Modeling, Control and Variational Integration for an inverted pendulum on $S^{1}$

The dynamics of an inverted pendulum naturally evolves on the nonlinear manifold
$S^{1}$
. The paper proposes the modeling of the dynamics of an inverted pendulum on the nonlinear manifold
$S^{1}$
. The paper also proposes a variational integrator for the dynamics of the inverted pendulum directly on
$S^{1}$
. The variational integration results in the conservation of configuration space as well as energy as compared to Runge-Kutta methods which destroys the configuration space
$S^{1}$
and is not able to conserve the energy. A control law is also proposed on
$S^{1}$
to stabilize the pendulum at a given reference configuration. These are illustrated with numerical simulation and comparison results in the paper.