## 2 Mathematical Dynamic Model of the IRW
### IRW Model Deduction  
We only consider the IRW rides on a straight, flat rail track at constant longitudinal speeds, so the coordinate system accompanying the IRW along the rail track center is an inertial reference system. Then, the dynamic equations can be set up in this coordinate system through the Euler-Lagrange approach. The kinetic energy of the carrier for lateral dynamics analysis is  
\[T_{\rm C}=\frac{1}{2}m_{\rm C}\dot{y}^{2}+\frac{1}{2}J_{\rm Cx}\dot{\alpha}^{2}+ \frac{1}{2}J_{\rm Cz}\dot{y}^{2}=\frac{1}{2}\bigg{(}\frac{\lambda^{2}}{b^{2}}J_ {\rm Cx}+m_{\rm C}\bigg{)}\dot{y}^{2}+\frac{1}{2}J_{\rm Cz}\dot{y}^{2} \tag{17}\]  
where \(J_{\rm Cx}\) and \(J_{\rm Cz}\) are carrier's inertia moments in the \(x\) and \(z\) directions, respectively. The angular kinetic energy of the left wheel in its rotation direction around its axle is  
\[T_{\rm L\beta}=\frac{1}{2}J_{Wy}\omega_{\rm L}^{2} \tag{18}\]  
Additionally, the wheel also moves with the carrier in translational and angular directions. The translational kinetic energy of the left wheel in the IRW's lateral direction is  
\[T_{\rm Ly}=\frac{1}{2}m_{\rm W}\dot{y}^{2} \tag{19}\]  
According to the [PERSON] theorem, the angular kinetic energy of the left wheel in the IRW's yaw direction is  
\[T_{\rm L\psi}=\frac{1}{2}J_{Wz}\dot{\psi}^{2}+\frac{1}{2}m_{\rm W}\Big{(}b\dot {\psi}\Big{)}^{2}=\frac{1}{2}\Big{(}J_{\rm Wz}+m_{\rm W}b^{2}\Big{)}\dot{\psi} ^{2} \tag{20}\]  
where \(J_{Wz}\) is the wheel's inertia moment in the \(z\) direction. Similarly, the angular kinetic energy of the left wheel in the IRW's roll direction is  
\[T_{\rm L\alpha}=\frac{1}{2}\Big{(}J_{\rm Wx}+m_{\rm W}b^{2}\Big{)}\dot{\alpha} ^{2} \tag{21}\]  
where \(J_{\rm Wx}\) is the wheel's inertia moment in the \(x\) direction. Then, the total kinetic energy of the left wheel is  
\[ \begin{array}{c}T_{\rm L}=T_{\rm Ly}+T_{\rm L\psi}+T_{\rm L\alpha}+T_{\rm L \beta}\\ =\frac{1}{2}\Big{(}\big{(}1+\lambda^{2}\big{)}m_{\rm W}+\frac{\lambda^{2}}{b^ {2}}J_{\rm Wx}\big{)}\dot{y}^{2}+\frac{1}{2}\big{(}J_{\rm Wz}+m_{\rm W}b^{2} \big{)}\dot{\psi}^{2}+\frac{1}{2}J_{\rm Wy}\omega_{\rm L}^{2}\end{array}  \tag{22}\]  
Similarly, the total kinetic energy of the right wheel is  
\[T_{\rm R}=\frac{1}{2}\bigg{(}\big{(}1+\lambda^{2}\big{)}m_{\rm W}+\frac{ \lambda^{2}}{b^{2}}J_{\rm Wx}\bigg{)}\dot{y}^{2}+\frac{1}{2}\Big{(}J_{\rm Wz}+ m_{\rm W}b^{2}\Big{)}\dot{\psi}^{2}+\frac{1}{2}J_{\rm Wy}\omega_{\rm R}^{2} \tag{23}\]  
The total kinetic energy of the IRW is  
Figure 4: Slight self-guidance ability under gyroscopic effects. \[ \begin{split} T=T_{\rm C}+T_{\rm L}+T_{\rm R}=\frac{1}{2}\Big{(}m_{ \rm C}+2\big{(}1+\lambda^{2}\big{)}m_{\rm W}+\frac{\lambda^{2}}{b^{2}}J_{\rm Cr }+\frac{2\lambda^{2}}{b^{2}}J_{\rm Wx}\Big{)}\dot{y}^{2}\\ +\frac{1}{2}\big{(}J_{\rm Cz}+2J_{\rm Wz}+2m_{\rm W}b^{2}\big{)} \dot{\psi}^{2}+\frac{1}{2}J_{\rm Wy}\omega_{\rm L}^{2}+\frac{1}{2}J_{\rm Wy} \omega_{\rm R}^{2}\end{split}  \tag{24}\]  
Assuming the linear conical wheel rides on a knife-edged rail and a yaw angular spring with stiffness coefficient \(K_{\psi}\) is installed, the total potential energy of the IRW is  
\[V=V_{\rm Cg}+V_{\rm Lg}+V_{\rm Rg}+V_{\psi}=0+m_{\rm Wg}(-\lambda y)+m_{\rm Wg} \lambda y+\frac{1}{2}K_{\psi}\psi^{2}=\frac{1}{2}K_{\psi}\psi^{2} \tag{25}\]  
Assuming a yaw damper with damping coefficient \(C_{\psi}\) is installed, the dissipation energy of the IRW is  
\[D=\frac{1}{2}C_{\psi}\dot{\psi}^{2} \tag{26}\]  
According to Euler-Lagrange mechanics theorem, the dynamic equations for a mechanical system can be expressed as follows  
\[\frac{d}{dt}\frac{\partial T}{\partial\dot{q}_{i}}-\frac{\partial T}{\partial q _{i}}+\frac{\partial V}{\partial q_{i}}+\frac{\partial D}{\partial\dot{q}_{i}} =Q_{i} \tag{27}\]  
where \(t\) represents time, \(q_{i}\) represents generalized coordinates, \(\dot{q}_{i}\) represents generalized velocities, and \(Q_{i}\) represents generalized forces. Then, the dynamic equations of the IRW for lateral dynamics analysis can be expressed in detail as follows  
\[