An offshore drilling vessel includes a floating hull subjected to heave motion. The hull includes a moonpool and a drilling tower near the moonpool. A drilling tubulars storage rack is provided for storage of drilling tubulars. The vessel includes a heave motion compensation support that is adapted to support a slip device whilst performing heave compensation motion relative to the heaving motion of the vessel. A racking device is provided with a heave motion synchronization system that is adapted to bring a tubular retrieved from the storage rack in heave motion into a vertical motion that is synchronous with the heave compensation motion of the string slip device. The racking device includes vertical rails and at least two separate motion arm assemblies mounted on said vertical rails. Each separate motion arm assembly includes its own vertical drive which is electrically connected to the heave motion synchronization system.