# A line measurement method for geometric error measurement of the vertical machining center

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## Abstract

This paper presents a new line-measurement method for three-axis CNC (computer numerical control) vertical machining center. The measurement method measures the twenty-one geometric error of the numerical control machining center through using the laser interferometer and ball bar. The objective of this research work was to illustrate the measurement method (a line measurement method for geometric error measurement), measurement system and testing results with specific experimental conditions that may provide an easy way to measure the geometric errors. To estimate the geometric errors of the machining center, both direct and indirect measurement methods are used to predict the errors. The ball bar is used to measure the squareness errors of the machining center. The linear, angular and straightness errors estimated by the laser interferometer provide the particular reference value at the same time on the numerical control machine tool and geometric precision detection. Therefore, the twenty-one geometric errors of the machining system that reflect on machine tool can be predicted by a line measurement method which comes up convenient and time-saving way as compared to others line methods such as 12th line, 13th line and 15th line etc.

## Keywords

Geometric errors Machining center Line measurement system Laser interferometer Ball bar## 1 Introduction

The micro-manufacturing technology industry today is based on higher dimensional accuracy, tight tolerance, surface finish, and the geometric accuracy of manufactured parts. Precision manufacturing has become a fundamental importance in modern manufacturing sectors [1]. Therefore, methodologies for producing specific components efficiently and cost-effectively are significant topics of interest for manufacturing development. New identification and evaluation methods are being continuously improved and developed to increase the precision and performance of machine tools [2].

Geometric errors are significant factors that cause inaccuracies in the machining centers [3]. There are in total of 21 position-dependent geometric errors (PDGEs) in machining centers. A linear displacement in the Y-axis leads to a position error in the Y-axis, two straightness errors between the X–Y and Z–Y axis, the squareness errors between every two axis X–Y, Y–Z and Z–X, and three rotational errors in each axis: a pitch error, yaw error and roll error [4].

Direct measurement and indirect measurement tend to be the main methods used to find errors. The direct measurement method finds the error for a single axis observed at one time. Interferometer, different artefact, and inclinometer are a few devices used to measure errors directly [5]. While in the indirect measurement method, the movement of the two axes takes place at the same time. The ball bar and laser tracker are the devices used to take the measurements [6].

The researchers have developed different methods to measure the geometric error. Eman et al. [7] presented a way to shape a general error model of a multi-axis machine of arbitrary configuration. Based on the author’s assumption, the machine tool has rigid body motion on which the homogeneous transformation matrices are developed. Yu et al. [8] presented the geometric error modelling based on screw theory that can be used to make a linear error model efficiently and accurately. They have demonstrated the model on the 2-DOF mechanism.

Xiang et al. [9] presented a systematic way to detect and compensate the geometric errors. An efficient test on the ball-bar with a 45° segment is intended to decouple the position-dependent and independent geometric errors for the non-orthogonal rotary axis. The geometric errors of translational axes have been measured by step-size body diagonal tests based on a laser interferometer. The volumetric error is anticipated and compensated based on the screw theory.

The 9-line measurement method has used the equations to measure the twenty-one geometric errors in the machining center which was quite complicated to understand. Mostly this method is used to find the linear position error as well as the straightness error. However, this method has influenced the squareness error. The disadvantage of this method is the influence of the straightness error on angular errors [10].

The 12-line measurement method has additional three face diagonals with 9 linear lines. The measurements have been taken along the X–Y, Y–Z and Z–X face diagonals. This method has a drawback; it did not consider the effect of the angular error on bi-axial movement on face diagonals [11] [12].

In the 13-line measurement method, the first 12 lines are measured the same way as in the 12-line measurement method. It has an additional body diagonal measurement that is taken having tri-axial motion. Until now, the 13-line method is the latest and most updated method, which discusses all the previous errors such as straightness and angular error. Moreover, pit error in guideway has also been addressed in this method [13] although the diagonal body measurement is not easy to measure in this method.

Chen et al. [14] presented a method to detect the 21 geometric error components by performing displacement measurements along 15 lines using laser interferometer. The authors provided a 15-line measurement method, which shortens the Zhang’s 22-line measurement method. Chapman [15] discussed the defects of diagonal body measurements as if the measurements of the diagonal body were along the four-body diagonals. The results produced on the machine would either give a better diagonal resulting in the poorest volumetric efficiency or a better volumetric efficiency producing the poorest diagonal result. Therefore, it carries volumetric error and influence of angular error in the linear axis as well as multi-axis motion, thus giving more error budget.

Zhang [16] presented the line method that measures displacement error along 22-lines in a volume. It is quite complicated to take values on 22-lines. In the 14-line measurement method, the squareness errors are considered equal to zero which makes it non-applicable, because it is impossible for a machine with zero squareness error. Therefore the 14-line method is obsolete and not applicable.

This paper measures the 21 geometric errors of the machine tool by using the line measurement method with the help of a laser interferometer and a ball bar apparatus. The method measures the geometric error in the linear axis of the CNC (computer numerical control) machining center. The previous work on the line method with equations is a little complex. It has influenced the squareness and angular errors that affect the results. The diagonal measurement is used in other methods, which has given good diagonal results in producing the poorest volumetric efficiency. To overcome the problems mentioned above, the volume of the machine tool has to be divided into four parts which provide the shape of four triangles in each axis. This paper presents a convenient and straightforward method to measure the errors of the machining center. This method is a software-based compensation in which the measurement of the geometric error has been taken by using the 9-lines at different places in the given volume of the machining center. The ball-bar apparatus is used to measure the squareness error between the X–Y, Z–Y and Y–Z planes.

