# 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 star tracker is a device that determines the satellite attitude in the celestial navigation system by taking pictures of stars in space. This paper presents the STM method based on the star tracker for achieving a long measurement arc of RSOs. Figure 1 shows the staring process of the proposed STM method for RSOs. Where the dotted line represents the initial orbit of the RSO estimated from the original TSA data, the blue point \(T_{0}\) represents the estimated RSO position. The red line is the real RSO orbit, and the red point \(T\) represents the actual position of the RSO. During the tracking process from time \(t_{1}\) to \(t_{3}\), the optical axis of the star tracker keeps tracking and staring at the real RSO, which effectively extends the measurement arcs. The staring-tracking law is the star tracker and should al ways keep the real RSO in the center of the FOV. However, due to the estimated initial orbit elements being inaccurate, the optical axis cannot keep pointing at the real RSO. Therefore, we first analyze the visibility conditions of RSOs to ensure that the proposed STM method could always observe the real RSOs during staring-tracking imaging. Figure 2 shows the observation area of the star tracker on the RSO orbital plane, since the FOV of the star tracker is assumed to be a circle, it is an ellipse. The blue and red points represent the estimated RSO and the actual RSO positions, respectively. This paper considers the estimated RSO orbital plane to be the same as the actual orbital plane to simplify the analysis of the visibility conditions. According to the geometric relationship in the observation area, the lengths of the major and minor semi-axes of the ellipse can be separately determined as:  
\[a_{e}=\frac{1}{2}\left(\frac{d\sin\theta_{fov/2}}{\cos\Bigl{(}\theta_{fov/2}+ \alpha\Bigr{)}}+\frac{d\sin\theta_{fov/2}}{\cos\Bigl{(}\theta_{fov/2}-\alpha \Bigr{)}}\right)b_{e}=d\tan\theta_{fov/2} \tag{1}\]  
where \(a_{e}\) is the semi-major axis of the observable region; \(b_{e}\) is the semi-minor axis; \(\theta_{fov/2}\) denotes the half FOV of the star tracker; \(\theta_{st}\) denotes the angle between the optical axis of the star tracker and the direction vector of the RSO; \(d\) is the distance between the RSO and the observation satellite. Additionally, \(\beta\) is defined as the angle between the orbital plane and the optical axis, and \(\alpha\) is the complement of \(\beta\). Figure 1: Schematic diagram of tacking imaging in the star tracker. When \(\theta_{st}\) is smaller than \(\theta_{fov/2}\), the real RSO is in the observable region that can be captured by the star tracker. Moreover, the observable region \(S\) can be expressed as:  
\[S=\pi a_{e}b_{e}=\frac{\pi d^{2}\sin^{2}\theta_{fov/2}\cos\alpha}{\cos^{2}\theta_ {fov}\cos^{2}\alpha-\sin^{2}\theta_{fov}\sin^{2}\alpha} \tag{2}\]  
Obviously, due to the inaccurately estimated RSO, the optical axis is not identical to the real observation direction. We defined the current optical axis direction of the estimated RSO in the center of the FOV as the initial observation direction (the star tracker is fixed on the satellite), and the observation direction that the real RSO is in the center of the FOV as the expected observation direction. To track and stare at the real RSO, the satellite should adjust its three-axis attitude to make the initial observation direction consistent with the expected observation direction, as shown in Figure 3.  
Figure 2: Schematic diagram of the observable region of the star tracker on the RSO orbit plane. Figure 3 shows the satellite maneuvering process from the initial direction to the expected direction. Before we calculate the expected satellite attitude, then two frames should be defined first.