# Performance Analyses of a RAIM Algorithm for Kalman Filter with GPS and NavIC Constellations
## 2 Overview of Existing KF RAIM
### Protection Levels (PLs)
#### 2.2.1 Mean Position Error Bounds
If \(\lambda_{\mathbf{L}\mathbf{\Lambda}^{-1}}^{\max}\) is the maximum eigenvalue of \(\mathbf{L}\mathbf{\Lambda}^{-1}\),  
\[\text{VPE}_{1,\,i}^{U}=\sqrt{\lambda_{\mathbf{L}\mathbf{\Lambda}^{-1}}^{\max} \Gamma} \tag{10}\]  
where \(\Gamma\) is calculated as follows:  
\[(T_{th,\,1}^{*})^{2}=\frac{\lambda_{\mathbf{\Theta}}^{\max}}{(\lambda_{ \mathbf{\Theta}}^{\min})^{2}}T_{th,\,1}^{2};\sqrt{\Gamma^{*}}=\frac{1}{\sqrt{ \lambda_{\mathbf{\Theta}}^{\max}}}(\sqrt{\Gamma}-T_{th,\,1})+T_{th,\,1}^{*} \tag{11}\]  
where \(\lambda_{\mathbf{\Theta}}^{\min}\) and \(\lambda_{\mathbf{\Theta}}^{\max}\) are the minimum and maximum eigenvalues of \(\mathbf{\widetilde{\Theta}}\). The block diagonal matrix, \(\mathbf{\widetilde{\Theta}}\), has two blocks. The first one is \(\mathbf{\Theta}_{2}\) (see Equation (5)), and the second one is the identity matrix. Hence, \(\lambda_{\mathbf{\Theta}}^{\max}=1\), and \(\lambda_{\mathbf{\Theta}}^{\min}=\lambda_{\mathbf{\Theta},\,2}^{\min}\) (minimum eigenvalue of \(\mathbf{\Theta}_{2}\)), which is shown to be equal to the minimum eigenvalue of \(\mathbf{\Omega}\), \(\lambda_{\mathbf{\Omega}}^{\min}\). It is also proved that all eigenvalues of \(\mathbf{\Omega}\) are real and positive. First, \(\Gamma^{*}\) is determined such that the probability of the non-central chi-squared distribution \(P_{ncr}((T_{th,\,1}^{*})^{2},\text{DOF},\Gamma^{*})=P_{MD,\,i}^{alloc}\); DOF stands for the required degrees of freedom. \(P_{MD,\,i}^{alloc}\) is the allocated probability of missed detection for the \(i\)th fault mode. Then, \(\Gamma\) is calculated from Equation (11) with \(\lambda_{\mathbf{\widetilde{\Theta}}}^{\max}=1\) and saved in a lookup table for different values of DOF, \(P_{MD,\,i}^{alloc}\), and \(1/\lambda_{\mathbf{\widetilde{\Theta}}}^{\min}\). Following [30], it is also justified that \(\Gamma\) increases as \(\lambda_{\mathbf{\widetilde{\Theta}}}^{\min}\) decreases. \(\Gamma\) for \(\alpha_{1}\) is obtained from the lookup table for a fault mode, the next entry of \(1/\lambda_{\mathbf{\widetilde{\Theta}}}^{\min}\) higher than the calculated value, and the required DOF. \(VPE_{2,\,i}^{U}\) for \(\alpha_{2}\) is calculated similarly to that of \(VPE_{1,\,i}^{U}\) but using a more computationally efficient approach. This is because it involves terms of \((N-M)\) epochs, often found to be greater than \(M\). \(VPE_{3}^{U}\) is the mean vertical position error (VPE) upper bound for the ignored terms and the same for all fault modes. It is  
\[VPE_{3}^{U}=U_{\text{max}}N_{\text{max}}\times\frac{1\times 10^{-6}}{1-0.1}( \hat{b}_{\text{max}}+\hat{b}_{\text{max}}) \tag{12}\]  
In order to have the north and east components, the factor \(1\times 10^{-6}\) is changed to \(\sqrt{2}\times 10^{-6}\) for the horizontal position. The maximum of all \(N\)s up to the current epoch \(t_{\text{g}}\) is denoted as \(N_{\text{max}}\). \(U_{\text{max}}\) is an upper bound obtained in a computationally efficient way at each epoch [26]. The maximum bias \(\hat{b}_{\text{max}}\) at epoch \(k\) is given as  
\[\Big{(}\min_{\forall\ell=1,\dots,k\in\mathbb{1},\ \Theta_{2}}\min_{i\in \text{fault modes}}\lambda^{\text{min}}_{\boldsymbol{\Theta},\ 3}\Big{)}\hat{b}_{\text{max}}^{2}=\Gamma_{\text{max}} \tag{13}\]  
where \(\lambda^{\text{min}}_{\boldsymbol{\Theta},\ 3}\) is the minimum eigenvalue of \(\boldsymbol{\Theta}_{3}\). Operation \"min\" represents the minimum of all terms underneath it. \(\Gamma_{\text{max}}\) is the maximum of all calculated \(\Gamma\)s. The maximum bias rate, \(\hat{b}_{\text{max}}\), can be determined in a similar way with the minimum eigenvalue of \(\boldsymbol{\Theta}_{4}\). \(\boldsymbol{\Theta}_{4}\) is a block-diagonal matrix, with a block having a dimension of one or two. It can be shown that for each two-dimensional block, the minimum eigenvalue is more than the ratio of its determinant to the sum of the diagonal elements. This lower bound of the minimum eigenvalue is used to find \(\hat{b}_{\text{max}}\). Next, the PL equations are briefly mentioned.