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Geometric Fault-Tolerant Neural Network Tracking Control of Unknown Systems on Matrix Lie Groups

Robin Chhabra, Farzaneh Abdollahi

TL;DR

This work develops a geometric neural network tracking controller for systems evolving on matrix Lie groups $G$, including unknown dynamics, actuator faults, and bounded disturbances. By leveraging a left-invariant tangent bundle and intrinsic Lie-group calculus, it formulates coordinate-free NN learning rules that operate directly on $G$ (and its Lie algebra), yielding global-like weight search and Lyapunov-based stability results. The authors prove uniform ultimate boundedness of both the configuration error $\tilde{g}$ and velocity error $\tilde{\xi}$, as well as NN weight errors $\tilde{W}, \tilde{V}$, and extend the framework to decentralized formation control on $SE(3)$. Numerical simulations on a multi-agent formation with unknown inertia and actuator faults demonstrate robust formation maintenance and bounded control effort, highlighting practical applicability to robotic systems with uncertain dynamics.

Abstract

We present a geometric neural network-based tracking controller for systems evolving on matrix Lie groups under unknown dynamics, actuator faults, and bounded disturbances. Leveraging the left-invariance of the tangent bundle of matrix Lie groups, viewed as an embedded submanifold of the vector space $\R^{N\times N}$, we propose a set of learning rules for neural network weights that are intrinsically compatible with the Lie group structure and do not require explicit parameterization. Exploiting the geometric properties of Lie groups, this approach circumvents parameterization singularities and enables a global search for optimal weights. The ultimate boundedness of all error signals -- including the neural network weights, the coordinate-free configuration error function, and the tracking velocity error -- is established using Lyapunov's direct method. To validate the effectiveness of the proposed method, we provide illustrative simulation results for decentralized formation control of multi-agent systems on the Special Euclidean group.

Geometric Fault-Tolerant Neural Network Tracking Control of Unknown Systems on Matrix Lie Groups

TL;DR

This work develops a geometric neural network tracking controller for systems evolving on matrix Lie groups , including unknown dynamics, actuator faults, and bounded disturbances. By leveraging a left-invariant tangent bundle and intrinsic Lie-group calculus, it formulates coordinate-free NN learning rules that operate directly on (and its Lie algebra), yielding global-like weight search and Lyapunov-based stability results. The authors prove uniform ultimate boundedness of both the configuration error and velocity error , as well as NN weight errors , and extend the framework to decentralized formation control on . Numerical simulations on a multi-agent formation with unknown inertia and actuator faults demonstrate robust formation maintenance and bounded control effort, highlighting practical applicability to robotic systems with uncertain dynamics.

Abstract

We present a geometric neural network-based tracking controller for systems evolving on matrix Lie groups under unknown dynamics, actuator faults, and bounded disturbances. Leveraging the left-invariance of the tangent bundle of matrix Lie groups, viewed as an embedded submanifold of the vector space , we propose a set of learning rules for neural network weights that are intrinsically compatible with the Lie group structure and do not require explicit parameterization. Exploiting the geometric properties of Lie groups, this approach circumvents parameterization singularities and enables a global search for optimal weights. The ultimate boundedness of all error signals -- including the neural network weights, the coordinate-free configuration error function, and the tracking velocity error -- is established using Lyapunov's direct method. To validate the effectiveness of the proposed method, we provide illustrative simulation results for decentralized formation control of multi-agent systems on the Special Euclidean group.
Paper Structure (20 sections, 9 theorems, 85 equations, 3 figures)

This paper contains 20 sections, 9 theorems, 85 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. NN-based Adaptive Control
  4. Control Systems on Lie Groups
  5. Adaptive Control of Multi-Agent Systems
  6. Preliminaries and Problem Statement
  7. Notation
  8. Matrix Lie Groups
  9. NNs as Function Approximators on Matrix Lie Groups
  10. Uncertain Euler-Poincaré Control Systems
  11. Neural Network Controller
  12. Neural Network Formation Control of Unknown Multi-Agents: A Case Study
  13. Decentralized Formation Control of Multi-agents on $\text{SE}(3)$
  14. Numerical Analysis
  15. Conclusions
  16. ...and 5 more sections

Key Result

Lemma 1

Let $f\in\mathcal{C}^0(G)$ be any continuous ($\mathbb{R}^k$-valued) function on the Lie group $G\subset \mathbb{R}^{N\times N}$. There exists a continuous extension of this function $\bar{f}\in\mathcal{C}^0(\mathbb{R}^{N\times N})$ such that $\bar{f} (g)= f(g)$ for all $g\in G$.

Figures (3)

  • Figure 1: Unknown time-dependent external disturbances
  • Figure 2: (a) Rotation angels of vehicles; (b) 3D position trajectories of vehicles. Curves corresponding to the virtual leader and desired offsets are black, Vehicle 1 is blue, Vehicle 2 is green, and Vehicle 3 is red.
  • Figure 3: (a) Velocity error for Vehicle 1 when compared with the nominal error dynamics; (b) Control input for vehicle 1.

Theorems & Definitions (31)

  • Lemma 1: Extension of continuous functions
  • proof
  • Lemma 2: Extension of smooth functions
  • proof
  • Remark 1
  • Definition 1
  • Remark 2
  • Proposition 1
  • proof
  • Definition 2
  • ...and 21 more