Analyzing the Impact of Wind on Trajectory Tracking for Unmanned Vehicles Based on Road Adhesion Coefficient Estimation

Abstract

This paper presents a novel approach to address the challenge of trajectory tracking control in unmanned vehicles (UVs) operating on roads with varying ad hesion coefficients and subject to wind disturbances. The proposed method integrates real-time estimation of road adhesion coefficients with a wind disturbance model, embedding both into the trajectory tracking control system. Specifically, the Unscented Kalman Filter (UKF) is employed to dynamically estimate the road adhesion coefficients. These estimates are then combined with the wind disturbance model and incorporated into a Model Predictive Control (MPC) framework, enhancing the robustness and accuracy of trajectory tracking under varying conditions. The integrated simulation results from CarSim/Simulink demonstrate high accuracy in road adhesion coefficient estimation for trajectory tracking control, with tracking errors consistently remaining below 0.1 across different road adhesion conditions. For example, the road adhesion coefficient estimation error for the docking surface is 0.5%, and for the split road surface, it is 0.75%, indicating the approach’s applicability across a wide range of road conditions. Notably, when the wind speed reaches level-five intensity and a shelter-type vehicle is used as a test case, the proposed controller significantly outperforms the controller that does not consider wind effects. Under both high and low adhesion conditions, the controller reduces the yaw angle by approximately 15% and 30%, respectively, and the yaw rate by approximately 25% and 20%. These results provide strong evidence for the effectiveness of the proposed trajectory tracking controller in enhancing the stability and control performance of unmanned vehicles.

1. Introduction

Unmanned driving technology has witnessed remarkable advancements in recent years, particularly in the domain of trajectory tracking control [1]. To enhance precision and reliability, various control methods have been extensively studied and applied. Among these methods are the Proportional–Integral–Derivative (PID) Control [2], Fuzzy Control [3,4], Pure Trajectory Tracking [5], Sliding Mode Control (SMC) [6], Adaptive Control [7], Deep Reinforcement Learning (DRL), Linear Quadratic Regulator (LQR), and Model Predictive Control (MPC). These approaches have demonstrated significant potential to improve the performance of autonomous vehicles under diverse conditions.

Building on this foundation, the estimation of road adhesion coefficients—a critical factor affecting vehicle dynamics—has become a focal point in advancing autonomous vehicle control. Typically, road adhesion coefficients are estimated using algorithms based on the Extended Kalman Filter (EKF) [8,9], Unscented Kalman Filter (UKF) [10,11], and Cubature Kalman Filter (CKF) [12,13].

Road adhesion coefficient estimation methods can be broadly categorized into cause-based and effect-based approaches. Among the cause-based methods, Wang X. et al. [14] employed machine learning algorithms such as Random Forest (RF), Extreme Learning Machine (ELM), and Radial Basis Function Neural Networks (RBF-NNs) for road type identification. Zhang L. et al. [15] advanced road surface classification accuracy using a segmentation method that combined DeeplabV3+ with MobileNetV2, leveraging lookup tables for refinement. Yuan C. et al. [16] proposed a real-time estimation technique for peak road adhesion coefficients based on texture features extracted from a gray-level co-occurrence matrix, demonstrating both reliability and computational efficiency. These methods, while effective in controlled scenarios, present limitations when integrated into dynamic vehicle control systems, particularly under external disturbances such as varying wind conditions or road speeds.

Effect-based methods have garnered attention for their robust performance, with Kalman Filter (KF)-based approaches [17,18] leading the field. Zhang X. et al. [19] employed adaptive Kalman filtering integrated with an extended state observer to estimate road adhesion coefficients and slip ratios with precision. Similarly, Qi J. [20] enhanced estimation accuracy by introducing an adaptive Cubature Kalman Filter algorithm, outperforming traditional techniques. Meanwhile, Li B. et al. [21] presented a novel SE-CNN framework that utilized LiDAR reflectance intensity for road adhesion classification, achieving high accuracy across diverse lighting and speed conditions. Despite these advancements, the majority of effect-based methods prioritize estimation accuracy without adequately addressing their seamless integration into control systems or their responsiveness to real-world environmental disturbances, such as crosswinds and sudden road surface changes.

The influence of wind presents additional challenges to vehicle stability and control, particularly for unmanned vehicles (UVs). Yakub F. et al. [22] demonstrated that Model Predictive Control (MPC) significantly enhances robustness and mitigates crosswind effects during path-following maneuvers. Tian M. et al. [23] further validated MPC strategies for improving stability and control accuracy in crosswind conditions, thereby increasing safety and responsiveness. Chen Y. et al. [24] conducted aerodynamic simulations to analyze the combined effects of wind and rain on vehicle stability, revealing increased drag and reduced lateral stability under adverse conditions. Keviczky et al. [25] explored the application of nonlinear MPC in active front steering to counteract side-wind effects on autonomous vehicles. While these studies highlight the effectiveness of MPC in addressing wind disturbances, they primarily consider wind as an isolated factor, overlooking its interaction with varying road adhesion conditions—a critical limitation in developing integrated control systems.

Experts and scholars worldwide have conducted investigations into the trajectory tracking challenges faced by unmanned vehicles. Hou Y. et al. [26] proposed an LQR-based active trailer steering (ATS) controller that significantly enhanced lateral stability and trajectory tracking during high-speed maneuvers. The proposed LQR-based [27] control strategy enhances path tracking accuracy and vehicle stability in four-wheel independent drive electric trucks under complex driving conditions. Yao J. et al. [28] introduced a DCN-DDPG algorithm, which outperformed traditional approaches like DDPG and pure pursuit in adaptability and tracking accuracy.

Trajectory tracking, a cornerstone of vehicle control, has been extensively studied, with MPC emerging as a preferred approach [29,30,31,32] due to its strengths in model prediction, rolling optimization, and real-time correction. Building on these efforts, Falcone P. et al. [33,34] advanced MPC applications by employing a linear time-varying model, while other researchers [35,36] explored Nonlinear MPC (NMPC) for articulated vehicles, achieving superior precision and safety. Ji J. et al. [37] integrated MPC with environment perception and trajectory planning, demonstrating robust performance across diverse scenarios. Tian J. et al. [38] proposed a control strategy using MPC and SMC for DSAGV, achieving precise trajectory tracking and enhanced body stability through differential steering. However, despite these advancements, limited efforts have been made to incorporate real-time road adhesion estimation or wind disturbance modeling into trajectory tracking frameworks, leaving a significant research gap.

This study addresses these gaps by developing a unified framework that integrates real-time road adhesion coefficient estimation with wind effect modeling within an MPC-based trajectory tracking system. Using the Unscented Kalman Filter (UKF), the proposed method ensures accurate, real-time estimation of road adhesion coefficients, which are seamlessly incorporated into the trajectory tracking controller alongside a wind effect model. This integrated approach enables robust trajectory tracking under diverse road and environmental conditions. The main contributions of this study are as follows:

1. This research bridges a significant gap by integrating road adhesion coefficient estimation and wind effect modeling into the trajectory tracking control framework for unmanned vehicles, addressing the limitations of previous studies that considered these factors in isolation.

2. The study employs a seven degrees of freedom vehicle dynamics model and uses the Unscented Kalman Filter (UKF) for real-time road adhesion coefficient estimation. A wind effect model is also developed to accurately account for environmental influences.

3. A Model Predictive Control (MPC)-based trajectory tracking controller is designed using a three degrees of freedom vehicle model and tire model. This controller is tailored to enhance the tracking performance of unmanned vehicles, particularly under the influence of varying road adhesion and wind conditions.

