## 1 Introduction  
In the modern manufacturing industry, ultra-precision multi-axis machining has become the most important development direction [1; 2]. In addition, along with the increasing need for workpieces with complex geometries, the demand for multi-axis CNC machine tools has significantly increased in recent years [3; 4]. With regard to the efficiency and convenience of manufacturing, multi-axis machine tools with rotary tables have superior characteristics, such as the ability to aid in manufacturing deep holes in two different orientations [5; 6; 7]. Moreover, many workpieces with precise angular specifications, such as polyhedrons, are machined through tilting and rotating the rotary tables to meet the requirements for precise positioning and orientation. Therefore, among the multi-axis machine tools, the five-axis CNC machine tools are more widely applied than three-axis CNC machine tools because they have two additional rotary axes. However, the performance of rotary tables immediately affects the position and orientation of the tools relative to the workpieces and the machining accuracy of the machine tools [7; 8; 9]. In general, 6-degree-of-freedom (6-DOF) geometric errors greatly affect the performance of rotary tables [10; 11]. Consequently, how to accurately measure and identify the 6-DOF geometric errors of rotary tables is particularly important for improving the accuracy of machine tools. As defined by the International Organization for Standardization (ISO) 230-7, the position-independent geometric errors (PIGEs) and position-dependent geometric errors(PDGEs) of a rotary axis are described as location and component errors, respectively [12; 13; 14]. In ISO 10791-1 to 10791-3, quasi-static tests to calibrate the static errors of the axis average line of rotary axes have been proposed [15; 16; 17]. Furthermore, for measuring the position and orientation errors of rotary axes in five-axis machine tools, ISO 10791-6 presents dynamic interpolation tests that use the double ball-bar or the R-test [18; 19]. Although the measurement methods mentioned in these ISO standards are effective for identifying the geometric errors of the rotary axes, they lack efficiency and the automated calibration procedures that embody the spirit of Industry 4.0 [7; 20]. Over the past few decades, many research studies for identifying the geometric errors of rotary axes have been reported that use double ball-bar (DBB), touch-trigger probes and their artifacts, and geometrical optics methods, as well as interferometry techniques [21; 22; 23]. For example, [PERSON] et al. used the double ball-bar (DBB) to identify PIGEs in rotary axes of a five-axis machine tool by establishing specific testing paths [24]. Furthermore, [PERSON] et al. presented a decoupled method to identify the PIGEs and PDGEs of a rotary axis based on a double ball-bar [25]. By using touch-trigger probes and their artifacts, [PERSON] et al. proposed a series of schemes to calibrate the error map and the location errors of rotary axes with on-the-machine measurements of test pieces [20; 26]. In our earlier research, a set of methodologies was also proposed for measuring the PIGEs and PDGEs of rotary axes by using a probe and artifacts. In addition, based on geometrical optics methods, [PERSON] et al. proposed a dual optical path measurement method (DOPMM) to identify error parameters in the rotary axis of a machine tool by using two precise laser displacement sensors and a self-designed fixture [27]. A single-mode fiber-coupled laser and two retroreflectors were adopted by [PERSON] et al. for simultaneously measuring 5-DOF motion errors of a rotary axis [28]. In recent years, with the rapid development of interferometry techniques, there has been many studies proposing geometric error methodologies for rotary axes that use laser interferometers. Using laser interferometers for measuring geometric errors in machine tools has two advantages: (1) a larger angle measurement range, and (2) minimizing the error influences during measurement [9; 29].