# 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  
The mean position error, \(\underline{\mu}_{k}^{\prime}\), of the navigation filter at epoch \(t_{k}\) in the local navigation frame can be written as  
\[\underline{\mu}_{k}^{\prime}=-\underbrace{\sum_{j=N-1}^{k-(m+1)}\mathbf{\mathsf{ B}}_{k-j}^{\prime}\underline{f}_{k-m-j}}_{\text{Ignored terms}}-\left(\sum_{j=0}^{N-2}\mathbf{\mathsf{B}}_{k-j}^{\prime} \underline{f}_{k-m-j}+\mathbf{K}_{s,\ k}^{\prime}\underline{f}_{k-m+1} \right) \tag{7}\]  
where \(\mathbf{K}_{s,\ k}^{\prime}=\mathbf{C}_{e}^{n}\mathbf{K}_{s,\ k}\left(1:2:5,:\right)\) and \(\mathbf{\mathsf{B}}_{k-j}^{\prime}=\mathbf{C}_{e}^{n}\mathbf{B}_{k-j}\left(1:2:5,:\right)\). \(\mathbf{B}_{k-j}=\left[\prod_{\ell=k}^{k-j}\mathbf{A}_{\ell}\right]\mathbf{K}_ {s,\ k-j-1}\), and \(\mathbf{A}_{\ell}=\left(\mathbf{I}-\mathbf{K}_{s,\ \ell}\mathbf{C}_{s,\ \ell}\right)\mathbf{F}_{s,\ \ell-1}\). \(\mathbf{F}_{s,\ \mathbf{C}_{s}}\), and \(\mathbf{K}_{s}\) are the state transition, measurement model, and Kalman gain matrices of SKF for the estimated states. For a matrix \(\mathbf{J},\mathbf{J}\left(1:2:5,:\right)\) comprises the first, third, and fifth rows of \(\mathbf{J}\). In \(\mathbf{B}_{k-j}\) and \(\mathbf{K}_{s,\ k}\), these rows belong to the position states of the filter. \(\mathbf{C}_{e}^{n}\) is the coordinate transformation matrix from ECEF to the north-east-down (NED) frame. The two test statistics, \(\alpha_{1}\) and \(\alpha_{2}\), together have terms of \(N\) recent epochs. Terms from other previous epochs are ignored. An upper bound of the contributions from these terms to the mean position error is derived mathematically. Fault is represented by a vector \(\underline{f}_{k-m+1}\) at \(t_{k}\); it started at epoch \(m\). Three or more simultaneous faults are not required to be monitored for the integrity risk considered. If \(s_{1}\) and \(s_{2}\) represent any two faulty satellite indices, then \(\underline{f}_{k-m+1}(s_{1})=b_{1,\,k-m+1},\underline{f}_{k-m+1}(s_{1}+n)= \dot{b}_{1,\,k-m+1}\), \(\underline{f}_{k-m+1}(s_{2})=b_{2,\,k-m+1}\), and \(\underline{f}_{k-m+1}(s_{2}+n)=\dot{b}_{2,\,k-m+1}\), with all other elements being zero. For single fault (SF) cases, only the \(s_{1}th\) satellite has a fault. The mean error bound of the vertical position for \(\alpha_{1}\) under a fault mode \(i\) (which can be SF or dual faults (DF), with \(i=1,...,\ N_{fault}\)), \(VPE_{1,\,i}^{U}\) is obtained as follows. For a fault mode, the square of the maximum FMS [25] for \(\alpha_{1}\) (having \(M\) recent epoch terms from \(t_{k}\) to \(t_{k-M+1}\)) is  
\[\text{(maximum \ FMS)}_{\alpha_{1},\,i}^{2}=\max_{\underline{f}_{w}^{T} \underline{\Gamma}_{w}^{T}\underline{\Gamma}_{w}} \tag{8}\]  
The FMS is maximized with the worst-case fault vector, \(\underline{f}_{w}\). \(\mathbf{L}\) is formed with the appropriate elements of \(\mathbf{B}^{\prime}\) matrices over \(M\) recent epochs for a fault mode. \(\underline{f}_{w}\) constitutes the corresponding \(b\) and \(\dot{b}\) terms. \(\mathbf{\Lambda}\) is  
\[\mathbf{\Lambda}= \begin{bmatrix}\mathbf{\Theta}_{3}&[\mathbf{0}]\\ [\mathbf{0}]&\mathbf{\Theta}_{4}\end{bmatrix}  \tag{9}\]  
where \([\mathbf{0}]\) denotes a matrix of all zero entries. \(\mathbf{\Theta}_{3}\) is determined with the terms of Equation (4) corresponding to a fault mode. \(\mathbf{\Theta}_{4}\) is obtained from the second term of Equation (6) on the right. It is a diagonal matrix for SF modes and a block diagonal matrix with each block of size \(2\times 2\) for DF modes. A computationally efficient method is used to calculate the inverse of each block of \(\mathbf{\Theta}_{4}\) without inverting it.