# The Symmetry of GPS Orbit Ascending Nodes
## 2 Materials and Methods
The last, i.e., seventh, module of the algorithm determined the symmetry parameters of the adjusted OP ANs relative to the reference values. The module 2 allocation procedure was applied using the position of the satellite orbit ANs indicated by the \(\Omega\) parameter. Assuming the theoretical symmetrical 60-degree distribution of the GPS OPs in the equator plane, six satellite subsets, A, B, C, D, E, and F, with similar \(\Omega\) parameter values were extracted. As an example, according to the last processed almanac from the 2123 rd (75 th in the current third cycle) GPS week and the319,488 th s (16 September 2020 16:44:48 UTC), the state of the constellation OPs was as follows: 4 (A), 5 (B), 5 (C), 6 (D), 6 (E), and 5 (F) satellite vehicles (SVs). In the set of all 31 almanacs' \(\Omega\) values, six subsets with the size of 4, 5, 5, 6, 6, and 5 elements were created with the GPS OPs' satellites. In the 24-year-long research timeframe, the cardinality and elements of the OP subsets were changing due to space segment modernization. The six \(\Omega\)-dependent subsets of satellites provided information on the spatial distribution of the orbit ANs in the equatorial plane. As the almanacs do not contain information on the satellite OP designation, a method to assign the six subsets with OP alphabetical indicators was developed. Based on the Current Operational Advisories published on the USCG website, it was established that the SVN46 (PRN11) satellite has been carrying out its space mission continuously since 2000 and belongs to OP D. According to the prograde movement direction of the GPS satellites, the other OP subsets were assigned as A, B, C, E, and F for the almanacs generated between 2000 and 2019. For the remaining almanacs from 1996 to 2000, an analogous procedure was used based on the SVN18 (PRN18) satellite from orbital plane F. The application of algorithm module 2 resulted in the creation of six GPS OP subsets containing the parameters of the satellites for each SEM almanac file. The sample result of the application of module 2 of the algorithm is presented in Table 1. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**OP** & \(\Omega\) [°] **(SVN)** & **Mean \(\Omega\) [°]** & **StDev \(\Omega\) [°]** \\ \hline A & 241.4338 (48), 237.1903 (65), 242.6358 (64), & 240.9030 & 2.1891 \\  & 242.3521 (52) & 240.9030 & 2.1891 \\ \hline B & 304.9308 (58), 306.0106 (56), 300.9003 (62), & 303.3641 & 2.9968 \\  & 298.7580 (71), 306.2207 (44) & 303.3641 & 2.9968 \\ \hline C & 359.9438 (72), 4.1978 (53), 6.8009 (59), & 3.3277 & 2.5694 \\  & 0.8136 (66), 4.8822 (57) & 3.3277 & 2.5694 \\ \hline D & 61.2960 (63), 56.7924 (61), 60.8234 (67), & 55.2791 & 9.8795 \\  & 33.6900 (46), 62.1870 (75), 56.8857 (45) & & \\ \hline E & 120.8952 (69), 118.9797 (50), 120.7238 (73), & 118.1296 & 2.8547 \\  & 113.012 (51), 115.7114 (47), 119.4557 (76) & & \\ \hline F & 182.7315 (74), 179.9369 (68), 188.2194 (43), & 181.1958 & 4.3990 \\  & 174.6445 (55), 180.4468 (70) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The assignment of satellites to orbital planes on the example of the GPS time third cycle almanac from epoch 75.319488. Figure 1: The workflow of the research algorithm. The values of the \(\Omega\) parameter in six subsets indicate the position of the satellite ANs at the almanac epoch. The subsets have a different degree of convergence of the satellite \(\Omega\) parameter. [PERSON] and [PERSON] (2013) indicated the existence of a significant deviation in the position of the SVN46 (PRN11) satellite ascending node in relation to other OP D satellites (Figure 2). The Earth-centered inertial (ECI) frame is a non-accelerating, non-rotating frame aligned with the Earth-centered Earth-fixed (ECEF) frame at a specific instant of time ([PERSON] and [PERSON], 2017). The presence of the indicated and potentially other outliers in OPs required applying a method that minimizes their impact on the estimators of the position of orbital plane ascending nodes.