# A Modified Residual-Based RAIM Algorithm for Multiple Outliers Based on a Robust MM Estimation
## 1 Introduction  
Integrity is a required feature for global navigation satellite system (GNSS) users. The snapshot receiver autonomous integrity monitoring (RAIM) algorithms based on consistency checks with redundant observations have been widely used especially in aviation [1]. The residual-based (RB) snapshot RAIM schemes are typical post-processing outlier detection methods. Outliers participate in the process of parameter estimation and take the sum of the squares of pseudo-range residuals as the test statistic for outlier detection [2; 3]. When a significant deviation exists between the estimated and the actual parameter, the test statistics of the RB RAIM detector can respond quickly and the calculation amount is required. RB RAIM schemes have been developed for aerial navigation where the presence of multiple simultaneous outliers is unlikely. Due to the modernization of American GPS, the full deployment of Russian GLONASS, the emergence of Chinese BDS and European Galileo and the development of regional navigation satellite systems NAVIC and QZSS, more satellites have become available, causing the possibility of double simultaneous satellite failures to be higher than the integrity risk requirement [4; 5; 6].  
For multi-constellation, the probability of three or more simultaneous faults remains an order of magnitude below the integrity requirement [7]. However, the GNSS signal can also be reflected by the multipath effect and non-line-of-sight (NLOS) receptions especially in urban city areas [8; 9]. The three or more outliers mode could not be neglected any more. In this case, traditional RB RAIM schemes are invalid as the residuals cannot be used as the basis for outlier identification.  
There are two kinds of modified methods for the detection and identification of multiple simultaneous outliers. One is a redesign of the RAIM detector and the other is the optimal parameter estimator. In this paper, we propose solutions aiming at an RB RAIM detector based on a robust estimator. The main contributions of this paper are as follows.  
(1) We propose an RB RAIM method based on the MM estimator, which contains a least trimmed squares (LTS) estimator with a high breakdown point and an M estimator with high efficiency, making residuals more consistent with the actual ranging errors. The advantages of the RB RAIM detector are preserved and the ability to detect multiple simultaneous outliers is significantly improved.  
(2) We present a fast subset selection method based on the characteristic slope. With an increasing amount of available observations, the number of subsets that an LTS estimator needs to estimate increases dramatically. The efficiency of robust estimation is improved as much as possible by removing the satellites that make little contribution to the geometry while ensuring a better position dilution of precision (PDOP) condition.  
The remainder of this paper is organized as follows. The related works on RAIM for a multi-outlier mode are given in Section 2. The camouflage effect of an RB RAIM in a multiple outlier mode is analyzed in Section 3. The proposed RB RAIM based on MM estimation and the fast subset selection method are detailed in Section 4. Section 5 describes the experiments used to demonstrate the practical utility of our approach. Finally, the discussion and conclusions are provided in Section 6.