# Robust Kalman Filter Aided GEO/IGSO/GPS Raw-PPP/INS Tight Integration
## 2 Methods
### Performance Analysis

Table 2: RMS of position offsets of PPP, PPP/INS tight integration (PPP/INS), and robust PPP/INS tight integration (robust PPP/INS) by comparing with GPS + BDS RTK/INS tight integration solutions (unit: m). Figure 6: Position offsets of PPP (**a**) and PPP/INS tight integration (**b**) by comparing with GPS + BDS RTK/INS tight integration solutions. Figure 7: Position offsets of robust PPP/INS tight integration (**a**) and position RMS of all data processing modes (**b**); in subfigure (**b**), 1–4 denote PPP using GPS, GPS + GEO, GPS + IGSO, and GPS + GEO + IGSO, 5–8 denote PPP/INS tight integration using GPS, GPS + GEO, GPS + IGSO, and GPS + GEO + IGSO, 9–10 denote robust PPP/INS tight integration using GPS + GEO, GPS + IGSO, and GPS + GEO + IGSO. decreasing the influence of GEO satellites. However, the real quality of GPS and BDS can be reflected directly by the posteriori residuals that are obtained from Equation (26). As plotted in Figure 9, the posteriori residuals of codes and phases of GEO, IGSO, and GPS were significantly different from each other. According to the statistics listed in Table 3, the quality of GEO codes in terms of RMS are 0.759 m and 1.406 m for B1 and B2 frequencies, which are about 1.5-4.5 times worse than that of IGSO (0.491 m and 0.292 m) and GPS (0.307 m and 0.424 m). GEO' phase residual RMS were 5 mm and 7 mm for B1 and B2 frequencies, which was about 1.0-1.5 times lower than IGSO (4 mm and 3 mm) and GPS (3 mm and 4 mm). As such the quality of IGSO was similar to that of GPS. The average variance of posteriori residuals of P1, P2, L1, and L2 of GPS and BDS were 0.004, 0.005, 0.494, and 0.731 m, respectively. By using Equation (26), the re-determined equivalent weight could reflect the real quality of GEO/IGSO/GPS. Then, based on such equivalent weight [39], the R-EKF could increase the weight of GPS and IGSO and decrease that of GEO. Then, the robust positioning solutions was obtained, which can be seen in Figure 7a. Figure 8: Azimuth-elevation based sky plot of the observed GEO, IGSO, and GEO satellite. Figure 9: Code residuals (**a**) and phase residuals (**b**) of GEO, IGSO, and GPS satellites. Besides the position, the velocity and attitude are also important for dynamic users. Therefore, the impacts of GEO, IGSO, INS, and R-EKF on velocity and attitude determination were accessed. In this part, the reference values for velocity and attitude were calculated from RTK/INS tight integration by using navigation grade IMU measurements. Shown in Figure 10 are velocity offsets from PPP, EKF based PPP/INS tight integration, and R-EKF based PPP/INS tight integration and attitude offsets from PPP/INS tight integration with and without robust EKF by using GPS, GPS + GEO, GPS + IGSO, and GPS + GEO + IGSO data, respectively. From Figure 10, it can be seen that INS could enhance velocity accuracy visibly. However, the improvements from GEO, IGSO, and even GEO + IGSO were not significant. This outcome was because the velocity accuracy in PPP mode was mainly dependent on the quality of Doppler measurements (about decimeter per second level), and had little relationship with positioning accuracy [26]. As listed in Table 4, the velocity RMS on average were 4.4 \(\pm\) 0.4 cm/s, 4.1 \(\pm\) 0.2 cm/s, and 5.9 \(\pm\) 0.5 cm/s in north, east, and vertical directions, respectively, in PPP mode. While applying INS, the effect of Doppler noises on velocity was decreased significantly. Then, velocity accuracy was up to around 4 mm/s in all three directions. In this case, the effects of GEO and IGSO on improving velocity were covered by INS, and no improvement could be seen when using GPS + BDS together compared with using only the GPS data. This was because the velocity accuracy in PPP/INS tight integration only depended on the IMU performance [26]. Meanwhile, from Figure 10a and the statistics in Table 4, it can be seen that the R-EKF also had slight effect on velocity.