Robust Shared Control for Four-Wheel Steering Considering Driving Comfort and Vehicle Stability

Abstract

Although the four-wheel steering system expands the flexibility of vehicle control, it also brings the problem of difficult coordination between driver comfort and vehicle stability. To this end, this paper proposes robust coordinated control for a four-wheel steering (4WS) vehicle considering driving comfort and vehicle stability. First, the vehicle dynamics model is constructed to reflect the lateral motion characteristics of a 4WS vehicle. Then, the driver model is coupled into the 4WS vehicle model to describe the driver’s handling characteristics. To suppress the system perturbation caused by the uncertainties of driver behavior and vehicle states, the Takagi-Sugeno fuzzy robust control method is developed to design the human-machine co-driving system. Moreover, the robust positive invariant set theory is used to guarantee the stability and safety constraints of the vehicle. Finally, the proposed human-machine shared robust control for 4WS vehicle is verified through the driving simulator platform. The results indicate that the fuzzy robust shared control approach comprehensively improves the driving comfort, vehicle stability, and path tracking.

1. Introduction

As X-by-wire technology continues to mature, its proliferation and integration within the automotive domain have become increasingly prevalent. Innovations such as steer-by-wire, brake-by-wire, and electronic throttle control have progressively evolved into standard configurations in electric vehicles (EVs) [1,2]. Among these applications, four-wheel steering (4WS) control systems have been regarded as the most promising technology to enhance vehicle stability during operation and improve driver maneuverability [3,4]. Although 4WS technology can provide a more flexible control interface for EVs, consumers’ higher comfort and stability requirements for vehicles have also brought design challenges to this technology. Therefore, how to balance driving comfort and vehicle stability is attracting widespread attention from academia and industry [5,6,7].

Driving comfort is an important indicator of vehicle performance. In particular, human-machine shared control systems will be the main technology route map before the realization of fully autonomous technologies [8,9,10]. To this end, many scholars have tried to understand the driver’s behavior to improve the driving comfort of the human-machine shared control system. In [11], the semi-supervised training learning method was adopted to label the driver’s longitudinal behavior as aggressive and conservative labels. To improve the efficiency of driving style recognition, Whang et al. [12] developed an online unsupervised driving style classification model. PCA-Kmeans was also proven to have better classification accuracy and efficiency in driving style recognition [13]. The above-mentioned studies could provide personalized driving characteristics for the design of steering assistance controller. Based on the driver behavior analysis, some studies designed a different control strategy for aggressive and conservative drivers to meet the different driving requirements. In fact, driving conditions may influence the driver’s driving styles [14,15]. The change of driving styles causes frequent switching of steering control strategies, which may result in the instability of the vehicle control system. Therefore, this paper aims to understand continuous driver behavior and model the human-machine steering interaction behavior under 4WS vehicles for the shared controller design.

In terms of vehicle stability, active front steering (AFS) is regarded as one of the most promising technologies to improve vehicle safety through fast motor response [16]. The fast terminal sliding algorithm with an extreme learning machine was adopted in the AFS to improve the yaw stability for EVs [17]. Reference [18] proposed a robust gain-scheduled steering assistance control method to enhance the vehicle stability. In [19], the resilient control strategy with random uncertainty was presented to guarantee different performance requirements of EVs. Nevertheless, for extreme conditions, the lateral force of the tire tends to enter the nonlinear saturation region, which limits the performance of AFS. Therefore, direct yaw-moment control (DYC) was proposed to compensate for under steer for vehicle stability. DYC can be calculated through different tire forces to change the yaw motion of EVs. The multiple disturbances were considered in the DYC-AFS system to guarantee the vehicle stability in [20]. An explainable weight function was defined to adjust the frequency responses of DYC-AFS system. To further improve further vehicle stability, reference [21] proposed an integral slide model torque-vector control method. Lu et al. [22] introduced vehicle communication delays into the DYC-AFS system to ensure the path-tracking ability and vehicle stability. Based on the hardware in the loop platform, Pugi et al. [23] optimized the allocation method of braking torque to improve computational efficiency.

However, the redistribution of tire longitudinal forces will bring greater energy consumption, which will undoubtedly exacerbate the range anxiety problem of EVs. To this end, 4WS has become the technology with the most potential commercialization in the vehicle stability control system of EVs. Yin et al. [24,25] applied the μ-synthesis theory into the 4WS controller to enhance the robustness of the vehicle against uncertain pavement incentives. Jin et al. [26,27] combined DYC and 4WS to achieve the tracking control of desired sideslip and yaw rate through the gain-scheduled H_∞ control method. Furthermore, the mixed H₂/H_∞ optimal control strategy was adopted into the 4WS system to improve handling stability performance in [28]. To balance energy saving and vehicle stability, the hierarchical chassis coordinated control strategy was proposed in [29]. The expert PID and model predictive control (MPC) were integrated to achieve the speed tracking and path tracking in the upper-level. In the low level, the mutant particle swarm optimization algorithm was used to track the desired steering angle of each wheel. In fact, the front and rear wheels can be regarded as four agents from the perspective of the game theory. Therefore, Zhu et al. [30] designed novelty a quadratic differential game-based 4WS controller to optimize the stability performance under the J-turn maneuver. The Pareto game optimal method was also applied into 4WS system to improve the execution efficiency of four steering motors in [31].

