Abstract: The finishing apparatus is a grinder or like device having a workpiece holder manipulator which is translatable along X and Y axes and rotatable about a Z axis, an axis of tilt with respect to the plane of the X and Y axes and an axis normal to the plane of tilt. The grinder head is mounted on an arm which moves toward and away from the workpiece holder manipulator. All of the above movements are hydraulically powered through servo valves driven by feedback amplifiers, the feedback coming from position and rate transducers on the apparatus. Signals from those transducers are supplied through analog-to-digital converters to a digital computer with random access memory. The grinder is operated in the Teach mode by manually controlling the movement of the workpiece manipulator and grinder head to cause the grinding element to transverse the faces of a finished workpiece or a pattern thereof and the successive positions are stored in the memory.