# A Staring Tracking Measurement Method of Resident Space Objects Based on the Star Tacker
## 2 Materials and Methods
### Three-Axis Attitude Model of the Satellite for RSOs Staring-Tracking Imaging
 \tag{10}\]  
Figure 4: Schematic of staring–tracking to the RSO. Additionally, the angle distance \(E_{i}\) in the ECI frame can be calculated as:  
\[\cos E_{i}=\cos(\frac{\mathbf{u}_{st}^{I}\cdot\mathbf{u}_{i}^{I}}{\|\mathbf{u}_{st}^{I}\| \cdot\|\mathbf{u}_{t}^{I}\|}) \tag{15}\]  
where \(\mathbf{u}_{st}^{I}\) and \(\mathbf{u}_{i}^{I}\) can be calculated by positions of the RSO and the star in the ECI frame:  
\[\mathbf{u}^{I}= \begin{bmatrix}x\\ y\\ z\end{bmatrix} = \begin{bmatrix}\cos\alpha\cdot\cos\delta\\ \sin\alpha\cdot\cos\delta\\ \sin\delta\end{bmatrix}  \tag{16}\]  
where \((\alpha,\delta)\) are the angles of the star or the RSO in the ECI frame. We define \((\alpha_{i},\delta_{i})\), \((\alpha_{t},\delta_{t})\) as the right ascension and the declination of the i-th star, and the RSO, respectively. Additionally, \((\alpha_{i},\delta_{i})\) can be obtained from the star catalogue by star identification [31]. Thus, \(\mathbf{u}_{i}^{I}\) can be expressed as:  
\[\mathbf{u}_{i}^{I}= \begin{bmatrix}x_{i}\\ y_{i}\\ z_{i}\end{bmatrix} = \begin{bmatrix}\cos\alpha_{i}\cdot\cos\delta_{i}\\ \sin\alpha_{i}\cdot\cos\delta_{i}\\ \sin\delta_{i}\end{bmatrix}  \tag{17}\]  
Figure 5: Schematic diagram of the star tracker imaging process. According to the ADI principle, \(E_{i}\) is equal to \(e_{i}\). Therefore, the solution of \(\mathbf{u}_{st}^{I}\) should meet the following conditions:  
\[J(u_{st})=\sum\left\|\cos E_{i}-\cos e_{i}\right\|_{2} \tag{18}\]  
where \(J(u_{st})\) is the loss function. The optimal solution of \(\mathbf{u}_{st}^{I}\) occurs when the loss function is minimized. Additionally, when the partial derivative of the loss function is equal to zero, the loss function has the minimum value. The partial derivative of the loss function is shown as follows:  
\[\left\{ \begin{array}{l}\frac{\partial{I(u_{st})}}{\partial x}=\sum\limits_{i= 1}^{k}x_{i}^{2}\cdot x_{st}+\sum\limits_{i=1}^{k}x_{i}\cdot y_{i}\cdot y_{st}+ \sum\limits_{i=1}^{k}x_{i}\cdot z_{i}\cdot z_{st}-\sum\limits_{i=1}^{k}x_{i} \cdot\cos e_{i}\\ \frac{\partial{I(u_{st})}}{\partial y}=\sum\limits_{i=1}^{k}x_{i}\cdot y_{i} \cdot x_{st}+\sum\limits_{i=1}^{k}y_{i}^{2}\cdot y_{st}+\sum\limits_{i=1}^{k} y_{i}\cdot z_{i}\cdot z_{st}-\sum\limits_{i=1}^{k}y_{i}\cdot\cos e_{i}\\ \frac{\partial{I(u_{st})}}{\partial z}=\sum\limits_{i=1}^{k}x_{i}\cdot z_{i} \cdot x_{st}+\sum\limits_{i=1}^{k}y_{i}\cdot z_{i}\cdot y_{st}+\sum\limits_{i= 1}^{k}z_{i}^{2}\cdot z_{st}-\sum\limits_{i=1}^{k}z_{i}\cdot\cos e_{i}\end{array} \right. \tag{19}\]  
When the partial derivative is equal to zero, it can be expressed as:  
\[\left[ \begin{array}{l}\sum\limits_{i=1}^{k}x_{i}\cdot\cos e_{i}\\ \sum\limits_{i=1}^{k}y_{i}\cdot\cos e_{i}\\ \sum\limits_{i=1}^{k}z_{i}\cdot\cos e_{i}\end{array} \right]=\left[ \begin{array} []{l}\sum\limits_{i=1}^{k}x_{i}^{2}&\sum\limits_{i=1}^{k}x_{i}y_{i}&\sum \limits_{i=1}^{k}x_{i}z_{i}\\ \sum\limits_{i=1}^{k}x_{i}y_{i}&\sum\limits_{i=1}^{k}y_{i}^{2}&\sum\limits_{i= 1}^{k}y_{i}z_{i}\\ \sum\limits_{i=1}^{k}x_{i}z_{i}&\sum\limits_{i=1}^{k}y_{i}z_{i}&\sum\limits_{i= 1}^{k}z_{i}^{2}\end{array} \right]\cdot\left[ \begin{array}{l}x_{st}\\ y_{st}\\ u_{st}\end{array} \right] \tag{20}\]  
Therefore, the optimal solution of \(\mathbf{u}_{st}^{I}\) can be obtained by:  
\[\mathbf{u}_{st}^{I}=\mathbf{A}^{-1}\mathbf{Y} \tag{21}\]  
Therefore, \(\mathbf{r}_{t}^{I}\) can be expressed as:  
\[\mathbf{r}_{t}^{I}=\mathbf{u}_{st}^{I}+\mathbf{r}_{s}^{I} \tag{22}\]  
Additionally, \((\alpha_{t},\delta_{t})\) of the RSO in the ECI frame can be expressed as:  
\[\left\{ \begin{array}{l}\alpha_{t}=\arctan\frac{y_{t}}{\sqrt{x_{t}^{2}+y_{t }^{2}}}=\arctan\frac{y_{s}+y_{st}}{x_{s}+x_{st}}\\ \delta_{t}=\arctan\frac{z_{t}}{\sqrt{x_{t}^{2}+y_{t}^{2}}}=\arctan\frac{\sqrt{ (x_{s}+x_{st})^{2}+(y_{s}+y_{st})^{2}}}{\sqrt{(x_{s}+x_{st})^{2}+(y_{s}+y_{st })^{2}}}\end{array} \right. \tag{23}\]  
After completing the coordinate measurement of the RSO, it is necessary to remove the aberration of light. This method in [32] is used to complete the correction of the aberration.