# A Modified Residual-Based RAIM Algorithm for Multiple Outliers Based on a Robust MM Estimation
## 3 Camouflage Effect of an RB RAIM Detector in Multi-Outlier Mode  
### Baseline of an RB RAIM Detector  
This section introduces notations describing the well-established least squares (LS) estimator and residual-based (RB) RAIM detector and analyzes the camouflage effect of pseudo-range residuals, which is the main factor in reducing the fault detection and exclusion (FDE) performance of an RB RAIM detector in a multiple outlier mode. Let \(m\) and \(n\) denote the number of states to be estimated and visible satellites, respectively. Assume the linearize functional model is [8]:  
\[\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{\varepsilon}+\mathbf{b} \tag{1}\]  
where \(\mathbf{y}\) is an \(n\times 1\) vector of observations containing the differences between the expected ranging values and the raw ones to all visible satellites, \(\mathbf{H}\) is an \(n\times(m+3)\) linear observation matrix, \(\mathbf{x}\) is an \((m+3)\times 1\) state vector that obtains \(m\) clock offsets and 3-dimensional (3D) positions in the local Cartesian coordinate (ENU), \(\mathbf{b}\) is an \(n\times 1\) bias vector that is a zero vector when there is no outlier among the observations and \(\mathbf{\varepsilon}\) is an \(n\times 1\) observation error vector, which is assumed to be normally distributed with the zero-mean and covariance matrix \(\mathbf{D}\)[8]:  
\[\mathbf{\varepsilon}\sim N(0,\mathbf{D}) \tag{2}\]  
where \(\mathbf{0}\) is an \(n\times 1\) zero vector and \(\mathbf{D}\) is assumed to be diagonal. The probability distribution of nominal observation noise can be reliably modeled using large amounts of experimental data and bounded by the Gaussian function described in Equation (2). The LS solution for the state vector \(\mathbf{x}\) in Equation (1) is:  
\[\mathbf{\hat{x}}=\left(\mathbf{H}^{\mathrm{T}}\mathbf{W}\mathbf{H}\right)^{- 1}\mathbf{H}^{\mathrm{T}}\mathbf{W}\mathbf{y},\mathbf{W}=\mathbf{D}^{-1}. \tag{3}\]  
The correction of the state parameter vector is:  
\[\delta\mathbf{\hat{x}}=\mathbf{\hat{x}}-\mathbf{x}=\left(\mathbf{H}^{\mathrm{ T}}\mathbf{W}\mathbf{H}\right)^{-1}\mathbf{H}^{\mathrm{T}}\mathbf{W}(\mathbf{ \varepsilon}+\mathbf{b}). \tag{4}\]  
Let \(\mathbf{A}=\left(\mathbf{H}^{\mathrm{T}}\mathbf{W}\mathbf{H}\right)^{-1} \mathbf{H}^{\mathrm{T}}\mathbf{W}\). We obtain \(\delta\mathbf{\hat{x}}=\mathbf{A}(\mathbf{\varepsilon}+\mathbf{b})\). The pseudo-range residual vector \(\mathbf{r}\) is:  
\[\mathbf{r}=\mathbf{y}-\mathbf{H}\mathbf{\hat{x}}. \tag{5}\]  
Here, we take a single GNSS as an example and the number of state parameters to be estimated should be added accordingly for multiple GNSSs. By substituting Equation (4) into Equation (5), the residual vector can be expressed as:  
\[\mathbf{r}=(\mathbf{\varepsilon}+\mathbf{b})-\mathbf{H}\mathbf{\hat{x}}=( \mathbf{\varepsilon}+\mathbf{b})-\mathbf{H}\mathbf{\left(H}^{\mathrm{T}} \mathbf{W}\mathbf{H}\right)^{-1}\mathbf{H}^{\mathrm{T}}\mathbf{W}(\mathbf{ \varepsilon}+\mathbf{b})=\left(\mathbf{I}-\mathbf{H}\mathbf{\left(H}^{\mathrm{ T}}\mathbf{W}\mathbf{H}\right)^{-1}\mathbf{H}^{\mathrm{T}}\mathbf{W}\right)( \mathbf{\varepsilon}+\mathbf{b}). \tag{6}\]  
Let \(\mathbf{S}=\mathbf{I}-\mathbf{H}\mathbf{\left(H}^{\mathrm{T}}\mathbf{W} \mathbf{H}\right)^{-1}\mathbf{H}^{\mathrm{T}}\mathbf{W}\) and \(\mathbf{r}=\mathbf{S}\cdot(\mathbf{\varepsilon}+\mathbf{b})\). The residual vector \(\mathbf{r}\) can be regarded as the projection of the error vector \((\mathbf{\varepsilon}+\mathbf{b})\) through the symmetric idempotent matrix \(\mathbf{S}\). The residual vector is used as a substitute for FDE because the error vector \((\mathbf{\varepsilon}+\mathbf{b})\) is unknown.