# Practical Approach for Developing Lateral Motion Control of Autonomous Lane Change System
### Lateral Motion Model in Plane Domain  
In this study, a generalized lateral dynamic motion model of a vehicle is considered. The lateral dynamics of the bicycle model as shown in Figure 2 are described by summing the forces \(F_{y}\) and moment \(M_{z}\) at the center of gravity of the vehicle:  
\[\begin{split}\sum F_{y}&=F_{yf}+F_{yr}=m(\dot{y}+ \dot{\psi}V_{x})\\ \sum M_{z}&=l_{f}F_{yf}+l_{r}F_{yr}=l_{z}\ddot{\psi}.\end{split} \tag{1}\]  
For small slip angles, the lateral tire force \(F_{y}\) can be approximated as a linear function of the tire slip angle \((\alpha_{f},\alpha_{r})\)[16; 17]. The tire slip angle is defined as the angle between the orientation of the tire and that of the velocity vector of the wheel \(\theta_{V}\):  
\[\begin{split} F_{yf}&=2C_{af}\alpha_{f}=2C_{af}( \delta-\theta_{V_{f}})\\ F_{yr}&=2C_{ar}\alpha_{r}=2C_{ar}(-\theta_{V_{r}}),\end{split} \tag{2}\]  
where \(C_{\alpha}\) denotes tire cornering stiffness. Using small angle approximations for both \(\theta_{V_{f}}\) and \(\theta_{V_{r}}\), we obtain  
\[\begin{split}\theta_{V_{f}}=\frac{\dot{y}+l_{f}\dot{\psi}}{V_{x}} =\frac{-V_{y}+l_{f}\dot{\psi}}{V_{x}}&=-\beta+\frac{l_{f}\dot{ \psi}}{V_{x}},\\ \theta_{V_{r}}=\frac{\dot{y}-l_{r}\dot{\psi}}{V_{x}}& =\frac{-V_{y}-l_{r}\dot{\psi}}{V_{x}}&=-\beta-\frac{l_{r}\dot{ \psi}}{V_{x}},\end{split} \tag{3}\]  
with  
\[\beta\approx\tan(\beta)=\frac{V_{y}}{V_{x}}. \tag{4}\]  
For the LKS, we define the following two state variables:  
* \(e_{y}\) is the lateral offset error at the center of gravity of the vehicle from the lane center.
* \(e_{\psi}\) denotes the heading error of the vehicle with respect to the desired road.  
Then, the dynamic model for the lateral control system can be described as follows in terms of state vector \(\mathbf{x}=\begin{bmatrix}e_{y}&\dot{e}_{y}&e_{\psi}&\dot{e}_{\psi}\end{bmatrix}^{T}\), control input \(u=\delta\), and external signal \(\varphi=\begin{bmatrix}\dot{\psi}_{des}\end{bmatrix}^{T}\)[12; 16; 18]:  
\[\dot{\mathbf{x}}=A\mathbf{x}+Bu+B_{\varphi}\varphi \tag{5}\]  
Figure 2: Bicycle model of the vehicle for lateral dynamics.  
where,  
\[A=\begin{bmatrix}0&1&0&0\\ 0&a_{22}&a_{23}&a_{24}\\ 0&0&0&1\\ 0&a_{42}&a_{43}&a_{44}\end{bmatrix},\ B=\begin{bmatrix}0\\ b_{21}\\ 0\\ b_{41}\end{bmatrix},\ B_{\varphi}=\begin{bmatrix}0\\ a_{24}-V_{x}\\ 0\\ a_{44}\end{bmatrix}\]  
with  
\[a_{22}=-\frac{2(C_{\alpha f}+C_{\alpha r})}{mV_{x}},\ \ a_{23}=-a_{22}V_{x},\]  
\[a_{24}=-\frac{2(C_{\alpha f}l_{f}-C_{\alpha r}l_{r})}{mV_{x}^{2}},\ \ a_{42}=- \frac{2(C_{\alpha f}l_{f}-C_{\alpha r}l_{r})}{l_{z}},\]  
\[a_{43}=-a_{42},\ \ a_{44}=-\frac{2(C_{\alpha f}l_{f}^{2}+C_{\alpha r}l_{r}^{2} )}{l_{z}V_{x}},\]  
\[b_{21}=\frac{2C_{\alpha f}}{mV_{x}},\ \ b_{41}=\frac{2C_{\alpha f}l_{f}}{l_{z}}.\]  
Equation (5) represents the lateral motion of the vehicle with respect to the road in the plane domain. \(e_{y}\), \(e_{\psi}\), and \(\dot{e}_{\psi}\) can be obtained using the camera sensor [11].  
**Remark 1**.: _The lateral offset, \(e_{y}\), is measured based on the current driving lane. When the vehicle changes the lane, \(e_{y}\) increases gradually until the vehicle crosses the lane. After the vehicle crosses the lane, a jump of \(e_{y}\) occurs because of the change of driving lane, and the error decreases gradually. In other words, the sign of the lateral offset suddenly changes during lane change in the plane motion model when the camera vision sensor begin obtaining the target lane information instead of the information of the original lane. That is, the lateral offset jumps during lane change. To overcome these problems in the plane motion model, we propose the cylinder domain approach, which can remove state jumps in the lateral offset._