## 3 Results and Discussion
### Relative Navigation Improvement  
The relative navigation robustness is very important when an SFF mission rigorously demands the separation accuracy among their distributed space systems. For instance, the precision of formation keep and relative state estimation for a 1000-m baseline SFF mission is sensitive to the spatial resolution of small-scale (e.g., \(\leq\)100 m) ionospheric irregularity or large-scale ionospheric morphology (e.g., size or boundary of plasma structure) observations.  
Besides ionospheric remote sensing, the new version of VTFFTB after incorporating Galileo also shows an improvement on relative state estimation. The baseline ESF scenario described in Section 3.1 is used for HIL simulations, in order to compare the EKF-based relative state estimations between a GPS scenario and a multi-constellation scenario. A number of qualitative comparisons were performed, and one comparison based on the 8 March 2019 almanac is shown here as an example. The relative state estimation errors presented in Table 2 are based on a simulation using the GPS (L1) constellation only for relative navigation. The relative state estimation errors in Table 3 is based on a simulation using both the GPS (L1) and Galileo (E1) constellations. The error is defined as the difference between the EKF estimated value and the simulator recorded \"truth\" after each simulation, and it is calculated including the initialization period before the EKF converges. All the error values in Tables 2 and 3 are rounded to three decimal places. Note that, the scintillation effect is not fully simulated and the induced cycle slips were not fully addressed in the HIL simulation for simplicity. This is likely the main factor degrading the overall relative estimation performance. However, several non-control simulations on the VTFFTB with more robust algorithms (to handle cycle slips) were used for formation flying relative navigation and their relative position errors were less than 10-cm after the EKF converges. Table 4 gives the computed error decrease percentages of Table 3 relative to Table 2, which quantifies how much relative state estimate errors were decreased by using multi-constellation GNSS (GPS L1 and Galileo E1) against using GPS only. The positive values indicate error decrease, while a negative value indicates an error increase. After a large ensemble of simulations, it is found that the average absolute errors in all ECEF (Earth-centered, Earth-fixed) components of relative position and relative velocity typically decreased. The standard deviation of errors decreased in most cases as well. The simulations based on a newer almanac usually showed better improvement as well due to a larger number of Galileo satellites. Also, the EKF-based relative estimation converge time is typically faster using multi-constellation, compared to using GPS-only. Therefore, relative navigation with both GPS and Galileo constellations can offer better estimation performance, as the dimension of differential GNSS measurement matrix becomes larger with more PRNs.  
Figure 12: Horizontal view of 1-Hz S4 level observed from different Galileo frequency bands on the chief’s receiver: (**a**) Galileo E1; (**b**) Galileo E5a; (**c**) Galileo E5b.