## 4 Simulation and Analysis  
To validate the feasibility of the proposed method for resonance suppression of the MISP, a numerical simulation is presented. The relative parameters are listed in Table 1.  
Figure 6: The adaptive notch filter structure diagram.  
To simulate the actual operation of the MISP, the experimental conditions were set as follows:  
1. Input 0.05 rad/s tilting signal to the frame, and the swing period lasted for 4 s.
2. The tilting direction was changed at 1 and 3 s. It is assumed that clearance occurs in the gear rack during the direction turn around, and the transmission structure stiffness changes when N(A) is assumed to be 0.8 s and 0.5 s, respectively, which will result in a resonance frequency variable. Theoretically, the initial resonance frequency of the gimbal frame is 216 rad, which was changed to 193 rad and 164 rad.
3. The RLS algorithm with a constant forgetting factor and an algorithm in [21] were used for comparison with the improved algorithm in the simulation. The constant forgetting factor is 0.99, and the forgetting factor in [21] is given as \[\lambda(k)=\frac{b}{c+|e(k)e(k-1)|}\] (35) where the value of \(b\), \(c\) is set according to the error \(e(k)\). The simulation results are shown in Figures 7 and 8.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Symbol** & **Value** & **Unit** \\ \hline Equivalent inertia of motor and & \(m_{0}\) & 0.0432 & kg/m\({}^{2}\) \\ reducer & \(M\) & 3.2 & kg/m\({}^{2}\) \\ \hline Inertia of glatform & \(m\) & 1.6 & kg/m\({}^{2}\) \\ \hline Negative stiffness & \(k\) & 50,000 & Nm/rad \\ \hline Parameter about forgetting & \(\alpha\) & 0.5 & – \\ factors & \(\beta\) & 0.67 & – \\ \hline Stiffness of transmission shaft & \(K\) & 2000 & – \\ \hline Damping of transmission shaft & \(C_{w}\) & 1 & – \\ \hline IIR filter & \(\rho\) & 0.9 & – \\ bandwidth parameter & \(\rho\) & 0.9 & – \\ \hline Time step & \(T_{s}\) & 0.001 & s \\ \hline \multirow{3}{*}{Coarse system controller} & \(k_{p}\) & 5833 & – \\ \cline{2-4}  & \(i\) & 2 & – \\ \cline{2-4}  & \(d\) & 0.03 & – \\ \hline \multirow{3}{*}{Fine system controller} & \(K_{p}\) & 300,000 & – \\ \cline{2-4}  & \(l\) & 100 & – \\ \cline{1-1} \cline{2-4}  & \(D\) & 0.003 & – \\ \hline \end{tabular}
\end{table}
Table 1: System parameters in simulation.  
As shown in Figure 7a,b, the adaptive notch filter can effectively suppress the resonance with variable frequency in the gimbal frame with the dual-stage system so that the stabilized platform can maintain a high pointing accuracy. From Figure 7a,c, the RLS algorithm with the proposed forgetting factor is better than that with a constant forgetting factor and that in [21] for resonance suppression, especially when the resonance frequency becomes lower, and the resonance amplitude intensifies. The improvement in the resonance suppression ability of the gimbal frame can reduce the relative tilting angle between the gimbal frame and the platform and avoid exceeding the saturated gap. As shown in Figure 8, the proposed forgetting factor RLS algorithm has a better tracking ability and stability, which results in better resonance suppression for the adaptive notch filter. From Figures 7a and 8, compared with that at 1 s, the resonance at 3 s is more severe, and the  
Figure 7: Ttiling angles of the MISP under adaptive resonance suppression: **(a)** The titling angle of the gimbal frame. **(b)** The titling angle of the stabilized platform. **(c)** The relative tilting angle of dualstage systems.  
adjustment time is longer. This means that the resonance intensifies with a decrease in the resonance frequency when the gear gap reduces the structural stiffness, which is consistent with the conclusion in Section 2.