[align=center] A Theoretical Analysis and Application of Sequential Step Volumetric Measurement Ni Lifeng, Zhang Hongtao, Wu Hao, Yang Jianguo (Shanghai Jiaotong University) Charles Wang (Optodyne Incorporation, United States)[/align] Abstract: This paper introduces a method of sequential step volumetric displacement measurement. Because these errors can be separated from the volumetric displacement error, it is convenient and fast to measure the volumetric positioning accuracy. Meanwhile, the software can automatically generate the compensation codes according to the measured data, and then these codes are loaded into the CNC controller to improve the volumetric positioning accuracy of the machine. Examples of machine error compensation verify the correctness and high efficiency of this method. Key words : Sequential step measurement, Volumetric Error, Error compensation. Keywords: multi-step measurement, volume error, error compensation 1. Introduction CNC machine tools have relatively complex motion, so the various errors generated during their motion are also relatively complex. Taking a three-axis machining center as an example, there are 21 error elements, including 3 linear errors, 6 straightness errors, 3 perpendicularity errors, 3 pitch angle errors, 3 yaw angle errors and 3 rotation angle errors. Traditional measurement does not consider pitch angle, yaw angle and rotation angle errors, so the accuracy is not high, and the complete detection of the volume positioning accuracy of the machine tool is very complicated and time-consuming. In view of the above reasons, many international standardization organizations have proposed a method of measurement along the body diagonal[1]. The so-called body diagonal refers to the diagonal of the cuboid formed by the maximum stroke of the three feed directions of the machine tool table in the spatial rectangular coordinate system. The main reason why the international standardization organizations recommend this method is that the measurement of the body diagonal is very sensitive to various error elements. However, a major drawback of this method is that it cannot obtain enough information to separate the error elements during the measurement process, so it is difficult to compensate for the error[2]. Multi-step volumetric positioning measurement takes into account angular errors, thus achieving higher accuracy. Furthermore, compensation is applied to the measured results, making it practically meaningful. Compared to traditional laser interferometry (parallel to the X, Y, or Z axes of the machine tool), this measurement method is simpler and more time-efficient. The volumetric positioning error is determined by 21 machine tool errors, and the measurement accuracy is determined by the machine tool's repeatability. This method was applied to test two CNC machine tools, and compensation was then used to improve machine tool accuracy. 2. Three-Axis Machine Tool Error Measurement For a 3-axis machine tool, 21 machine tool errors can be represented as follows: Linear displacement error: Dx(x), Dy(y), and Dz(z) Straightness error in the horizontal plane: Dy(x), Dx(y), and Dx(z) Straightness error in the vertical plane: Dz(x), Dz(y), and Dy(z) Rotation angle error: Ax(x), Ay(y), and Az(z) Pitch angle error: Ay(x), Ax(y), and Ax(z) Yaw angle error: Az(x), Ax(y), and Ay(z) Perpendicularity error: Φxy, Φyz, and Φxz Here, D refers to linear error, the subscript indicates the direction of the error, the coordinates in parentheses are the position coordinates, A is the angular error, and the subscript is the rotation axis. Machine tool working volume: x from x=0 to X, y from y=0 to Y, z from z=0 to Z. The traditional body diagonal measurement method was once highly recommended due to its speed of detection. Body diagonal measurement involves moving the spindle along the body diagonal to a new point on the body diagonal, at which point a displacement error can be measured. Assuming the displacement of the spindle along the body diagonal is R, the measured error is the displacement error dR of the displacement R. Then the spindle can continue to be moved to measure the displacement error until the spindle moves to the other corner of the diagonal. The same measurement is performed on the other three diagonals. The positioning accuracy of each position measured above actually depends on the positioning accuracy of the three axes, and is usually also affected by the geometric accuracy of the machine tool [3]. It should be said that the body diagonal measurement method is a relatively good measurement method, but it cannot identify the source of error, and of course it cannot be used to compensate for the machine tool [4][5]. The displacement error measured along the body diagonal