## 2 Materials and Methods
### Design of Proposed Control System
#### 2.3.2 Design of Biased State-Dependent Weighting (BSDW) Matrix of Input \(\mathbf{R}\)
 \tag{26}\]  
Figure 6: Relationship between angular momentum of SGCMGs and the gimbal angle. Figure 7: Overview of suitable angular momentum of SGCMGs for singularity avoidance/escape considering the direction of the roll axis maneuver. \[ \begin{split}& if\ 0\leq\delta_{2}^{m}<90\\ &\qquad\left\{\begin{array}{l}R_{2}=w_{R}^{+},\ \ R_{4}=w_{R}^{-} \qquad\left(\delta_{2}^{m}\leq\delta_{4}^{m}\leq 360-\delta_{2}^{m}\right)\\ R_{2}=w_{R}^{-},\ \ R_{4}=w_{R}^{-}\qquad\left(\delta_{4}^{m}<\delta_{2}^{m}\ \sigma\ 360-\delta_{2}^{m}<\delta_{4}^{m}\right) \end{array}\right.\\ & if\ 90\leq\delta_{2}^{m}<180\\ &\qquad\left\{\begin{array}{l}R_{2}=w_{R}^{+},\ \ R_{4}=w_{R}^{-} \qquad\left(\delta_{4}^{m}<\delta_{2}^{m}\ \sigma\ 360-\delta_{2}^{m}<\delta_{4}^{m}\right) \\ R_{2}=w_{R}^{-},\ \ R_{4}=w_{R}^{+}\qquad\left(\delta_{2}^{m}\leq\delta_{4}^{m}\leq 36 0-\delta_{2}^{m}\right)\end{array}\right.\\ & f\ 180\leq\delta_{2}^{m}<270\\ &\qquad\left\{\begin{array}{l}R_{2}=w_{R}^{+},\ \ R_{4}=w_{R}^{-} \qquad\left(\delta_{4}^{m}<360-\delta_{2}^{m}\ \sigma\ \delta_{2}^{m}<\delta_{4}^{m}\right) \\ R_{2}=w_{R}^{-},\ \ R_{4}=w_{R}^{+}\qquad\left(360-\delta_{2}^{m}\leq\delta_{4}^{m} \leq\delta_{2}^{m}\right)\end{array}\right.\\ & f\ 270\leq\delta_{2}^{m}<360\\ &\qquad\left\{\begin{array}{l}R_{2}=w_{R}^{+},\ \ R_{4}=w_{R}^{-} \qquad\left(360-\delta_{2}^{m}\leq\delta_{4}^{m}\leq\delta_{2}^{m}\right) \\ R_{2}=w_{R}^{-},\ \ R_{4}=w_{R}^{+}\quad\left(\delta_{4}^{m}<360-\delta_{2}^{m}\ \sigma\ \delta_{2}^{m}<\delta_{4}^{m}\right) \end{array}\right.\end{split}  \tag{27}\]  
where \(\delta_{2}^{m}\) and \(\delta_{4}^{m}\) are gimbal angle of SGCMG2 and SGCMG4 when \(m=0.3\). It is assumed that the magnitude and sign of the gimbal angle are normalized from 0 to 360 degrees using the periodicity of trigonometric function. The weighting functions \(w_{R}^{+}\) and \(w_{R}^{-}\), as shown in Equations (28) and (29), are designed by a sigmoid function where the difference between \(w_{R}^{+}\) and \(w_{R}^{-}\) becomes significantly larger when \(m<0.3\). When the difference between the weighting functions is large, the control input is calculated such that the other steering amount increases so that one of the steering amounts of SGCMG 2 and SGCMG 4 is decreased. Thus, \(w_{R}^{+}\) and \(w_{R}^{-}\) make it possible to realize the concept of singularity avoidance/escape suitable for the direction of the attitude maneuver axis described above. \[w_{R}^{+}(\delta_{e})=\varepsilon+w_{R0}\left(1+\frac{2}{1+\exp\left(\alpha m \left(\delta_{e}\right)^{2}\right)}\right) \tag{28}\]  
\[w_{R}^{-}(\delta_{e})=\varepsilon+w_{R0}\left(-1+\frac{2}{1+\exp\left(-\alpha m \left(\delta_{e}\right)^{2}\right)}\right) \tag{29}\]  
where \(\varepsilon\) is a very small positive scalar for \(\mathbf{R}>\mathbf{0}\), \(w_{R0}\) is a positive scalar of basic weighting function, and \(\alpha\) is a positive scalar to determine a sensitivity of \(w_{R}^{+}\) and \(w_{R}^{-}\). In this paper, we determine \(w_{R0}=1\) and \(\alpha=50\). Therefore, \(w_{R}^{+}\) and \(w_{R}^{-}\) are shown in Figure 8.