Coordinated Control Method for Lateral Stability and Differential Power-Assisted Steering of In-Wheel Motor Drive Electric Vehicles

Abstract

In order to improve the lateral stability and handling performance of in-wheel motor drive electric vehicles, a coordinated control method considering lateral stability and differential power-assisted steering performance is proposed. A vehicle dynamics model with two degrees of freedom is established, in which the influence of system disturbance is considered. At the same time, the effect of differential torque on vehicle power-assisted steering control was analyzed, and a differential power-assisted steering control method of vehicle was designed based on referenced steering wheel torque. In response to the coupling relationship and dynamic game problem between the lateral stability control system and the differential power-assisted steering control system, a coordinated control system with a vehicle lateral stability module and a differential power-assisted steering module was designed based on the Nash equilibrium game theory, achieving comprehensive optimization of multi-objective performance. Corresponding simulation tests were conducted in the co-simulation vehicle model, and the results show that the proposed coordinated control method can achieve the differential power-assisted steering control function of vehicle while ensuring lateral stability.

1. Introduction

The development of the automobile industry scale and the accumulation of technical capabilities have greatly promoted the production capacity and popularity of the automobile industry. However, the accompanying problems of exhaust emissions and energy shortages have made alternative fuel vehicles, represented by electric vehicles, a hot issue in the industry [1,2,3]. In recent years, many countries have introduced a series of favorable policies at the macro policy level to demonstrate strong support for the innovative development of electric vehicle technology. These measures also reflect the importance of electric vehicles for healthy development and innovative growth of the automotive industry structure. The in-wheel motor drive electric vehicle is a special form of electric vehicle driving architecture [4,5,6]. Due to its advantages of precise and fast motor torque response, this driving form of electric vehicles has higher control mobility. In addition, the structural design of using in-wheel motors to directly drive the wheels can effectively reduce transmission losses and improve power transmission efficiency. At the same time, the torque of each in-wheel motor is also directly controllable. Thus, all of these driving advantages give the entire motion control system of electric vehicles a higher degree of freedom, thereby providing more design potential and room for improvement in vehicle motion control and stability [7,8,9].

With the unique driving advantages of in-wheel motor drive electric vehicles, it can provide a huge design space and optimization potential for the improvement in driving capability and stability of electric vehicles by coordinated control system design [10,11]. The electric-drive chassis of the vehicle has the integrated control advantages of multiple degrees of freedom, fast response, and electro-mechanical collaborative driving, greatly enhancing the lateral stability control effect [12,13]. At present, there have been many studies on vehicle lateral stability control, aimed at the real-time tracking of referenced values of the yaw rate and vehicle sideslip angle through vehicle steering and drive control in order to achieve the goal of vehicle stability control [14,15]. YU et al. designed a hierarchical control method for the direct yaw moment of distributed drive electric vehicles in which feed-forward control and feed-back control methods were designed in the upper level yaw stability controller and used to calculate the required yaw control moment, and in the lower level control, an optimal distribution method of wheel and tire force was designed to dynamically control the four-wheel tire force [16]. LU et al. designed an integrated control method for the vehicle lateral deviation and yaw rate to improve the ultimate control performance of vehicles during cornering maneuver. Using multi-variable feedback control theory, the controllability, limitations, and potential advantages of the yaw rate and center of mass lateral deviation were analyzed [17].

With the in-depth research and expansion of in-wheel motor drive electric vehicles, the concept of differential power-assisted steering technology has gradually been proposed and received widespread attention [18,19,20,21]. This driving method generates a torque difference around the kingpin by controlling the driving torque of the left and right in-wheel motors on the front axle of the vehicle in real time. By relying on the torque difference of the left and right in-wheel motors, the steering system tie rod is driven to achieve a steering assistance effect [22,23]. Chen et al. proposed a coordinated control strategy of differential speed and differential power-assisted steering for electric vehicles, which adjusts the weights of the two control systems through real-time coefficient allocation, achieving the coordination effect of yaw stability control and differential power-assisted steering control [24]. Jin et al. proposed a coordinated control method for power-assisted steering and stability based on vehicle state estimation, a stability controller was designed using a sliding mode control algorithm, and a power-assisted coefficient correction factor was designed in the power-assisted steering system. The dynamic adjustment was based on the vehicle state and actual operating conditions, achieving vehicle stability while also considering the portability of steering and driving [25].

For in-wheel motor drive electric vehicles, there is a phenomenon of mechanical coupling connection and electro-mechanical composite drive between the steering system assembly and the electric drive wheels [26,27]. In fact, the vehicle lateral stability control system relies on steering and drive control, and at the same time, the differential power-assisted steering system of the vehicle is also based on the steering motor and the left-right in-wheel motors of front wheels to achieve the coupled drive control. For the two types of systems mentioned above, different control objectives are achieved through parallel control of the same set of steering actuators. Therefore, it is inevitable that both control systems have varying degrees of coupling interference and dynamic game phenomena at the execution control level [28,29]. The multi-objective dynamic game control strategy and comprehensive optimization method contribute to the dynamic coordination of vehicle lateral stability control and differential power-assisted steering control objectives, thereby effectively improving the overall control performance of the vehicle, which has important research significance and value.

