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Globally exponentially convergent observer for systems evolving on matrix Lie groups

Soham Shanbhag, Dong Eui Chang

TL;DR

This work tackles state estimation for systems evolving on matrix Lie groups when velocity bounds are unknown. It introduces a globally exponentially convergent observer designed in ambient Euclidean space, with a Lyapunov-based proof ensuring convergence of both the state estimate and velocity bias. The approach is demonstrated on SE(3) and compared against a literature observer, showing global convergence and robust bias estimation even under noise and bound-unknown conditions. The method enables robust, continuous state estimation for rigid-body and more general Lie-group dynamics without requiring prior bounds on velocity.

Abstract

We propose a globally exponentially convergent observer for the dynamical system evolving on matrix Lie groups with bounded velocity with unknown bound. We design the observer in the ambient Euclidean space and show exponential convergence of the observer to the state of the system. We show the convergence with an example of a rigid body rotation and translation system on the special Euclidean group. We compare the proposed observer with an observer present in the literature.

Globally exponentially convergent observer for systems evolving on matrix Lie groups

TL;DR

This work tackles state estimation for systems evolving on matrix Lie groups when velocity bounds are unknown. It introduces a globally exponentially convergent observer designed in ambient Euclidean space, with a Lyapunov-based proof ensuring convergence of both the state estimate and velocity bias. The approach is demonstrated on SE(3) and compared against a literature observer, showing global convergence and robust bias estimation even under noise and bound-unknown conditions. The method enables robust, continuous state estimation for rigid-body and more general Lie-group dynamics without requiring prior bounds on velocity.

Abstract

We propose a globally exponentially convergent observer for the dynamical system evolving on matrix Lie groups with bounded velocity with unknown bound. We design the observer in the ambient Euclidean space and show exponential convergence of the observer to the state of the system. We show the convergence with an example of a rigid body rotation and translation system on the special Euclidean group. We compare the proposed observer with an observer present in the literature.
Paper Structure (5 sections, 1 theorem, 13 equations, 2 figures)

This paper contains 5 sections, 1 theorem, 13 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proposed Observer
  4. Numerical Simulation
  5. Conclusion

Key Result

Theorem 1

Define the observer error terms where $A = Fg$. Suppose that Assumptions assum:vel_bound and assum:traj hold. Then, with $k_1, k_2 > 0$ and the update laws as given in equations sys_main and sys_est, the error terms $\| E_A(t) \|$ and $\| E_b(t) \|$ converge exponentially to $0$ as $t \to \infty$ for all $(\bar{A}(0), \bar{b}(0))

Figures (2)

  • Figure 1: Simulation of the proposed observer in presence of noise
  • Figure 2: Comparison of proposed observer with literature

Theorems & Definitions (2)

  • Theorem 1
  • proof