The above-mentioned studies about 4WS indeed improved the stability and path tracking of the vehicle to a certain extent. However, these studies are limited to a single driving condition or even when the vehicle dynamics parameters are fixed, which brings huge limitations to the application of a 4WS shared controller. For instance, under different driving conditions, vehicle speed and tire cornering stiffness are time-varying and uncertain, respectively. The uncertainty of these two parameters will bring parameter perturbation to the vehicle dynamics model, which will seriously affect the lateral stability of the vehicle [32]. Furthermore, for the shared co-driving vehicle, the human’s manipulation intention will also affect the performance of the controller. The driver’s sudden intervention may cause the vehicle body to vibrate. To this end, the driver’s manipulation uncertainty and vehicle dynamic parameters are taken into account in the 4WS shared controller to suppress the system perturbation.

The Takagi-Sugeno (T-S) fuzzy robust approach is a common approach to addressing the parameter uncertainties in engineering applications. To achieve lower energy consumption, Nonami et al. [33] adopted a fuzzy output feedback method for the semi-active suspension system in the real-vehicle. Based on the driving simulator, Cabello et al. [34] designed a fuzzy speed-tracking controller to ensure excellent speed control effect. In terms of a four-wheel distributed electric vehicle, Pugi et al. [35] introduced the fuzzy logic method into the electric traction/braking controller to improve the vehicle stability. These studies showed that the fuzzy control method could effectively be applied in the real vehicle platform. Chen et al. [36] presented a composite nonlinear feedback steering controller to assist the impaired drivers for lane keeping. To solve the model mismatch problem caused by driver distraction, Nguyen et al. [37,38] adopted the fuzzy method to reconstruct the control-oriented vehicle dynamics. Zhang et al. [39] proposed a T-S fuzzy tire lateral dynamics model used for steering controller design under extreme conditions. To improve the versatility of the vehicle control system, Hu et al. [40] designed a steering assistance system with time-varying vehicle speed control based on the T-S fuzzy method. The above-mentioned studies reveal the effectiveness of robust control methods in handling parametric uncertainty issues. Despite this, fewer studies have introduced varying driver characteristics into 4WS vehicle systems.

This paper proposes the shared steering control method for 4WS to suppress the parametric uncertainties and improve the vehicle stability and driving comfort. Therefore, this paper’s contributions can be summarized as follows:

• To understand the driver’s continuous steering behavior, a driver model with adaptive preview distance is proposed. Meanwhile, to solve the model mismatch problem caused by vehicle parameter perturbation, the fuzzy shared driver-vehicle dynamics model is constructed for steering control.
• The shared steering control method for 4WS is proposed to suppress parametric uncertainties caused by time-varying driver characteristics, cornering stiffness, and vehicle speed. Moreover, constraints on the driver-vehicle system and actuators are considered by using the robust invariant set to enhance the safety of EVs.

The rest of this paper is structured as follows. In the Section 2, the mathematical model of 4WS vehicle system is established. In the Section 3, the proposed robust shared 4WS control algorithm method is introduced. In the Section 4, driver-in-the-loop experiments are conducted, and the results are analyzed. In the Section 5, the full text is summarized.

2. Mathematical Model of 4WS Vehicle System

2.1. 4WS Vehicle Dynamics Model

To facilitate the design of the controller, this paper only considers the lateral motion and yaw motion of the vehicle. In this paper, the longitudinal vehicle speed is assumed as a fixed value. Therefore, the lateral vehicle dynamics can be described as shown in Figure 1. Based on the Ackerman steering principle, the steering angle of 4WS can be obtained as follows [3]:

$$\begin{array}{l}\tan {\delta }_{fl}=\frac{\tan {\delta }_f}{1-\frac{B}{2l}\left(\tan {\delta }_f-\tan {\delta }_r\right)},\tan {\delta }_{fr}=\frac{\tan {\delta }_f}{1+\frac{B}{2l}\left(\tan {\delta }_f-\tan {\delta }_r\right)}\\ \tan {\delta }_{rl}=\frac{\tan {\delta }_r}{1-\frac{B}{2l}\left(\tan {\delta }_f-\tan {\delta }_r\right)},\tan {\delta }_{rr}=\frac{\tan {\delta }_r}{1+\frac{B}{2l}\left(\tan {\delta }_f-\tan {\delta }_r\right)}\end{array}$$

(1)

where B and l are the body width and wheelbase, respectively. δ_f and δ_r are the desired front and rear wheel angles, respectively. δ_i (i = fl, fr, rl, rr) represent the steering angles of each wheel. As shown in Figure 1, according to Newton’s second law, the single-track dynamic equation of the vehicle can be derived as [3]:

$$\left\{\begin{array}{l}{\dot{v}}_y=\frac{1}{m}\left(F_{yf}\cos {\delta }_f+F_{yr}\cos {\delta }_r\right)-v_{x}r\hfill \\ \dot{r}=\frac{1}{I_z}\left(l_f{F}_{yf}\cos {\delta }_f-l_r{F}_{yr}\cos {\delta }_r\right)+\frac{1}{I_z}{M}_z\hfill \end{array}\right.$$

