# Robust Kalman Filter Aided GEO/IGSO/GPS Raw-PPP/INS Tight Integration
## 2 Methods
### Performance Analysis
Similarly, works in [26] have also proven that attitude determination accuracy had a weak relationship to GNSS data but was highly determined by IMU sensors' level. The results in this test also indicated the same conclusion. From the statistics in Table 5, the attitude error RMS on average were about 0.007\({}^{\circ}\), 0.015\({}^{\circ}\), and 0.202\({}^{\circ}\) for roll, pitch, and heading, respectively. Compared to roll and pitch, poorer accuracy in heading direction was due to the weaker observability in IMU's vertical (z-axis) gyroscope [42] and the inadequate maneuver motions (as shown in Figure 4). In contrast, the heading computed by PPP velocity using \(heading=\tan^{-1}(v_{E}/v_{N})\)[25] was much worse than that of PPP/INS integration, as is shown in Figure 11. As shown, such heading angles were sensitive to horizontal velocity. Therefore, the accuracy of the heading changed regularly along with the velocity, especially when the velocity direction was changed. Such phenomenon can also be found by making a comparation between Figures 11 and 4b. According to the results above, it can be concluded that GEO and IGSO have different influences on improving PPP performance and the R-EKF that is based on posteriori residuals can effectively adjust the optimal weight for GPS + BDS data processing. Then, more reliable and accurate positioning solutions are provided. However, there is little relationship between these methods and the accuracy of velocimetry and attitude determination. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Residuals** & **P1(m)** & **P2(m)** & **L1/B1(m)** & **L2/B2(m)** \\ \hline GEO & 0.759 & 1.406 & 0.005 & 0.007 \\ IGSO & 0.491 & 0.292 & 0.004 & 0.003 \\ GPS & 0.307 & 0.424 & 0.003 & 0.004 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: RMS of residuals of codes (P1 and P2) and phases (GPS: L1 and L2, BDS: B1 and B2) of GEO, IGSO, GPS. Figure 11: Heading angles calculated by velocity (**top**) from PPP and the corresponding velocity in horizontal (**bottom**). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Data**} & \multicolumn{3}{c}{**PPP**} & \multicolumn{3}{c}{**PPP/INS**} & \multicolumn{3}{c}{**Robust PPP/INS**} \\ \cline{2-9}  & **North** & **East** & **Down** & **North** & **East** & **Down** & **North** & **East** & **Down** \\ \hline GPS & 0.0485 & 0.0438 & 0.0629 & 0.0044 & 0.0040 & 0.0039 & - & - & - \\ GPS + GEO & 0.0449 & 0.0399 & 0.0619 & 0.0043 & 0.0040 & 0.0038 & 0.0043 & 0.0040 & 0.0038 \\ GPS + IGSO & 0.0430 & 0.0418 & 0.0589 & 0.0042 & 0.0040 & 0.0036 & 0.0042 & 0.0040 & 0.0036 \\ GPS + GEO + IGSO & 0.0403 & 0.0392 & 0.0533 & 0.0040 & 0.0040 & 0.0036 & 0.0040 & 0.0040 & 0.0036 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Velocity RMS of PPP, PPP/INS tight integration, and robust PPP/INS tight integration (unit: m). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Data**} & \multicolumn{3}{c}{**PPP/INS**} & \multicolumn{3}{c}{**Robust PPP/INS**} \\ \cline{2-9}  & **Roll** & **Pith** & **Heading** & **Roll** & **Pith** & **Heading** \\ \hline GPS + GEO & 0.0065 & 0.0154 & 0.2044 & 0.0065 & 0.0154 & 0.2030 \\ GPS + IGSO & 0.0065 & 0.0153 & 0.2028 & 0.0065 & 0.0152 & 0.2023 \\ GPS + GEO + IGSO & 0.0065 & 0.0153 & 0.2025 & 0.0064 & 0.0152 & 0.1983 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Attitude RMS of PPP/INS tight integration with and without robust extended Kalman filter (unit: \({}^{\circ}\)).