Enhanced Anti-Rollover Control for Commercial Vehicles Under Dynamic Lateral Interferences

Abstract

Commercial vehicles frequently experience lateral interferences, such as crosswinds or side slopes, during extreme maneuvers like emergency steering and high-speed driving due to their high centroid. These interferences reduce vehicle stability and increase the risk of rollover. Therefore, this study takes a bus as the carrier and designs an anti-rollover control strategy based on mixed-sensitivity and robust $H_{\infty }$ controller. Specifically, a 7-DOF vehicle dynamics model is introduced, and the factors influencing vehicle rollover are analyzed. Based on this, to minimize excessive intervention in the vehicle’s dynamic characteristics, the lateral velocity, roll angle, and roll rate are recorded at the vehicle’s rollover threshold as desired values. The lateral load transfer rate (LTR) is chosen as the evaluation index, and the required additional yaw moment is determined and distributed to the wheels for anti-rollover control. Furthermore, to verify the effectiveness of the proposed anti-rollover control strategy, a co-simulation platform based on MATLAB/Simulink and TruckSim is developed. Various dynamic lateral interferences (side winds with different changing trends and wind speeds) are introduced, and the fishhook and J-turn maneuvers are selected to analyze and compare the proposed control strategy with a fuzzy logic algorithm. The results indicate that the maximum LTR of the vehicle is reduced by 0.11. Additionally, the lateral acceleration and yaw rate in the steady state are reduced by more than 1.8 m/s² and 15°, respectively, enhancing the vehicle’s lateral stability.

1. Introduction

With the rapid development of transportation, commercial vehicles play an indispensable role in modern society. Due to the higher centroid and large unsprung mass, commercial vehicles overturn more easily, especially in extreme maneuvers such as high-speed driving, emergency steering, or avoiding obstacles. As is well known, rollover will not only seriously threaten the safety of drivers and passengers but also cause irreparable social impacts. Statistical data show that more than 520,000 traffic accidents occurred in China in the past three years, of which about 19.2% were caused by vehicle rollover, and buses and heavy commercial vehicles accounted for nearly 50% of these accidents [1]. Therefore, how to effectively prevent commercial vehicle rollover and ensure driving safety is significantly important.

To ensure driving safety, anti-rollover control has gradually become the focus of research. In general, the anti-rollover control can be divided into two parts: The upper layer controller decides the expected control effects according to the driver’s operations and the vehicle dynamics states; the lower controller monitors the vehicle status based on the rollover evaluation indexes, utilizes active steering, active suspension, and differential braking to transform the expected control effects into the target tire forces, and improves the vehicle dynamics states by adjusting the tire force. Previous studies have extensively explored how to determine the expected control effects. Fu, Zhou, Fu, Zhang et al. decided on the expected control effect based on the model predictive control (MPC) algorithm, they proposed an active suspension control method based on distributed MPC and took the improved lateral load transfer rate as the rollover evaluation index. The simulation results show that the proposed anti-rollover control strategy can effectively inhibit vehicle rollover [2]. Zhou, Yu, Li et al. adopted adaptive MPC with time-varying weights and constraints to ensure vehicle stability and realized the anti-rollover control by engine torque limitation, differential braking, and active steering [3]. To enhance the lateral and roll stabilities under extreme driving maneuvers, Fu, Wan, Liu et al. proposed an anti-rollover control method combining adaptive sliding mode controller (ASMC) and MPC to improve the roll stability of commercial vehicles on rough roads by using a nonlinear 14-DOF vehicle model and a modified LuGre tire model [4]. Zhang, Zhao, Luan et al. studied the rollover risk of three-axle commercial vehicles under extreme driving maneuvers and proposed an integrated anti-rollover control strategy by steering–braking coordinated with multi-coupled DOF method and time-varying non-linear MPC. Simulation and hardware-in-the-loop (HIL) results confirm its effectiveness [5]. Chen, Zheng, Zhang et al. designed anti-rollover control of electromagnetic active suspension based on a linear quadratic regulator (LQR). Simulation results show that this method can significantly improve vehicle stability and reduce rollover risk under extreme working maneuvers [6]. In order to improve the anti-rollover performance of counterweight forklift trucks, Xia, Li, Tang et al. established an anti-rollover Takagi–Sugeno (TS) fuzzy system. Simulation and real vehicle tests show that the control strategy can effectively and quickly reduce the impact after actuator failure, so as to improve the safety and reliability of forklift trucks [7]. Sun, Nair et al. combined a genetic algorithm with an optimization function to identify the key parameters of the 10-DOF vehicle dynamics model and designed an anti-rollover controller for commercial vehicles based on the LQR method; the effectiveness of the proposed strategy is verified by simulation [8,9]. For improving the rollover stability of heavy trucks during lifting and unloading, Fu, Hu, Guo et al. took the gravity ratio of wheels with lower carrying capacity as the rollover evaluation index and proposed an anti-rollover control method for active suspension based on sliding mode control, and the results show that the proposed anti-rollover control method has better anti-rollover ability and vehicle stability [10]. To mitigate the effects of crosswind on vehicle driving, Yamamoto, Ejiri, and Oya proposed a rollover prevention controller using front and rear wheel steering that adjusts to variations in vehicle velocity. The method tracks lateral acceleration to reduce rollover risk, and numerical simulations confirm the controller’s effectiveness in enhancing vehicle stability under crosswind interferences [11]. Zhou, Qiu, and He developed a 3-DOF yaw-plane model to improve the lateral stability of the center-axle-trailer (TCAT) under crosswind interferences. Through numerical simulations. The simulation results indicate that without the intervention of the direct yaw moment control (DYC) system, the TCAT exhibits unstable yaw and lateral behavior even with a crosswind speed as low as 3 m/s. The articulation angle and yaw rate curves oscillate violently, reaching a peak value of 0.5 rad/s before eventually diverging, which signals a loss of stability. With the DYC system engaged, however, the vehicle remains stable even at a crosswind speed of 12 m/s, with the yaw rate peaking at 0.28 rad/s before stabilizing. This demonstrates that the DYC system effectively enhances the vehicle’s stability [12].

After literature research, it was found that previous studies about anti-rollover control of commercial vehicles have made remarkable progress in control methods and rollover evaluation indexes. However, there are relatively few studies that focus on anti-rollover control considering lateral interferences. In fact, crosswinds and side slopes have a significant impact on high-speed commercial vehicles, as these lateral interferences affect vehicle stability and increase the risk of rollover. For example, references [11,12] discussed these issues. However, these studies do not comprehensively address the combined effects of various lateral interferences on vehicle stability. Therefore, further research into the impact of lateral interferences and the development of anti-rollover control strategies that specifically incorporate these effects are essential steps to ensure the driving safety of commercial vehicles.

