Analysis of stable model inversion methods for constrained underactuated mechanical systems

Inverse dynamics methods are presented and analyzed in detail for the feedforward control of underactuated manipulators. Thereby, two different exact strategies are discussed: the standard stable inversion method, which relies on the formulation of a two-point boundary value problem, and an alternative optimization problem formulation, which does not require any boundary conditions. This paper proves that the solution of the optimization problem converges to the solution of the boundary value problem in the limit case where the lengths of the pre- and post-actuation phases tend to infinity. It is also shown that both strategies can be directly based on the DAE form of the equations of motion, so that there is no need to explicitly derive the input-output normal form, and standard multibody modeling codes can be used. The developments are illustrated using examples of manipulators with passive and compliant joints.