“…Based on the Abbe principle, the measuring axis of the position sensor, typically a linear encoder or a laser interferometer, is required to be aligned coaxially with the axis of motion of the object to avoid the Abbe error caused by angular error motions of the stage [4][5][6]. The positioning systems for ultra-precision metrological applications, such as the line scale comparator and the metrological scanning probe microscope, are basically designed to obey the Abbe principle [7,8]. Many other positioning systems, such as those for machine tools and coordinate measuring machines (CMMs), however, do not satisfy the Abbe principle [9,10].…”