面向误差补偿的数控机床空间误差建模与辨识方法研究进展

doi:10.3901/JME.2025.19.202

机械工程学报 ›› 2025, Vol. 61 ›› Issue (19): 202-228.doi: 10.3901/JME.2025.19.202

• 数字化设计与制造 • 上一篇

扫码分享

面向误差补偿的数控机床空间误差建模与辨识方法研究进展

孙光明^1,2, 韩冰¹, 张大卫², 田文杰^2,3, 郭鑫^2,4, 赵坚¹, 何改云², 高卫国², 苏喆^2,4

1. 天津城建大学控制与机械工程学院天津 300384;
2. 天津大学机构理论与装备设计教育部重点实验室天津 300072;
3. 天津大学海洋科学与技术学院天津 300072;
4. 沈阳机床中捷友谊厂有限公司沈阳 110142

收稿日期:2024-08-30 修回日期:2025-02-06 发布日期:2025-11-24
作者简介:孙光明，男，1987年出生，博士，副教授，硕士研究生导师。主要研究方向为高档数控机床设计、数控机床误差测量分析与补偿技术。E-mail：gmsun@tju.edu.cn
韩冰，男，2000年出生，硕士研究生。主要研究方向为数控机床误差测量分析与补偿技术。E-mail：1507341034@qq.com
张大卫，男，1962年出生，博士，教授，博士研究生导师。主要研究方向为高档数控机床精度设计与补偿技术等。E-mail：medzhang@tju.edu.cn
田文杰，男，1986年出生，博士，副教授，博士研究生导师。主要研究方向为数控机床与工业机器人的误差建模、检测与补偿等。E-mail：wenjietian@tju.edu.cn
郭鑫，男，1981年出生，正高级工程师，博士研究生。主要研究方向为高档数控机床结构正向设计、金属切削机床性能测试标准及自动化检测技术。E-mail：guoxin7@gt.cn
赵坚(通信作者)，女，1966年出生，博士，教授，硕士研究生导师。主要研究方向数控机床误差检测与故障诊断、智能制造等研究。E-mail：zhaojiantcu@163.com
何改云，女，1965年出生，博士，教授，博士研究生导师。主要研究方向为加工质量在机检测与控制、智能制造装备误差检测与补偿。E-mail：hegaiyun@tju.edu.cn
高卫国，男，1974年出生，博士，教授，博士研究生导师。主要研究方向为智能机床设计与制造关键技术等。E-mail：gaowg@tju.edu.cn
苏喆，男，1987年出生，工程师，硕士研究生。主要研究方向为高档数控机床加工及装配工艺技术设计，数控机床切削标准及仿形技术。E-mail：sz13704030181@163.com
基金资助:
天津市自然科学基金青年项(22JCQNJC01000)、国家自然科学基金联合基金(U23B20102)、国家自然科学基金青年基金(52205535)和沈阳市科技重大专项(23-401-1-01)资助项目。

Research Progress on Spatial Error Modeling and Identification Methods for CNC Machine Tools Based on Error Compensation

SUN Guangming^1,2, HAN Bing¹, ZHANG Dawei², TIAN Wenjie^2,3, GUO Xin^2,4, ZHAO Jian¹, HE Gaiyun², GAO Weiguo², SU Zhe^2,4

1. School of Control and Mechanical Engineering, Tianjin Chengjian University, Tianjin 300384;
2. Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University, Tianjin 300072;
3. School of Marine Science and Technology, Tianjin University, Tianjin 300072;
4. Shenyang Machine Tool Zhongjie Friendship Factory Co., Ltd., Shenyang 110142

Received:2024-08-30 Revised:2025-02-06 Published:2025-11-24

摘要/Abstract

摘要： 数控机床空间误差的建模分析与辨识一直是误差补偿的重要步骤。首先，论述了机床空间误差常用建模理论和辨识方法的研究历程及技术发展；其次，机床空间误差建模分析是进行误差补偿的重要前提。对机床空间误差建模理论进行详细的阐述和分析，包括刚体运动学理论、齐次坐标变化理论、D-H变换理论、多体理论、以及旋量理论等方法。再次，误差元素的准确测量和精确识别是实现有效控制的关键。对机床空间误差重点辨识方法的现状与发展趋势进行具体介绍和综合评述，包括激光干涉仪多线法、体对角线法、以及球杆仪、激光跟踪仪测量等方法。最终，综合机床空间误差的建模、检测、辨识三个方面，系统性地分析了现有数控机床空间精度改善中尚需解决的问题，强调了技术创新对于提高测量效率和准确性的重要性；并对未来的发展方向进行展望，对数控机床精度的提升具有一定的指导意义。