To solve these issues, this study takes a bus as the carrier and builds an anti-rollover control strategy for suppressing the adverse effects of lateral interferences on stable driving of commercial vehicles. Specifically, a 7-DOF vehicle dynamics model is introduced and the factors affecting vehicle rollover are explored. Second, the lateral load transfer rate (LTR) is selected as an evaluation index to judge the rollover risk of the vehicle. Then, based on the simplified 3-DOF vehicle dynamics model that considers the lateral, yaw, and roll motions of the vehicle body, a mixed sensitivity and robust $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ controller is established to decide the expected additional yaw moment for stable driving to reduce the rollover risk. Furthermore, differential braking is used for realizing the expected vehicle dynamics states. Finally, the fuzzy logic algorithm is used as a comparison, and the fishhook maneuver and the J-turn maneuvers are selected as typical maneuvers for analyzing the effectiveness of the proposed anti-rollover control under lateral interference such as crosswind based on MATLAB/Simulink and TruckSim.

The main contributions of this study are as follows:

• To achieve anti-rollover control while minimizing excessive intervention in the vehicle’s dynamic characteristics, the lateral velocity, roll angle, and roll rate at the vehicle’s rollover threshold were recorded as target values based on the simulation results of the 7-DOF vehicle dynamics model.
• The adverse effects of lateral interferences on stable driving of high-speed commercial vehicles are analyzed, and an anti-rollover control strategy considering the lateral interferences is established based on the mixed sensitivity and algorithm. Various dynamic lateral interferences (side winds with different changing trends and wind speeds) are introduced, and fuzzy control is used as a control comparison to verify the effectiveness of the anti-rollover control strategy under multiple operating maneuvers.

The rest of this manuscript is arranged as follows: Section 2 analyzes the main factors that adversely affect the vehicle rollover. An anti-rollover control strategy based on $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ algorithm is established in Section 3. The co-simulation platform is built and the co-simulation analyses are carried out in Section 4. The conclusions are made in Section 5.

2. Analysis of Vehicle Rollover

A non-tripping rollover occurs when the vertical force on the wheels becomes zero due to the vehicle’s lateral acceleration exceeding its critical limit. To investigate a reasonable and effective anti-rollover control strategy, it is necessary to analyze the lateral stability of the vehicle. Hence, the lateral, yaw, and roll motions of the vehicle body and the rotation of four wheels are focused, and the factors affecting the lateral stability of the vehicle are analyzed based on a 7-DOF vehicle dynamics model in this section.

2.1. Vehicle Dynamics Model

The principle of the 7-DOF vehicle dynamics model is shown in Figure 1. It includes the lateral motion of the body along the y-axis, the yaw motion of the body around the z-axis, the roll motion of the body around the x-axis, and the rotation of the four wheels. The differential equations can be obtained by decomposing each force along the coordinate axis.

Lateral motion of the vehicle body:

$$m({\dot{v}}_y+v_x{\omega }_r)-m_s{h}_s\ddot{\phi }=(F_{yfl}+F_{yfr})\cos {\delta }_f+F_{yrl}+F_{yrr}+(F_{xfl}+F_{xfr})\sin {\delta }_f$$

(1)

Yaw of the vehicle body:

$$\begin{array}{l}{\dot{\omega }}_r{I}_z=F_{xfl}(a\sin {\delta }_f-{\displaystyle \frac{B_1}{2}}\cos {\delta }_f)+F_{xfr}(a\sin {\delta }_f+{\displaystyle \frac{B_2}{2}}\cos {\delta }_f)-(F_{xrl}-F_{xrr}){\displaystyle \frac{B_2}{2}}\\ +F_{yfl}(a\cos {\delta }_f+{\displaystyle \frac{B_2}{2}}\sin {\delta }_f)+F_{yfr}(a\cos {\delta }_f-{\displaystyle \frac{B_2}{2}}\sin {\delta }_f)-(F_{yrl}+F_{yrr})b\end{array}$$

(2)

Roll of the vehicle body:

$$I_x\ddot{\phi }-I_{xz}\ddot{\phi }=m_s{h}_s(a_y\cos \phi +g\sin \phi )-(C_{\phi }\dot{\phi }+K_{\phi }\phi )$$

(3)

and the rotation of the wheels:

$$J_w{\dot{\omega }}_w=-F_{x}R-T_b+T_d$$

(4)

where $\hspace{0.17em}m\hspace{0.17em}$is the vehicle weight, $\hspace{0.17em}{m}_s\hspace{0.17em}$ is the suspension weight; ${\delta }_f$ is the front wheel angle; $\hspace{0.17em}{v}_x\hspace{0.17em}$,$\hspace{0.17em}{v}_y\hspace{0.17em}$ represent the longitudinal and lateral velocity, respectively; $\hspace{0.17em}{a}_y\hspace{0.17em}$ is the lateral acceleration; $\hspace{0.17em}{\omega }_r\hspace{0.17em}$ is the yaw rate; $\hspace{0.17em}{F}_{xfl},F_{xfr},F_{xrl},F_{xrr}\hspace{0.17em}$ are the longitudinal force on the left front, right front, left rear, and right rear wheels, respectively; $\hspace{0.17em}{F}_{yfl},F_{yfr},F_{yrl},F_{yrr}\hspace{0.17em}$ are the lateral force on the left front, right front, left rear, and right rear wheels, respectively; $\hspace{0.17em}{B}_1\hspace{0.17em}$ and $\hspace{0.17em}{B}_2\hspace{0.17em}$ are the wheelbase of the front and rear axles, respectively; $\hspace{0.17em}{h}_s\hspace{0.17em}$ is the distance from the roll center to the centroid; $\hspace{0.17em}a\hspace{0.17em}$ and $\hspace{0.17em}b\hspace{0.17em}$ are the distances from the centroid to the front and rear axles, respectively; $\hspace{0.17em}{I}_x\hspace{0.17em}$ and $\hspace{0.17em}{I}_z\hspace{0.17em}$ are the inertia moment of the vehicle body around the coordinate axes of $\hspace{0.17em}x\hspace{0.17em}$ and $\hspace{0.17em}z\hspace{0.17em}$; $\hspace{0.17em}\phi \hspace{0.17em}$ is the roll angle of the vehicle body; $\hspace{0.17em}g\hspace{0.17em}$ is the gravitational acceleration; $\hspace{0.17em}{K}_{\phi }\hspace{0.17em}$ is the equivalent roll stiffness; $\hspace{0.17em}{C}_{\phi }\hspace{0.17em}$ is the equivalent damping coefficient of suspension; $\hspace{0.17em}{I}_{xz}\hspace{0.17em}$ is the inertia product around the $\hspace{0.17em}x\hspace{0.17em}$ and $\hspace{0.17em}z\hspace{0.17em}$ axes, and the $\hspace{0.17em}{I}_{xz}\hspace{0.17em}$ is ignored due to the small value. Furthermore, $\hspace{0.17em}{T}_d\hspace{0.17em}$ is the driving torque; $\hspace{0.17em}{T}_b\hspace{0.17em}$ is the braking torque; $\hspace{0.17em}{F}_x\hspace{0.17em}$ is the longitudinal force on the wheels; $\hspace{0.17em}R\hspace{0.17em}$ is the rolling radius of the wheels; $\hspace{0.17em}{J}_w\hspace{0.17em}$ is the inertia moment of the wheels; ${\omega }_w$ is the angular speed of wheels.

