# Performance Analyses of a RAIM Algorithm for Kalman Filter with GPS and NavIC Constellations
## 7 Conclusions
This is because \(\mathbf{\Psi}_{\text{sub, max}}^{i}\) is % pre-determined for \(\alpha_{2}^{\prime}\) for different \((N-M)\) for its relatively large size, and % loaded in the RAIM algorithm, while \(\mathbf{\Psi}_{\text{sub, max}}^{i}\) for \(\alpha_{1}^{\prime}\) is recursively % calculated at each time epoch % Calculate \(\Delta\mathbf{\Psi}~{}=~{}(\mathbf{\Psi}_{\text{max}}~{}-~{}\mathbf{\Psi})\) and minimum eigenvalue of \(\mathbf{D}\) for \(\alpha_{1}^{\prime}\), % where \(\mathbf{D}=(c\mathbf{I}~{}+~{}\Delta\mathbf{\Psi})\mathbf{\Psi}^{-1}\) (see the last paragraph of Section 4.3 and % Appendix B) % Form \(\mathbf{H}\) matrix. Pick three components of each satellite LOS vector from a % uniform distribution between -1 and 1 such that the sum of their squares is % one. Since geometry changes slowly, keep \(\mathbf{H}\) constant over the number of % epochs % Form \(\mathbf{\hat{S}}^{\prime}\) (see after Equation (4)) % Calculate \(c\) adaptively for \(\alpha_{1}^{\prime}\) so that the maximum eigenvalue of \(\mathbf{\Omega}_{1}\), \(\lambda_{\mathbf{\Omega},1}^{\text{max}}\) % \(<1\) % Alternatively, calculate \(c\) as \((\lambda_{\mathbf{\Psi},\text{max}}^{\text{max}}-\lambda_{\mathbf{\Psi},\text{max}}^{ \text{min}})\) for \(\alpha_{2}^{\prime}\) (see after % Equation (6)) % Calculate \(\mathbf{\Theta}\) as \((c\mathbf{I}~{}+~{}\mathbf{\Psi}_{\text{max}})^{-1}\) % Calculate the minimum and maximum eigenvalues of \(\mathbf{\Omega}_{1}\) and \(\mathbf{\Omega}\) % Check if the maximum eigenvalue of \(\mathbf{\Omega}\), \(\lambda_{\mathbf{\Omega}}^{\text{max}}\) is \(\leq\lambda_{\mathbf{\Omega},1}^{\text{max}}\) % Check if the minimum eigenvalue of \(\mathbf{\Omega}\), \(\lambda_{\mathbf{\Omega}}^{\text{min}}\) is greater than or equal to % that of \(\mathbf{\Omega}_{1}\times\mathcal{K}_{\text{min}}\) (for \(\alpha_{1}^{\prime}\)) (and greater than or equal to \(\lambda_{\mathbf{\Omega},1}^{\text{min}}\times\frac{\mathcal{K}_{\text{min}}}{( \mathcal{K}_{\text{wh}}~{}+~{}1)}\) % for \(\alpha_{2}^{\prime}\) ) % Check if \(\lambda_{\mathbf{\Omega}}^{\text{min}}\) is greater than zero, and if \(\mathbf{\Omega}\) and \(\mathbf{\Omega}_{1}\) are full rank matrices % If any of the previous four conditions is violated, display a message and % halt end % End of simulation loop % Plot results of a batch end % End of batch loop
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**Algorithm A1:**\(Cont\). ## Appendix B Determination of \(c_{\text{min}}\) for \(\alpha_{1}^{\prime}\)

During the adaptive calculation of \(c\), a minimum value, \(c_{\text{min}}\) is needed to ensure that \(\lambda_{\mathbf{\Omega}}^{\text{max}}\) is less than \(\lambda_{\mathbf{\Omega},1}^{\text{max}}\) and one, where \(\mathbf{\Omega}=\mathbf{\tilde{S}}^{\prime T}\mathbf{\Psi}\mathbf{\tilde{S}}^{\prime S}\mathbf{ \Omega}\mathbf{\tilde{S}}^{\prime}\) and \(\mathbf{\Omega}_{1}=\mathbf{\tilde{S}}^{\prime T}\mathbf{\Psi}_{\text{max}}\mathbf{\tilde{S}} ^{\prime}\mathbf{\tilde{S}}^{\prime}\mathbf{\Omega}\mathbf{\tilde{S}}^{\prime}\). \(c_{\text{min}}\) is considered to be the initial guess of \(c\). It is determined in this section. First, the modeled \(a_{\ell q}\) of Equation (20) form the elements of \(\mathbf{\Psi}\) (or sub-matrix \(\mathbf{\Psi}_{\text{sub}}^{i}\) for satellite \(i\)). Then, uncertainties in the knowledge of the actual \(a_{\ell q}\) are taken into account. From the definition, the matrix 2-norm of \(\mathbf{\tilde{S}}^{\prime}\) is one and \(\mathbf{\tilde{S}}^{\prime T}\mathbf{\tilde{S}}^{\prime}=\mathbf{I}\). It is found through simulations that \(\lambda_{\mathbf{\Omega}}^{\text{max}}\) is close to the maximum eigenvalue of \(\mathbf{\Psi\Theta}\), \(\lambda_{\mathbf{\Omega}}^{\text{max}}\) and generally lower than the latter. This can be verified using the Matlab code, matlab_program_algorithm_1_ts - 1_figure_3.m, provided in the Supplementary Materials.