# The Symmetry of GPS Orbit Ascending Nodes
## 2 Materials and Methods
Due to the perturbations in the movement of GPS satellites and the modernization of the space segment occurring in the 24-year research timeframe, the position of six OP ANs did not meet the theoretical division of the full angle into six equal parts. The assessment of hexagonal symmetry required the calculation of reference values that met the condition of the angular separation between successive nodes amounting to 60\({}^{\circ}\). The adjustment procedure is abbreviated here as hex-LS. Based on the robust estimators of the position of OP ANs determined in module 3 of the algorithm, observational equations of the angles between the ANs were created and then adjusted using the least-squares method. The  
Figure 3: Comparison of the determination of the orbital plane (OP) D ascending node (AN) using the arithmetic mean and the robust estimation method with the weight values of the observations. design matrix **A** defining the theoretical hexagonal geometry and the vector of intercepts **y** took the following forms:  
\[\textbf{A}=\left[ \begin{array}{cccccc}-1&1&0&0&0&0\\ 0&-1&1&0&0&0\\ 0&0&-1&1&0&0\\ 0&0&0&-1&1&0\\ 0&0&0&0&-1&1\\ 1&0&0&0&0&-1\\ 1&1&1&1&1&1\end{array} \right] \tag{13}\]  
\[\textbf{y}=\left[ \begin{array}{cccc}\delta_{AB}-60^{\circ}\\ \delta_{BC}-60^{\circ}\\ \delta_{CD}-60^{\circ}\\ \delta_{DE}-60^{\circ}\\ \delta_{EF}-60^{\circ}\\ \delta_{FA}-60^{\circ}\\ \delta_{AB}+\delta_{BC}+\delta_{CD}+\delta_{DE}+\delta_{EF}+\delta_{FA}-360^{ \circ}\end{array} \right] \tag{14}\]  
where \(\delta_{AB},\delta_{BC},\delta_{CD},\delta_{DE},\delta_{EF},\delta_{FA}\) are the angles between the values of robust estimators determined in module 3 of the algorithm, e.g., \(\delta_{AB}=LAN_{OPB}-LAN_{OPA}\). The weights of individual observations were set as equal due to the earlier application of the Huber function that minimized the impact of outliers. The weight matrix was a unit matrix, \(\textbf{P}=diag[1,1,1,1,1,1]\). The vector of corrections to the parameters was determined by adopting the optimization criterion of the least-squares method:  
\[d\textbf{X}=-\left(\textbf{A}^{T}\textbf{P}\textbf{A}\right)^{-1}\textbf{A}^{ T}\textbf{P}\textbf{y} \tag{15}\]  
Estimators obtained from the hex-LS adjustment were the reference quantities for a given epoch of the SEM almanacs. Based on them, the d\(\Omega\) deviations of the position of the satellite ANs relative to the OP were determined in algorithm module 5. Comparing the estimators of the robust method and the hex-LS method in module 7, the angular separations of OPs relative to the theoretical geometric conditions of the GPS were determined. The sample results of the estimation procedures from modules 3 and 4 are shown in Figure 4.  
The figure shows the position of the OP ANs obtained at the consecutive stages of calculations. The arithmetic mean of the OPs' \(\Omega\) parameter is marked in black. The result of the robust method is shown in green. The final reference position of the OP ANs meeting the condition of hexagonal symmetry is marked in red. The ascending nodes of the orbits of individual GPS satellites are shown in gray with pink highlighting the anomalous orbit of the SVN46 satellite. The orthogonal view with a longitude of 90\({}^{\circ}\) N allows for a preliminary assessment of the correctness of the algorithm result on the example of the  
Figure 4: Position estimators of OP ANs determined by the robust estimation method and hex-LS on 13 September 2020 in top (**a**), and perspective view (**b**). most characteristic OP, D. The perspective view shows fragments of the ECI orbits located near the ascending nodes. Module 6 of the algorithm determined the trend of separation of the GPS satellite AN from the OP. The parameters of the regression lines were calculated using the least-squares method presented in [PERSON] et al.