## 2 Mathematical Dynamic Model of the IRW
### Steady States Analysis  
If the IRW system described by Equations (28)-(31) can be stabilized by the yaw spring and yaw damper, the steady-state solution of the dynamics equations will represent the steady characteristics of the IRW. When steady, the translational and angular accelerations of the IRW are zero, that is, \(\ddot{y}=0\), \(\ddot{\psi}=0\), \(\dot{\omega}_{\rm L}=0\), and \(\dot{\omega}_{\rm R}=0\). Whether \(\dot{y}\) is zero or not and whether \(\dot{\psi}\) is zero or not is unknown at the beginning of the analysis. Assuming that \(\dot{\psi}\
eq 0\), which means that \(\dot{\psi}=\text{const}\), noted as \(\dot{\psi}=\dot{\psi}_{0}\), then if the IRW reaches steady states after time \(t_{s}\), the yaw angle \(\psi\) varying with time \(t\) can be expressed as follows  
\[\psi(t)=\dot{\psi}_{0}\cdot(t-t_{s})+\psi(t_{s}) \tag{32}\]  
Substituting Equation (32) into Equation (28), we can find that the lateral speed \(\dot{y}\) varies with time \(t\), which contradicts with \(\ddot{y}=0\). Then, \(\dot{\psi}=0\) can be deduced. According to Equation (30), when \(\dot{\omega}_{\rm L}=0\), we can deduce that  
\[\omega_{\rm L}(r_{0}-\lambda y)-\left(v+b\dot{\psi}\right)=0 \tag{33}\]  
which can be rearranged as  
\[\omega_{\rm L}=\frac{v+b\dot{\psi}}{r_{0}-\lambda y}=\frac{v}{r_{0}-\lambda y} \tag{34}\]  
Similarly, we can also obtain  
\[\omega_{\rm R}=\frac{v-b\dot{\psi}}{r_{0}+\lambda y}=\frac{v}{r_{0}+\lambda y} \tag{35}\]  
The [PERSON] expansion of Equations (34) and (35) at \(y=0\) are expressed as follows, respectively  
\[ \begin{cases}\omega_{\rm L}=\frac{v}{r_{0}-\lambda y}=\frac{v}{r_{0}}\left(1+ \frac{\lambda y}{r_{0}}+\left(\frac{\lambda y}{r_{0}}\right)^{2}+\left(\frac{ \lambda y}{r_{0}}\right)^{3}+\cdots\right)\\ \omega_{\rm R}=\frac{v}{r_{0}+\lambda y}=\frac{v}{r_{0}}\left(1-\frac{\lambda y }{r_{0}}+\left(\frac{\lambda y}{r_{0}}\right)^{2}-\left(\frac{\lambda y}{r_{0 }}\right)^{3}+\cdots\right)\end{cases}  \tag{36}\]  
Generally, \(\lambda y<0.1r_{0}\), so the higher-order items can be omitted, and then the differential rotational speed of the two wheels \(\Delta\omega\) can be deduced as  
\[\Delta\omega=\omega_{\rm L}-\omega_{\rm R}=\frac{2\lambda v}{r_{0}^{2}}y \tag{37}\]  
Based on the analysis above, Equation (29) can be simplified as  
\[K_{\psi}\psi-((m_{\rm C}+2m_{\rm W})g+F_{\rm Lz}+F_{\rm Rz})\lambda b\psi+ \frac{\lambda J_{\rm Wy}}{b}\dot{y}(\omega_{\rm L}+\omega_{\rm R})=0 \tag{38}\]  
Consider the situation where \(K_{\psi}\gg((m_{\rm C}+2m_{\rm W})g+F_{\rm Lz}+F_{\rm Rz})\lambda b\), then the yaw angle \(\psi\) can be obtained as  
\[\psi=-\frac{\lambda J_{\rm Wy}(\omega_{\rm L}+\omega_{\rm R})}{b\left(K_{\psi }-((m_{\rm C}+2m_{\rm W})g+F_{\rm Lz}+F_{\rm Rz})\lambda b\right)}\dot{y} \tag{39}\]  
When steady, the lateral speed \(\dot{y}\) should be quite low. Additionally, because the magnitude of the yaw stiffness is usually large enough, the coefficient in front of \(\dot{y}\) is generally less than 0.1. Then, yaw angle \(\psi\) can be treated as zero when the IRW reaches steady states, that is  
\[\psi=0 \tag{40}\]  
Then, Equation (28) can be simplified as  
\[k_{22}\frac{\dot{y}}{v}+k_{22}\frac{\dot{y}}{v}+((m_{\rm C}+2m_{\rm W})g+F_{ \rm Lz}+F_{\rm Rz})\frac{\lambda}{b}y=0 \tag{41}\]  
The relation between the lateral speed \(\dot{y}\) and the lateral displacement \(y\) can be deduced as  
\[\dot{y}=-\frac{\lambda v((m_{\rm C}+2m_{\rm W})g+F_{\rm Lz}+F_{\rm Rz})}{2 bk_ {22}}y \tag{42}\]  
Equation (42) is an ordinary differential equation, and if the IRW's dynamic oscillation is suppressed after time \(t_{s}\), its solution can be expressed as  
\[y(t)=y(t_{s})\text{exp}\left(-\frac{\lambda v((m_{\rm C}+2m_{\rm W})g+F_{ \rm Lz}+F_{\rm Rz})}{2 bk_{22}}(t-t_{s})\right) \tag{43}\]  
In Equation (43), the coefficient in front of time duration \(t-t_{s}\) is negative, which means that the lateral displacement \(y(t_{s})\) will decay and finally converge.