## 3 Results and Discussion
### Results of Numerical Simulation of Comparison with the Conventional Method  
Figure 9 and Figure 11 show the results of the conventional method, and Figure 10 and Figure 12 show the results of the proposed method. Moreover, results of attitude settling time and terminal gimbal angle are shown in Table 3. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Symbols** & **Values** \\ \hline moment of inertia of satellite & \(\mathbf{I}\) & \(\text{diag}\big{(}\begin{array}{cccc}5000&5000&3000\end{array}\big{)}\) kgm\({}^{2}\) \\ angular momentum of SGCMG & \(h_{0}\) & & 75 Nms \\ target angular velocity of satellite & \(\mathbf{\omega}_{r}\) & \(\big{[}\begin{array}{cccc}0&0&0&0\end{array}\big{]}^{T}\text{ deg/s}\) \\ initial gimbal angle & - & \(\big{[}\begin{array}{cccc}0&0&0&0\end{array}\big{]}^{T}\text{ deg}\) \\ target gimbal angle & \(\delta_{r}\) & \(\big{[}\begin{array}{cccc}-60&60&-60&60\end{array}\big{]}^{T}\text{ deg}\) \\ initial satellite euler angle & - & \(\big{[}\begin{array}{cccc}0&0&0\end{array}\big{]}^{T}\text{ deg}\) \\ target satellite euler angle & - & \(\big{[}\begin{array}{cccc}60&0&0\end{array}\big{]}^{T}\text{ deg}\) \\ target gimbal angular velocity & \(\dot{\delta}_{r}\) & \(\big{[}\begin{array}{cccc}0&0&0&0\end{array}\big{]}^{T}\text{ deg/s}\) \\ target angular momentum & \(\mathbf{h}_{r}\) & \(\big{[}\begin{array}{cccc}0&0&0\end{array}\big{]}^{T}\text{ Nms}\) \\ very small negative scalar & \(\kappa\) & \(-10^{-9}\) \\ very small positive scalar & \(\varepsilon\) & \(10^{-5}\) \\ time constant of gimbal motor & \(\tau\) & 0.3 s \\ skew angle of four SGCMGs & \(\beta\) & 54.74 deg \\ Maximum gimbal angular velocity & - & 1 rad/s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters and values for the numerical simulation. First, it can be seen from Figures 9 and 10 that the both conventional and proposed method can avoid an internal singularity. After the avoidance, it is confirmed that it reaches the external singularity which is the performance limit. However, in the conventional method, the amount of movement in the angular momentum space is larger than in the proposed method. This is an undesirable behavior because the possibility of falling into another singularity after the singularity avoidance/escape becomes high. Next, as shown in Figures 11e and 12e, the proposed method has a longer time to hold the maximum angular momentum that SGCMGs can have. As a result, since it is possible to lengthen the attitude maneuver time at the maximum angular velocity, this is advantageous for an agile attitude maneuver. Furthermore, from Figures 11c and 12c, the time to reach the maximum satellite angular velocity is about 1.5 s faster in the proposed method. This is because gimbal steering for singularity avoidance/escape suitable for the direction of the attitude maneuver axis by the BSDW (in Figure 12g) is performed near the singularity. From the above, it is possible to shorten the attitude maneuver time of the proposed method by 3.8 s with respect to the comparison method, as shown in Table 3. Therefore, it was shown that the proposed method is more suitable for agile maneuvering. Figure 10: Results of the trajectory of SGCMG angular momentum (proposed method) with a singular surface: (**a**) \(H_{y}\)-\(H_{z}\) plane; (**b  
Figure 11: Results of the conventional method as: (**a**) Time history of the Euler angle; (**b**) Time history of torque by SGCMGs; (**c**) Time history of the satellite angular velocity; (**d**) Time history of the gimbal angle; (**e**) Time history of the angular momentum of SGCMGs; (**f**) Time history of the gimbal angular velocity; (**g**) Time history of biased state-dependent weighting (BSDW); (**h**) Time history of the singularity parameter.