# Enhancement of Yaw Moment Control for Drivers with Excessive Steering in Emergency Lane Changes
## 2 Vehicle Lateral Control Model  
### Lateral Control Model and Steady State  
As shown in Figure 1, consider the following lateral control model for a ground vehicle traveling at speeds \(V_{0}\in[V_{\text{min}},V_{\text{max}}]\) (see [23] and references therein):  
\[\dot{\mathbf{x}}_{v}=\mathbf{A}_{v}(V_{0})\mathbf{x}_{v}+\mathbf{B}_{v}(V_{0} )\delta_{f}+\mathbf{B}_{m}M_{z}+\mathbf{d}_{v},\ V_{0}\in[V_{\text{min}},V_{ \text{max}}] \tag{1a}\] \[a_{y}=\mathbf{C}_{v}(V_{0})\mathbf{x}_{v}+D_{v}\delta_{f}\] (1b) \[\alpha_{f}=\mathbf{E}_{f}(V_{0})\mathbf{x}_{v}+\delta_{f},\ \alpha_{r}= \mathbf{E}_{r}(V_{0})\mathbf{x}_{v} \tag{1c}\]\[\mathbf{x}_{v}=\left[ \begin{array}{cc}\beta&\gamma\end{array} \right]^{T}\] \[\mathbf{A}_{v}(V_{0})=\left[ \begin{array}{cc}-\frac{C_{f}+C_{r}}{ mV_{0}}&\frac{-C_{f}l_{f}+C_{r}l_{r}}{mV_{0}^{2}}-1\\ -\frac{C_{f}l_{f}+C_{r}l_{r}}{l_{z}}&-\frac{C_{f}l_{f}^{2}+C_{r}l_{r}^{2}}{l_{z }V_{0}}\end{array} \right],\ \ \ \mathbf{B}_{v}(V_{0})=\left[ \begin{array}{c}\frac{C_{f}}{ mV_{0}}\\ \frac{C_{f}l_{r}}{l_{z}}\end{array} \right],\ \ \ \mathbf{B}_{m}=\left[ \begin{array}{c}0\\ \frac{1}{l_{z}}\end{array} \right]\] \[\mathbf{C}_{v}(V_{0})=\left[ \begin{array}{cc}-\frac{C_{f}+C_{r}}{ m}&\frac{-C_{f}l_{f}+C_{r}l_{r}}{mV_{0}}\end{array} \right],\ D_{v}=\frac{C_{f}}{m},\] \[\mathbf{E}_{f}(V_{0})=\left[ \begin{array}{cc}-1&-\frac{l_{f}}{V_{0}} \end{array} \right],\ \mathbf{E}_{r}(V_{0})=\left[ \begin{array}{cc}-1&\frac{l_{r}}{V_{0}} \end{array} \right]\]  
where \(\beta\), \(\gamma\), and \(a_{y}\) are the side slip angle, yaw rate, and lateral acceleration of the vehicle body measured at the center of gravity (CG), respectively; \(\alpha_{f}\) and \(\alpha_{r}\) are the wheel slip angles of the front and rear wheels; \(\delta_{f}\) is the front-wheel steering angle; \(C_{f}\) and \(C_{r}\) denote the cornering stiffness of the front and rear axles, respectively; \(m\) represents the mass of the vehicle; and \(l_{z}\) denotes the inertia about the \(z\) axis. \(l_{f}\) and \(l_{r}\) are the distances from the center of gravity (CG) to the front and rear axles, while \(l_{d}\) is the distance from the CG to the wheel side. The yaw moment \(M_{z}\), which is mainly generated by the longitudinal tire/road friction forces \(f_{xj}\), \(j=1,2,\ldots,4\) of the four wheels, is calculated using  
\[M_{z}=-l_{d}f_{x1}\cos(\delta_{f})+l_{d}f_{x2}\cos(\delta_{f})-l_{d}f_{x3}+l_{ d}f_{x4}\]  
Since the front-wheel steering angle \(\delta_{f}\) is small in operation, the yaw moment \(M_{z}\) can approximate to  
\[M_{z}=-l_{d}f_{x1}+l_{d}f_{x2}-l_{d}f_{x3}+l_{d}f_{x4} \tag{2}\]  
The yaw moment \(M_{z}\) is restricted due to the limitation on the tire/road friction. We assume that the parameters above are nominal values, and any errors resulting from model linearization and parameter estimation are represented as the uncertain disturbance \(\mathbf{d}_{v}\in\Re^{2}\). When the vehicle is driving at a constant speed \(V_{0}\) and operating normally, the system matrix \(\mathbf{A}_{v}(V_{0})\) is asymptotically stable and the matrix pair \((\mathbf{A}_{v}(V_{0}),\mathbf{B}_{m})\) is controllable with the augmented matrix \([\mathbf{B}_{v}(V_{0})\mathbf{B}_{m}]\) of full column rank. Additionally, it is typical in vehicle system analysis to consider the driver steering angle \(\delta_{d}\) as the front-wheel steering angle \(\delta_{f}\), i.e., \(\delta_{f}=\delta_{d}\). Therefore, a reference vehicle lateral model traveling at speed \(V_{0}\) from (1) with \(\delta_{f}=\delta_{d}\), \(M_{z}=0\) and \(\mathbf{d}_{v}=\mathbf{0}\), is provided as  
\[\dot{\mathbf{x}}_{v}=\mathbf{A}_{v}(V_{0})\mathbf{x}_{v}+\mathbf{B}_{v}(V_{0}) \delta_{d} \tag{3a}\] \[a_{y}=\mathbf{C}_{v}(V_{0})\mathbf{x}_{v}+D_{v}\delta_{d}\] (3b) \[\alpha_{f}=\mathbf{E}_{f}(V_{0})\mathbf{x}_{v}+\delta_{d},\ \alpha_{r}=\mathbf{E}_{r}(V_{0})\mathbf{x}_{v} \tag{3c}\]  
Figure 1: Planar vehicle model.