# Enhancement of Yaw Moment Control for Drivers with Excessive Steering in Emergency Lane Changes
## 5 Simulations  
The control scheme is evaluated using a nonlinear vehicle model (D-Class Sedan) within the CarSim-Simulink co-simulation environment where the wet asphalt road (\(\mu_{\max}\ =\ 0.8\)) is used for the test. The vehicle parameters provided by CarSim are listed in Table 1. In the scenario, assume the driver executes an emergency lane change on the wet asphalt road while the vehicle is traveling in an initial speed \(V_{0}=120\ \mathrm{km/h}\). The overreacting response of the driver steering angle (in degrees) is provided as follows:  
\[\delta_{d}(t)=\begin{cases}0,&if\ 0\leq t<0.375\\ \min\{3.75,\ 5\sin(2(t-0.375))\},if\ 0.375\leq t<0.375+\pi\\ 0,&if\ 0.375+\pi\leq t\end{cases} \tag{63}\]  
Without any control assistance, i.e., \(\delta_{sat}=0\), \(\delta_{f}=\delta_{d}\), and \(M_{z}=0\), Figure 5(a) depicts the history of the driver steering angle \(\delta_{d}\) and the related steering angle limit \(\widetilde{\delta}_{\lim}\) which is updated with the vehicle speed \(V_{0}\) (in black) as shown in Figure 5(b). The responses of the four wheel speeds \(\omega_{j}\) are also shown in Figure 5(b). Figure 5(c),d displays the subsequent responses of the vehicle state \((\beta,\gamma)\). It is noticeable that the vehicle state \((\beta,\gamma)\) rapidly deviates from their reference \((\beta_{ref},\gamma_{ref})\). Figure 5(e),f illustrates the history of the vehicle acceleration \((a_{x},a_{y})\) and the combined slip magnitude \(\left\|\mathbf{\mathrm{S}}_{j}\right\|\) of the four wheels. At around \(0.8\ \mathrm{s}\), the lateral acceleration \(a_{y}\) begins to saturate with \(a_{y}\approx\widetilde{a}_{y,\lim}=0.85\times 0.8\ \mathrm{g}=\ 0.68\ \mathrm{g}\), and the combined slips of the two front wheels go beyond their linear region, i.e., \(\left\|\mathbf{\mathrm{S}}_{j}\right\|\ \geq\ 0.08,j=1,2\), resulting in the vehicle skidding and losing maneuverability from about \(0.8\ \mathrm{s}\).  
The performance of the vehicle control system under (I) the integration control (25), (27), and (51); (II) the enhanced integration control (25), (27), and (56); and (III) the acting-alone enhanced yaw moment control (56) is demonstrated and compared in simulations. The related parameters for the saturation angle scheme (27), the yaw moment control (51), (56), LMIs (52), and the wheel control torque (40) are chosen below:  
\begin{table}
\begin{tabular}{l l l} \hline \hline \(m=1530\ \mathrm{kg}\) & \(l_{z}=2315.3\ \mathrm{kg}\)-\(\mathrm{m}^{2}\) & \(l_{f}=1.110\ \mathrm{m}\) \\ \hline \(l_{r}=1.67\ \mathrm{m}\) & \(l_{d}=0.775\ \mathrm{m}\) & \(r_{e}=0.325\ \mathrm{m}\) \\ \hline \(C_{f}\)= \(11.6130\ \times\ 10^{4}\ \mathrm{Nt/rad}\) & \(C_{r}\)= \(8.3900\times 10^{4}\ \mathrm{Nt/rad}\) & \(\mu_{\max}=0.8\) \\ \hline \(\widetilde{a}_{y,\lim}=0.68\ \mathrm{g}\ (6.6708\ \mathrm{m}/\mathrm{s}^{2})\) & \(\widetilde{\alpha}_{f,\lim}=0.0528\ \mathrm{(rad)}\) & \(\widetilde{\alpha}_{r,\lim}=0.0486\ \mathrm{(rad)}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Vehicle parameters.  
\(\alpha=30\), \(\widetilde{s}_{Res,allow}=0.08\), \(\widetilde{s}_{Sallow}=0.0528\), \(\widetilde{s}_{L,allow}=0.0601\), \(k=14\), \(\widetilde{M}_{z,allow}=9781.2\), \(\alpha_{c}=7\), \(\gamma_{c}=0.3\), \(\mu_{c}=0.2\), \(\rho_{c}=0.044\), \(\rho_{\xi}=5868.73\), \(g_{c}=1.5\), \(V_{\max}=34\) m/s, \(V_{\min}=20\) m/s \(\mathbf{Q}=\left[\begin{array}{cc}0.08152&0.00082\\ 0.00082&0.08535\end{array}\right]\), \(\mathbf{Y}=\left[\begin{array}{cc}-797.97698&-1832.24857\end{array}\right]\), \(\mathbf{P}=\left[\begin{array}{cc}12.26665&-0.11762\\ -0.11762&11.71726\end{array}\right]\), \(\mathbf{K}=\left[\begin{array}{cc}-9572.98975&-21375.07610\end{array}\right]\), \(\gamma_{H}\mathbf{B}_{m}^{T}\mathbf{P}=\left[-508.0318\ \ 50607.9673\right]\), \(\gamma_{H}=1\times 10^{7}\) All the cases use the same parameter values as provided above.