# LEO Precise Orbit Determination with Inter-satellite Links
## 4 Results and Analyses
### POD with Reference Provided by Navigation Satellites  
According to the above analysis, POD with ISLs gets bad performance in the case of center of gravity reference when link ranging accuracy is poor and numbers of links are small. When the link accuracy is 40 cm, average 3D orbital errors of 60-LEO constellation are 0.358 m, which can only meet the demand of autonomous navigation. Compared to a center of gravity reference, a spatial reference which is decided using GNSS data from onboard LEO receiver is more common in use and it is confirmed to have better POD potential. In our study, two satellites in each constellation are considered as \"reference satellites\" and their 3D coordinates, which are assumed to be decided by navigation satellites, are simulated by adding 3 cm noises to the x, y, and z components of their real coordinates.  
The accuracy of POD can be improved with constraint of orbital parameters [43]. Previous studies have shown that the predicted accuracy of RAAN is very high for the MEO and GEO satellites [25], thus the orbit determination accuracy can be effectively improved by using the RAAN predicted value  
Figure 10: Average orbital errors under 20 cm ranging accuracy of the 60-LEO (**red**), 120-LEO (**blue**) and 192-LEO (**green**) constellations.  
Figure 9: 192-LEO constellation orbital errors in radial, along-track, cross-track and 3D under 20 cm ranging accuracy, in which satellite of PRN 1, 10, 19,..., 190 (divided at interval 9 of PRN number) are displayed.  
to impose constraints to the observation functions. Though the orbital parameters of the LEO satellites are more changeable because of relatively low orbit altitudes, RAAN values of LEO satellites are still stable within a few hours [44; 45]. Therefore, it is feasible to constrain RAAN to improve the orbit determination accuracy of LEO satellites during short periods.  
Since more numbers of links bring better POD performance as mentioned above, 120- and 192-LEO constellations are selected for test and the link ranging accuracies are both 20 cm. RAANs of each orbit in both constellations are regarded as known values when computing the link ranging residuals which form the matrix \(l\) in the network adjustment process as shown in Equation (13). Average link ranging residuals and total numbers of links of both constellations are displayed in Table 4. It can be calculated that when using known RAAN values as constraints, average ranging residuals get 68.0% reduction for 120-LEO constellation and 48.6% for 192-LEO constellation, which means the elements that make up the matrix \(l\) in Equation (13) have smaller absolute values. Then, the orbital errors of experimental constellations are shown in Figures 11 and 12. It can be clearly seen from the figures in both constellations, orbital errors of all the satellites displayed are very small. For further detailed analysis, average orbital errors of both constellations are shown in the Table 5. We can conclude from the table that average orbital errors of both constellations in every direction are less than 1 cm and 192-LEO constellation achieves slightly better POD performance than 120-LEO constellation. Obviously, with RAAN as the constraint, POD by using the reference provided by navigation satellites reaches an extremely high precision. It could have resulted from the