is a comprehensive reflection of the 21 errors of the machine tool. At the same time, we can regard the displacement error measured along the body diagonal as the projection of the position error generated when the three motion axes move respectively onto the body diagonal [6]. There are three displacement errors along each axis. The errors along the X axis are: Dx(x), Dy(x), Dz(x), and along the Y and Z axes are: Dx(y), Dy(y), Dz(y), Dx(z), Dy(z), Dz(z). The above 9 position errors actually include all 21 errors generated when the three axes move (linear displacement error, straightness error, rotation error, perpendicularity error, and even some other non-rigid body motion errors). Therefore, the 9 position errors reflect the spatial position accuracy of the machine tool. From the perspective of error compensation, for CNC systems with spatial position error compensation functions, compensating for these nine position errors is equivalent to compensating for the influence of all geometric error elements of the machine tool on the machine tool's position accuracy. For example, when compensating for the motion error of the X-axis, Dx(x) is compensated by the X-axis, while Dy(x) and Dz(x) can be compensated by the Y and Z axes respectively. Therefore, as long as the nine position error data are processed and transmitted to the CNC system according to the compensation format, geometric error compensation of the machine tool can be achieved, thereby improving the volume positioning accuracy of the machine tool. This leads to the proposal of multi-step volume positioning measurement. The biggest advantage of multi-step volume positioning measurement is that its measurement direction and motion direction can be different. Thus, the measurement results are sensitive to errors in multiple directions, and errors in multiple directions are included. By separating the errors from the whole to each direction, we can obtain more data than with traditional measurement methods, allowing for error separation and compensation. 3. Multi-step Measurement Method The difference between the multi-step measurement method and the traditional body diagonal measurement method lies in the use of multi-step measurement. The measurement process is shown in Figure 1. To perform multi-step measurements, the starting point (0, 0, 0) and ending point (X, Y, Z) of the diagonal must first be defined. Therefore, the machine tool's workspace range is X×Y×Z. Assuming n measurement points per axis, the total number of measurement points is 3n, and the increments for each axis are Dx, Dy, and Dz, where: Dx=X/n, Dy=Y/n, Dz=Z/n. As shown in Figure 2, the machine tool has four body diagonals. Here, we take one as an example, i.e., a→g. The path for measuring this diagonal using the multi-step measurement method is as follows: The moving optical target (plane mirror) mounted on the spindle starts from point a (0, 0, 0), moves Dx, pauses, and during the pause, the software automatically collects data. Then, it moves Dy in the Y-axis with the same feed rate and pause time, and finally moves Dz in the Z-axis with the same feed rate and pause time. These steps are repeated until the other point g on the body diagonal is reached. For the other three diagonals, the starting point and the increment of each axis need to be changed separately for measurement. [align=center] Figure 1 Multi-step measurement using a plane mirror as the target Figure 2 Four body diagonals[/align] As can be seen from the above process, each time the spindle moves to a new position in the direction of the body diagonal, the multi-step measurement method can measure three displacement errors. Moreover, the data measured along each axis direction is only generated independently by the spindle's movement along that axis direction. In this way, the measured error data can be separated into three independently generated axial directions, thereby achieving the purpose of error separation. For the cuboid formed by the maximum stroke of the machine tool table along the three guide rails, there are four body diagonals. In a Cartesian coordinate system, these can be represented as follows: From (0, 0, 0) to (X, Y, Z), denoted as PPP; From (X, 0, 0) to (0, Y, Z), denoted as NPP; From (0, y, 0) to (X, 0, Z), denoted as PNP; From (0, 0, Z) to (X, Y, 0), denoted as PPN; Where PPP indicates that the feed along the x, y, and z directions is along the positive directions of each coordinate axis; NPP indicates that the feed along the x direction is along the negative x-axis, while the feed along the y and z axes is along the positive y and z axes; PNP indicates that the feed along the y direction is along the negative y-axis, while the feed along the x and z axes is along the positive x and z