# Design and Analysis of the Composite Stability Control of the Reflective Optoelectronic Platform
## 2 Problem Description
### The Preparation of System Motion  
To simplify analysis, this study adopts a different approach than previous methods [3; 4; 9; 10], setting \(x\), \(y\), and \(z\) to correspond to the rolling, pitch, and azimuth axes of the carrier, respectively. Physical quantities are denoted by subscripts \(b\), \(A\), \(E_{1}\), \(E_{2}\), \(a\), and \(e\), corresponding to the coordinate systems of the carrier, coarse-stable azimuth framework, coarse-stable pitch driving framework, coarse-stable pitch driven framework, fine-stable azimuth framework, and fine-stable pitch framework. The coordinate systems for each axis are defined as shown in Figure 2, and the motion transmission relationships between the frameworks are shown in Figures 3 and 4. Figure 1: The constitution diagram of system schematic. (**a**) Schematic diagram of basic principles. (**b**) Arrangement diagram of systematic shafting. to \(\lambda\). A two-degree freedom gyro is installed on the active framework of the coarse-stable pitch, corresponding to the coordinate system \(E_{1}\). The notation \(\bar{\omega}_{b}\) is used to represent the carrier's angular velocity vector, with components \(\left[\omega_{bx},\omega_{by},\omega_{bz},1\right]^{T}\). The coarse-stable azimuth frame's angular velocity vector is denoted by \(\bar{\omega}_{A}\), with components \(\left[\omega_{Ax},\omega_{Ay},\omega_{Az},1\right]^{T}\). Similarly, the coarse-stable pitch driving frame's angular velocity vector is denoted as \(\bar{\omega}_{E1}\), with components \(\left[\omega_{E1x},\omega_{E1y},\omega_{E1z},1\right]^{T}\), and that of the coarse-stable pitch driven frame as \(\bar{\omega}_{E2}\), with components \(\left[\omega_{E2x},\omega_{E2y},\omega_{E2z},1\right]^{T}\). The LOS coarse-stable azimuth frame's angular velocity vector is \(\bar{\omega}_{LA}\), with components \(\left[\omega_{ax},\omega_{ay},\omega_{az},1\right]^{T}\), and that of the LOS fine-stable pitch frame is \(\bar{\omega}_{Le}\), with components \(\left[\omega_{ex},\omega_{ey},\omega_{ez},1\right]^{T}\). Therefore, Equation (1) from references [11; 12] can be used to describe the kinematic model of the line of sight (LOS), where \(\bar{\omega}_{La}\) represents the angular velocity vector of the LOS fine-stable azimuth frame, and \(\bar{\omega}_{LE}\) represents that of the LOS coarse-stable pitch frame. \[\bar{\omega}_{Le} =W_{ea}\bar{\omega}_{La}=W_{ea}\bar{\cdot}W_{aE}\bar{\cdot}\bar{ \omega}_{LE} \tag{1}\] \[=W_{ea}\bar{\cdot}W_{aE}\bar{\cdot}W_{EA}\bar{\cdot}\bar{\omega}_ {LA}\]  
Figure 5: The transfer relation of the LOS coordinate caused by coarse-stable angular motion. (**a**) The transfer relation of the LOS caused by coarse-stable azimuth motion. (**b**) The transfer relation of the LOS caused by coarse-stable pitch driven motion. Figure 6: The transfer relation of the LOS coordinate caused by fine-stable angular motion. (**a**) The transfer relation of the LOS caused by fine-stable azimuth motion. (**b**) The transfer relation of the LOS caused by fine-stable pitch motion. where  
\[W_{ea}= \begin{bmatrix}cos(\lambda\theta_{e})&0&-sin(\lambda\theta_{e})&0\\ 0&1&0&\lambda\theta_{e}\\ sin(\lambda\theta_{e})&0&cos(\lambda\theta_{e})&0\\ 0&0&0&1\end{bmatrix} ,\]  
\[W_{aE}= \begin{bmatrix}cos(\theta_{a})&sin(\theta_{a})&0&0\\ -sin(\theta_{a})&cos(\theta_{a})&0&0\\ 0&0&1&\theta_{a}\\ 0&0&0&1\end{bmatrix} , \tag{2}\]  
\[W_{EA}= \begin{bmatrix}cos(\lambda\theta_{E2})&0&-sin(\lambda\theta_{E2})&0\\ 0&1&0&\lambda\theta_{E2}\\ sin(\lambda\theta_{E2})&0&cos(\lambda\theta_{E2})&0\\ 0&0&0&1\end{bmatrix} \]  
Due to the limited range of motion in \(\theta_{a}\) and \(\theta_{e}\), representing the real-time angular velocities of the fine-stable azimuth frame and the fine-stable pitch frame, their sine and cosine values can be considered negligible, resulting in Equation (1) being simplified to Equation (3).