(2)

where v_x and v_y mean the longitudinal vehicle speed and lateral vehicle speed, respectively. r and m are the yaw rate and the vehicle mass, respectively. F_yi (i = fl, fr, rl, rr) denotes the tire lateral forces. F_yf = F_yfl + F_yfr. F_yr = F_yrl + F_yrr. l_f and l_r represent front wheelbase and rear wheelbase, respectively. I_z is the yaw moment of inertia of the vehicle. M_z can be calculated by the following equation:

$$M_z=\frac{B}{2}\left(-F_{xfl}+F_{xfr}-F_{xrl}+F_{xrr}\right)$$

(3)

where F_xi (i = fl, fr, rl, rr) denotes the tire longitudinal forces. Under the premise that the tire slip angle is small at high speed, the tire lateral of front wheels and rear wheels can be approximated as follows:

$$F_{yf}=-k_f{\alpha }_f,\hspace{1em}{F}_{yr}=-k_r{\alpha }_r$$

(4)

where α_f and α_r are tire slip angles, calculated by [3]:

$${\alpha }_f=\left(v_y+l_{f}r\right)/v_x-{\delta }_f,{\alpha }_r=\left(v_y-l_{r}r\right)/v_x-{\delta }_r$$

(5)

According to the geometric relationship in Figure 1, the sideslip angle β can be expressed as follows:

$$\beta =\arctan \left(v_y/v_x\right)$$

(6)

The sideslip angle CoG is usually small, thus Equation (7) can be approximated as follows:

$$\beta \approx v_y/v_x$$

(7)

In this paper, v_x can be regarded as the constant value in specific condition. Therefore, substituting Equations (3)–(5) and (7) into Equation (2), the control-oriented dynamics are rewritten as follows:

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

(8)

2.2. Driver Model

In path tracking [8,9,10], it has been demonstrated that the driver is guided to steer the steering wheel by gazing at two key viewpoints on the road, which is known as two-point driver model. As shown in Figure 2, the near point and far point form the near vision angle θ_near and the far vision angle θ_far. θ_far is used to predict the upcoming road, while θ_near is adopted to correct vehicle position.

According to the geometric relationship shown in Figure 2, the explicit mathematic formulae of θ_far and θ_near are expressed by [6]:

$${\theta }_{near}\approx \arctan (\frac{y_L}{l_s})+{\psi }_L=\frac{y_L}{l_s}+{\psi }_L,\hspace{1em}{\theta }_{far}\approx \frac{l_{far}}{R_{ref}}=l_{far}{\rho }_{ref}$$

(9)

where l_far is the distance between the far point and CoG. l_s is the distance between the near point and CoG. Note that l_s = 0.4 l_far [6]. y_l and φ_L are defined as look-ahead path-tracking error and heading error, respectively, which are expressed as:

$$\begin{array}{cc}{\dot{y}}_L=\beta v_x+l_s\gamma +{\phi }_L{v}_x,& {\dot{\phi }}_L=\gamma -{\rho }_{ref}{v}_x\end{array}$$

(10)

where ρ_ref =1/R_ref is curvature value of the given desired path, which can be obtained in real time with on-board sensors.

Through the two-point tracking strategy, the human steering behavior is constructed by the decision-making system and neuromuscular system as shown in Figure 3. The driver’s decision-making process mainly consists of feedforward and feedback behaviors. K_p and K_c can be defined feedforward and feedback gains, respectively, to adjust the path-tracking errors. The PD control is used to represent the compensatory behavior of the driver for reducing the tracking error, in which τ_L is a differential constant. Considering the brain’s delay, the steering decision behavior is simulated. Then, the arm neuromuscular system is approximated as 1/(1 + τ_d²s) [8,9,10]. In order to facilitate controller design, e^−τd1s is approximated as 1/(1 + τ_d1s). Note that T_d = τ_d1 + τ_d² represents the driver’s total delay time, and a₀ = τ_d1τ_d²/T_d². a₀ = 0.21 is selected as a constant for simplification. Therefore, a simplified driver model is built for control purposes as follows:

$$\dot{\upsilon }=A_d\upsilon +B_d{u}_d$$

(11)

where $\upsilon ={\left[\begin{array}{cc}\epsilon & {\delta }_{fd}\end{array}\right]}^T,u_d={\left[\begin{array}{cc}{\theta }_{near}& {\theta }_{far}\end{array}\right]}^T,A_d=\left[\begin{array}{cc}0& -\frac{1}{a_0{T}_d{}^2}\\ 1& -\frac{1}{a_0{T}_d}\end{array}\right],B_d=\left[\begin{array}{cc}-\frac{R_g{K}_c}{a_0{T}_d{}^2}& \frac{R_g{K}_p}{a_0{T}_d{}^2}\\ -\frac{R_g{K}_c{\tau }_L}{a_0{T}_d{}^2}& 0\end{array}\right]$, $\epsilon ={\displaystyle \int (-\frac{1}{a_0{T}_d{}^2}{\delta }_{fd}+\frac{R_g{K}_p}{a_0{T}_d{}^2}{\theta }_{far}-\frac{R_g{K}_c}{a_0{T}_d{}^2}{\theta }_{near})dt}$. δ_fd represents the front wheel steering output from the driver model, R_g means the transmission coefficient of the steering-by-wire system.

2.3. Driver-Vehicle Model

The driver model (11) is integrated into the vehicle dynamics model (8) in order to better describe the driver-in-the-loop handling characteristics. Therefore, the driver-vehicle differential equation can be obtained:

$$\dot{x}=Ax+B_{u}u+B_{\varpi }\varpi$$

(12)

