Nonlinear control and reduction of underactuated systems with symmetry. I. Actuated shape variables case

We address nonlinear control and reduction of a class of high-order underactuated mechanical systems with kinetic symmetry called Class-I systems. Class-I systems are underactuated systems with actuated shape variables, decoupled inputs, and integrable normalized momentums. We show that all Class-I underactuated systems can be transformed into cascade systems in strict feedback form with a zero-dynamics that is a Lagrangian system itself.