# A Modified Residual-Based RAIM Algorithm for Multiple Outliers Based on a Robust MM Estimation
## 5 Algorithm Verification and Simulation Results  
### Simulation Conditions  
To compare the RB RAIM detectors, 24 h of GPS and BDS data were simulated. The orbit data of the 210 th day of 2020 were download from CELESTRACK with a 10\({}^{\circ}\) masking angle and the observation update rate was 10 Hz. The simulated receiver was set on the roof of the Aerospace Information Research Institute, Chinese Academy of Sciences (Lat. 40\({}^{\circ}\)4\({}^{\prime}\)12\({}^{\prime\prime}\), Lon. 116\({}^{\circ}\)16\({}^{\prime}\)12\({}^{\prime\prime}\), height 100 m). As simulation hardware an Intel Core i5-10210U CPU @2.60 GHz with 16 GB RAM was used and all simulations were performed within MATLAB 2019a. The number of visible satellites during simulation is displayed in Figure 6.  
The pseudo-range noises follow the same uncorrelated Gaussian distribution and the \(i\)th element of diagonal covariance matrices \(\sigma_{i,i}^{2}\) satisfies:  
\[\sigma_{i,i}^{2}=\sigma_{URA,i}^{2}+\sigma_{tropo,i}^{2}+\sigma_{user,i}^{2} \tag{29}\]  
where \(\sigma_{URA,i}^{2}\) is the user range accuracy (URA) of the \(i\)th satellite, which was set to 0.75 m [2]. The tropospheric delay model and the user error model are shown in Table 2.  
Four RB RAIM detectors were compared:  
* Detector 1: Baseline RB RAIM detector based on the LS estimator.
* Detector 2: RB RAIM detector based on the M estimator with IGG III function with the constant values \(k_{0}=1.5\) and \(k_{1}=3.0\).
* Detector 3: RB RAIM detector based on the MM estimator; the trimmed parameter was \(h=n-4\).
* Detector 4: RB RAIM detector proposed in this paper; the trimmed parameter was \(h=n-4\) and the termination criteria were \(n_{\sin}=6\), \(n_{mult}=12\) and \(\lambda=0.03\).  
Figure 6: The number of satellites during simulation.  
The PDOP value and the reduction of subsets to be calculated during simulation by the fast subset selection method based on the characteristic slope are shown in Figure 7. The mean value of the PDOP for the all-in-view and selected were 1.1928 and 1.3617, respectively, which increased by 14.16% after subset selection but this was still within a relatively good range. However, at this time, the number of corresponding subsets to be calculated decreased over 90%, which considerably reduced the calculation amount for the LTS estimator.  
The simulation consisted of the following three cases:  
Case 1: Each combination of two outliers was considered with a fixed bias in the first epoch of simulation.  
Case 2: A comparison of the detectors for the double outliers combination with the largest characteristic slope in each sampling epoch with a fixed \(\beta=1\).  
Assume that the \(i\)th and \(j\)th satellite are two failures and \(b_{j}=\beta b_{i}\) according to the definition of the characteristic slope:  
\[Slope_{ij}=\sqrt{\frac{\left(\mathbf{A}_{1i}+\beta\mathbf{A}_{1j}\right)^{2}+ \left(\mathbf{A}_{2i}+\beta\mathbf{A}_{2j}\right)^{2}+\left(\mathbf{A}_{3i}+ \beta\mathbf{A}_{3j}\right)^{2}}{\mathbf{S}_{ii}+(1+\beta)\mathbf{S}_{ij}+ \beta\mathbf{S}_{jj}}}. \tag{30}\]  
Case 3: A comparison of the detectors for random one, two, three and four outliers with a random bias in each epoch separately.