The lateral stability control and differential power-assisted steering control of electric vehicles, as well as the coordinated control of the two systems, are of great significance for improving the lateral stability and handling performance of vehicles. To address the above issues, and motivated by above analysis, a coordinated control method for lateral stability and differential power-assisted steering of electric vehicles based on game control theory was proposed. Aiming at the problem of lateral stability control, a vehicle lateral stability controller was designed based on a two degrees of freedom vehicle model and a fast power reaching law sliding mode control algorithm. A differential power-assisted steering control method was designed based on the referenced steering wheel torque to address the issue of differential power-assisted steering. On this basis, in order to balance the performance of lateral stability and power-assisted steering control, a coordinated control strategy for lateral stability and differential power-assisted steering was designed based on the Nash equilibrium game control method, achieving comprehensive optimization of the control objectives for lateral stability and differential power-assisted steering.

The rest of this paper is organized as follows. The vehicle dynamics modeling is presented in Section 2. The coordinated control method for lateral stability and differential power-assisted steering is analyzed in Section 3. The analysis and verification are provided in Section 4, and the conclusive remarks are presented in Section 5.

2. Vehicle Dynamics Modeling

In order to design the yaw stability controller of the vehicle, it is necessary to mathematically model the vehicle powertrain relationship. Thus, a single-track vehicle model with two degrees of freedom is established, and the vehicle coordinate system has been set where the origin of the coordinate system coincides with the mass center of vehicle. At the same time, the x-axis direction of the coordinate system is the same as the vehicle travel direction, and the y-axis is perpendicular to the x-axis and its direction is the same as the lateral motion direction of vehicle. The dynamic relationship of the vehicle in the horizontal motion plane is considered, and the differences in wind resistance and tire mechanical characteristics are ignored. Thus, the vehicle dynamics model is established and can be shown in Figure 1. The vehicle dynamics equation can be expressed as

$$m{v}_x\dot{\beta }=F_{yf}+F_{yr}-m{v}_x\gamma ,$$

(1)

$$I_z\dot{\gamma }=l_f{F}_{yf}-l_r{F}_{yr}+M_z,$$

(2)

where m is the whole vehicle mass; v_x is the longitudinal vehicle speed; γ is the yaw rate; I_z is the inertia moment of the whole vehicle around axis z; β is the vehicle sideslip angle and can be denoted as $\beta =\frac{v_y}{v_x}$; l_f and l_r, respectively, represent the distance from the gravity center of the vehicle to the front and rear axle of the vehicle; F_yf and F_yr, respectively, represent the generalized lateral forces of front and rear tires; and $M_z$ is the external yaw moment of vehicle body. Assuming that the front-wheel steering angle of the vehicle is relatively small, and the tire forces are in a linear working area, the lateral tire force of the front and rear wheels can be expressed as

$$\begin{array}{l}{F}_{yf}=2C_f{\alpha }_f\\ F_{yr}=2C_r{\alpha }_r\end{array},$$

(3)

where C_f and C_r, respectively, represent the cornering stiffness of front and rear tires, α_f and α_r, respectively, represent the tire slip angle of front and rear tires. The tire cornering stiffness of the vehicle can be expressed as

$$\begin{array}{l}{\alpha }_f={\delta }_f-\frac{l_f\gamma +v_y}{v_x}\\ {\alpha }_r=\frac{l_r\gamma -v_y}{v_x}\end{array},$$

(4)

where δ_f is the front-wheel steering angle of vehicle. Combining the Equations (1)–(4), it can be concluded that the vehicle dynamics model is

$$\dot{\gamma }=\frac{C_r{l}_r-C_f{l}_f}{I_z}\beta -\frac{C_f{l}_f^2+C_r{l}_r^2}{I_z{v}_x}\gamma +\frac{C_f{l}_f}{I_z}{\delta }_f+\frac{1}{I_z}{M}_z,$$

(5)

$$\dot{\beta }=-\frac{C_f+C_r}{m{v}_x^2}\beta -\left(1+\frac{C_f{l}_f-C_r{l}_r}{m{v}_x^2}\right)\gamma +\frac{C_f}{m{v}_x}{\delta }_f,$$

(6)

3. Coordinated Control Method for Lateral Stability and Differential Power-Assisted Steering

3.1. Overall Control Strategy

To achieve comprehensive optimization control of lateral stability and differential power-assisted steering for electric vehicles, a coordinated control strategy was designed as shown in Figure 2. The overall control strategy is divided into the control layer and the coordination layer. In the control layer, there are two modules: the lateral stability control module and the differential power-assisted steering control module. These two modules, respectively, focus on the lateral stability control effect of the vehicle body and the power-assisted steering control effect of the vehicle steering system, and then output the required front-wheel steering angle and yaw torque for vehicle motion control. In the coordination layer, a vehicle coordination controller of lateral stability and differential power-assisted steering was designed based on the Nash equilibrium game theory. The final output results of the front-wheel steering angle ${\delta }_{fN}$ and yaw moment control $M_{zN}$ were obtained through dynamic game theory and optimization calculation, thereby effectively improving the comprehensive control performance of in-wheel motor drive electric vehicle.