The lateral force of the tire is not only the source of the vehicle steering but also the significant cause that affects the vehicle rollover. Assume that the cornering characteristic of the wheels are linear, and the lateral force in the front and rear wheels are as follows:

$$\left\{\begin{array}{l}{F}_f=k_f{\alpha }_f\\ F_r=k_r{\alpha }_r\end{array}\right.$$

(5)

where $\hspace{0.17em}{F}_f\hspace{0.17em}$ and $\hspace{0.17em}{F}_r\hspace{0.17em}$ are the lateral forces in the front and rear wheels, respectively; $\hspace{0.17em}{k}_f\hspace{0.17em}$ and $\hspace{0.17em}{k}_r\hspace{0.17em}$ are the lateral stiffness of the wheels; $\hspace{0.17em}{\alpha }_f\hspace{0.17em}$ and $\hspace{0.17em}{\alpha }_r\hspace{0.17em}$ are the slip angle of the wheels.

2.2. Model Validation

In order to verify the accuracy of the model, under the same simulation maneuvers (fishhook maneuvers, maximum steering wheel angle of 90°, vehicle speed of 60 km/h, and road surface adhesion coefficient of 0.85) and corner inputs, the outputs of center of mass lateral deflection, transverse pendulum angular velocity, and body camber from both the TruckSim model and 7-DOF dynamics model were compared, and the results are shown in Figure 2.

As can be seen from Figure 2b–d, the 7-DOF vehicle dynamics model has a good similarity with the TruckSim vehicle model in terms of lateral inclination angle, lateral acceleration, and transverse angular velocity curves. The accuracy of the established 7-DOF vehicle dynamics model is better, and it can be used for the study of the anti-rollover control of commercial vehicles in this paper.

2.3. Crucial State Variables of Vehicle Rollover

This study takes a coach as the carrier to analyze the main factors affecting the vehicle’s lateral stability, and the vehicle parameters are shown in

Table 1. Vehicle parameters.

Symbol	Value	Unit	Symbol	Value	Unit
$m$	7690	kg	$L$	4.490	m
$m_s$	6360	kg	$B_1$	2.030	m
$h_s$	0.642	m	$B_2$	1.863	m
$a$	3.102	m	$R$	0.510	m
$b$	1.388	m	$k_f$	150,000	N$\cdot$rad−1
$I_z$	30,782.4	kg·m2	$k_r$	350,000	N$\cdot$rad−1
$g$	9.8	m/s2	$C_{\phi }$	487,050	N$\cdot$m$\cdot$s−1
$I_x$	7695.6	kg$\cdot$m2	$K_{\phi }$	400,000	N$\cdot$m$\cdot$rad−1

.

The road adhesion coefficient and the initial velocity are set to 0.85 and 75 km/h according to the GB/T 6323-2014 [13] The front wheel angle and vehicle velocity are set as variables, and the simulation is carried out under fishhook maneuver based on the 7-DOF vehicle dynamics model.

2.3.1. Wheel Angle

The steering wheel angle is set to 60°, 90°, and 180°, and the front wheel angle is 2.4°, 3.6°, and 7.2°, correspondingly, and the simulation results are shown in Figure 3.

The above figures show the variations in lateral velocity, yaw rate, roll angle, and roll rate with steering wheel angle under the fishhook maneuver. As can be seen from Figure 3a,b, when the steering wheel angles are 60° and 90°, the lateral velocity and yaw rate fluctuate slightly in the initial stage, and then they tend to stabilize. When the steering wheel angle is 180°, the lateral velocity increases rapidly after 6 s and reaches a maximum of 20 km/h, and the yaw rate also reaches a peak of 25 deg/s.

As can be seen from Figure 3c,d, when the steering wheel angle is 60° and 90°, the roll angle and roll rate fluctuate slightly and maintain a stable value. When the steering wheel angle is 180°, the roll rate sharply increases to 140 deg/s after 4 s, and the roll angle exceeds 75° in a short period, indicating that the vehicle overturned.

2.3.2. Vehicle Velocity

The vehicle velocity is set to 40 km/h, 80 km/h, and 120 km/h, respectively. The simulation results under the steering wheel angle of 90° are shown in Figure 4.

The above figures show the variations in lateral velocity, yaw rate, roll angle, and roll rate with vehicle velocity under the fishhook maneuver. As can be seen from Figure 4a,b, when the vehicle velocity is at low speeds (40 km/h and 80 km/h), the lateral velocity and yaw rate responses are stable and fluctuate less. When the vehicle velocity is at high speed (120 km/h), the vehicle exhibits more obvious lateral and yaw responses, and the lateral velocity and yaw rate change significantly. The lateral velocity increases rapidly after 2.3 s and reaches a maximum of 40 km/h, and the yaw rate also reaches a peak of 24 deg/s.

As can be seen from Figure 4c,d, when the vehicle velocity is at low speed (40 km/h and 80 km/h), the roll angle and roll rate fluctuate slightly and maintain a low value in the steady state. When the vehicle velocity is at high speed (120 km/h), the roll rate reaches the peak of 120 deg/s after 5.8 s, and the roll angle exceeds 77° in a short time; the lateral stability of the vehicle deteriorates, indicating that the vehicle overturned.

It can be seen from the results that the variables such as lateral acceleration, yaw rate, roll angle, and roll rate all increased with the vehicle velocity and the front wheel angle, where the lateral acceleration reflects the change in centrifugal force on the vehicle, the roll angle reflects the inclination of the vehicle, and the yaw rate indicates the steering severity of the vehicle. Therefore, the above variables play crucial roles in characterizing the dynamic states of the vehicle and affect the vehicle’s stability profoundly.

3. Anti-Rollover Control Strategy

The anti-rollover control architecture proposed in this study is shown in Figure 5. In general, the control architecture is divided into four modules: expected control effects decision, vehicle dynamics monitoring, additional yaw moment calculation, and anti-rollover control. The expected control effects decision module takes the vehicle velocity and front wheel angle as inputs and determines the desired lateral velocity, the roll angle, and the roll rate based on a 3-DOF vehicle dynamics model, considering the lateral, yaw, and roll motion of the vehicle body. The vehicle dynamics monitoring module determines the possibility of vehicle rollover by calculating the lateral load transfer rate (LTR) of suspension mass. When the high possibility of a rollover is identified, the additional yaw moment decision module takes errors of the lateral velocity, the yaw rate, the roll angle, and rate of the vehicle, the front wheel angle and velocity as inputs, and calculates the additional yaw moment based on the mixed sensitivity $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ controller to make the vehicle meet the expected control effects. Furthermore, the anti-rollover control module reasonably distributes the additional yaw moment to each wheel by differential braking according to the vehicle states and finally realizes the anti-rollover control.

