Swirling Pendulum Dynamics and Control: A Pedagogical Perspective

The swirling pendulum is an underactuated, two-link, two-degree-of-freedom mechanism consisting of a pair of swirling and swinging links, that on account of its unusual out-of-the plane inertial coupling exhibits interesting dynamics and dynamical properties like multiple (eight) isolated equilibria (both stable as well as unstable), regions with loss of relative degree, loss of controllability and loss of inertial coupling. This work is an attempt to highlight the utility of the swirling pendulum as an attractive as well as a simple system for demonstrating a range of concepts/notions a learner would typically encounter in academic courses related to controls and dynamics. To achieve this goal, we start by presenting a (yet incomplete) summary of modeling, analysis, and control design concepts that can be studied in linear or nonlinear frameworks in various courses. We discuss as examples, the problems of input-output linearization and partial feedback linearization picked up from this summary. Furthermore, we also present inversion-based selective tracking control for tracking trajectories on the swirling pendulum outputs. In doing so, we explain how the problem of system zeros on the imaginary axis (that results in an unstable inversion-based feedforward controller) is avoided. We conclude by mentioning the problems that can be taken up as future work.