axes; PPN indicates that the feed along the z direction is along the negative z-axis, while the feed along the x and y axes is along the positive x and y axes. 4. Measurement Application and Compensation Examples Figure 3 shows the multi-step volume measurement device. [align=center] Figure 3 Multi-step volume measurement device[/align] 4.1 Volume positioning error detection was performed on a Siemens 840 CNC machine tool, and corresponding compensation was performed based on the measured data. The machine tool working space is 800mm×600mm×600mm. During the measurement process, the test point increments in the X, Y, and Z directions were 20mm, 15mm, and 15mm, respectively, n=40. First, the measurement was performed without any compensation. The maximum volume error on the body diagonal was 60μm, as shown in Figure 4. [align=center] Figure 4 Measured displacement error of the four diagonals before compensation[/align] After processing the collected error data, the corresponding error compensation code was automatically generated according to the model of the control system used by the CNC machine tool. The error compensation code was input into the control system to compensate for its position error. To verify the compensation effect, the measurement was performed with volume error compensation. The error on the four diagonals after compensation is shown in Figure 5. [align=center]Figure 5 Measured displacement error of the four diagonals after compensation[/align] As can be seen from the figure, the volumetric error was relatively large before compensation, reaching a maximum of 60μm. After compensation, the volumetric error was significantly reduced, with the maximum volumetric error being only 6.8μm. It can be seen that the volumetric positioning accuracy of the machine tool was greatly improved after compensation. 4.2 Volumetric positioning error was detected on a JOBS-LINK five-axis CNC machine tool, and compensation was performed based on the measured data. Its working space was 900mm×720mm×720mm, n=30. During the measurement process, the test point increments in the X, Y, and Z directions were 30mm, 24mm, and 24mm, respectively. First, the measurement was performed without any compensation. The maximum volumetric error of the body diagonal was 58μm, as shown in Figure 6. [align=center]Figure 6 Measured displacement error of the four diagonals before compensation[/align] After processing the collected error data, the model of the control system used by the CNC machine tool is selected, and then the error data is input into the compensation software to generate error compensation code. Inputting the error compensation code into the control system can compensate for its position error. To verify the compensation effect, measurements were taken with volume error compensation. The errors on the four diagonals after compensation are shown in Figure 7. [align=center]Figure 7 Measured displacement error of the four diagonals after compensation[/align] As can be seen from the figure, the volume error was relatively large before compensation, reaching a maximum of 58μm. After compensation, the volume error was significantly reduced, with a maximum volume error of only 8.7μm. It is evident that the volume positioning accuracy of the machine tool was greatly improved after compensation. 5. Summary and Conclusion Multi-step volume positioning measurement is a rapid measurement method. Since the data measured along each axis direction is solely due to movement along that axis direction, the measured error data can be separated into those generated independently along the three axis directions, thus achieving error separation. Since these nine positional errors encompass all 21 errors generated during the movement of the three axes, we improve the accuracy of the machine tool by measuring and compensating for these nine positional errors. References [1] ISO230-6: 2002 Test code for machine tools - Part 6: Determination of positioning accuracy on body and face diagonal (Diagonal displacement tests), an International Standard by International Standards organization, 2002. [2] Methods for Performance Evaluation of Computer Numerically Controlled Machining Centers, An American National Standard, ASME B5.54 -1992 by the American Society of Mechanical Engineers, P69, 1992 [3] An American National Standard, ASME B5.54-1992, of the American Society of Mechanical Engineers, 1992 [4] Sun Changku Ye Shenghua Laser Measurement Technology Tianjin University Press 2001, 7 [5] Yin Chunyong Modern Interferometric Measurement Technology Tianjin University Press 1999, 7 [6] Ren Yongqiang Research on Efficient Measurement, Modeling and Compensation Application of CNC Machine Tool Error Shanghai Jiaotong University Doctoral Dissertation 2005