4. The proposed framework is demonstrated on a shelter-type unmanned vehicle through extensive simulations involving complex road conditions and diverse wind scenarios, highlighting its effectiveness and robustness.

The structure of this paper is as follows: Section 2 establishes a seven-degrees-of-freedom vehicle dynamics model and employs the Unscented Kalman Filter (UKF) for online estimation of road adhesion coefficients. Section 3 develops a wind effect model for the vehicle and designs an MPC-based trajectory tracking controller using a three-degrees-of-freedom vehicle model and tire model. Section 4 outlines the simulation conditions for this study, including the settings for parameters such as friction coefficients and wind speed, as well as the construction of the simulation environment in CarSim. Section 5 validates the effectiveness of the improved MPC trajectory tracking controller through CarSim/Simulink co-simulation, which accurately estimates varying road adhesion coefficients under the influence of wind. Finally, concluding remarks are given in Section 6.

2. The Estimation Mode of Road Adhesion Coefficient

The road adhesion coefficients are estimated using the UKF, leveraging a seven-degrees-of-freedom vehicle dynamics model and the Dugoff tire model, which requires fewer parameters for computing tire forces and wheel states.

2.1. The Seven-Degrees-of-Freedom Vehicle Dynamics Model

The seven-degrees-of-freedom vehicle dynamics model, as illustrated in Figure 1, is established in this study. This model encompasses longitudinal, lateral, yaw, and four-wheel rolling motions. Assumptions regarding vehicle motion are made: the vehicle adopts front-wheel steering, with a zero-steering angle for the rear wheels. Effects of the steering trapezoidal mechanism are disregarded, and it is assumed that the steering angles of the left and right wheels of the front axle remain consistent.

In this paper, we use the following nomenclature: The vehicle mass is denoted by $m$, where $v_x$ and $v_y$ represent the longitudinal and lateral velocities of the center of mass, respectively. $a$ and $b$ denote the distances from the center of mass to the front and rear axles. $F_{x\_ij}$ and $F_{y\_ij}$ signify the longitudinal and lateral forces acting on the tires, with subscripts $i$ indicating front or rear tires (where $f$ represents front tires and $r$ represents rear tires), and $j$ denoting left or right tires (where $l$ signifies left tires and $r$ signifies right tires). $\delta$ is the steering angle of the front wheels, ${\alpha }_{ij}$ denotes the tire slip angle, $\dot{\phi }$ represents the yaw rate, $B_f$ is the front track width, $B_r$ is the rear track width, and $I_z$ denotes the moment of inertia about the z-axis of the vehicle body.

Its dynamic equations are as follows:

(1)$m\left({\dot{v}}_x-\dot{\phi }{v}_y\right)=F_{x_{fl}}\cos \delta +F_{x_{fr}}\cos \delta -F_{y_{fl}}\sin \delta -F_{y_{fr}}\sin \delta +F_{x_{rl}}+F_{x_{rr}}$

(2)$m\left({\dot{v}}_y+\dot{\phi }{v}_x\right)=F_{x_{fl}}\sin \delta +F_{x_{fr}}\sin \delta +F_{y_{fl}}\cos \delta +F_{y_{fr}}\cos \delta +F_{y_{rl}}+F_{y_{rr}}$

(3)$\begin{array}{r}{I}_z\ddot{\phi }={\displaystyle \frac{B_f}{2}}\left(F_{x_{fr}}\cos \delta -F_{x_{fl}}\cos \delta -F_{y_{fr}}\sin \delta +F_{y_{fl}}\sin \delta \right)+{\displaystyle \frac{B_r}{2}}\left(F_{x_{rr}}-F_{x_{rl}}\right)\\ +a\left(F_{x_{fl}}\sin \delta +F_{y_{fl}}\cos \delta +F_{x_{fr}}\sin \delta +F_{y_{fr}}\cos \delta \right)-b\left(F_{y_{rl}}+F_{y_{rr}}\right)\end{array}$

2.2. The Tire Dynamics Model

The Dugoff tire model [39], selected for this study, offers independent lateral and longitudinal stiffness with high computational accuracy and efficiency. Its simple equations, excluding self-aligning torque, enable easy isolation of the road adhesion coefficient, thus facilitating research on road adhesion coefficient estimation.

The Dugoff tire dynamics model is depicted in Figure 2. In the figure, $F_x$, $F_y$, and $F_z$ represent the longitudinal force, lateral force, and vertical force of the tire, respectively. $\omega$ denotes the tire angular velocity, and $\alpha$ represents the slip angle. The expressions are as follows:

(4)$\left\{\begin{array}{l}{F}_x=f(L)\mu F_z{C}_l{\displaystyle \frac{s}{1-s}}\\ F_y=f(L)\mu F_z{C}_c{\displaystyle \frac{\tan \alpha }{1-s}}\end{array}\right.$

(5)$f(L)=\left\{\begin{array}{l}L(2-L),L<1\\ 1,L\ge 1\end{array}\right.$

(6)$L={\displaystyle \frac{1}{2\sqrt{C_l^2{s}^2+C_c^2{\tan }^2\alpha }}}(1-s)(1-\epsilon v_x\sqrt{C_l^2{s}^2+C_c^2{\tan }^2\alpha })$

In the equation, $C_l$ and $C_c$ represent the longitudinal and lateral stiffness of the tire, respectively. $\epsilon$ is the velocity influence factor, the magnitude of which is influenced by tire material and structure, allowing for the correction of the effect of tire slip speed on longitudinal and lateral forces. $L$ represents the boundary value, which can describe the nonlinear characteristics of tire longitudinal and lateral forces.

To facilitate road adhesion coefficient estimation, Equation (4) has been normalized. $F_x^0$ and $F_y^0$ are normalized longitudinal and lateral tire forces, independent of the road adhesion coefficient. The processed formula [40] is as follows:

(7)$\left\{\begin{array}{l}{F}_x=\mu F_x^0=f(L)\mu F_z{C}_l{\displaystyle \frac{s}{1-s}}\\ F_y=\mu F_y^0=f(L)\mu F_z{C}_c{\displaystyle \frac{\tan \alpha }{1-s}}\end{array}\right.$

The parameters required to calculate the longitudinal and lateral forces of the tires can be determined using sensors within the CarSim vehicle model. $\beta$ is the vehicle’s sideslip angle at the center of mass, and $h$ is the height of the vehicle’s center of mass. The calculation formulas are as follows:

• (1). Vertical tire force $F_{zi}$

  (8)$\left\{\begin{array}{l}{F}_{zfl,zfr}={\displaystyle \frac{mg{l}_r}{2l}}-{\displaystyle \frac{m{a}_{x}h}{2l}}\pm {\displaystyle \frac{m{a}_{y}h{l}_r}{b_{f}l}}\\ F_{zrl,zrr}={\displaystyle \frac{mg{l}_f}{2l}}+{\displaystyle \frac{m{a}_{x}h}{2l}}\pm {\displaystyle \frac{m{a}_{y}h{l}_f}{b_{r}l}}\end{array}\right.$

• (2). Tire slip angle ${\alpha }_i$

  (9)$\left\{\begin{array}{l}{\alpha }_{fl,fr}=\delta -\arctan {\displaystyle \frac{v_y+a\dot{\phi }}{v_x\pm \frac{B_f}{2}\dot{\phi }}}\\ {\alpha }_{rl,rr}=\arctan {\displaystyle \frac{-v_y+b\dot{\phi }}{v_x\pm \frac{B_r}{2}\dot{\phi }}}\end{array}\right.$

• (3). Slip ratio $S_i$

  (10)$s_{ij}=\left\{\begin{array}{l}{\displaystyle \frac{{\omega }_{ij}R}{v_{ij}}}-1,v_{ij}>{\omega }_{ij}R\\ 1-{\displaystyle \frac{v_{ij}}{{\omega }_{ij}R}},v_{ij}<{\omega }_{ij}R\end{array}\right.$

