## 6 Simulations
### Simulation I
#### 6.1.2 Results of Simulation I  
Figure 12 shows the time history of Euler angle of satellite, satellite angular velocity, antenna angle, antenna angular velocity, RW angular acceleration, Torque from antenna motor, RW wheel speed and SDW matrix \((\mathbf{Q}_{qcl},\mathbf{Q}_{qcl}^{int})\) of proposed method. A solid line represents the time history of state and a dotted line represents the reference time history of state. From Figure 12a,c, we can confirm that proposed method can track a reference attitude. Figure 13a-d show the time history of the Euler angle around roll axis magnified near settling time. Figure 13e,f shows the RW angular acceleration of proposed method and SDRE servo. Figure 13g,h show the integral error of state vector of each controller.  
Table 11 shows the settling time of the satellite and antenna of each method. From Table 11, we confirmed that the proposed method could track the reference attitude and reduced the settling time when compared with other controller. On the other hand, the SDRE servo controller the SDW of which is fixed increased settling time when compared with LQR controller and SDRE controller. Figure 13b shows the SDRE servo controller could track the reference attitude until 43 s, however the error of Euler angle around roll axis increased from 43 s to 47.8 s. This is because the torque from RW increased from 43 s to 47.8 s in order to reduce the error of integral state vector as shown in Figure 13h. In contrast, proposed controller could be track the reference attitude at all times as shown in Figure 13a. This is because we transformed the SDW matrix from a servo controller to a regulator when the reference trajectory is fixed, and steady error is shown in Figure 13h. Consequently, the responsiveness of the proposed controller is raised by prioritizing the reduction of the state vector error over the integral state vector error, as shown in Figure 13g.  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Parameters** & **Symbols** & **Values** & **Unit** \\ \hline Inertia of satellite main body & \(I_{s}\) & diag(166.7, 166.7, 66.67) & kg m\({}^{2}\) \\ Inertia of antenna (coordinates of the antenna) & \(I_{d}\) & diag(2.96, 3.71, 2.96) & kg m\({}^{2}\) \\ Real inertia of antenna (coordinates of the antenna) & \(I_{a}^{real}\) & diag(3.26, 4.08, 3.26) & kg m\({}^{2}\) \\ Inertia of Reaction wheel & \(I_{RW}\) & diag(0.2383, 0.2383, 0.2383) & kg m\({}^{2}\) \\ Initial attitude angle of satellite & \(\theta^{0}\) & \([-45\ 0\ 0]^{T}\) & deg \\ Maximum value of RW wheel speed & \(\Gamma^{lim}\) & 2200 & rpm \\ Target attitude angle of satellite & \(\mathbf{\theta}^{i}\) & \([45\ 0\ 0]^{T}\) & deg \\ Initial joint angle of antenna & \(\mathbf{\phi}^{0}\) & \([114.8\ 22.8]^{T}\) & deg \\ Target joint angle of antenna & \(\mathbf{\phi}^{t}\) & \([65.2\ 22.8]^{T}\) & deg \\ Elements of dividing the control input & \(N\) & 6 & pes \\ Control cycle & \(\Delta t\) & 0.02 & s \\ Settling conditions of satellite & & \([45\pm 0.003,\pm 0.003]^{T}\) & deg \\ Settling conditions of antenna & & \([65.2\pm 0.3,22.8\pm 0.3]^{T}\) & deg \\ \hline \hline \end{tabular}
\end{table}
Table 9: Simulation parameters.