Approximate feedback linearization of the Acrobot tracking error dynamics with application to its walking-like trajectory tracking

The purpose of this paper is to provide theoretical framework enabling to design tracking feedback for a general Acrobot trajectory which allows rigorous convergence proof. It is based on the partial exact feedback linearization of the Acrobot model followed by further approximate feedback linearization of the tracking error dynamics for arbitrary target trajectory. The approach presented here enables to prove the convergence in a rigorous way at least for small initial tracking errors. The slight novelty here is that neglecting is made with respect to tracking error along any general trajectory to be tracked, not just in some neighborhood of fixed working point. To demonstrate viability of this approach, simulations of a tracking of a walking-like cyclic trajectory are presented. The walking includes several steps including impacts between them. As a matter of fact, the exponentially stable tracking during the swing phase only is capable to stabilize overall walking, including the effect of the impacts.