Machine learning based state observer for discrete time systems evolving on Lie groups

TL;DR

This work addresses state estimation for discrete-time systems evolving on Lie groups by introducing a chart-free, data-driven observer that keeps the estimate on the manifold. The core idea is to predict a Lie-algebra residual $v_k\in T_eG$ with a neural network and update the state via $\bar{x}_{k+1}=\exp_{\bar{x}_k}(v_k)$, using a Whitney embedding $\pi$ to define a training loss that measures embedding-distance between true and estimated states. The approach eliminates the need for charts or switching between local models and is demonstrated on a rigid-body system evolving on $SO(3)\times\mathbb{R}^3\times\mathbb{R}^3\times\mathbb{R}^3$, achieving significant error reductions and maintaining a negligible deviation from the Lie group manifold even under measurement noise. The results indicate that geometry-respecting, data-driven observers can provide robust state estimates for manifold-valued systems without requiring explicit system models, with practical implications for robotics and control where Lie-group dynamics are prevalent.

Abstract

In this paper, a machine learning based observer for systems evolving on manifolds is designed such that the state of the observer is restricted to the Lie group on which the system evolves. Conventional techniques involving machine learning based observers on systems evolving on Lie groups involve designing charts for the Lie group, training a machine learning based observer for each chart, and switching between the trained models based on the state of the system. We propose a novel deep learning based technique whose predictions are restricted to a measure 0 subset of Euclidean space without using charts. Using this network, we design an observer ensuring that the state of the observer is restricted to the Lie group, and predicting the state using only one trained algorithm. The deep learning network predicts an ``error term'' on the Lie algebra of the Lie group, uses the map from the Lie algebra to the group, and uses the group action and the present state to estimate the state at the next epoch. This model being purely data driven does not require the model of the system. The proposed algorithm provides a novel framework for constraining the output of machine learning networks to a measure 0 subset of a Euclidean space without chart specific training and without requiring switching. We show the validity of this method using Monte Carlo simulations performed of the rigid body rotation and translation system.

Figures (2)

• Figure 1: Proposed observer model for state estimation.
• Figure 2: State estimation results for the machine learning based observer for the rigid body system.