关键词: 数控机床, 误差建模, 误差检测辨识, 空间误差补偿

Abstract: The modeling analysis and identification of the spatial errors of CNC machine tools have always been important steps in error compensation. Firstly, the research history and technological development of the modeling theories and identification methods for machine tool spatial errors are discussed. Secondly, the modeling and analysis of machine tool spatial errors is an important prerequisite for error compensation. The modeling theory of machine tool spatial errors has been comprehensively reviewed and analyzed, including methods such as rigid body kinematics theory, homogeneous coordinate change theory, D-H transformation theory, multi-body theory, and screw theory. Thirdly, the accurate measurement and precise identification of spatial error elements in machine tools are key to achieving effective control. The current status and development trends of key measurement and identification methods for machine tool spatial errors are specifically introduced and comprehensively evaluated, including laser interferometer multi line method, body diagonal method, as well as ball bar method, laser tracker method, and other methods. Finally, the modeling, detection, and identification of spatial errors in integrated machine tools are systematically analyzed to identify the problems that still need to be solved in improving the spatial accuracy of existing CNC machine tools. The importance of technological innovation in improving measurement efficiency and accuracy is emphasized; And prospects for future development directions have certain guiding significance for improving the accuracy of CNC machine tools.

Key words: CNC machine tool, error modeling, error detection and identification, space error compensation

中图分类号:

TG659

孙光明, 韩冰, 张大卫, 田文杰, 郭鑫, 赵坚, 何改云, 高卫国, 苏喆. 面向误差补偿的数控机床空间误差建模与辨识方法研究进展[J]. 机械工程学报, 2025, 61(19): 202-228.

SUN Guangming, HAN Bing, ZHANG Dawei, TIAN Wenjie, GUO Xin, ZHAO Jian, HE Gaiyun, GAO Weiguo, SU Zhe. Research Progress on Spatial Error Modeling and Identification Methods for CNC Machine Tools Based on Error Compensation[J]. Journal of Mechanical Engineering, 2025, 61(19): 202-228.

