# Multi-Body Dynamics Modeling and Simulation of Maglev Satellites
## 3 Design of Three Body Multi Closed Loop Control Law  
The payload-centered control method consists of high-precision rapid attitude pointing of the payload, position tracking of the platform subsystem, and MM tracking. We design a position-tracking control law based on Equation (23), and control the position tracking through a certain allocation strategy using OLFMBs and LLFMBs. On the basis of considering the influence of the position follow-up of the platform subsystem on the coupling of the payload attitude, combined with the theory of multivariable super-twisting, an extended state observer is designed to estimate and compensate for the uncertainty factors of the load attitude. A non-singular fixed-time fast terminal sliding mode surface (NFFTSMS) law is designed, which directly controls the imaging of the satellite load to the ground without additional disturbance force through OLFMBs and two-stage RLFMB, achieving high-precision and fast attitude stability of the payload. Furthermore, considering the interference effect of electromagnetic reactions from OLFMBs and LLFMBs, an MM driving control law is designed based on Equation (26). The specific control scheme is shown in the following Figure 5.  
The expected attitude quaternion \(\mathbf{q}_{pd}\) of the load subsystem and the expected angular velocity \(\mathbf{\omega}_{pd}\) of the load subsystem can be determined by their expected attitude. For the MM and SM, in the load-centered control method, it is necessary to follow the motion of the load subsystem. Therefore, the control objectives of the maglev cabin and satellite platform come from the pose information of the rotating load. If the angle at which the load subsystem rotates around the \(Y\)-axis is \(\phi\), then the quaternion of rotation is \(\mathbf{q}_{\phi}=\begin{bmatrix}q_{\phi 0}&\mathbf{q}_{\phi v}^{\mathsf{T}}\end{bmatrix}^{ \mathsf{T}}\), where \(q_{\phi 0}=\cos(\phi/2)\), \(\mathbf{q}_{\phi v}=\begin{bmatrix}0&\sin(\phi/2)&0\end{bmatrix}^{\mathsf{T}}\) hold.  
The expected attitude quaternion of the MM can be expressed as:  
\[\mathbf{q}_{rd}=\mathbf{q}_{p}\otimes\mathbf{q}_{\phi}^{*} \tag{29}\]  
where \(\mathbf{q}_{p}\) is the quaternion of the current attitude of the PM. If the expected angle of rotation of the MM relative to the SM is \(\theta_{rd}\), then its expected attitude quaternion is \(\mathbf{q}_{\theta}=\begin{bmatrix}q_{\theta 0}&\mathbf{q}_{\theta v}^{\mathsf{T}} \end{bmatrix}^{\mathsf{T}}\), where \(q_{\theta 0}=\cos(\theta_{rd}/2)\), \(\mathbf{q}_{\theta v}=\begin{bmatrix}\sin(\theta_{rd}/2)&0&0\end{bmatrix}^{ \mathsf{T}}\) hold. The expected attitude quaternion of the satellite platform is:  
\[\mathbf{q}_{bd}=\mathbf{q}_{rd}\otimes\mathbf{q}_{\theta}^{*} \tag{30}\]  
The attitude transformation matrix of the payload subsystem relative to the expected MM system is denoted as \(\mathrm{A}_{\phi}\). Therefore, the projection \(\mathbf{\omega}_{rd}\) of the expected angular velocity vector of the MM in this system is:  
\[\mathbf{\omega}_{rd}=\mathrm{A}_{\phi}(\mathbf{\omega}_{p}-\begin{bmatrix}0&\dot{\phi }&0\end{bmatrix}^{\mathsf{T}}) \tag{31}\]  
The expected position of the centroid of the platform subsystem is:  
\[\mathbf{X}_{pfd}=\mathbf{X}_{p}+\mathbf{A}_{hd}\mathbf{r}_{pf} \tag{32}\]  
where \(\mathbf{X}_{p}\) is the projection of the origin of the payload subsystem in the inertial frame; \(\mathbf{r}_{pf}\) is the vector pointing from the origin of the MM to the center of mass of the platform subsystem; and \(\mathbf{A}_{hd}\) is the attitude transformation matrix of the SM relative to the inertial frame.  
Figure 5: Three-body High-precision attitude control scheme centered on payload.