In the lateral stability control module, the sliding mode controller is designed for vehicle lateral stability based on the fast power reaching law, and a compensation control method is designed for vehicle front-wheel steering angle rectification. The vehicle lateral sliding mode controller can calculate the required front-wheel steering angle and yaw moment for lateral control based on the known vehicle state, and its output is expressed as ${\delta }_{f1}$ and $M_{z1}$. In addition, for lateral stability control systems, the dynamic-game-based coordination control of the front-wheel steering angle and yaw moment will cause the actual yaw moment to deviate to a certain extent from the optimal front-wheel steering angle required for stability control objectives. Therefore, in order to compensate for the weakening effect of coordinated control on lateral stability control, as shown in Figure 2, a front-wheel steering angle compensation control module was designed to correct vehicle steering control through dynamic compensation of the front-wheel steering angle, thereby achieving more accurate and reliable lateral stability control effects. The result of the front-wheel steering angle compensation control based on the PI controller can be expressed as

$${\delta }_{fc}={\delta }_{f1}+k_{P1}\left(M_{z1}-M_{zN}\right)+k_{I1}{\displaystyle {\int }_0^t\left(M_{z1}-M_{zN}\right)dt},$$

(7)

where, ${\delta }_{fc}$ is the compensation control result of the front-wheel steering angle, $k_{P1}$ and $k_{I2}$, respectively, represents the proportional coefficient and integral coefficient of the PI controller for front-wheel steering angle compensation.

In the differential power-assisted steering control module, there is a lookup table module based on the referenced steering wheel torque and a referenced steering wheel torque tracking module based on PI controller. The table lookup module calculates the current referenced steering wheel torque $T_{sf}$ based on real-time vehicle driving speed $v_x$ and front-wheel steering angle ${\delta }_{st}$. To achieve the power-assisted steering control, a referenced steering wheel torque tracking method is designed, and then the PI controller dynamically calculates the required differential torque based on the difference between the referenced steering wheel torque and the actual steering wheel torque, which can be represented as $M_{z2}$.

3.2. Vehicle Lateral Stability Sliding-Mode Controller (LSC) Based on Fast Power Reaching Law

To achieve the yaw stability control of the vehicle, it is necessary to design a yaw controller based on the vehicle model. To facilitate controller design, the vehicle system model parameters can be set to $a_1=\frac{C_r{l}_r-C_f{l}_f}{I_z}$, $a_2=-\frac{C_f{l}_f^2+C_r{l}_r^2}{I_z{v}_x}$, $a_3=-\frac{C_f+C_r}{m{v}_x}$, $a_4=-\left(1+\frac{C_f{l}_f-C_r{l}_r}{m{v}_x^2}\right)$, $b_1=\frac{C_f{l}_f}{I_z}$, $b_2=\frac{1}{I_z}$, $b_3=\frac{C_f}{m{v}_x}$. Thus, the vehicle models in Equations (5) and (6) can be represented as the following state space equation:

$$\dot{x}=f_x+bu+\epsilon ,$$

(8)

where, $x={\left[\begin{array}{cc}{x}_1& x_2\end{array}\right]}^T={\left[\begin{array}{cc}\gamma & \beta \end{array}\right]}^T$, $u={\left[\begin{array}{cc}{u}_1& u_2\end{array}\right]}^T={\left[\begin{array}{cc}{\delta }_f& M_z\end{array}\right]}^T$, $f_x={\left[\begin{array}{c}{a}_1{x}_2+a_2{x}_1\\ a_3{x}_2+a_4{x}_1\end{array}\right]}^T$, $b=\left[\begin{array}{cc}{b}_1& b_2\\ b_3& 0\end{array}\right]$, $\epsilon$ is the system interference amount.

