Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Equivariant Filter (EqF)

Pieter van Goor, Tarek Hamel, Robert Mahony

TL;DR

The paper presents the Equivariant Filter (EqF), a nonlinear observer for systems on homogeneous spaces that leverages Lie group symmetry by placing the observer on the symmetry group and deriving global error dynamics via a lifted model. A Riccati-based correction, together with an equivariant lift and optionally an equivariant output linearisation (EqF^*), reduces linearisation error and improves transient performance relative to the EKF. The approach subsumes the invariant EKF as a special case and delivers significant gains in a challenging bearing-estimation example on $S^2$ by exploiting both state and output equivariance. The results indicate broad potential for EqF in robotics and aerospace applications where states naturally reside on manifolds with transitive group actions, enabling robust, intrinsic state estimation on homogeneous spaces.

Abstract

The kinematics of many systems encountered in robotics, mechatronics, and avionics are naturally posed on homogeneous spaces; that is, their state lies in a smooth manifold equipped with a transitive Lie group symmetry. This paper proposes a novel filter, the Equivariant Filter (EqF), by posing the observer state on the symmetry group, linearising global error dynamics derived from the equivariance of the system, and applying extended Kalman filter design principles. We show that equivariance of the system output can be exploited to reduce linearisation error and improve filter performance. Simulation experiments of an example application show that the EqF significantly outperforms the extended Kalman filter and that the reduced linearisation error leads to a clear improvement in performance.

Equivariant Filter (EqF)

TL;DR

The paper presents the Equivariant Filter (EqF), a nonlinear observer for systems on homogeneous spaces that leverages Lie group symmetry by placing the observer on the symmetry group and deriving global error dynamics via a lifted model. A Riccati-based correction, together with an equivariant lift and optionally an equivariant output linearisation (EqF^*), reduces linearisation error and improves transient performance relative to the EKF. The approach subsumes the invariant EKF as a special case and delivers significant gains in a challenging bearing-estimation example on by exploiting both state and output equivariance. The results indicate broad potential for EqF in robotics and aerospace applications where states naturally reside on manifolds with transitive group actions, enabling robust, intrinsic state estimation on homogeneous spaces.

Abstract

The kinematics of many systems encountered in robotics, mechatronics, and avionics are naturally posed on homogeneous spaces; that is, their state lies in a smooth manifold equipped with a transitive Lie group symmetry. This paper proposes a novel filter, the Equivariant Filter (EqF), by posing the observer state on the symmetry group, linearising global error dynamics derived from the equivariance of the system, and applying extended Kalman filter design principles. We show that equivariance of the system output can be exploited to reduce linearisation error and improve filter performance. Simulation experiments of an example application show that the EqF significantly outperforms the extended Kalman filter and that the reduced linearisation error leads to a clear improvement in performance.

Paper Structure

This paper contains 28 sections, 5 theorems, 86 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivating Example: Single Bearing Estimation
  3. Preliminaries
  4. Problem Description
  5. Systems on Homogeneous Spaces
  6. Equivariant Systems
  7. Equivariant Error System
  8. Observer Architecture
  9. Global Error Dynamics
  10. Linearisation
  11. Error Dynamics
  12. System Output
  13. Equivariant Output Linearisation
  14. Equivariant Filter (EqF)
  15. EqF Gain Tuning
  16. ...and 13 more sections

Key Result

Proposition 3.1

Any right action $\phi: \mathbf{G} \times \mathcal{M} \to \mathcal{M}$ induces a right action on the vector fields over $\mathcal{M}$, denoted $\Phi: \mathbf{G} \times \mathfrak{X}(\mathcal{M}) \to \mathfrak{X}(\mathcal{M})$, and defined by for any $f \in \mathfrak{X}(\mathcal{M})$ and $X \in \mathbf{G}$. For a fixed $X \in \mathbf{G}$, $\Phi_X$ is a linear map on $\mathfrak{X}(\mathcal{M})$.

Figures (3)

  • Figure 1: The angle error \ref{['eq:absolute_angle_error']} and Lyapunov value \ref{['eq:linearised_lyapunov_func']} for each of the filters without noise added to any of the gyroscope or magnetometer measurements. The EqF$^\star$ (red solid line) shows faster initial convergence than both the EqF (green dot-dashed line) and the EKF (blue dashed line).
  • Figure 2: The median angle error \ref{['eq:absolute_angle_error']} and Lyapunov value \ref{['eq:linearised_lyapunov_func']} for each of the filters over 500 trials with noise generated for each trial independently. The EqF$^\star$ (red solid line) outperforms both the EqF (green dot-dashed line) and the EKF (blue dashed line) in terms of both angle error and Lyapunov value. The coloured areas show the 25th and 75th percentile for each filter's angle error and Lyapunov value.
  • Figure 3: The error of linearising the output residual $\tilde{y}$ for the EKF, EqF, and EqF$^\star$. The EKF and EqF are similar in terms of overall quality, with the EqF performing better close to the linearisation point. The EqF$^\star$ shows clearly superior performance to both the EKF and EqF as expected from Lemma \ref{['lem:equivariant_output']}.

Theorems & Definitions (12)

  • Proposition 3.1
  • proof
  • Remark 4.1
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • proof
  • Remark 6.1
  • ...and 2 more