# Performance Analyses of a RAIM Algorithm for Kalman Filter with GPS and NavIC Constellations
## 6 Simulation Studies
### Fault Detection Performance
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Fault Type**} & \multicolumn{2}{c}{**SKF**} & \multicolumn{2}{c}{**Full-State EKF**} & \multirow{2}{*}{**Change in PE**} \\  & \multicolumn{2}{c}{**Fault Detection Time**} & & & **If PE \(>\) PL** & **Fault Detection Time** \\  & **1 sat (N2)** & **2 sat (G28 and N2)** & & **1 sat (N2)** & **2 sat (G28 and N2)** \\ \hline \(-5\) m step & No & No & No; constant offset & 100 s & 100 s & Constant offset \\  & detection & detection & 1 sat: PE \(<\) 5 m & & 1 sat: PE \(<\) 6.5 m \\  & & & 2 sat: PE \(<\) 10 m & & 2 sat: PE \(<\) 8.55 m \\ \multirow{2}{*}{\begin{tabular}{c} Ramp \\ with slope \\ \(-0.05\) m/s \\ \end{tabular} } & \multirow{2}{*}{563 s} & \multirow{2}{*}{614 s} & \multirow{2}{*}{1 sat: from 1100 s} & \multirow{2}{*}{\begin{tabular}{c} No \\ 2 sat: from 661 s \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} No \\ 2 sat: from 661 s \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} No \\ 2 sat: from 661 s \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} No \\ 2 sat: from 661 s \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} No \\ 2 sat: from 661 s \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} No \\ 2 sat: PE \(<\) 10 m \\ \end{tabular} } \\ \hline \multirow{2}{*}{\
\begin{tabular}{c} Sinusoidal \\ 5\(\sin{(2\pi t/3600)}\) m \\ \end{tabular} & No & No & No; sinusoidal PE & 1 sat: PE \(<\) 5 m \\  & detection & detection & 1 sat: PE \(<\) 5 m & & 1 sat: PE \(<\) 6.5 m \\ \multirow{2}{*}{
\begin{tabular}{c} Sinusoidal \\ 5\(\cos{(2\pi t/3600)}\) m \\ \end{tabular} } & No & No & No; sinusoidal PE & 1 sat: PE \(<\) 5 m \\  & detection & detection & 2 sat: PE \(<\) 10 m & & 2 sat: PE \(<\) 10 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fault detection performance of SKF and full-state EKF. Fault detection time is calculated from the beginning of simulation. PE stands for position error. It can be horizontal or vertical PE. The corresponding PL is HPL or VPL. Maximum of horizontal PE and VPE is noted under PE. “sat” stands for satellite. Next, it is checked if the SKF test statistics can detect step faults in G28 and N2, for which the VPE just crosses the VPL in different simulation runs and at different time instants. It is observed that its first statistic \(\alpha_{1}\) exhibits a sharp peak at the onset time of faults. The magnitude of the peak does cross the threshold and is near 45-50, resulting in instant detection. Similar analysis is also performed with WLS. It detects the corresponding step faults within 10 s of PE crossing the PL during initial aircraft descent and before the crossing afterwards. Thus, detection occurs within the required time to alert [41]. The same observation is made with cosine faults. Table 1 indicates that the full-state EKF does not detect the ramp faults, but its position error grows with time. With dual faults, the position error increases faster. Although faults were introduced into two satellites of GPS and NavIC--one each--as they maximize the FMS for mean VPE, it is unlikely that satellites from different constellations would fail simultaneously. Therefore, it is investigated what would occur when two satellites of NavIC fail. With NavIC being a new constellation, such a possibility cannot be ruled out. It is noted that if N2 and N6 fail at 10 s into simulation, both with a ramp fault of \(-0.05\) m/s, the EKF VPE crosses the vertical alert limit during the final approach phase without any detection. This is illustrated in Figure 12. The performance of KF RAIM with SKF is also shown for comparison purposes. The EKF test statistic does not display any abnormal behavior because the filter confuses such a slowly growing fault with a residual measurement error that it is supposed to estimate. SKF can detect such faults because it does not estimate measurement errors, but treats them as \"consider parameters\" instead. Missed detection with full-state EKF is also observed in all of 100 different simulation runs.