# A Modified Residual-Based RAIM Algorithm for Multiple Outliers Based on a Robust MM Estimation
## 2 Related Works  
Over the past decade, scholars have done some related work on a modified RAIM algorithm for a multi-outlier mode. The multiple hypothesis solution separation (MHSS) algorithm detects outliers via comparing the position estimate made with all satellites in view with estimates from subsets that have removed some hypothetical outliers [10; 11; 12; 13; 14]. The range consensus (RANCO) algorithm calculates position solutions based on the best four-satellite subset and compares this estimate with the pseudo-ranges of all the satellites not contributing to this solution [15; 16]. The core idea of this kind of method is to find the best subset that does not contain outliers by relying on a combinatorial search, which could be intractable as the number of visible satellites and hypothetical outliers increases [17; 18; 19]. [PERSON] et al. [17] modified the MHSS algorithm using a fixed set of subsets to reduce the computational load. [PERSON] et al. [18] presented a modified method that evaluated multiple outlier cases using subsets that excluded entire constellations to narrow the search range for outliers. Other approaches neglect many of the less likely scenarios to simplify the RAIM detector based on the MHSS algorithm [19; 20]. These methods only consider outliers caused by satellite failure; in that scenario, the probability of three or more simultaneous outliers is lower than the integrity risk requirement, which can be ignored. The computational burden of these methods increases dramatically as the number of simultaneous outliers increases.  
[PERSON] et al. [21] designed a detector based on a non-least-squares (NLS) estimator to reduce the integrity risk at the cost of lower accuracy. The concept aimed to improve the combined availability of accuracy and integrity. [PERSON] and [PERSON] [22] proposed NIORAIM to better balance accuracy and integrity. The weights were designed both in the position domain and in the measurement domain. [PERSON] et al. [23] proposed the correlation-weighted least squares residual (CW-LSR) algorithm in which the pseudo-range residuals were weighted with the characteristic slope, making it approximate to the optimal test statistic. However, the characteristic slope was valid only in a single outlier mode. The robust estimators could also improve the robustness and effectiveness of residuals although the accuracy was not optimal. [PERSON] [24] presented an RB RAIM detector based on a robust M estimation with an IGG III function for multiple outlier detection and exclusion, which could also control the influence of near-failure observations. However, the robustness of the M estimation built on the relatively reliable iterative initial value, which was generally the LS estimation, thus having a breakdown point tending to zero especially when the number of simultaneous outliers increased [25; 26; 27]. [PERSON] and [PERSON] [28] compared the outlier test and robust methods and identified MM estimators [29; 30] and the L1 norm obtained the highest rates of normal exclusion as the number of outliers increased. Thus, it is the objective of this paper to improve the ability of an RB RAIM to detect and identify simultaneous outliers based on MM estimation.