## 1 Introduction  
The method of mounting the workpiece onto a mandrel and rotating the mandrel by a two-center pivotal clamp to achieve angular positioning of the workpiece has been widely used in many machines, such as gear and cam grinding and measuring machines. One center is to be clamped and rotated by the spindle of a rotary table, and the other center is normally fixed to the rail. In an ideal case, the mandrel is coaxial, and the spindle axis and the spindle is in pure rotation. In real cases, however, due to manufacturing and assembly errors, both the spindle and the mandrel have certain inevitable systematic errors. On one hand, the spindle has inherent geometric errors in six degrees of freedom (6-DOF), as mentioned in ISO230-1, ISO230-7, and ASME B89.3.4 [1; 2; 3] as well as in the book by [PERSON] [4], which includes three translational errors and three rotational errors. On the other hand, the mandrel has inherent coaxiality errors with respect to the spindle axis due to misalignment of the two centers. All these errors induce an angular positioning error of the workpiece,resulting in dimensional error or measurement error. In the past, many researchers studied the techniques of multi-degree-of-freedom measurement (MDFM) on the geometric errors of the rotary stage. [PERSON] developed the error motion and the angular indexing measuring devices using position sensitive detectors (PSDs) and diffraction grating [5]. [PERSON] established a novel technique using diffractive grating and PSDs to calibrate the 4-DOF errors of a rotary table [6]. [PERSON] proposed a new measurement system based on a laser diode, PSDs, and a homogeneous transformation approach to separately measure six geometric errors of a rotary axis [7]. [PERSON] et al. proposed a method of using a dual optical path measurement method (DOPMM) to identify all 6-DOF errors of a rotary stage one by one [8]. [PERSON] proposed a novel measuring method for geometric error identification of the rotary table on five axis machine tools using a double ball bar [9]. [PERSON] proposed concentric circle grating and phase modulation interferometers to achieve concurrent measurement of all 6-DOFerrors of a rotary table [10]. [PERSON] et al. expressed a general methodology of self-calibration in dimensional metrology using a reversal or error separation method, including the spindle errors [11].  
Although many methods for measuring the error motions of spindles or rotary tables have been investigated, the angular positioning error caused by the coaxility error of two cascade axes of rotation has not been paid much attention. Previous studies by [PERSON] et al. on gear measurement were on a specially designed test rig, called a double-disc instrument, for measuring the involute profile [12; 13]. For the pitch measurement on a commercial gear measuring machine (GMM), [PERSON] et al. used a multi-step method to measure the pitch deviation of the gear [14]. [PERSON] et al. considered the offset of the probing position due to alignment error of the gear axis (same as the mandrel axis in this report) and proposed a method to compensate the induced pitch deviation in measurement [15]. 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. Their influences on the angular positioning error of the mandrel at the gear measuring plane was theoretically derived. This theoretically predicated error was validated by experimental results. It was found that the measurement accuracy of any GMM can be largely improved after the predicted angular positioning error is compensated.