# Intelligent Environment-Adaptive GNSS/INS Integrated Positioning with Factor Graph Optimization
## 5 Further Analysis and Evaluation  
### Comparisons between Receivers and across Integration Methods (KF-Based vs. FGO-Based)  
This section compares the positioning results across the three GNSS receivers in terms of the RMSE of the solutions and STD of the 2D error. With reference to KF, the positioning improvement on the RMSE and STD of the other three integration methods are introduced in:  
\[RMSE_{improvement\%}=\frac{RMSE_{method}-RMSE_{KF}}{RMSE_{KF}} \times 100\%, \tag{31}\] \[STD_{improvement\%}=\frac{STD_{method}-STD_{KF}}{STD_{KF}} \times 100\%, \tag{32}\]  
where the subscript \"method\" may denote AKF, FGO, or AFGO. It is noted that the RMSE and STD of the GNSS solution are not taken as the reference because those should have been infinity given the missing GNSS epochs; thus, further improvement based on GNSS is unreasonable to evaluate. Combined with Figure 13a showing the GNSS availability defined in Equation (21), graphs on the RMSE and STD of different integration methods are given in Figures 13b and 13c, respectively. It is displayed that GNSS availability decreases from U-Blox (92.61%) to Xiaomi (86.41%) to NovAtel (67.61%), whereas the bar charts about RMSE and STD improvement for each receiver basically show the opposite relationship, which rises from U-Blox (red) to Xiaomi (magenta) to NovAtel (blue). Overall, compared with KF, the less the GNSS availability, the greater the improvements in both RMSE and STD tend to happen for each of the methods (i.e., AKF/FGO/AFGO). Despite the anomaly for the AKF method where Xiaomi has greater improvements in the RMSE and STD than NovAtel, the results could reflect that the receiver with less GNSS availability would potentially benefit more from GNSS/INS integration in terms of positioning accuracy and smoothness as greater improvements would be achieved according to Figure 13b,c. Moreover, in Figure 13, the improvements from AKF (all below 25%) appear less significant than those given by FGO-based methods (all over 65%). Possible explanations may lie in that the GNSS unavailability causes free propagation using INS data. In contrast from KF-based methods, which accumulate the errors during INS transition, FGO-based solutions are less constrained on the missing GNSS epochs and are less likely to produce positioning drifts, thus giving higher improvement in both the RMSE and STD. This is also revealed in the bar chart in Figure 14, which summarizes the 2D RMSE and STD of  
Figure 13: (**a**) GNSS availability of the three receivers; improvement of solution performance compared with KF in terms of (**b**) 2D RMSE and (**c**) 2D STD. The only anomaly is that for the AKF method, Xiaomi has greater improvements than NovAtel. This is expected because greater improvement comes with lower GNSS availability, which is based on empirical observation rather than strict rules. the four integration methods during all 920 s. The FGO and AFGO methods have less 2D RMSE and less 2D STD for all three receivers, which means a higher accuracy and smoother solutions compared with KF and AKF. Especially for Xiaomi and NovAtel, as shown in Figure 14, FGO-based solutions have a significant decrease (approximately ranging from 70% to 95%) in both the RMSE and STD from KF-based ones, and these are also reflected in the results of Figure 13b,c. Additionally, according to Figures 7, 9 and 11, at junctions where large trajectory drifts exist, the 2D positioning error results in Figure 8 (Junction C), Figure 10 (junction C), and Figure 12 (junction B and C) roughly reveal that the FGO and AFGO curves are far below those of KF and AKF. For Xiaomi and NovAtel, which have more extreme positioning drifts than U-Blox, this situation seems more evident. It should also be mentioned that AFGO (FGO-based) proves to be the most accurate and stable method amongst all integration methods, having the lowest 2D RMSE and STD for all grades of GNSS receivers.