• (4). Tire center velocity $r_{ij}$

  (11)$\left\{\begin{array}{l}{v}_{fl,fr}=v_{cog}+\dot{\phi }(\pm {\displaystyle \frac{b_f}{2}}-l_f\beta )\\ v_{rl,rr}=v_{cog}+\dot{\phi }(\pm {\displaystyle \frac{b_f}{2}}-l_r\beta )\\ v_{cog}=\sqrt{v_x^2+v_y^2}\\ \beta =\arctan {\displaystyle \frac{v_x}{v_y}}\end{array}\right.$

2.3. A Road Surface Adhesion Coefficient Estimation Model Based on the UKF

The mathematical relationship between the road surface adhesion coefficient and vehicle motion state parameters can be described using the constructed seven-degrees-of-freedom vehicle dynamics model and the Dugoff tire model. In this study, the UKF is employed for algorithm design. The state Equation (12) and the observation Equation (13) can be represented as follows:

(12)${\dot{X}}_2\left(t\right)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\cdot \left[\begin{array}{c}{u_f}_l\\ {u_f}_r\\ u_{rl}\\ u_{rr}\end{array}\right]+W_2\left(t\right)$

(13)$Y_2\left(t\right)=\left[\begin{array}{cccc}{\displaystyle \frac{F_{x_{fl}}^0\cdot \cos \delta -F_{y_{fl}}^0\cdot \sin \delta }{m}}& {\displaystyle \frac{F_{x_{fr}}^0\cdot \cos \delta -F_{y_{fr}}^0\cdot \sin \delta }{m}}& {\displaystyle \frac{F_{x_{rl}}^0}{m}}& {\displaystyle \frac{F_{x_{rr}}^0}{m}}\\ {\displaystyle \frac{F_{x_{fl}}^0\cdot \sin \delta +F_{y0fl}\cdot \cos \delta }{m}}& {\displaystyle \frac{F_{x_{fr}}^0\cdot \sin \delta +F_{y_{fr}}^0\cdot \cos \delta }{m}}& {\displaystyle \frac{F_{y_{rl}}^0}{m}}& {\displaystyle \frac{F_{y_{rr}}^0}{m}}\\ H\left(3,1\right)& H\left(3,2\right)& H\left(3,3\right)& H\left(3,4\right)\end{array}\right]\cdot \left[\begin{array}{c}{u}_{fl}\\ u_{fr}\\ u_{rl}\\ u_{rr}\end{array}\right]+V_2\left(t\right)$

where $H\left(\mathrm{3,1}\right)$, $H\left(\mathrm{3,2}\right)$, $H\left(\mathrm{3,3}\right)$ and $H\left(\mathrm{3,4}\right)$ are defined as follows:

$\begin{array}{l}H\left(3,1\right)={\displaystyle \frac{a\left(F_{x_{fl}}^0\cdot \sin \delta +F_{y_{fl}}^0\cdot \cos \delta \right)-\frac{T_f}{2}\left(F_{x_{fl}}^0\cdot \cos \delta -F_{y_{fl}}^0\cdot \sin \delta \right)}{I_z}};\\ H\left(3,2\right)={\displaystyle \frac{a\left(F_{x_{fr}}^0\cdot \sin \delta +F_{y_{fr}}^0\cdot \cos \delta \right)+\frac{T_f}{2}\left(F_{x_{fr}}^0\cdot \cos \delta -F_{y_{fr}}^0\cdot \sin \delta \right)}{I_z}};\\ H\left(3,3\right)={\displaystyle \frac{-\frac{T_r}{2}\cdot F_{x_{rl}}^0-b\cdot F_{y_{rl}}^0}{I_z}};\\ H\left(3,4\right)={\displaystyle \frac{\frac{T_r}{2}\cdot F_{x_{rr}}^0-b\cdot F_{y_{rr}}^0}{I_z}}\end{array}$

where the state vector is $X_2\left(t\right)=[u_{fl},u_{fr},u_{rl},u_{rr}]$, the observation vector is $Y_2\left(t\right)=[a_x,a_x,\dot{\phi }]$, the control input is the front wheel steering angle ${\delta }_{\mu }$, and the normalized tire force is $F_{x_{ij}}^0$, $F_{y_{ij}}^0$. The covariance of the process noise $W_2\left(t\right)$ for state estimation is$R_2$, and the covariance of the observation noise $V_2\left(t\right)$ is $R_2$. Here, $u_{ij}$ represents the road surface adhesion coefficients for the four tires, $a_x$ denotes the vehicle’s longitudinal acceleration, and $a_y$ denotes the vehicle’s lateral acceleration.

3. Design of Tracking Control

3.1. Three-Degrees-of-Freedom Vehicle Model

To simplify the vehicle dynamics model, it is assumed that the wheel steering angle, slip angle, and roll angle are small, and the effects of the suspension and vertical motion of the vehicle are neglected. A three-degrees-of-freedom model, including longitudinal, lateral, and yaw motions, is established as shown in Figure 3. In the figure, $F_{xf}$ and $F_{xr}$ represent the longitudinal forces on the front and rear wheels, respectively, while $F_{yf}$ and $F_{yr}$ represent the lateral forces on the front and rear wheels, respectively. $F_{wx}$, $F_{wy}$, and $M_{wz}$ represent the wind forces in the x and y directions and the yaw moment caused by the wind, respectively. The meanings of other symbols are the same as in Figure 1. The dynamic and kinematic equations [41] are as follows:

(14)$m{\dot{v}}_x-m{v}_y\dot{\phi }=F_{xf}\cos {\delta }_f+F_{xr}-F_{yf}\sin {\delta }_f+F_{wx}$

(15)$m{\dot{v}}_y+m{v}_x\dot{\phi }=F_{yf}\cos {\delta }_f+F_{xf}\sin {\delta }_f+F_{yr}+F_{wy}$

(16)$I_z\ddot{\phi }=a(F_{yf}\cos {\delta }_f+F_{xf}\sin {\delta }_f)-b{F}_{yr}+M_w$

(17)$\dot{X}=\dot{x}\sin \phi +\dot{y}\cos \phi$

(18)$\dot{Y}=\dot{x}\cos \phi -\dot{y}\sin \phi$

3.2. The Tire Model

Among the existing nonlinear tire models, the Pacejka tire model [42] is widely used in the industry. This model is developed by determining its structure and parameters through the measurement of tire data, which effectively describes tire behavior and performance.

According to the assumptions in this study, the Pacejka tire model can be described under linear conditions. Its dynamic model is shown in Figure 2. The relationship between the longitudinal and vertical forces of the wheel, as well as the longitudinal slip ratio, can be expressed by Equations (19) and (20) [43].

(19)$F_x=\mu F_z$

(20)$s={\displaystyle \frac{r\omega -v_x}{v_x}}$

where $F_{zf}={\displaystyle \frac{bmg}{L}}$ and $F_{zr}={\displaystyle \frac{amg}{L}}$ denote the normal forces on the front and rear wheels, respectively, $r$ represents the rolling radius of the wheel, and $L=a+b$ denotes the wheelbase.

The lateral forces on the front and rear wheels are as follows:

(21)$F_{yf}=C_f{\alpha }_f$

(22)$F_{yr}=C_r{\alpha }_r$

Under the assumption of small wheel angles, ${\alpha }_f={\displaystyle \frac{v_y+a\dot{\phi }}{v_x}}-{\delta }_f$ and ${\alpha }_r={\displaystyle \frac{v_y-b\dot{\phi }}{v_x}}-{\delta }_r$, respectively, where $C_f$ and $C_r$ represent the lateral stiffness of the front and rear wheels, respectively.