where $x=\left[\begin{array}{c}\beta \\ \gamma \\ y_L\\ {\phi }_L\\ \epsilon \\ {\delta }_{fd}\end{array}\right]$, $A=\left[\begin{array}{cccccc}-\frac{l_f{C}_f-l_r{C}_r}{m{v}_x}& \frac{l_f{C}_f-l_r{C}_r}{m{v}_x^2}+1& 0& 0& 0& \frac{k_f}{m{v}_x}\\ -\frac{l_f{C}_f-l_r{C}_r}{I_z}& -\frac{l_f^2{C}_f+l_r^2{C}_r}{I_z{v}_x}& 0& 0& 0& \frac{l_f{k}_f}{I_z}\\ v_x& l_s& 0& v_x& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& -\frac{R_g{K}_c}{l_s{a}_0{T}_d{}^2}& \frac{R_g{K}_p}{a_0{T}_d{}^2}& 0& -\frac{1}{a_0{T}_d{}^2}\\ 0& 0& -\frac{R_g{K}_c{\tau }_L}{l_s{a}_0{T}_d{}^2}& 0& 1& -\frac{1}{a_0{T}_d}\end{array}\right]$, $B_u=\left[\begin{array}{cc}\frac{C_f}{m{v}_x}& \frac{C_r}{m{v}_x}\\ \frac{l_f{C}_f}{I_z}& \frac{l_r{C}_r}{I_z}\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\end{array}\right]$, $B_{\varpi }=\left[\begin{array}{cc}0& 0\\ \frac{1}{I_z}& 0\\ 0& 0\\ 0& v_x\\ 0& 0\\ 0& 0\end{array}\right]$, $u=\left[\begin{array}{c}{\delta }_f\\ {\delta }_r\end{array}\right]$, $\varpi =\left[\begin{array}{c}{M}_z\\ {\rho }_{ref}\end{array}\right]$.

2.4. Validation of the Driver-Vehicle Model

This section is to identify the parameters of driver characteristics and validate the DVR model (13). Firstly, the RBF neural network is adopted to identify the driver parameters k = [K_p K_c τ_L T_d]^T. More details of the identification algorithm can be referred to in [6]. Thirteen drivers, denoted as Driver 1 to Driver 13, were invited to participate in the tests, driving along an identical reference trajectory by using a driving simulator. The reference trajectory is depicted in Figure 4. The 13 drivers’ personal information is given in Appendix A.

The identified parameters of the 13 drivers are shown in Figure 5. It can be observed that the gain characteristics K_p and K_c are relatively dispersed, reflecting the different driving styles and skills of these drivers, while τ_L and T_d, which represent driver’s reflection time, are relatively concentrated. In this paper, the average values of these drivers’ characteristic parameters are applied as the default parameters of the driver-vehicle model for the controller design.

One extra driver was invited in the validation test. The performance of the driver-vehicle model is compared with the maneuvers of this human driver on the driving simulator under the identical driving condition. The steering inputs of the human driver and the driver-vehicle model are compared in Figure 6a. It can be found that the steering angle and yaw rate response of the driver-vehicle model are very close to those of the human driver, and the fit goodness reaches about 85% Besides, as shown in Figure 6b, the yaw rate response of the model closely matches the that of human driver.

3. Robust Shared Control Design for 4WS Vehicle

3.1. T-S Fuzzy Description of Parameter Uncertainty

The measured output and control performance output for the closed-loop system are designed as follows:

$$\left\{\begin{array}{l}\dot{x}=Ax+{{\rm B}}_{u}u+B_{\varpi }\varpi \\ y=C_{yx}x+D_{yu}u+E_{y\varpi }\varpi \\ z=Qy\end{array}\right.$$

(13)

where $y={\left[\begin{array}{cccc}\gamma & {\theta }_{near}& {\dot{\theta }}_{near}& \begin{array}{cc}{\delta }_{fd}& \begin{array}{cc}{\dot{\delta }}_{fd}& a_y\end{array}\end{array}\end{array}\right]}^T$ is the measured output. Additionally, $Q=diag(\begin{array}{cccc}{Q}_{\gamma }& Q_{{\theta }_{near}}& Q_{{\dot{\theta }}_{near}}& \begin{array}{cc}{Q}_{{\delta }_{fd}}& \begin{array}{cc}{Q}_{{\dot{\delta }}_{fd}}& Q_{a_y}\end{array}\end{array}\end{array})$ is the weighting matrix, which is used to balance different performance indices. γ is an effective state variable of EVs to describe the stability performance of the vehicle. ${\theta }_{near}$ and ${\dot{\theta }}_{near}$ can describe the path-tracking performance of the vehicle. Refencing [25], the driver’s workload can be reflected by ${\delta }_{fd}$ and ${\dot{\delta }}_{fd}$. Therefore, we design the performance output of the controller:

$$z=C_{zx}x+D_{zu}u+E_{z\varpi }\varpi$$

(14)

where $C_{zx}=Q{C}_{yx},D_{zu}=Q{D}_{yui},E_{z\varpi }=Q{E}_{y\varpi }$. Due to the existence of 1/v_x in Equation (13), the closed-loop system (13) is nonlinear. Therefore, we linearize the variable 1/v_x based on the following equation:

$$\begin{array}{cc}{v}_x=v_0+v_1\Delta v,& \frac{1}{v_x}=\frac{1}{v_0}-\frac{v_1}{v_0^2}\Delta v\end{array},\Delta v\in \left(-1,1\right)$$

(15)

where $\Delta v\in \left(-1,1\right)$ represents the variation of v_x. The constants v₀ and v₁ are obtained as and $v_0=\left(v_{\max }+v_{\min }\right)/2$, respectively. Equation (15) is brought into Equation (13). Then, the T-S fuzzy method is used to handle the time-varying parameters l_s, C_f, C_r, and Δv.

