# GNSS RTK Positioning Augmented with Large LEO Constellation
## 4 Performance of Different LEO Constellations Augmented GPS RTK
In order to analyze the baseline solution precision after fixing ambiguity, the mean RMS errors in the east, north, and up baseline components for different LEO constellation schemes are given in Table 5. For the GPS-only scheme, the baseline solution precision is 0.77, 0.93, and 1.73 cm in east, north, and up baseline components, respectively. By introducing the different scales of LEO  
Figure 3: Comparisons of different scales of LEO constellations augmented GPS RTK solutions in east, north, and up components, respectively, for ‘P339-P341’ baseline scheme. The visible satellite numbers and position dilution of precision (PDOP) values are also presented. constellations, the RMS errors can be best reduced to 0.58, 0.65, and 1.37 cm, respectively. Obviously, whether or not to introduce LEO satellites, the RTK positioning precisions can all attain millimeter level in horizontal directions and centimeter level in the elevation direction after fixing DD ambiguities. Nevertheless, the larger LEO constellation is conducive to achieve more stable and better positioning accuracy. Such as 288-LEO constellation scheme, the solution precisions of east, north, and up baseline components can be increased by 25, 30, and 21%, respectively. Figure 5 shows the average convergence time, TTFF and fixing ratio for different LEO constellations augmented GPS RTK solutions, respectively. We can find that the augmented performances are closely related to the numbers of LEO satellites. Along with the increasing LEO satellites, the average convergence time and TTFF of RTK positioning will be much shorter, as well as higher fixing ratio. Compared with the GPS-only scheme, the convergence time can be reduced from 4.94 to 2.73, 1.47, 0.92, and 0.73 min by combination with observations from 60, 96, 192, and 288 LEO satellites, respectively. And the TTFF can be shortened from 7.28 to 3.33, 2.38, 1.22, and 0.87 min, respectively. Furthermore, the corresponding fixing ratio can be increased by 11.85, 16.33, 19.38, and 20.16%, respectively. Especially for 192- or 288-LEO constellation scheme, the improvements on RTK convergence and fixing speeds are more significant contrast to that of GPS-only, the convergence time can be shortened from about 5 min to within 1 min, where the TTFF is also realized from about 7 to about 1 min. Meanwhile, both schemes can reach above 95% average fixing ratio. Figure 6 shows 1-h sky plots (azimuth (0-360\({}^{\circ}\)) versus elevation (0-90\({}^{\circ}\))) of GPS with the augmentation of 60, 96, 192, and 288 polar-orbiting LEO satellites, respectively. As can be observed, the circular markers illustrate the positions of visible satellites at the initial epoch. Compared with the GPS-only scheme, the number of visible satellites is gradually increasing with the introduction of LEO satellites. At meanwhile, the LEO constellations consisting of 96, 162, and 288 satellites, as augmentations or complements to GPS satellites, can nearly fill the coverage gap above rover station, except that of 60 satellites. Moreover, noting that the GPS elevation mostly varies from 0\({}^{\circ}\) to 30\({}^{\circ}\) over an hour, while LEO satellites can streak much longer tracks with the elevation variations from 0\({}^{\circ}\) to  
Figure 4: The local enlarged picture of the Figure 3 (dotted box) shows the first 10-min comparison of different scales of LEO constellations augmented GPS RTK solutions at three components, for P339-P341\({}^{\prime}\) baseline scheme. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**RMS[cm]** & **GPS-Only** & **G+60L** & **G+96L** & **G+192** & **G+288L** \\ \hline East & 0.77 & 0.76 & 0.71 & 0.62 & 0.58 \\ North & 0.93 & 0.90 & 0.83 & 0.69 & 0.65 \\ Up & 1.73 & 1.71 & 1.67 & 1.55 & 1.37 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Statistics of the baseline solution precisions after fixing ambiguity in three directions. 90 during the same period, which contributes for improving GPS RTK positioning as a result of the rapid geometric change.