## 2 Mathematical Models of the 4 WS System
### The 2 DOF Vehicle Model for the Controller Design  
A 2 DOF vehicle model is widely used as a reference model for the controller design of the vehicle steering system [30; 31]. Both theory and experiment prove that the controllers based on the 2 DOF vehicle model can work normally and represent the actual physical process of the vehicle steering system with great accuracy [32]. Thus, a 2 DOF vehicle dynamics model was constructed to design the H2/H\(\infty\) robust controller of the 4 WS system, as shown in Figure 1. As the reference coordinate system of the vehicle, the inertial coordinate frame \((x_{e},y_{e},z_{e})\) is fixed on the ground, where the direction represented by the \(z_{e}\) axis is perpendicular to the \((x_{e},y_{e})\) plane. The body coordinate system is represented by \((x,y,z)\) to describe the vehicle movement, where CG is the coordinate origin. The \((x,y,z)\) coordinate system rotates the yaw angle \(\varphi\) towards the \((x_{e},y_{e},z_{e})\). The vehicle runs at a constant speed \(V\) measured at the CG point, and it rotates at a certain yaw velocity \(r\). \(\beta\) is the sideslip angle of the CG point.  
Only considering lateral motion and yaw motion, the vehicle kinematics equation is derived from [PERSON]'s second law, which is shown below:  
\[\begin{array}{c}mV(\dot{\beta}+r)cos\beta=F_{f}cos\delta_{f}+F_{r}cos\delta_ {r}\\ I_{z}\dot{r}=l_{f}F_{f}cos\delta_{f}-l_{r}F_{r}cos\delta_{r}\end{array} \tag{1}\]  
where \(m\) denotes the vehicle mass; \(F_{f}\) and \(F_{r}\) are the lateral forces between the tire and the road surface of the front and rear wheels; \(\delta_{f}\) and \(\delta_{r}\) represent the front wheel angle and the rear wheel angle of the vehicle, respectively; \(l_{z}\) denotes the moment of inertia of the vehicle around the \(z\) axis; \(l_{f}\) (\(l_{r}\)) is the distance between the gravity center and the front (rear) axle.  
In general, \(\delta_{f}\) and \(\delta_{r}\) are quite small:  
\[cos\delta_{f}\approx 1,cos\delta_{r}\approx 1. \tag{2}\]  
Figure 1: The 2 DOF vehicle model.  
If \(\beta\) is small and \(V\) changes slowly, then  
\[\alpha_{f}=\delta_{f}-\beta-\frac{l_{f}}{V}r,\alpha_{r}=\delta_{r}-\beta+\frac{l _{r}}{V}r \tag{3}\]  
where \(\alpha_{f}\) and \(\alpha_{r}\) represent the slip angles of the front wheel and the rear wheel, respectively.  
Based on the consideration of the cornering characteristics of tires, the following approximate linear models are adopted.  
\[F_{f}=C_{f}\alpha_{f},F_{r}=C_{r}\alpha_{r} \tag{4}\]  
where \(C_{f}\) and \(C_{r}\) denote the cornering stiffness of the front wheel and the rear wheel, respectively.  
Combined with Equations (1)-(4) mentioned above, the vehicle dynamics model is shown as follows.  
\[\begin{array}{c}mV(\dot{\beta}+r)=-\left(C_{f}+C_{r}\right)\beta-\frac{l_{f} C_{f}-l_{r}C_{r}}{V}r+C_{f}\delta_{f}+C_{r}\delta_{r}\\ I_{z}\dot{r}=-\left(l_{f}C_{f}-l_{r}C_{r}\right)\beta-\frac{l_{f}C_{f}+l_{r}^{ 2}C_{r}}{V}r+l_{f}C_{f}\delta_{f}-l_{r}C_{r}\delta_{r}.\end{array} \tag{5}\]  
The state space model is shown below:  
\[\left\{\begin{array}{c}\dot{x}=Ax+Bu\\ y=Cx+Du\end{array}\right. \tag{6}\]  
where  
\[\begin{array}{c}x=\left[\begin{array}{cc}\beta&r\end{array}\right]^{T},u= \left[\begin{array}{cc}\delta_{f}&\delta_{r}\end{array}\right]^{T},y=\left[ \begin{array}{cc}\beta&r\end{array}\right]^{T}\\ A=\left[\begin{array}{cc}\frac{-\left(C_{f}+C_{r}\right)}{mV}&-1-\frac{l_{f}C_{ r}-l_{r}C_{r}}{mV^{2}}\\ \frac{-l_{f}C_{f}-l_{r}C_{r}}{I_{z}}&-\frac{l_{f}^{2}C_{f}+l^{2}C_{r}}{I_{z}V} \end{array}\right]\\ B=\left[\begin{array}{cc}\frac{C_{f}}{mV}&\frac{C_{r}}{mV}\\ \frac{l_{f}C_{f}}{I_{z}}&-\frac{l_{r}C_{r}}{I_{z}}\end{array}\right],C=\left[ \begin{array}{cc}1&0\\ 0&1\end{array}\right],D=\left[\begin{array}{cc}0&0\\ 0&0\end{array}\right].\end{array}\]