# 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
The Earth-centered (ECI) internal frame: the original point is the Earth center, the z axis points to the north pole, the x axis points to the spring equinox in the equator plane, and the y axis obeys the right-hand rule in the equator plane. The orbit frame: the original point is the satellite center, the z axis points to the Earth center in the orbit plane, the x axis is the same as the satellite moving direction in the orbit plane, and the y-axis obeys the right-hand rule. In the orbit frame, the satellite changes the yaw \(\psi\), roll \(\varphi\), pitch \(\theta\) angel around the z, x, and y axes, which meets the requirements of staring-tracking. Additionally, the expected three-axis attitude can be expressed as the expected quaternion \(\mathbf{Q}_{c}\)[30]:  
\[\mathbf{Q}_{c}= \begin{bmatrix}\cos\frac{\delta}{2}&\mathbf{e}\sin\frac{\delta}{2} \end{bmatrix} ^{T}= \begin{bmatrix}q_{1}&q_{2}&q_{3}&q_{4}\end{bmatrix} ^{T} \tag{3}\]  
where \(\delta\) is the included angle between the optical axis unit vector \(\mathbf{u}^{o}\) and the expected observation unit vector \(\mathbf{u}^{o}_{st}\) in the orbit frame, the vector \(\mathbf{e}\) is perpendicular to \(\mathbf{u}^{o}\) and \(\mathbf{u}^{o}_{st}\) at the same time, expressed separately as:  
\[\delta=\arccos\frac{\mathbf{u}^{o}\cdot\mathbf{u}^{o}_{st}}{|\mathbf{u}^{o}|\cdot|\mathbf{u}^{ o}_{st}|} \tag{4}\]  
\[e=\frac{\mathbf{u}^{o}\times\mathbf{u}^{o}_{st}}{|\mathbf{u}^{o}\times\mathbf{u}^{o}_{st}|} \tag{5}\]  
The unit vector \(\mathbf{u}^{o}\) could be determined by the satellite position and the installation angle of the star tracker. Additionally, the unit vector \(\mathbf{u}^{o}_{st}\) can be expressed as:  
\[\mathbf{u}^{o}_{st}=A^{o}_{1}\mathbf{u}^{I}_{st}=A^{o}_{I} \begin{pmatrix}\mathbf{r}^{I}_{ t}-\mathbf{r}^{I}_{s}\end{pmatrix}  \tag{6}\]  
where \(A^{o}_{I}\) is the transformation matrix from the ECI frame to the orbital frame, \(\mathbf{r}^{I}_{t}\) represents the unit observation vector of the real RSO in the ECI frame, and \(\mathbf{r}^{I}_{s}\) represents the unit position vector of satellites in the ECI frame, \(\mathbf{u}^{I}_{st}\) represents the unit observation vector in  
Figure 3: The process of satellite attitude maneuver in staring–tracking imaging mode. the ECI frame. Figure 4 shows the positions of the observation satellite and the RSO in the ECI frame during staring-tracking imaging. \(\mathbf{r}_{t}^{l}\) can be expressed by the right ascension \(\alpha_{t}\) and the declination of the real RSO \(\delta_{t}\) is ECI frame as:  
\[\mathbf{r}_{t}^{l}= \begin{bmatrix}x_{t}\\ y_{t}\\ z_{t}\end{bmatrix} = \begin{bmatrix}\cos\alpha_{t}\cos\delta_{t}\\ \sin\alpha_{t}\cos\delta_{t}\\ \sin\delta_{t}\end{bmatrix}  \tag{7}\]  
where \(\alpha_{t},\delta_{t}\) can be determined through the measurement method based on the principle of angular distance invariance proposed in Section 2.2. The specific observation satellite position can be determined by Keplerian elements. Where \((a,e,i,f,\omega,\Omega)\) can be defined as: \(a\) is the semi-major axis of the orbit, \(e\) is the eccentricity of the orbit, \(i\) is the orbit inclination, \(f\) is the true anomaly, \(\omega\) is the argument of periapsis, and \(\Omega\) is the longitude of the ascending node. Therefore, the observation satellite position in the ECI frame can be expressed as  
\[\mathbf{r}_{s}=a(\cos E-e)\cdot\mathbf{P}+a\sqrt{1-e^{2}}\sin E\cdot\mathbf{Q} \tag{8}\]  
where \(E\) is the near-point angle, \(\mathbf{P}\) is the matrix of the unit vector along the semi-major axis, and \(\mathbf{Q}\) represents the matrix of the unit vector along the semi-minor axis of the orbit in the ECI system:  
\[\mathbf{P}= \begin{bmatrix}\cos\omega\cos\Omega-\sin\omega\sin\Omega\cos i\\ \cos\omega\sin\Omega+\sin\omega\sin\Omega\cos i\\ \sin\omega\sin i\end{bmatrix}  \tag{9}\]  
\[\mathbf{Q}= \begin{bmatrix}-\sin\omega\cos\Omega-\cos\omega\sin\Omega\cos i\\ -\sin\omega\sin\Omega+\cos\omega\cos\Omega\cos i\\ \cos\omega\sin i\end{bmatrix}