$$\left\{\begin{array}{l}{l}_s={\rho }_1(l_s)l_s{}_{\min }+{\rho }_2(l_s)l_s{}_{\max }\\ C_f={\rho }_1(C_f)C_f{}_{\min }+{\rho }_2(C_f)C_f{}_{\max }\\ C_r={\rho }_1(C_r)C_r{}_{\min }+{\rho }_2(C_r)C_r{}_{\max }\\ \Delta v={\rho }_1(\Delta v)\Delta v_{\min }+{\rho }_2(\Delta v)\Delta v_{\max }\end{array}\right.$$

(16)

where ${\rho }_i(\ast )$ (i = 1, 2) is the membership function as follows:

$$\begin{array}{l}{\rho }_1(l_s)=\frac{l_s{}_{\max }-l_s}{l_s{}_{\max }-l_s{}_{\min }},{\rho }_2(l_s)=\frac{l_s-l_s{}_{\min }}{l_s{}_{\max }-l_s{}_{\min }}\\ {\rho }_1(C_f)=\frac{C_f{}_{\max }-C_f}{C_f{}_{\max }-C_f{}_{\min }},{\rho }_2(C_f)=\frac{C_f-C_f{}_{\min }}{C_f{}_{\max }-C_f{}_{\min }}\\ {\rho }_1(C_r)=\frac{C_r{}_{\max }-C_r}{C_r{}_{\max }-C_r{}_{\min }},{\rho }_2(C_r)=\frac{C_r-C_r{}_{\min }}{C_r{}_{\max }-C_r{}_{\min }}\\ {\rho }_1(\Delta v)=\frac{\Delta v_{\max }-\Delta v}{\Delta v_{\max }-\Delta v_{\min }},{\rho }_2(\Delta v)=\frac{\Delta v-\Delta v_{\min }}{\Delta v_{\max }-\Delta v_{\min }}\end{array}$$

(17)

Therefore, the schematic diagram of the fuzzy model can be obtained as shown in Figure 7.

Table 1 shows all combinations of the premise variables, where B and S represent “big” and “small”, respectively. Combining the fuzzy rules with Equations (13) and (16), the fuzzy closed-loop system is rewritten:

$$\left\{\begin{array}{l}\dot{x}={\displaystyle \sum _{i=1}^{16}{h}_i\left(A_{i}x+{{\rm B}}_{ui}u+B_{\varpi i}\varpi \right)}\\ y={\displaystyle \sum _{i=1}^{16}{h}_i\left(C_{yxi}x+D_{yui}u+E_{y\varpi i}\varpi \right)}\\ z={\displaystyle \sum _{i=1}^{16}{h}_i\left(C_{zxi}x+D_{zui}u+E_{z\varpi i}\varpi \right)}\end{array}\right.$$

(18)

where the system matrix in Equation (18) can be expressed by:

• If l_s is S, C_f is S, C_r is S, and v_x is S, then l_smin, C_fmin, C_rmin, and v_xmin replace l_s, C_f, C_r, and v_x of A₁, B_u1, and B_w1 in Equation (18);
• If l_s is S, C_f is S, C_r is S, and v_x is B, then l_smin, C_fmin, C_rmin, and v_xmax replace l_s, C_f, C_r, and v_x of A₂, B_u2, and B_w2 in Equation (18);
• If l_s is S, C_f is S, C_r is B, and v_x is S, then l_smin, C_fmin, C_rmax, and v_xmin replace l_s, C_f, C_r, and v_x of A₃, B_u3, and B_w3 in Equation (18);
• …
• If l_s is B, C_f is B, C_r is B, and v_x is S, then l_smax, C_fmax, C_rmax, and v_xmin replace l_s, C_f, C_r, and v_x of A₁₅, B_u15, and B_w15 in Equation (18);
• If l_s is B, C_f is B, C_r is B, and v_x is B, then l_smax, C_fmax, C_rmax, and v_xmax replace l_s, C_f, C_r, and v_x of A₁₆, B_u16, and B_w16 in Equation (18).

Table 1. List of the fuzzy rules.

Rule No.	Premise Variables
	ls	Cf	Cr	vx
1	S	S	S	S
2	S	S	S	B
3	S	S	B	S
4	S	S	B	B
…	…	…	…	…
15	B	B	B	S
16	B	B	B	B

Table 1. List of the fuzzy rules.

Rule No.	Premise Variables
	ls	Cf	Cr	vx
1	S	S	S	S
2	S	S	S	B
3	S	S	B	S
4	S	S	B	B
…	…	…	…	…
15	B	B	B	S
16	B	B	B	B

3.2. Robust Shared Controller Design for 4WS

By considering the multi-objective H_∞ robust control theory, the optimized objective function is designed as follows [41]:

$$J={\displaystyle {\int }_0^{+\infty }z{(t)}^{T}z(t)dt}={\displaystyle {\int }_0^{+\infty }\chi {(t)}^T\mathbb{Q}\chi (t)dt}$$

(19)

with $\chi (t)=\left[\begin{array}{c}x\\ u\\ \varpi \end{array}\right],\mathbb{Q}=\left[\begin{array}{ccc}{C}_{zx}^T{C}_{zx}& C_{zx}^T{D}_{zu}& C_{zx}^T{E}_{z\varpi }\\ \ast & D_{zu}^T{D}_{zu}& D_{zu}^T{E}_{z\varpi }\\ \ast & \ast & E_{z\varpi }^T{E}_{z\varpi }\end{array}\right]$.

From Equations (14) and (19), it can be seen that z can be improved by tuning the weighting $\mathbb{Q}$. This paper adopts the state-feedback control to design control law since it provides sufficient system information. Hence, the control law u_i in each fuzzy system can be expressed as:

$$u_i=K_{i}x$$

(20)

where K_i is the gain vector to be calculated.

