# Distributed Localization of Non-Cooperative Targets in Non-Coplanar Rendezvous Processes
## 4 Numerical Simulation and Analysis
### Configuration Optimization Based on Improved NSGA-III  
The simulation step was set to 1 s. The measurement precision of the optical sensors was 0.0057\({}^{\circ}\) for both the azimuth and pitch angles (\(1\sigma\)). The initial position error of the filter was 1000 m in all three axes (\(1\sigma\)), and the velocity error was 10 m/s in all three axes (\(1\sigma\)). The simulation scenario was as follows: the target was in a circular orbit with an orbital inclination of 53\({}^{\circ}\), a right ascension of the ascending node of 0\({}^{\circ}\), and an orbital altitude of 1110 km. The tracking satellites 1 and 2 were located in circular orbits with an orbital inclination of 58\({}^{\circ}\), with different orbital altitudes, and the specific parameters are shown in Table 2. Satellite 3 did not participate in the rendezvous, and satellites 1 and 2 performed maneuvers at different times to approach the target. The altitudes of satellites 1 and 2 and the true anomaly at the time of maneuver were selected as the decision variables for configuration optimization. In the middle of the whole rendezvous process, parallel experiments were conducted. One group simulated the signal loss of satellite 2, resulting in a degradation from a three-satellite positioning system to a two-satellite system. The other group maintained the three-satellite configuration. The results of the two groups were compared to calculate the RMSE loss in position estimation when the three-satellite system degraded to a two-satellite system. After optimization by improved NSGA-III, 56 groups of configurations were obtained. In these data, some solutions were suboptimal (there exist better solutions in all four objectives), while others were similar solutions that could be categorized into a class. After review, five representative groups of results were selected, as shown in Table 3. The first configuration had the shortest rendezvous time but incurred high energy costs. The third configuration exhibited extremely low maneuvering energy costs but had a high  
Figure 9: Solution comparations for three algorithms. (**a**) Improved NSGA-III and NSGA-III. (**b**) Improved NSGA-III and NSGA-II. The blue solid line represents the estimation errors of the collaborative positioning system before configuration optimization, while the red solid line indicates the errors after optimization. As the figures show, the total rendezvous time cost before optimization was 1868 s, which decreased to 1662 s after optimization, representing an 11.02% reduction in rendezvous time. The position estimation RMSE calculations began from the midpoint of the rendezvous. The distributed positioning system before optimization converged around 1400 s, with a position estimation RMSE of 45.2 m. In contrast, the optimized system converged around 800 s, achieving a position estimation RMSE of 6.7 m. Compared to the pre-optimization results, the convergence speed of the optimized distributed positioning system improved by approximately 42.85%, while the positioning accuracy was enhanced by 85.17%. After the configuration optimization, the target position estimation errors of the three stars of the cooperative positioning system are shown in Figure 11, below, where the blue solid line is the position estimation error of the three-satellite positioning system and the red dashed line is the position estimation error after degradation to two-satellite positioning in the parallel experiment, where the signal of satellite 2 was lost. The part of the figure that is boxed by the rectangle is the image within 800-1800 s after the convergence of the filtering, and the small figure is the zoomed-in detail of the boxed part of the rectangle. Figure 11: Co-localization system position estimation error. (**a**) Estimation error of satellite 1; (**b**) Estimation error of satellite 2; (**c**) Estimation error of satellite 3.