A carrier head that has a housing, a base assembly, a retaining ring, a carrier ring, and a flexible membrane is described. The base assembly is vertically movable relative to the housing. The retaining ring is connected to and vertically movable relative to the base assembly and has a lower surface configured to contact a polishing pad and an inner surface configured to circumferentially surround the edge of a substrate to retain the substrate. The carrier ring is connected to and vertically fixed relative to the base assembly, circumferentially surrounds the retaining ring to prevent lateral motion of the retaining ring, and has a bottom surface configured to contact a polishing pad.