## 4 Simulations and Experiments
### Experiments  
#### 4.2.1 Experimental Setup  
The experimental device is shown in Figure 9. It is an optoelectronic platform composed of 3-axis servo unit motors with encoders, inertial measurement unit (IMU) with 3-axis gyros and accelerometers, and pay load. The velocity acceleration of the motors are measured by gyros and accelerometers of IMU. The measurement bandwidth and accuracy of encoder can be sure to be convinced for control and the system will not be unstable due to its measurement problems. Therefore, it is reliable to take the encoder measurement data as the standard experimental result. The IMU used in this study is WT931. The accuracy of the gyro is 0.05\({}^{\circ}\)/s and the accuracy of the accelerometers is 0.1 m/s\({}^{2}\). The encoder applied in this study is AS5048 and its resolution is 0.05\({}^{\circ}\). The sampling time is 1.5 ms. The cut-off frequency \(g\) of phase-lag-free LPF of roll, pitch, and yaw attitude are 150 Hz, 100 Hz, and 150 Hz, respectively. Other parameters are shown in Tables 4 and 5. In the experiments, the angular attitude calculated by gyro, the angular attitude calculated by DFBGA, the angular attitude calculated by DFBGA and SLAIM, the angular attitude calculated by the proposed method and the angular attitude calculated by 3-axis encoders are compared. This experiment includes two parts. In part 1, the motion carrier moves only with translational motion in the horizontal direction and without 3-axis rotation motion. In part 2, the translational motion in horizontal direction and 3-axis rotation exist in the motion of moving carrier at the same time. The horizontal acceleration is accompanied by slight jitter to simulate the maneuvering motion of UAV. The amplitude of jitter is about 1 cm and the frequency is about 2 to 3 times per second. Due to the limitation of the number of data acquisition software channels, the attitude information of roll, pitch and yaw directions are output, respectively. Figure 8: Application in motion control. The first subfigure is the simulated original-gyro signal feedback, the second subfigure is the feedback signal processed by DFBGA, the third subfigure is the feedback signal processed by proposed method. The output of the three-axis encoder is used as the reference of angular attitude through coordinate transformation. The coordinate transformation from body coordinate system to geographical coordinate system can be decomposed into three basic rotations around Z-axis, X-axis, and Y-axis, respectively, and each basic rotation corresponds to a transformation matrix. The transformation matrix corresponding to rotation around Z-axis, X-axis and Y-axis are shown in Equation (29), respectively. \[A_{z,\varphi}=\left[ \begin{array}{ccc}\cos\psi&\sin\psi&0\\ -\sin\psi&\cos\psi&0\\ 0&0&1\end{array} \right],A_{x,\gamma}=\left[ \begin{array}{ccc}1&0&0\\ 0&\cos\gamma&-\sin\gamma\\ 0&\sin\gamma&\cos\gamma\end{array} \right],A_{y,\theta}=\left[ \begin{array}{ cccc}\cos\theta&0&\sin\theta\\ 0&1&0\\ -\sin\theta&0&\cos\theta\end{array} \right]. \tag{29}\]  
The output data of the motor encoders rotating around the ZXY-axis are defined as \(\phi_{1},\phi_{2},\phi_{3}\). According to the mechanical structure of the experimental device, define  
\[\left[ \begin{array}{ccc}a_{z1}&a_{z2}&a_{z3}\end{array} \right]^{T}=A_{z, \varphi}\cdot\left[ \begin{array}{ccc}0&0&\phi_{1}\end{array} \right]^{T}, \tag{30}\]  
\[\left[ \begin{array}{ccc}b_{x1}&b_{x2}&b_{x3}\end{array} \right]^{T}=A_{z, \varphi}\cdot A_{x,\gamma}\cdot A_{y,60^{\circ}}\cdot\left[ \begin{array}{ cccc}\phi_{2}&0&0\end{array} \right]^{T}, \tag{31}\]