Machine learning based state observer for discrete time systems evolving
  on Lie groups
Soham Shanbhag Dong Eui Chang
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.