# Multi-Body Dynamics Modeling and Simulation of Maglev Satellites
## 3 Design of Three Body Multi Closed Loop Control Law
### Design of Payload-Centered Attitude Pointing Control Law  
Unlike traditional platform-centered control methods, in the payload-centered control method, payload attitude control dominates and changes the direction of control transmission. The payload is not affected by the satellite body and reduces error transmission links. Therefore, it can improve control accuracy and isolate platform vibration effects, achieving a disturbance-free space for payload attitude and providing an ultra-stable and ultra-static working environment. The main differences and comparisons between the two control methods are shown in Table 2.  
On the basis of the Equation (28), simplify the attitude dynamics equation of the payload subsystem as follows: \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Category**} & **Platform-Centered** & **Payload-Centered** \\  & **Control Method** & **Control Method** \\ \hline \multirow{4}{*}{Payload attitude control expectation} & \multirow{4}{*}{Expected trajectory undergoes two-stage rotation} & On the basis of ground \\  & & orientation, two-stage rotation \\  & & according to the \\  & & expected trajectory \\  & Non-contact eddy current sensor & Star sensors, gyroscopes, and \\  & & position sensors \\ Satellite platform control expectations & Ground orientation (only unilateral posture) & Follow the movement of the \\  & & payload (in terms of position \\  & & and posture) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of two control methods. \[\mathbf{J}_{pl}\dot{\mathbf{\omega}}_{p}+\widetilde{\mathbf{\omega}}_{p}\mathbf{J}_{pl}\mathbf{\omega}_ {p}=\mathbf{u}_{pl}+\mathbf{D}+\mathbf{D}_{1} \tag{37}\]  
where \(\mathbf{u}_{pl}\) is the attitude control torque of the payload subsystem; \(\mathbf{D}\) is the interference of the position control force of the platform subsystem on the attitude control of the payload subsystem; and \(\mathbf{D}_{1}\) is the modeling error. The error quaternion \(\mathbf{q}_{pe}= \begin{bmatrix}q_{pe0}&\mathbf{q}_{pev}^{\mathsf{T}}\end{bmatrix} ^{ \mathsf{T}}\) of the load subsystem can be expressed as:  
\[\mathbf{q}_{pe}=\mathbf{q}_{pd}^{*}\otimes\mathbf{q}_{p} \tag{38}\]  
where \(\mathbf{q}_{pd}^{*}\) is the conjugate quaternion of the expected quaternion for the payload subsystem. Definition \(\mathbf{A}(q_{pe})\) is the attitude transformation matrix of the load system relative to the expected frame, and the error angular velocity \(\mathbf{\omega}_{pe}\) of the load can be expressed as \(\mathbf{\omega}_{pe}=\mathbf{\omega}_{p}-\mathbf{\omega}_{pr}=\mathbf{\omega}_{p}-\mathbf{A}(q_{ pe})^{\mathsf{T}}\mathbf{\omega}_{pd}\). The platform position control force is provided by OLFMBs and LLFMBs:  
\[\mathbf{D}=\widetilde{\tau}_{su}\mathbf{C}_{p}^{r}\mathbf{I}_{ur1}+\widetilde{\tau}_{sa} \mathbf{C}_{p}^{r}\mathbf{I}_{ar}+\widetilde{\tau}_{sl}\mathbf{C}_{p}^{r}\mathbf{I}_{lr1} \tag{39}\]  
Convert the attitude dynamics equation of the payload subsystem into error form:  
\[\dot{\mathbf{\omega}}_{pe}=\mathbf{J}_{pl}^{-1}(\widetilde{\tau}_{su}\mathbf{C}_{p}^{r} \mathbf{I}_{ur1}+\widetilde{\tau}_{sa}\mathbf{C}_{p}^{r}\mathbf{I}_{ar}+\widetilde{\tau}_{ sl}\mathbf{C}_{p}^{r}\mathbf{I}_{lr1})+\mathbf{J}_{p}^{l-1}\mathbf{D}_{1}-\dot{\mathbf{\omega}}_{pr}+ \mathbf{J}_{p}^{l-1}\mathbf{u}_{pl} \tag{40}\]  
The kinematic relationship between error quaternion \(\mathbf{q}_{pe}\) and error angular velocity \(\mathbf{\omega}_{pe}\) is:  
\[\dot{q}_{pe0}=-\frac{1}{2}\mathbf{\omega}_{pe}^{\mathsf{T}}\mathbf{q}_{pev}\dot{\mathbf{q} }_{pev}=\mathbf{Q}(\mathbf{q}_{pe})\mathbf{\omega}_{pe} \tag{41}\]  
where \(\mathbf{Q}(\mathbf{q}_{pe})\) is a matrix about the error quaternion, satisfying \(\mathbf{Q}(\mathbf{q}_{pe})=(\widetilde{\mathbf{q}}_{pev}+q_{po0}\mathbf{I}_{3})/2\). Combining Equation (40) and Equation (41), organize the equations as follows:  
\[