Citation: Zhao, G.; Jiang, S.; Dong, K.;
Xu, Q.; Zhang, Z.; Lu, L. Influence
Analysis of Geometric Error and
Compensation Method for Four-Axis
Machining Tools with Two Rotary
Axes. Machines 2022, 10, 586.
https://doi.org/10.3390/
machines10070586
Academic Editors: Fang Cheng,
Qian Wang, Tegoeh Tjahjowidodo
and Ziran Chen
Received: 18 June 2022
Accepted: 12 July 2022
Published: 19 July 2022
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Article
Influence Analysis of Geometric Error and Compensation
Method for Four-Axis Machining Tools with Two Rotary Axes
Guojuan Zhao
1
, Shengcheng Jiang
2
, Kai Dong
2
, Quanwang Xu
1
, Ziling Zhang
1
and Lei Lu
2,3,4,
*
1
Logistics Engineering College, Shanghai Maritime University, Shanghai 201306, China;
gjzhao070906@126.com (G.Z.); 202030210016@stu.shmtu.edu.cn (Q.X.); zhangziling1119@126.com (Z.Z.)
2
Boneng Transmission (Suzhou) Co., Ltd., Suzhou 215021, China; jesec2006@163.com (S.J.);
dkmech@163.com (K.D.)
3
Jiangsu Provincial Key Laboratory of Advanced Robotics, Soochow University, Suzhou 215021, China
4
Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University,
Suzhou 215021, China
* Correspondence: jluc2ll@163.com
Abstract:
Four-axis machine tools with two rotary axes are widely used in the machining of complex
parts. However, due to an irregular kinematic relationship and non-linear kinematic function with
geometric error, it is difficult to analyze the influence the geometry error of each axis has and to
compensate for such a geometry error. In this study, an influence analysis method of geometric
error based on the homogeneous coordinate transformation matrix and a compensation method was
developed, using the Newton iterative method. Geometric errors are characterized by a homogeneous
coordinate transformation matrix in the proposed method, and an error matrix is integrated into the
kinematic model of the four-axis machine tool as a means of studying the influence the geometric
error of each axis has on the tool path. Based on the kinematic model of the four-axis machine tool
considering the geometric error, a comprehensive geometric error compensation calculation model
based on the Newton iteration was then constructed for calculating the tool path as a means of
compensating for the geometric error. Ultimately, the four-axis machine tool with a curve tool path
for an off-axis optical lens was chosen for verification of the proposed method. The results showed
that the proposed method can significantly improve the machining accuracy.
Keywords:
geometric error; error compensation; homogeneous coordinate transformation matrix;
Newton iteration; four-axis machining tools
1. Introduction
Multi-axis CNC machine tools have high efficiency and exhibit excellent performance.
They are widely used in manufacturing, particularly for complex surface machining tasks,
and have become a crucial part of modern manufacturing equipment [
1
,
2
]. Machining
accuracy is essential for the evaluation of machine tool performance. It is affected by
geometry, heat, motion, stiffness, vibration, and several other factors. Wu et al. [
3
] proposed
a robust design method for optimizing the static accuracy of a vertical machining center
to make the machining accuracy meet design requirements. Niu et al. [
4
] provided a
new analysis method for evaluating machining accuracy reliability based on the nonlinear
correlation between errors. Li et al. [
5
] overviewed the thermal error modeling methods
that had been researched and applied in the past ten years. Geometric error is a factor that
has a significant impact on machining accuracy and accounts for approximately 40% of all
errors. To improve the accuracy of error recognition, Wei et al. [
6
] provided an overview
of the current research algorithms. Lin et al. [
7
] provided a geometric error modeling
method for five-axis CNC machine tools based on the differential transformation method.
Geng et al. [8]
summarized state-of-the-art research in the calibration of geometric errors of
ultra-precision machine tools (UPMTs). Compared to the mature method for traditional
Machines 2022, 10, 586. https://doi.org/10.3390/machines10070586 https://www.mdpi.com/journal/machines
