A robotic surgery system includes an orienting platform, a support linkage movably supporting the orienting platform, a plurality of surgical instrument manipulators, and a plurality of set-up linkages. Each of the manipulators includes an instrument holder and is operable to rotate the instrument holder around a remote center of manipulation (RC). At least one of the manipulators includes a reorientation mechanism that when actuated moves the attached manipulator through a motion that maintains the associated RC in a fixed position.