## 3 Vehicle Dynamics Model  
In the process of lane-changing, in addition to maintaining the original longitudinal movement, the vehicle also needs to make appropriate lateral movement according to the road conditions. A schematic diagram of vehicle kinematic model is shown in Figure 2. The motion of the two dimensions is not simply superimposed. Considering generality, the dynamics model of vehicle in lane change process is modeled. In order to facilitate the problem analysis and the realization of control objectives, in the inertial coordinate system **OXY**, the 2-DOF vehicle dynamic equation of the vehicle is established as follows:  
\[\left[\begin{array}{c}\dot{y}\\ \ddot{y}\\ \dot{\delta}\\ \ddot{\delta}\\ \end{array}\right]=\left[\begin{array}{cccc}0&1&0&0\\ 0&-\frac{2\,k_{f}+2\cdot k_{b}}{m}&\frac{2\,k_{f}+2\cdot k_{b}}{m}&\frac{2\,k_ {f}-\,l_{f}-2\cdot k_{b}\cdot l_{b}}{m}\\ 0&0&0&1\\ 0&\frac{2\,k_{f}-\,l_{f}+2\cdot k_{b}\cdot l_{b}}{l_{z}v}&\frac{2\,k_{f}-\,l_ {f}-2\cdot k_{b}\cdot l_{b}}{l_{z}}&-\frac{2\,k_{f}\,l_{f}^{2}+2\cdot k_{b}\cdot l _{b}^{2}}{l_{z}v}\\ \end{array}\right]u \tag{1}\]  
where \(y\) and \(\delta\) denote the lateral offset and the yaw angle of the vehicle, respectively. \(m\) is the quality of the vehicle. \(l_{z}\) represents the moment of inertia around the vertical-axis of the vehicle. \(k_{f}\) and \(k_{b}\) stand for the front and rear tire cornering stiffness, respectively. \(l_{f}\) and \(l_{b}\) are the distance between the front and rear axles and the center of the vehicle mass, respectively. \(v\) is the longitudinal speed. \(y\) is the distances from the center of mass \(P\) of the vehicle to the \(x\)-axis. \(\varpi\) is the front wheel angle of the vehicle which is described using the controller input variable \(u\).  
The vehicle dynamic Equation (1) can be simplified by taking into account the system's internal parameter inaccuracy and external interference. The model can be shortened to:  
Figure 2: Schematic diagram of the Vehicle Kinematic Model.  
\[\left[\begin{array}{c}\ddot{y}\\ \ddot{\delta}\end{array}\right]=\left[\begin{array}{cc}k_{11}&k_{12}\\ k_{21}&k_{22}\end{array}\right]\left[\begin{array}{c}\ddot{y}\\ \ddot{\delta}\end{array}\right]+\left[\begin{array}{cc}0&1\\ 0&1\end{array}\right]\left[\begin{array}{c}y\\ \delta\end{array}\right]+\left[\begin{array}{c}\ell_{1}\\ \ell_{2}\end{array}\right]u+\Delta f \tag{2}\]  
where \(\left[\begin{array}{cc}k_{11}&k_{12}\\ k_{21}&k_{22}\end{array}\right]=K_{\prime}\left[\begin{array}{cc}0&1\\ 0&1\end{array}\right]=F\), \(\left[\begin{array}{c}\ell_{1}\\ \ell_{2}\end{array}\right]=\Im\). Simplify the definition of two of the key variables to form the expression \(\theta=\left[\begin{array}{c}y\\ \delta\end{array}\right]\).  
\[\ddot{\theta} = K\dot{\theta}+F\theta+\Im u+\Delta f \tag{3}\] \[=f+\Im u+\Delta f\]  
where \(\Delta f\) represents the lumped error of the system.  
According to the state equation of the system, the lateral displacement, lateral velocity, yaw angle, and yaw velocity of the vehicle can be analyzed under the given input of the front wheel angle. The purpose of lateral lane change control is to reduce the deviation of the trajectory designed for tracking. It is required that the controller can analyze the deviation of the vehicle tracking at a given front wheel angle through the state equation.