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

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. Designing machine learning based observers for systems evolving on Lie groups using charts would require training a machine learning based observer for each chart of the Lie group, 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 certain measure 0 subsets of the 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, the group operation, and the present state to estimate the state at the next epoch. This approach, being purely data driven, does not require a model of the system. The proposed algorithm provides a novel framework for constraining the output of machine learning networks to certain measure 0 subsets of a Euclidean space without training on each specific chart and without requiring switching. We show the validity of this method using Monte Carlo simulations performed of the rigid body rotation and translation system.