3.3. The Wind Model

It is assumed in this study that the vehicle does not experience slip under the influence of wind. The resultant force exerted by the wind on the vehicle is decomposed into components along the $x$, $y$, and $z$ axes in the vehicle coordinate system, denoted as $F_{wx}$, $F_{wy}$, and $F_{wz}$, respectively. When the point of application of the resultant wind force is not at the vehicle’s center of mass, the vehicle will also experience moments of yaw, roll, and pitch. The distances of the point of application of the resultant wind force from the center of mass in the x, y, and z directions are denoted as Δ$x$, $∆y$, and $\mathsf{\Delta }z$ respectively. According to aerodynamic principles and theoretical mechanics, the formulas for calculating the forces and moments generated by the wind are given by Equations (23) and (24).

(23)$F_{wk}={\displaystyle \frac{C_{Dk}\rho A{v}_{rk}^2}{2}}$

(24)$M_{wk}={\displaystyle \frac{C_{Dk}\rho A\Delta k{v}_{rk}^2}{2}}$

where $\rho$ is the air density, $v_{rk}$ is the relative wind speed in the $k$ direction, $C_{Dk}$ is the aerodynamic coefficient in the $k$ direction, $A$ is the frontal area of the vehicle, and $\mathsf{\Delta }k$ represents the distance of the point of application of the resultant wind force from the center of mass in the $k$ direction, where $k=x,y,z$.

The influence of the lateral and longitudinal components $F_{wx}$ and $F_{wy}$ of the resultant wind force, as well as the yaw moment $M_{wz}$ caused by the wind, is incorporated into the three-degrees-of-freedom vehicle dynamics model in this paper. Since the vertical load, roll moment, and pitch moment are not considered in the three-degrees-of-freedom vehicle model, this study represents the effects of these factors by varying the tire lateral stiffness. The average lateral stiffness of the left and right tires is taken as the lateral stiffness of the front and rear tires in the model shown in Figure 3. The relationship between the tire’s lateral stiffness and the vertical load is depicted in Figure 4, as described in [43].

When calculating the variation in vertical load on the tires, it is assumed that the vertical load, roll moment, and pitch moment induced by the wind are positive (with opposite signs when negative). The change in vertical load on the left and right tires due to the wind-induced roll moment is denoted as $\Delta F_{wx}={\displaystyle \frac{M_{wx}}{B}}$, while the change in vertical load on the front and rear tires due to the wind-induced pitch moment is denoted as $\Delta F_{wy}={\displaystyle \frac{M_{wy}}{L}}$. The variation in vertical load on the front and rear tires caused by the vertical load itself is denoted as $\Delta F_{zf}={\displaystyle \frac{b\Delta F_z}{L}},\Delta F_{zr}={\displaystyle \frac{a\Delta F_z}{L}}$. The changes in vertical load on the four vehicle tires are expressed in Equations (25)–(28).

(25)$\Delta F_{fl}=-{\displaystyle \frac{M_{wx}}{2B}}+{\displaystyle \frac{M_{wy}}{2L}}-{\displaystyle \frac{b\Delta F_{zf}}{2L}}$

(26) $\Delta F_{fr}={\displaystyle \frac{M_{wx}}{2B}}+{\displaystyle \frac{M_{wy}}{2L}}-{\displaystyle \frac{b\Delta F_{zf}}{2L}}$

(27) $\Delta F_{rl}=-{\displaystyle \frac{M_{wx}}{2B}}-{\displaystyle \frac{M_{wy}}{2L}}-{\displaystyle \frac{a\Delta F_{zr}}{2L}}$

(28) $\Delta F_{rr}={\displaystyle \frac{M_{wx}}{2B}}-{\displaystyle \frac{M_{wy}}{2L}}-{\displaystyle \frac{a\Delta F_{zr}}{2L}}$

3.4. The MPC Controller Under Wind Model

MPC is a sophisticated algorithm widely utilized in various engineering fields, including autopilot systems, aerospace, and the chemical industry. Over time, through its extensive application in industrial settings, MPC has advanced and become increasingly refined. It incorporates strategies such as multi-step prediction, rolling optimization, and feedback correction, offering significant advantages such as enhanced control effectiveness, robust performance, reduced dependence on model accuracy, and the capability to manage multiple real-time constraints [44]. The control principle of MPC is depicted in Figure 5.

MPC employs rolling optimization to dynamically control the system, with the outputs from the CarSim vehicle model acting as state inputs for the MPC controller, thus forming a closed-loop system. At each time step $k$, the system’s future outputs are forecasted for $N_p$ steps ahead, based on the measured outputs and the system model. The optimal inputs at time $k$ are then determined by minimizing the discrepancy between the predicted and desired outputs. Throughout the trajectory tracking task, this prediction and optimization process is continually updated by using the system state at $k+i$ moments to calculate new control inputs, simultaneously shifting both the control and prediction horizons until the task is completed.

In this section, the MPC controller is applied to solve the trajectory tracking problem. Deviations in state quantities are incorporated into the prediction model to continually update the state operating points and linearize the system model. The objective of MPC is to compute a series of control actions within the prediction horizon that minimizes the difference between the reference and actual outputs, while also narrowing the gap between the predicted and reference trajectories, utilizing the system dynamics during the tracking process [45].

In CarSim, wind is assumed to act in the vehicle’s Y-direction, provided by a fan. To compensate for wind-induced errors, the wind model is integrated into the MPC controller, as illustrated in Figure 6.

In Figure 6, the MPC controller predicts variables like vehicle speed and steering angle using system input parameters, optimizing through objective functions and constraints. To enhance control accuracy and robustness amid environmental uncertainties, models for road adhesion and wind effects are incorporated, correcting vehicle state parameters input into the MPC. This transforms the optimal motion planning and control problem into a quadratic optimization problem. MPC computes real-time, trajectory-tracking trajectories and control sequences that satisfy vehicle dynamics and kinematic constraints, achieving precise, stable, and reliable control in dynamic environments and under external disturbances.

• (1). Predictive Modeling

The predictive model is based on a discretized time-varying linear model, from which the system’s predictive output expression can be derived. The expression [36] is as follows:

