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Affine EKF: Exploring and Utilizing Sufficient and Necessary Conditions for Observability Maintenance to Improve EKF Consistency

Yang Song, Liang Zhao, Shoudong Huang

TL;DR

A novel affine EKF (Aff-EKF) framework is proposed to overcome the inconsistency of standard EKF by affine transformations, which not only naturally satisfies the observability constraint but also has a clear design procedure.

Abstract

Inconsistency issue is one crucial challenge for the performance of extended Kalman filter (EKF) based methods for state estimation problems, which is mainly affected by the discrepancy of observability between the EKF model and the underlying dynamic system. In this work, some sufficient and necessary conditions for observability maintenance are first proved. We find that under certain conditions, an EKF can naturally maintain correct observability if the corresponding linearization makes unobservable subspace independent of the state values. Based on this theoretical finding, a novel affine EKF (Aff-EKF) framework is proposed to overcome the inconsistency of standard EKF (Std-EKF) by affine transformations, which not only naturally satisfies the observability constraint but also has a clear design procedure. The advantages of our Aff-EKF framework over some commonly used methods are demonstrated through mathematical analyses. The effectiveness of our proposed method is demonstrated on three simultaneous localization and mapping (SLAM) applications with different types of features, typical point features, point features on a horizontal plane and plane features. Specifically, following the proposed procedure, the naturally consistent Aff-EKFs can be explicitly derived for these problems. The consistency improvement of these Aff-EKFs are validated by Monte Carlo simulations.

Affine EKF: Exploring and Utilizing Sufficient and Necessary Conditions for Observability Maintenance to Improve EKF Consistency

TL;DR

A novel affine EKF (Aff-EKF) framework is proposed to overcome the inconsistency of standard EKF by affine transformations, which not only naturally satisfies the observability constraint but also has a clear design procedure.

Abstract

Inconsistency issue is one crucial challenge for the performance of extended Kalman filter (EKF) based methods for state estimation problems, which is mainly affected by the discrepancy of observability between the EKF model and the underlying dynamic system. In this work, some sufficient and necessary conditions for observability maintenance are first proved. We find that under certain conditions, an EKF can naturally maintain correct observability if the corresponding linearization makes unobservable subspace independent of the state values. Based on this theoretical finding, a novel affine EKF (Aff-EKF) framework is proposed to overcome the inconsistency of standard EKF (Std-EKF) by affine transformations, which not only naturally satisfies the observability constraint but also has a clear design procedure. The advantages of our Aff-EKF framework over some commonly used methods are demonstrated through mathematical analyses. The effectiveness of our proposed method is demonstrated on three simultaneous localization and mapping (SLAM) applications with different types of features, typical point features, point features on a horizontal plane and plane features. Specifically, following the proposed procedure, the naturally consistent Aff-EKFs can be explicitly derived for these problems. The consistency improvement of these Aff-EKFs are validated by Monte Carlo simulations.

Paper Structure

This paper contains 38 sections, 7 theorems, 108 equations, 5 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Inconsistency Issue
  4. Consistent Estimators
  5. Preliminary Knowledge
  6. Manifolds
  7. Extended Kalman Filter on Manifolds
  8. Observability Analysis
  9. Sufficient and Necessary Conditions for Observability Maintenance
  10. Motivating Example
  11. Sufficient and Necessary Conditions
  12. Affine EKF
  13. Theoretical Foundations
  14. Affine EKF through Affine Transformations
  15. Affine EKF through Manipulating Covariance
  16. ...and 23 more sections

Key Result

Lemma 1

Suppose $\mathcal{M}$ is a manifold, and $\mathcal{A}^\eta=\{\phi_{\hat{\textbf{X}}}\}$, $\mathcal{A}^\xi=\{\psi_{\hat{\textbf{X}}}\}$ are two arbitrary atlases on it, then where the matrices with superscript of $\eta$ and $\xi$ are computed based on $\mathcal{A}^\eta=\{\phi_{\hat{\textbf{X}}}\}$ and $\mathcal{A}^\xi=\{\psi_{\hat{\textbf{X}}}\}$, respectively.

Figures (5)

  • Figure 1: Representations of a Mapping on Manifold.
  • Figure 2: Monte Carlo Results of SLAM with Point Features in Environment 1 with Noise Settings of $(0.003,0.01,0.1)$
  • Figure 3: Monte Carlo Results of SLAM with Known Height Horizontal Plane Constrained Point Features
  • Figure 4: Monte Carlo Results of SLAM with Unknown Height Horizontal Plane Constrained Point Features
  • Figure 5: Monte Carlo Results of SLAM with Plane Features

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Definition 3
  • Remark
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • ...and 9 more