A novel method of angular positioning error analysis of rotary stages based on the Abbe principle

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 lateral displacement relative to its sensing head that is fixed inside the stage housing. The actual measured angle is, therefore, different from the commanded angle, causing the angular positioning error of 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 error of the spindle, the offset and the size of encoder is analyzed and experimentally verified.

An Analysis of Angular Indexing Error of a Gear Measuring Machine

“…From Figure 8, the radial error motion (δ) of the tip of the lower center was estimated to be about 1.6 μm. The theoretical angular indexing error of the mandrel relative to the table top of the rotary stage can be calculated by Equation (16). The angular indexing error of the rotary stage is shown in Figure 10.…”

“…The measured angular indexing error is cumulated angular errors from the encoder to the polygon, including (1) the angular indexing error of the spindle at the top plane of the rotary stage, as given by Figure 10 and (2) the induced angular indexing error caused by the radial error motion of the tip of the lower center and the coaxiality error of the mandrel to the spindle, as given by Equation (16). It is noted that the polygon's angular accuracy was calibrated in advance by the above-mentioned autocollimator and the multi-tooth indexing table.…”

“…As pointed out in Figure 1 that the spindle itself also has tilt and radial errors that would result in the radial motion of the tip of lower center, denoted by δ in Figure 4, during the spindle rotation. Such a radial motion has to be measured in order to verify Equation (16). Figure 7 shows the setup for measuring the radial error motion of the lower center.…”

“…It is quite close to the goal of this study, but they only verified the error model by computer simulation and neglected error motions of the rotating shaft. It is known that the radial motion and tilt of the spindle will cause an angular positioning error at the rotary stage [16]. Here, not only the coaxiality and spindle errors but also the angular positioning error of the rotary table of an investigated GMM was measured.…”

“…The mandrel has a coaxiality error with respect to the ideal spindle axis. The spindle errors are generated from the bearing plane from which the radial error motion and tilt error motion of the shaft, or spindle, start [16]. It is obvious that the mandrel will rotate around a cone contour because the tip of the lower center has radial run-out error motion due to combined error sources at the tip's height.…”