Nonlinear control and reduction of underactuated systems with symmetry. II. Unactuated shape variables case

For pt. I see ibid., p.4158-63 (2001). We address nonlinear control and reduction of two classes of high-order underactuated mechanical systems with kinetic symmetry called Class-II and Class-III systems. Class-II systems are underactuated systems with unactuated shape variables, decoupled inputs, and integrable normalized momentums (all to be defined). We show that all Class-II underactuated systems can be transformed into cascade systems in nontriangular quadratic form using an explicit change of coordinates and control. In addition,, we characterize a subclass of Class-II systems, called Class-III systems, that can be explicitly transformed into cascade systems in strict feedforward form. This allows application of existing nonlinear control design methods like nested saturations and feedforwarding to control of underactuated systems