Global Dynamics and Control Problems for the 3D Pendulum

N. Harris McClamroch
University of Michigan

The classical planar pendulum has long been the paradigm for characterizing dynamic regularity, and its influence on technology has been profound. The planar pendulum has been a key academic example, motivating a vast amount of research on dynamics and control. The spherical pendulum, a 2D generalization of the planar pendulum, has also motivated dynamics and control research. A further generalization is the 3D pendulum, which consists of a rigid body, supported at a fixed, frictionless pivot, with three rotational degrees of freedom. A gravitational force and, perhaps, control forces and moments act on the 3D pendulum. Several different 3D pendulum models are introduced and used to analyze nonlinear dynamics properties of the uncontrolled 3D pendulum: equilibria, relative equilibria, local stability properties, and global dynamics properties. The 3D coupling and the global dynamics of the 3D pendulum give rise to new control challenges. A sample of stabilization results and attitude maneuver results are presented. An experimental laboratory facility, the triaxial attitude control testbed, is described as a physical implementation of the 3D pendulum. Comments are made about the importance of 3D pendulum models as academic benchmarks and as surrogates for new classes of models in robotics and space vehicles.