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Nonlinear Attitude Estimation Using Intermittent and Multi-Rate Vector Measurements

Miaomiao Wang, Abdelhamid Tayebi

TL;DR

This work addresses nonlinear attitude estimation on $SO(3)$ from high-rate angular velocity and intermittent, multi-rate vector measurements. It introduces two geometric hybrid observers: one achieving almost global asymptotic stability (AGAS) and a second with a switching mechanism to obtain global asymptotic stability (GAS), both preserving continuous attitude estimates. The AGAS design yields global exponential convergence of the vector estimates and almost global convergence of the attitude error, while the GAS design uses a switching variable $\theta$ and rotation $\mathcal{R}_u(\theta)$ to eliminate undesired equilibria and guarantee GAS. Stability analyses rely on almost global ISS on manifolds and hybrid Lyapunov methods, complemented by invariance arguments, and the approach is validated through simulations and real-time experiments with IMU and RGB-D data. The results demonstrate robust, globally convergent attitude estimation with intermittent/multi-rate sensing, offering practical benefits for robotics and aerospace systems.

Abstract

This paper considers the problem of nonlinear attitude estimation for a rigid body system using intermittent and multi-rate inertial vector measurements as well as continuous (high-rate) angular velocity measurements. Two types of hybrid attitude observers on Lie group $SO(3)$ are proposed. First, we propose a hybrid attitude observer where almost global asymptotic stability is guaranteed using the notion of almost global input-to-state stability on manifolds. Thereafter, this hybrid attitude observer is extended by introducing a switching mechanism to achieve global asymptotic stability. Both simulation and experimental results are presented to illustrate the performance of the proposed hybrid observers.

Nonlinear Attitude Estimation Using Intermittent and Multi-Rate Vector Measurements

TL;DR

This work addresses nonlinear attitude estimation on from high-rate angular velocity and intermittent, multi-rate vector measurements. It introduces two geometric hybrid observers: one achieving almost global asymptotic stability (AGAS) and a second with a switching mechanism to obtain global asymptotic stability (GAS), both preserving continuous attitude estimates. The AGAS design yields global exponential convergence of the vector estimates and almost global convergence of the attitude error, while the GAS design uses a switching variable and rotation to eliminate undesired equilibria and guarantee GAS. Stability analyses rely on almost global ISS on manifolds and hybrid Lyapunov methods, complemented by invariance arguments, and the approach is validated through simulations and real-time experiments with IMU and RGB-D data. The results demonstrate robust, globally convergent attitude estimation with intermittent/multi-rate sensing, offering practical benefits for robotics and aerospace systems.

Abstract

This paper considers the problem of nonlinear attitude estimation for a rigid body system using intermittent and multi-rate inertial vector measurements as well as continuous (high-rate) angular velocity measurements. Two types of hybrid attitude observers on Lie group are proposed. First, we propose a hybrid attitude observer where almost global asymptotic stability is guaranteed using the notion of almost global input-to-state stability on manifolds. Thereafter, this hybrid attitude observer is extended by introducing a switching mechanism to achieve global asymptotic stability. Both simulation and experimental results are presented to illustrate the performance of the proposed hybrid observers.
Paper Structure (18 sections, 6 theorems, 57 equations, 7 figures, 1 table, 1 algorithm)

This paper contains 18 sections, 6 theorems, 57 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary Material
  3. Notations and Definitions
  4. Hybrid Systems Framework
  5. Problem Statement
  6. Hybrid Observer with AGAS Guarantees
  7. Observer Design
  8. Stability Analysis
  9. Hybrid Observer with GAS Guarantees
  10. Observer Design
  11. Stability Analysis
  12. Simulation and Experimental Results
  13. Simulation Results
  14. Experiments Results
  15. Conclusion
  16. ...and 3 more sections

Key Result

Proposition 1

Consider the following system on $SO(3)$: where $\tilde{R}, \hat{R}\in SO(3)$, $k_o>0$, $\bar{\Gamma}:SO(3) \to \mathbb{R}^{3\times m}$ and $u\in \mathcal{D}_u\subset \mathbb{R}^m$ with $\mathcal{D}_u$ being a closed subset of $\mathbb{R}^m$. Suppose that $A$ is positive definite with three distinct eigenvalues, and $\|\bar{\Gamma}(X)\|_F\l

Figures (7)

  • Figure 1: The architecture of the proposed hybrid observer \ref{['eqn:hybrid_observer0']}.
  • Figure 2: The architecture of the proposed hybrid observer \ref{['eqn:hybrid_observer1']}.
  • Figure 3: Time evolution of the attitude and vector estimation errors for different attitude observers with noise-free measurements ($\sigma=0$).
  • Figure 4: Time evolution of the attitude estimation errors for the proposed attitude observer with different gain parameter $k_r$ in the presence of measurements noise ($\sigma=0.08$).
  • Figure 5: Time evolution of the attitude estimation errors for the attitude observers with different values of the noise covariance, angular velocity amplitude and sampling rate.
  • ...and 2 more figures

Theorems & Definitions (18)

  • Remark 1
  • Proposition 1
  • Remark 2
  • Proposition 2
  • proof
  • Remark 3
  • Theorem 1
  • proof
  • Remark 4
  • Remark 5
  • ...and 8 more