# Enhancement of Yaw Moment Control for Drivers with Excessive Steering in Emergency Lane Changes
## 2 Vehicle Lateral Control Model
### Vehicle Safety Reference State and Reference Steering Angle  
Figure 2 illustrates the flow diagram depicting the relationship between the steady-state solution (4)-(9), the vehicle speed \(V_{0}\), and the driver steering angle \(\delta_{d}\). As shown in the figure, these steady-state solutions are not independent but coupled. By using the steady-state yaw rate \(\gamma_{ss}\) as a design parameter, we can reverse the flow in Figure 2 and derive a new relationship as follows:  
\[\beta_{ss}(V_{0})=\Bigg{(}l_{r}-\frac{ml_{f}}{C_{r}(l_{f}+l_{r})}V_{0}^{2} \Bigg{)}\frac{\gamma_{ss}(V_{0})}{V_{0}} \tag{10}\]  
\[a_{y,ss}(V_{0})=V_{0}\gamma_{ss}(V_{0}) \tag{11}\]  
\[\delta_{d}(V_{0})=(l_{f}+l_{r})\Big{(}1+KV_{0}^{2}\Big{)}\frac{\gamma_{ss}(V_ {0})}{V_{0}} \tag{12}\]  
\[\alpha_{f,ss}(V_{0})=\frac{l_{r}mV_{0}}{(l_{f}+l_{r})C_{f}}\gamma_{ss}(V_{0})= \frac{l_{r}m}{(l_{f}+l_{r})C_{f}}a_{y,ss}(V_{0}) \tag{13}\]  
Figure 2: Diagram of the steady state relationship (4)–(8), vehicle speed \(V_{0}\), and driver steering angle \(\delta_{d}\). \[\alpha_{r,ss}(V_{0})=\frac{l_{f}mV_{0}}{(l_{f}+l_{r})C_{r}}\gamma_{ss}(V_{0})= \frac{l_{f}m}{(l_{f}+l_{r})C_{r}}a_{y,ss}(V_{0}) \tag{14}\]  
In the active safety design, the steady-state yaw rate \(\gamma_{ss}\) is considered to be limited as below (see [24; 25; 26] and references therein):  
\[|\gamma_{ss}(V_{0})|\leq\widetilde{\gamma}_{\text{lim}}(V_{0})\ \ =0.85\frac{l_{\text{max}}g}{V_{0}} \tag{15}\]  
where \(\mu_{\text{max}}\) is the maximum tire/road friction coefficient and \(g\) is the gravity constant. The limitation (15) is used to maintain the lateral acceleration \(a_{y}\) unsaturated. By using the relationship (10)-(14) and the restriction (15), the allowable operating ranges of these values can be obtained as follows:  
\[|\beta_{ss}(V_{0})|\leq\widetilde{\beta}_{\text{lim}}(V_{0})\ \ =\left|\frac{l_{r}}{V_{0}}-\frac{ml_{f}V_{0}}{C_{r}(l_{f}+l_{r})}\right| \widetilde{\gamma}_{\text{lim}}(V_{0}) \tag{16}\]  
\[|a_{y,ss}(V_{0})|\leq\widetilde{a}_{y,\text{lim}}(V_{0})=V_{0}\widetilde{ \gamma}_{\text{lim}}(V_{0})=0.85\mu_{\text{max}}g \tag{17}\]  
\[|\delta_{d}|\leq\widetilde{\delta}_{\text{lim}}(V_{0})=\left(l_{f}+l_{r} \right)\left|\frac{1+KV_{0}^{2}}{V_{0}}\right|\widetilde{\gamma}_{\text{lim}} (V_{0}) \tag{18}\]  
\[a_{y,ss}(V_{0})=V_{0}\gamma_{ss}(V_{0}) \tag{11}\]0  
\[a_{y,ss}(V_{0})=V_{0}\gamma_{ss}(V_{0}) \tag{11}\]1  
By using the restriction \((\widetilde{\gamma}_{\text{lim}},\widetilde{\beta}_{\text{lim}},\widetilde{ \delta}_{\text{lim}})\) above, we can establish the safety reference state \((\beta_{ref},\gamma_{ref})\) and the reference driver steering angle \(\delta_{d,ref}\) as the following functions of the vehicle speed \(V_{0}\):  
\[\beta_{ref}(V_{0})= \begin{cases}\beta_{ss}\ (V_{0})\ \ \ \ \ \ \ \ \,if\ |\beta_{ss}(V_{0})|\leq\widetilde{\beta}_{\text{lim}}(V_{0})\\ \widetilde{\beta}_{\text{lim}}(V_{0})\ \ sign(\ \beta_{ss}\ (V_{0})),if\ |\beta_{ss}(V_{0})|> \widetilde{\beta}_{\text{lim}}(V_{0})\end{cases}  \tag{21}\]  
\[\gamma_{ref}(V_{0})= \begin{cases}\gamma_{ss}(V_{0})\ \ \ \ \ \ \,if\ |\gamma_{ss}(V_{0})|\leq \widetilde{\gamma}_{\text{lim}}(V_{0})\\ \widetilde{\gamma}_{\text{lim}}(V_{0})\ sign(\gamma_{ss}(V_{0})),if\ |\gamma_{ss}(V_{0})|> \widetilde{\gamma}_{\text{lim}}(V_{0})\end{cases}  \tag{22}\]  
\[\delta_{d,ref}(V_{0},\delta_{d})= \begin{cases}\delta_{d}\ \ \ \ \,if\ |\delta_{d}|\leq \widetilde{\delta}_{\text{lim}}(V_{0})\\ \widetilde{\delta}_{\text{lim}}(V_{0})sign(\delta_{d}),if\ |\delta_{d}|> \widetilde{\delta}_{\text{lim}}(V_{0})\end{cases}  \tag{23}\]  
The steady state solution \((\beta_{ss},\gamma_{ss},\delta_{d})\) and the restriction parameter \((\widetilde{\beta}_{\text{lim}},\widetilde{\gamma}_{\text{lim}},\widetilde{ \delta}_{\text{lim}})\) have the relationship  
\[\mathbf{A}_{v}(V_{0}) \begin{bmatrix}\beta_{ss}\\ \gamma_{ss}\end{bmatrix} +\mathbf{B}_{v}(V_{0})\delta_{d}=0\]  
\[\mathbf{A}_{v}(V_{0}) \begin{bmatrix}\widetilde{\beta}_{\text{lim}}sign(\beta_{ ss})\\ \widetilde{\gamma}_{\text{lim}}sign(\gamma_{ss})\end{bmatrix}