# Robust Kalman Filter Aided GEO/IGSO/GPS Raw-PPP/INS Tight Integration
## 2 Methods
### Performance Analysis  
Shown in Figures 6 and 7 are the time-series of position offsets in n-frame (i.e., north-east-down), which were obtained by making differences between PPP, EKF based PPP/INS tight integration, and R-EKF based PPP/INS tight integration using GPS, GPS + GEO, GPS+IGSO, and GPS + GEO + IGSO data with the reference values. According to the statistics in Figure 6a, the positioning accuracy of GPS PPP could be upgraded while using GPS + GEO and GPS + IGSO. While using both GEO and IGSO together, the position offsets reduced even more. According to the Root Mean Square (RMS) values listed in Table 2, the position RMSs were upgraded from 0.265, 0.160, and 0.225 m in GPS PPP mode to 0.078, 0.114, and 0.095 m in GPS + GEO + IGSO PPP mode, with improvements of 70.5%, 28.7%, and 57.8% in the north, east, and down directions, respectively. The corresponding accuracy enhancements were about 40.0%, 17.5%, and 8.4% by adding GEO and 64.2%, 23.7%, and 46.2% by adding IGSO. This means that the IGSO satellites provided more enhancement than GEO, which may be due to the almost static locations of GEO satellites over the equator. As shown in Figure 8, all the four observed GEO satellites were almost in a straight line in the south, while the IGSO satellites moved around the mission area. Such a constellation means that IGSO can provide much more user-satellite geometry structure improvements than GEO. Besides, a small mutation in the north component of GPS PPP emerged at around 0.95 h, which may be because there were only four visible GPS satellites (as shown in Figure 5b). However, such mutations were rather small when no cycle slips occurred because the receiver DCB, ionosphere delays, and troposphere delay should have been constants in the short term, which could have provided strong constraints for PPP calculations. According to the RMS results in Figure 7b and Table 2, both INS and R-EKF presented positive influences on PPP position accuracy. For the PPP/INS tight integration case, the average improvements from INS were 19.6%, 17.6%, and 14.5% in the three directions compared to the PPP solutions. The enhancements from GEO and IGSO were 33.0%-13.1%-11.1% and 65.1%-14.6%-49.2% in north-east-down components. For the R-EKF based PPP/INS tight integration case, the average enhancements from R-EKF were 17.8%, 50.2%, and 22.4% compared to the solutions from EKF, and 32.9%, 60.3%, and 34.5% compared to PPP solutions. Visibly, both INS and R-EKF could make the solutions more stable and accurate. Figure 5: Satellite sky plot (**a**) and the corresponding position dilution of precision (PDOP) (**b**) during the land-borne test. The contribution of INS to enhancing PPP was largely due to its low-pass filter characteristics and the enhanced observability in the vertical direction, which was been presented in [10; 20; 24; 40; 41]. Although the enhancement from BDS on GPS has been proven by previous works [11; 13; 14], the effect of different constellations of BDS satellites on GPS has not been demonstrated. Most of those works only adopted the empirical priori weight and the satellite elevation dependent weight function [6] to decrease GEO's influence. However, observation's real quality would change along with users' environments, especially in the dynamic cases. Then, the empirical method may not work well in \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Data**} & \multicolumn{2}{c}{**PPP**} & \multicolumn{3}{c}{**PPP/INS**} & \multicolumn{3}{c}{**Robust PPP/INS**} \\ \cline{2-10}  & **North** & **East** & **Down** & **North** & **East** & **Down** & **North** & **East** & **Down** \\ \hline GPS & 0.265 & 0.160 & 0.225 & 0.212 & 0.137 & 0.199 & - & - & - \\ GPS + GEO & 0.159 & 0.132 & 0.206 & 0.142 & 0.119 & 0.177 & 0.149 & 0.068 & 0.119 \\ GPS + IGSO & 0.095 & 0.122 & 0.121 & 0.074 & 0.117 & 0.101 & 0.046 & 0.044 & 0.080 \\ GPS + GEO + IGSO & 0.078 & 0.114 & 0.095 & 0.058 & 0.066 & 0.080 & 0.046 & 0.036 & 0.069 \\ \hline \hline \end{tabular}
\end{table}