# Multi-Probe Measurement Method for Error Motion of Precision Rotary Stage Based on Reference Plate
## 1 Introduction  
The precision rotary stage is a mechanical device engineered for highly accurate rotational positioning and motion control, enabling controlled angular displacements with minimal errors for tasks demanding ultra-fine adjustments. It typically consists of components such as a rotor and stator and is extensively utilized in the semiconductor industry and the domain of precise measurement. The error motion of its axis refers to the undesired deviation in the position or orientation of a moving component from its ideal path during rotary displacement. The error motion, especially the tilt error motion, can significantly affect the machining and measuring accuracy of products. Even minimal tilt error motion can, through amplification, diminish the motion accuracy of the rotary stage in both axial and radial directions. Therefore, in order to evaluate and improve the performance of the precision rotary stage, it is particularly important to measure the error motion [1, 2]. Typically, the measurement method for the error motion of a rotary stage is similar to that of a machine tool spindle, achieved by affixing a standard artifact (such as astandard sphere or cylinder) to the spindle [3]. The sensor measurement data include both the spindle error motion and the form error of the standard artifact. By utilizing suitable error separation methods, these two components can be efficiently separated. The most commonly employed error separation techniques may be classified into three main categories based on various separation principles: the reversal method [4; 5], the multi-step method [6; 7], and the multi-probe method [8; 9]. Each category may be further subdivided based on certain application contexts. Recent scholarly focus has increasingly centered on in situ measurement of machine tool spindles [10] and the application of measurement methodologies in ultra-precision machining. By integrating error motion measurement with error compensation techniques, researchers have developed feasible solutions for rapid calibration in ultra-precision machining processes [11; 12]. Nevertheless, the majority of the aforementioned measuring techniques primarily concentrate on radial and axial error motion, while investigation into the measurement of tilt error motion is comparatively limited. The spindle error analyzer (SEA) developed by Lion Precision Company is extensively utilized to measure error motion in precision spindles. It employs capacitance sensors to assess the double-ball standard rod. It integrates with dedicated data acquisition and processing software to execute the measurement of the 5 degrees-of-freedom (DOF) error motion of the spindle [13]. [PERSON] devised a spindle error motion measurement system and ascertained the radial and tilt error motions of the spindle by evaluating data gathered from various cross-sections [14]. In another study, [PERSON] performed trials employing the Estler reversal method to separate tilt and face error motions from the form error of the typical artifact [15]. Additionally, [PERSON] employed the reverse method to quantify the radial, axial, and tilt error motions of a rotary stage and developed a specialized sensor fixture to improve the efficiency and convenience of the measuring process [1]. The aforementioned measurements of tilt error motion are predominantly derived from the reverse method. Throughout the measurement procedure, both the standard artifact and the sensor must be precisely rotated by 180\({}^{\circ}\). This process may introduce errors due to reverse installation. Moreover, due to the necessity of the reversal process, in situ measurement is not feasible [16]. These disadvantages have constrained the widespread implementation of the reverse method in measuring tilt error motion. In recent years, laser interferometry-based measurement systems have seen increasing application in spindle error measurement [17; 18; 19; 20; 21; 22].