## 2 Problem Formulation and System Modeling
### General Flexible Chassis Kinetic Model  
A simplified chassis dynamics model (Figure 4) is adopted to establish the FC MMS state equation. This study only considers the vehicle's longitudinal motion (\(x\)-direction), lateral motion (\(y\)-direction), and yaw motion (rotation around the \(z\)-axis). It is also assumed that there is no suspension system effect or air resistance. Changes in vehicle longitudinal speed are ignored. The coordinate system of the model takes the centroid (\(CG\)) of the FC as the origin. The \(x\)-axis is the longitudinal direction of the FC and the \(y\)-axis is the lateral direction.  
In the dynamic model, the kinetic equations for longitudinal, lateral, and yaw motion are expressed using Equations (7)-(9), respectively:  
\[m(\dot{u}-v\gamma) =\sum_{i=1}^{4}\left(F_{xi}\cos\delta_{i}-F_{yi}\sin\delta_{i} \right), \tag{7}\] \[m(\dot{u}+v\gamma) =\sum_{i=1}^{4}\left(F_{xi}\sin\delta_{i}+F_{yi}\cos\delta_{i} \right),\] (8) \[I\dot{\gamma} =M_{Z}, \tag{9}\]  
where \(u\) is longitudinal speed, \(v\) is lateral speed, \(I\) is the moment of inertia of the FC, \(m\) is the mass of the FC, and \(i=1\)(\(f\)), \(2\)(\(f\)), \(3\)(\(r\)), \(4\)(\(rr\)).  
According to the linear tire model [33], the steering angle, side-slip angle, longitudinal speed, and lateral speed have the following relationship:  
\[\tan(\alpha_{i}+\delta_{i})=\frac{v_{j}}{u_{j}}, \tag{10}\]  
where \(\alpha_{i}\) denotes a side-slip angle, \(v_{j}\) denotes longitudinal speed, and \(u_{j}\) denotes lateral speed (\(j=\) CM, IM, DM, SM).  
The side-slip angle of each motion is calculated as:  
\[\left\{\begin{array}{ll}\alpha_{fl}=\delta_{fl}-\tan^{-1}(\frac{v+l_{f}\gamma }{u-.5 Sb}),&\alpha_{fr}=\delta_{fr}-\tan^{-1}(\frac{v-l_{f}\gamma}{u-.5 Sb})\\ \alpha_{rl}=\delta_{rl}-\tan^{-1}(\frac{v+l_{f}\gamma}{u+.5 Sb}),&\alpha_{rr}= \delta_{rr}-\tan^{-1}(\frac{v-l_{f}\gamma}{u+.5 Sb})\\ \end{array}\right.. \tag{11}\]  
Figure 4: Simplified flexible chassis dynamics model. \(CG\) is the center of chassis gravity; \(l\), and \(l_{f}\) are the distances from the front and rear off-center shafts to \(CG\), respectively; \(M_{Z}\) represents the yaw moment generated by the four electric wheels; \(F_{xi}\) represents the longitudinal tire force of each wheel; \(F_{yi}\) is the lateral tire force of each wheel; and \(\alpha_{yi}\) is the slip angle of each wheel; \(\gamma\) is the yaw rate.  
The tire side-slip angle generally is very small in normal driving. It can be assumed that there is a linear relationship between the tire side force and the side-slip angle of the FC. The relationship between tire side force \(F_{y}\), side-slip angle \(\alpha\), and tire cornering stiffness \(C_{\alpha}\) is then given by:  
\[F_{yj}=-C_{\alpha j}\alpha_{j}. \tag{12}\]  
Finally, the state equation of FC movement is derived as follows:  
\[\left[\begin{array}{c}\dot{v}\\ \dot{\gamma}\end{array}\right]=\left[\begin{array}{cc}-\frac{C_{\alpha f}+C_{ \alpha f}+C_{\alpha f}+C_{\alpha f}}{mu}&\frac{I_{\gamma}(C_{\alpha f}-C_{ \alpha f})-I_{f}(C_{\alpha f}-C_{\alpha f})}{mu}-u\\ \frac{I_{\gamma}(C_{\alpha f}+C_{\alpha f})-I_{f}(C_{\alpha f}+C_{\alpha f})}{ li}&-\frac{I_{\gamma}(C_{\alpha f}-C_{\alpha f})+I_{f}^{2}(C_{\alpha f}-C_{ \alpha f})}{lu}\end{array}\right]\left[\begin{array}{c}v\\ \gamma\end{array}\right] \tag{13}\]  
Similarly, based on the above derivation, the state equations for CM, IR, DM, and SM can be derived according to Equation (13) and will not be derived here again.