## 1 Introduction
However, these models did not consider the influence of the motion of one rotating axis on the position error of another rotating axis after the long-term operation of five-axis machines. In the study of the geometric error measurement and identification methods of five-axis machine tools, the positioning error of each linear axis, two straightness errors, three corner errors, and the perpendicularity between the two axes can be directly measured by a laser interferometer or other optical measuring systems [19; 20; 21]. Laser tracer technology is used for the volumetric error mapping and compensation of geometric errors of machine tools [22]. [PERSON] et al. [23] used a tracking interferometer to measure and compensate the geometric error of a small CNC machine. [PERSON] et al. [24] proposed a novel concept of an \"open-loop\" tracking interferometer for machine tool volumetric error measurement. [PERSON] et al. [25] proposed a new methodology to measure thermal distortion in large machine tools using a single tracking interferometer and designed a thermal distortion compensation system [26]. However, the direct measurement of the geometric error of a rotating axis is difficult and measurement standards has not yet been unified. In a study of the geometric error detection of rotating axis, [PERSON] and [PERSON] [27] developed a measuring instrument called the \"3D probe-ball\", which could be used to measure the overall position error of a five-axis machine tool. [PERSON] et al. [28] used the two measuring paths of double ball bar (DBB) along the rotating axis to determine the structural error of rotating axis, including two perpendicularity errors and two position errors. [PERSON] et al., [29] also introduced a set of DDBs to calibrate the measurement mode of the rotary axis position error. The measurement capabilities of DBB are limited because of the limited adjustment range of DBB rod length, which necessitates multiple switching cycles in the measurement mode of a system when decoupling linear and angular errors. In order to solve the problem that the measuring instruments such as laser interferometer and DBB only detect the comprehensive error of single direction at a time. [PERSON] [30] developed a contact type measuring device named the R-test for the geometric error of a rotating axis. The device could simultaneously detect combined displacement errors along three directions, which greatly improved measurement efficiency. However, in order to avoid collision between the three displacement sensors and the precision ball, the measuring requirements of R-test are relatively high. Operators who use such measuring instruments for the identification of position errors must be experienced, so it cannot be widely used. [PERSON] et al. [31] improved the measurement equipment of the contact R-test and developed rotating axis position error detection equipment using a laser displacement sensor instead of a contact displacement sensor, avoiding the occurrence of collision, but did not solve the problem of high preset accuracy. For the first time, an automatic error calibration scheme was adapted by [PERSON] [32], which used the trigger probe installed on the spindle of the machine tool to measure the space position of four standard blocks which were clamped on the C-axis worktable, and then the geometric error of the rotary axis was calibrated. [PERSON] [33] proposed a location error identification method using a touch-trigger probe and a test-piece on a five-axis machine tool with a tilting head. In their nature, all of these methods involve the contact detection of block standard parts. When the rotating attitude changes, the detection position and range are limited by the influence of sample geometry. Touch-trigger probes have become popular because they can improve detection quality and efficiency [34; 35]. However, current research results are not comprehensive. For example, the commercial software AxiSet Check-Up provided by Renishaw can only identify two linear offset errors for each rotary axis.