## 3 Model of Image Motion
### Coordinate Systems
Following the procedures shown in Figure 3, the coordinate transformation from the geocentric inertial coordinate system \(O_{I}-x_{I}y_{I}z_{I}\) to the camera image plane coordinate system \(O_{p}-x_{p}y_{p}z_{p}\) can be given as follows. 1. Geocentric inertial coordinate system \(O_{I}-x_{I}y_{I}z_{I}\) to spacecraft orbit coordinate system \(O_{b}-x_{b}y_{b}z_{b}\): The spacecraft orbital coordinate system is obtained by the geocentric inertial coordinate system after translation and then rotation, where \(\gamma_{0}\) is the initial orbit angle, \(\Omega\) is the orbital angular velocity, \(i_{0}\) is the orbit inclination angle, \(R\) is the Earth radius, and \(h\) is the orbit height. Therefore, the transformation matrix between the two coordinate systems at time \(t\) can be expressed as \[\mathbf{T}_{1}=\left[ \begin{array}{cccc}\cos(i_{0})&\sin(i_{0})&0&0\\ -\sin(i_{0})&\cos(i_{0})&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \  
\[\mathbf{T}_{2}=\left[ \begin{array}{cccc}\cos\psi&0&-\sin\psi&0\\ 0&1&0&0\\ \sin\psi&0&\cos\psi&0\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{cccc}1&0&0&0\\ 0&\cos\theta&\sin\theta&0\\ 0&-\sin\theta&\cos\theta&0\\ 0&0&0&1\\ \end{array} \right]\left[ \begin{array}{cccc}\cos\varphi&\sin\varphi&0&0\\ -\sin\varphi&\cos\varphi&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]. \tag{2}\]  
\[\mathbf{3}.\]  
Spacecraft coordinate system \(O_{s}-x_{s}y_{s}z_{s}\) to camera installation coordinate system \(O_{c}-x_{c}y_{c}z_{c}\):  
The installation angle errors in the pitch, yaw, and roll directions are \(\theta_{p}\), \(\psi_{p}\), and \(\varphi_{p}\), respectively. The transformation matrix is as follows:  
\[\mathbf{T}_{3}=\left[\begin{array}{cccc}\cos\varphi_{p}&\sin\varphi_{p}&0&0 \\ -\sin\varphi_{p}&\cos\varphi_{p}&0&0\\ 0  
the lateral magnification of the optical systems is \(\beta\), and the transformation matrix can be expressed as follows:  
\[\mathbf{T}_{5}=\left[ \begin{array}{cccc}\beta&0&0&0\\ 0&\beta&0&0\\ 0&0&-\frac{f}{f}\beta&0\\ 0&0&0&1\end{array} \right]. \tag{5}\]  
The coordinate transformation matrix \(\mathbf{T}\) from the geocentric inertial coordinate system \(O_{I}-x_{I}y_{I}z_{I}\) to the object plane coordinate system \(O_{d}-x_{d}y_{d}z_{d}\) is  
\[\mathbf{T}=\mathbf{T}_{5}\mathbf{T}_{4}\mathbf{T}_{3}\mathbf{T}_{2}\mathbf{T} _{1}. \tag{6}\]  
Supposing the right ascension and declination coordinates of the target star are \((\alpha_{p},\lambda_{p})\), then the homogeneous coordinates of the unit vector of the star's direction in the geocentric inertial coordinate system are  
\[\mathbf{R}_{I}=\left[ \begin{array}{c}\cos(\lambda_{p})\cos(\alpha_{p})\\ \cos(\lambda_{p})\sin(\alpha_{p})\\ \sin(\lambda_{p})\\ 1\end{array} \right]. \tag{7}\]  
During the exposure time, the image of the target star is always maintained at the center of the image plane through the motion of a two-axis pointing platform, and the coordinates of the image in the camera image coordinate system are  
\[\mathbf{R}_{p}=[0,0,f,1]^{\mathrm{T}}. \tag{8}\]  
Given the satellite orbital motion parameters, satellite attitude parameters, and camera installation matrix, as well as by guaranteeing the matching of objects and images during in-orbit exposure by transforming the matrix \(\mathbf{T}\), the following equation is satisfied  
\[\mathbf{R}_{p}=\mathbf{T}\mathbf{R}_{I}, \tag{9}\]  
where the azimuth and the elevation angle of the two-axis pointing platform \(A_{t}\) and \(E_{t}\) can be obtained.