## 1 Introduction  
Because they possesses the advantages of high efficiency and strong capability in machining complex surfaces [1], multi-axis computerized numerical control (CNC) machine tools are widely used in industrial production such as in the automobile, ship manufacturing, and aerospace industries. Rotary axes are important moving parts of a multi-axis CNC machine tool, which has six degrees-of-freedom (DOF) geometric motion errors when it moves. The geometric motion errors of a rotary axis are important factors affecting the accuracy of the machined parts [2]. In order to improve the machining accuracy of multi-axis CNC machine tools and machining centers, scholars worldwide have conducted a considerable amount of research to compensate the geometric motion errors of the rotating shaft [3; 4; 5; 6]. Geometrical error compensation technology is generally divided into three steps: error measurement, error model establishment, and error compensation [5]. Therefore, error measurement is the basis of improving the motion accuracy of a rotary axis. This paper focuses on precisely measuring the geometric motion errors of a rotary axis. Compared to the linear axis, research on the measurement of multi-DOF geometric errors of a rotary axis is relatively late [6]. At present, the main measurement methods are divided into contact measurement and noncontact measurement. The contact measurement methods include two kinds of methods based on capacitance sensor and circular trajectory, respectively. Both of them cannot measure the angular positioning error which is the most important parameter of a rotary axis. [PERSON] et al. used a built-in capacitance sensor to measure radial motion error [7]. [PERSON] et al. used a cylindrical capacitance sensor to measure motion error of milling machine spindle. The sensing range would be \(\pm 30\)\(\upmu\)m [8]. [PERSON] et al. used T-type capacitive sensor to measure five DOF motion errors [9]. The contact measurement method, based on capacitive sensors, has the advantages of compact structure and low cost, but its measurement range is small. The measurement methods of a circular trajectory include the ball-bar and R-test. [PERSON] et al. put forward the measuring principle of measuring the motion error of rotary axis by using the ball-bar in 1982, and it has been used to measure the geometric error of three-coordinate machine and CNC machine tools [10]. [PERSON] et al. use a double ball-bar clockwise and anticlockwise measured rotary axis and obtained five DOF motion errors [11]. [PERSON] et al. used an R-test to measure the trajectories of the rotary axis at different positions and heights, and the five DOF motion errors were obtained [12]. The measurement methods of a circular trajectory have the advantages of using simple instruments, convenient portability, and low system costs; however, error decoupling is complex, the measurement of angle positioning error cannot be realized, and the measurement results are not given in real time. Moreover, because the length of the ball bar and R-test measuring rod is short and can only vary according to limited spacing, there are great limitations in measuring large CNC machine tools and compensating spatial errors. There are many kinds of noncontact optical measurement methods, including interferometry-based, laser tracker-based, diffraction-based, laser collimation-based, and polyhedral prism-based methods. The interference method usually uses a single-frequency or dual-frequency laser interferometer as the core device, and its measurement has the advantage of high precision. [PERSON] et al. used two laser interferometers to measure the six DOF motion errors in three steps [13]. [PERSON] et al. used laser interferometers to measure two rotary axis Angular position error in 5-axis CNC machine tool in two steps [14]. In most cases, the laser interferometer can only measure a single parameter.