Abstract: The invention is concerned with performing grinding operations on a workpiece at predetermined locations along its length and is described in connection with a machine for sharpening broaches in which a broach to be sharpened is mounted on a fixed bed, the positions of all the teeth on the broach are sensed by a probe during a first or forward pass of a work head along the whole length of the broach, the information so obtained is fed to a micro-processor and the broach teeth are sharpened in turn by a grinding wheel during a return pass of the work head along the broach using the information obtained by the sensing probe and input data fed initially into the micro-processor by means of a keyboard on a control panel.