## 6 Positioning Algorithm Analysis  
The satellite tool kit (STK) platform was used to build the observation model. The observation platform was located in the solar synchronous orbit, and the orbit parameters refer to the observation data of the union of American Yousi scientists. The space debris position vector \(r_{deb}\) was generated by simulation to traverse the low orbit and high orbit. The instantaneous position height was from 6800 km to 40,000 km, and the orbit inclination was 0\({}^{\circ}\)-2\({}^{\circ}\). The orbital inclination range of the observation platform was 96.8\({}^{\circ}\)-98.5\({}^{\circ}\)and the eccentricity was 0. The positions of three star trackers were randomly generated according to the space debris position vector, and the observation distance from the space debris was guaranteed. A total of 200 cooperative positioning of space debris were carried out, and the positioning errors of star trackers at different heights were calculated at the observation distances of 200 km, 100 km, and 50 km. In order to simulate the attitude determination error of the star tracker and the position error of the platform itself in the task, the position errors of 10 m and 100 m were introduced into the position of the star tracker in the geocentric coordinate system according to the performance of the existing star tracker, and the pointing accuracy of the star tracker was set to 2\({}^{\prime\prime}\), which is more in line with the practical application. The code had been verified on computers with an Intel Core i7-6500u (dual cores) CPU up to 3.1 GHz, 8 GB Ram, and operating system Windows 10. The simulation results are shown in Figures 13 and 14. It can be seen from the above curve data that the positioning accuracy of the debris also changed with the change in the positioning accuracy of the satellite itself. When the positioning accuracy of the satellite itself was 10 m, the positioning accuracy of the debris was also about 10 m, as shown in Figure 13. It can be found from Figure 14 that when the positioning accuracy of the satellite itself was 100 m, the positioning accuracy of the debris was about 100 m. For the positioning of space debris with a plurality of optical sensors, the positioning accuracy varied with the positioning accuracy of the observation satellite itself, and the positioning accuracy of the space debris was limited by the positioning accuracy of the satellite itself. Figure 13: The positioning error of space debris varies with the height of the debris position at different observation distances when the star tracker’s own positioning error was 10 m. (**a**) The space debris observation distance of 50 km. (**b**) The space debris observation distance of 100 km. (**c**) The space debris observation distance of 200 km. By calculating the positioning algorithm, a higher accuracy position determination of the target can be accomplished. If the spatial positions of the target at two moments can be obtained, the initial orbital parameters of the space debris can be calculated by Equations (37)-(43). If there are multiple measurements of space debris in one orbital arc, the initial orbital elements can be further obtained with higher accuracy by calculating the average orbital elements. The space debris was in different orbits, and the positioning accuracy of space debris changed due to the influence of the positioning accuracy of the observation star tracker itself, and the positioning accuracy of the high-orbit satellite itself was poor compared with that of the low-orbit satellite. The following simulation calculations were completed by using 20 sets of position measurements in one orbital arc segment to solve the orbital elements, the simulation results are shown in Figures 15 and 16. As shown in Figures 15 and 16, the orbital parameters were estimated by using the arc segments distributed at different positions and the 20-position information of each arc segment with a position error of 10 m, where there were 20 sets of measurements for each arc segment.