## 2 Measurement method

Detection content on each line in the line measurement method

Line no. | The initial point of the line on the coordinate | The final point of the line on the coordinate | Measurement content of errors |
---|---|---|---|

1 | 0, 0, 0 | 1000, 0, 0 | Positional displacement error of x-axis |

2 | 0, − 255,270 | 1000, − 255, 270 | Pitch, yaw and rolling error of x-axis |

3 | 0, 0, 540 | 1000, 0, 540 | Straightness errors of Y and Z axis |

4 | 0, 0, 0 | 0, − 510, 0 | Positional displacement error of y-axis |

5 | 500, 0, 270 | 500, − 510, 270 | Pitch, yaw and rolling error of y-axis |

6 | 0, 0, 540 | 0, − 510, 540 | Straightness errors of X and Z axis |

7 | 0, 0, 0 | 0, 0, 540 | Positional displacement error of z-axis |

8 | 500, − 255, 0 | 500, − 255, 540 | Pitch, yaw and rolling error of z-axis |

9 | 0, − 510, 0 | 0, − 510, 540 | Straightness errors of X and Y axis |

- 1.
The new line measurement method developed based on the diagram of the line measurement method shown in Fig. 1. In which the specific lines have been drawn in the volume of the machine tool, according to lines, which has mainly synchronised with the actual movement of the machine tool. Before starting the experiment, the machine tools needed to be on idle running for at least 30–45 min [17], to test the actual movement of the machine tool. Therefore, consider the required space for mounting the optical instrument on the machine tool, the feasibility of the environment and make sure the required method of line measurement produced [10].

- 2.
After, conducting the test of the actual movement of the machine tool. Check the mounting feasibility for the measurement setup. Assembled the laser optics on the machine tool and made the alignment correctly in such a way, so that there would be no misalignment error in the results (especially removed slope for long distance while taking straightness value).

- 3.
To make the program on Renishaw software for capturing data while measuring the required line concerning desired error. In a similar manner developed the program using G codes on the NC controller for measuring the desired error on lines and set the target point for each required line.

- 4.The measuring instrument optics and software assembled, which shown in Figs. 2 and 3. The alignment made between the optics was checked and corrected by using the signal strength of the laser beam. Subsequently, the positional, straightness and angular error data were obtained. In the process of capturing data, repeatability, backlash, effectiveness and correction, to make sure the following data collections was checked, the detail about the errors on each line as shown in Table 1. The measurement arrangement of the experiment was determined as the positional error, angular and straightness error. The squareness error would be measured by using the ball bar apparatus.
- 5.
The Experimental setup for ball bar is easy and convenient by following these simple steps; install the center mount on the machine table and mount the magnetic tool cup into a suitable tool holder with the spindle. The setting ball used to set the offset of the machine tool, which was applied for alignment between the mount center and tool cup on the desired position. Synchronise the QC20-W wireless ball bar with computer system after that follow step 3 mentioned above.

## 3 Experiment and results

The data collected from the laser interferometer and ball bar, the measurement for each axis saved and analysed by the Renishaw analysis software. The error as shown below in Y-axis similarly, to find the 21 geometric error of three-axis NC machining center. The values were taken five times in bidirectional to minimize the chance of various other factors, which could influence results after that calculated mean of those values, could use for the compensation of NC machining center.

### 3.1 The positional errors in a vertical machine center

### 3.2 The angular errors in a vertical machine center

### 3.3 The straightness errors in a vertical machine center

The overall accuracy of the x-axis straightness error of the machine tool in the y-axis was observed to be 0.2252 mm. The maximum positive repeatability was detected as 0.1473 mm at target 18, maximum reverse repeatability is recorded as 0.1137 mm at target 13 and maximum bi-directional repeatability was measured as 0.1473 at target 18. Figure 7a shows the mean deviation of the x-axis straightness error in the forward and reverse direction of the y-axis. The graph shows the results of straightness errors in z-axis concerning the y-axis, which is in the range of 0.005 mm to − 0.01 mm; the behaviour of the chart indicated that straightness error increased in the negative direction as its moving away from the origin. The overall accuracy of the z-axis straightness error of the machine tool in the y-axis observed as 0.0285 mm, which is calculated from the measured values. The maximum positive repeatability detected as 0.0099 mm at target 22, maximum reverse repeatability reported as 0.0197 mm at target 15 and maximum bi-directional repeatability observed as 0.0211 at target 15. Figure 8a shows the mean deviation of the z-axis straightness error in the forward and reverse direction of the y-axis. The values had taken three times in bidirectional on the y-axis to get straightness in x-axis and z-axis. The graph indicated that the machine has a straightness error that could eliminate by doing composition.

### 3.4 The squareness error in between X–Y plane

## 4 Conclusion

This paper has presented a geometric error measurement method for the computer numerical control three-axis machine tools. The twenty-one geometric errors items of the machine tools were measured based on the laser Interferometer and ball-bar. The ball-bar apparatus measured the squareness error. The positional error, angular error and straightness error of the geometric error were measured by laser interferometer based on the line measurement method. The proposed line measurement method verified on a CNC (computer numerical control) three-axis vertical machine tool.

The measurement method has certain advantages as compared to other geometric errors measurement methods (time-saving, accuracy improve by applying for compensation, calculate errors separately and simple way to get precise results). The experimental results were shown in Figs. 4, 5, 6, 7, 8 and 9. The findings from the above experiments have indicated that it can sufficiently measure the geometric errors of the machine tool, which may be helpful for the mechanical industry.

## Notes

### Funding

This research was funded by the National Natural Science Foundation of China (Grant No. 51375100), Guangdong Science and Technology Project (Grant No. 2013B011304009).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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