## 4 Experimental Platform and Result Analysis
### Dynamic Test Analysis  
The working bandwidth of the MHD ARS is 2-1000 Hz. We used the azimuth motor of the turntable to generate angular vibrations of different frequencies to simulate the harmonic vibrations of a spacecraft. In order to verify the effectiveness of the proposed method under the dynamic condition, several single-frequency signals were tested. We acquired the angular vibration signals with different frequencies of 5 Hz, 10 Hz, and 15 Hz with a sampling frequency of 2 kHz and filtered the noisy MHD ARS signals. The angular vibration can be expressed as follows:  
\[\theta(t)=A\sin(2\pi ft) \tag{20}\]  
where \(A\) is the amplitude of the angular vibration and \(f\) is the frequency. The angular rate of the angular vibration can be expressed as follows:  
\[w(t)=2\pi f\cdot A\cos(2\pi ft) \tag{21}\]  
In this test, the amplitude of the angular vibration is at the microradian level, and the output of the MHD ARS is at the millivolt level. In CEEMDAN, mode mixing results in relatively small magnitudes for the relevant modes. For the above reasons, the parameters \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Methods** & **Q (\({}^{\circ}\))** & **N (\({}^{\circ}\)/\(\sqrt{\mathbf{h}}\))** & **B (\({}^{\circ}\)/h)** & **K (\({}^{\circ}\)/h\({}^{32}\))** & **R (\({}^{\circ}\)/h\({}^{2}\))** \\ \hline Noisy MHD ARS signal & 7.171 \(\times\) 10\({}^{-4}\) & 2.088 \(\times\) 10\({}^{-3}\) & 0.205 & 3.151 & 18.056 \\ CEEMDAN & 2.225 \(\times\) 10\({}^{-3}\) & 6.289 \(\times\) 10\({}^{-4}\) & 0.337 & 6.539 & 36.058 \\ CEEMDAN-Soft Thresholding & 1.185 \(\times\) 10\({}^{-3}\) & 4.941 \(\times\) 10\({}^{-4}\) & 0.242 & 4.687 & 25.829 \\ CEEMDAN-Hard Thresholding & 2.230 \(\times\) 10\({}^{-3}\) & 8.403 \(\times\) 10\({}^{-4}\) & 0.386 & 7.474 & 41.206 \\ Proposed Method & 5.400 \(\times\) 10\({}^{-5}\) & 6.466 \(\times\) 10\({}^{-5}\) & 0.117 & 2.249 & 12.370 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Allan variance analysis results of MHD ARS signal in static condition. Figure 9: Allan variance analysis of MHD ARS signal in static condition. were the same as the settings of the static experiment. We used the angular rate measured by an optical encoder system mounted on the azimuth axis as the true angular rate. We acquired single-frequency angular vibration signals at 5 Hz, 10 Hz, and 15 Hz with different angular rates. As shown in Figure 10, the vibration signals at each frequency contain three amplitudes. At 5 Hz, the angular rates of the vibrations are 0.300\({}^{\circ}\)/s, 0.200\({}^{\circ}\)/s, and 0.100\({}^{\circ}\)/s. At 10 Hz, the angular rates of the vibrations are 0.150\({}^{\circ}\)/s, 0.100\({}^{\circ}\)/s, and 0.050\({}^{\circ}\)/s. At 15 Hz, the angular rates of the vibrations are 0.100\({}^{\circ}\)/s, 0.050\({}^{\circ}\)/s, and 0.025\({}^{\circ}\)/s. Figure 11 shows the IMFs and frequency spectrum of the 0.100\({}^{\circ}\)/s signal at 15 Hz. Figure 11: IMFs and frequency spectrum of the 0.100\({}^{\circ}\)/s signal at 15 Hz. Figure 10: Denoising results of MHD ARS signals at 15 Hz. It is observed that the denoised signal obtained by the proposed method is much smoother. Figure 11 shows the IMFs of the 0.100\({}^{\circ}\)/s signal at 15 Hz. Figure 12 shows the \(R_{RMS}\) of the relevant modes. As shown in Figure 11, mode mixing appears in the relevant modes, which results in a larger magnitude of IM7 than the other relevant modes. It is due to the fact that noise affects the selection of the local minimum and maximum values, the average values are obtained from the combination of the noise and the real signal [24]. Noise in the relevant modes IMF5 and IMF6 can be observed. It can be seen from the frequency spectrum in Figure 11 that IMF10 and IMF11 are ineffective modes in low-frequency without the 15 Hz signal, and IMF1-IMF4 are noise modes in high-frequency. RMSE and SNR are commonly used indicators to evaluate the effect of signal denoising [36].