# LEO Precise Orbit Determination with Inter-satellite Links
## 4 Results and Analyses
### POD with Center of Gravity Reference  
Using ISL measurements, POD with center of gravity reference mentioned in Section 3.4 can be achieved. In order to study the factors affecting the POD results, we carry out comparative experiments with several sets of satellite numbers and different observation accuracy. Because the numbers of satellites are very large in our experiments, for the sake of convenience the satellites are displayed at intervals 3, 6, 9 of PRN number for 60-, 120-, 192-LEO constellations, respectively. This means there are 20, 20, 22 satellites to be shown in the 60-, 120-, 192-LEO constellations, respectively.  
First, in order to verify the impact of link ranging accuracy on the orbit determination, we select 60-LEO constellation detailed in the Table 1 for test. Experiments are carried out with 40, 20, and 10 cm ISL ranging accuracy and the orbital errors in the radial, along-track, cross-track, and 3D directions are shown in the Figures 3-6. From these figures, it can be clearly concluded that higher ranging accuracy leads to lower orbital errors in every direction. It is because that observation errors which compose \(l\) in Equation (15) are smaller during adjustment process. Average orbital errors of 60-LEO constellation after POD with 40, 20, and 10 cm ranging accuracy are shown in the Table 3. When the ranging accuracy is 40 cm, POD result is poor with average 3D error of 0.358 m but it turns to be much better when the ranging accuracy is 20 or 10 cm. It can be calculated that compared to the ranging accuracy of 40 cm, the 20 and 10 cm schemes bring about 76.0 and 93.9% 3D orbital error reductions, respectively.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Errors** & **\(\Delta\)x/m** & **\(\Delta\)y/m** & **\(\Delta\)z/m** & **3D/m** \\ \hline
4-connected & 0.095 & 0.045 & 0.038 & 0.112 \\ all-connected & 0.052 & 0.002 & 0.045 & 0.069 \\ \hline \end{tabular}
\end{table}
Table 2: Average orbital errors of 60-LEO constellation with different link topologies.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Direction**} & \multicolumn{3}{c}{**Ranging Accuracy**} \\ \cline{2-4}  & **40 cm** & **20 cm** & **10 cm** \\ \hline Radial [m] & 0.179 & 0.043 & 0.011 \\ Along-track [m] & 0.084 & 0.020 & 0.005 \\ Cross-track [m] & 0.272 & 0.065 & 0.016 \\
3D [m] & 0.358 & 0.086 & 0.022 \\ \hline \end{tabular}
\end{table}
Table 3: Average orbital errors of 60-LEO constellation under different ranging accuracies.  
Figure 4: 60-LEO constellation orbital errors in the along-track direction under different ranging accuracies.  
Figure 3: 60-LEO constellation orbital errors in the radial direction under different ranging accuracies.  
Then, several experiments are carried out to verify the impact of link numbers on the orbit determination. It is obvious that the more satellites in a constellation, the more ISLs will be formed, so we select 60-, 120-, and 192-LEO constellations detailed in the Table 1 for further study and the 20 cm ISL ranging accuracy is chosen for all constellations. The orbital errors of these constellations are shown in the Figures 7-9. Figure 10 shows the average orbital errors in different directions of these constellations. Average numbers of links per satellite in these constellations are 16.7, 33.0, and 51.5, respectively. Obviously, the more satellites, the more links there are, which contributes to the smaller orbital errors in all directions. It is because more link observations, which means the number of rows in matrix \(l\) in Equation (15) are larger, brings stronger geometric constraints to the coordinates of the  
Figure 5: 60-LEO constellation orbital errors in the cross-track direction under different ranging accuracies.  
Figure 6: 60-LEO constellation 3D orbital errors under different ranging accuracies.