Flat-surfaced workpieces such as semiconductor wafers or sapphire disks are attached to a rotatable floating or rigid workpiece holder carrier that is supported by a pressurized-air flexible elastomer disk sealed air-chamber device and is rotationally driven by a flexible arm or a lug-pin device. The rotating wafer carrier rotor is horizontally restrained by a set of idlers that are attached to a stationary housing to provide rigid support against abrading forces. The abrading system can be operated at the very high abrading speeds used in high speed flat lapping with raised-island abrasive disks. A wide range of abrading pressures are applied uniformly across the surface of the workpiece. Vacuum can also be applied to the elastomer chamber to quickly move the wafer away from the abrading surface. Internal constraints limit the axial, lateral and circumferential motion of the workholder. Wafers can be quickly attached to the workpiece carrier with vacuum.