## 4 Numerical Simulation
The attitude controller of the proposed method in this paper has 9 control coefficients in total, and their values used in the simulation are shown in Table 2.  
The simulation results of the sunlight occlusion where the satellite is located and the satellite control phase are shown in Figure 4. The initial damping phase lasts for nearly 1130 s. The 3-axis angular rates of the satellite are between \(-0.2^{\circ}\)/s and \(0.2^{\circ}\)/s at the end of this phase (Figure 5a). Then, ADCS enters the sun-aligned phase and begins to search for the sun. At about 3060 s, the sun enters the field of view of the sun sensor (Figure 6a). After about 130 s, ADCS enters the spin-up phase and begins to spin up around the \(-O_{b}Y_{b}\) axis. However, the satellite flies into the shadow zone of the orbit at 3980 s, and the satellite must return to the sun-aligned phase. At 6370 s, the sun enters the field of view of the sun sensor again (Figure 6a). After a series of attitude control, the satellite finally enters the spin-stabilized phase at about 9170 s, which is the long-term operating mode of the satellite. The simulation results show that the satellite can keep the sun-oriented attitude steadily after entering the spin-stabilized phase. Most notably, with the difference of satellite characteristic parameters, orbit parameters, initial attitude, and control coefficients, the variation curve of the control phase will be different. \begin{table}
\begin{tabular}{c c c} \hline
**Site** & **Parameter** & **Value** \\ \hline \multirow{6}{*}{Environmental torques} & Gravity gradient torque & \(1.81\times 10^{-6}\) N\(\cdot\)m \\ \cline{2-3}  & Aerodynamic torque & \(4.92\times 10^{-7}\) N\(\cdot\)m \\ \cline{2-3}  & Solar radiation torque & \(1.58\times 10^{-7}\) N\(\cdot\)m \\ \cline{2-3}  & Geomagnetic torque & \(7.89\times 10^{-7}\) N\(\cdot\)m \\ \hline Geomagnetic field model & Accuracy & 8-order IGRF [34] \\ \cline{1-1} \cline{2-3} Orbit injection parameters & Separation angular rate & \([-1.5\)\(-1.5\)\(-1.5]^{\circ}/\)s \\ \cline{1-1}  & Inertial attitude & \([160\)\(20\)\(60]^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._ \begin{table}
\begin{tabular}{c c} \hline
**Control Coefficient** & **Value** \\ \hline \(K_{1}\), \(K_{9}\) & 0.01 \\ \(K_{2}\), \(K_{5}\) & 0.0005 \\ \(K_{3}\), \(K_{6}\) & 0.001 \\ \(K_{4}\), \(K_{7}\) & 0.02 \\ \(K_{8}\) & 0.006 \\ \hline \end{tabular}
\end{table} 
Table 2: The control coefficients of the controller in the simulation. Figure 4: The variation curve of the sunlight occlusion where the satellite is located and the satellite control phase. Figure 5: Simulation results of (**a**) 3-axis angular rates of the satellite estimated by rate filter; (**b**) angular rates estimation errors of rate filter. The 3-axis angular rates of the satellite estimated by the rate filter and its estimation errors are shown in Figure 5. According to statistics, the mean value and standard deviation of rate filter estimation errors are \([-0.00105-0.134-0.00388]^{\circ}/\)s and \([0.0797\ 0.0785\ 0.0681]^{\circ}/\)s.  
Figure 6 shows the solar vector measured by the sun sensor and the true solar vector; both are represented in the coordinate system \(O_{b}X_{b}Y_{b}Z_{b}\). The \(O_{b}X_{b}\)-axis and \(O_{b}Z_{b}\)-axis components of the solar vector have obvious oscillations, which is mainly due to the nutation of the satellite while spinning. Especially in the shadow zone of the orbit, the satellite cannot obtain the solar vector and control the direction of the spin axis, so nutation becomes more intense during this period. Figure 7 shows the control torque outputted by the 3-axis MTQs. When the satellite flies from the shadow zone to the sunlit zone, the control torque increases to reduce the satellite nutation. Figure 6: Measured (**a**) and true (**b**) solar vectors.