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Polynomial Updates for the Unscented Kalman Filter

Chiran Cherian, Simone Servadio

Abstract

Most nonlinear filters used in spacecraft navigation are based on a linear approximation of the optimal minimum mean square error estimator. The Unscented Kalman Filter (UKF) handles nonlinear dynamics through a sigma-point transform, but the resulting state estimate remains a linear function of the measurement. This paper proposes a polynomial approximation of the optimal Bayesian update, leading to a Polynomial Unscented Kalman Filter that retains the structure of the standard UKF but enriches the measurement update with higher-order (polynomial) terms. To compute the moments required by this polynomial estimator, we employ a Conjugate Unscented Transformation (CUT), which accurately captures higher-order central moments of the state and measurement. Numerical examples, including Clohessy-Wiltshire and Circular Restricted 3-Body dynamics with non-Gaussian measurement noise, illustrate that the resulting polynomial-CUT filters improve both state estimation accuracy and covariance consistency when compared with their linear counterparts.

Polynomial Updates for the Unscented Kalman Filter

Abstract

Most nonlinear filters used in spacecraft navigation are based on a linear approximation of the optimal minimum mean square error estimator. The Unscented Kalman Filter (UKF) handles nonlinear dynamics through a sigma-point transform, but the resulting state estimate remains a linear function of the measurement. This paper proposes a polynomial approximation of the optimal Bayesian update, leading to a Polynomial Unscented Kalman Filter that retains the structure of the standard UKF but enriches the measurement update with higher-order (polynomial) terms. To compute the moments required by this polynomial estimator, we employ a Conjugate Unscented Transformation (CUT), which accurately captures higher-order central moments of the state and measurement. Numerical examples, including Clohessy-Wiltshire and Circular Restricted 3-Body dynamics with non-Gaussian measurement noise, illustrate that the resulting polynomial-CUT filters improve both state estimation accuracy and covariance consistency when compared with their linear counterparts.
Paper Structure (17 sections, 82 equations, 9 figures, 1 table)

This paper contains 17 sections, 82 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Optimal Polynomial Estimator
  3. The Optimal Quadratic Estimator
  4. The Optimal Polynomial Estimator
  5. High Order Central Moments
  6. The Quadratic Unscented Kalman Filter (QUKF)
  7. The Quadratic Augmented Unscented Kalman Filter (QAUKF)
  8. Generalization of Quadratic Updates
  9. Generalization of the UKF to Arbitrary Order
  10. The Conjugate Unscented Transformation for the Polynomial Update
  11. The Conjugate Unscented Transformation
  12. The Polynomial Augmented Conjugate Unscented Kalman Filter (PACUKF-$c$)
  13. Numerical Applications
  14. Scalar Problem
  15. Clohessy-Wiltshire Relative Motion
  16. ...and 2 more sections

Figures (9)

  • Figure 1: Symmetric set of CUT points and axes in (a) 2D space and (b) 3D first octant space
  • Figure 2: Representation of the state-measurement joint distribution and of the relative estimates from different estimators.
  • Figure 3: RMSE among selected estimators
  • Figure 4: Target's relative motion with respect to the chief satellite.
  • Figure 5: Monte Carlo Consistency Analysis for the Convergence of the QUKF.
  • ...and 4 more figures