# A Modified Residual-Based RAIM Algorithm for Multiple Outliers Based on a Robust MM Estimation
## 4 An RB RAIM Based on a Robust MM Estimation
### Fast Subsets Selection Based on the Characteristic Slope  
The LTS estimation requires more operations for sorting the squared residuals than MHSS, which means that LTS estimation creates a larger computational burden for the receiver especially for multiple GNSS users. When the number of visible satellites reaches a certain value, the improvement of the PDOP by adding additional satellites gradually decreases but the growth of the number of subsets to be calculated gradually increases. A fast subset selection method was designed to determine the satellite with the least contribution to the PDOP in the current set based on the characteristic slope. These satellites are eliminated until the termination condition is reached. The observation matrix is denoted by \(\mathbf{H}_{n}=\left[ \begin{array}{cc}\mathbf{h}_{1}^{\mathrm{T}}&\cdots& \mathbf{h}_{n}^{\mathrm{T}}\end{array} \right]^{\mathrm{T}}\) and \(\mathbf{h}_{i},i=1,\ldots,n\) is the row vector corresponding to each satellite. After removing the \(k\)th one of all satellites, the observation matrix is \(\mathbf{H}_{n-1}^{k}=\left[ \begin{array}{cc}\mathbf{h}_{1}^{\mathrm{T}}& \cdots&\mathbf{h}_{k-1}^{\mathrm{T}}&\mathbf{h}_{k+1}^{\mathrm{T}}&\cdots& \mathbf{h}_{n}^{\mathrm{T}}\end{array} \right]^{\mathrm{T}}\) and the relationship between \(\mathbf{H}_{n}\) and \(\mathbf{H}_{n-1}^{k}\) is:  
\[\mathbf{H}_{n}^{\mathrm{T}}\mathbf{H}_{n}=\mathbf{H}_{n-1}^{k\mathrm{T}} \mathbf{H}_{n-1}^{k}+\mathbf{h}_{k}^{\mathrm{T}}\mathbf{h}_{k}. \tag{22}\]  
Let \(\mathbf{G}_{n}=\left(\mathbf{H}_{n}^{\mathrm{T}}\mathbf{H}_{n}\right)^{-1}\); the PDOP could be calculated as:  
\[PDOP=\sqrt{\sum\limits_{i=1}^{3}\mathbf{G}_{n}(i,i)} \tag{23}\]  
and let \(\mathbf{G}_{n-1}=\left(\mathbf{H}_{n-1}^{k\mathrm{T}}\mathbf{H}_{n-1}^{k} \right)^{-1}\). The Sherman-Morrison formula provides a numerical relationship between \(\mathbf{G}_{n-1}^{k}\) and \(\mathbf{G}_{n}\):  
\[\mathbf{G}_{n-1}^{k}=\left(\mathbf{H}_{n}^{\mathrm{T}}\mathbf{H}_{n}-\mathbf{ h}_{k}^{\mathrm{T}}\mathbf{h}_{k}\right)^{-1}=\mathbf{G}_{n}+\frac{\mathbf{G}_{n} \mathbf{h}_{k}^{\mathrm{T}}\mathbf{h}_{k}\mathbf{G}_{n}}{1-\mathbf{h}_{k} \mathbf{G}_{n}\mathbf{h}_{k}^{\mathrm{T}}}. \tag{24}\]  
Let \(K_{k}=\left(1-\mathbf{h}_{k}\mathbf{G}_{n}\mathbf{h}_{k}^{\mathrm{T}}\right)\) and \(\mathbf{P}_{k}=\mathbf{G}_{n}\mathbf{h}_{k}^{\mathrm{T}}\mathbf{h}_{k}\mathbf{ G}_{n}/K_{k}\). The relationship between \(PDOP_{n}\) and \(PDOP_{n-1}\) is:  
\[PDOP_{n-1}^{k2}=PDOP_{n}^{2}+\sum\limits_{i=1}^{3}\mathbf{P}_{k}(i,i). \tag{25}\]  
The contribution of the \(k\)th satellite to the PDOP is then:  
\[\sqrt{PDOP_{n-1}^{k2}-PDOP_{n}^{2}}=\sqrt{\sum\limits_{i=1}^{3}\mathbf{P}_{k} (i,i)}=\sqrt{\frac{\sum\limits_{i=1}^{m+3}\left(\mathbf{A}_{n}(k,i)\right)^{2} }{\mathbf{S}_{n}(k,k)}}=Slope_{k}. \tag{26}\]  
Figure 4: Huber weight function. Following [PERSON], the characteristic slope [32] is defined as:  
\[Slope_{k}=\sqrt{\frac{\sum\limits_{i=1}^{3}\left(\mathbf{A}_{n}(k,i)\right)^{2}} {\mathbf{S}_{n}(k,k)}} \tag{27}\]  
The characteristic slope is an indicator used to measure the degree of leverage in RB RAIM. Equation (27) shows that it also represents the degree of contribution of each satellite to the PDOP. The matrices \(\mathbf{A}\) and \(\mathbf{S}\) here are calculated by the observation matrix \(\mathbf{H}\). Although there is an error between the matrix \(\mathbf{H}\) used here and the real value, the position error of the receiver is negligible here because the distance between the receiver and the satellite is usually tens of thousands of kilometers. In this way, we remove the satellite with the lowest characteristic slope from the set that contributes the least to the PDOP but can effectively reduce the number of subsets to be estimated in the LTS estimator. The main advantage is that we do not need to calculate the PDOP of all (n\(-\)1) satellites combinations to identify the satellite that has the least impact on the PDOP, which avoids a large number of repeated complex matrix operations.