# 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
### Robust MM Estimation
#### 4.2.1 Least Trimmed Squares Estimator  
The LTS estimator was developed by [PERSON], which is close to ordinary least squares except that the largest weighted squared residuals are excluded from the summation to be minimized [31]. The robust LTS estimation is based on the subset of \(h\) cases (out of \(n\)) whose least squares fit possesses the smallest sum of squared residuals:  
\[\text{minimize}\left(\sum_{i=1}^{h}r_{(i)}^{2}\right) \tag{19}\]  
where \(r_{(i)}^{2}\) is the squared residuals in ascending order. Only one subset with the \(h\) smallest squared residuals would be considered so that \((n-h)\) pseudo-ranges would not be used in the robust estimation. In this way, the LTS estimation has sufficient robustness when the number of outliers does not exceed \((n-h)\).  
The LTS estimation is somewhat similar to the MHSS in principle. Given the trimmed parameter \(h\), the observation model in Equation (1) could be partitioned to distinguish the subset obtaining \(h\) observations (subscript A) from the subset, obtaining \((n-h)\) observations (subscript B):  
\[\left[\begin{array}{c}\mathbf{y}_{A}\\ \mathbf{y}_{B}\end{array}\right]=\left[\begin{array}{c}\mathbf{H}_{A}\\ \mathbf{H}_{B}\end{array}\right]\mathbf{x}+\left[\begin{array}{c}\varepsilon _{A}\\ \varepsilon_{B}\end{array}\right]+\left[\begin{array}{c}\mathbf{b}_{A}\\ \mathbf{b}_{B}\end{array}\right]. \tag{20}\]  
We obtain \(C_{n}^{h}=\frac{n!}{h!(n-h)!}\) possible subsets and obtain the corresponding state parameter vector estimation based on LS estimation by the subset with subscript A. The corresponding residuals of all \(n\) pseudo-ranges are then computed and the \(h\) smallest squared residuals are arranged in ascending order for each subset. The flow chart of the LTS estimator is shown in Figure 3.  
The scale factor estimated by the LTS estimator is  
\[\hat{\sigma}_{LTS}=\left(\sum_{i=1}^{h}|v|_{i}^{2}\right)^{1/2}. \tag{21}\]  
Figure 2: Flow chart of an RB detector based on an MM estimator.  
From Equation (21), we determine that the breakdown point of the LTS estimator is determined by \(h\). In general, the range of \(h\) is \(n/2\leq h<n\) and the breakdown point is \(h/n\), which should be set to 0.5 to achieve maximum robustness.  
We can adjust the balance between robustness and efficiency by setting an appropriate value for \(h\). Considering the requirements for the number of visible satellites in GNSS positioning:  
* At least \((m+3)\) satellites are required to complete the positioning solution. Defining the basic lower limit of the trimmed parameter is the maximum of \((n-m-3)\) and \(n/2\);
* In general, the PDOP decreases as the number of visible satellites increases. The accuracy deteriorates, which affects the effectiveness of the corresponding residuals;
* The value of the trimmed parameter should be as large as possible to avoid too many subsets to be calculated; and
* All possible multiple outlier modes need to be within the effective robust range to ensure the robustness of the estimation, which defines the upper limit of the trimmed parameter according to the integrity requirements.