Stabilization of a 3D Axially Symmetric Rigid Pendulum Nalin A. Chaturvedi, Fabio Bacconi, Dennis Bernstein, N. Harris McClamroch Department of Aerospace Engineering University of Michigan Ann Arbor, MI 48109 Abstract— 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 equi- librium. 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. I. I NTRODUCTION Pendulum models have provided a rich source of examples that have motivated and illustrated many recent developments in nonlinear dynamics and control. Much of the published research treats 1D planar pendulum models or 2D spherical pendulum models or some multi-body version of these. In a recent paper [1], we summarized a large part of this published research, emphasizing control design results. In addition, we introduced a new 3D pendulum model that, seems not to have been studied in the prior literature. A closely related paper [2], obtained controllers for a 3D asymmetric rigid pendulum. Controllers were introduced that were shown to provide asymptotic stabilization of a reduced attitude equilibrium. The reduced attitude of the 3D rigid pendulum is defined as the attitude or orientation of the 3D rigid pendulum, modulo rotation about a vertical axis. Stabilization results are provided in [2] for the hanging equilibrium, and for the inverted equilibrium. The present paper continues our research on control of 3D rigid pendula. The pendulum is supported at a pivot that This research has been supported in part by NSF under grant ECS- 0140053 and grant ECS-0244977. is assumed to be frictionless and inertially fixed. The rigid body is axially symmetric. The location of its center of mass is distinct from the location of the pivot. Forces that arise from uniform and constant gravity act on the pendulum. Two independent control moments are assumed to act about the two principal axes of the pendulum that are not the axis of symmetry; in other words, there is no control moment about the axis of symmetry of the pendulum. This is in contrast to the assumption of three independent control moments in [2]. We follow the development and notation introduced in [1]. In particular, the formulation of the models depends on construction of a Euclidean coordinate frame fixed to the pendulum with origin at the pivot and an inertial Euclidean coordinate frame with origin at the pivot. We assume that the pendulum fixed coordinate frame is selected to be coincident with the principal axes of the pendulum, and that the center of mass of the pendulum lies on the axis of symmetry of the rigid pendulum. We also assume that the inertial coordinate frame is selected so that the first two axes lie in a horizontal plane and the “positive” third axis points down. These assumptions are shown to guarantee that the angular velocity component about the axis of symmetry of the rigid pendulum is always constant. This conservation property allows development of reduced equations of motion for the 3D axially symmetric pendulum. The resulting reduced model is expressed in terms of two components of the angular velocity vector of the pendulum and the reduced attitude vector of the pendulum. The control problems that are treated in this paper involve asymptotic stabilization of an equilibrium of the reduced equations of motion of the 3D pendulum; this corresponds to stabilization of a relative equilibrium of the 3D pendulum. The relative equilibrium corresponds to either the hanging reduced equilibrium or the inverted reduced equilibrium with a pure spin about the pendulum’s axis of symmetry. The main contributions of this paper are the develop- ment of controllers that asymptotically stabilize the hanging relative equilibrium, the development of controllers that asymptotically stabilize the inverted relative equilibrium, and for the special case that there is zero angular velocity about the axis of symmetry of the pendulum, development of controllers that asymptotically stabilize either the hanging reduced equilibrium or the inverted reduced equilibrium. If the angular velocity component about the axis of symmetry Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005 WeB13.2 0-7803-9568-9/05/$20.00 ©2005 IEEE 5287