Abstract: The machine (20) comprises: a supporting structure (24, 26); a movable unit (28) mounted on the supporting structure (24, 26) so that it can translate along a first working direction (Z) and along a second direction (X) towards and away from a stationary workpiece-carrying structure (88); a tool-carrying unit (10, 11, 12) carried by the movable unit (28); and a driving system for controlling the movement of the movable unit (28) in the working direction (Z). The driving system includes a first motor unit (60) for controlling the rotation of a driving shaft (62) and a mechanism for converting the rotational movement of the shaft (62) into the translational movement of the movable unit (28). The motion conversion mechanism comprises a cam member (76), which is rotatably mounted on the movable unit (28) and the rotation of which is controlled by the driving shaft (62), and a roller member (78) which is rotatably mounted on the supporting structure (24, 26) and on which the cam member (76) rests.