Diversities in the Inverse Dynamics Problem for Underactuated Mechanical Systems Subject to Servo-constraints

Underactuated mechanical systems are featured with less control inputs than degrees of freedom. Their performance goal may then be realization of specified in time outputs whose number coincides the number of inputs. A solution to the inverse simulation problem (servo-constraint problem), that is, determination of an input control strategy that forces the underactuated system to complete the partly specified motion, is a challenging task. Since systems may be “underactuated” in several ways and the servo-constraint realization may range from orthogonal to tangential, diverse formulations and analysis methods of the servo-constraint problem arise. The diversity is discussed with reference to some simple case studies. The governing equations are handled in two ways. A direct formulation in configuration coordinates is first motivated and is then compared to a setting in which the actuated coordinates are replaced with the outputs. The governing equations arise either as ODEs (ordinary differential equations) or DAEs (differential-algebraic equations). Some computational issues related to the ODE and DAE formulations are discussed, and simulation results for the sample case studies are reported.