3.1. Reference Model

To simplify the calculation burden on the premise of reflecting the vehicle dynamics, the wheel’s rotation is ignored based on Formulas (1)~(4), and a 3-DOF vehicle dynamics model, which considers the lateral, yaw, and roll of the vehicle body, is further established for controller designing. The following assumptions were made:

• The influence of unsprung mass during the movement is neglected.
• The effects of the wheel toe angle and steering trapezoid are neglected, and it is assumed that the left and right wheel angles are the same.
• The cornering characteristics of non-steering wheels are neglected, and it is assumed that the cornering characteristics of all steering wheels are the same.

$$\left\{\begin{array}{l}m({\dot{v}}_y+v_x{\omega }_r)-m_s{h}_s\ddot{\phi }=F_f\cos \delta +F_r\hfill \\ I_z{\dot{\omega }}_r-I_{xz}\ddot{\phi }=a{F}_f\cos \delta -b{F}_r\hfill \\ I_x\ddot{\phi }-I_{xz}\ddot{\phi }=m_s{h}_s(a_y\cos \phi +g\sin \phi )-(C_{\phi }\dot{\phi }+K_{\phi }\phi )\hfill \end{array}\right.$$

(6)

Based on the third assumption, set $\hspace{0.17em}{\alpha }_r$ = 0. The meanings of the symbols in the formula are consistent with the previous definition.

According to the analysis in Section 2, the lateral velocity $\hspace{0.17em}{v}_y\hspace{0.17em}$, the yaw rate $\hspace{0.17em}{\omega }_r\hspace{0.17em}$, the roll angle $\hspace{0.17em}\phi \hspace{0.17em}$, and the roll rate $\hspace{0.17em}\dot{\phi }\hspace{0.17em}$ of the vehicle body are the key parameters that significantly affect the dynamics of the vehicle. Therefore, the above parameters are selected as the state variables of the reference model, and the lateral velocity $\hspace{0.17em}{v}_y\hspace{0.17em}$, the yaw rate $\hspace{0.17em}{\omega }_r\hspace{0.17em}$and the roll angle $\hspace{0.17em}\phi \hspace{0.17em}$ are selected as the output variables. Based on the anti-rollover control principle, the additional yaw moment $\hspace{0.17em}\Delta M_z\hspace{0.17em}$ is selected as the control variable.

In addition, lateral interferences usually act on the vehicle in the form of a crosswind or side slope, and the vehicle dynamics will be affected due to the lateral force of steering wheels altered by the interferences. When the cornering characteristics of non-steering wheels are neglected, the lateral interferences will change the sideslip angle by affecting the lateral force of the wheels, and then, the steering wheel angles are altered. Therefore, the steering wheel angle $\hspace{0.17em}{\delta }_f\hspace{0.17em}$ was selected as the interference input for the reference model.

Accordingly, the linearized reference model is shown in Equation (7):

$$\left\{\begin{array}{l}{\dot{x}}_p=A_p{x}_p+B_{1p}w+B_{2p}u\hfill \\ y=C_{p}x+D_{1p}w+D_{2p}u\hfill \end{array}\right.$$

(7)

where

• $X={\left[\begin{array}{cccc}{v}_{\mathrm{y}}& {\omega }_r& \dot{\phi }& \phi \end{array}\right]}^T$,
• $A_p=\left[\begin{array}{cccc}{\displaystyle \frac{(k_f+k_r)}{m{v}_x}}+{\displaystyle \frac{m_s{h}_s(k_f+k_r)}{m^2{v}_x{I}_x}}& {\displaystyle \frac{(a{k}_f-b{k}_r)}{m{v}_x}}-vx+{\displaystyle \frac{m_s{h}_s(k_f+k_r)}{m^2{v}_x{I}_x}}& -C_{\phi }{\displaystyle \frac{m_s{h}_s}{m{I}_x}}& (m_{s}g{h}_s-K_{\phi }\phi ){\displaystyle \frac{m_s{h}_s}{m{I}_x}}\\ {\displaystyle \frac{(a{k}_f-b{k}_r)}{v_x{I}_z}}& {\displaystyle \frac{(a^2{k}_f+b^2{k}_r)}{v_x{I}_z}}& 0& 0\\ {\displaystyle \frac{m_s{h}_s(k_f+k_r)}{m{v}_x{I}_x}}& {\displaystyle \frac{m_s{h}_s(a{k}_f-b{k}_r)}{m{v}_x{I}_x}}& -{\displaystyle \frac{C_{\phi }}{I_x}}& {\displaystyle \frac{m_{s}g{h}_s-K_{\phi }\phi }{I_x}}\\ 0& 0& 1& 0\end{array}\right]$
• $B_{1p}=\left[\begin{array}{c}-{\displaystyle \frac{k_f}{m}}+{\displaystyle \frac{-m_s{}^{2}h_s{}^{2}k_f}{m^2{I}_x}}\\ -{\displaystyle \frac{a{k}_f}{I_z}}\\ {\displaystyle \frac{-m_s{h}_s{k}_f}{m{I}_x}}\\ 0\end{array}\right]$,$B_{2p}=\left[\begin{array}{c}0\\ {\displaystyle \frac{1}{I_z}}\\ 0\\ 0\end{array}\right]$,$C_p=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$, $D_{1p}=D_{2p}={\left[0\right]}_{3\times 1}$.

3.2. Expected Control Effects Decision

3.2.1. Yaw Rate

When the velocity is fixed and the cornering characteristics of the wheel are linear, the yaw rate is approximately linear to the front wheel angle. Therefore, the reference of the yaw rate could be [14] as follows:

$${\omega }_{rd}={\displaystyle \frac{v_x{\delta }_f}{L(1+K{v}_x{}^2)}}$$

(8)

where $\hspace{0.17em}L\hspace{0.17em}$ is the wheelbase and $\hspace{0.17em}K\hspace{0.17em}$ is the stability factor. In addition, the vehicle’s motion is also limited by the road adhesion maneuvers, for lateral movement:

$${\omega }_{r,\max }={\displaystyle \frac{\mu g}{v_x}}$$

(9)

where $\hspace{0.17em}\mu \hspace{0.17em}$ is the road adhesion coefficient.

In summary, the yaw velocity reference value is as follows:

$${\omega }_{rd}{}^{\ast }=\min \left\{\left|{\displaystyle \frac{v_x{\delta }_f}{L(1+K{v}_x{}^2)}}\right|,\left|{\displaystyle \frac{\mu g}{v_x}}\right|\right\}\mathrm{sign}({\delta }_f)$$

(10)

3.2.2. Lateral Velocity, Roll Angle, and Roll Rate

To ensure anti-rollover control without affecting the vehicle’s motion to the extent, the lateral velocity, roll angle, and roll rate of the vehicle under the critical state of rollover are taken as references according to the simulation results in Section 2.

