The Use of Servo-Constraints in the Inverse Dynamics Analysis of Underactuated Multibody Systems

Underactuated mechanical systems have fewer control inputs than degrees of freedom. The specified in time outputs, equal in number to the number of inputs, lead to servo-constraints on the system. The servo-constraint problem is then a specific inverse simulation problem in which an input control strategy (feedforward control) that forces an underactuated system to complete the partly specified motion is determined. Since mechanical systems may be “underactuated” in several ways, and the control forces may be arbitrarily oriented with respect to the servo-constraint manifold, this is, in general, a challenging task. The use of servo-constraints in the inverse dynamics analysis of underactuated systems is discussed here with an emphasis on diverse possible ways of the constraint realization. A formulation of the servo-constraint problem in configuration coordinates is compared with a setting in which the actuated coordinates are replaced with the outputs. The governing equations can then be set either as ordinary differential equations (ODEs) or differential-algebraic equations (DAEs). The existence and nonexistence of an explicit solution to the servo-constraint problem is further discussed, related to so-called flat systems (with no internal dynamics) and nonflat systems (with internal dynamics). In case of nonflat systems, of paramount importance is stability of the internal dynamics. Simple case studies are reported to illustrate the discussion and formulations.