## 3 Experiments  
The measurement process involved placing the hole plate at the center of the machine coordinate system and measuring it in three different directions on the machine bed. The hole plate, shown in Figure 4, is a length standard used for two-dimensional geometric measurements. It has a total of 44 holes with a nominal diameter of 20 mm, spaced at a nominal distance of 50 mm between each other. The plate has dimensions of 550 mm \(\times\) 550 mm and is made of NEXCERA, a material with a low thermal expansion coefficient of 0.03 \(\times\) 10\({}^{-6}\)/\({}^{\circ}\)C. Geometric errors in CMMs can result from imperfections  
Figure 3: Administrative information of CMM in XML type. in the faulty assembly in the system, leading to inaccurate measurements and affecting overall accuracy. To mitigate this issue, using a standard such as the hole plate enables more precise measurements and improved accuracy in analyzing geometric errors. The hole plate is a two-dimensional standard used to measure orthogonal directions or planar measurements. As the plate is rotated, the X and Y coordinates are changed in the CMM coordinate frame. By using this standard as a traceability reference, the features of the workpiece can be accurately measured, and their positions' coordinates can be obtained. For the measurement process, the CMM software automatically generated measurement paths for all the holes, and the inner diameter was calculated using a four-point circle [35; 41; 42]. Figure 5 displays the hole plate measurement in different directions. For each direction, the stylus calibration was performed as the probe direction was different. Furthermore, the axis alignment between the hole plate and the bed was adjusted, the touch-trigger probe was calibrated, and forward and reverse measurements were conducted, as required by ISO 230-2 [43]. ISO 230 comprises a broad spectrum of assessments that can be utilized to assess the comprehensive accuracy and functionality of machine tools. For example, [PERSON] et al. [44] employed specialized equipment such as LaserTRACER to gauge the geometric errors of CMMs. The positioning accuracy and repeatability of linear axes in machine tools are determined using ISO 230-2 and ISO 230-6 [45]. To measure the three-dimensional coordinate position, the hole plate was set up in three different directions. According to ISO 230-1, three-axis machine tools utilize a right-handed Cartesian coordinate system with three linear axes: the \(X\)-axis, \(Y\)-axis, and \(Z\)-axis. There are six errors in a linear axis, including a position error, two straightness errors, three angular errors, and there is a squareness error between axes. The coordinate system made up of the three linear axes encompasses a total of 21 errors. As displayed in Table 1, the table presents the 21 geometric errors of three linear axes and shows all geometric errors of three-axis CMM. Figure 4: The length standard known as the hole plate. Figure 5: Schematic diagram of the hole plate measurement in different direction (**a**) XY plane, (**b**) YZ plane, (**c**) XZ plane. In order to achieve accurate results, the axis alignment between the hole plate and the bed was adjusted before the measurement. In the previous article by [PERSON] and [PERSON], they described the evaluation of 21 geometric errors using CMM to measure the hole plate, which we will only briefly discuss here. To evaluate position errors, measurements are taken from both sides of the plate at the same orientation or in the upper and lower positions, and then averaged to eliminate the effects of rotational errors. For squareness errors, data sets are prepared in the same way as for the position error analysis. The nominal u values of the calibrated plate are subtracted from the measurement data set and a regression line is calculated. Similarly, the nominal v value is subtracted from the same set and a regression line is calculated.