# Performance Analyses of a RAIM Algorithm for Kalman Filter with GPS and NavIC Constellations
## 7 Conclusions  
This paper integrates realistic error models of the carrier-smoothed pseudorange measurements into the RAIM algorithm with an SKF navigation filter. Error models include appropriate time constants of individual error sources, thus overcoming an important limitation of the prior work on KF RAIM. Uncertainties of the model parameters are accounted for. A provision is also made for adding artificial white noise to deal with initial transients in the position error. The suitability of the adopted models for RAIM with SKF is justified through extensive simulations. Performance of the modified KF RAIM is then analyzed with simulated GPS L1 and NavIC L5 signals for aircraft navigation during different phases of flight. The following important conclusions are drawn from the analyses of the results. WLS RAIM has lower PLs than the developed KF RAIM, but it is not appropriate for advanced navigation systems such as vector tracking and sensor integration, where KF is an integral part. A full-state EKF mistakes slow ramp faults for the residual measurement errors it needs to estimate, which results in missed detection. An SKF implementation, on the other hand, can successfully detect such faults. Hence, it offers a promising solution to developing KF RAIM algorithms in the range domain. With realistic error models, the modified KF RAIM is shown to complete processing well within time. Some salient features are also studied to gain insights into its working principles. It is noted that the mean position error bound is mainly determined by a few recent epoch measurements. However, contributions of all epochs must be taken into account for rigorous calculation of the PL for high-reliability applications. Future work aims at determining error model parameters specific to NavIC measurements, and subsequently validating the KF RAIM with real GPS and NavIC signals and different geometries. Constellation-wide faults need to be handled in the range-based formulation. Furthermore, a fault isolation methodology is required to successfully exclude faulty satellites upon detection. The performance of KF RAIM in weak signal environments will be studied. Its suitability for sensor integration applications will be explored. As ionospheric scintillation prevalent in low latitude regions such as the Indian subcontinent can degrade GNSS measurements and introduce non-Gaussian errors [42; 43], it is also important to analyze KF RAIM performance in such scenarios. **Supplementary Materials:** The following are available online at [[https://www.mdpi.com/article/10.3390/s21248441/s1](https://www.mdpi.com/article/10.3390/s21248441/s1)]([https://www.mdpi.com/article/10.3390/s21248441/s1](https://www.mdpi.com/article/10.3390/s21248441/s1)), matlab_program_figures_1_2.m: Matlab program to reproduce Figures 1 and 2 and simulations described at the end of Section 3; matlab_program_algorithm_1_ts1_figure_3.m: Matlab program for Algorithm A1 in Appendix A for \(\alpha^{\prime}_{1}\) and subplots 1 and 2 of Figure 3; matlab_program_algorithm_1_ts2_figure_3.m: Matlab program for Algorithm A1 in Appendix A for \(\alpha^{\prime}_{2}\) and subplots 3 and 4 of Figure 3; matlab_program_appendix_figure_m: Matlab program to reproduce the first three subplots of Figure A1; aircraft_trajectory_long_duration.png: 54.48 min aircraft trajectory \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Fault Slope (m/s)**} & \multicolumn{2}{c}{**Full-State EKF**} & \multicolumn{2}{c}{**SKF**} \\  & **Fault Detection Time (s)** & **Time When VPE \(>\) alert Limit (s)** & **Fault Detection Time (s)** & **Time When VPE \(>\) VPL (s)** \\ \hline \(-0.5\) & 1055 & \(\geq\)975 & 925 & \(\geq\)999 \\ \(-0.75\) & 937 & \(\geq\)948 & 914 & \(\geq\)963 \\ \(-1\) & 904 & \(\geq\)934.8 & 909 & \(\geq\)946 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Ramp fault detection performance of full-state EKF and SKF with different slopes. Time is noted from the beginning of simulation.