In the actual driving process of electric vehicles, system interference is difficult to avoid due to various unknown comprehensive factors. Therefore, when designing controllers, the interference needs to be taken into account to improve the control accuracy. In order to achieve lateral stability control of vehicles, a sliding mode controller can be designed to track the referenced values of the yaw rate and sideslip angle of the vehicle in real time. Among them, the referenced value of the yaw rate can be calculated by the referenced vehicle model and recorded as ${\gamma }_d$, and the referenced value of the vehicle sideslip angle is 0. If the referenced state of the sliding mode controller is $x_d={\left[\begin{array}{cc}{\gamma }_d& \beta \end{array}\right]}^T$, its tracking error can be expressed as $e=x-x_d$. Based on the tracking error, the sliding surface can be designed as

$$s=k_{s}e+\dot{e},$$

(9)

where, $k_s$ is a sliding mode control parameter, and we have $k_s>0$. The selection of the sliding mode approach law has a significant impact on the effectiveness of sliding mode control. Although the convergence effect of the commonly used exponential approach law can meet the overall control requirements, its chattering phenomenon near the sliding mode surface is more obvious, which will affect the lateral stability control effect of vehicles. To suppress the chattering phenomenon of the sliding mode controller, the fast power reaching law is designed and represented as

$$\dot{s}=-k_{1}s-k_2{\left|s\right|}^{m_s}\mathrm{sgn}\left(s\right),$$

(10)

where, both $k_1$, $k_2$, and $m_s$ are sliding mode parameters greater than 0. By using the fast power reaching law, when the sliding mode motion is far from the sliding mode, the sliding mode motion speed is relatively higher, and the system converges faster. When the sliding mode motion is closer to the sliding mode, the sliding mode motion speed is relatively lower, and the adjustment amplitude is smaller. Compared to the exponential sliding mode approach law, the fast power-law approach law has a significantly faster approach speed and can effectively reduce the chattering phenomenon near the sliding mode surface. By combining the vehicle system model, the sliding mode control rate under fast power reaching law can be obtained as

$$u=\frac{1}{b}\left(\frac{1}{k_s}\left(-k_{1}s-k_2{\left|s\right|}^{m_s}\mathrm{sgn}\left(s\right)+k_s{\dot{x}}_d-\ddot{e}\right)-f_x-\epsilon \right).$$

(11)

To demonstrate the stability of the sliding mode controller for vehicle lateral stability, the Lyapunov function is selected as $V=\frac{1}{2}{s}^2$. Then, taking the derivative of Lyapunov function, we have

$$\dot{V}=s\dot{s}=s\left(-k_{1}s-k_2{\left|s\right|}^{m_s}\mathrm{sgn}\left(s\right)\right)=-k_1{s}^2-k_2{\left|s\right|}^{m_s+1}\le 0.$$

(12)

According to $\dot{V}\le 0$, the stability of the sliding mode controller can be proved.

3.3. Differential Power-Assisted Steering Control (DPASC) Based on Referenced Steering Wheel Torque

The differential power-assisted steering system of the distributed drive electric vehicle directly actuated by four in-wheel motors has independent and controllable function characteristics, and its structural principle is shown in Figure 3. When the front wheel of the vehicle is turning, due to the negative feedback of resistance torque such as tire lateral force, tire righting torque, and steering friction torque, a load force opposite to the output torque of the steering motor will be generated through the transmission of the steering trapezoid. When the load force is fed back to the driver’s steering wheel, the driver needs to apply greater force to induce the vehicle to turn, thus affecting the sensitivity and portability of the steering to a certain extent.

Considering the unique steering configuration of electric vehicles driven by in-wheel motors, the differential power-assisted steering function can be achieved by reasonably distributing the driving force of the left and right wheels on the front axle. As shown in Figure 3, if the driving torque distributed to the left and right wheels of the front axle is different, an independent torque difference will be generated between the left and right wheels, which will force the left and right wheels to generate a torque around the steering kingpin. The torque around the kingpin is transmitted through the steering trapezoid to push the steering tie rod of the steering system. Then, the steering tie rod can generate a force vector in the same direction as the output torque of the steering motor through the gear rack structure to reduce the steering wheel torque demand, thereby achieving the steering assistance function of electric vehicles.

Research shows that the assistance torque required for vehicle steering is positively correlated with vehicle speed and steering wheel torque, and the magnitude of steering wheel torque also increases with the increase in the steering wheel angle. Therefore, the vehicle speed and steering wheel angle are two key parameters that affect the magnitude of power-assisted steering torque. To obtain the differential torque required for different vehicle driving states, it is necessary to first obtain the referenced steering wheel torque. The referenced steering wheel torque is positively correlated with the vehicle speed and steering wheel angle and has a boundary value. When the steering wheel angle exceeds the boundary value, the reference steering wheel torque reaches saturation. Based on the above analysis, a referenced steering wheel torque MAP based on vehicle speed and steering wheel angle was fitted using simulation software, as shown in Figure 4. According to Figure 4, combined with the current vehicle speed and steering wheel angle, the referenced steering wheel torque in the current state can be obtained by looking up the table.