To minimize the performance index J, we firstly defined the Lyapunov function as follows:

$$\begin{array}{cc}\vartheta \left(x\right)=x^{T}Px,& P>0\end{array}$$

(21)

Equation (21) satisfies Hamilton-Jacobi’s inequality

$$\begin{array}{cc}\dot{\vartheta }\left(x\right)+z^{T}z<\mathsf{{\rm Y}}{\varpi }^T\varpi -2\varsigma \vartheta ,& \mathsf{{\rm Y}},\varsigma >0\end{array}.$$

(22)

Furthermore, the following inequality can be obtained:

$${\displaystyle {\int }_0^{+\infty }z{(t)}^{T}z(t)dt}<\vartheta (0)-\vartheta (\infty )+\mathsf{{\rm Y}}{\displaystyle {\int }_0^{+\infty }\varpi {(t)}^T\varpi (t)dt}$$

(23)

Then, the above Equation (23) can be rewritten as the following format:

$$J<\vartheta \left(0\right)+\mathsf{{\rm Y}}\rho$$

(24)

where the positive scalar ρ is the upper bound of the system disturbance w. For the given initial values of $\vartheta \left(0\right)$ and ρ, the objective function is optimized by adjusting $\mathsf{{\rm Y}}$. Equations (18) and (20) are brought into Equation (22). Equation (22) can be converted as

$$\begin{array}{l}\mathsf{\Omega }={\dot{x}}^{T}Px+x^{T}P\dot{x}+z^{T}z-\mathsf{{\rm Y}}{\varpi }^T\varpi +2\varsigma \vartheta \left(x\right)\\ ={\displaystyle \sum _{i=1}^{16}{h}_i}He({\left[(A_i+B_{ui}{K}_i)x+B_{\varpi }\varpi \right]}^{T}Px+\varsigma x^{T}Px)\\ +{(C_{zxi}x+D_{zui}u+E_{z\varpi i}\varpi )}^T(C_{zxi}x+D_{zui}u+E_{z\varpi i}\varpi ))-\mathsf{{\rm Y}}{\varpi }^T\varpi <0\end{array}$$

(25)

where $He(X)=X+X^T$. Inequality Equation (25) can be rewritten as

$$\mathsf{\Omega }={\displaystyle \sum _{i=1}^{16}{h}_i}{\left[\begin{array}{c}x\\ \varpi \end{array}\right]}^T{\mathsf{\Phi }}_{ij}\left[\begin{array}{c}x\\ \varpi \end{array}\right]\prec 0$$

(26)

The only condition for Equation (26) to hold is:

$${\mathsf{\Phi }}_i=\left[\begin{array}{cc}He\left(\left[P{A}_i+P{B}_{ui}{K}_i+\varsigma P\right]\right)& P{B}_{\varpi i}\\ B_{\varpi i}^{T}P& -\mathsf{{\rm Y}}I\end{array}\right]+\left[\begin{array}{c}{C}_{zxi}^T+K_i^T{D}_{zui}^T\\ E_{z\varpi i}^T\end{array}\right]\left[\begin{array}{cc}{C}_{zxi}+D_{zui}{K}_i& E_{z\varpi i}\end{array}\right]\prec 0$$

(27)

By Schur complement lemma [41], Equation (27) can be equivalent to

$$\left[\begin{array}{ccc}He(\left[P{A}_i+P{B}_{ui}{K}_i+\varsigma P\right])& P{B}_{\varpi i}& C_{zxi}^T+K_i^T{D}_{zui}^T\\ B_{\varpi i}^{T}P& -\mathsf{{\rm Y}}I& E_{z\varpi i}^T\\ C_{zxi}+D_{zui}{K}_{ij}& E_{z\varpi i}& -I\end{array}\right]\prec 0$$

(28)

Introducing a new variable by pre-multiplying and post-multiplying Equation (28) with a diagonal matrix (X, I, I). X is the inverse of P as X = P⁻¹.

$$F_i=K_{i}X$$

(29)

The condition (28) equals:

$${\mathsf{\Psi }}_i=\left[\begin{array}{ccc}{A}_{i}X+B_{ui}{F}_i+\varsigma X+{\left(A_{i}X+B_{ui}{F}_i+\varsigma X\right)}^T& B_{\varpi i}& C_{zxij}^T+F_i^T{D}_{zui}^T\\ B_{\varpi i}^T& -\mathsf{{\rm Y}}I& E_{\varpi i}^T\\ C_{zxi}X+D_{zui}{F}_i& E_{z\varpi i}& -I\end{array}\right]\prec 0.$$

(30)

3.3. Robust Positive Invariant Set Constraint Design

To enhance both vehicle safety and tracking performance, it is essential to take into account the limitations on system output and control input. The vehicle’s states must adhere to the specified bounded region.

$$\left|\beta \right|<{\beta }_{\max },\left|\gamma \right|<{\gamma }_{\max },\left|y_L\right|<y_L{}_{\max },\left|{\phi }_L\right|<{\phi }_L{}_{\max }$$

(31)

Theorem 1.

Under the given initial condition x(0), the constraint $‖x_i‖\le {\epsilon }_i$ is always satisfied when the following inequality is satisfied [41].

$$\left[\begin{array}{cc}1& x{(0)}^T\\ x(0)& X\end{array}\right]\ge 0,\left[\begin{array}{cc}X& X{d}_i^T\\ d_{i}X& {\epsilon }_i{}^{2}I\end{array}\right]\ge 0,\mathrm{with} x_i(t)=d_{i}x(t)$$

(32)

Furthermore, taking into account the actuator’s saturation constraint, it is necessary to ensure that the control input remains within specified bounds.

$$\left|{\delta }_c\right|<{\delta }_c{}_{\max }$$

(33)

Theorem 2.

