Kalman Filter Error Bound Computation via Multivariate t-Distribution
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Solution Overview
Problem
Current methods for computing Protection Levels in GNSS navigation, particularly for Kalman filters, face challenges due to restrictive hypotheses about measurement noise behavior and the impact of temporal correlations, which affect the accuracy and reliability of position and velocity estimates.
Innovation Solution
A method that autonomously computes error bounds using t-distributions, adjusting for the sum of errors from current and previous epochs, and accounts for temporal correlations through a fitting process that decomposes errors by measurement type and considers the effects of temporal correlations in the measurement noise.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If standard Protection Level methods are used for Kalman filter-based GNSS navigation, then computational simplicity is maintained, but accuracy and reliability of error bounds deteriorate due to restrictive hypotheses about measurement noise behavior and temporal correlations
Solution Approach 1:
The patent segments the error bound computation by decomposing it into contributions from different measurement types (pseudorange, carrier phase, Doppler) and different error sources (measurement noise, satellite clock errors, ephemeris errors). This segmentation allows for more accurate error characterization while maintaining computational efficiency by processing each segment separately and combining results.
Solution Approach 2:
The patent changes the parameters used in error bound computation from standard Gaussian assumptions to more sophisticated models that account for temporal correlations and measurement-specific error characteristics. It introduces time-varying parameters such as autocorrelation coefficients and measurement-specific standard deviations that adapt to actual error behavior, thereby improving accuracy without excessive computational burden.
2Reliability
If restrictive hypotheses about measurement noise are made, then computational simplicity is maintained, but reliability of position and velocity estimates deteriorates
Solution Approach 1:
The patent applies dynamic modeling to error bounds by making them time-varying rather than static. It incorporates temporal correlations through autocorrelation coefficients that adapt to changing error characteristics over time. The error bounds are updated at each epoch based on current measurement quality and correlation with previous epochs, providing reliable estimates that adapt to dynamic error conditions without requiring complex fixed models.
3Measurement precision
If temporal correlations in measurement noise are not accounted for, then computational simplicity is maintained, but accuracy of error bounds deteriorates
Solution Approach 1:
The patent performs preliminary computation of autocorrelation coefficients and their impact on error bounds before final error bound calculation. By pre-computing these correlation effects and incorporating them into the error model, the patent achieves accurate error bounds that account for temporal correlations without adding excessive computational complexity to the main navigation solution.
Data Source
AI summary
The invention relates to a method for computing a bound B up to a given confidence level 1-α, of an error in a state vector estimation KSV of a state vector TSV of a physical system as provided by a Kalman filter. The method decomposes the errors of the Kalman solution as a sum of the errors due to each of the measurement types used in the filter. In addition, the contribution of each type of measurement is bounded by a multivariate t-distribution that considers the error terms from all the epochs processed. Then, the method implements three main operations: - computing a probability distribution of the measurement errors for each epoch and measurement type; - summing the previous distributions to obtain a global distribution that models the Kalman solution error; and - computing the error bound B for a given confidence level from the resulting distribution.