Specifically, we set the simulation conditions as fishhook and the J-turn conditions (steering angle and vehicle velocity are the same as in Section 4.2.1 and Section 4.3.1; no velocity wind is acted, and other maneuvers remain unchanged), record the critical moment (the moment when the sudden change in the roll angle occurs in the simulation results), when the vehicle is about to roll over under these two conditions, and use the lateral velocity, roll angle, and roll rate at this time as our control effects required for the controller design.

$$\left\{\begin{array}{l}{v}_y{}^{\ast }=3.8\hspace{0.33em}\mathrm{km}/\mathrm{h}\hfill \\ {\phi }^{\ast }\hspace{0.33em}\hspace{0.17em}=4.2\hspace{0.33em}\mathrm{deg}\hfill \\ {\dot{\phi }}^{\ast }\hspace{0.33em}\hspace{0.17em}=6.25\hspace{0.33em}\mathrm{deg}/s\hfill \end{array}\right.$$

Using these reference values, the control system is calibrated to maintain the roll angle within a safe range, minimizing unnecessary interventions. The aim of this approach is to achieve anti-rollover control while minimizing the impact on the vehicle’s dynamic characteristics.

In summary, the expected control effect decision is$\hspace{0.17em}r={\left(\begin{array}{cccc}{{\omega }_{rd}}^{\ast }& {v_y}^{\ast }& {\phi }^{\ast }& {\dot{\phi }}^{\ast }\end{array}\right)}^T$.

3.3. Vehicle Dynamics Monitoring

The sprung mass will be transferred laterally when the vehicle rolls over, so the LTR is used for monitoring the vehicle dynamics in this study. The LTR is defined as follows [15,16]:

$$LTR={\displaystyle \frac{F_{zr}-F_{zl}}{F_{zl}+F_{zr}}}$$

(11)

where $F_{zl}$ and $F_{zr}$ are the vertical loads of the left and right wheels, respectively.

According to (11), the range of LTR is [−1, 1]. Previous studies show that the critical value of LTR is usually ±0.8 [17]. When the LTR is greater than 0.8, the vehicle has a high risk of rollover to the right; when the LTR is less than −0.8, the vehicle has the risk of rollover to the left; when the LTR is within the range of (−0.8, 0.8), there is basically no risk of rollover.

3.4. Additional Yaw Moment Decision

To improve the robustness of the vehicle’s motion state under lateral interferences and prevent vehicle rollover, an additional yaw moment decision module based on $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ algorithm is designed. The $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ algorithm is a type of robust control. It introduces the generalized system and takes both controlled objectives $\hspace{0.17em}{G}_0(s)\hspace{0.17em}$ and performance targets of the traditional feedback system as generalized controlled objectives $\hspace{0.17em}{G}_p(s)\hspace{0.17em}$. Moreover, the controller $\hspace{0.17em}K(s)\hspace{0.17em}$ is designed to stabilize the generalized system which is composed of $\hspace{0.17em}{G}_p(s)\hspace{0.17em}$ and $\hspace{0.17em}K(s)\hspace{0.17em}$, and the $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ norm of the transfer function from the interference $\hspace{0.17em}W(s)\hspace{0.17em}$ to controlled outputs $\hspace{0.17em}Y(s)\hspace{0.17em}$ in the generalized system meets the requirements.

3.4.1. Generalized System

Mixed sensitivity $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ control achieves a balance between system performances (such as tracking and anti-jamming) and stabilities (such as model uncertainty) by minimizing the mixed sensitivity function, and this method allows the dynamic characteristics of the system to be considered and analyzed more intuitively. Hence, this section built the generalized system based on the mixed sensitivity method and outputs feedback according to the reference model established in Section 3.1, as shown in Figure 6.

The generalized system, shown in Figure 6, contains the controlled objectives $\hspace{0.17em}{G}_0(s)\hspace{0.17em}$ described by the (7) and the weighting function $\hspace{0.17em}W(s)\hspace{0.17em}$ for evaluating the system’s performance. According to Figure 6, the state space of the generalized system is as follows:

$$\Sigma :\left\{\begin{array}{c}\dot{x}=Ax+B1W+B2u\\ \begin{array}{l}Z=C_{1}x+D_{11}W+D_{12}u\\ Y=C_{2}x+D_{21}W+D_{22}u\end{array}\end{array}\right.$$

(12)

where $\hspace{0.17em}W={\left(\begin{array}{cc}w& r\end{array}\right)}^T\hspace{0.17em}$ is the generalized interference, and it contains interference $w={\delta }_f$ and expected control effects $r={\left(\begin{array}{cccc}{{\omega }_{rd}}^{\ast }& {v_y}^{\ast }& {\phi }^{\ast }& {\dot{\phi }}^{\ast }\end{array}\right)}^T$, $u=\Delta M_z$ is the generalized control variable, $x={\left(\begin{array}{cccc}{v}_{\mathrm{y}}& {\omega }_r& \dot{\phi }& \phi \end{array}\right)}^T$ is the generalized states, and the states equation is as follows:

$$\dot{x}=A_{p}x+\left(\begin{array}{cc}{B}_{1p}& 0\end{array}\right)\left(\begin{array}{c}w\\ r\end{array}\right)+B_{2p}u$$

(13)

The generalized outputs $\hspace{0.17em}Z\hspace{0.17em}$ is as follows:

$$Z=\left(\begin{array}{c}z1\\ z2\\ z3\end{array}\right)=\left(\begin{array}{c}e\\ u\\ y\end{array}\right)=\left(\begin{array}{c}-Cp\\ 0\\ Cp\end{array}\right)xp+\left(\begin{array}{cc}0& 1\\ 0& 0\\ 0& 0\end{array}\right)\left(\begin{array}{c}w\\ r\end{array}\right)+\left(\begin{array}{l}0\hfill \\ 1\hfill \\ 0\hfill \end{array}\right)u$$

(14)

The measurement output $\hspace{0.17em}Y\hspace{0.17em}$ is as follows:

$$Y=e=-C_p\cdot xp+\left(\begin{array}{cc}0& 1\end{array}\right)\cdot \left(\begin{array}{c}w\\ r\end{array}\right)+0\cdot u$$

