# Multi-Body Dynamics Modeling and Simulation of Maglev Satellites
## 2 Modeling of Three Body Dynamics of Maglev Satellite
### Platform Subsystem Kinetic Energy
\begin{split}\overline{\mathbf{v}}_{r}&=\dot{\mathbf{X}}_{pf }+\ddot{\mathbf{r}}_{r}+\ddot{\mathbf{\bar{g}}}_{r}=\ddot{\mathbf{X}}_{pf}+\overset{\circ} {\overline{r}}_{r}+\overset{\circ}{\overline{\omega}}_{b}\times\overline\mathbf{{r}}_{r}+(\overline{\omega}_{b}+\overline{\omega}_{r})\times\overline{\mathbf{g}}_ {r}\\ &=\ddot{\mathbf{X}}_{pf}+\overset{\circ}{\overline{r}}_{r}+\overline{ \omega}_{b}\times(\overline{\mathbf{r}}_{r}+\overline{\mathbf{g}}_{r})+\overline{ \omega}_{r}\times\overline{\mathbf{g}}_{r}\end{split}  \tag{6}\]  
where \(\overline{\omega}_{r}\) is the angular velocity of MM relative to the platform rotation. Represent the Equation (6) in the MM system as:  
\[\mathbf{v}_{r}=C_{r}^{\mathrm{T}}A_{b}^{\mathrm{T}}\dot{\mathbf{X}}_{pf}+C_{r}^{ \mathrm{T}}\dot{\mathbf{r}}_{r}+(C_{r}^{\mathrm{T}}\dot{\mathbf{r}}_{r}^{\mathrm{T}}+ \ddot{\mathbf{\bar{g}}}_{r}^{\mathrm{T}}C_{r}^{\mathrm{T}})\mathbf{\omega}_{b}+\ddot{ \mathbf{\bar{g}}}_{r}^{\mathrm{T}}\mathbf{\omega}_{r} \tag{7}\]  
where \(C_{r}\) is the attitude transformation matrix of the MM system relative to the platform system, and \(\mathbf{v}_{r},\mathbf{r}_{r},\mathbf{g}_{r},\mathbf{\omega}_{r}\) are the projections of \(\overline{\mathbf{v}}_{r},\overline{\mathbf{r}}_{r},\overline{\mathbf{g}}_{r},\overline{ \mathbf{\omega}}_{r}\) under the MM system, respectively. The kinetic energy of the MM can be expressed as:  
\[\begin{split} T_{r}=\frac{1}{2}\underset{r}{\int}\left( \begin{array} []{c}\dot{\mathbf{X}}_{pf}^{\mathrm{T}}\dot{\mathbf{X}}_{pf}+2\dot{\mathbf{X}}_{pf}^{ \mathrm{T}}A_{b}\mathbf{C}_{r}(\mathbf{C}_{r}^{\mathrm{T}}\dot{\mathbf{r}}_{r}^{\mathrm{T} }+\ddot{\mathbf{\bar{g}}}_{r}^{\mathrm{T}}C_{r}^{\mathrm{T}})\mathbf{\omega}_{b}\\ +2\dot{\mathbf{X}}_{pf}^{\mathrm{T}}A_{b}\mathbf{C}_{r}\ddot{\mathbf{\bar{g}}}_{r}^{ \mathrm{T}}\mathbf{\omega}_{r}+2\dot{\mathbf{X}}_{pf}^{\mathrm{T}}(\widetilde{\mathbf{r}}_ {r}\mathbf{C}_{r}+C_{r}\widetilde{\mathbf{\bar{g}}}_{r})\ddot{\mathbf{\bar{g}}}_{r}^{ \mathrm{T}}\mathbf{\omega}_{r}\\ +\dot{\mathbf{v}}_{r}^{\mathrm{T}}\ddot{\mathbf{\bar{g}}}_{r}\ddot{\mathbf{\bar{g}}}_{r}^{ \mathrm{T}}\mathbf{\omega}_{r}+\mathbf{\omega}_{b}^{\mathrm{T}}(\widetilde{\mathbf{r}}_{r} \mathbf{C}_{r}+C_{r}\widetilde{\mathbf{\bar{g}}}_{r})(C_{r}^{\mathrm{T}}\dot{\mathbf{r}}_{r}^{ \mathrm{T}}+\ddot{\mathbf{\bar{g}}}_{r}^{\mathrm{T}}C_{r}^{\mathrm{T}})\mathbf{\omega}_{ b}\\ +2\dot{\mathbf{r}}_{r}^{\mathrm{T}}A_{b}^{\mathrm{T}}\dot{\mathbf{X}}_{pf}+\dot{\mathbf{r}}_{ r}^{\mathrm{T}}\dot{\mathbf{r}}_{r}+2\dot{\mathbf{r}}_{r}^{\mathrm{T}}C_{r}\ddot{\mathbf{\bar{g}}}_{r}^{ \mathrm{T}}\mathbf{\omega}_{r}+2\dot{\mathbf{r}}_{r}^{\mathrm{T}}(\widetilde{\mathbf{r}}_ {r}^{\mathrm{T}}+\mathbf{C}_{r}\widetilde{\mathbf{\bar{g}}}_{r}^{\mathrm{T}}C_{r}^{ \mathrm{T}})\mathbf{\omega}_{b}\end{array} \right)dm\\ \mathbf{J}_{r}=\int\overset{\circ}{\overline{r}}_{r}\delta\widetilde{\mathbf{\bar{g}}}_{r}^{ \mathrm{T}}dm\text{ is the projection of the moment of inertia of the MM relative to its origin in this system, and the relevant terms in the equation are expanded as follows:  
\[ \begin{split}&\mathbf{\omega}_{b}^{\mathrm{T}}\underset{r}{\int} \left(\widetilde{\mathbf{r}}_{r}C_{r}+C_{r}\widetilde{\mathbf{\bar{g}}}_{r}\right)(C_{r}^{ \mathrm{T}}\dot{\mathbf{r}}_{r}^{\mathrm{T}}+\ddot{\mathbf{\bar{g}}}_{r}^{\mathrm{T}}C_{r }^{\mathrm{T}})dm\mathbf{\omega}_{b}\\ &=\mathbf{\omega}_{b}^{\mathrm{T}}\underset{r}{\int}\left(\widetilde{\mathbf{r}}_{r} ^{\mathrm{T}}\dot{\mathbf{r}}_{r}+\widetilde{\mathbf{r}}_{r}C_{r}\widetilde{\mathbf{\bar{g}}}_{r}^{ \mathrm{T}}C_{r}^{\mathrm{T}}+C_{r}\widetilde{\mathbf{\bar{g}}}_{r}C_{r}^{\mathrm{T}} \widetilde{\mathbf{r}}_{r}^{\mathrm{T}}\right.\\ &\left.+C_{r}\widetilde{\mathbf{\bar{g}}}_{r}^{\mathrm{T}}C_{r}^{\mathrm{T}} \right)dm\end{split}  \tag{9}\]The physical meaning of the integral sign on the right side of the equal sign in Equation (9) is the moment of inertia of the magnetic levitation module relative to the center of mass of the platform subsystem, denoted as \(\mathbf{J}_{rc}\), which can be organized as:  
\[