## 4 Simulations and Experiments
### Simulations
#### 4.1.1 Simulation Setup  
(a) Simulation Setup of Signal Processing  
A standard sinusoidal signal \(sin(10t)\) is assumed to be an angular attitude reference signal. The random noise \(n_{a}(t)\in[-0.5,0.5]\) is added to the reference signal as the angular attitude information calculated by the accelerometer and a noise with amplitude \(A_{a}=1\) is introduced every 0.1 s to simulate the maneuvering acceleration interference. The random noise \(n_{v}(t)\in[-0.1,0.1]\) is added to the differential of the signal as the angular velocity information calculated by the gyroscope. These two signals are applied in DFBGA and SLAIM. The random noise \(n(t)\in[-0.2,0.2]\) is added to the reference signal and Butterworth filter is carried out to introduce phase lag. This signal is applied in Phase-lag-free LPF. The cut-off frequency \(g\) of phase-lag-free LPF is 40 Hz. Other parameters are shown in Tables 1 and 2. In this simulation, RMSE is applied to compare the signal processed by corresponding algorithm and reference signal. \[q_{RMSE}=\sqrt{\frac{1}{m}\sum_{k=1}^{m}\left(y_{k}-\hat{y}_{k}\right)^{2}}, \tag{27}\]  
where \(y_{k}\) is reference signal \(\hat{y}_{k}\) is the signal processed by corresponding algorithm and \(m\) is number of sampling points. This section constructs a simulated single-axis close-loop control of pitch direction for optoelectronic platform, composing of proportion-differentiation (PD) control and disturbance observer (DOB). According to the physical characteristics, the dynamic electromechanical model of each motor can be simplified as  
\[J_{n}\ddot{\vartheta}+B_{n}\dot{\vartheta}=u+d, \tag{28}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Value** \\ \hline State transition matrix \(A_{0}\) & 1 \\ Measurement transition matrix \(H_{0}\) & \(\left[\begin{array}{cc}1&0\\ \end{array}\right]^{T}\) \\ Process noise \(Q_{0}\) & 0.1 \\ Measurement noise \(R_{0}\) & 0.000001 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of data fusion method based on gyro and accelerometer in simulation. where \(J_{n}\) is nominal mass, \(B_{n}\) is nominal damping, \(u\) is control variable, \(d\) is system disturbance, \(\hat{\vartheta}\) is angular position response which processed by proposed method and \(\hat{\vartheta}\) is angular velocity response which measured by gyro. The control system diagram is shown in Figure 5. \(\hat{\vartheta}_{1}\) is the angular position command, \(\hat{\vartheta}_{0}\) is the angular position response and \(\hat{\vartheta}_{0}\) is the angular velocity response. \(d\) is system disturbance and \(\hat{d}\) is disturbance estimation. The actual position response is compared with the position command to verify the algorithm. A standard sinusoidal signal \(sin(10t)\) is assumed to be the position command. The motor speed response with random noise \(n_{v}(t)\in[-0.1,0.1]\) and inertial DC offset which amplitude \(A_{v}=1\) is used as the simulated original-gyro signal. The motor position response with random noise \(n_{a}(t)\in[-0.5,0.5]\) is simulated as the angular attitude calculated by accelerometer. A noise with amplitude \(A_{a}=1\) is introduced every 0.1 s to simulate the maneuvering acceleration interference. The speed response feedback, the feedback signal proposed by DFBGA and the proposed feedback signal are applied in control system. Table 3 shows the parameters of the control system. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Value** \\ \hline State transition matrix \(A_{0}\) & 1 \\ Measurement transition matrix \(H_{0}\) & 1 \\ Process noise \(Q_{0}\) & 0.001 \\ Measurement noise \(R_{0}\) & 0.001 \\ Adjustable parameter \(\delta_{j}\) & 0.3 \\ Random process noise \(\omega\) & 0.01 \\ Adjustable parameter \(a_{j}\) & 0.8 \\ Adjustable parameter \(b_{j}\) & 0.1 \\ Adjustable parameter \(c_{j}\) & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of attitude estimation for suppressing maneuvering acceleration interference in simulation. Figure 5: Control system diagram.