On this basis, a PI controller is designed to achieve the differential power-assisted steering control of the vehicle. The difference between the referenced steering wheel torque and the actual steering wheel torque is used as input to the PI controller, and the required differential torque can be obtained by adjusting the control parameters. This differential torque is also the yaw torque obtained from the torque distribution between the left and right in-wheel motors on the front axle. Under the action of this yaw torque, the power-assisted steering function of the vehicle can be achieved, and its calculation formula can be expressed as

$$M_{z2}=k_{P1}\left(T_{sf}-T_s\right)+k_{I1}{\displaystyle {\int }_0^t\left(T_{sf}-T_s\right)dt}.$$

(13)

where $k_{P1}$ and $k_{I1}$ are the proportional and integral coefficient of the PI controller, respectively.

3.4. Coordinated Control Method Based on the Nash Equilibrium Game Theory

According to Figure 2, it can be seen that both the lateral stability control module and differential power-assisted steering module can achieve control objectives through the output of the front-wheel steering angle and yaw moment. The tracking control objectives of the two modules exhibit dynamic game behavior in terms of outputs. At the same time, lateral stability control and differential power-assisted steering control are independent of each other and do not constrain each other, so the coordinated control belongs to the category of non-cooperative games. At the coordination control level, the lateral stability control module is game object 1, and the differential power-assisted steering control module is game object 2. The coordination control goal is to obtain the Nash equilibrium couple with the best comprehensive performance based on the Nash equilibrium game theory. According to game theory and coordinated control strategy, the two sub-control modules make the game result approach the Nash equilibrium point through dynamic game, so as to obtain the optimal front-wheel steering angle and yaw moment output. The Nash equilibrium couple characteristics can be represented as

$$\left(U_1^{\ast },U_2^{\ast }\right)=\left({\delta }_{fN},M_{zN}\right),$$

(14)

where, $U_1^{\ast }={\delta }_{fN}$ and $U_2^{\ast }=M_{zN}$, respectively, represents the optimal game solution of stability control and power-assisted steering control at the Nash equilibrium point. The optimization objective functions of sub-control modules can be represented as

$$\left\{\begin{array}{l}{J}_1={\displaystyle {\int }_0^{\infty }\left(x^T{Q}_{1}x+U_1^T{R}_{11}{U}_1+U_2^T{R}_{12}{U}_2\right)dt}\\ J_2={\displaystyle {\int }_0^{\infty }\left(x^T{Q}_{2}x+U_1^T{R}_{21}{U}_1+U_2^T{R}_{22}{U}_2\right)dt}\end{array}\right.,$$

(15)

where, $J_1$ and $J_2$, respectively, represents objective functions of game object 1 and game object 2; x is the state variable of game objects; $U_1$ and $U_2$ are the control variables of game objects; and $Q_1$, $Q_2$, $R_{11}$, $R_{12}$, $R_{21}$, $R_{22}$ are the weight matrices of game objects. In the dynamic game process, both parties pursue the maximization of their respective interests by obtaining the minimum objective function. Assuming that the final solution for both parties to reach an equilibrium state is $U_1^{\ast }$ and $U_2^{\ast }$, the relationship between the cost functions of both parties in the game can be expressed as

$$\left\{\begin{array}{l}{J}_1\left(U_1,U_2^{\ast }\right)\ge J_1\left(U_1^{\ast },U_2^{\ast }\right)\\ J_2\left(U_1^{\ast },U_2\right)\ge J_2\left(U_1^{\ast },U_2^{\ast }\right)\end{array}\right..$$

(16)

According to (16), at the optimal solution, both game object 1 and game object 2 can achieve the optimal extremum of their own objective function. By adopting a closed-loop Nash game strategy, the Nash equilibrium solution of the coordinated control layer can be represented as

$$\left\{\begin{array}{l}{u}_1^{\ast }=-R_{11}^{-1}{B}_1^T{K}_{1}x\\ u_2^{\ast }=-R_{22}^{-1}{B}_2^T{K}_{2}x\end{array}\right.,$$

(17)

where $B_1$ and $B_2$ are the input matrices and $K_1$ and $K_2$ are the coefficient matrices. The coefficient matrix $K_1$ and $K_2$ is the solution of the coupled Riccati equation and belongs to the symmetric positive-definite matrix, then the coefficient matrix $K_1$ and $K_2$ can be obtained by solving the following coupled Riccati equation:

$$\left\{\begin{array}{l}{K}_1\left(A-S_2{K}_2\right)+{\left(A-S_2{K}_2\right)}^T{K}_1+Q_1+K_2{S}_{12}{K}_2-K_1{S}_1{K}_1=0\\ K_2\left(A-S_1{K}_1\right)+{\left(A-S_1{K}_1\right)}^T{K}_2+Q_2+K_1{S}_{21}{K}_1-K_2{S}_2{K}_2=0\end{array}\right.,$$

