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.