# Adaptive Global Fast Sliding Mode Control for Steer-by-Wire System Road Vehicles
## 2 Vehicle Dynamics Modeling  
Figure 1 illustrates the simplest bicycle model of a vehicle, which has a central front wheel and a central rear wheel, in place of two front and two rear wheels. The vehicle has two degrees of freedom, represented by the lateral motion \(y\) and the yaw angle \(\psi\). According to Figure 1, the dynamics along the \(y\) axis and yaw axis are described as [38; 39]:  
\[m\Big{(}\ddot{y}+V_{x}\dot{y}\Big{)}=F_{yf}\cos\delta_{fw}+F_{xf}\sin\delta_{fw }+F_{yr} \tag{1}\]  
\[I_{x}\ddot{\psi}=I_{f}\Big{(}F_{yf}\cos\delta_{fw}+F_{xf}\sin\delta_{fw}\Big{)} -I_{r}F_{yr}, \tag{2}\]  
where \(\ddot{y}\), \(\dot{\psi}\), and \(\ddot{\psi}\). are the acceleration with respect to the \(y\) axis motion, yaw rate, and yaw acceleration, respectively. \(I_{f}\) and \(I_{r}\) represent the distance of front and rear axes from the center of gravity, respectively. \(m\) and \(I_{z}\) are the mass of vehicle and the moment of inertia along the yaw axis, respectively. \(V_{x}\) denotes the longitudinal vehicle velocity at the center of gravity. \(F_{xf}\) and \(F_{yr}\) are the longitudinal forces of the front and rear wheels, respectively. \(F_{yf}\) and \(F_{yr}\) are the lateral frictional forces of front and rear wheels, respectively, as shown in Figure 1.  
In order to simplify the model, it is assumed that longitudinal forces \(Fx_{f},F_{xr}\) are equal to zero and by using the small angle approximation, i.e., \(\cos\delta_{fw}\approx 1\), the simplified dynamics can be modeled as follows:  
\[m\Big{(}\ddot{y}+V_{x}\dot{\psi}\Big{)}=F_{yf}+F_{yr} \tag{3}\]  
\[I_{z}\ddot{\psi}=I_{f}F_{yf}-I_{r}F_{yr}. \tag{4}\]  
For small slip angles, the lateral frictional forces are proportional to slip-angle \(\alpha_{f}\) and \(\alpha_{r}\) at the front and rear wheels, respectively. Therefore, lateral forces are defined as:  
\[F_{yf}=2C_{f}.\alpha_{f} \tag{5}\]  
\[F_{yr}=2C_{r}.\alpha_{r}, \tag{6}\]  
where  
\[\alpha_{f}=\delta_{fw}-\frac{V_{y}+I_{f}\dot{\psi}}{V_{x}} \tag{7}\]  
Figure 1: Bicycle model of vehicle.  
\[\alpha_{r}=-\frac{V_{y}-l_{f}\dot{\psi}}{V_{x}}, \tag{8}\]  
where \(C_{f}\) and \(C_{r}\) are the front and rear tires' cornering stiffness coefficients. \(\delta_{fw}\) denotes the steering angle of front wheels, which is considered the same for both front wheels, and factor 2 accounts for two front and two rear wheels, respectively.  
By using the small angle approximation \(V_{y}=\dot{y}\)[38], Equations (7) and (8) can be written as:  
\[\alpha_{f}=\delta_{fw}-\frac{\dot{y}+l_{f}\dot{\psi}}{V_{x}} \tag{9}\]  
\[\alpha_{r}=-\frac{\dot{y}-l_{f}\dot{\psi}}{V_{x}}. \tag{10}\]  
Substituting Equations (5), (6), (9) and (10) into Equations (3) and (4), the state space model is represented as:  
\[\dot{x}=Ax+B\delta_{fw}, \tag{11}\]  
where  
\[A=\left[\begin{array}{cc}-2\Big{(}\frac{C_{f}+C_{r}}{mV_{x}}\Big{)}&-\Big{(} V_{x}+2\Big{(}\frac{l_{f}C_{f}-l_{r}C_{r}}{mV_{x}}\Big{)}\Big{)}\\ -2\Big{(}\frac{l_{f}C_{f}-l_{r}C_{r}}{l_{z}V_{x}}\Big{)}&-2\Big{(}\frac{l_{f} ^{2}C_{f}+l_{r}^{2}C_{r}}{l_{z}V_{x}}\Big{)}\end{array}\right],B=\left[ \begin{array}{c}\frac{2C_{f}}{m}\\ \frac{2l_{f}C_{f}}{l_{z}}\end{array}\right]. \tag{12}\]