(29)$Y(t)={\psi }_t\xi (t\left|t\right.)+{\Theta }_t\Delta U(t)$

$\begin{array}{cc}Y(t)=\left[\begin{array}{l}\eta (t+1\left|t)\right.\\ \eta (t+1\left|t)\right.\\ \cdots \\ \eta (t+N_c\left|t)\right.\\ \cdots \\ \eta (t+N_p\left|t)\right.\end{array}\right]{\psi }_t=\left[\begin{array}{l}{\tilde{C}}_{(t,t)}{A}_{(t,t)}\\ {\tilde{C}}_{(t,t)}{A^2}_{(t,t)}\\ \cdots \\ {\tilde{C}}_{(t,t)}{A^{N_c}}_{(t,t)}\\ \cdots \\ {\tilde{C}}_{(t,t)}{A^{N_p}}_{(t,t)}\end{array}\right];& {\Theta }_t=\left[\begin{array}{cccc}\begin{array}{l}{\tilde{C}}_{(t,t)}{\tilde{B}}_{(t,t)}\\ {\tilde{C}}_{(t,t)}{\tilde{A}}_{(t,t)}{\tilde{B}}_{(t,t)}\\ \cdots \\ {\tilde{C}}_{(t,t)}{{\tilde{A}}^{N_{\mathrm{c}}}}_{(t,t)}{\tilde{B}}_{(t,t)}\\ ⋮\\ {\tilde{C}}_{(t,t)}{{\tilde{A}}^{N_p-1}}_{(t,t)}{\tilde{B}}_{(t,t)}\end{array}& \begin{array}{l}0\\ {\tilde{C}}_{(t,t)}{\tilde{B}}_{(t,t)}\\ \cdots \\ {\tilde{C}}_{(t,t)}{{\tilde{A}}^{N_{\mathrm{c}}-1}}_{(t,t)}{\tilde{B}}_{(t,t)}\\ ⋮\\ {\tilde{C}}_{(t,t)}{{\tilde{A}}^{N_p-2}}_{(t,t)}{\tilde{B}}_{(t,t)}\end{array}& \begin{array}{l}\cdots \\ \cdots \\ \ddots \\ \cdots \\ \ddots \\ \cdots \end{array}& \begin{array}{l}0\\ 0\\ \cdots \\ {\tilde{C}}_{(t,t)}{\tilde{A}}_{(t,t)}{\tilde{B}}_{(t,t)}\\ ⋮\\ {\tilde{C}}_{(t,t)}{{\tilde{A}}^{N_p-N_c-1}}_{(t,t)}{\tilde{B}}_{(t,t)}\end{array}\end{array}\right];\\ \Delta U(t)=\left[\begin{array}{l}\Delta u(t)\left|t\right.)\\ \Delta u(t+1)\left|t\right.)\\ ⋮\\ \Delta u(t+N_c)\left|t\right.)\end{array}\right]& \end{array}$

where $Y\left(t\right)$ is the control output vector, ${\psi }_t$ and ${\mathsf{\Theta }}_t$ are coefficient matrices, $\xi \left(t|t\right)$ is the system state vector at time $t$ based on the reference at time $t$, $∆U\left(t\right)$ is the control increment of the system, and $A$, $B$, and $C$ are coefficient matrices represented as follows:

$A=\left[\begin{array}{cccccc}{\displaystyle \frac{\left(C_{cf}+C_{cr}\right)}{m{v}_x}}& {\displaystyle \frac{\partial f_{v_y}}{\partial \dot{x}}}& 0& -v_x+{\displaystyle \frac{-\left(b{C}_{cr}-a{C}_{cf}\right)}{m{v}_x}}& 0& 0\\ \dot{\phi }-{\displaystyle \frac{C_{cf}{\delta }_f}{m{v}_x}}& {\displaystyle \frac{\partial f_{v_x}}{\partial \dot{x}}}& 0& v_y-{\displaystyle \frac{a{C}_{cf}{\delta }_f}{m{v}_x}}& 0& 0\\ 0& 0& 0& 1& 0& 0\\ {\displaystyle \frac{-\left(b{C}_{cr}-a{C}_{cf}\right)}{I_z{v}_x}}& {\displaystyle \frac{\partial f_{\dot{\phi }}}{\partial \dot{x}}}& 0& {\displaystyle \frac{\left(b^2{C}_{cr}+a^2{C}_{cf}\right)}{I_z{v}_x}}& 0& 0\\ \cos \left(\phi \right)& \sin \left(\phi \right)& v_x\cos \left(\phi \right)-v_y\sin \left(\phi \right)& 0& 0& 0\\ \begin{array}{c}-\sin \left(\phi \right)\\ 0\end{array}& \begin{array}{c}\cos \left(\phi \right)\\ 0\end{array}& \begin{array}{c}-v_y\cos \left(\phi \right)-v_x\sin \left(\phi \right)\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ {\displaystyle \frac{C_{Dk}\rho A{v}_{rk}^2}{2}}\end{array}\end{array}\right];$

$B={\left[\begin{array}{lllll}{\displaystyle \frac{C_{cf}}{m}}\hfill & {\displaystyle \frac{C_{cf}\left(\frac{v_y+a\dot{\phi }}{v_x}-{\delta }_f\right)}{m}}\hfill & 0\hfill & {\displaystyle \frac{a{C}_{cf}}{I_z}}\hfill & \begin{array}{cc}0& \begin{array}{cc}0& 1\end{array}\end{array}\hfill \end{array}\right]}^T;$

$C=\left[\begin{array}{cccc}\begin{array}{c}0\\ \begin{array}{c}0\\ 0\end{array}\end{array}& \begin{array}{c}0\\ \begin{array}{c}0\\ 0\end{array}\end{array}& \begin{array}{c}1\\ \begin{array}{c}0\\ 0\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}\begin{array}{c}0\\ \begin{array}{c}0\\ 0\end{array}\end{array}& \begin{array}{c}0\\ \begin{array}{c}1\\ 0\end{array}\end{array}\end{array}& \begin{array}{c}\begin{array}{cc}0& 0\end{array}\\ \begin{array}{cc}\begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 1\end{array}\end{array}\end{array}\end{array}\end{array}\right]$

• (2). Feedback Correction

The optimization objective function aims to minimize vehicle state error and control input increments, ensuring smooth actuator outputs and maintaining control input continuity, as detailed in Equation (30).

(30)$J(k)={{{\displaystyle \sum _{i=1}^{N_P}‖\eta (k+i\left|t\right.)-{\eta }_{ref}(k+i\left|t\right.)‖}}^2}_Q+{{\displaystyle \sum _{i=1}^{N_e-1}‖\Delta U(k+i\left|t\right.)‖}}_R^2+\rho {\epsilon }^2$

One of the advantages of the MPC is its ability to handle multiple constraints. To ensure the control process aligns with real driving conditions and remains stable, the following constraints have been incorporated in this study.

In Equation (30), $\eta \left(k+i|t\right)$ and $∆U\left(k+i|t\right)$ represent the system output state variables and reference output state variables at time $t+i$ relative to time $t$ respectively. $Q$, $R$, and $\rho$ are weighting coefficients, $N_p$ is the prediction horizon, $N_c$ is the control horizon, and $\epsilon$ is a relaxation factor. The first term in Equation (30) reflects the system’s ability to follow the reference trajectory, while the second term reflects the requirement for smooth changes in control inputs. The overall function of the expression is to enable the system to rapidly and smoothly track the desired trajectory from the previous period.

• (1). Control Input Constraints

(31) $u_{\min }\le u(t+i|t)\le u_{\max },i=0,1,\cdots ,N_c-1$

• (2). Control Increment Constraints

(32) $\Delta u_{\min }\le \Delta u(t+i|t)\le \Delta u_{\max },i=0,1,\cdots ,N_c-1$

• (3). Output Constraints

(33) ${\eta }_{\min }-\epsilon 1_p\le \eta (t+i|t)\le {\eta }_{\max }+\epsilon 1_p,i=1,\cdots ,N_p$

In the equation, $1_p$ represents a $p$-dimensional column vector, and $\epsilon$ is the relaxation factor, which is introduced to prevent the occurrence of infeasible solutions during the optimization process.

After establishing the objective function and constraints, the trajectory tracking problem is transformed into an optimization problem.