(15)

where

$$\begin{gathered} A=\left[\begin{array}{cccc}{\displaystyle \frac{(k_f+k_r)}{m{v}_x}}+{\displaystyle \frac{m_s{h}_s(k_f+k_r)}{m^2{v}_x{I}_x}}& {\displaystyle \frac{(a{k}_f-b{k}_r)}{m{v}_x}}-vx+{\displaystyle \frac{m_s{h}_s(k_f+k_r)}{m^2{v}_x{I}_x}}& -C_{\phi }{\displaystyle \frac{m_s{h}_s}{m{I}_x}}& (m_{s}g{h}_s-K_{\phi }\phi ){\displaystyle \frac{m_s{h}_s}{m{I}_x}}\\ {\displaystyle \frac{(a{k}_f-b{k}_r)}{v_x{I}_z}}& {\displaystyle \frac{(a^2{k}_f+b^2{k}_r)}{v_x{I}_z}}& 0& 0\\ {\displaystyle \frac{m_s{h}_s(k_f+k_r)}{m{v}_x{I}_x}}& {\displaystyle \frac{m_s{h}_s(a{k}_f-b{k}_r)}{m{v}_x{I}_x}}& -{\displaystyle \frac{C_{\phi }}{I_x}}& {\displaystyle \frac{m_{s}g{h}_s-K_{\phi }\phi }{I_x}}\\ 0& 0& 1& 0\end{array}\right] \\ {B}_1=\left[\begin{array}{cccc}-{\displaystyle \frac{k_f}{m}}+{\displaystyle \frac{-m_s{}^{2}h_s{}^{2}k_f}{m^2{I}_x}}& 0& 0& 0\\ -{\displaystyle \frac{a{k}_f}{I_z}}& 0& 0& 0\\ {\displaystyle \frac{-m_s{h}_s{k}_f}{m{I}_x}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right], B_2=B_{2p}=\left[\begin{array}{c}0\\ {\displaystyle \frac{1}{I_z}}\\ 0\\ 0\end{array}\right], C_1=\left[\begin{array}{cccc}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right], \\ D11=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right], D_{12}=\left[\begin{array}{c}0\\ 0\\ 0\\ 1\\ 0\\ 0\\ 0\end{array}\right], C_2=C_p=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right], D_{21}=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],D_{22}=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]. \end{gathered}$$

the definition of the symbols is consistent with the previous texts.

3.4.2. Weighting Functions

The crucial technique for deducing the $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ controller based on mixed sensitivity method is to make the generalized system have closed-loop stability by properly selecting the weighting coefficients for sensitivity functions. Furthermore, the transfer function from $\hspace{0.17em}W(s)\hspace{0.17em}$ to $\hspace{0.17em}Z(s)\hspace{0.17em}$ should be satisfied with the following constraints:

$$\min {‖T_{ZW}(s)‖}_{\infty }=\min {‖\left[\begin{array}{c}{W}_S(s)S\\ W_R(s)R\\ W_T(s)T\end{array}\right]‖}_{\infty }<1$$

(16)

where $\hspace{0.17em}S={\displaystyle \frac{1}{1+GK}}\hspace{0.17em}$ is the sensitivity function, $\hspace{0.17em}R={\displaystyle \frac{K}{1+GK}}\hspace{0.17em}$ is the open-loop transfer function, $\hspace{0.17em}T={\displaystyle \frac{GK}{1+GK}}\hspace{0.17em}$ is the complement sensitivity function; $\hspace{0.17em}{W}_S(s)\hspace{0.17em}$, $\hspace{0.17em}{W}_R(s)\hspace{0.17em}$, and $\hspace{0.17em}{W}_T(s)\hspace{0.17em}$ are the weighting coefficients of the functions $\hspace{0.17em}S\hspace{0.17em}$, $\hspace{0.17em}R\hspace{0.17em}$, and $\hspace{0.17em}T\hspace{0.17em}$, respectively. The calibration principle of each weighting coefficient is as follows:

• The $\hspace{0.17em}S\hspace{0.17em}$ is a transfer function from reference inputs to tracking errors, which is used to characterize the tracking performance and the ability to resist the interferences. Appropriately reducing the$\hspace{0.17em}{W}_S(s)\hspace{0.17em}$ can suppress the influence of interferences on system performances [18].
• The $\hspace{0.17em}T\hspace{0.17em}$ is a transfer function from reference inputs to system outputs, which characterizes the robust stability of the system. Appropriately reducing the $\hspace{0.17em}{W}_T(s)\hspace{0.17em}$ can enhance the system stability under parameter changes and modeling errors. Due to $\hspace{0.17em}S+T=1\hspace{0.17em}$, it is necessary to reasonably select $\hspace{0.17em}{W}_S(s)\hspace{0.17em}$ and $\hspace{0.17em}{W}_T(s)\hspace{0.17em}$ based on practical requirements and make a compromise in frequency division [19].

Specifically, the tracking performance for references and the ability to suppress the interferences are mainly reflected in the low-frequency range. Hence, the higher the gain of the weighting coefficients at low frequency, the lower the gain of the function $\hspace{0.17em}S\hspace{0.17em}$, and the stronger the tracking performance and anti-interference ability of the system.

Hence, the $\hspace{0.17em}{W}_S(s)\hspace{0.17em}$ and $\hspace{0.17em}{W}_T(s)\hspace{0.17em}$ are obtained through repeated verification as follows:

$$\begin{gathered} W_S(s)=\left[\begin{array}{ccc}{W}_{svy}& & \\ & W_{swr}& \\ & & W_{s\phi }\end{array}\right]=\left[\begin{array}{ccc}{\displaystyle \frac{40}{1500s+2}}& & \\ & {\displaystyle \frac{1}{s+0.5}}& \\ & & {\displaystyle \frac{1}{s+0.5}}\end{array}\right] \\ {W}_T(s)=\left[\begin{array}{ccc}{W}_{Tvy}& & \\ & W_{Twr}& \\ & & W_{T\phi }\end{array}\right]=\left[\begin{array}{ccc}{\displaystyle \frac{s+0.4}{0.01s+0.01}}& & \\ & {\displaystyle \frac{50s+1200}{s+5000}}& \\ & & {\displaystyle \frac{50s+1200}{s+5000}}\end{array}\right] \end{gathered}$$

where $W_{svy}$, $W_{swr}$, and $W_{s\phi }$ are components of the sensitivity weighting function matrix, representing the sensitivity weighting terms for lateral velocity, yaw rate, and roll angle, respectively. $W_{Tvy}$, $W_{Twr}$, and $W_{T\phi }$ are components of the complementary sensitivity weighting function matrix, representing the complementary sensitivity weighting terms for lateral velocity, yaw rate, and roll angle, respectively.

• If the control inputs are too large, the signal amplitudes are limited by the function $\hspace{0.17em}{W}_R(s)\hspace{0.17em}$, which is usually a minor constant to guarantee the bandwidth. For this, the $\hspace{0.17em}{W}_R(s)\hspace{0.17em}$is set to 10⁻⁵ [20].

3.4.3. $H_{\infty }$ Controller Solving

This part is based on the output feedback method to solve the controller. For a linear time-invariant system, the sufficient maneuver for the existence of an output feedback $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ controller $\hspace{0.17em}u=Ky\hspace{0.17em}$ is that there exists a symmetric positive definite matrix $\hspace{0.17em}X\hspace{0.17em}$ and a matrix $\hspace{0.17em}Y\hspace{0.17em}$ (positive definite but asymmetric), such as:

$$\begin{array}{c}\min {\gamma }_0\\ s.t.\left[\begin{array}{cccc}AX+X{A}^T+B_{2}Y+Y^T{B_2}^T& X{B}_1& A^T+B_{2}Y& {C_1}^T\\ B_1{}^{T}X& -I& 0& {D_{11}}^T\\ A+Y^T{B_2}^T& 0& -X& 0\\ C_1& D_{11}& 0& -\gamma I\end{array}\right]<0\\ \left[\begin{array}{cc}X& I\\ I& Y\end{array}\right]>0\end{array}$$

(17)

where $\hspace{0.17em}\gamma \hspace{0.17em}$ is the $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ norm index and $\hspace{0.17em}{\gamma }_0\hspace{0.17em}$ is the optimal $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ norm index ($\gamma \ge {\gamma }_0$) [21].

