Geometric formulations of Furuta pendulum control problems

Abstract

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.