(18)

where $S_i=B_i{R}_{ii}^{-1}{B}_i^T$, $S_{ij}=B_j{R}_{jj}^{-1}{R}_{ij}{R}_{jj}^{-1}{B}_j^T$, i = 1, 2, j = 1, 2. Considering Lyapunov stability, the energy function with positive-definite quadratic form is designed as

$$V=x^T{K}_{1,2}x.$$

(19)

It is known that $V\ge 0$. Taking the derivative of the Lyapunov function, we have

$$\begin{array}{l}\dot{V}={\dot{x}}^T{K}_{1,2}x+x^T{K}_{1,2}\dot{x}\\ =x^T\left(-Q_{1,2}-{\left(R_{ii}^{-1}{B}_{1,2}^T{K}_{1,2}\right)}^T{B}_{1,2}^T{K}_{1,2}\right)x\le 0\end{array},$$

(20)

Thus, it can be demonstrated that the stability of the dynamic game coordination controller meets the requirements of vehicle motion control.

At the coordination level, when the difference between the actual vehicle yaw rate and the referenced yaw rate is too large, the weight $M_{z1}$ of the coordinated control results increases. At this point, the vehicle coordinated control results focus on increasing the weight of lateral stability control. When the difference between the actual steering wheel torque and the referenced steering wheel torque of the vehicle is too large, the weight $M_{z2}$ of the yaw moment coordination result increases. At this time, the vehicle coordination control result focuses on increasing the weight of the power-assisted steering control. Thus, the coordination control layer can dynamically coordinate the weight between the two control modules based on the vehicle lateral stability and power-assisted steering demand, thereby achieving the best coordination control effect.

4. Analysis and Verification

In order to verify the application effectiveness of the proposed vehicle coordinated control strategy in this paper in improving vehicle lateral stability and ensuring power-assisted steering function, the simulation analysis and verification was conducted. The CarSim software is a widely used vehicle dynamics simulation software. With CarSim software and Matlab/Simulink software, a co-simulation model is built and used for simulation testing. In simulation, the CarSim software can be used to provide an entire dynamic model of vehicle and set vehicle model parameters. The corresponding coordinated control strategy and input/output ports are built in Simulink. The vehicle parameters involved in the co-simulation model are shown in

Table 1. Vehicle parameters.

Symbol	Parameters	Value and Units
m	Vehicle mass	850 kg
r	Effective radius of wheel	0.25 m
lf	Distances from vehicle gravity center to the front axle	0.815 m
lr	Distances from vehicle gravity center to the rear axle	0.985 m
bf, br	Half treads of the front(rear) wheels	0.78 m
Cf	Equivalent cornering stiffness of front wheel	65,000 N/rad
Cr	Equivalent cornering stiffness of rear wheel	45,000 N/rad
Iz	Moment of inertia	1000 kg·m2

.

As shown in Figure 4, there is a direct positive correlation between the steering wheel angle and the steering wheel torque demand. Therefore, it is necessary to study the relationship between steering wheel torque changes under different vehicle speeds and weight coefficients. By setting the steering wheel angle to 25 deg and using Simulink and CarSim for co-simulation of coordinated control strategy, the steering wheel torque under different vehicle speeds and weight coefficients can be obtained, as shown in Figure 5. As shown in the figure, with a fixed steering wheel angle, when the weight coefficient decreases, that is, the proportion of differential power-assisted steering control increases, and the steering wheel torque correspondingly decreases. This indicates that the differential power-assisted steering system has a clear functional response at this time, effectively reducing the steering wheel torque.

Similarly, the differential torque under different vehicle speeds and weight coefficients is shown in Figure 6. As shown in the figure, under a fixed weight coefficient, the magnitude of differential torque shows a positive correlation with the magnitude of vehicle speed, which increases linearly with the increase in vehicle speed. When the vehicle speed is fixed, the magnitude of differential torque basically increases with the increase in weight coefficient. But when the weight coefficient approaches 1, the differential torque reaches its maximum value and decreases with the increase in the weight coefficient. This is because when the vehicle approaches a single differential power-assisted steering mode, the vehicle control system does not perform yaw stability control and does not require differential torque to suppress the control demand of vehicle yaw stability torque. At this point, the differential torque of the vehicle is correspondingly reduced, and the yaw stability of the vehicle is also correspondingly reduced, which also reflects the necessity of coordinated control system design.

The demand for the referenced yaw rate is also directly positively correlated with the size of the front-wheel steering angle. Therefore, this study investigates the variation relationship of the yaw rate under different vehicle speeds and weight coefficients. Using the same co-simulation method and setting the steering wheel angle size to 25 deg, the yaw rate under different vehicle speeds and weight coefficients is shown in Figure 7. As shown in the figure, the magnitude of the yaw rate is directly affected by the vehicle speed, and as the vehicle speed increases, the yaw rate also increases accordingly. As the weight coefficient changes, there is no significant fluctuation in the magnitude of the yaw rate. This indicates that the designed coordinated control strategy and corresponding front-wheel steering angle compensation method can achieve real-time balance between stability control and power-assisted steering control objectives through dynamic adjustment of the front-wheel steering angle and additional yaw moment, and then achieve power-assisted steering function without sacrificing vehicle stability control effectiveness.

