## 2 SRT Structure and Dynamics Model
### CG Generalized Forces  
Finally, the equivalent CG generalized forces of every module generated by the wheel torque and steering angle can be calculated, called the CG generalized force model. This model is mainly used for the control allocation of the wheel states after calculating the CG generalized forces required for path tracking. The tire force of the \(j^{th}\) wheel of the \(i^{th}\) module can be expressed as  
\[f_{x}^{i,j}=\frac{Q^{i,j}}{R^{i,j}}, \tag{21}\]  
\[f_{y}^{i,j}=c_{a}^{i,j}(\delta^{i,j}-\frac{v_{y}^{i}+l_{j}^{i}\gamma^{i}}{v_{x }^{i}}), \tag{22}\]  
where \(f_{x}^{i,j}\) is the longitudinal force, \(f_{y}^{i,j}\) is the lateral force, \(Q^{i,j}\) is the wheel torque input, \(\delta^{i,j}\) is the wheel steering angle, \(R^{i,j}\) is the effective radius, \(c_{a}^{i,j}\) is the cornering stiffness, and \(l_{j}^{i}\) is the longitudinal distance between the wheel and the CG. As shown in Figure 3, due to the wheel steering angle, the tire force in the module coordinate system can be expressed as  
\[ \begin{bmatrix}F_{x}^{i,j}\\ F_{y}^{i,j}\end{bmatrix} = \begin{bmatrix}\cos\delta^{i,j}&-\sin\delta^{i,j}\\ \sin\delta^{i,j}&\cos\delta^{i,j}\end{bmatrix} \cdot \begin{bmatrix}F_{x}^{i,j}\\ F_{y}^{i,j}\end{bmatrix} . \tag{23}\]  
Therefore, the CG longitudinal force \(F_{x}^{i}\), lateral force \(F_{y}^{i}\), and yaw moment \(M_{z}^{i}\) generated by tire forces are represented as follows with \(d^{i}\) the wheelbase:  
\[F_{x}^{i}=F_{x}^{i,1}+F_{x}^{i,2}+F_{x}^{i,3}+F_{x}^{i,4}, \tag{24}\]  
\[F_{y}^{i}=F_{y}^{i,1}+F_{y}^{i,2}+F_{y}^{i,3}+F_{y}^{i,4}, \tag{25}\]  
\[M_{z}^{i}=d^{i}\cdot(-F_{x}^{i,1}+F_{x}^{i,2}-F_{x}^{i,3}+F_{x}^{i,4})/2+l_{1,2 }^{i}\cdot(F_{y}^{i,1}+F_{y}^{i,2})+l_{3,4}^{i}\cdot(-F_{y}^{i,3}-F_{y}^{i,4}). \tag{26}\]  
Since every module of the VTT may have two or four wheels, only some can provide driving torque. Therefore, the Boolean matrix for the tire force of the \(i^{th}\) module is defined as  
\[B_{m}^{i}=diag( \begin{bmatrix}b_{mx}^{i,1}&b_{my}^{i,1}&b_{mx}^{i,2}&b_{my}^{i, 2}&b_{mx}^{i,3}&b_{my}^{i,3}&b_{mx}^{i,4}&b_{my}^{i,4}\end{bmatrix} ), \tag{27}\]  
where \(b_{mx}^{i,j}=1\) represents the wheel and can provide driving torque and \(b_{my}^{i,j}=1\) represents the wheel and can provide lateral force; otherwise, the value is 0. This matrix configures the wheels' number and the torques providing ability. Concretely, this matrix is changeable based on the module's running gears, for example, double axle with hub-motor driving, double axle with only one driving axle, or single axle without a motor. Thus, it leads to different CG generalized force models for each module in the SRT. By arranging the Equations (21)-(26) in matrix form, the CG generalized forces of the \(i^{th}\) module can be expressed as  
Figure 3: Tire forces and equivalent CG generalized forces of the \(i^{th}\) module. \[F^{i}_{CG}=L^{i}\cdot B^{i}_{m}\cdot DCM^{i}\cdot(A^{i}_{av}\cdot X^{i}+B^{i}_{av} \cdot U^{i}), \tag{28}\]  
where  
\[F^{i}_{CG}= \begin{bmatrix}F^{i}_{x}\\ F^{i}_{y}\\ M^{i}_{x}\end{bmatrix} _{3\times 1},L^{i}= \begin{bmatrix}1&0&1&0&1&0&1\\ 0&1&0&1&0&1&0\\ -d^{i}/2&l^{i}_{1,2}&d^{i}/2&l^{i}_{1,2}&-d^{i}/2&l^{i}_{3,4}&d^{i}/2&l^{i}_{3, 4}\\ \end{bmatrix} _{3\times 8},\] \[DCM^{i}= \begin{bmatrix}\cos\delta^{i,1}&-\sin\delta^{i,1}&0&0&0&0 &0&0\\ \sin\delta^{i,1}&\cos\delta^{i,1}&0&0&0&0&0&0\\ 0&0&\cos\delta^{i,2}&-\sin\delta^{i,2}&0&0&0&0\\ 0&0&\sin\delta^{i,2}&\cos\delta^{i,2}&0&0&0&0\\ 0&0&0&\cos\delta^{i,3}&-\sin\delta^{i,3}&0&0\\ 0&0&0&0&\sin\delta^{i,3}&\cos\delta^{i,3}&0&0\\ 0&0&0&0&0&\cos\delta^{i,4}&-\sin\delta^{i,4}\\ 0&0&0&0&0&\sin\delta^{i,4}&\cos\delta^{i,4}&\cos\delta^{i,4}\\ \end{bmatrix} _{8\times 8},L^{i}= \begin{bmatrix}\dot{Q}^{i,1}\\ \dot{\beta}^{i,1}\\ \dot{Q}^{i,2}\\ \dot{Q}^{i,3}\\ \dot{Q}^{i,4}\\ \end{bmatrix} _{8\times 1},\] \[A^{i}_{av}=