Special Issue Article Proc IMechE Part B: J Engineering Manufacture 1–8 Ó IMechE 2017 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0954405416688936 journals.sagepub.com/home/pib A novel method of angular positioning error analysis of rotary stages based on the Abbe principle Yi-Tsung Li and Kuang-Chao Fan Abstract Abbe error is the inherent systematic error in linear displacement measurement due to the measuring axis being out of line with the moving axis. The resulting gap is called the Abbe offset, which will multiply the angular pitch error of the moving stage to become the positioning error of the linear stage along the moving axis. Analogous to the Abbe principle, in the rotary stage, the rotary encoder is used to detect the worktable’s rotational angle. The encoder is normally mounted at a distance from the bearing. This distance can be also regarded as Abbe offset. Due to the inherent tilt and radial motions of the axis of rotation, the encoder’s rotating component, that is, the circular grating, would result in a lat- eral displacement relative to its sensing head that is fixed inside the stage housing. The actual measured angle is, there- fore, different from the commanded angle, causing the angular positioning errorof the rotary stage in machine tools and open-loop controlled system. In this article, the angular positioning error of the rotary stage caused by the tilt motion error and radial motion errorof the spindle, the offset and the size of encoder is analyzed and experimentally verified. Keywords Rotary stage, Abbe error, angular positioning error, tilt error, radial error Date received: 30 April 2016; accepted: 21 December 2016 Introduction In 1890, Ernst Abbe 1 proposed a measuring principle for the design of dimensional measuring instruments which was translated into English as: ‘‘The measuring apparatus is to be arranged in such a way that the dis- tance to be measured is a straight line extension of the graduation used as a scale.’’ Bryan 2 recognized this so- called Abbe principle as being the first principle of machine tool design and dimensional metrology and extended the concept to the straightness measurement to propose the Bryan principle as ‘‘The effective point of a straightness measuring system should lie along a line which is perpendicular to the direction of slideway travel and passes the functional point whose straight- ness is to be measured. If it is not possible, either the slideways that transfer the straightness must be free of angular motion or angular motion data must be used to calculate the consequences of the offset.’’ Zhang 3 later modified the Abbe principle with a new definition: ‘‘The line connecting the reference point and sensing point should lie in the sensitive direction.’’ He applied the principle to all cases of dimensional measurements, including one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) measurements; straightness and roundness measurements; and run-out measure- ments. Jaeger et al. 4 applied the Abbe principle in 3D design of an ultra-high-precision nanopositioning and nanomeasuring machines. Today, engineers and researchers worldwide use both the Abbe and Bryan principles as guidelines in the design of linear motion machines, such as linear positioning stages, machine tools and coordinate measuring machines (CMMs). 5–8 In addition to linear machines, modern industry uses a variety of rotary indexing devices, such as rotary stages, C-axis in machine tools, CMMs and robots. The rotational accuracy of the rotary stage is as impor- tant as the positional accuracy of the linear stage. Due to design, manufacturing and assembly errors, both Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan Corresponding author: Kuang-Chao Fan, Department of Mechanical Engineering, National Taiwan University, Room 139, College of Engineering Building, No. 1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan (R.O.C.). Email: fan@ntu.edu.tw