Dynamics and control of a multi-body planar pendulum

The explicit equations of motion for a general n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-body planar pendulum are derived in a simple and concise manner. A new and novel approach for obtaining these equations using mathematical induction on the number bodies in the pendulum system is used. Assuming that the parameters of the system are precisely known, a simple method for its control that is inspired by analytical dynamics is developed. The control methodology provides closed-form nonlinear control and makes no approximations/linearizations of the nonlinear system. No a priori structure is imposed on the controller. Globally, asymptotic Lyapunov stability is achieved along with the minimization of a user-provided control cost at each instant of time. This control methodology is then extended to include uncertainties in the parameters of the system through the use of an additional continuous controller. Simulations showing the simplicity and efficacy of the approach are provided for a 10-body pendulum system whose model is only known imprecisely. The ease with which the uncertain system can be controlled to move from any initial state to various final so-called inverted configurations is demonstrated.