Tracking fast inverted trajectories of the underactuated Acrobot

The Acrobot is a simple underactuated system consisting of a double pendulum with an actuator at the second joint only. We derive a set of exact trajectories of the nonlinear equations of motion, which involve inverted periodic motions. The trajectories can be made arbitrarily fast by an appropriate choice of the Acrobot mass and length parameters. Next, we present a nonlinear control law and show how it can be applied to the Acrobot to track these trajectories. In simulations we compare tracking results for our controller and one based on pseudo-linearization. The pseudo-linearizing controller produces significant error for a 1 Hz trajectory, while ours produces none. Finally, we present experimental results which demonstrate that the assumptions of the theory were not overly restrictive. In particular, peak-to-peak oscillations of joints as large as 850 were obtained, despite real-world effects, such as joint friction, inexact parameter values, and noisy and delayed joint velocity data.