In order to verify the effectiveness of coordinated control strategies in achieving stability control and power-assisted steering functions, further comparative verification was conducted. Considering that there is currently no research on coordinated control of LSC and DPASC based on game control theory, we selected the typical control methods and compared them with the proposed coordinated control method. The simulation working condition is selected as a relatively common double lane change working condition, as shown in Figure 8. At the same time, the vehicle speed is set to 20 m/s to verify the effectiveness of the proposed coordinated control method during the process of the common lane-change maneuver.

In order to fully demonstrate the role of designed coordinated control strategy in ensuring vehicle stability and power-assisted steering effectiveness, the simulation tests were conducted under the same operating conditions for only lateral control and only differential power-assisted steering control, and the corresponding simulation results were compared with the coordinated control results. The stability control results of the vehicle are shown in Figure 9, including the control results of the vehicle yaw rate and the vehicle sideslip angle. Compared to the exponential approach law, the fast power approach law has more advantages in suppressing jitter amplitude values and tracking speed. This paper focuses more on presenting the advantages of the proposed coordinated control strategy compared to typical single-objective control methods. The key point lies in reflecting the effectiveness of LSC and DPASC coordinated control in improving comprehensive control performance. From Figure 9a, it can be seen that under only lateral control and coordinated control, the actual yaw rate of the vehicle can generally track the referenced yaw rate well. Among them, the yaw rate of the vehicle under coordinated control shows some shaking, and its tracking effect is consistent with the overall trend of the yaw rate under lateral control. The overall control effect of the two is not significantly different, and both can meet the requirements of yaw stability control. However, if only the differential power-assisted steering control mode is used, the vehicle yaw rate will deviate significantly from the referenced yaw rate. Similarly, in Figure 9b, the vehicle sideslip angle under lateral control and coordinated control are significantly smaller than the control results of differential power-assisted steering.

In order to more intuitively reflect the effectiveness of the yaw rate tracking control under different control methods, the comparison results of yaw rate tracking errors under different control modes are shown in Figure 10. The comparison results in this figure clearly demonstrate the superiority of the coordinated control method in ensuring the vehicle yaw stability. Among them, the tracking control deviation of the yaw rate under individual DPASC control mode is the largest, while the change trend and size of the tracking error under the coordinated control and LSC control mode are relatively similar. The above results indicate that in terms of vehicle lateral stability control, both the coordinated control method and LSC control method can effectively ensure vehicle stability, while the single DPASC control mode cannot meet the requirements of vehicle lateral stability control.

Figure 11 shows the comparative results of differential power-assisted steering control. From Figure 11, it can be seen that under differential power steering control and coordinated control, the actual steering wheel torque of the vehicle can track the reference steering wheel torque in real time, indicating that the vehicle has the function of differential power steering under both control methods. In addition, compared to differential power steering control, there is some fluctuation in steering wheel torque under coordinated control, which is caused by dynamic weight changes under coordinated control. However, the overall power steering control effect is basically consistent with differential control. From the perspective of overall control effect, the coordinated control of steering wheel torque changes can track the trend of reference steering wheel torque changes, and the overall control effect can meet the needs of vehicle power steering. However, if only lateral control is used, the actual steering wheel torque is significantly greater than the reference steering wheel torque, indicating that the power steering ability of the vehicle under lateral control is weaker than differential control and coordinated control. Figure 12 shows the error of steering wheel torque under different control modes. From Figure 12, it is possible to more intuitively compare the steering wheel torque tracking effect and differential assistance control effect under different control modes. As shown in the Figure 12, the steering wheel torque tracking error under DPASC control mode is the smallest, while the error under LSC control mode is significantly larger. At the same time, the steering wheel torque error under coordinated control mode is generally maintained at a relatively small level. Figure 13 shows the process of adjusting the adaptive weight coefficient of LSC over time under coordinated control. As shown in Equations (14) and (17), the Nash equilibrium solution of the coordinated control system can be obtained by combining the coefficient matrix. Due to the fact that the DPASC control system only inputs the yaw moment M_z2, the coordinated result of the front-wheel steering angle is actually equivalent to the output value of stability control module. And the dynamic game module actually only needs to coordinate M_z1 and M_z2 through weight coefficients. The M_zN can be obtained through the weight coefficient in Figure 13 and $M_{zN}=k{M}_{z1}+\left(1-k\right)M_{z2}$. We intend to use changes in weight coefficients to reflect the dynamic adjustment ability of coordinated control methods to weights as the vehicle status changes (including the vehicle yaw rate, sideslip angle, and steering wheel torque). Based on Figure 13, it can be seen that the steering wheel torque fluctuation under coordinated control is caused by the real-time adjustment effect of the adaptive weight coefficient, aiming to better coordinate and meet the two control objectives by dynamically adjusting the weight occupied by lateral control and differential power steering control. By synthesizing the performance of the above three control methods in vehicle stability control and power steering, it can be found that lateral control and differential power steering control have good control performance in vehicle stability and power steering, respectively, but they are far from meeting the required control target requirements in another indicator. In contrast, the coordinated control strategy effectively balances lateral stability and power steering control objectives while sacrificing vehicle stability and power steering effectiveness to a lesser extent. Although there are slight fluctuations in individual control performance compared to lateral control and differential power-assisted steering control, the overall coordinated control effect is good.