Given the known initial condition x(0), the constraint is guaranteed to be satisfied for as long as the Linear Matrix Inequalities (LMIs) specified in reference [41] are valid.

$$\left[\begin{array}{cc}1& x{(0)}^T\\ x(0)& X\end{array}\right]\ge 0,\left[\begin{array}{cc}X& F_{ij}{}^T\\ F_{ij}& {\mu }^{2}I\end{array}\right]\ge 0,\mathrm{with} \mu ={\delta }_{c\max }$$

(34)

Theorem 3.

Given system (17) and a positive matrix X, the control law can achieve asymptotic stability of (30) by optimizing (26), provided there exists X, K_i (i = 1, 2, …, 16), and $\mathsf{{\rm Y}}$ that satisfy the following optimization solution issue:

$$\begin{array}{l}\begin{array}{cc}Min& \mathsf{{\rm Y}}\end{array}\\ s.t. (30),(32),\mathrm{and} (34)\end{array}$$

(35)

Additionally, the feedback gains can be obtained from (29):

$$K_{ij}=F_{ij}{X}^{-1}$$

(36)

Each fuzzy robust controller firstly is calculated offline. Then, the weight of each controller is updated online. Ultimately, by satisfying the above linear matrix inequality (30), the Lyapunov stability of the driver-vehicle system (21) can be guaranteed. Furthermore, by satisfying linear matrix inequalities (32) and (34), the state constraints of the driver-vehicle system can be guaranteed.

4. Driver-in-the-Loop Experiment

4.1. Experiment Settings

As depicted in Figure 8, the experimental setup for driver-in-the-loop experiments primarily consists of three components: the traffic environment, high-fidelity vehicle dynamics, and control unit. The PC hosts the PreScan, which primarily serves to provide the traffic states. The human driver manipulates the vehicle based on road information. The real-time control unit receives the vehicle states and computes the assisted steering angle. For an accurate representation of the EV, the Carsim software platform 8.0.2 is used as a simulated vehicle. Details of the parameters of the simulated vehicle can be found in

Table 2. Vehicle parameters.

Symbol	Meaning	Value
m	Vehicle total mass	1705 kg
Mz	Vehicle yaw moment of inertia	3048 kg·m2
lf	Distance between CG and front axle	1.035 m
lr	Distance between CG and rear axle	1.665 m
Cf	Equivalent Cornering Stiffness of front tire	103,130 N/rad
Cr	Equivalent Cornering Stiffness of rear tire	73,854 N/rad

[6].

To verify the performance of the proposed approach, three different drivers, denoted by Driver I–III, perform the path-tracking maneuvers. The reference trajectory to follow is depicted in Figure 4. To identify the characteristic parameters and driving styles of the testing drivers, we ask the three drivers to drive along the reference trajectory without assistance of the shared controller. The phase portraits with respect to the steering wheel angle ${\delta }_d$ and steering rate ${\dot{\delta }}_d$ are shown in Figure 9a. Vehicle trajectories are shown in Figure 9b. It can be seen that the steering phase of Driver I is in the middle area and Driver I’s trajectory is close to the reference. Thus, Driver I is regarded as an experienced driver. The steering wheel angle and steering rate of Driver II are significantly greater than other drivers. Moreover, Driver II exhibits curvature-cutting behavior [8] as shown in Figure 9b. Therefore, we consider Driver II as an aggressive driver. In contrast, Driver III is a conservative driver. The characteristic parameters of these three drivers are identified with the method in [6] and are shown in

Table 3. Driver model parameters.

Parameter	KP	Kc	τL	Td	Driving Style
Driver I	2.70	1.50	0.18	0.18	Experienced
Driver II	3.60	2.50	0.18	0.18	Aggressive
Driver III	1.90	1.10	0.19	0.19	Conservative

.

The indexes are introduced to quantify the performances of driver-automation cooperation, driving comfort, vehicle stability, and path tracking. The performance of the driver-automation cooperation for the proposed shared controllers is quantified as [5]

$$J_1=\frac{w_1{\displaystyle \int {\delta }_d{\delta }_c}dt}{T}$$

(37)

where T is testing time. A negative value of J₁ indicates that the steering input of the driver is opposite to that of the controller, i.e., the conflict occurs. As J₁ decreases, the level of conflict between humans and machines increases. Conversely, a significantly positive value of J₁ indicates a high degree of cooperation. The metrics for driving comfort, vehicle stability, and path tracking performance are denoted as J₂, J₃, and J₄, respectively [6]:

$$J_2=\frac{{\displaystyle \int \left(w_2\left|{\dot{\delta }}_d\right|+w_3\left|{\dot{\delta }}_c\right|\right)}dt}{T},J_3=\frac{{\displaystyle \int (w_4\left|a_y\right|+w_5\left|v_y\right|)}dt}{T},J_4=\frac{{\displaystyle \int (w_6\left|y_L\right|+w_7\left|{\phi }_L\right|)}dt}{T}$$

(38)

where w_i (i = 1, 2, …, 7) are weight coefficients. Note that J₂ is defined as the driving comfort of the vehicle with the shared steering control. Large and constant changes of the driver steering angle and the assistance steering angle may lead to an uncomfortable driving experience.

