# The Symmetry of GPS Orbit Ascending Nodes
## 2 Materials and Methods  
According to the ICD-GPS-240 interface [20], the GPS almanac is available in two formats: YUMA and System Effectiveness Model (SEM). The format name YUMA comes from the Yuma Proving Ground in Arizona (USA), where numerous tests on the GPS accuracy were performed. Apart from the differences in units and the number of parameters, the YUMA format is distinguished by the content of text descriptions. The SEM format has a tabular structure with greater variables precision than YUMA. For this reason, the SEM format was used in this work. The almanac files since 1997 are available on the United States Coast Guard website [21]. Previous almanacs (since 1991) can be obtained from The Center for Space Standards & Innovation website [22]. Due to the different initial orbit configurations, the analysis omitted the almanacs containing the parameters of the block I satellites. In the almanac generated in the 846 th week and the 32,768 th s of GPS time, SVN10 (PRN12) appears for the last time. The first almanac used in this study represents the epochs of the 847 th week and the 61,440 th s of GPS time on 24 March 1996. In total, 7035 SEM almanacs were used in this study. A total of 39 files were downloaded from the CSSI (years: 1996-1997) and 6996 files were downloaded from the USCG (years: 1997-2020). Until the year 2000, the GPS control segment published almanac files with a weekly frequency with several periods of longer breaks, e.g., in 1999. Currently, the almanacs are updated daily. Two parameters in the GPS almanac parameters were used in the research: \(\Omega\) and the rate of the longitude of the ascending node (dot(\(\Omega\))). The position of GPS orbit ANs in the almanac model is referenced to the Greenwich meridian at the beginning of the GPS week [23]. Therefore, in the description of the trajectory of satellites, \(\Omega\) is used instead of the classic Keplerian right ascension of the ascending node related to the First Point of Aries. In astronomy, the position of stars and other celestial objects is defined in the equatorial coordinate system using angular coordinates. The coordinates are not fixed to the Earth but determined on the celestial sphere. The reference direction in the celestial equator plane is the First Point of Aries, also called the Vernal (March) Equinox. The GPS inertial system, based on the Greenwich meridian, defines the orientation of the coordinate axes with respect to the Earth, not the celestial sphere. To ensure the recurrence of Earth-centered Earth-fixed (ECEF) tracks of the GPS satellites, an additional force is applied to the satellites in the process of placing the satellite in the orbit. Its direction and magnitude compensate for the rotation of the Earth and result in the delay of the ANs' drift. That additional angular velocity of the GPS satellites is expressed in the almanac by the dot(\(\Omega\)) parameter. As the angular degrees were the unit used in the calculations, the conversion of the SEM almanacs' semicircles was introduced. To achieve the aim of the study, the algorithm with seven modules was developed (Figure 1). The first module selected the \(\Omega\) and dot(\(\Omega\)) values from the SEM almanacs. The second module of the algorithm assigned the satellites to the six GPS orbital planes. The third module used a robust estimation method to determine the position of the ascending nodes of the orbital planes based on the \(\Omega\) parameter. The fourth module of the algorithm performed a least-squares method adjustment and determined the reference estimators satisfying the condition of the hexagonal symmetry of the ascending nodes of the orbital planes. The fifth module compared the positions of the ascending nodes of the individual satellites with the reference values of the orbital planes. After all the SEM almanacs had been processed, the last two steps of the calculation were run. The sixth module determined the satellite AN separation trend in relation to the OP based on the calculated values of deviations d\(\Omega\).