Nonlinear control and reduction of underactuated systems with symmetry.III. Input coupling case

In this paper, we address nonlinear control and reduction of high-order underactuated mechanical systems with kinetic symmetry and input coupling. In some aerospace vehicles, the effects of the control of the attitude dynamics appears in the translational dynamics. We present a general framework for decoupling of these effects. The decoupling is done by applying a change of coordinates in explicit form that transforms the original system into a cascade nonlinear system. We obtain three types of cascade normal forms, namely, nonlinear systems in strict feedback form, strict feedforward form, and nontriangular quadratic form. For a class of underactuated systems that are differentially flat, we obtain the flat outputs automatically as a by-product of the decoupling change of coordinates