## 2 Integrated Modeling
### Optical Model  
The payload of the satellite is an optical camera of Cassegrain type. The optical system is shown in Figure 6. It is made up of a primary mirror (PM), a secondary mirror (SM), a calibration mirror (CM), and an image plane (IP). The disturbance force and tongue will deform the optical component, which are usually small because it is very small. The deformation of each optical components can be considered a rigid-body motion. The best-fit rigid-body motion of the \(i\)th optical element is defined as a vector \(\mathbf{T}_{i}\) of 6 terms as:  
\[\mathbf{T}_{i}= \begin{bmatrix}T_{ix}&T_{iy}&T_{iz}&R_{ix}&R_{iy}&R_{iz}\end{bmatrix} ^ {\mathrm{T}}, \tag{13}\]  
Figure 6: The optical system of the satellite. where \(T_{ix}\), \(T_{iy}\), and \(T_{iz}\) are translations along coordinate axes and \(R_{ix}\), \(R_{iy}\), and \(R_{iz}\) are rotations about coordinate axes. Then, the displacement of the node \(j\) due to these motions is [36]  
\[\left\{ \begin{array}{l}u_{ij}=T_{ix}+z_{ij}R_{iy}-y_{ij}R_{iz}\\ v_{ij}=T_{iy}-z_{ij}R_{ix}+x_{ij}R_{iz}\\ w_{ij}=T_{iz}+y_{ij}R_{ix}-x_{ij}R_{iy},\end{array} \right. \tag{14}\]  
where \(x_{ij}\), \(y_{ij}\), and \(z_{ij}\) are the coordinates of the node \(j\), and \(u_{ij}\), \(v_{ij}\), and \(w_{ij}\) are the displacements of the node \(j\). When there are \(n\) nodes on each optical component, Equation (14) is expressed as follows:  
\[\mathbf{A}_{i}T_{i}=\mathbf{q}_{i}, \tag{15}\]  
where \(\mathbf{q}_{i}=[u_{i1},\ v_{i1},\ w_{i1},\cdots,\ u_{ij},\ v_{ij},\ w_{ij},\cdots,\ u_{in},\ v_{in},\ w_{in}]^{\rm T}\), and matrix \(\mathbf{A}_{i}\) is expressed as follows:  
\[\mathbf{A}_{i}= \begin{bmatrix}1&0&0&0&z_{i1}&-y_{i1}\\ 0&1&0&-z_{i1}&0&x_{i1}\\ 0&0&1&y_{i1}&-x_{i1}&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 1&0&0&0&z_{ij}&-y_{ij}\\ 0&1&0&-z_{ij}&0&x_{ij}\\ 0&0&1&y_{ij}&-x_{ij}&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 1&0&0&0&z_{in}&-y_{in}\\ 0&1&0&-z_{in}&0&x_{in}\\ 0&0&1&y_{in}&-x_{in}&0\end{bmatrix} . \tag{16}\]  
The vector \(\mathbf{T}_{i}\) can be obtained from Equation (15) as follows:  
\[\mathbf{T}_{i}=\mathbf{C}_{i}\mathbf{q}_{i}, \tag{17}\]  
where \(\mathbf{C}_{i}=(\mathbf{A}_{i}^{\rm T}\mathbf{A}_{i})^{-1}\mathbf{A}_{i}^{\rm T}\). The matrix \(\mathbf{C}_{i}\) and \(\mathbf{q}_{i}\) are functions of the coordinates and displacements of node \(i\), respectively. In order to ensure the accuracy of the fitting, \(n\) is usually greater than or equal to 3. The image motion \(\mathbf{p}(t)\) can be obtained by summing the product of the rigid-body motions \(\mathbf{T}_{i}\) and the optical sensitivity coefficients \(\mathbf{L}_{i}\) of the \(i\)th optical element. \[\mathbf{p}(t)=\sum_{i=1}^{m}\mathbf{L}_{i}^{\rm T}\mathbf{T}_{i}, \tag{18}\]  
where \(m\) is the number of optical components. \(\mathbf{L}_{i}\) is defined as follows:  
\[\mathbf{L}_{i}=\left[ \begin{array}{c}\mathbf{L}_{ix}^{\rm T}\\ \mathbf{L}_{iy}^{\rm T}\end{array} \right]^{\rm T}= \begin{bmatrix}l_{11}&l_{12}&l_ {13}&l_{14}&l_{15}&l_{16}\\ l_{21}&l_{22}&l_{23}&l_{24}&l_{25}&l_{26}\end{bmatrix} ^{\rm T}, \tag{19}\]  
where \(l_{11}\), \(l_{12}\), and \(l_{13}\) represent the image motions in the \(x\) direction by translating the \(i\)th optical component one unit displacement along the \(x\), \(y\), and \(z\) axes, respectively. Further, \(l_{14}\), \(l_{15}\), and \(l_{16}\) represent the image motions in the \(x\) direction by rotating the \(i\)th optical component one unit angle about the \(x\), \(y\), and \(z\) axes, respectively. Moreover, \(l_{21}\), \(l_{22}\), and \(l_{23}\) represent the image motions in the \(y\) direction by translating the \(i\)th optical component one unit displacement along the \(x\), \(y\), and \(z\) axes, respectively. Finally, \(l_{24}\), \(l_{25}\), and \(l_{26}\) represent the image motions in the \(y\) direction by rotating the \(i\)th optical component one unit angle about the \(x\), \(y\), and \(z\) axes, respectively.