The present work aims to incorporate control (or servo) constraints into finite-dimensional mechanical systems subject to holonomic constraints. In particular, we focus on underactuated systems, defined as systems in which the number of degrees of freedom exceeds the number of inputs. The corresponding equations of motion can be written in the form of differential-algebraic equations (DAEs) with a mixed set of holonomic and control constraints. Apart from closed-loop multibody systems, the present formulation accommodates the so-called rotationless formulation of multibody dynamics. The rotationless formulation has proven to be especially well-suited for the design of energy and momentum conserving schemes, which typically exhibit superior numerical stability properties (see [4, 7, 10]). Subsequent to the incorporation of the servo constraints, we deal with a reformulation of the underlying DAEs, which is amenable to a direct numerical discretization. To this end, we apply a specific projection method to the DAEs in terms of redundant coordinates. A similar projection approach has been previously developed in the framework of generalized coordinates by Blajer & Kolodziejczyk [12]. A numerical example is presented, which deals with a 3D rotary crane.