# Multi-Body Dynamics Modeling and Simulation of Maglev Satellites
## 2 Modeling of Three Body Dynamics of Maglev Satellite
### Platform Subsystem Kinetic Energy
\begin{split}\mathbf{J}_{rc}&=M_{r}\tilde{\mathbf{r}}_{r} \tilde{\mathbf{r}}_{r}^{\mathrm{T}}+M_{r}\tilde{\mathbf{r}}_{r}C_{g}\tilde{\mathbf{r}}_{rc}^ {\mathrm{T}}C_{r}^{\mathrm{T}}\\ &+M_{r}C_{r}\tilde{\mathbf{g}}_{rc}C_{r}^{\mathrm{T}}\tilde{\mathbf{r}}_{ r}^{\mathrm{T}}+C_{r}\mathbf{J}_{r}C_{r}^{\mathrm{T}}\end{split}  \tag{10}\]  
where \(M_{r}\) is the mass of the MM, \(\mathbf{g}_{rc}\) is the vector from the origin of the MM to the center of mass, denoted as \(\mathbf{\omega}_{r}=\mathbf{D}_{r}\hat{\theta}_{r}\), \(\mathbf{D}_{r}=[1,0,0]^{\mathrm{T}}\), and \(J_{rx}\) are the \(O_{r}X_{r}\)-axis principal moments of inertia \(\mathbf{J}_{r}\). Therefore, the kinetic energy of the MM body is further expressed as:  
\[ \begin{split} T_{r}&=\tfrac{1}{2}M_{r}\dot{\mathbf{X}}_{pf }^{\mathrm{T}}\dot{\mathbf{X}}_{pf}+\tfrac{1}{2}\mathbf{\omega}_{b}^{\mathrm{T}}\mathbf{J} _{rc}\omega_{b}+\tfrac{1}{2}J_{rx}\hat{\theta}_{r}^{2}+M_{r}\dot{\mathbf{X}}_{pf}^ {\mathrm{T}}\mathbf{A}_{b}\tilde{\mathbf{r}}_{r}^{\mathrm{T}}\mathbf{\omega}_{b}\\ &+M_{r}\dot{\mathbf{X}}_{pf}^{\mathrm{T}}A_{b}C_{r}\tilde{\mathbf{g}}_{rc }^{\mathrm{T}}C_{r}^{\mathrm{T}}\mathbf{\omega}_{b}+M_{r}\dot{\mathbf{X}}_{pf}^{ \mathrm{T}}\mathbf{A}_{b}C_{g}\tilde{\mathbf{r}}_{rc}^{\mathrm{T}}\mathbf{D}_{r}\hat{\theta }_{r}+M_{r}\mathbf{\omega}_{b}^{\mathrm{T}}\tilde{\mathbf{r}}_{r}C_{g}\tilde{\mathbf{r}}_{ r}^{\mathrm{T}}\mathbf{D}_{r}\hat{\theta}_{r}\\ &+\mathbf{\omega}_{b}^{\mathrm{T}}C_{r}\mathbf{J}_{r}D_{r}\hat{\theta}_{r} +M_{r}\dot{\mathbf{r}}_{r}^{\mathrm{T}}\dot{\mathbf{A}}_{b}^{\mathrm{T}}\dot{\mathbf{X}}_{pf }+M_{r}\dot{\mathbf{r}}_{r}^{\mathrm{T}}\dot{\mathbf{r}}_{r}^{\mathrm{T}}+C_{r}\tilde{ \mathbf{g}}_{rc}^{\mathrm{T}}\mathbf{D}_{r}\hat{\theta}_{r}+\tilde{\mathbf{r}}_{r}^{ \mathrm{T}}\mathbf{\omega}_{b}+C_{r}\tilde{\mathbf{g}}_{rc}^{\mathrm{T}}C_{r}^{\mathrm{ T}}\mathbf{\omega}_{b})\end{split}  \tag{11}\]  
Based on the above kinetic energy expression, the kinetic energy of the platform subsystem is:  
\[T_{pf}=T_{b}+T_{r} \tag{12}\]  
Further considering the characteristics of the system center of mass, the following equation holds:  
\[M_{b}\mathbf{r}_{bc}+M_{r}(\mathbf{r}_{r}+C_{r}\mathbf{g}_{rc})=\mathbf{0} \tag{13}\]  
Therefore, in the kinetic energy expression of the platform subsystem, the following equation holds:  
\[M_{b}\dot{\mathbf{X}}_{pf}^{\mathrm{T}}\mathbf{A}_{b}\tilde{\mathbf{r}}_{ bc}^{\mathrm{T}}\mathbf{\omega}_{b}+M_{r}\dot{\mathbf{X}}_{pf}^{\mathrm{T}}\mathbf{A}_{b} \tilde{\mathbf{r}}_{r}^{\mathrm{T}}\mathbf{\omega}_{b} \tag{14}\] \[+M_{r}\dot{\mathbf{X}}_{pf}^{\mathrm{T}}\mathbf{A}_{b}C_{g}\tilde{\mathbf{r}} _{rc}^{\mathrm{T}}C_{r}^{\mathrm{T}}\mathbf{\omega}_{b}=\mathbf{0}\] (15) \[\dot{\mathbf{X}}_{pf}^{\mathrm{T}}\mathbf{A}_{b}M_{b}\dot{\mathbf{r}}_{b}+M_{r }\dot{\mathbf{X}}_{pf}^{\mathrm{T}}\mathbf{A}_{b}C_{r}\tilde{\mathbf{g}}_{rc}^{\mathrm{T}} \mathbf{D}_{r}\hat{\theta}_{r}=\mathbf{0} \tag{16}\]  
Considering that \(\dot{\mathbf{r}}_{b}=\dot{\mathbf{r}}_{bc}=\dot{\mathbf{r}}_{r}\) holds, the following Equation (14) holds for the combination equation:  
\[ \begin{split}& M_{b}\dot{\mathbf{r}}_{b}^{\mathrm{T}}\tilde{\mathbf{r}}_{ bc}^{\mathrm{T}}\mathbf{\omega}_{b}+M_{r}\dot{\mathbf{r}}_{r}^{\mathrm{T}}\tilde{\mathbf{r}}_{ r}^{\mathrm{T}}\mathbf{\omega}_{b}+M_{r}\dot{\mathbf{r}}_{r}^{\mathrm{T}}C_{g}\tilde{\mathbf{r}}_{ rc}^{\mathrm{T}}C_{r}^{\mathrm{T}}\mathbf{\omega}_{b}\\ &=\dot{\mathbf{r}}_{b}^{\mathrm{T}}\Big{(}M_{b}\tilde{\mathbf{r}}_{bc}^{ \mathrm{T}}+M_{r}C_{g}\tilde{\mathbf{r}}_{rc}^{\mathrm{T}}C_{r}^{\mathrm{T}}\Big{)} \mathbf{\omega}_{b}=\mathbf{0}\end{split}  \tag{17}\]  
In summary, Equation (12) can be simplified as:  
\[