(34)$\left\{\begin{array}{l}\underset{\Delta U(t),\epsilon }{\min }J(\chi (t),u(t-1),\Delta U(t),\epsilon )\\ u_{\min }\le u(k)\le u_{\max },k=t,t+1,\cdots ,t+N_c-1\\ s.t.\Delta u_{\min }\le \Delta u(k)\le \Delta u_{\max },k=t,t+1,\cdots ,t+N_c-1\\ {\eta }_{\min }-\epsilon 1_p\le \eta (k)\le {\eta }_{\max }+\epsilon 1_p,k=t+1,t+2,\cdots ,t+N_p\\ 0\le \epsilon \le {\epsilon }_{\max }\end{array}\right.$

To simplify the calculations, Equation (34) is converted into a standard quadratic form, allowing for the solution of a quadratic programming problem. The following definitions are made:

(35)$\left\{\begin{array}{l}{Q}_e={\left[\begin{array}{ccccc}Q& 0& \cdots & \cdots & 0\\ 0& Q& 0& \cdots & 0\\ 0& 0& Q& \ddots & 0\\ ⋮& ⋮& \ddots & \ddots & 0\\ 0& 0& \cdots & 0& Q\end{array}\right]}_{N_p\times N_p}\\ R_e={\left[\begin{array}{ccccc}R& 0& \cdots & \cdots & 0\\ 0& R& 0& \cdots & 0\\ 0& 0& R& \ddots & 0\\ ⋮& ⋮& \ddots & \ddots & 0\\ 0& 0& \cdots & 0& R\end{array}\right]}_{N_c\times N_c}\end{array}\right.$

In the equation, $Q_e\in {ℝ}^{p{N}_p\times p{N}_p}$, $R_e\in {ℝ}^{m{N}_c\times m{N}_c}$.

(36)$M={\left[\begin{array}{ccccc}{I}_m& 0& \cdots & \cdots & 0\\ I_m& I_m& 0& \cdots & 0\\ I_m& I_m& I_m& \ddots & 0\\ ⋮& ⋮& \ddots & \ddots & 0\\ I_m& I_m& \cdots & I_m& I_m\end{array}\right]}_{N_c\times N_c}$

In the equation, $I_m$ represents an $m$-dimensional identity matrix, $M\in {ℝ}^{m{N}_c\times m{N}_c}$.

(37)$U(t-1)={\left[\begin{array}{l}u(t-1|t)\\ u(t-1|t)\\ ⋮\\ u(t-1|t)\end{array}\right]}_{N_c\times 1}$

In the equation, $U(t-1)\in {ℝ}^{m{N}_c\times 1}$, the control input can be derived from Equations (35)–(37) as follows:

(38)$U(t)=M\Delta U(t)+U(t-1)$

Substituting Equation (38) into Equation (30) transforms the objective function into a quadratic form.

(39)$\left\{\begin{array}{l}J(\chi (t),u(t-1),\Delta U(t),\epsilon )={\left[\begin{array}{l}\Delta U(t)\\ \epsilon \end{array}\right]}^T{H}_t\left[\begin{array}{l}\Delta U(t)\\ \epsilon \end{array}\right]+G_t\left[\begin{array}{l}\Delta U(t)\\ \epsilon \end{array}\right]+P_t\\ H_t=\left[\begin{array}{cc}{\Theta }_t^T{Q}_e{\Theta }_t+R_e& 0\\ 0& \rho \end{array}\right]\\ G_t=\left[\begin{array}{cc}2E^T(t)Q_e{\Theta }_t& 0\end{array}\right]\\ P_t=E^T(t)Q_{e}E(t)\\ E_t={\Psi }_t\chi (t|t)-Y_{ref}(t)\\ Y_{ref}=\left[{\eta }_{ref}(t+1|t),\cdots ,{\eta }_{ref}(t+N_p|t)\right]\end{array}\right.$

Thus, the optimization problem represented by Equation (34) can be reformulated as a quadratic programming problem.

(40)$\left\{\begin{array}{l}\underset{\Delta U(t),\epsilon }{\min }{\left[\begin{array}{l}\Delta U(t)\\ \epsilon \end{array}\right]}^T{H}_t\left[\begin{array}{l}\Delta U(t)\\ \epsilon \end{array}\right]+G_t\left[\begin{array}{l}\Delta U(t)\\ \epsilon \end{array}\right],s.t.\left[\begin{array}{l}\Delta U_{\min }\\ 0\end{array}\right]\le \left[\begin{array}{l}\Delta U(t)\\ \epsilon \end{array}\right]\le \left[\begin{array}{l}\Delta U_{\max }\\ \epsilon \end{array}\right]\\ \left[\begin{array}{cc}M& 0\\ -M& 0\\ {\Theta }_t& -1_{p{N}_p\times 1}\\ -{\Theta }_t& 1_{p{N}_p\times 1}\end{array}\right]\left[\begin{array}{l}\Delta U(t)\\ \epsilon \end{array}\right]\le \left[\begin{array}{l}{U}_{\max }-U(t)\\ -U_{\max }+U(t)\\ Y_{\max }-{\Psi }_t\chi (t|t)\\ -Y_{\max }+{\Psi }_t\chi (t|t)\end{array}\right]\end{array}\right.$

By solving Equation (40), the control increment and the relaxation factor can be determined as follows:

(41)$\left[\begin{array}{c}\Delta U(t)\\ \epsilon \end{array}\right]=\left[\begin{array}{c}\Delta u(t)\\ \Delta u(t+1)\\ ⋮\\ \Delta u(t+N_c-1)\\ \epsilon \end{array}\right]$

• (3). Roll Optimization

The first term in Equation (41) is taken as the actual control input. Subsequently, the control input for the current control time domain is determined based on the control inputs from the previous time domain. Since the reference input includes trajectory point information and the reference control input is set to zero, the feedback control input can be expressed as follows:

(42)$u(t)=u(t-1)+\Delta u(t)$

The above solving process is repeated in each control cycle to obtain the real-time feedback control input. This feedback control input is then fed into the controlled system in a continuous loop, thereby achieving trajectory tracking control for the vehicle.

Let the state variables of the vehicle be denoted as $x_d={[\dot{x}\dot{y}\phi \dot{\phi }XY∆F_{ij}]}^T$, where $\dot{x}$ represents the longitudinal velocity, $\dot{y}$ represents the lateral velocity, $\phi$ represents the yaw angle at the vehicle’s center of mass, $\dot{\phi }$ represents the yaw rate at the vehicle’s center of mass, $X$ represents the longitudinal displacement, $Y$ represents the lateral displacement, and $∆F_{ij}$ represents the vertical load variation on the four wheels. The output control variables of the controller incorporating the wind model are denoted as $u_d={\delta }_f$ in the trajectory tracking controller, the established wind model is integrated into the vehicle dynamics model as a predictive model.

4. Simulation Conditions

4.1. Input Parameter Settings

The selection of simulation parameters aimed to replicate real-world driving conditions as accurately as possible, ensuring a comprehensive and realistic validation of the proposed algorithms. The parameters were carefully chosen to evaluate the performance of the system under diverse scenarios, including varying road surface conditions and lateral wind influences. The following key parameters were used:

4.1.1. Road Adhesion Coefficient

Table 1 lists the common road adhesion coefficients for different road surface conditions, providing a reference for the selected simulation parameters. The specific parameters used are detailed in Table 1 [43].

The adhesion coefficient was varied across three distinct values to represent different types of road surfaces: optimal (0.8), intermediate (0.5), and low adhesion (0.3). These values were selected based on typical real-world scenarios where road conditions can vary significantly due to weather, road type, and wear.

4.1.2. Wind Conditions

The simulation incorporated a lateral wind model to replicate the impact of crosswinds, which are prevalent in real-world scenarios and can notably influence vehicle stability and trajectory tracking performance. A wind speed of 10 m/s was selected, equivalent to the effect of a level-six sustained wind typically experienced on bridges. This intensity of lateral wind was introduced as a disturbance to assess the control system’s robustness in maintaining trajectory tracking accuracy and stability under dynamic environmental forces.

4.1.3. Vehicle Dynamics Parameters

A shelter-type vehicle model was selected as the basis for the simulation. This model was parameterized with realistic vehicle dynamics values to ensure accuracy in the representation of physical behavior. The specific parameters used are detailed in Table 2, which were obtained from CarSim.

