Geometric Formulations of Furuta Pendulum Control Problems

A Furuta pendulum is a serial connection of two thin, rigid links, where the first link is actuated by a vertical control torque while it is constrained to rotate in a horizontal plane; the second link is not actuated. The second link of the conventional Furuta pendulum is constrained to rotate in a vertical plane orthogonal to the first link, under the influence of gravity. Methods of geometric mechanics are used to formulate a new global description of the Lagrangian dynamics on the configuration manifold (S1)2. In addition, two modifications of the Furuta pendulum, viewed as double pendulums, are introduced. In one case, the second link is constrained to rotate in a vertical plane that contains the first link; global Lagrangian dynamics are developed on the configuration manifold (S1)2. In the other case, the second link can rotate without constraint; global Lagrangian dynamics are developed on the configuration manifold S1×S2. The dynamics of the Furuta pendulum models can be viewed as under-actuated nonlinear control systems. Stabilization of an inverted equilibrium is the most commonly studied nonlinear control problem for the conventional Furuta pendulum. Nonlinear, under-actuated control problems are introduced for the two modifications of the Furuta pendulum introduced in this paper, and these problems are shown to be extremely challenging.