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Constructive Equivariant Observer Design for Inertial Velocity-Aided Attitude

Pieter van Goor, Tarek Hamel, Robert Mahony

TL;DR

This work tackles inertial velocity-aided attitude estimation for RPAS by designing a nonlinear, equivariant observer on the SE(3) group that uses both IMU data and an inertial-frame velocity measurement. By constructing a synchronous error dynamics via lifted SE(3) representations and carefully chosen correction terms, the authors prove almost-global asymptotic stability for the attitude error and global exponential stability for the velocity error, without assuming constant acceleration. They demonstrate almost global convergence and robustness to large initial errors through simulations, validating the theoretical results. The approach leverages Lie-group symmetry to provide a globally valid correction mechanism, offering a principled alternative to high-gain or uniformly linearised observers in inertial VAA problems.

Abstract

Inertial Velocity-Aided Attitude (VAA), the estimation of the velocity and attitude of a vehicle using gyroscope, accelerometer, and inertial-frame velocity (e.g. GPS velocity) measurements, is an important problem in the control of Remotely Piloted Aerial Systems (RPAS). Existing solutions provide limited stability guarantees, relying on local linearisation, high gain design, or assuming specific trajectories such as constant acceleration of the vehicle. This paper proposes a novel non-linear observer for inertial VAA with almost globally asymptotically and locally exponentially stable error dynamics. The approach exploits Lie group symmetries of the system dynamics to construct a globally valid correction term. To the authors' knowledge, this construction is the first observer to provide almost global convergence for the inertial VAA problem. The observer performance is verified in simulation, where it is shown that the estimation error converges to zero even with an extremely poor initial condition.

Constructive Equivariant Observer Design for Inertial Velocity-Aided Attitude

TL;DR

This work tackles inertial velocity-aided attitude estimation for RPAS by designing a nonlinear, equivariant observer on the SE(3) group that uses both IMU data and an inertial-frame velocity measurement. By constructing a synchronous error dynamics via lifted SE(3) representations and carefully chosen correction terms, the authors prove almost-global asymptotic stability for the attitude error and global exponential stability for the velocity error, without assuming constant acceleration. They demonstrate almost global convergence and robustness to large initial errors through simulations, validating the theoretical results. The approach leverages Lie-group symmetry to provide a globally valid correction mechanism, offering a principled alternative to high-gain or uniformly linearised observers in inertial VAA problems.

Abstract

Inertial Velocity-Aided Attitude (VAA), the estimation of the velocity and attitude of a vehicle using gyroscope, accelerometer, and inertial-frame velocity (e.g. GPS velocity) measurements, is an important problem in the control of Remotely Piloted Aerial Systems (RPAS). Existing solutions provide limited stability guarantees, relying on local linearisation, high gain design, or assuming specific trajectories such as constant acceleration of the vehicle. This paper proposes a novel non-linear observer for inertial VAA with almost globally asymptotically and locally exponentially stable error dynamics. The approach exploits Lie group symmetries of the system dynamics to construct a globally valid correction term. To the authors' knowledge, this construction is the first observer to provide almost global convergence for the inertial VAA problem. The observer performance is verified in simulation, where it is shown that the estimation error converges to zero even with an extremely poor initial condition.
Paper Structure (8 sections, 3 theorems, 37 equations, 2 figures)

This paper contains 8 sections, 3 theorems, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Description
  4. Observer Design
  5. Equivariant Observer Architecture
  6. Observer Design
  7. Simulations
  8. Conclusion

Key Result

Lemma 4.1

Define an error $\bar{E} := \hat{Z}^{-1} X \hat{X}^{-1} \hat{Z}$. The system dynamics eq:system_dynamics_se3 and the observer internal model eq:observer_architecture are $\overline{E}$-synchronous 2021_vangoor_AutonomousErrorConstructive; i.e. the dynamics of $\overline{E}$ depend only on the chosen

Figures (2)

  • Figure 1: A comparison of the true (solid blue) and estimated (dashed red) velocity and attitude of the example system over time. The pitch and roll components of the attitude converge more quickly than the yaw for the observer.
  • Figure 2: The evolution of the error metrics of the observer state over time. The attitude error, velocity error, and Lyapunov value all show a sharp initial decrease, followed by a slower second phase of convergence.

Theorems & Definitions (5)

  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • Lemma 6.1