Figure 14 shows the vehicle control variables in the simulation test, including the front wheel angle and yaw moment of the vehicle. As shown in Figure 14a, the designed front wheel angle compensation control plays a role in adjusting the front wheel angle. Based on the results of vehicle stability control, it can be seen that compared to lateral control, the front wheel angle compensation method further ensures vehicle stability through compensation when the yaw moment control quantity changes. Based on Figure 13 and Figure 14b, it can be seen that in the yaw moment control, vehicle lateral control still occupies a relatively large weight, while differential power steering control plays a more auxiliary role in lateral control. This indicates that in situations where the steering conditions change sharply, the coordinated control strategy further utilizes coordinated control methods to achieve the function of power steering control while ensuring vehicle lateral stability.

In the above content, the control performance of the proposed coordinated control method in stability control and differential power-assisted steering was verified through visual comparison and qualitative analysis. Then, it intends to further verify the effectiveness of the coordinated control method through quantitative comparison. The vehicle yaw rate tracking control error and referenced steering wheel torque tracking error were selected for error statistics and analysis, and at the same time, the average value of error (denoted as e_a) and root mean square of error (denoted as e_r) were used to quantitatively reflect the control effect. The computational formulas of e_a and e_r can be expressed as

$$\begin{array}{l}{e}_a=\frac{1}{N}{\displaystyle \sum _{i=1}^n\left|{\lambda }_i-{\lambda }_R\right|}\\ e_r=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^n{\left({\lambda }_i-{\lambda }_R\right)}^2}}\end{array},$$

(21)

where N is the total number of samples, ${\lambda }_i$ represents the actual control results in different control modes at the ith sampling time, and ${\lambda }_R$ represents the referenced control quantities in different control modes at the ith sampling time. The error quantification calculation results under different control modes obtained according to Equation (21) are shown in

Table 2. Error quantitative analysis under different control modes.

	Yaw Rate			Steering Wheel Torque
	ea	er	Accuracy	ea	er	Accuracy
LSC	0.0073	0.0699	93.39%	0.1136	0.1669	56.89%
DPASC	0.0245	0.1076	64.55%	0.0081	0.0386	97.32%
Coordination control	0.0055	0.0683	94.76%	0.0240	0.0501	93.68%

. In terms of stability control indicators, the e_a and e_r under LSC and coordinated control are significantly lower than the control results under DPASC. The stability control accuracy under LSC and coordinated control mode exceeds 90%, while the accuracy under DPASC mode clearly does not meet the stability control requirements. Similarly, in terms of differential power-assisted steering indicators, DPASC and coordinated control modes have significantly lower e_a and e_r than LSC, and DPASC and coordinated control accuracy can be maintained at a high level. However, the accuracy under LSC control mode is significantly lower, and it basically does not have the required differential power-assisted steering function. Through comprehensive comparison, it can be seen that the LSC and DPASC control modes can only meet the control requirements of a single indicator, while the coordinated control method can effectively balance vehicle stability and differential power-assisted steering function.

5. Conclusions

This paper presents a dynamic game-based coordinated control method that integrates vehicle lateral stability control and differential power-assisted steering control. A vehicle dynamics model with two degrees of freedom is established, and the system interference is considered and incorporated into the vehicle model. Then, a sliding mode controller was designed based on the fast power reaching law to achieve lateral stability control of vehicle. At the same time, the effect of differential torque on vehicle power-assisted steering control was analyzed, and a PID controller was designed to calculate the required differential torque based on the referenced steering wheel torque, thereby achieving differential power-assisted steering control of vehicle. Considering the coupling effect between lateral stability control and differential power-assisted steering control in the steering actuator, a Nash equilibrium game theory-based coordinated control strategy for simultaneous lateral stability and differential power-assisted steering control was designed with adaptive weight coefficients. The co-simulation results show that the proposed coordinated control strategy can balance the two control objectives of lateral stability and differential power-assisted steering system, which can effectively achieve the power-assisted steering function of the vehicle while ensuring the lateral stability, thus achieving optimization of comprehensive control performance.