Stabilization of a 3D Axially Symmetric Rigid Pendulum

Models for a 3D pendulum, consisting of a rigid body that is supported at a frictionless pivot, were introduced in a recent 2004 CDC paper [1]. Control problems were posed based on these models. A subsequent paper, in the 2005 ACC [2], developed stabilizing controllers for a 3D rigid pendulum assuming three independent control inputs. In the present paper, stabilizing controllers are developed for a 3D rigid pendulum assuming that the pendulum has a single axis of symmetry that is uncontrollable. This assumption allows development of a reduced model that forms the basis for controller design and closed loop analysis; this reduced model is parameterized by the constant angular velocity component of the 3D pendulum about its axis of symmetry. Several different controllers are proposed. The first controller, based on angular velocity feedback only, asymptotically stabilizes the hanging equilibrium. Then controllers are introduced, based on angular velocity and reduced attitude feedback, that asymptotically stabilize either the hanging equilibrium or the inverted equilibrium. These problems can be viewed as stabilization of a Lagrange top. Finally, if the angular velocity about the axis of symmetry is assumed to be zero, controllers are introduced, based on angular velocity and reduced attitude feedback, that asymptotically stabilize either the hanging equilibrium or the inverted equilibrium. This problem can be viewed as stabilization of a spherical pendulum.