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 $(\S^1)^2$. 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 $(\S^1)^2$. In the other case, the second link can rotate without constraint; global Lagrangian dynamics are developed on the configuration manifold $\S^1 \times \S^2$. 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.