# A Modified Residual-Based RAIM Algorithm for Multiple Outliers Based on a Robust MM Estimation
## 6 Discussion and Conclusions  
In this paper we first analyzed the residuals that could not be used as an effective basis for outlier identification when there were multiple simultaneous outliers, which is called the camouflage effect of residuals. In this case, replacing the LS estimator by the M estimator could not effectively improve the identification rate of multiple outliers as the robustness of the M estimation is built on the basis of a relatively reliable iterative initial value. We proposed an RB RAIM algorithm based on the MM estimator, which contained an LTS estimator with a high breakdown point and an M estimator with a Huber function with high efficiency. We designed a fast subset selection method based on the characteristic slope to improve the practicability of the algorithm as the computation amount of the LTS estimator increased sharply with the increase in the number of visible satellites. The simulated experimental results demonstrated that the proposed RB RAIM detector based on the MM estimator maintained a more robust performance than the LS estimator and the M estimator with an IGG III function especially with the increase of the number of outliers. The fast subset selection method based on the characteristic slope reduced the calculation time by more than 80%, which indicated the practical application value of the algorithm. Notably, categories (d) and (e) could be regarded as a correct exclusion; however, a high amount of category (d) was generally undesirable as it could lead to the positioning solution being unavailable if the number of satellites was small. The exclusion of normal observations was less than desirable because all normal observations would not be used. Although the probabilities of categories (d) and (e) were still more than 90%, with the increase of outliers, the actual detection effect of detector 3 and 4 gradually deteriorated. However, this became less of a concern as the number of visible satellites increased. Future work can focus on further reducing the computational burden. The subset selection process is not necessary in each epoch so we can design a judgment threshold to determine the validity period of each selected subset. In addition, the inertial information of the receiver can be used to provide support to determine the reliable initial iteration value for a robust estimation [33]. However, this may cause a new potential integrity risk, which requires further research and experimentation. **Author Contributions:** Conceived the main idea, W.W.; conceptualization, W.W. and Y.X. ; original draft preparation, W.W.; review and editing, Y.X. Both authors have read and agreed to the published version of the manuscript. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by \"Youth Innovation Promotion Association, Chinese Academy of Science\" (No. Y50301A1 BY). **Acknowledgments:** The authors greatly thank the reviewers' suggestions and comments to improve the manuscript. **Conflicts of Interest:** The authors declare no conflict of interest. \begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline  & \multicolumn{4}{c}{**One Outlier**} & \multicolumn{4}{c}{**Two Outliers**} \\ \hline  & (a) & (b) + (c) & (d) + (e) & (e) & (a) & (b) + (c) & (d) + (e) & (e) \\ Detector 1 & 4\% & 0\% & 96\% & 96\% & 0\% & 7\% & 93\% & 91\% \\ Detector 2 & 1\% & 0\% & 99\% & 99\% & 0\% & 2\% & 98\% & 98\% \\ Detector 3 & 0\% & 0\% & 100\% & 99\% & 0\% & 1\% & 99\% & 99\% \\ Detector 4 & 0\% & 0\% & 100\% & 99\% & 0\% & 1\% & 99\% & 98\% \\ \hline  & \multicolumn{4}{c}{**Three Outliers**} & \multicolumn{4}{c}{**Four Outliers**} \\ \hline  & (a) & (b) + (c) & (d) + (e) & (e) & (a) & (b) + (c) & (d) + (e) & (e) \\ Detector 1 & 0\% & 14\% & 86\% & 71\% & 0\% & 29\% & 71\% & 43\% \\ Detector 2 & 0\% & 6\% & 94\% & 91\% & 0\% & 13\% & 87\% & 69\% \\ Detector 3 & 0\% & 2\% & 98\% & 97\% & 0\% & 4\% & 96\% & 92\% \\ Detector 4 & 0\% & 2\% & 98\% & 95\% & 0\% & 4\% & 96\% & 91\% \\ \hline \hline \end{tabular}
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