## 3 Build the Controller Model
### The Driver Model Based on Preview  
In the process of trajectory tracking of intelligent vehicles, the lateral error of the vehicle and the heading error of the vehicle need to be considered. If only the front wheel angle is used as a variable, only one error can be controlled. Therefore, for the heading error of the intelligent vehicle when it drives to the preview point, this paper introduces a total error \(y_{m}\) to integrate the two errors, as shown in Figure 8.  
When the vehicle is driving, the longitudinal speed \(v_{x}\) is much greater than the lateral speed \(v_{y}\). Therefore, it is assumed that in the time \(\Delta t\), the yaw rate of the vehicle remains unchanged, the velocity of the center of mass remains unchanged, and the vehicle moves in a circle. In order to drive to the desired trajectory, the driver previews point \(D\). The actual trajectory follows the arc \(\dot{EC}\), and runs to point \(D\) within \(\Delta t\) time. Because the longitudinal velocity \(v_{x}\) is much larger than the lateral velocity \(v_{y}\), the point \(C\) coincides with the point \(D\) within \(\Delta t\) time. So, \(AC=AD=y_{m}\). Therefore, a preview-based vehicle trajectory error model is established [28], as shown in Figure 8. \(x_{m}\) is the preview distance, \(y_{m}\) is the total error, \(y_{m1}\) is the lateral error, \(\mu_{1}\) is the actual track heading angle of the intelligent vehicle, and \(\mu_{2}\) is the expected track heading angle of the intelligent vehicle.  
According to the vehicle trajectory error model based on preview in Figure 8, the formula obtained is as follows:  
\[\left\{\begin{array}{l}y_{m}=y_{m1}+x_{m}sin(\Delta\mu)\\ \dot{y}_{m1}=v_{y}cos(\Delta\mu)-v_{x}sin(\Delta\mu)\\ \dot{x}_{m1}=v_{y}sin(\Delta\mu)+v_{x}cos(\Delta\mu)\\ \Delta\mu=\mu_{1}-\mu_{2}\end{array}\right. \tag{22}\]  
The heading angular velocity \(\dot{\mu}_{2}\) of the desired trajectory is as follows:  
\[\dot{\mu}_{2}=\frac{\dot{x}_{m1}}{R}=\rho\dot{x}_{m1} \tag{23}\]  
Figure 8: The driver model based on a preview.  
where \(R\) is the radius of the actual trajectory within \(\Delta t\), \(\rho\) is the radius of curvature of the actual trajectory, and \(x_{m1}\) is the distance traveled by the intelligent vehicle along the reference trajectory arc \(\hat{EC}\) within \(\Delta t\) time.  
At time \(\Delta t\), since the longitudinal velocity \(v_{x}\) is much larger than the lateral velocity \(v_{y}\), \(\Delta\mu\) is very small, according to the small-angle approximation:  
\[\left\{\begin{array}{l}y_{m}=y_{m1}+x_{m}\Delta\mu\\ \dot{y}_{m1}=v_{y}-v_{x}\Delta\mu\\ \dot{x}_{m1}=v_{y}\Delta\mu+v_{x}\\ \Delta\mu=\mu_{1}-\mu_{2}\\ \dot{\mu}_{1}\approx\dot{B}\\ \dot{\mu}_{2}=\frac{\dot{x}_{m1}}{R}=\rho\dot{x}_{m1}\end{array}\right. \tag{24}\]  
where \(\dot{B}\) is the yaw rate.  
From Formulas (21) and (24), the following formula is obtained:  
\[\left\{\begin{array}{l}\ddot{y}_{m}=\ddot{y}_{m1}+x_{m}\Delta\ddot{\mu}\\ \ddot{y}_{m1}=\ddot{v}_{y}-\dot{v}_{x}\Delta\mu-v_{x}\Delta\dot{\mu}\\ \Delta\ddot{\mu}=\ddot{B}-\rho\ddot{x}_{m1}\\ \ddot{x}_{m1}=\dot{v}_{y}\Delta\mu+v_{y}\Delta\dot{\mu}+\dot{v}_{x}\end{array}\right. \tag{25}\]  
The trajectory error \(\ddot{y}_{m}\) is obtained according to Equations (14), (15) and (21):  
\[\ddot{y}_{m}=\dot{v}_{y}-\dot{v}_{x}\Delta\mu-v_{x}\Delta\dot{\mu}+x_{m}\left( \frac{2L_{f}C_{\alpha f}\Big{(}\dot{\delta}_{f}-\theta_{vf}\Big{)}+2L_{r}C_{ \alpha r}\theta_{vr}}{I_{z}}-\rho\ddot{x}_{m1}\right) \tag{26}\]  
From Formulas (18), (19) and (26), the following formula is obtained:  
\[\left\{\begin{array}{l}\ddot{y}_{m}=\dot{\varphi}_{1}+\varphi_{2}\delta_{f}+ d\\ \varphi_{1}=\dot{v}_{y}-\dot{v}_{x}\Delta\mu-v_{x}\Delta\dot{\mu}+\varphi_{3}\\ \varphi_{2}=x_{m}\frac{2L_{r}C_{\alpha f}}{I_{z}}\\ \varphi_{3}=x_{m}\Big{(}2L_{r}C_{\alpha r}\Big{(}v_{y}-L_{r}\dot{B}\Big{)}-2L_ {f}C_{\alpha f}\Big{(}v_{y}+L_{f}\dot{B}\Big{)}\Big{)}/v_{x}I_{z}\end{array}\right. \tag{27}\]  
where \(d\) is the disturbance error which represents the external disturbance.