## 2 10-DOF Vehicle Dynamics Model  
In this section, a 10-DOF vehicle model is established considering the lateral, longitudinal, yaw, vertical, roll, pitch, and vertical runout of four wheels. The modeling process makes the following assumptions: ignore the influence of the road slope, cross wind, and the impact of a suspension deformation and wheel alignment on the steering performance and assume that the front wheels are the only steering wheels, and the angles of the left and right front wheel are basically the same. The vehicle parameters are as Table 1. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Description** & **Parameter** & **Unit** & **Value** \\ \hline Total mass & \(m\) & kg & 1395 \\ Sprung mass & \(m_{s}\) & kg & 1245 \\ Distance between front axle and center & \(l_{f}\) & m & 1.080 \\ of gravity (CG) & \(l_{r}\) & m & 1.620 \\ Distance between rear axle and CG & \(l_{r}\) & m & 1.535 \\ Wheel track & \(d\) & m & 1.535 \\ Rotating mass conversion factor & \(\zeta\) & - & 1.06 \\ Frontal area & \(A\) & m\({}^{2}\) & 2.01 \\ Air resistance coefficient & \(C_{D}\) & - & 0.29 \\ Roll moment of inertia & \(l_{x}\) & Kgm\({}^{2}\) & 480 \\ Yaw moment of inertia & \(l_{z}\) & Kgm\({}^{2}\) & 1356 \\ Pitch moment of inertia & \(l_{y}\) & Kgm\({}^{2}\) & 1356 \\ Front suspension stiffness & \(k_{1},k_{2}\) & N/m & 27,358 \\ Front suspension damping & \(c_{1},c_{2}\) & Ns/m & 1695 \\ Rear suspension stiffness & \(k_{3},k_{4}\) & N/m & 19,600 \\ Rear suspension damping & \(c_{3},c_{4}\) & Ns/m & 1965 \\ Rolling resistance coefficient & \(f\) & - & 0.009 \\ Tire radius & \(R\) & m & 0.298 \\ Vertical stiffness of front tire & \(k_{11},k_{t2}\) & N/m & 87,000 \\ Vertical stiffness of rear tire & \(k_{13},k_{t4}\) & N/m & 92,000 \\ Wheel moment of inertia & \(J\) & Kgm\({}^{2}\) & 2.2 \\ Altitude difference between roll center & \(h_{r}\) & m & 0.254 \\ and sprung mass CG & \(h_{p}\) & m & 0.104 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Vehicle parameters in this paper. The expressions of lateral, longitudinal, and yaw motion of the whole vehicle are:  
\[\left\{ \begin{matrix}m(\dot{V}_{y}+rV_{x})-m_{s}h_{r}\dot{\phi}=F_{y}\\ F_{y}=F_{y3}+F_{y4}+\left(F_{y1}+F_{y2}\right)\cos\delta+\left(F_{x1}+F_{x2} \right)\sin\delta\end{matrix} \right. \tag{1}\]  
\[\left\{ \begin{matrix}m(\dot{V}_{x}-rV_{y})+m_{s}h_{r}\dot{\phi}=F_{x}\\ F_{x}=F_{x3}+F_{x4}+(F_{x1}+F_{x2})\cos\delta-\left(F_{y1}+F_{y2}\right)\sin \delta-mgf-\frac{C_{D}AV_{x}^{2}}{21.15}-\xi m\dot{V}_{x}\\ \end{matrix} \right. \tag{2}\]  
\[ \begin{matrix}I_{z}\dot{r}=\left(F_{y1}+F_{y2}\right)I_{f}\cos\delta+\left(F_{ x1}+F_{x2}\right)I_{f}\sin\delta-\left(F_{y3}+F_{y4}\right)I_{r}+\\ \left(F_{y1}-F_{y2}\right)\frac{\delta}{2}\sin\delta-\left(F_{x1}-F_{x2}\right) \frac{\delta}{2}\cos\delta+\left(F_{x3}-F_{x4}\right)\frac{\delta}{2}.\end{matrix}  \tag{3}\]  
where \(m\) is the total mass, \(m_{s}\) is the sprung mass, \(r\) is the yaw rate, \(\phi\) is the body roll angle, \(V_{x}\) and \(V_{y}\) are the longitudinal and lateral speed of the CG, \(h_{r}\) is the altitude difference between the roll center and sprung mass CG, \(h_{p}\) is the altitude difference between the pitch center and sprung mass CG, \(F_{xi}\) and \(F_{yi}\) are the longitudinal and lateral force of the \(i\)th wheel, \(i=1\), \(2\), \(3\), \(4\), representing the left front, the right front, the right rear, and the left rear wheel, \(\delta\) is the front wheel steering angle, \(f\) is the rolling resistance coefficient, \(A\) is the frontal area, \(C_{D}\) is the air resistance coefficient, \(\xi\) is the rotating mass conversion factor, \(g\) is the acceleration of gravity, \(l_{f}\) and \(l_{r}\) are the distance between the front, rear axles, and vehicle CG, \(I_{z}\) is the yaw moment of inertia, and \(d\) is the wheel track. Vertical motion of sprung mass:  
\[\left\{ \begin{matrix}m_{s}\ddot{z}=\sum_{i=1}^{4}F_{si}\\ F_{si}=k_{i}(z_{ui}-z_{i})+c_{i}\big{(}\dot{z}_{ui}-\dot{z}_{i}\big{)}+u_{i} \end{matrix} \right.