Servo-constraint realization for underactuated mechanical systems

The paper deals with underactuated mechanical systems, featured by less control inputs m than degrees of freedom f, m < f, subject to m servo-constraints (specified in time outputs) on the system. The arising servo-constraint problem (inverse dynamics analysis) is discussed with an emphasis on the way the servo-constraints are realized, varying from orthogonal to tangential, and a geometrical illustration of the different realization types is provided. Depending on the way the servo-constraints are realized, the governing equations are formulated either as ordinary differential equations (ODEs) or differential-algebraic equations (DAEs), and some computational issues for the ODEs and DAEs are discussed. The existence or non-existence of an explicit solution to the governing equations is further discussed, related to so-called differentially flat problems (without internal dynamics) and non-flat problems (with internal dynamics). It is shown that in case of non-flat problems with orthogonal realization of servo-constraints, stability of the internal dynamics must be assured. Simple case studies are reported to illustrate the proposed formulations and methodologies.

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