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CT-ESKF: A General Framework of Covariance Transformation-Based Error-State Kalman Filter

Jiale Han, Wei Ouyang, Maoran Zhu, Yuanxin Wu

TL;DR

This paper formalizes an equivalence framework for error-state Kalman filters, showing that different error-state definitions yield equivalent state and covariance propagation under appropriate transformations. It distinguishes covariance transformation from covariance switch and introduces CT-ESKF as a general, unifying framework that can adjust updated covariances to better reflect true uncertainty, thereby improving performance in multi-sensor navigation that uses both global- and body-frame observations. The authors demonstrate theoretically that InEKF's trajectory-independence does not guarantee superior covariance propagation compared to EKF, and they validate the CT-ESKF approach with land-vehicle and aircraft experiments, where covariance-transformed EKF and CT-EKF outperform InEKF and standard EKF in terms of convergence, consistency, and accuracy. The results suggest practical benefits for integrated navigation systems by carefully choosing error-state representations and applying covariance transformation to align covariances with actual uncertainty, enhancing robustness in heterogeneous sensing scenarios.

Abstract

Invariant extended Kalman filter (InEKF) possesses excellent trajectory-independent property and better consistency compared to conventional extended Kalman filter (EKF). However, when applied to scenarios involving both global-frame and body-frame observations, InEKF may fail to preserve its trajectory-independent property. This work introduces the concept of equivalence between error states and covariance matrices among different error-state Kalman filters, and shows that although InEKF exhibits trajectory independence, its covariance propagation is actually equivalent to EKF. A covariance transformation-based error-state Kalman filter (CT-ESKF) framework is proposed that unifies various error-state Kalman filtering algorithms. The framework gives birth to novel filtering algorithms that demonstrate improved performance in integrated navigation systems that incorporate both global and body-frame observations. Experimental results show that the EKF with covariance transformation outperforms both InEKF and original EKF in a representative INS/GNSS/Odometer integrated navigation system.

CT-ESKF: A General Framework of Covariance Transformation-Based Error-State Kalman Filter

TL;DR

This paper formalizes an equivalence framework for error-state Kalman filters, showing that different error-state definitions yield equivalent state and covariance propagation under appropriate transformations. It distinguishes covariance transformation from covariance switch and introduces CT-ESKF as a general, unifying framework that can adjust updated covariances to better reflect true uncertainty, thereby improving performance in multi-sensor navigation that uses both global- and body-frame observations. The authors demonstrate theoretically that InEKF's trajectory-independence does not guarantee superior covariance propagation compared to EKF, and they validate the CT-ESKF approach with land-vehicle and aircraft experiments, where covariance-transformed EKF and CT-EKF outperform InEKF and standard EKF in terms of convergence, consistency, and accuracy. The results suggest practical benefits for integrated navigation systems by carefully choosing error-state representations and applying covariance transformation to align covariances with actual uncertainty, enhancing robustness in heterogeneous sensing scenarios.

Abstract

Invariant extended Kalman filter (InEKF) possesses excellent trajectory-independent property and better consistency compared to conventional extended Kalman filter (EKF). However, when applied to scenarios involving both global-frame and body-frame observations, InEKF may fail to preserve its trajectory-independent property. This work introduces the concept of equivalence between error states and covariance matrices among different error-state Kalman filters, and shows that although InEKF exhibits trajectory independence, its covariance propagation is actually equivalent to EKF. A covariance transformation-based error-state Kalman filter (CT-ESKF) framework is proposed that unifies various error-state Kalman filtering algorithms. The framework gives birth to novel filtering algorithms that demonstrate improved performance in integrated navigation systems that incorporate both global and body-frame observations. Experimental results show that the EKF with covariance transformation outperforms both InEKF and original EKF in a representative INS/GNSS/Odometer integrated navigation system.

Paper Structure

This paper contains 22 sections, 7 theorems, 160 equations, 12 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Fundamentals of ESKF
  3. Relationship Between Error-State Representations
  4. Covariance Switch and its Effect in Kalman Filters
  5. Covariance Transformation
  6. Preliminaries of EKF and InEKF
  7. Linearization of the EKF Model
  8. Lie Group and State Representation in InEKF
  9. Linearization of the InEKF Model
  10. Observation Model
  11. Relationship Between EKF and InEKF
  12. Experimental Results and Analysis
  13. Test on Land Vehicle
  14. Covariance Propagation Analysis
  15. Evaluation of Covariance Transformation
  16. ...and 7 more sections

Key Result

Theorem 1

Suppose at time $t_s$, the following equalities hold and the state propagation satisfies Then, for $t \in [t_s, t_f]$, the following equivalences hold That is, the error states and covariance matrices of ESKF(a) and ESKF(b) remain equivalent throughout the propagation interval.

Figures (12)

  • Figure 1: An example of covariance switch.
  • Figure 2: Transformation relationships (0-6) among different ESKF algorithms.
  • Figure 3: Trajectory of the vehicle during the test period in the Wuhan University Dataset.
  • Figure 4: Covariance propagation comparison among EKF, L-InEKF, and R-InEKF.
  • Figure 5: RMSE comparison of attitude estimation errors under GNSS-only observations in land vehicle experiments.
  • ...and 7 more figures

Theorems & Definitions (22)

  • Remark 1
  • Remark 2
  • Definition 1: Equivalent Error States
  • Definition 2: Equivalent Covariance Matrices
  • Theorem 1: Equivalence of Error State and Covariance Kept through State Propagation
  • proof
  • Lemma 1: Ineffectiveness of Covariance Switch Han2024
  • proof
  • Lemma 2: Relationship of State Update Under Different Error State Definitions
  • proof
  • ...and 12 more