Abstract: Apparatus and method for abrading or grinding a workpiece with great accuracy. An abrasive support unit is mounted on a vertical column for vertical and horizontal movement. The abrasive support unit is driven automatically along a linear horizontal path and transversely thereof while being driven rotationally. The abrasive support unit has an abrasive mounted thereon on a universal element that provides for universal movement of the abrasive. Servo means are provided for applying the abrasive support unit to the workpiece at a preset pressure. A detector in the form of a differential transformer detects variations in the pressure of application, due to variations in the uneveness of the workpiece, and develops a feedback input to the servo means for applying the abrasive support unit at the preset pressure.