For the generalized system described in Equation (12), after approximate linearization, it can be considered fixed at each sampling period. Therefore, according to (17), the MATLAB toolbox can be used to solve at each sampling period under certain maneuvers ($\hspace{0.17em}{v}_x\hspace{0.17em}$ and $\hspace{0.17em}{\delta }_f\hspace{0.17em}$). If the controller has a solution, the optimal $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ norm index $\hspace{0.17em}{\gamma }_0\hspace{0.17em}$ can be obtained. If there is no solution to this problem, the suboptimal $H_{\infty }$ norm index can be further solved [21,22].

Using $\hspace{0.17em}{\gamma }_0\hspace{0.17em}$ or $\hspace{0.17em}\gamma \hspace{0.17em}$ as the norm index for the $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ controller, feasible solutions $\hspace{0.17em}{X}^{*}\hspace{0.17em}$ and $\hspace{0.17em}{Y}^{*}\hspace{0.17em}$ of the matrix inequalities can be obtained based on the bounded real lemma [21,22,23,24]. Accordingly, the control variable of the generalized system is $\hspace{0.17em}u=Y^{*}{(X^{*})}^{-1}y\hspace{0.17em}$, which is the additional yaw moment that needs to be applied to the body of the car to avoid rolling over.

3.5. Anti-Rollover Control

The principle of differential braking is as follows: When the vehicle is in motion and a correction of the vehicle’s posture is needed, the control system outputs a braking pressure signal. This signal activates the brake force distribution system, which applies braking control to different wheels of the vehicle accordingly. Due to this braking action, an additional yaw moment is generated, which significantly influences the vehicle’s posture. If this additional yaw moment opposes the roll moment, it counteracts the roll tendency, helping to reduce vehicle body roll and induce understeer. This contributes positively to enhancing the vehicle’s lateral stability. Therefore, differential braking is used in this paper to enhance the lateral stability of the vehicle.

After the additional yaw moment is obtained by solving the $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ controller, it is necessary to further consider how to regulate the yaw moment by reconstructing the braking force. In fact, the magnitude and direction of the yaw moment are affected by the brakes, and the relationship between wheel braking force and vehicle yaw moment is shown in Figure 7.

As shown in Figure 7, when braking is applied solely to the front outer wheel, the resulting yaw moment is beneficial for vehicle stability, increasing the understeer tendency. Its direction is consistent with the yaw moment caused by the reduction in lateral force, and both directions are opposite to the vehicle’s turning direction. Conversely, when braking is applied only to the rear outer wheel, the resulting yaw moment exacerbates vehicle instability and increases the oversteer tendency. When braking is applied individually to the other two wheels, the curves in the figure indicate that the effects on the vehicle are not significant in either case [25,26].

Therefore, in this study, only the front outer wheel is selected for braking control during the anti-rollover process. When the vehicle is at risk of rolling over to the left, the left front wheel should be braked to generate an understeer tendency for stable driving. Similarly, when the vehicle is at risk of rolling over to the right, the right front wheel should be braked. The wheel brake distribution scheme is shown in

Table 2. Wheel brake distribution scheme.

Steering	LTR	Vehicle Status	Brake Wheel	Additional Yaw Moment
Left steering	LTR $\ge$ 0.8	Rollover to the right	Right front wheel	Clockwise
Right steering	LTR $\le$ −0.8	Rollover to the left	Left front wheel	Counterclockwise

.

The relationship between wheel braking torque and vehicle yaw moment is as follows:

$$T={\displaystyle \frac{2R{M}_z}{B_1}}$$

(18)

where $\hspace{0.17em}T\hspace{0.17em}$ is the braking torque of a single wheel. The additional yaw moment determined by the $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ controller is converted into single-wheel braking torque. Based on this, the target braking pressure of the wheel can be obtained according to Figure 8; the obtained wheel cylinder pressure represents the total braking force, which is then distributed to either the left front wheel or the right front wheel.

4. Simulation Analyses

To verify the effectiveness of the proposed anti-rollover control method, a co-simulation platform based on MATLAB/Simulink and TruckSim is built in this section, as shown in Figure 9.

The maximum velocity of crosswind is set as 20 m/s to simulate the lateral interference when the vehicle turns according to GB/T 41722-2022 [27]. In accordance with the requirements of GB/T 6323-2014 [13], the steering wheel J-turn and fishhook maneuvers were selected as simulation scenarios to evaluate the effectiveness of the vehicle’s anti-rollover control capabilities. The J-turn maneuver, which is characterized by a sudden steering input in one direction, tests the vehicle’s lateral stability and its response to abrupt directional changes. In contrast, the fishhook maneuver involves rapid, successive sharp turns that generate high lateral acceleration and increase the risk of rollover. These maneuvers are suitable for assessing the vehicle’s dynamic response under extreme maneuvers. Furthermore, the anti-rollover control based on a fuzzy logic algorithm is selected as a comparison to verify the feasibility of the proposed control method.

4.1. Scheme for Comparison: Anti-Rollover Control Based on Fuzzy Logic Algorithm

Fuzzy control does not rely on the precise dynamic model of the system; it is relatively simple, easy to implement, and has shown effective application in commercial vehicle stability control, making it a valuable reference. By establishing fuzzy rules based on experience, the fuzzy controller demonstrates strong robustness and adaptability under various operating maneuvers. Therefore, fuzzy control is selected as the benchmark for comparison. The error of vehicle body roll angle $\hspace{0.17em}E\hspace{0.17em}$ and its change rate $\hspace{0.17em}EC\hspace{0.17em}$ are treated as inputs and the cylinder pressure $\hspace{0.17em}U\hspace{0.17em}$ is defined as output to build the fuzzy logic controller, and the fuzzy discourse domain of the inputs and output are set as [−1, 1], and the fuzzy subset {NB, NM, NS, ZE, PS, PM, PB} is selected to represent the negative large, negative medium, negative small, zero, positive small, middle, and high. The fuzzy domain and membership function are shown in Figure 10.

For this, 49 fuzzy rules are generated, as shown in

Table 3. Fuzzy control rules.

