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Design Guidelines for Nonlinear Kalman Filters via Covariance Compensation

Shida Jiang, Jaewoong Lee, Shengyu Tao, Scott Moura

Abstract

Nonlinear extensions of the Kalman filter (KF), such as the extended Kalman filter (EKF) and the unscented Kalman filter (UKF), are indispensable for state estimation in complex dynamical systems, yet the conditions for a nonlinear KF to provide robust and accurate estimations remain poorly understood. This work proposes a theoretical framework that identifies the causes of failure and success in certain nonlinear KFs and establishes guidelines for their improvement. Central to our framework is the concept of covariance compensation: the deviation between the covariance predicted by a nonlinear KF and that of the EKF. With this definition and detailed theoretical analysis, we derive three design guidelines for nonlinear KFs: (i) invariance under orthogonal transformations, (ii) sufficient covariance compensation beyond the EKF baseline, and (iii) selection of compensation magnitude that favors underconfidence. Both theoretical analysis and empirical validation confirm that adherence to these principles significantly improves estimation accuracy, whereas fixed parameter choices commonly adopted in the literature are often suboptimal. The codes and the proofs for all the theorems in this paper are available at https://github.com/Shida-Jiang/Guidelines-for-Nonlinear-Kalman-Filters.

Design Guidelines for Nonlinear Kalman Filters via Covariance Compensation

Abstract

Nonlinear extensions of the Kalman filter (KF), such as the extended Kalman filter (EKF) and the unscented Kalman filter (UKF), are indispensable for state estimation in complex dynamical systems, yet the conditions for a nonlinear KF to provide robust and accurate estimations remain poorly understood. This work proposes a theoretical framework that identifies the causes of failure and success in certain nonlinear KFs and establishes guidelines for their improvement. Central to our framework is the concept of covariance compensation: the deviation between the covariance predicted by a nonlinear KF and that of the EKF. With this definition and detailed theoretical analysis, we derive three design guidelines for nonlinear KFs: (i) invariance under orthogonal transformations, (ii) sufficient covariance compensation beyond the EKF baseline, and (iii) selection of compensation magnitude that favors underconfidence. Both theoretical analysis and empirical validation confirm that adherence to these principles significantly improves estimation accuracy, whereas fixed parameter choices commonly adopted in the literature are often suboptimal. The codes and the proofs for all the theorems in this paper are available at https://github.com/Shida-Jiang/Guidelines-for-Nonlinear-Kalman-Filters.
Paper Structure (14 sections, 3 theorems, 43 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 14 sections, 3 theorems, 43 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Background and Literature Review
  3. Contributions
  4. COVARIANCE COMPENSATION IN NONLINEAR KALMAN FILTERS
  5. General Framework for Nonlinear Kalman Filters
  6. The Moment Estimation Problem
  7. Formal definition of covariance compensation
  8. Covariance Compensation in Second-Order EKFs
  9. Covariance Compensation in Unscented Kalman Filters
  10. GUIDELINES FOR ROBUST AND ACCURATE STATE ESTIMATION
  11. Applications of Covariance Compensation
  12. Optimization of Covariance Compensation
  13. EXPERIMENTAL VALIDATION
  14. CONCLUSIONS

Key Result

Theorem 1

With the conditions in (setup2), and when the nonlinear mapping $f$ is real analytic at the origin, we have $P_{com}^{\textnormal{CKF}}\succeq 0$. (Proof in the Supplementary Material on Github.)

Figures (4)

  • Figure 1: The selected sigma points in different variants of unscented Kalman filters when $n=2, \bm{x}\sim(\bm{0},I_{2\times 2})$.
  • Figure 2: The sensitivity of the estimated covariance $P_{z}^\textnormal{est}$ and $P_{xz}^\textnormal{est}=[P_{xz}(1) \quad P_{xz}(2)]^T$ to rotation in different nonlinear KFs.
  • Figure 3: Estimation error of a simple system under different magnitudes of covariance estimation fluctuations and covariance compensation.
  • Figure 4: The actual and estimated geometric mean of the state estimation RMSE under different magnitudes of covariance compensation. (a) 3D target tracking, (b) terrain-referenced navigation, (c) generator state estimation.

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Remark 2