4.2. Performance Analysis of the Proposed Shared Controller

To validate the performance of the proposed controller, the robust shared 4WS controller is used as benchmark without considering driver characteristics. For convenience, the proposed fuzzy robust controller considering driver characteristics is denoted as FRC while the benchmark is denoted as RC. In addition, the model predictive control (MPC) method is also used as a benchmark to validate the robustness of the proposed method. The path-tracking results are shown in Figure 10. It can be found that the three controllers can track the reference path well. However, the proposed controller FRC is superior to RC and MPC controller in the large-curvature path-tracking performance.

In Figure 11, the steering wheel angles are depicted under different controllers. As observed in Figure 11a,b, the steering angles of Drivers I and II exhibit an opposing pattern to those of the RC. This discrepancy arises because the design of the RC does not account for driver characteristics, potentially resulting in a conflict state between the driver and RC controller. In contrast, the proposed controller, which considers driver characteristics, demonstrates an enhanced cooperative performance.

Table 4. Results of performance index.

Driver Type	Controller Type	J1	J2	J3	J4
Driver I	FRC	7.5	1.2	5.2	2.0
	RC	−6.3	6	6.8	2.5
	MPC	−3	8	7.0	2.2
Driver II	FRC	−5.3	4.1	7.5	2.4
	RC	−40	40	8.4	3.2
	MPC	−36	42	8.6	2.7
Driver III	FRC	5.5	1.8	4.4	2.1
	RC	4.2	3.2	5.6	2.3
	MPC	5.1	5	5.8	2.6

presents the J₁ index for evaluating human-machine conflict. When compared to the RC and MPC, the FRC can realize a remarkable reduction in steering conflicts, surpassing 59%.

In Figure 11b, the steering inputs from RC and FRC exhibit an opposite behavior compared to those of Driver II. When the steering angle of Driver II increases, it results in a substantial escalation of the human-machine conflict with the RC. An aggressive driving style is not beneficial to the cooperative performance of controllers. The human-conflict index J₁ is positive among both controllers for Driver III due to the conservative behavior. The cooperative performance of Driver III is obviously superior to Drivers I and II with RC. However, as shown in Table 4, experienced Driver I has a better cooperative performance with the proposed controller considering driver-automation interaction level.

With the assistance of the FRC, the driving comfort performance indices J₂ for all three drivers experience an improvement of more than 70% when compared to the RC and MPC (seen Table 4). Notably, Driver II exhibits the lowest driving comfort, primarily due to their aggressive driving behavior, which hinders effective human-machine cooperation. Conversely, Driver III’s driving comfort fares relatively better because their conservative driving style allows the controller to have a more dominant role in control authority. However, there exists a large steering jitter when driving in and out of the curve (see Figure 11c). This causes rapid changes of assistance steering inputs with RC, which is not beneficial to driving comfort and safety. Nonetheless, the proposed controller can improve the driving comfort of Driver III because it considers the driver’s characteristics. Since Driver I has excellent driving skills, the driving comfort is better with both controllers. In summary, the proposed controller can improve driving comfort and reduce human-machine conflicts.

In Figure 12, the β-γ phase portrait, commonly employed to illustrate vehicle stability, is depicted. In contrast to the RC, the FRC exhibits a smaller β-γ phase, indicating a better vehicle stability. Furthermore, constraints outlined in Equation (31) can be met. As demonstrated in Table 4, the vehicle stability performance J₃ of the FRC surpasses that of the RC and MPC.

Due to the aggressive driving style, the β-γ phase of Driver II is more stable than the other drivers. For the conservative Driver III, the β-γ phase is relatively small. However, it becomes larger when the sideslip angle is between 0.1° and 0.2° with the assistance of FRC. This is because the assistance steering wheel angle and its rate are relatively larger for compensating the understeer behavior of Driver III when the vehicle drives in and out the curve. In comparison with the other two drivers, Driver I can cooperate better with the automation, which makes the β-γ curve smoother.

Finally, Figure 13 shows the lateral error under FRC and RC. The tracking error of the proposed controller is smaller. From Table 4, the path tracking performance J₄ with FRC can be enhanced by 18% compared with RC. Driver I has the best path-tracking ability, with the maximum path tracking error smaller than 0.3 m. Instead, the path-tracking ability of Driver II is the worst with both controllers. However, the FRC yields a better lane-keeping performance than RC. The conservative Driver III has a better path-tracking ability. This is because the control authority is dominated by the controllers in this case.

5. Conclusions

This study introduces a robust coordinated control strategy for four-wheel steering (4WS) vehicles, focusing on enhancing driving comfort and vehicle stability. We developed a comprehensive vehicle dynamics model that captures the unique lateral motion characteristics of 4WS vehicles and incorporated a driver model to accurately represent driver handling behavior. To address uncertainties resulting from variations of driver manipulation, we adopted a Takagi-Sugeno fuzzy robust control method to establish the control-oriented human-machine dynamics model. Additionally, we applied robust positive invariant set theory to maintain stability and ensure vehicle safety. The driving simulator platform is used to verify the effectiveness of the proposed shared steering controller for 4WS vehicles. The findings showcase that this control strategy not only enhances driving comfort but also elevates vehicle stability and fine-tunes path tracking precision. In the future, we intend to incorporate more intricate vehicle dynamics models, including the vertical and longitudinal motion behaviors of the vehicle, into the controller design process to further enhance vehicle stability and driving comfort. Furthermore, we will explore aspects such as driver distraction, fatigue, and other states to enhance the safety of human-machine systems.