## 2 Modeling
### Dynamics of the Satellite Mounted with DRC Antenna  
A satellite model considered in this study is shown in Figure 5. We assume that DRC antenna has 2 gimbal axis and the Reaction wheel (RW) of the satellite are installed in each three axis of satellite and antenna joint motors can generate ideal torques. \(\mathcal{F}_{B}^{T}=\{\mathbf{b}\}=\{\mathbf{e}_{B1},\mathbf{e}_{B2},\mathbf{e}_{B3}\}\) represents the satellite coordinate system where \(\mathbf{e}_{B_{1}}\), \(\mathbf{e}_{B_{2}}\), and \(\mathbf{e}_{B_{3}}\) are the unit vectors along the satellite coordinate axis. \(\mathcal{F}_{A}^{T}=\{\mathbf{a}\}=\{\mathbf{e}_{A1},\mathbf{e}_{A2},\mathbf{e}_{A3}\}\) represents the antenna coordinate system, where \(\mathbf{e}_{A_{1}}\), \(\mathbf{e}_{A_{2}}\), and \(\mathbf{e}_{A_{3}}\) are unit vectors along the antenna coordinate axis. \(\mathbf{\phi}=[\phi_{1},\phi_{2}]^{T}\) represents a rotation joint angle vector of the antenna relative to the satellite. \(\mathbf{e}_{\phi 1},\mathbf{e}_{\phi 2}\) represent the unit direction vectors of the joint angle \(\phi_{1}\), \(\phi_{2}\). \(m_{s}\) represents the mass of the satellite, and \(m_{a}\) represents the mass of the antenna. \(\mathbf{I}_{s}=\mathrm{diag}(i_{s_{1}},i_{s_{2}},i_{s_{3}})\) represents the inertia matrix of the satellite. \(\mathbf{I}_{a}=\mathrm{diag}(i_{a_{1}},i_{a_{2}},i_{a_{3}})\) represents the inertia matrix of the antenna. \(d_{2}\) represents the distance between the center mass of the antenna and the axis \(\phi_{2}\). \(\mathbf{p}_{s}\) represents the position vector of the center of mass of the satellite. \(\mathbf{p}_{a}\) represents the position vector of the center of mass of the antenna. \(\mathbf{p}_{c}\) represents the position vector of the center of mass of the whole satellite mounted with DRC antenna. \(\mathbf{p}_{h}\) represents the position vector of the joint rotation hinge. Each position vector is expressed by the following equation:  
Figure 4: Control structure of proposed method  
Figure 3: Image of satellite operation. \[\mathbf{p}_{h}-\mathbf{p}_{s}=\mathcal{F}_{B} \begin{bmatrix}0\\ 0\\ -d_{3}\end{bmatrix} ,\ \mathbf{p}_{a}-\mathbf{p}_{s}=\mathcal{F}_{B} \begin{bmatrix}-d_{2} \sin\phi_{1}\cos\phi_{2}\\ -d_{2}\cos\phi_{1}\cos\phi_{2}\\ -d_{3}-d_{2}\sin\phi_{2}\end{bmatrix} ,\ \mathbf{p}_{c}=\frac{m_{s}\mathbf{p}_{s}+m_{a}\mathbf{p}_{a}}{m_{s}+m_{a}} \tag{2}\]  
The inertia matrix for the antenna \(\mathbf{I}_{a}\) can be expressed in the satellite coordinate system as following equation  
\[\mathbf{I}_{a}=\mathcal{F}_{A} \begin{bmatrix}i_{a1}&0&0\\ 0&i_{a2}&0\\ 0&0&i_{a3}\end{bmatrix} =\mathcal{F}_{B}\mathbf{C}_{AB} \begin{bmatrix}i_{a1}&0&0\\ 0&i_{a2}&0\\ 0&0&i_{a3}\end{bmatrix} \mathbf{C}_{AB}^{T},\mathbf{C}_{AB}= \begin{bmatrix}\cos\phi_{1 }&-\sin\phi_{1}\cos\phi_{2}&\sin\phi_{1}\sin\phi_{2}\\ -\sin\phi_{1}&-\cos\phi_{1}\cos\phi_{2}&\cos\phi_{1}\sin\phi_{2}\\ 0&-\sin\phi_{2}&-\cos\phi_{2}\end{bmatrix}  \tag{3}\]  
Considering that \(\mathbf{E}\) represents the unit matrix, and the inertia of the whole satellite around the center of mass of the whole satellite is represented by \(\mathbf{I}_{c}\), then the inertia of the antenna around the center of mass of the whole satellite, \(\mathbf{I}_{ac}\) can be expressed by following equation [17]:  
\[\mathbf{I}_{c}= \mathbf{I}_{s}+m_{s}\{(\mathbf{p}_{s}-\mathbf{p}_{c})^{T}(\mathbf{p}_{s}-\mathbf{p}_{ c})\mathbf{E}-(\mathbf{p}_{s}-\mathbf{p}_{c})(\mathbf{p}_{s}-\mathbf{p}_{c})^{T}\} \tag{4}\] \[+\mathbf{I}_{a}+m_{a}\{(\mathbf{p}_{a}-\mathbf{p}_{c})^{T}(\mathbf{p}_{a}-\mathbf{p}_{ c})\mathbf{E}-(\mathbf{p}_{a}-\mathbf{p}_{c})(\mathbf{p}_{a}-\mathbf{p}_{c})^{T}\}\] \[\mathbf{I}_{ac}=\mathbf{I}_{a}+m_{a}\{(\mathbf{p}_{a}-\mathbf{p}_{c})^{T}(\mathbf{p}_{ a}-\mathbf{p}_{h})\mathbf{E}-(\mathbf{p}_{a}-\mathbf{p}_{h})(\mathbf{p}_{a}-\mathbf{p}_{c})^{T}\} \tag{5}\]  
\(\mathbf{e}_{\phi_{1}},\mathbf{e}_{\phi_{2}}\) can be expressed by the following equation:  
\[\mathbf{e}_{\phi_{1}}=\mathcal{F}_{B} \begin{bmatrix}0&0&-1\end{bmatrix} ^{T},\ \ \mathbf{e}_{\phi_{2}}=\mathcal{F}_{B}