# Adaptive Global Fast Sliding Mode Control for Steer-by-Wire System Road Vehicles
## 3 Steer-by-Wire System Modeling  
Figure 2 depicts the standard model of SbW system for road vehicles. As shown, the steering wheel angle sensor is used to detect the driver's reference angle and the feedback motor is used to provide the artificial steering feel.  
Similarly, the front wheel angle is detected by the pinion angle sensor. Based on the error between the reference angle and the front wheel angle, the control signal is provided to the front wheel steering motor to closely steer the front wheels according to the driver's reference angle.  
Figure 2: Steer-by-wire model.  
The equivalent second-order dynamics of the front wheels' steering motor is expressed as follows [10; 15]:  
\[J_{eq}\ddot{\delta}_{fw}+B_{eq}\dot{\delta}_{fw}+\tau_{F}+\tau_{a}=ku, \tag{13}\]  
where \(J_{eq}\) and \(B_{eq}\) are the equivalent moment of inertia and the equivalent damping of the SbW system, respectively. \(u\) is the front wheels' steering motor control input and \(k\) is the steering ratio between the steering wheel angle and the front wheels' angle, given by \(\delta_{fw}=\delta_{sw}/k\).  
It is known that many modern road vehicles use the variable steering ratio. Therefore, dividing both sides of Equation (13) by \(k\) eliminates the impact of the variable steering ratio from the proposed control scheme without compromising the steering performance. Thus, the dynamics of SbW system can be written as:  
\[J_{ek}\ddot{\delta}_{fw}+B_{ek}\dot{\delta}_{fw}+\tau_{Fk}+\tau_{ak}=u, \tag{14}\]  
where  
\[J_{ek}=\frac{J_{eq}}{k}=\frac{J_{fw}}{k}+J_{fm}k \tag{15}\]  
\[B_{ek}=\frac{B_{eq}}{k}=\frac{B_{fw}}{k}+B_{fm}k \tag{16}\]  
\[\tau_{Fk}=\frac{\tau_{F}}{k} \tag{17}\]  
\[\tau_{ak}=\frac{\tau_{a}}{k}, \tag{18}\]  
where \(J_{fw}\) and \(J_{fm}\) are the moment of inertia of the front wheels and the front wheel steering motor, respectively. \(B_{fw}\) and \(B_{sm}\) are the damping factors of the front wheels and the front wheel steering motor, respectively.  
When the vehicle is turning, the steering system experiences torque that tends to resist the attempted turn, known as self-aligning torque \(\tau_{a}\). It can be seen from Figure 3 that the resultant lateral force developed by the tire manifests the self-aligning torque. The lateral force is acting behind the tire center on the ground plane and tries to align the wheel plane with the direction of wheel travel. Therefore, the total self-aligning torque is given by [14]:  
\[\tau_{a}=F_{yf}\big{(}t_{p}+t_{m}\big{)}, \tag{19}\]  
where \(t_{p}\) is the pneumatic trail (the distance between the application point of lateral force \(F_{yf}\) to the center of tire), \(t_{m}\) is the mechanical trail, also known as the caster offset, which is the distance between the tire center and the point where the steering axis intersects with the ground plane.  
Figure 3: Self-aligning torque on front wheel.  
By substituting Equations (5) and (9) into Equation (19), \(\tau_{a}\) can be written as:  
\[\tau_{a}=2C_{f}\Bigg{(}\delta_{fw}-\frac{\dot{y}+l_{f}\dot{\psi}}{V_{x}}\Bigg{)} \big{(}t_{p}+t_{m}\big{)}. \tag{20}\]  
Moreover, \(\tau_{F}\) is the coulomb frictional torque acting on SbW system, expressed as [23]:  
\[\tau_{F}=F_{zf}\mu t_{p}\text{sign}\Big{(}\dot{\delta}_{fw}\Big{)} \tag{21}\]  
\[F_{zf}=\frac{mgl_{r}}{l_{f}+l_{r}}, \tag{22}\]  
where \(F_{zf}\) is the normal load on front axle, \(\mu\) is the coeffient of friction, \(mg\) is the vehicle's weight without any external load, and \(\text{sign}()\) signum function is used to identify the direction of the frictional torque.  
The stability of the SbW system mainly depends on the road and environmental conditions. The uncertain road surface such as dry, wet, or icy can produce considerable variations in the tire cornering stiffness coefficients \(C_{f},\ C_{r}\), which can adversely affect the controller performance. Therefore, to estimate the vehicle states and the uncertain parameter variation, the cooperative ASMO and KF are designed in the next section.