参考文献

[1] 杨建国，范开国，杜正春. 数控机床误差实时补偿技术[M]. 北京：机械工业出版社，2020. YANG Jianguo，FAN Kaiguo，DU Zhengchun. Real time error compensation technology for CNC machine tools[M]. Beijing：China Machine Press，2020.
[2] 朱赤洲. 数控机床三维空间误差建模及补偿技术研究[D]. 南京：南京航空航天大学，2013. ZHU Chizhou. Research on modeling and compensation technique for 3-dimension volumetric errors of CNC machine tools[D]. Nanjing：Nanjing University of Aeronautics and Astronautics，2013.
[3] 杜正春，杨建国，冯其波. 数控机床几何误差测量研究现状及趋势[J]. 航空制造技术，2017(06)：34-44. DU Zhengchun，YANG Jianguo，FENG Qibo. Research status and trend of geometrical error measurement of CNC machine tools[J]. Aeronautical Manufacturing Technology，2017(06)：34-44.
[4] 朱少堃. 数控机床误差补偿技术及其应用[D]. 长沙：湖南大学，2014. ZHU Shaokun. Error compensation technology for CNC machine tool and its application[D]. Changsha：Hunan University，2014.
[5] SCHWENKE W，KNAPP H，HAITJEMA H. et al. Geometric error measurement and compensation of machines—an update[J]. CIRP Annals-Manufacturing Technology，2008，57(2)：660-675.
[6] CHENG Q，ZHAO H W，ZHANG G J，et al. An analytical approach for crucial geometric errors identification of multi-axis machine tool based on global sensitivity analysis[J]. The International Journal of Advanced Manufacturing Technology，2014，75(1-4)：107-121.
[7] 张亚飞. 高度非线性系统结构可靠性优化设计方法及应用[D]. 兰州：兰州理工大学，2020. ZHANG Yafei. Structural reliability-based design optimization method and its application to highly nonlinear system[D]. Lanzhou：Lanzhou University of Technology，2020.
[8] KIRIDENA V，FERREIRA P M. Mapping the effects of positioning errors on the volumetric accuracy of five-axis CNC machine tools[J]. International Journal of Machine Tools and Manufacture，1993，33(3)：417-437.
[9] 王伟文. 精密多轴数控机床误差建模研究[D]. 长春：长春工业大学，2021. WANG Weiwen. Research on error modeling of precision multi-axis CNC machine tools[D]. Changchun：Changchun University of Technology，2021.
[10] 寇锦. 数控机床误差辨识与诊断新方法研究[D]. 大连：大连理工大学，2013. KOU Jin. New methods for error identification and diagnosis of CNC machine tools[D]. Dalian：Dalian University of Technology，2013.
[11] MAKEOMN P A，LOXHAM J. Some aspects of the design of high precision measuring machines[J]. Annals of the CIRP，1973，22(1)：139-141.
[12] 康方. 数控机床制造精度分配优化方法的研究[D]. 北京：北京工业大学，2008. KANG Fang. A method for manufacturing accuracy distribution and optimization of a CNC machine tool[D]. Beijing：Beijing University of Technology，2008.
[13] 卢红星. 高档数控机床误差实时补偿关键技术及应用研究[D]. 上海：上海交通大学，2020. LU Hongxing. Critical techniques and applications of real-time error compensation on high-end CNC machine tools[D]. Shanghai：Shanghai Jiaotong University，2020.
[14] FERREIRA P M，LIU C R. An analytical quadratic model for the geometric error of a machine tool[J]. Journal of Manufacturing Systems，1986，5(1)：51-63.
[15] KIM K，KIM M K. Volumetric accuracy analysis based on generalized geometric error model in multi-axis machine tools[J]. Mechanism & Machine Theory，1991，26(2)：207-219.
[16] OKAFOR A C，ERTEKIN Y M. Derivation of machine tool error models and error compensation procedure for three axes vertical machining center using rigid body kinematics[J]. International Journal of Machine Tools and Manufacture，2000，40(8)：1199-1213.
[17] SRIVASTAVA A K，VELDHUIS S C，ELBESTAWIT M A. Modelling geometric and thermal errors in a five-axis CNC machine tool[J]. International Journal of Machine Tools and Manufacture，1995，35(9)：1321-1337.
[18] 杨建国，潘志宏，薛秉源. 数控机床几何和热误差综合的运动学建模[J]. 机械设计与制造，1998(5)：31-32. YANG Jianguo，PAN Zhihong，XUE Bingyuan. Kinematic modeling of geometric and thermal error synthesis in CNC machine tools[J]. Mechanical Design and Manufacturing，1998(5)：31-32.
[19] RAHMAN M，HEIKKALA J，LAPPALAINEN K. Modeling, measurement and error compensation of multi-axis machine tools. Part I: Theory[J]. International Journal of Machine Tools and Manufacture，2000，40(10)：1535-1546.
[20] UDDIN M S，IBARAKI S，MATSUBARA A，et al. Prediction and compensation of machining geometric errors of five-axis machining centers with kinematic errors[J]. Precision Engineering，2009，33(2)：194-201.
