# Multi-Body Dynamics Modeling and Simulation of Maglev Satellites
## 2 Modeling of Three Body Dynamics of Maglev Satellite
### Platform Subsystem Kinetic Energy  
The absolute velocity \(\overline{v}_{b}\) of a certain quality element on the platform is:  
\[\overline{v}_{b}=\dot{\overline{X}}_{pf}+\dot{\overline{r}}_{b}=\dot{\overline {X}}_{pf}+\dot{\overline{r}}_{b}+\dot{\overline{\omega}}_{b}\times\dot{ \overline{r}}_{b} \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbolic** & **Significance** \\ \hline \(\overline{X}_{pf}\) & The origin of the inertial frame points towards the center of mass of the platform subsystem \\ \(\overline{r}_{b}\) & The centroid of the platform subsystem points to a certain quality element on the platform \\ \(\overline{r}_{r}\) & The center of mass of the platform subsystem points towards the origin of the MM \\ \(\overline{g}_{r}\) & The origin of the MM system points to a certain mass element of the MM \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Position vector definition. Figure 4: Position vector definition diagram. Figure 3: Coordinate system definition diagram. where \(\overline{\omega}_{b}\) is the platform angular velocity, and \(\overset{\circ}{\overline{r}}_{b}\) represents the relative derivative of \(\overline{r}_{b}\) relative to the platform system. By transferring Equation (1) to the platform system, it can be expressed as follows:  
\[\mathbf{v}_{b}=A_{b}^{\mathrm{T}}\dot{\mathbf{X}}_{pf}+\dot{\mathbf{r}}_{b}+\ddot{\mathbf{r}}_{b} +\ddot{\mathbf{r}}_{b}^{\mathrm{T}}\mathbf{\omega}_{b} \tag{2}\]  
where \(\mathbf{v}_{b}\) is the projection of vector \(\overline{v}_{b}\) under the platform system, \(A_{b}\) is the attitude transformation matrix of the platform system relative to the inertial frame, \(\mathbf{X}_{pf}\) is the projection of \(\overline{\mathbf{X}}_{pf}\) under the inertial frame, \(\mathbf{r}_{b}=[r_{bx},r_{by},r_{bz}]^{\mathrm{T}}\) is the projection of \(\overline{\mathbf{r}}_{b}\) under the platform system, \(\mathbf{\omega}_{b}\) is the projection of \(\overline{\omega}_{b}\) under the platform system, and \(\widetilde{\mathbf{r}}_{b}\) represents the matrix operation as follows:  
\[\widetilde{\mathbf{r}}_{b}= \begin{bmatrix}0&-r_{bz}&r_{by}\\ r_{bz}&0&-r_{bx}\\ -r_{by}&r_{bx}&0\end{bmatrix}  \tag{3}\]  
So the kinetic energy of the platform can be expressed as:  
\[T_{b}=\frac{1}{2}\underset{b}{\int}\mathbf{v}_{b}^{\mathrm{T}}\mathbf{v}_{b}dm \tag{4}\]  
Through simplification, it can be concluded that:  
\[ \begin{split} T_{b}&=\frac{1}{2}\underset{b}{\int}\left[ \begin{array}{c}\dot{\mathbf{X}}_{pf}^{\mathrm{T}}\dot{\mathbf{X}}_{pf}+2\dot{\mathbf{X} }_{pf}^{\mathrm{T}}A_{b}\dot{\mathbf{r}}_{b}+2\dot{\mathbf{r}}_{b}^{\mathrm{T}}\dot\mathbf{{r}}_{b}^{\mathrm{T}}\mathbf{\omega}_{b}\\ +\dot{\mathbf{r}}_{b}^{\mathrm{T}}\mathbf{r}_{b}+\mathbf{\omega}_{b}^{\mathrm{T}}\dot{\mathbf{r }}_{b}^{\mathrm{T}}\mathbf{\omega}_{b}+2\dot{\mathbf{X}}_{pf}^{\mathrm{T}}A_{b}\dot{\mathbf{ r}}_{b}^{\mathrm{T}}\mathbf{\omega}_{bb}\end{array}\right]dm\\ &=\frac{1}{2}M_{b}\dot{\mathbf{X}}_{pf}^{\mathrm{T}}\dot{\mathbf{Y}}_{pf}+\frac{1}{2} \mathbf{\omega}_{b}^{\mathrm{T}}\mathbf{\gamma}_{b}^{\mathrm{T}}J_{bc}\mathbf{\omega}_{b}+ \dot{\mathbf{Y}}_{pf}^{\mathrm{T}}A_{b}M_{\dot{\mathbf{r}}}\dot{\mathbf{r}}_{b}\\ &+\frac{1}{2}M_{b}\dot{\mathbf{r}}_{b}^{\mathrm{T}}\dot{\mathbf{r}}_{b}+M_{b}\dot{\mathbf{ X}}_{pf}^{\mathrm{T}}A_{b}\dot{\mathbf{r}}_{bc}^{\mathrm{T}}\mathbf{\omega}_{b}+M_{b}\dot{\mathbf{ r}}_{b}^{\mathrm{T}}\dot{\mathbf{r}}_{bc}^{\mathrm{T}}\mathbf{\omega}_{b}\end{split}  \tag{5}\]  
where \(M_{b}\) is the mass of the platform, \(\mathbf{J}_{bc}\) is the projection of the moment of inertia of the platform relative to its center of mass in this system, and \(\mathbf{r}_{bc}\) is the position vector from the center of mass of the platform subsystem to the center of mass of the platform. The absolute velocity of a certain mass element on the MM is:  
\[