# A Staring Tracking Measurement Method of Resident Space Objects Based on the Star Tacker
## 3 Results
### Simulation Settings of the Proposed STM Method
#### 3.1.2 Star Tracker Parameters Settings  
The main purpose of the proposed STM method is RSO measurement, which means that the RSO should always be within the observation area. Because of the difference between the observation orbit plane and the RSO orbit plane, the observable region of the star tracker on the RSO orbit plane is an ellipse area. According to Equation (1), the semi-major \(a_{e}\) and the semi-minor axis \(b_{e}\) of the observable region are simulated and analyzed, respectively. There are three factors that affect the length of the major axis: the distance \(d\), the half FOV \(\theta_{fov/2}\), and the angle \(\alpha\) between the orbital plane and the observation plane. The curves of \(a_{e}\) and \(b_{e}\) under different conditions are shown in Figure 8.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Orbital Elements** & **Observation Satellite Orbit** & **Actual RSO Orbit** & **Estimated RSO Orbit** \\ \hline Semi-major axis/km & 6928.14 & 7000.14 & 6980.14 \\ Eccentricity & 0 & 0 & 0.1 \\ Inclination/(deg) & 97.5521 & 97.7141 & 97.7141 \\ Longitude of the ascending node/(deg) & 203.282 & 163.536 & 163.536 \\ Argument of periapsis/(deg) & 0 & 0 & 0 \\ True anomaly/(deg) & 0.108 26 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Orbital elements of the observation satellite and the RSO.  
Figure 8: The curves of the major axes \(a_{e}\) and \(b_{e}\): (**a**) curves with the increase in the detection distance \(d\); (**b**) curves with the increase in half FOV \(\theta_{fov/2}\); and (**c**) curves with the increase in angle \(\alpha\).  
Figure 7: Planned satellite attitude angle and angular velocity: (**a**) three-axis satellite attitude angle; and (**b**) three-axis satellite angular velocity.  
Apparently, the curves of \(a_{e}\) and \(b_{e}\) both increase linearly with the increase in \(d\) and \(\theta_{fov/2}\), as shown in Figure 8a,b, respectively. The curve of \(b_{e}\) is independent with the increase in \(\alpha\) as shown in Figure 8c. Meanwhile, \(a_{e}\) increases nonlinearly when \(\alpha\) is in the range of 30\({}^{\circ}\)-70\({}^{\circ}\), and increases rapidly when \(\alpha\) is greater than 70\({}^{\circ}\). Among the three factors, \(\alpha\) is determined by the relative position of the RSO and the star tracker, which should be larger than 70\({}^{\circ}\) to enlarge the observation area. As for the distance \(d\) and the half FOV \(\theta_{fov/2}\), these are determined by the parameters of the star tracker. For a considerable observation area under consideration with the star tracker performance, the parameters of the star tracker in the STM method are listed in Table 3.  
In contrast with static star images, the energy of star images captured in the staring-tracking method are distrusted as streaks. The planned satellite attitude angle and angular velocity in Section 3.1.1 are used to generate star images. A more detailed discussion of the simulation for simulating dynamic star images can be found in [33]. The magnitudes of stars are added with Gaussian noise to test the accuracy of the proposed observation method. The standard deviation of 0.05 pixels and 5000 dynamic star images are generated for the simulation of measurement accuracy, as shown in Figure 9.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Focal length & 35.5 mm \\ Field of view & 20\({}^{\circ}\times 20^{\circ}\) \\ Resolution & 2048 \(\times\) 2048 \\ Pixel length & 5.5 μm \\ Exposure time & 100 ms \\ Star magnitude limit & 5.2 Mv \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters of star tracker.  
Figure 9: A dynamic smearing star image simulated in the staring–tracking imaging.