## 5 Simulation and Validation
### Disscussion  
From results in Figure 6, it can be seen that under identical vertical loads, when slip angle was smaller than 0.08 rad, lateral force on front rear tires increases linearly with increase in slip angle. However, when slip angle was larger than 0.08 rad, rate of lateral force increment gradually slows down, and relationship with slip angle is no longer linear. In addition, for identical slip angles, a larger vertical load was accompanied by a larger lateral force. Thus, for the tire model, lateral force is a function of slip angle and vertical load. Relationships between slip angle and vertical load were nonlinear [18]. This leads to nonlinear behavior of tire lateral forces under various steering wheel inputs. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Road Excitation** & \multicolumn{3}{c}{**Level ISO-\(A\) Excitation**} & \multicolumn{3}{c}{**Level ISO-\(C\) Excitation**} \\ \hline \multicolumn{6}{c}{Accuracy of state calculation/\%} \\ Roll angle & Roll rate & Yaw rate & Roll angle & Roll rate & Yaw rate \\ \hline \(F_{1}\) (Kurtosis) & 100 & 81.5 & 98.0 & 99.5 & 89.1 & 70.2 \\ \(F_{2}\) (Crest factor) & 91.8 & 88.1 & 83.9 & 94.5 & 97.7 & 54.8 \\ \(F_{3}\) (Clearance factor) & 89.2 & 94.9 & 73.1 & 91.1 & 80.9 & 45.9 \\ \(F_{4}\) (Shape factor) & 99.8 & 98.7 & 99.6 & 99.6 & 97.4 & 94.1 \\ \(F_{5}\) (Impulse factor) & 92.0 & 87.0 & 84.6 & 94.9 & 99.8 & 55.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Simulation accuracy of comparing 9-DOF model and CARSIM® with steering wheel 10\({}^{\circ}\) input. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & **Feature Name** & **Expression** & **Parameters** & **Feature Name** & **Expression** \\ \hline \multirow{4}{*}{\(F_{1}\)} & \multirow{4}{*}{Kurtosis} & \(\frac{N}{\sum\limits_{n=1}^{N}(x(n)-\frac{\max(x(n))}{N})}\) & \multirow{4}{*}{\(F_{2}\)} & \multirow{4}{*}{Crest factor} & \(\frac{\max(x(n))}{\sqrt{\frac{N}{\sum\limits_{n=1}^{N}(x(n))^{2}}}}\) \\  & & \(\frac{(N-1)(\frac{\max(x(n))}{N})}{N-1}\) & & & \\ \multirow{4}{*}{\(F_{3}\)} & Clearance factor & \(\frac{\max(x(n))}{\left(\frac{\sum\limits_{n=1}^{N}(x(n))^{2}}{N}\right)}\) & \multirow{4}{*}{\(F_{4}\)} & \multirow{4}{*}{Shape factor} & \(\sqrt{\frac{\sum\limits_{n=1}^{N}(x(n))^{2}}{N-1}}\) \\  & & \(\frac{\max(x(n))}{N}\) & & & \\ \multirow{4}{*}{\(F_{5}\)} & \multirow{4}{*}{Impulse factor} & \(\frac{\max(x(n))}{N}\) & & & \\  & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Statistics feature parameters. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Road Excitation** & \multicolumn{3}{c}{**Level ISO-\(A\) Excitation**} & \multicolumn{3}{c}{**Level ISO-\(C\) Excitation**} \\ \hline \multicolumn{6}{c}{Accuracy of state calculation/\%} \\ Roll angle & Roll rate & Yaw rate & Roll angle & Roll rate & Yaw rate \\ \hline \(F_{1}\) (Kurtosis) & 97.7 & 46.5 & 97.0 & 93.5 & 11.7 & 70.2 \\ \(F_{2}\) (Crest factor) & 86.8 & 73.2 & 83.9 & 81.5 & 43.3 & 54.1 \\ \(F_{3}\) (Clearance factor) & 88.4 & 14.3 & 65.0 & 79.2 & 1.98 & 40.8 \\ \(F_{4}\) (Shape factor) & 95.5 & 45.3 & 99.6 & 95.6 & 31.1 & 94.1 \\ \(F_{5}\) (Impulse factor) & 88.4 & 33.2 & 83.6 & 80.4 & 13.4 & 50.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Simulation accuracy of comparing 6-DOF model and CARSIM® with steering wheel 10\({}^{\circ}\) input. To further illustrate superiority of the proposed model, \(5^{\circ}\) and \(15^{\circ}\) steering wheel input situations were also simulated. Similar results were obtained without road excitation, i.e., 6-DOF half-car model and 9-DOF full-car model are approximately equivalent under these conditions. However, 6-DOF model and 9-DOF model have significant differences with road input. Corresponding simulation results for different steering wheel inputs and various road excitations are shown in Tables 7-10, which are described as shown in Figure 12. In Figure 12a, feature \(F_{1}\), comparison of 6-DOF and 9-DOF models for road level ISO-\(A\)/\(C\) excitation is shown.