Nonlinear stabilizing control for a class of underactuated mechanical systems with multi degree of freedoms

An underactuated mechanical system with n(n≥3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n~(n\ge 3)$$\end{document} degree of freedoms (DOFs) is a complicated nonlinear system. This paper develops a new strategy to solve the nonlinear stabilizing control problem for this kind of mechanical systems. First, we introduce a coupled relationship between control torques. It changes the n-DOF underactuated system into a cascade-connected system, which has a 2-DOF driven subsystem and a (n-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n-2)$$\end{document}-DOF stable driving subsystem. And then, we analyze the passivity of the driven subsystem and discuss how to design an passivity-based controller that stabilizes the driven subsystem at the origin. Finally, the stabilization of the n-DOF underactuated system is achieved by the triangle lemma. Our proposed strategy transforms the stabilization of the n-DOF underactuated system into that of the 2-DOF driven subsystem. This makes the structure of the control system simple and also makes the problem of stabilizing a multi-DOF underactuated system easy to handle. As an application of the strategy, we give detailed statements of using it to achieve the global stabilization of a 3-DOF underactuated mechanical system called spring-coupled horizontal three-link underactuated manipulator. Simulation results demonstrate its validity.