## 2 System Modeling  
### Dynamic Model of Railway Vehicles with IRWs  
In this paper, we consider a two-axle prototype IRW vehicle, as shown in Figure 1. This vehicle comprises a car body and two sets of independently rotating wheelsets, and the corresponding parameters are detailed in Table 1. It has ten degrees of freedom, including the rotational dynamics of each IRW and the lateral position and yaw movements of both the car body and the wheelsets. Each IRW is driven by a motor providing guidance torque, typically using a permanent magnet synchronous motor (PMSM) in engineering practice. The wheel profile of each IRW is LMA, and the corresponding rail profile is UIC60. The dynamic equations of lateral and yaw movements for the car body are shown in Equations (1) and (2), respectively. \[ \begin{array}{l}m_{b}\ddot{y}_{b}+2C_{y}(2\dot{y}_{b}-\dot{y}_{w1}-\dot{y}_{ w2})+2K_{y}(2y_{b}-y_{w1}-y_{w2})=\\ \frac{m_{b}V_{0}^{2}}{2}(\frac{1}{R_{c1}}+\frac{1}{R_{c2}})-\frac{m_{b}S}{2}( \theta_{1}+\theta_{2})\end{array}  \tag{1}\]  
\[ \begin{array}{l}l_{b}\ddot{\Psi}_{b}+4(L_{b}^{2}C_{y}+L_{a}^{2}C_{x})\dot{ \Psi}_{b}+4(L_{b}^{2}K_{y}+L_{a}^{2}K_{x})\psi_{b}-\\ 2L_{b}C_{y}(\dot{y}_{w1}-\dot{y}_{w2})-2L_{b}K_{y}(y_{w1}-y_{w2})-2L_{a}^{2}K_{ x}(\dot{y}_{w1}+\dot{y}_{w2})-2L_{a}^{2}C_{x}(\dot{y}_{w1}+\dot{y}_{w2})=0\end{array}  \tag{2}\]where \(L_{a}\) is half of the lateral length of the primary suspension, \(L_{b}\) is half of the wheelbase, \(m_{b}\) is the mass of the car body, and \(I_{b}\) is the rotational inertia of the car body around the \(z\)-axis. \(K_{x}\) and \(K_{y}\) are the longitudinal and lateral stiffnesses of the primary suspension, respectively. \(C_{x}\) and \(C_{y}\) are the longitudinal and lateral damping of the primary suspension, respectively. \(R_{ci}\) is the radius of the curved track, and \(\theta_{i}\) is the cant angle of the curve. \(y_{b}\) and \(\psi_{b}\) are the lateral displacement and yaw angle of the car body, respectively. \(y_{ui}\) and \(\psi_{wi}\) are the lateral displacement and yaw angle of each wheelset, respectively. The dynamic equations of the IRWs are shown in Equations (3)-(5), where \(i=1\) and \(2\) represent the front and rear, respectively, of the independently rotating wheelsets. \[ \begin{array}{l}m_{w}\ddot{y}_{wi}+\frac{2f_{22}}{V_{0}}\dot{y}_{wi}+2C_{y}( \dot{y}_{wi}-\dot{y}_{b})+2K_{y}(y_{wi}-y_{b})+\\ K_{y}y_{wi}-2f_{22}\dot{y}_{wi}+(-1)^{i}\cdot 2L_{b}C_{y}\dot{y}_{b}+(-1)^{i} \cdot 2L_{b}K_{y}\dot{y}_{b}=m_{w}\bigg{(}\frac{V_{0}^{2}}{R_{ci}}-g\dot{g} _{i}\bigg{)}\end{array}  \tag{3}\]  
\[ \begin{array}{l}I_{wz}\ddot{\psi}_{wi}+\frac{2f_{11}L_{1}^{2}}{V_{0}}\dot{ \psi}_{wi}+\frac{2f_{11}\lambda L_{y}}{r_{0}}y_{wi}+2L_{a}^{2}C_{x}(\dot{\psi} _{wi}-\dot{\psi}_{b})+\\ 2L_{a}^{2}K_{x}(\psi_{wi}-\psi_{b})-K_{y}\psi_{wi}-\frac{2r_{0}f_{11}L_{1}}{V_ {0}}\dot{\beta}_{wi}=\frac{2f_{11}L_{1}^{2}}{R_{ci}}+\frac{2f_{11}\lambda L_{t }}{r_{0}}y_{ti}\\ I_{wy}\ddot{\beta}_{wi}+\frac{r_{0}f_{11}}{V_{0}}\dot{\beta}_{wi}+f_{11}\lambda y _{wi}+\frac{r_{0}f_{11}L_{1}^{2}}{V_{0}}\dot{\psi}_{wi}=\frac{r_{0}f_{11}L_{1 }}{R_{ci}}+f_{11}\lambda y_{ti}+T_{wi}\end{array}  \tag{4}\]  
where \(L_{t}\) is half of the lateral distance between the rolling circles and \(m_{w}\) is the mass of the wheelset. \(I_{wz}\) and \(I_{wy}\) are the wheelset yaw and pitch moments of inertia, respectively. \(V_{0}\) is the longitudinal running speed of the vehicle, \(\lambda\) denotes the wheel conicity of the IRWs, and \(r_{0}\) is the nominal rolling radius of each IRW. \(K_{g}\) and \(K_{\psi}\) are the gravity stiffness and angular stiffness of the wheelset, respectively. \(f_{11}\) and \(f_{22}\) are the creep coefficients in the longitudinal and lateral directions, respectively. \(y_{ti}\) is the irregularity of the track. \(\dot{\beta}_{wi}\) represents the speed difference of each wheelset, and \(T_{wi}\) is half of the steering torque difference between the wheels, as defined in Equations (6) and (7).