Navigation State Estimation With Observability-Constrained Kalman Filtering
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Solution Overview
Problem
Extended Kalman filters used in inertial navigation units often face coherence issues due to linearization errors, leading to inaccuracies in error estimation and reduced precision, particularly along non-observable axes, which can result in substantial errors over time.
Innovation Solution
The process involves adjusting the transition and observation matrices to verify an observability condition, specifically by calculating primary bases of non-observable vectors and offsetting the matrices to prevent reduction of the error region's dimension along non-observable axes, thereby improving coherence and precision of estimations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If extended Kalman filter is used for state estimation, then navigation state can be estimated in real time, but coherence issues arise due to linearization errors causing inaccuracy in error estimation
Solution Approach 1:
The patent segments the error estimation process by separately identifying and handling observable and non-observable error components. The covariance matrix is decomposed to isolate the non-observable subspace, allowing targeted correction of linearization errors in that specific dimension without affecting the overall estimation process.
Solution Approach 2:
The patent modifies the covariance matrix parameters by adding a correction term that accounts for linearization errors. This parameter change adjusts the error envelope along non-observable axes to maintain coherence between the true error distribution and the estimated error distribution.
2Adaptability or versatility
If linearization is performed in extended Kalman filter, then non-linear calculations can be executed, but errors accumulate over time particularly along non-observable axes
Solution Approach 1:
The patent performs preliminary correction of the covariance matrix before the standard Kalman update step. By pre-compensating for linearization errors in the covariance matrix, the system prevents error accumulation along non-observable axes before it occurs, rather than correcting it after accumulation.
Solution Approach 2:
The patent introduces a feedback mechanism where the observability analysis of the system is continuously used to adjust the covariance matrix. The correction term is computed based on the current state and observability properties, creating a closed-loop system that actively compensates for linearization errors.
3Productivity
If standard extended Kalman filtering is applied, then computation can be performed efficiently, but the 3 sigma envelope may not contain the true error cloud along non-observable axes
Solution Approach 1:
The patent segments the covariance matrix into observable and non-observable components using observability analysis. This segmentation allows the correction to be applied only where needed (non-observable axes) without requiring complete re-computation of the entire estimation process, maintaining computational efficiency.
Solution Approach 2:
The patent applies local correction to the covariance matrix by adding a correction term that specifically targets the non-observable subspace. This local quality approach ensures that computational resources are focused only on correcting the specific dimension where coherence issues arise, rather than re-computing the entire matrix.
Data Source
AI summary
There is proposed a method of estimating a navigation state with several variables of a mobile carrier according to the extended Kalman filter method, comprising the steps of:—acquisition of measurements of at least one of the variables,—extended Kalman filtering (400) producing a current estimated state and a covariance matrix delimiting in the space of the navigation state a region of errors, with the help of a previous estimated state, of an observation matrix, of a transition matrix and of the measurements acquired, the method being characterized in that it comprises a step (310, 330) of adjustment of the transition matrix and of the observation matrix before their use in the extended Kalman filtering in such a way that the adjusted matrices satisfy an observability condition which depends on at least one of the variables of the state of the carrier, the observability condition being adjusted so as to prevent the Kalman filter from reducing the dimension of the region along at least one non-observable axis of the state space, in which the observability condition to be satisfied by the adjusted transition and observation matrices is the nullity of the kernel of an observability matrix associated therewith and in which the adjustment comprises the steps of:—calculation (301) of at least one primary basis of non-observable vectors with the help of the previous estimated state—for each matrix to be adjusted, calculation (306, 308) of at least one matrix deviation associated with the matrix with the help of the primary basis of vectors, shifting (330) of each matrix to be adjusted according to the matrix deviation associated therewith so as to satisfy the observability condition.