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A Transformation-based Consistent Estimation Framework: Analysis, Design and Applications

Ning Hao, Chungeng Tian, Fenghua He

TL;DR

The paper addresses inconsistency in nonlinear state estimation caused by observability mismatch and proves that the EKF's unobservable subspace is state-independent and contained in the underlying system's unobservable subspace, enabling observability matching under certain conditions. It introduces linear time-varying transformations that yield a transformed system with a state-independent unobservable subspace and presents two equivalent, consistent transformation-based EKF estimators, T-EKF 1 and T-EKF 2, with distinct propagation/update strategies. The framework is validated across multi-robot cooperative localization, multi-source target tracking, and 3D visual-inertial navigation, demonstrating improved accuracy, consistency, and computational efficiency over traditional baselines. These results offer a principled, versatile approach to achieving consistent estimation in partially observable nonlinear systems with broad practical impact.

Abstract

In this paper, we investigate the inconsistency problem arising from observability mismatch that frequently occurs in nonlinear systems such as multi-robot cooperative localization and simultaneous localization and mapping. For a general nonlinear system, we discover and theoretically prove that the unobservable subspace of the EKF estimator system is independent of the state and belongs to the unobservable subspace of the original system. On this basis, we establish the necessary and sufficient conditions for achieving observability matching. These theoretical findings motivate us to introduce a linear time-varying transformation to achieve a transformed system possessing a state-independent unobservable subspace. We prove the existence of such transformations and propose two design methodologies for constructing them. Moreover, we propose two equivalent consistent transformation-based EKF estimators, referred to as T-EKF 1 and T-EKF 2, respectively. T-EKF 1 employs the transformed system for consistent estimation, whereas T-EKF 2 leverages the original system but ensures consistency through state and covariance corrections from transformations. To validate our proposed methods, we conduct experiments on several representative examples, including multi-robot cooperative localization, multi-source target tracking, and 3D visual-inertial odometry, demonstrating that our approach achieves state-of-the-art performance in terms of accuracy, consistency, computational efficiency, and practical realizations.

A Transformation-based Consistent Estimation Framework: Analysis, Design and Applications

TL;DR

The paper addresses inconsistency in nonlinear state estimation caused by observability mismatch and proves that the EKF's unobservable subspace is state-independent and contained in the underlying system's unobservable subspace, enabling observability matching under certain conditions. It introduces linear time-varying transformations that yield a transformed system with a state-independent unobservable subspace and presents two equivalent, consistent transformation-based EKF estimators, T-EKF 1 and T-EKF 2, with distinct propagation/update strategies. The framework is validated across multi-robot cooperative localization, multi-source target tracking, and 3D visual-inertial navigation, demonstrating improved accuracy, consistency, and computational efficiency over traditional baselines. These results offer a principled, versatile approach to achieving consistent estimation in partially observable nonlinear systems with broad practical impact.

Abstract

In this paper, we investigate the inconsistency problem arising from observability mismatch that frequently occurs in nonlinear systems such as multi-robot cooperative localization and simultaneous localization and mapping. For a general nonlinear system, we discover and theoretically prove that the unobservable subspace of the EKF estimator system is independent of the state and belongs to the unobservable subspace of the original system. On this basis, we establish the necessary and sufficient conditions for achieving observability matching. These theoretical findings motivate us to introduce a linear time-varying transformation to achieve a transformed system possessing a state-independent unobservable subspace. We prove the existence of such transformations and propose two design methodologies for constructing them. Moreover, we propose two equivalent consistent transformation-based EKF estimators, referred to as T-EKF 1 and T-EKF 2, respectively. T-EKF 1 employs the transformed system for consistent estimation, whereas T-EKF 2 leverages the original system but ensures consistency through state and covariance corrections from transformations. To validate our proposed methods, we conduct experiments on several representative examples, including multi-robot cooperative localization, multi-source target tracking, and 3D visual-inertial odometry, demonstrating that our approach achieves state-of-the-art performance in terms of accuracy, consistency, computational efficiency, and practical realizations.

Paper Structure

This paper contains 62 sections, 89 equations, 11 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Inconsistency Analysis
  4. Observability Constraint-Based Estimators
  5. Robot-centric Estimators
  6. Matrix Lie Group-Based Estimators
  7. Transformation-Based Estimators
  8. Observability Mismatch Issue Analysis
  9. Observability Mismatch Issue Statement
  10. Observability Matching Conditions
  11. Transformation-based Consistent Estimation Approach
  12. Design of Linear Time-Varying Transformations
  13. Transformation Design Approach 1
  14. Transformation Design Approach 2
  15. Transformed Extended Kalman Filter 1
  16. ...and 47 more sections

Figures (11)

  • Figure 1: The plots of statistical RMSE and NEES (position and orientation) of different estimators over Monte Carlo simulation for CL applications.
  • Figure 2: The statistical RMSE (position and orientation) under different detection probabilities of relative measurements for CL applications.
  • Figure 3: The five robots' trajectories in sub-dataset $9$ from $\rm 0$ to $\rm 600s$.
  • Figure 4: Statistical error distributions of these estimators over nine sub-datasets for CL applications.
  • Figure 5: The plots of statistical RMSE and NEES (position and orientation) over 200 Monte Carlo simulation trials for target tracking application.
  • ...and 6 more figures

Theorems & Definitions (11)

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  • ...and 1 more