# The Symmetry of GPS Orbit Ascending Nodes
## 4 Conclusions  
This work presents a novel method of assessing the symmetry of the ascending nodes of the GPS orbital planes. Previous space segment studies focused mainly on the accuracy of determining the position of satellites. This study provides a more general perspective and is an extension of the conclusions drawn from the previous authors' research on the anomalous orbit of one of the GPS satellites. The research material consisted of 7035 SEM almanacs containing crucial Keplerian orbital parameters and published between 1996 and 2020. Based on the longitude of the ascending node (\(\Omega\)) and rate of the longitude of the ascending node (\(\mathrm{dot}(\Omega)\)) parameters, the symmetry properties of the satellite orbits' ascending nodes were determined. Figure 9: Deviations of OP AN position estimators in relation to the hexagonal symmetry reference. The research algorithm included robust estimation and least-squares adjustment methods. The reference values estimated for each almanac epoch met the requirement of the initial GPS hexagonal symmetry of orbital plane ascending nodes in the equatorial plane. The estimators were determined in two stages. In the first stage, the Huber weight function eliminated the influence of outliers in the ascending node \(\Omega\) datasets. The second stage introduced the condition of hexagonal symmetry in the least-squares method. The reference \(\Omega\) values of the orbital plane ascending nodes were used to determine the d\(\Omega\) deviations and the separation trend values of the ascending nodes of individual GPS satellites. The symmetry of the GPS orbital planes was also analyzed. The study showed that for five out of six orbital planes, the hexagonal ascending nodes symmetry error value did not exceed 2\({}^{\circ}\). An exception was orbital plane D, where three times higher values of deviations were recorded. In this case, the robust estimation method was of particular importance as it minimized the impact of the anomalous SVN46 satellite orbit on the adjustment results. The analysis showed an average trend of separation of satellite ascending nodes from the orbital planes of 0.25\({}^{\circ}\)/year with an average space mission duration of 11.7 years. The highest value of the separation trend was obtained for orbital plane D with the anomalous SVN46. The study also assessed the impact of the space mission duration on the ascending node separation trends. With a correlation coefficient of 0.36, the highest mean trend value of 0.54\({}^{\circ}\)/year occurred in the case of satellites operating for over 20 years. Satellites with less than 5 years of mission duration achieved four times lower values. The developed method for assessing the symmetry of the position of the orbital plane ascending nodes is a novel solution to obtaining information about the current and past states of the satellite constellation. Adaptation of thousands of GPS almanacs to a statistical sample increased the reliability of the analysis. The trend of satellite ascending node separation from orbital planes determined in the 24-year-long research timeframe additionally enables the prediction of the future constellation state and can significantly support the actions of the GPS control segment. This research was funded from the statutory activities of Gdynia Maritime University, grant number WN/2020/PZ/05. Not applicable. Data Availability Statement: Publicly available datasets were analyzed in this study. This data can be found here: [[https://www.navcen.uscg.gov](https://www.navcen.uscg.gov)]([https://www.navcen.uscg.gov](https://www.navcen.uscg.gov)), and [[https://celestrak.com](https://celestrak.com)]([https://celestrak.com](https://celestrak.com)). Conflicts of Interest: The author declares no conflict of interest. Figure 14: Separation of satellite ANs from the hex-LS-estimated position of the OP A AN with satellite dot(\(\Omega\)) values. Figure 13: Separation of satellite ANs from the hex-LS-estimated position of the OP A AN with satellite dot(\(\Omega\)) values.