## 4 Vehicle Dynamics Model
### Full-Car Dynamics Model
#### 4.2.1 Kinematics of Vehicle System  
In this section, the major coordinate system is established, i.e., \(S_{G}\) is the earth-fixed inertial reference coordinate frame and \(S_{v}\) is the vehicle fixed non-inertial reference coordinate frame. Rotating angular velocity of \(S_{v}\) is \(\mathbf{w}_{v}\), and Translating velocity of \(S_{v}\) is \(\mathbf{v}_{v}\). The corresponding equations of the 9-DOF model are stated as follows. More details can be found in [15; 17]. Translation of Sprung Mass C.G. Position of C.G. of sprung mass in the reference coordinate frame \(S_{v}\) can be expressed as:  
\[\overset{\rightarrow}{l}_{os}=R_{y}(\theta)R_{x}(\phi)\left[ \begin{array}{ c}0\\ 0\\ -h_{roll}\end{array} \right]+\left[ \begin{array}{c}0\\ 0\\ z_{b}\end{array} \right]=\left[ \begin{array}{c}-h_{roll}\cos\phi\sin\theta\\ h_{roll}\sin\theta\\ -h_{roll}\cos\phi\sin\theta+z_{b}\end{array} \right] \tag{42}\]  
And,  
\[R_{y}(\theta)=\left[ \begin{array}{ccc}\cos\theta&0&\sin\theta\\ 0&1&0\\ -\sin\theta&0&cos\theta\end{array} \right]\text{; }R_{x}(\phi)=\left[ \begin{array}{ ccc}1&0&0\\ 0&\cos\phi&-\sin\phi\\ 0&\sin\phi&\cos\phi\end{array} \right] \tag{43}\]  
Then,  
\[\left(\frac{d}{dt}\right)_{G}\overset{\rightarrow}{l}_{os}=\left(\frac{d}{dt }\right)\overset{\rightarrow}{v}_{l}^{\prime}{}_{os}+\mathbf{w}_{v}\overset{ \rightarrow}{l}_{os} \tag{44}\]  
where '\(\times\)' represents the cross-product operator, and \(\mathbf{w}_{v}\) = [0 0 \(w\)]T.  
Velocity of sprung mass C.G. with respect to \(S_{G}\) is denoted as the sum of translational \(\mathbf{v}_{v}\) of the vehicle reference coordinate frame \(S_{v}\) and the time derivative of position \(\overset{\rightarrow}{l}_{os}\). \[\mathbf{v}_{s}=\mathbf{v}_{v}+\left(\frac{d}{dt}\right)_{G}=\mathbf{v}_{v}+\left(\frac{d }{dt}\right)_{v}\overset{\rightarrow}{l}_{os}+\mathbf{w}_{v}\times\overset{ \rightarrow}{l}_{os}=\mathbf{v}_{v}+\overset{\rightarrow}{l}_{os}+\mathbf{w}_{v} \times\overset{\rightarrow}{l}_{os} \tag{45}\]  
where \"\(\times\)\" represents the cross-product operator, and \(\mathbf{v}_{v}\) = [\(V_{x}\)\(V_{y}\)\(0\)]T.  
The acceleration of sprung mass C.G. with respect to \(S_{G}\) is calculated as:  
\[\mathbf{a}_{s}=\left(\frac{d}{dt}\right)_{G}\mathbf{v}_{s}=+\left(\frac{d}{dt}\right) _{p}\mathbf{v}_{v}+\mathbf{w}_{v}\times\mathbf{v}_{v}=\dot{\mathbf{v}}_{v}+\mathbf{w}_{v}\times\bm {v}_{v} \tag{46}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Vehicle Model Parameters** & **Symbol** & **Unit** \\ \hline Unsprung mass vertical displacement of left front side & \(z_{w1}\) & m \\ Unsprung mass vertical displacement of right front side & \(z_{w2}\) & m \\ Unsprung mass vertical displacement of left rear side & \(z_{w3}\) & m \\ Unsprung mass vertical displacement of right rear side & \(z_{w4}\) & m \\ Sprung mass of vehicle body & \(m_{s}\) & kg \\ Unsprung mass of vehicle left front side & \(m_{w1}\) & kg \\ Unsprung mass of vehicle right front side & \(m_{w2}\) & kg \\ Unsprung mass of vehicle left rear side & \(m_{w3}\) & kg \\ Unsprung mass of vehicle right rear side & \(m_{w4}\) & kg \\ Road profile of left front side & \(z_{r1}\) & m \\ Road profile of right front side & \(z_{r2}\) & m \\ Road profile of left rear side & \(z_{r3}\) & m \\ Road profile of right rear side & \(z_{r4}\) & m \\ Distance between roll center to C.G. of sprung mass & \(h_{\text{roll}}\) & m \\ Roll angle of sprung mass around \(x\)-axis of \(S_{v}\) & \(\varphi\) & deg (\({}^{\circ}\)) \\ Pitch angle of sprung mass around \(y\)-axis of \(S_{v}\) & \(\theta\) & deg (\({}^{\circ}\)) \\ Yaw rate of sprung mass around \(z\)-axis of \(S_{v}\) & \(\omega\) & deg/s(\({}^{\circ}\)/s) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._Translation of Unsprung Mass C.G. As mentioned in Section 4.2.1, position of unsprung mass C.G. in the reference coordinate frame \(S_{v}\) can be expressed as:  
\[\overset{\rightarrow}{l}_{w1}=\left[ \begin{array}{c}a\\ -\frac{B}{2}\\ z_{w1}-h_{w1}\end{array} \right];\overset{\rightarrow}{l}_{w2}=\left[