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

arXiv: https://arxiv.org/abs/2505.04725 · HTML: https://mlsanity.com/paper/2505.04725/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 66 references that are in the map.
- First posted on arXiv 7 May 2025.

## Builds on (most cited first, 11 of 11)

- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Stability Analysis using Quadratic Constraints for Systems with Neural Network Controllers](https://mlsanity.com/paper/2006.07579/) (2020)
- [Hamiltonian-based Neural ODE Networks on the SE(3) Manifold For Dynamics Learning and Control](https://mlsanity.com/paper/2106.12782/) (2021)
- [Deep Lagrangian Networks for end-to-end learning of energy-based control for under-actuated systems](https://mlsanity.com/paper/1907.04489/) (2019)
- [Reliably-stabilizing piecewise-affine neural network controllers](https://mlsanity.com/paper/2111.07183/) (2021)
- [Optimal Control Problems with Symmetry Breaking Cost Functions](https://mlsanity.com/paper/1701.06973/) (2017)
- [Integral Control on Lie Groups](https://mlsanity.com/paper/1410.7614/) (2014)
- [Adaptive Neural Network Stochastic-Filter-based Controller for Attitude Tracking with Disturbance Rejection](https://mlsanity.com/paper/2206.12476/) (2022)
- [Screw and Lie Group Theory in Multibody Kinematics -- Motion Representation and Recursive Kinematics of Tree-Topology Systems](https://mlsanity.com/paper/2306.17415/) (2023)

## Abstract (arXiv)

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 RN× 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.