## 2 Materials and Methods  
### Modeling  
#### 2.1.1 Formulation of Satellite Attitude Maneuvers Using SGCMG  
The satellite attitude in the body coordinate system is represented by quaternion \(\mathbf{q}\), and time derivative \(\dot{\mathbf{q}}\) is expressed by the following equation:  
\[\dot{\mathbf{q}}=\frac{1}{2}\left[\begin{array}{cccc}0&-\omega_{1}&-\omega_{2}&- \omega_{3}\\ \omega_{1}&0&\omega_{3}&-\omega_{2}\\ \omega_{2}&-\omega_{3}&0&\omega_{1}\\ \omega_{3}&-\omega_{2}&-\omega_{1}&0\end{array}\right]\mathbf{q}=\frac{1}{2}\tilde {\mathbf{c}}q \tag{1}\]  
where \(\mathbf{\omega}=\left[\begin{array}{cc}\omega_{1}&\omega_{2}&\omega_{3}\end{array} \right]^{T}\) is the satellite angular velocity.  
The rotation of a rigid-body satellite in the body coordinate system is defined by the Euler equation including the cross product calculation \"\(\times\)\" as follows:  
\[\mathbf{I}\dot{\mathbf{\omega}}+\mathbf{\omega}\times\mathbf{I}\mathbf{\omega}=\mathbf{T} \tag{2}\]  
where \(\mathbf{I}\) is the moment of inertia matrix of the satellite. Control torque vector \(\mathbf{T}\) is produced by the SGCMG momentum exchange. In this paper, disturbance torques are assumed to be negligible, and the nonlinear rotating dynamics of the satellite are expressed as  
\[\left[\begin{array}{c}\dot{\mathbf{q}}(t)\\ \dot{\mathbf{\omega}}(t)\end{array}\right]=\left[\begin{array}{c}\frac{1}{2} \widetilde{\mathbf{\omega}}(t)\mathbf{q}(t)\\ -\mathbf{I}^{-1}(\mathbf{\omega}(t)\times\mathbf{I}\mathbf{\omega}(t))\end{array}\right]+ \left[\begin{array}{c}\mathbf{0}\\ \mathbf{I}^{-1}\end{array}\right]\mathbf{T}(t) \tag{3}\]  
Now, a relation between control torque \(\mathbf{T}\) and generated torque \(\mathbf{T}_{c}\) by the SGCMG is expressed as  
\[\mathbf{T}=-\mathbf{T}_{c}-\mathbf{\omega}\times\mathbf{h} \tag{4}\]  
where \(\mathbf{h}\) is the angular momentum vector of the SGCMG. The time derivative of the angular momentum of the SGCMG is equal to the output torque \(\mathbf{T}_{c}\), and is expressed by the following equation.  
\[\dot{\mathbf{h}}=h_{0}\mathbf{A}(\delta)\dot{\delta}\equiv\mathbf{T}_{c} \tag{5}\]  
where \(h_{0}\) is the angular momentum of the SGCMG wheel, \(\dot{\delta}\) is the gimbal angular velocity vector of the SGCMGs, and \(\mathbf{A}(\delta)\) is a Jacobian matrix. The size of the Jacobian matrix representing how to steer the SGCMGs differs depending on the number of mountings. In this research, from the viewpoint of redundancy and the symmetry of the angular momentum, the SGCMGs should be equipped on a four-skew array like a pyramid, as shown in Figure 1b.  
Therefore, the Jacobian matrix \(\mathbf{A}(\delta)\) is expressed as  
\[\mathbf{A}(\delta)=\left[\begin{array}{cccc}-c\beta\cos\delta_{1}&\sin\delta_{ 2}&c\beta\cos\delta_{3}&-\sin\delta_{4}\\ -\sin\delta_{1}&-c\beta\cos\delta_{2}&\sin\delta_{3}&c\beta\cos\delta_{4}\\ s\beta\cos\delta_{1}&s\beta\cos\delta_{2}&s\beta\cos\delta_{3}&s\beta\cos \delta_{4}\end{array}\right] \tag{6}\]  
where \(\beta\) is the skew angle of four SGCMGs, \(s\beta=\sin\beta\), and \(c\beta=\cos\beta\). From Equation (3) to Equation (5), the rotation of a satellite equipped with SGCMGs is expressed as  
\[\left[\begin{array}{c}\dot{\mathbf{q}}(t)\\ \dot{\mathbf{\omega}}(t)\end{array}\right]=\left[\begin{array}{c}\frac{1}{2} \widetilde{\mathbf{\omega}}(t)\mathbf{q}(t)\\ -\mathbf{I}^{-1}\{\mathbf{\omega}(t)\times(\mathbf{I}\omega(t)+\mathbf{h}(\delta))\}\end{array} \right]+\left[\begin{array}{c}\mathbf{0}\\ -\mathbf{I}^{-1}\end{array}\right]\mathbf{T}_{c}(t) \tag{7}\]