## 6 Experiments and Discussion
### Semi-Physical Simulation Experiments
#### 6.2.2 Navigation Accuracy  
Table 3 presents the root mean square errors (RMSE) of different navigation parameters of the entire simulation. The latitude and longitude errors are converted to northward and eastward position errors in meters. The RMCNE-TC owns an overall better performance than those of STC, ATC and MATC. To further evaluate the navigation accuracy of the different frameworks, we compared the three-dimensional (3D) positioning error, which is defined as:  
\[d(k)\ =\ \sqrt{\left[x_{E}(k)\ -\ x_{T}(k)\right]^{2}\ +\ \left[y_{E}(k)\ -\ y_{T}(k)\right]^{2}\ +\ \left[z_{E}(k)\ -\ z_{T}(k)\right]^{2}} \tag{36}\]  
where \(\left[x_{E}(k),y_{E}(k),z_{E}(k)\right]^{T}\) is the ECEF position at the \(k\)-th epoch calculated by the different schemes, and \(\left[x_{T}(k),y_{T}(k),z_{T}(k)\right]^{T}\) is the true ECEF position at the \(k\)-th epoch. In what follows, the 3D positioning errors obtained by the different schemes are analyzed over three typical segments, including [730 s, 750 s], [1900 s, 2500 s] and [2900 s, 2960 s]. (1) 3D Positioning Errors During [730 s, 750 s]  
Figure 6 shows the positioning errors of the selected schemes from the 730-th second to the 750-th second, where the measurements of satellites #10, #13, and #24 were contaminated by large errors. The comparison result shows that RMCNE-TC is more robust in this scenario than other schemes. This superiority is mainly owing to the satellite selection procedure and the expanded **R** design. Table 4 lists the satellite selections, GDOP values, and positioning errors of these schemes at the 750-th epoch. In contrast to STC and ATC, the contaminated satellites #10, #13, and #24 are detected according to the threshold \(\Delta\rho_{thrd}\) in MATC and RMCNE-TC. But the requirement for selecting five satellites necessitates including satellite #10 in the candidate list. Here, the expanding scale \(\beta\) employed in the adaptive UKF plays a vital role to effectively suppresses the negative impact of large error of satellite #10. (2) 3D Positioning Errors During [1900 s, 2500 s]  
Figure 7 shows the positioning errors from the 1900-th second to the 2500-th second, during which the pseudo-range measurement noise was increased. Particularly the standard deviations of the pseudo-range measurement noise of satellites #10, #13, and #24, were set to 5 m. The fixed **R** employed in STC cannot adapt to the changes, which results in large positioning errors. ATC provides considerable improvement due to its adaptive strategy based on RAE. MATC provides a second smallest positioning error due to a better satellite selection. RMNCE-TC provides the smallest positioning error. Table 5 shows the GDOP of different schemes at the 2050-th second. Although the GDOP of RMNCE-TC is not the smallest, its 3D positioning error is the minimum. (3) 3D Positioning Errors During [2900 s, 2960 s]  
Figure 8 presents the performances of the tested schemes when only one satellite is visible. The result shows that ATC and MATC even performs worse than STC, but RMNCE-TC still holds a better performance. Figure 9 compares the residual sequence and SOMD sequence of satellite #24. We note that the residual sequence is clearly biased during GNSS outage, which contradicts the conventional assumption that the residual sequence is zero mean white noise. Consequently, the RAE based **R** estimation becomes larger and generates harmful influence on the Kalman filter. In contrast, the SOMD sequence is more robust owing to its decoupling from the state estimation error. Hence, RMNCE-based noise estimation is more accurate than RAE. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Selected Satellites ID** & **GDOP** & **3D Positioning Error** \\ \hline STC & 10, 13, 29, 21 & 2.766 & 13.580 m \\ ATC & 15, 10, 18, 13, 29 & 2.408 & 3.115 m \\ MATC & 15, 10, 18, 21, 29 & 2.452 & 2.215 m \\ RMNCE-TC & 15, 10, 18, 21, 29 & 2.452 & 0.152 m \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Satellite selection results of the different schemes at the 740-th second.