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Equivariant Observer for Bearing Estimation with Linear and Angular Velocity Inputs

Gil Serrano, Marcelo Jacinto, Bruno J. Guerreiro, Rita Cunha

TL;DR

This work extends equivariant observer design to bearing estimation on the unit sphere by incorporating a tangent-space linear velocity input arising from optical flow. By lifting the dynamics to the rotation group SO(3) and crafting a consistent right action and input map, the authors derive an equivariant lift and a correction term that yield almost global asymptotic stability, with the estimator recoverable on the original manifold. The approach is validated numerically under realistic noise and outliers, showing robust bearing convergence and improved stability over naive on-manifold observers. The results have practical implications for image-based visual servoing and relative localization where both rotational and translational motions influence bearing estimates.

Abstract

This work addresses the problem of designing an equivariant observer for a first order dynamical system on the unit-sphere. Building upon the established case of unit bearing vector dynamics with angular velocity inputs, we introduce an additional linear velocity input projected onto the unit-sphere tangent space. This extended formulation is particularly useful in image-based visual servoing scenarios where stable bearing estimates are required and the relative velocity between the vehicle and target features must be accounted for. Leveraging lifted kinematics to the Special Orthogonal group, we design an observer for the bearing vector and prove its almost global asymptotic stability. Additionally, we demonstrate how the equivariant observer can be expressed in the original state manifold. Numerical simulation results validate the effectiveness of the proposed algorithm.

Equivariant Observer for Bearing Estimation with Linear and Angular Velocity Inputs

TL;DR

This work extends equivariant observer design to bearing estimation on the unit sphere by incorporating a tangent-space linear velocity input arising from optical flow. By lifting the dynamics to the rotation group SO(3) and crafting a consistent right action and input map, the authors derive an equivariant lift and a correction term that yield almost global asymptotic stability, with the estimator recoverable on the original manifold. The approach is validated numerically under realistic noise and outliers, showing robust bearing convergence and improved stability over naive on-manifold observers. The results have practical implications for image-based visual servoing and relative localization where both rotational and translational motions influence bearing estimates.

Abstract

This work addresses the problem of designing an equivariant observer for a first order dynamical system on the unit-sphere. Building upon the established case of unit bearing vector dynamics with angular velocity inputs, we introduce an additional linear velocity input projected onto the unit-sphere tangent space. This extended formulation is particularly useful in image-based visual servoing scenarios where stable bearing estimates are required and the relative velocity between the vehicle and target features must be accounted for. Leveraging lifted kinematics to the Special Orthogonal group, we design an observer for the bearing vector and prove its almost global asymptotic stability. Additionally, we demonstrate how the equivariant observer can be expressed in the original state manifold. Numerical simulation results validate the effectiveness of the proposed algorithm.

Paper Structure

This paper contains 22 sections, 46 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Special Orthogonal Group and Related Properties
  4. Smooth Manifolds and the 2-Sphere
  5. Invariance, Equivariance and Group Actions
  6. Problem Statement
  7. Equivariant System Formulation
  8. Symmetry of the Bearing System
  9. Transitive Right Group Action
  10. Coordinate System's Origin
  11. Equivariance Condition and Right Input Action
  12. Equivariant System Lift onto the Group
  13. Dynamics of the Lifted System
  14. Equivariant observer design
  15. Observer on the Group
  16. ...and 7 more sections

Figures (4)

  • Figure 1: Bearing system: a camera translating and rotating in space, while tracking a target point also moving in space.
  • Figure 2: Observer design on the Lie group.
  • Figure 3: Representative simulation with inputs and measurements corrupted by noise. The true bearing is depicted in dashed blue lines, while the estimates provided by the equivariant and naive observers are provided in solid blue and orange lines, respectively.
  • Figure 4: Evolution of the estimation error and measurement outliers.

Theorems & Definitions (2)

  • proof
  • proof