## 2 Mathematical Dynamic Model of the IRW
### Gyroscopic Moment  
Strictly analyzing the rotation of the wheels, it is a relative rotational motion in the carrier reference system. Due to the carrier's yaw and roll movements, although the IRW rides at a constant forward velocity on the rail, the carrier coordinate system is a non-inertial reference coordinate. In addition, the three angular motions of the IRW are orthometric to each other. Then, some coupling effects exist between the wheels and the IRW. When the wheels rotate at high speeds, we find that the IRW will regain slight self-guidance ability to resist the derailment tendency caused by the gravitational restoring moment. This phenomenon has the same pattern of mechanisms as the gyroscopic effects on a top spinning at high speeds, which makes the spin top resist the gravity effects and not fall down to the ground. Figure 3 demonstrates a tiny rolling angle change \(d\alpha\) of the IRW from position A to position B after a little time gap \(dt\). Let \(\overrightarrow{a}\), \(\overrightarrow{b}\), and \(\overrightarrow{c}\) denote the unit direction vectors of the wheel axle at position A, the wheel axle at position B, and the rolling axis of the IRW, respectively. The unit direction vector of the wheel axle at position B can be derived as  
\[\overrightarrow{b}=\overrightarrow{a}+\dot{a}dt\left(\overrightarrow{c} \times\overrightarrow{a}\right) \tag{8}\]  
where \(\dot{\alpha}\) is the instantaneous rolling angular speed. Because both sides of the wheels rotate in the same direction when the IRW is riding forward, the angular momentum magnitude of the wheels is given by  
\[H=H_{\text{L}}+H_{\text{R}}=J_{\text{Wy}}(\omega_{\text{L}}+\omega_{\text{R}}) \tag{9}\]  
where \(J_{\text{Wy}}\) is the inertia moment of wheel in the \(y\) axle. The variation in this angular momentum can be deduced as  
\[\dot{d}\overrightarrow{H}=(H+dH)\overrightarrow{b}-H\overrightarrow{a}=H \dot{a}dt\left(\overrightarrow{c}\times\overrightarrow{a}\right)+dH \overrightarrow{b} \tag{10}\]  
Then, the derivation of \(\overrightarrow{H}\) with respect to time \(t\) can be obtained as  
Figure 3: The IRW running on the rail track with its roll angle changing from position A to B.  
Figure 2: Derailment motion caused by gravitational restoring moment \(M_{g\psi}\). \[\frac{d\overset{\rightarrow}{H}}{dt}=\left(\dot{\alpha}\overset{\rightarrow}{c} \right)\times\left(H\overset{\rightarrow}{a}\right)+\frac{dH}{dt}\overset{ \rightarrow}{b}=\overset{\rightarrow}{\dot{\alpha}}\times\overset{ \rightarrow}{H}+\frac{dH}{dt}\overset{\rightarrow}{b} \tag{11}\]  
When the IRW rides at a constant longitudinal forward speed, the angular momentum magnitude \(H\) does not change, which means that  
\[\frac{dH}{dt}=0 \tag{12}\]  
Then, the derivation of \(\overset{\rightarrow}{H}\) with respect to time \(t\) can be written as  
\[\frac{d\overset{\rightarrow}{H}}{dt}=\overset{\rightarrow}{\dot{\alpha}} \times\overset{\rightarrow}{H} \tag{13}\]  
Equation (13) shows that the existence of the rolling angular speed results in the variation in the angular momentum of the two wheels. Furthermore, the direction of this cross product is coincidentally the same as the yaw angular direction of the IRW.