## 2 Materials and Methods
### Design of Proposed Control System
#### 2.3.2 Design of Biased State-Dependent Weighting (BSDW) Matrix of Input \(\mathbf{R}\)  
The proposed SDLR of the SISM can explicitly design a gimbal angular velocity as a control input, and each gimbal angular velocity input can be individually designed. Therefore, we propose the BSDW matrix of \(\mathbf{R}\) for the singularity avoidance/escape via the singularity parameter \(m\), which is dependent on the gimbal angle. Here, we use the fact that the singularity parameter gets closer to 0 as it approaches the singularity. First, the relation between the attitude maneuver axis and the number of SGCMGs that contribute greatly to the attitude maneuver is shown in Table 1. For the four-skew array SGCMG system, the SGCMGs steer symmetrically owing to the symmetry of the angular momentum. From Figure 4b, a singularity exists symmetrically, and it can be seen that the center of this figure is in a zero-momentum state. Moreover, SGCMG with a number which greatly contributes to attitude maneuver is prone to a singularity state. The BSDW of the roll axis is shown in this paper because agile satellites such as the Earth observation satellites usually maneuver with respect to the roll axis. In the roll axis maneuver, SGCMG1 and SGCMG3 symmetrically steer, and SGCMG2/SGCMG4 do not steer by basic pseudo inverse steering law. Therefore, when the proposed BSDW is close to a singularity, it gives a bias to the weightings between the SGCMG2 and SGCMG4, which are not steered. Thus, singularity avoidance/escape is performed by outputting the torque for other attitude maneuver axes. It is noted that stability can always be guaranteed because it is taken into account explicitly in the proposed control system. On the other hand, the conventional avoidance/escaping laws also steer the second and fourth SGCMG, but such steering laws cannot guarantee the stability of attitude control because it is not considering by attitude controller of satellite. Next, for formulation of the BSDW, a relationship between the angular momentum that can be held by asymmetrically steering SGCMG 2 and SGCMG 4 is shown in Figure 6. From Figure 6, it can be seen that through the asymmetric steering of SGCMGs, which does not contribute to the attitude maneuver, angular momentum can be held on the attitude maneuver axis. In the positive direction of the roll axis maneuver, the angular momentum of SGCMGs is negative, and in the negative direction of the roll axis maneuver, the angular momentum of SGCMGs is positive. This is because the SDRE control system using SISM calculates the optimal control input that maximizes the resource of the \begin{table}
\begin{tabular}{c c} \hline \hline
**Maneuver Axis** & **Gimbal Steering Number** \\ \hline roll & \#1 and \#3 \\ pitch & \#2 and \#4 \\ yaw & all SGCMGs \\ roll and pitch & all SGCMGs \\ pitch and yaw & all SGCMGs \\ yaw and roll & all SGCMGs \\ roll, pitch, and yaw & all SGCMGs \\ \hline \hline \end{tabular}  
* single-gimbal control moment gyros (SGCMGs).  
\end{table} 
Table 1: Relationship between the maneuver axis and gimbal steering number 1.  
output torque. Therefore, by designing the BSDW in consideration of the relationship between the direction of the attitude maneuver and the gimbal angles of SGCMG 2 and SGCMG 4, singularity avoidance/escape steering suitable for agile attitude maneuvers becomes possible. Figure 7 shows a brief summary of this relationship. Although not shown in this paper, in the case of the yaw axis and multiple axis maneuvers there is no symmetry in steering, but BSDW with SISM can also avoid/escape the singularity. This is because the BSDW can control the output of the singularity avoidance/escaping steering suitable for each attitude maneuver axis as shown in Figure 6.  
Finally, we show a selection method of BSDW satisfying the above relationship:  
\[ \begin{array}{llll}R(\delta)&=\text{diag}\Big{(}\begin{array}{cccc}R_{1}&R_ {2}&R_{3}&R_{4}\end{array}\Big{)}\\ &=\text{diag}\Big{(}\begin{array}{cccc}w_{R0}&R_{2}(\delta)&w_{R0}&R_{4}( \delta)\end{array}\Big{)}\end{array}