EC	E
	NB	NM	NS	ZE	PS	PM	PB
NB	PB	PB	PB	PB	PM	ZO	ZO
NM	PB	PB	PB	PB	PM	ZO	ZO
NS	PM	PM	PM	PM	ZO	NS	NS
ZE	PM	PM	PS	ZO	NS	NM	NM
PS	PS	PS	ZO	NM	NM	NM	NM
PM	ZO	ZO	NM	NB	NB	NB	NB
PB	ZO	ZO	NM	NB	NB	NB	NB

.

4.2. Simulation Under Fishhook Maneuver

4.2.1. Without Lateral Interference

Assuming that the vehicle is driving at a velocity of 75 km/h on a horizontal road with an adhesion coefficient of 0.85, the maximum steering angle was set to 180° (the corresponding front wheel angle is 7.2°). The simulation under the fishhook maneuver is carried out, and the results are shown in Figure 11.

From Figure 11a, the steering wheel angle under the fishhook maneuver changes from left 180° to right 180° within 1 s. From Figure 11b,c, the LTR is limited to the range of ±0.8, and the lateral acceleration is also within the range of ±0.6 g attributed to the anti-rollover control based on the $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ algorithm. In comparison, the LTR is close to 1, and the maximum lateral acceleration exceeds 0.8 g based on the fuzzy logic algorithm. Therefore, the anti-rollover control method proposed in this study has a more significant control effect.

From Figure 11d,e, the maximum roll angle exceeds 75° without control, and the vehicle overturned at 5 s can be verified from the figure. Contrarily, the roll angle remains stable at around 4° under anti-rollover control. Furthermore, the steady state of yaw rate is decreased by 52% based on the$\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$algorithm, and the time to steady state is shorter. From Figure 11f, the trend of velocity is more gentle compared with the result which is controlled by the fuzzy logic algorithm; so the control method based on the $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ algorithm has less impact on the vehicle velocity.

4.2.2. Lateral Interference Intervention

A crosswind is acted on the vehicle to simulate the lateral interference, and other maneuvers remain unchanged, and the results are shown in Figure 12.

It can be seen from Figure 12a, to maximize the effects of crosswind on the vehicle under turning maneuver, the direction and velocity of the crosswind are set to vary with the steering wheel angle. When the steering wheel is turned left, a leftward crosswind is applied. The wind speed increases linearly over time, starting from 0 m/s and reaching its peak value of 20 m/s at 1.5 s. The increase in lateral acceleration when the vehicle turns right can cause it to roll to the right. Similarly, when the steering wheel is turned right, a rightward crosswind is applied, and the wind speed changes over time, reaching a peak of −20 m/s at 2.5 s.

It can be seen from Figure 12b,c, although the LTR and the lateral acceleration are disturbed by the crosswind, the LTR is limited to the range of ±0.9, and the lateral acceleration is also within the range of ±0.3 g by the anti-rollover control based on $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ algorithm. In comparison, the LTR reaches 1, and the maximum lateral acceleration exceeds 0.8 g based on the fuzzy logic algorithm.

From Figure 12d,e, the maximum roll angle exceeds 80°, and the vehicle overturned at 5.2 s under the effect of the fuzzy logic algorithm. On the contrary, the steady state of the roll angle and yaw rate is about 6° and 2°, respectively, and the $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ control can reduce the adverse effects caused by lateral interference so that the vehicle is not overturned. From Figure 12f, the vehicle velocity almost maintains a constant rate of descent, and the impact on velocity is less than fuzzy logic control.

4.3. Simulation Under J-Turn Maneuver

4.3.1. Without Lateral Interference

Assuming that the vehicle velocity is 80 km/h, the other maneuvers remain unchanged, the J-turn maneuver is simulated, and the results are shown in Figure 13.

From Figure 13a, the steering wheel angle under the J-turn maneuver changes from 0° to positive 180° within 1 s. From Figure 13b,c, the LTR is limited to the range of ±0.8, and the lateral acceleration is also within the range of ±0.7 g based on the $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ algorithm. In comparison, the LTR is close to 0.9 and the maximum lateral acceleration exceeds 0.8 g based on the fuzzy logic algorithm. Hence, the lateral stability of the vehicle is enhanced, attributed to the anti-rollover control proposed in this study.

Figure 13d,e shows that the maximum roll angle exceeds 80° without anti-rollover control and the vehicle overturned at 4.7 s, while the roll angle remains stable at around 4°attributed to the anti-rollover control. Furthermore, the steady state of the yaw rate is decreased by 60%, and the anti-rollover control is more effective based on $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ algorithm. From Figure 13f, the vehicle velocity keeps a constant rate of decline based on $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ algorithm.

4.3.2. Lateral Interference Intervention

A crosswind is acted on the vehicle to simulate the lateral interference, and other maneuvers remain unchanged, and the results are shown in Figure 14

The direction and velocity of the crosswind are set to vary with the steering wheel angle, as shown in Figure 14a. From Figure 14b,c, the LTR is limited to the range of ±0.8, and the lateral acceleration is also within the range of ±0.32 g attributed to the $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$algorithm. In comparison, the LTR is close to 1, and the maximum lateral acceleration exceeds 0.4 g based on the fuzzy logic algorithm.

From Figure 14d,e, the roll angle remains stable at around 3°~4° under anti-rollover control. Furthermore, although the vehicle does not overturn in the maneuver, the roll angle and yaw rate are lower and stable based on $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ algorithm. From Figure 14f, the vehicle velocity based on $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ control changes more gently, compared with the fuzzy logic algorithm. Hence, the effectiveness of the anti-rollover control proposed in this study is verified.

5. Conclusions

To the issue of rollover control for commercial vehicles under lateral interferences, the main factors that adversely affected the vehicle rollover are analyzed, an anti-rollover control strategy is built based on mixed sensitivities and $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$ algorithm, and the co-simulations based on MATLAB/Simulink and TruckSim are carried out in this study. The results show that:

• The variables such as lateral velocity, roll angle, yaw rate, and roll angle rate play a crucial role and profoundly affect vehicle stability, and they provide a basis for establishing an accurate and effective anti-rollover control strategy.
• The braking reconstruction is dynamically and flexibly adjusted based on the proposed anti-rollover control based on the mixed sensitivity and $\hspace{0.17em}{H}_{\infty }\hspace{0.17em}$algorithm, so the anti-rollover effect is more obvious. Under fishhook maneuvers and steering wheel angular step maneuver, the maximum LTR of the vehicle is reduced by more than 0.11, and the lateral acceleration and yaw rate in steady state are reduced by more than 1.8 m/s² and 15° compared with the anti-rollover control based on the fuzzy logic algorithm, and the lateral stability of the vehicle is improved.

In future research, we will focus on improving the accuracy of rollover evaluation indices and minimizing adverse effects on passenger comfort while ensuring the effectiveness of the anti-rollover control. We will also conduct hardware-in-the-loop (HIL) testing to verify the suitability of the control algorithms in practical applications. Additionally, we plan to integrate friction braking and regenerative braking in electro-mechanical braking (EMB) synergistic control to achieve anti-rollover through differential braking and enhance energy recovery efficiency.