Kalman Filter-Based RAIM for Reliable Aircraft Positioning with GPS and NavIC Constellations
Abstract
:1. Introduction
2. Prior Work and Contributions
3. Overview of Existing KF RAIM Algorithms
3.1. Solution Separation KF RAIM
3.2. Existing Range-Based KF RAIM
4. Schmidt KF
5. Range-Based KF RAIM Algorithm
5.1. Fault Detection Method
5.1.1. Formulation of Ψ for First Test Statistic
5.1.2. Formulation of Θ for First Test Statistic
5.1.3. Formulation of Θ for Second Test Statistic
5.1.4. Test Statistics and Thresholds
5.2. Mean Position Error Bounds under Faults
5.2.1.
5.2.2.
5.2.3.
5.3. Protection Levels (PLs)
6. Simulation Studies
7. Conclusions
Supplementary Materials
Author Contributions
Funding
Conflicts of Interest
Appendix A. Solution Separation KF RAIM
Appendix B. SKF Process and Measurement Models and Key Equations
Appendix C. Algorithm to Find Θ for α 1
Algorithm A1: Determination of for |
% Add/remove previous epoch terms as needed in so that it consists of terms of % epochs excluding current epoch k. This accounts for change in M at every epoch. % Reference [50] details a computationally efficient algorithm of how to update the inverse of % a matrix, if a number of rows and columns of the matrix are removed. This is used when M % reduces at k. For addition of terms when M increases, part of the steps described next is % used to update after minor modifications. Based on the updated , is found as % follows. % Form (i.e., rows of for the current epoch) index = 1 % Set a flag for condition check c = 0.05 % Set initial c as 0.05 factor = 9 % It is used to find c = 1.5 % Set to a value greater than one while ( > 1) if (index == 1) % Find by adding current epoch terms; is partitioned as = + = = = inv( = = = inv() % Find the inverse of = − = + = % Use the matrix inversion lemma index = 0 % If maximum eigenvalue is still greater than one, do not enter here else % Algorithm under if condition generally does not work when the number of satellites % changes at k % Hence, the following algorithm is used to find then c = ( + )/factor % Calculate c again for index1 = 1:length(vis_sat_info) % vis_sat_info holds all visible PRN IDs % over M epochs % Find epochs over which PRN(index1) is visible. Suppose the number % of epochs is = inv % Dimension is . See before Equation (30) for the % definition of for satellite i % Place elements of in appropriate entries of corresponding to PRN(index1) end % end of for index1 = 1:length(vis_sat_info) end % end of if(index == 1) factor = factor - 0.2 % Update factor to calculate a new c % Calculate with new end % end of while( > 1) |
Appendix D. Justification That Γ is Overbounded for
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Position | WLS Range- | KF Range- | KF Solution |
---|---|---|---|
Based RAIM | Based RAIM | Separation | |
RAIM | |||
Vertical | 99.63 | 97.4 | 99.68 |
Horizontal | 99.63 | 97.4 | 99.63 |
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Bhattacharyya, S.; Mute, D. Kalman Filter-Based RAIM for Reliable Aircraft Positioning with GPS and NavIC Constellations. Sensors 2020, 20, 6606. https://doi.org/10.3390/s20226606
Bhattacharyya S, Mute D. Kalman Filter-Based RAIM for Reliable Aircraft Positioning with GPS and NavIC Constellations. Sensors. 2020; 20(22):6606. https://doi.org/10.3390/s20226606
Chicago/Turabian StyleBhattacharyya, Susmita, and Dinesh Mute. 2020. "Kalman Filter-Based RAIM for Reliable Aircraft Positioning with GPS and NavIC Constellations" Sensors 20, no. 22: 6606. https://doi.org/10.3390/s20226606
APA StyleBhattacharyya, S., & Mute, D. (2020). Kalman Filter-Based RAIM for Reliable Aircraft Positioning with GPS and NavIC Constellations. Sensors, 20(22), 6606. https://doi.org/10.3390/s20226606