## 4 Experiment Setup and Result Analysis
### Result Analysis
#### 4.2.1 Simulation Test
Figure 7: Position errors in 3D direction corresponding to gradual fault: (**a**) before using a robust algorithm; (**b**) after using a robust algorithm. in Figure 7a, it is the incorrect positioning results that lead to inaccurate code phases. In Figure 8b, the code phase errors of eight satellites always stay normal even after fault injection. The propagation of fault information is successfully suppressed by the robust algorithm. In addition, since the fault injection object is a pseudo-range error rather than a code phase error, the code phase of satellite 14 is not contaminated by fault. On the contrary, this code phase is accurate owing to the prediction based on corrected navigation solution. In Figure 9a, there are four satellites, whose detection results exceed the alarm threshold due to the fault propagation problem. The results make it difficult not only to determine the number of fault satellites, but also to distinguish the actual fault satellite. In Figure 9b, only the detection values of satellite 14 gradually exceed the alarm threshold after 150 s. Thus, it is easy to confirm that satellite 14 is the only fault satellite. For further comparison, the Monte Carlo test is carried out. The accuracy of fault identification is increased from 77% to 98% after improvement. It indicates that employing a robust algorithm is beneficial to identify the gradual fault. The second test is to inject the abrupt fault into the collected data. In order to facilitate the comparison with a gradual fault, satellite 14 is still the object of fault injection. The injection method is to add an outlier of 30 m in the pseudo-range error from 150 s.  
In Figure 10a, the position errors in a three-dimensional direction quickly deviate after fault injection. However, in Figure 10b, there is no great deviation except for a surge in the period of 151 s  
Figure 8: Code phase errors of eight satellite signals corresponding to gradual fault: (**a**) before using a robust algorithm; (**b**) after using a robust algorithm. Figure 9: The graph of fault identification statistics corresponding to gradual fault: (**a**) before using a robust algorithm; (**b**) after using a robust algorithm. to 154 s. By comparing the two figures, it demonstrates that the proposed robust algorithm is also effective in dealing with the abrupt fault. Even though the fault type changes, the robust algorithm based on CNN is still superior to that based on Huber by comparing the variances shown in Table 4. The proposed algorithm that doesn't need any adjustment can output the desired factor adaptively according to the changes of fault, while the performance of Huber is restricted by its parameter. However, since the adjustment of the robust algorithm lags behind the rapid occurrence of abrupt fault, its variance is much larger than that without fault. As shown in Figure 11a, different deviations occur instantly in the code phase errors of all eight satellites after fault injection. The result proves once again that the fault propagation problem does exist in deep integration. Compared with the gradual fault, the abrupt fault is exposed faster and spreads faster to the other satellites. In Figure 11b, there is no deviation, proving that the fault propagation problem is solved by using the proposed robust algorithm. \begin{table}
\begin{tabular}{c c} \hline \hline
**Method** & **Variance (m)** \\ \hline No fault & 1.802 \\ CNN & 3.157 \\ Huber (\(d=1\)) & 3.606 \\ Huber (\(d=1.35\)) & 3.411 \\ Huber (\(d=1.5\)) & 3.725 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Variance of position error corresponding to abrupt fault. Figure 10: Position error in 3D direction corresponding to abrupt fault: (**a**) before using a robust algorithm; (**b**) after using a robust algorithm. In Figure 12a, not only do the detection values of four satellites quickly exceed the alarm threshold, but also the time of exceeding the threshold is very close. These results make it difficult to identify the actual fault satellite, thus increasing the probability of misjudgment.