Variable Length Pendulum

Abstract

This chapter concerns a trajectory tracking control problem for a pendulum with variable length, which is an underactuated mechanical system of 2-DOF (degree of freedom) with a single input of adjusting the length of the pendulum. The goal of this chapter is to design a time-invariant control law to pump appropriate energy into the variable length pendulum for achieving a desired swing motion (trajectory) with given desired energy and length of the pendulum. First, this chapter shows that it is difficult to avoid singular points in the controller designed by using the conventional energy-based control approach in which the total mechanical energy of the pendulum is controlled. Second, it presents a tracking controller free of singular points by using only the kinetic energy of rotation and the potential energy of the pendulum and not using the kinetic energy of the motion along the rod. Third, it analyzes globally the motion of the pendulum and clarifies the stability issue of two closed-loop equilibrium points. Finally, this chapter shows numerical simulation results to validate the presented theoretical results.