# Rapid Measurement and Identification Method for the Geometric Errors of CNC Machine Tools
## 1 Introduction  
The machining accuracy of CNC MTs decreases with the increase of their service life, the loss of their parts and so on. It is costly to maintain and improve the machining accuracy of MTs by upgrading the hardware. Alternatively, error compensation technology offers a significant means for improving the geometric accuracy of MTs and extending their service life [1]. Modeling, measurement and identification are important prerequisites for geometric error compensation [2]. In recent years, two methods: direct and indirect measurement [3], have focused on the geometric error measurement and identification of MTs. The direct measurement method involves obtaining all geometric errors directly through the measurement equipment, such as a laser interferometer [4]. [PERSON] et al. [5] designed a geometric error measurement system, including one pentaprism module, one PSD module, as well as one laser source, which is capable of obtaining both horizontal and straightness errors in one measurement. However, the system depends on the laser source, so it is necessary to ensure the stability of the laser source during the whole measurement process. In addition, the measurement results are affected by the coupling errors. [PERSON] et al. [6; 7] proposed a measurement system, which is mainly composed of two sinusoidal XY-grids and a two-dimensional slope sensor. The simultaneous measurement of fivedegrees-of-freedom motion, including three-dimensional displacement and a two-dimensional angle, is realized. Nevertheless, there are some disadvantages of the measurement system, such as the complex machining of a sinusoidal angle grid and the nonlinear errors in the displacement measurement. The indirect measurement method involves obtaining a part of the geometric error through the measurement system, then the remaining geometric errors are decoupled and identified through the integrated error model. The 3-axis MT has 21 geometric error parameters, so the indirect measurement method is widely applied because of its high efficiency. The classical transfer models of geometric errors include the vector method [8], error matrix method [9], rigid body kinematics method [10], multi-body system (MBS) theory [11], and so on. In the practical application environment, any mechanical system can be outlined and abstracted into MBS, so it is widely used in the kinematics analysis and precision modeling of complex mechanical systems, such as robots, MTs and CMM. [PERSON] et al. [12; 13] proposed a new probe-ball device, which is qualified to measure the overall position errors of a five-axis MT. Furthermore, the unknown and not directly measurable link errors can be estimated through the error model. However, due to the contact measurement, the accuracy of its measurement system is affected by the contact deformation, wear and other factors. [PERSON] et al. [14] revised the geometric integrated error modeling of 3-axis MTs. Furthermore, the measurement and identification method, based on a cross grid encoder, is proposed. It is suitable for \"on machine\" measurement. However, this method must be supplemented with a variety of inspection tools, so the detection and installation process is quite complex. The ball board is one of the recommended methods of ISO. [PERSON] et al. [15] designed a measurement system, a ball-bar test with a 45\({}^{\circ}\) wedge, to obtain the systematic geometric error of five-axis MTs. [PERSON] et al. [16] proposed a calibration method for the positional errors of CMM, which can be applied in the detection of the linear guideway of MTs as well. It can reduce the cyclic errors by collecting measurement datum in a dynamic mode. [PERSON] et al. [17] shows a compensation method of MTs, according to the error model based on the Abbe principle. However, these systems ([PERSON] et al. [15], [PERSON] et al. [16], [PERSON] et al. [17]) rely on a laser interferometer to identify the geometric error of the linear guideway.