[21] ZHU S，DING G，QIN S，et al. Integrated geometric error modeling, identification and compensation of CNC machine tools[J]. International Journal of Machine Tools and Manufacture，2012，52(1)：24-29.
[22] FAN K，YANG J，YANG L. Unified error model based spatial error compensation for four types of CNC machining center: Part I—Singular function based unified error model[J]. Mechanical Systems and Signal Processing，2015，60-61：656-667.
[23] XIANG S，YANG J，FAN K，et al. Multi-machine tools volumetric error generalized modeling and Ethernet-based compensation technique[J]. Proceedings of the Institution of Mechanical Engineers，Part B: Journal of Engineering Manufacture，2016，230(5)：870-882.
[24] DING S，HUANG X，YU C，et al. Actual inverse kinematics for position-independent and position-dependent geometric error compensation of five-axis machine tools[J]. International Journal of Machine Tools and Manufacture，2016，111：55-62.
[25] 冯文龙. 大型数控机床多误差元素建模及综合补偿[D]. 上海：上海交通大学，2016. FENG Wenlong. Multi error element modeling and comprehensive compensation for large CNC machine tools [D]. Shanghai：Shanghai Jiaotong University，2016.
[26] JHA B K，KUMAR A. Analysis of geometric errors associated with five-axis machining center in improving the quality of cam profile[J]. International Journal of Machine Tools and Manufacture，2003，43(6)：629-636.
[27] 吕永军，刘峰，郑飂默，等. 通用和修正D-H法在运动学建模中的应用分析[J]. 计算机系统应用，2016，25(5)：197-202. LÜ Yongjun，LIU Feng，ZHENG Liaomo，et al. Application analysis of generalized and modified D-H method in kinematic modeling[J]. Computer Systems Applications，2016，25(05)：197-202.
[28] DONEMZ M A. A general methodology for machine tools accuracy enhancement by error compensation[J]. Precision Engineering，1986，8(4)：187-196.
[29] SUH S，LEE J. Five-axis part machining with three-axis CNC machine and indexing table[J]. Journal of Manufacturing Science and Engineering，1998，120(5)：120-127.
[30] KIRIDENA V S B，FERREIRA P M. Kinematic modeling of quasistatic errors of three-axis machining centers[J]. International Journal of Machine Tools and Manufacture，1994，34(1)：85-100.
[31] LIN P D，TZENG C S. Modeling and measurement of active parameters and workpiece home position of a multi-axis machine tool[J]. International Journal of Machine Tools and Manufacture，2008，48(3-4)：338-349.
[32] WANG K F，EHMANN S M. Measurement methods for the position errors of a multi-axis machine. Part 1: principles and sensitivity analysis[J]. International Journal of Machine Tools and Manufacture，1999，39(6)：951-964.
[33] 刘守法，王晋鹏，李勇，等. 基于D-H法的5-DOF串并联机床运动学分析[J]. 制造技术与机床，2018(11)：110-115. LIU Shoufa，WANG Jinpeng，LI Yong，et al. Kinematics analysis of 5-DOF series-parallel machine tool based on D-H method[J]. Manufacturing Technology & Machine Tool，2018(11)：110-115.
[34] 单鹏，谢里阳，田万禄，等. 基于D-H矩阵的Stewart型并联机床位姿误差计算模型[J]. 机械工程学报，2010，46(17)：186-191. SHAN Peng，XIE Liyang，TIAN Wanlu，et al. Modeling of position and pose error of Stewart machine tool based on D-H transforming matrix[J]. Journal of Mechanical Engineering，2010，46(17)：186-191.
[35] 李伟. 五轴数控机床综合误差补偿关键技术研究[D]. 南京：南京航空航天大学，2021. LI Wei. Research on key technologies of comprehensive error compensation for five-axis CNC machine tools[D]. Nanjing：Nanjing University of Aeronautics and Astronautics，2021.
[36] 田文杰. 精密卧式加工中心几何精度设计及误差补偿方法研究[D]. 天津：天津大学，2014. TIAN Wenjie. Investigation into accuracy design and error compensation of high-precision horizontal machining centers[D]. Tianjin：Tianjin University，2014.
[37] 刘又午，章青，王国锋，等. 数控机床误差补偿技术及应用发展动态及展望[J]. 制造技术与机床，1998(12)：8-9，24. LIU Youwu，ZHANG Qing，WANG Guofeng，et al. Development trends and prospects of error compensation technology and application in CNC machine tools[J]. Manufacturing Technology and Machine Tools，1998(12)：8-9，24.
[38] 刘丽冰，王广彦，刘又午. 复杂机械系统运动误差自动建模技术研究[J]. 中国机械工程，2000，11(6)：642-646. LIU Libing，WANG Guangyan，LIU Youwu. Research on automatic modeling technology for motion errors of complex mechanical systems[J]. China Mechanical Engineering，2000，11(6)：642-646.
