## 4 Validation of the Proposed Disturbance Torque Model
### Simulation Results  
According to the disturbance torque model established in Section 2, the rotation process of the SADA-driven solar array is simulated by using the Simulink toolbox in the commercial software Matlab (MathWorks, Natick, MA, USA). The schematic diagram of the simulation model is shown in Figure 13.  
The disturbance torque generated by the SADA-driven SAS of state 1 is simulated. The input parameters are mainly composed of two parts: (a) The SADA and its control system parameters; (b) parameters of the SAS. Table 3 shows the parameters of the SADA and its control system, which are provided by the manufacturer after test or calculation. Parameters of the SAS are stiffness matrix, mass matrix, and damping information. These parameters are extracted from the accurate finite element model of the SAS, as shown in Figure 14a. Figure 14b-d show the first three modes of the SAS; the moment of inertia of the finite element model along the rotation axis was 8.46 kg\(\cdot\)m\({}^{2}\). On the other hand, the natural frequency and moment of inertia of the SAS were meas  
Figure 13: Schematic diagram of the disturbance torque simulation. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Parameter** & **Value (Unit)** \\ \hline \(z\) & Number of rotor teeth & 50 \\ \(b\) & Beat number & 4 \\ \(n\) & Subdivisions & 16 \\ \(I\) & Current Amplitude & 0.6 A \\ \(K_{m}\) & Motor torque constant & 0.3 Nm/A \\ \(K_{d}\) & Detent torque constant & 0.01 Nm/A \\ \(J_{r}\) & Moment of inertia of SADA rotor & 0.2 kg\(\cdot\)cm\({}^{2}\) \\ \(J_{wg}\) & Moment of inertia of wave generator & 0.052 kg\(\cdot\)cm\({}^{2}\) \\ \(R\) & Winding resistance & 6 \(\Omega\) \\ \(L\) & Self-induction coefficient & 13 mH \\ \(h\) & Reduction ratio & 100 \\ \(\omega\) & Cruising speed of solar array & 0.06\({}^{\circ}\)/s \\ \(K_{p}\) & Proportional coefficient & 500 \\ \(K_{l}\) & Integral coefficient & 6000 \\ \(K_{D}\) & Differential coefficient & 0 \\ \(T_{c}\) & Coulomb friction coefficient for motor & 0.0246 \\ \(T_{s}\) & Static friction coefficient for motor & 0.0462 \\ \(\omega_{s}\) & Stribeck velocity for motor & 0.003\({}^{\circ}\)/s \\ \(\sigma\) & Viscous friction coefficient for motor & 0.0255 \\ \(N_{c}\) & Number of contact pairs of ring and brush & 320 \\ \(p\) & Pressure of a single point & 0.042 N \\ \(\mu\) & The friction factor & 1.53 \\ \(r\) & The radius of conductive ring & 0.0265 m \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the SADA. The tested moment of inertia of the SAS about the rotation axis, which is 8.19 kg\(\cdot\)m\({}^{2}\), was taken as reference; the error of the moment of inertia calculated by the finite element model was 0.27 kg\(\cdot\)m\({}^{2}\), and the relative error was only 3.3%. Table 4 shows the frequencies of the first three modes obtained by finite element method and test. The table shows that the frequency error is no more than 3.15%, which indicates that the finite element model is an accurate model of the SAS under test state 1. Figure 16 shows the frequency domain curve comparison between the simulation results and the test results. Table 5 shows the frequency values of the peaks of the simulation results and the test results. The frequency error between the simulation results and the test results is no more than 0.684%. Table 6 shows the amplitudes of simulation results and test results at each peak; the relative errors of the amplitude of the disturbance torque between simulation results and test results at each frequency component are not greater than 22.33%. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Test (Hz)** & **Finite Element Model (Hz)** & **Relative Error** \\ \hline
1 & 1.189 & 1.187 & 0.419\% \\
2 & 18.72 & 18.16 & 2.99\% \\
3 & 18.76 & 18.27 & 3.15\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of modal frequency. Figure 14: Finite element model and the first three modes of the SAS. Figure 15: Modal test and moment of inertia test of the SAS.