Covariance Matrix Update for GNSS Integrity Monitoring
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Solution Overview
Problem
Current GNSS systems face high computational complexity in computing covariance matrices, particularly for Advanced Receiver Autonomous Integrity Monitoring (ARAIM) and geometry screening, which becomes inefficient with a large number of satellites, and existing methods like the rank-one update formula can be imprecise when removing all satellites from a constellation.
Innovation Solution
The method involves precomputing a vector and weighting factors based on the original covariance and geometry matrices, using symmetry to efficiently compute a subset of modified covariance matrix elements, reducing the need for matrix inversion and improving accuracy by reflecting computed values symmetrically, especially focusing on diagonal values of the upper left 3×3 submatrix.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If the standard matrix inversion method is used to compute covariance matrices for each subsolution, then the computation is straightforward and reliable, but the computational complexity becomes prohibitively high when dealing with a large number of satellites
Solution Approach 1:
The patent precomputes the inverse of the full geometry matrix once before generating any subsolutions. This preliminary computation stores the results in a form that can be efficiently reused for all subsequent subsolution covariance matrix calculations, avoiding repeated matrix inversions and significantly reducing overall computational complexity
Solution Approach 2:
The patent segments the covariance matrix computation by separating the full geometry matrix inversion from the individual subsolution calculations. By dividing the computation into a preliminary full-matrix inversion stage and subsequent efficient subsolution stages using precomputed components, the method avoids redundant calculations across multiple subsolutions
2Productivity
If the rank-one update formula is used to compute modified covariance matrices, then the computational burden is reduced, but the precision deteriorates when removing all satellites from a constellation
Solution Approach 1:
The patent introduces an intermediary approach by using precomputed components from the full geometry matrix inversion as building blocks for subsolution calculations. This intermediary method avoids the precision issues of direct rank-one updates while maintaining computational efficiency through reusable precomputed data structures
Solution Approach 2:
The patent performs preliminary computation of the full geometry matrix inverse and stores it in a reusable format. This preliminary action ensures that all subsequent subsolution calculations start from a precise, precomputed foundation, preventing the accumulation of numerical errors that occur with iterative rank-one updates
3Reliability
If all subsolution covariance matrices are computed using standard methods, then complete integrity monitoring is achieved, but the processing time becomes unacceptable for real-time applications
Solution Approach 1:
The patent performs a preliminary computation of the full geometry matrix inverse once, storing the results in a format that enables rapid generation of all subsolution covariance matrices. This single preliminary computation serves as the foundation for all subsequent integrity monitoring calculations, dramatically reducing the time required to process multiple subsolutions
Solution Approach 2:
The patent merges the computation of multiple subsolution covariance matrices by using a unified precomputed foundation from the full geometry matrix. Instead of independently computing each subsolution, the method combines them all through shared precomputed components, reducing redundant operations and processing time
Data Source
AI summary
An efficient covariance matrix computation method is disclosed in connection with certain GNSS applications, including Advanced Receiver Autonomous Integrity Monitoring (ARAIM) and geometry screening. The system and method of the present application enable computation of multiple covariance matrices with substantially greater efficiency than previous approaches, including the rank-one update formula. For example, the system and method of the present application advantageously involves substantially fewer and simpler arithmetic operations than previous approaches. In addition, unlike the rank-one update formula, the system and method of the present application can be used to compute the subsolution in which all the satellites of a given constellation are removed.