4.1.4. Model Predictive Control Parameters

The Model Predictive Control (MPC) algorithm relies on several critical parameters to ensure optimal control performance in trajectory tracking tasks. These parameters were chosen based on the specific requirements of the simulation scenario and are detailed in Table 3.

4.2. Simulation Environment Construction

In order to effectively validate the proposed algorithms and evaluate the trajectory tracking performance under varying conditions, the simulation environment was carefully constructed. The environment was designed to closely replicate real-world driving conditions, incorporating multiple dynamic factors such as road surface characteristics, wind effects, and vehicle dynamics. This ensured comprehensive testing of the control system’s robustness and adaptability in a variety of scenarios.

4.2.1. Estimation of Road Adhesion Coefficient

Different road adhesion coefficient simulation scenarios were set up in CarSim software (2016 for USA). The accuracy of road adhesion coefficient estimation based on the UKF was analyzed through joint simulation using CarSim and Simulink (R2019a for USA). The principle of the road adhesion coefficient estimation algorithm is illustrated in Figure 7.

4.2.2. Trajectory Tracking Control Under Windy Conditions

The double lane-change test [46], also known as the dynamic swerve test, is primarily employed to assess the rapid steering reversal of vehicles during sharp cornering maneuvers, and it is an internationally recognized test procedure. Due to its capability to effectively evaluate the trajectory tracking performance of control algorithms and reflect the handling stability of vehicles, it is commonly utilized in automotive dynamic simulations and stability testing. In this study, the double lane-change reference trajectory and the position of the fan in the CarSim software are depicted in Figure 8. Simulation experiments involve placing fans at the side of the road to simulate crosswind conditions, conducted through joint simulation using CarSim/Simulink, and compared with simulations conducted without the inclusion of crosswind effects.

The experimental vehicle utilizes front-wheel steering control, and the study employs a wind model developed for this purpose. Assuming a constant wind speed of V = 10 m/s, simulating conditions comparable to a level-five wind, the wind is directed consistently along the Y-axis as illustrated in Figure 8. It is further assumed that the vehicle’s travel speed remains constant. As the vehicle navigates the double lane-change trajectory, both the wind direction and the torque exerted on the vehicle’s body fluctuate. The CarSim software is used to monitor wheel forces and to calculate the variations in forces and torques resulting from the wind.

5. Simulation and Analysis

5.1. Simulation of Estimation of Road Adhesion Coefficient

To validate the road adhesion coefficient estimation algorithm, joint simulations were performed using CarSim and Simulink, with different road adhesion scenarios set to 0.8, 0.5, and 0.3 to represent optimal, intermediate, and low-adhesion road surfaces. The accuracy of the estimation was evaluated based on the road adhesion coefficient values predicted by the system in comparison with the ground truth values used for simulation.

5.1.1. Optimal Road Conditions (Adhesion Coefficient: 0.8)

The road adhesion coefficient was set to 0.8 to simulate optimal road conditions, with a simulated vehicle speed of 96 km/h to replicate high-speed driving. The simulation duration was 10 s, and the results are illustrated in Figure 9.

In Figure 9, the estimated road adhesion coefficients at the contact points of the four wheels are depicted, providing vital reference values for the vehicle’s operating surface. Analysis of these coefficients disclosed several key findings. Firstly, the estimated coefficients for the left front and left rear wheels stabilize at 0.79, while those for the right front and right rear wheels stabilize at 0.81. Given the actual set road adhesion coefficient of 0.8, this results in a maximum deviation of merely 0.01, indicating a high degree of estimation accuracy. Furthermore, the estimates converge rapidly, achieving stability within approximately 0.1 s. This rapid convergence underscores the system’s fast estimation response.

5.1.2. Intermediate Road Conditions (Adhesion Coefficient: 0.5)

The road adhesion coefficient was set to 0.5 to simulate typical road conditions, with a simulated vehicle speed of 72 km/h to replicate urban driving speeds. The simulation duration was 10 s. The simulation results are illustrated in Figure 10.

Based on the simulation results depicted in Figure 9, it can be observed that within approximately 0.1 s after the start of the simulation, the final estimated road adhesion coefficients for the left front and left rear wheels stabilize at 0.45, with an error of 0.05 compared to the actual set road adhesion coefficient of 0.5. Similarly, the final estimated road adhesion coefficients for the right front and right rear wheels stabilize at 0.54, with a difference of 0.04 compared to the actual road adhesion coefficient. The estimation accuracy is relatively high.

5.1.3. Low Road Adhesion Conditions (Adhesion Coefficient: 0.3)

The road adhesion coefficient was set to 0.3 to simulate typical road conditions, with a simulated vehicle speed of 54 km/h to replicate urban driving speeds. The simulation duration was 10 s. The simulation results are illustrated in Figure 11.

Based on the simulation results illustrated in Figure 11, it is evident that the final estimated road adhesion coefficients for the left front and left rear wheels stabilize at 0.23 within approximately 0.1 s after the start of the simulation. This stabilization represents an error of 0.07 when compared to the actual set road adhesion coefficient of 0.3. Similarly, the estimated coefficients for the right front and right rear wheels settle at 0.39, exhibiting a difference of 0.09 from the actual coefficient. Despite these initial discrepancies, the error range remains within ±0.1. Notably, after 0.1 s, the road adhesion coefficients for all four wheels quickly converge to the actual value of 0.3.

5.1.4. Docking Road Condition and Split Road Condition

Maintaining all other parameters constant, a simulation was performed at a vehicle speed of 60 km/h under two distinct road surface conditions illustrated in Figure 12 and Figure 13.

For the docking road surface scenario shown in Figure 12, the road adhesion coefficient was set to 0.8 for the first 7.5 s. At 7.5 s, the adhesion coefficient abruptly dropped from 0.8 to 0.4, simulating a sudden change in road conditions. The simulation was then continued to evaluate the system’s response under these altered conditions, with the results presented in Figure 14.

For the split road surface scenario shown in Figure 13, the left-side tires (front and rear) were assigned a road adhesion coefficient of 0.8, while the right-side tires (front and rear) were set to 0.4. This configuration simulates uneven road surface conditions. The simulation was continued to validate the system’s performance, and the corresponding results are depicted in Figure 15.

Figure 14 demonstrates that the system responds rapidly to abrupt changes in road adhesion coefficients, with minimal oscillation in the estimated adhesion coefficient curve. The final average estimated adhesion coefficient is 0.4035, with an error of approximately 0.875%. As shown in Figure 15, the average estimated adhesion coefficient for the left-side tires is 0.804, with an error of about 0.5%, while the right-side tires yield an average estimated adhesion coefficient of 0.403, with an error of approximately 0.75%. The results indicate minimal fluctuation in the estimated road adhesion coefficients, highlighting the system’s robustness and accuracy.

5.1.5. Discussion of Results

Comprehensive simulation analysis demonstrates that the estimated road adhesion coefficients for all four wheels closely match the actual set values and converge rapidly within 0.1 s under varying road adhesion coefficients and vehicle speed conditions. Additionally, the algorithm effectively tracks road adhesion coefficients in complex scenarios, such as docking and split road surfaces. Notably, a comparison of estimation results under different road adhesion conditions reveals superior performance on high-adhesion surfaces. These findings provide strong evidence of the effectiveness and reliability of the road adhesion coefficient estimation algorithm.

5.2. Simulation of Trajectory Tracking Control Under Windy Conditions

In this section, the estimated values of the road adhesion coefficients for the three types of road surfaces from the previous section are input into the controller. These estimated values are essential for the trajectory tracking control system, enabling it to adapt to varying road conditions. By incorporating the road adhesion coefficients, which were estimated under different surface conditions, the controller can dynamically adjust the control inputs, ensuring the stability and accuracy of the vehicle’s trajectory under diverse road conditions.

