate the algorithmic robustness of vehicles during lane changes on transitioning road surfaces. The host vehicle travels at a speed of 100 km/h, with a road surface adhesion coefficient of 0.8. Upon completing the first lane change, the road surface adhesion coefficient changes to 0.5 at X = 120 m, where the host vehicle then completes a second lane change. The verification data are presented in Figure 18.

Figure 18a indicates that at X = 120 m, when the vehicle initiates the second lane change, Controllers 1 and 2 fail to closely follow the desired path. Figure 18b shows that at 6 s, Controllers 1 and 2 exhibit poorer speed control, unable to maintain the desired velocity. Figure 18c–f depict lateral deviation, heading angle deviation, yaw rate, and sideslip angle data, respectively, revealing that Controller 3 outperforms Controllers 1 and 2 during the second lane change. Controllers 1 and 2 exhibit a sharp increase in yaw rate and sideslip angle after 6 s, resulting in an inability to maintain vehicular stability. There are mainly two reasons for this. One is that the real-time estimated cornering stiffness can enhance the adaptability and robustness of Controller 3 when the road friction coefficient changes; the other is the adaptive adjustment between the weight coefficient and the normalized stability factor.

Figure 19 illustrates the output data for each controller. In Figure 19a, the front wheel angle data for vehicles equipped with Controllers 1 and 2 undergo sudden changes after 6 s, while the data for vehicles equipped with Controller 3 exhibit minimal variation. Figure 19b–d demonstrate that stability control only intervenes when the vehicle becomes unstable, and the output active yaw moment is minimal, greatly avoiding interference with trajectory tracking control. Therefore, it can be concluded that Controller 3, based on the proposed cooperative strategy, is significantly superior to Controller 1, which only considers trajectory tracking, and Controller 2, in which the active yaw moment is always involved, in terms of adaptability and robustness.

The verification results for Case 3, where the road adhesion coefficient transitions from 0.3 to 0.8 at X = 120 m under otherwise unchanged conditions, are depicted in Figure 20. As shown in Figure 20a, vehicles equipped with Controller 1 and Controller 2 fail to track the reference trajectory and exhibit instability during lane-changing maneuvers, while Controller 3 achieves accurate trajectory tracking. Figure 20b–d present key vehicle dynamics parameters: lateral acceleration, yaw rate, and sideslip angle. The proposed control strategy minimizes trajectory deviations and confines critical parameters (sideslip angle, yaw rate, and lateral acceleration) within predefined safety thresholds. Figure 20e illustrates the sideslip angle–yaw rate phase plane, while Figure 20f presents the sideslip angle–sideslip angle velocity phase plane. Vehicles equipped with Controller 3 exhibit superior convergence in phase plane trajectories compared to Controllers 1 and 2. The stability control intervention ensures that phase trajectories return to the origin.

5. Conclusions

In order to solve the problem of mutual interference between stability control and trajectory tracking control of an autonomous vehicle based on distributed drive under extreme working conditions, a coordinated control strategy is proposed.

(1) A tire cornering stiffness estimation model was established based on the Forgetting Factor Recursive Least Squares method, achieving accurate identification of tire parameters. The trajectory tracking control module, integrated with real-time updates of tire cornering stiffness, utilized a Model Predictive Control (MPC) algorithm to enhance time-varying characteristics during trajectory tracking.

(2) For stability control, a fuzzy sliding mode algorithm was developed. An integral sliding-mode controller generated additional yaw moments, while a fuzzy controller optimized the system to suppress chattering effects inherent in sliding mode control.

(3) In the coordination mechanism, a normalized stability index was introduced to evaluate vehicle instability levels. An adaptive coordination strategy between trajectory tracking and stability control was designed, dynamically adjusting control priorities based on real-time stability assessments.

Validation through hardware-in-the-loop (HIL) testing under three typical scenarios demonstrated that the proposed strategy minimized deviations from the ideal trajectory while maintaining critical parameters (e.g., sideslip angle and yaw rate) within safe thresholds. Comparative simulation data confirmed superior performance in balancing tracking accuracy and stability margins.