The system comprises a headstock (1) in which is established a rotary shaft (10) driven via a cone (8) that can be coupled to a drive means, for example a robot, the shaft (10) being linked to a spring (11) which permits axial displacement of the shaft (10) in the upward and downward direction, while coupled to the other end, with the interposition of an eccentric (21), is a plate (2) for carrying an abrasive disc (3) intended for sanding the pins of the fasteners, following the operation of shaving said pins. The eccentric (21) transforms the rotation of the shaft (10) into an orbital motion so that, in combination with the upward and downward displacements by means of the spring (11), the abrasive disc (3) is successfully adapted on the surface on which the pin of the fasteners intended to sand is located.