Nonlinear Stabilization of the Rotational Inverted Pendulum using the Controlled Lagrangians Method

Control systems that have more degrees of freedom than the number of
actuators are called underactuated systems. The method of controlled
Lagrangians has been developed as a technique to stabilize
underactuated Lagrangian mechanical systems with symmetry. The basic
idea of this method is to modify the kinetic energy and the potential
energy to produce a new Lagrangian which describes closed-loop
dynamics. This method has the advantage that stabilization can be
understood in terms of energy-based Lyapunov functions which give
information on how to design control inputs to achieve closed-loop

The rotational inverted pendulum, which has two rotational degrees of
freedom and one actuator, is an example of an underactuated
system. The main purpose of this paper is to stabilize the rotational
inverted pendulum at the straight up state using the method of
controlled Lagrangians.  The control input designed by the present
method provides Lyapunov stability. Further another controller which
provides asymptotic stability to the inverted pendulum at the straight
up state is designed as well.