A normal form for energy shaping: application to the Furuta pendulum

We derive a nonlinear control law for stabilization of the Furuta pendulum system with the pendulum in the upright position and the rotating rigid link at rest at the origin. The control law is derived by first applying feedback that makes the Furuta pendulum look like a planar pendulum on a cart plus a gyroscopic force. The planar pendulum on a cart is an example of a class of mechanical systems which can be stabilized in full state space using the method of controlled Lagrangians. We consider this class of systems as our normal form and for the case of the Furuta pendulum, we add to the first transforming feedback law, the energy-shaping control law for the planar pendulum system. The resulting system looks like a mechanical system plus feedback-controlled dissipation and an external force that is quadratic in velocity. Using energy as the Lyapunov function we prove local exponential stability and demonstrate a large region of attraction.