The simulation conditions for trajectory tracking control considering wind effects align with those used for road adhesion coefficient estimation in this study. Specifically, the simulations were conducted under high-speed travel on well-maintained roads, with a road adhesion coefficient set at 0.8, a vehicle speed of 96 km/h, and a wind speed of 10 m/s. The results of these simulations are illustrated in Figure 16. Compared to the controller without the inclusion of the wind model, the trajectory tracking control incorporating the wind model demonstrated superior performance by closely following the reference trajectory. Additionally, the vehicle’s yaw angle decreased by approximately 2°, which represents a reduction of about 15%, and reached a stable state after 7 s with faster convergence. The yaw rate also showed improvement, being reduced to around 12°/s, a decrease of approximately 25%, and stabilized after 8 s.

Under conditions that simulate typical urban road surfaces and driving speeds, the simulation model was set with a road adhesion coefficient of 0.5, a vehicle speed of 72 km/h, and a wind speed of 10 m/s. The results of these simulations are depicted in Figure 17. The data indicate that, when comparing the controller integrated with a wind model to one without, the trajectory tracking capability of the former closely aligns with the reference trajectory, signifying enhanced control effectiveness. Additionally, the vehicle’s yaw angle is contained within 10°, and the yaw rate is limited to 15°/s. Specifically, the integration of the wind model results in a reduction of approximately 2° in the yaw angle, which amounts to a 16% decrease, and a reduction of about 6°/s in the yaw rate, equating to a 23% decrease, when compared to the controller lacking wind model integration. Furthermore, both the yaw angle and yaw rate achieve stable values more rapidly with the wind model-integrated controller, indicating improved dynamic stability.

To evaluate the effectiveness of trajectory tracking control on icy and snowy road surfaces, simulations were conducted with specific parameters: a road adhesion coefficient set to 0.3, a vehicle speed of 54 km/h, and a wind speed of 10 m/s. The simulation results are illustrated in Figure 18. Compared to the controller without wind model integration, the trajectory controlled with the wind model closely aligns with the reference trajectory. Additionally, the yaw angle remains within 7°, and the yaw rate is controlled within 4°/s. Furthermore, the controller with wind model integration shows a reduction of approximately 4° in yaw angle, representing a decrease of around 30%, and a decrease of approximately 4°/s in yaw rate, indicating a reduction of about 20%, compared to the controller without wind model integration.

The simulation analysis reveals that under dual-track conditions, and when accounting for varying road adhesion coefficients and vehicle speeds, the trajectory tracked with the incorporation of a wind model demonstrates superior alignment with the reference trajectory compared to a controller devoid of wind model integration. Furthermore, the inclusion of the wind model results in reduced yaw angle and yaw rate, which stabilize more quickly. These findings substantiate the efficacy of integrating the wind model into trajectory tracking control systems.

6. Conclusions

This study addresses the issue of low trajectory tracking accuracy and poor driving safety of autonomous vehicles under the influence of winds on different road surfaces and at varying speeds. A seven-degrees-of-freedom model and Dugoff tire model were established, and the UKF was employed for real-time online estimation of road adhesion coefficients. Based on a three-degrees-of-freedom dynamic model, a trajectory tracking controller incorporating a wind model was designed. The main conclusions of the research are as follows.

Under varying road adhesion coefficients and vehicle speeds, the proposed algorithm demonstrated excellent performance in both adhesion coefficient estimation and trajectory tracking. The road adhesion coefficient estimation algorithm exhibited high accuracy, with estimation errors of approximately 0.01, 0.05, and 0.08 for adhesion coefficients of 0.8, 0.5, and 0.3, respectively, and achieved rapid convergence within 0.1 s. The algorithm effectively adapted to complex road conditions such as docking and split-road surfaces, with superior performance observed on surfaces with higher adhesion coefficients. Similarly, in dynamic lane-change simulations, the trajectory tracking controller incorporating the wind model achieved closer alignment with the reference trajectory, reduced lateral angle and angular velocity, and faster convergence to stable speeds compared to the controller without the wind model. The trajectory tracking performance was notably better on surfaces with higher adhesion coefficients. These findings validate the effectiveness of the proposed algorithm in road adhesion coefficient estimation and trajectory tracking control under diverse operating conditions.

Author Contributions

Validation, Z.L.; Investigation, Z.L.; Writing—original draft, Z.L.; Writing—review & editing, Z.L.; Supervision, Y.Z.; Funding acquisition, Y.Z. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

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Figures and Tables

Figure 1. The seven-degrees-of-freedom vehicle dynamics model.

Figure 2. Tire dynamics model.

Figure 3. The three-degrees-of-freedom vehicle dynamics model.

Figure 4. The relationship between tire lateral stiffness and vertical load.

Figure 5. MPC schematic diagram.

Figure 6. Predictive model for Model Predictive Control with the Wind Force Model.

Figure 7. Principle of road adhesion coefficient estimation based on the UKF.

Figure 8. Double lane-change reference trajectory and fan position during vehicle lane-changing maneuvers.

Figure 9. Four-wheel adhesion coefficient detection with a road adhesion coefficient of 0.8.

Figure 10. Four-wheel adhesion coefficient detection with a road adhesion coefficient of 0.5.

Figure 11. Four-wheel adhesion coefficient detection with road adhesion coefficient of 0.3.

Figure 12. Schematic diagram of docking road.

Figure 13. Schematic diagram of split road.

Figure 14. Adhesion coefficient of docking road surface.

Figure 15. Adhesion coefficient of split road surface.

Figure 16. Trajectory tracking performance under the conditions of 10 m/s crosswind velocity and a road adhesion coefficient of 0.8.

Figure 16. Trajectory tracking performance under the conditions of 10 m/s crosswind velocity and a road adhesion coefficient of 0.8.

Figure 17. Trajectory tracking performance under the conditions of 10 m/s crosswind velocity and a road adhesion coefficient of 0.5.

Figure 17. Trajectory tracking performance under the conditions of 10 m/s crosswind velocity and a road adhesion coefficient of 0.5.

Figure 18. Trajectory tracking performance under the conditions of 10 m/s crosswind velocity and a road adhesion coefficient of 0.3.

Table 1

Reference values for typical road surface adhesion coefficients.

Road Surface Condition	Adhesion Coefficient (μ)
Dry Asphalt	0.8–0.9
Wet Asphalt	0.5–0.6
Icy Road (Compacted Snow or Ice)	0.2–0.3

Table 2

Vehicle dynamics parameters.

Parameter/Unit	Value
Car mass m/kg	1100
Inertia around z axis Iz/(kg·m2)	1859
Inertia around x axis Ix/(kg·m2)	539
Distance of front wheels from centre of gravity a/m	1.35
Distance of rear wheels from centre of gravity b/m	1.23
Mass center height h/m	0.71
Lateral stiffness of front and rear axles Cf, Cr/(N·rad−1)	56,900, 52,700

Table 3

Model Predictive Control parameters.

Parameter/ Unit	Value	Description
Predictive horizon $N_p$	20	The number of future steps considered for prediction in the MPC optimization.
Control horizon $N_c$	10	The number of steps over which the control input is optimized.
Sampling time/(s)	0.05	The interval at which the control inputs are updated.
Simulation time/(s)	20	The total duration of the simulation for each test scenario.
Weight matrix Q	Diag [300; 700]	Penalizes deviations from the desired trajectory.
Weight matrix R	[500]	Penalizes excessive control efforts.
Relaxation factor ε	1000	A typical value used in this study, balancing flexibility in control adjustments while ensuring stability and performance.

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