## 1 Introduction
This approach helped to map orbits that had a high potential to require less fuel consumption for station-keeping maneuvers [11]. In terms of constellation configuration maintenance, [PERSON] et al. proposed a method of calculating the major semi-axis deviations indirectly from the change in the relative phase and deduced the evolution of the relative phase with a limit cycle control method, which met the requirements of long-term orbit preservation [12]. [PERSON] et al. analyzed the orbital perturbations and configurational phase drift of constellation satellites, and they presented a numerical differential correction method for constellation configurations [13]. [PERSON] et al. employed a relative configuration preservation method to minimize the average drift deviation and implemented a free-segment-maintenance strategy. This approach effectively decreased the control frequency and achieved orderly and decentralized constellation management in both temporal and spatial dimensions [14]. [PERSON] et al. derived a formula to calculate the maintenance control amount for the semi-major axis by considering the phase drift caused by the attenuation of the satellite's orbital semi-major axis. They also analyzed the maintenance control rules for satellites with varying surface mass ratios during periods of both low and high solar activity. The orbital altitude changes of the on-orbit satellites in the Starlink constellation have verified that this method can effectively maintain the satellite phase within the desired interval and ensure a stable and consistent control frequency [15; 16]. [PERSON] et al. developed a second-order consistent control methodology for maintaining a constellation's relative configuration, which effectively compensated for the configuration drift caused by variations in the surface mass ratios. This method effectively compensates for the configuration drift caused by the changes in the ratio of the cross-sectional area to the mass [17; 18]. [PERSON] et al. introduced a dual-bias strategy to significantly mitigate the relative drift. The initial bias was derived by fitting the relative drift of the constellation in its unbiased state. Subsequently, the second bias was obtained by fitting the residual relative drift after applying the first bias. By superimposing these two bias components, the overall relative drift was substantially reduced [19]. [PERSON] et al. employed the principle of satellite initial parameter bias compensation to model and adjust for the cumulative effects of various perturbations, thereby mitigating long-term orbital deviations caused by these factors and preserving the stability of the constellation configuration [20]. [PERSON] et al. conducted an in-depth analysis of the laws of motion of satellites and the evolution characteristics of constellation configurations under perturbations. They proposed a relative phase-maintenance strategy anchored on a reference satellite, which effectively reduced the complexity and frequency of the constellation configuration maintenance, conserved fuel, and shortened the orbit-control time [21]. [PERSON] et al. developed a relative phase-maintenance strategy for a constellation based on dynamic adjustments to this reference orbit. By computing the relative phase deviation and the rate of change for each satellite with respect to the reference orbit, they ensured that the phase deviations of all the satellites remained within allowable limits [22]. [PERSON] et al. developed a method for calculating the phase-holding period and changes in the semi-major axis using limit cycle theory. They subsequently proposed a high-precision phase-holding method grounded in limit cycle principles [23]. [PERSON] et al. utilized high-precision latitude angle data obtained from satellite-based global navigation satellite system (GNSS) receivers to approximate the changes in the relative semi-major axis during constellation configuration maintenance.