[39] 赵小松，方沂，章青，等. 四轴联动加工中心误差补偿技术的研究[J]. 中国机械工程，2000，11(6)：637-639. ZHAO Xiaosong，FANG Yi，ZHANG Qing，et al. Research on error compensation technology for four axis link age machining centers[J]. China Mechanical Engineering，2000，11(6)：637-639.
[40] 粟时平. 多轴数控机床精度建模与误差补偿方法研究[D]. 长沙：国防科学技术大学，2002. SU Shiping. Study on the methods of precision modeling and error compensation for multi-axis CNC machine tools[D]. Changsha:：National University of Defense Technology，2002.
[41] 粟时平，李圣怡. 五轴数控机床综合空间误差的多体系统运动学建模[J]. 组合机床与自动化加工技术，2003(5)：15-18. SU Shiping，LI Shengyi. Modeling the volumetric synthesis error of five-axis machine tools based on multi -body system kinematics[J]. Modular Machine Tool & Automatic Manufacturing Technique，2003(5)：15-18.
[42] 陈英姝. 数控误差补偿新技术研究[D]. 天津：河北工业大学，2006. CHEN Yingshu. Research on new error compensation techniques for CNC machine tools[D]. Tianjin：Hebei University of Technology，2006.
[43] 王小平，姚英学，荆怀靖. 数控机床几何误差建模及误差补偿的研究[J]. 机械工程师，2005(9)：18-20. WANG Xiaoping，YAO Yingxue，JING Huaijing. Investigation of geometric error modeling and error compensation of CNC machine tools[J]. Mechanical Engineer，2005(9)：18-20.
[44] 王秀山，杨建国，闫嘉钰. 基于多体系统理论的五轴机床综合误差建模技术[J]. 上海交通大学学报，2008，42(5)：761-764. WANG Xiushan，YANG Jianguo，YAN Jiayu. Synthesis error modeling of the five-axis machine tools based on multi-body system theory[J]. Journal of Shanghai Jiaotong University，2008，42(5)：761-764.
[45] 杨枝. 高档数控机床几何误差建模与参数溯源优化技术及其应用[D]. 杭州：浙江大学，2014. YANG Zhi. Research and application on geometric error modeling and parameters traceability of CNC machine tools[D]. Hangzhou：Zhejiang University，2014.
[46] MURRAY R M，LI Z，SASTRY S S. A mathematical introduction to robotic manipulation[M]. Boca Raton CRC Press，2017.
[47] MOON Y M，KOTA S. Generalized kinematic modeling method for reconfigurable machine tools[C]//International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers，1998，80302：V01AT01A080.
[48] YANG J，ALTINTAS Y. Generalized kinematics of five-axis serial machines with non-singular tool path generation[J]. International Journal of Machine Tools & Manufacture，2013，75：119-132.
[49] FU G，FU J，Xu Y，et al. Product of exponential model for geometric error integration of multi-axis machine tools[J]. The International Journal of Advanced Manufacturing Technology，2014，71(9-12)：1653-1667.
[50] TIAN W，GAO W，ZHANG D，et al. A general approach for error modeling of machine tools[J]. International Journal of Machine Tools and Manufacture，2014，79：17-23.
[51] CHENG Q，FENG Q，LIU Z，et al. Sensitivity analysis of machining accuracy of multi-axis machine tool based on POE screw theory and Morris method[J]. The International Journal of Advanced Manufacturing Technology，2016，84：2301-2318.
[52] 韩伟. 五轴联动数控机床空间误差建模及补偿技术研究[D]. 大连：大连理工大学，2022. HAN Wei. Research on volumetric error modeling and compensation of five-axis CNC machine tools[D]. Dalian：Dalian University of Technology，2022.
[53] 钟学敏. 考虑误差耦合的数控机床空间误差建模、辨识及多层误差溯源研究[D]. 武汉：华中科技大学，2020. ZHONG Xuemin. Research on the modelling，identification of volumetric error considering error coupling and traceability of multi-level geometric errors for CNC machine tools[D]. Wuhan：Huazhong University of Science and Technology，2020.
[54] LUO W，HU T，ZHU W，et al. Digital twin modeling method for CNC machine tool[C]//2018 IEEE 15th International Conference on Networking，Sensing and Control (ICNSC). IEEE，2018：1-4.
[55] 秦秀，张文武，茹浩磊，等. 基于数字孪生技术的五轴联动坐标转换[J]. 工具技术，2020，54(9)：52-55. QIN Xiu，ZHANG Wenwu，RU Haolei，et al. Five-axis coordinate conversion based on digital twin technology[J]. Tool Engineering，2020，54(9)：52-55.
[56] 张雷，刘检华，庄存波，等. 基于数字孪生的多轴数控机床轮廓误差抑制方法[J]. 计算机集成制造系统，2021，27(12)：3391-3402. ZHANG Lei，LIU Jianhua，ZHUANG Cunbo，et al. Contour error reduction method for multi axis CNC machine tools based on digital twin[J]. Computer Integrated Manufacturing Systems，2021，27(12)：3391-3402.
[57] 黄华，支晓波，李嘉然，等. 基于数据-模型驱动的数控机床综合误差动态预测方法[J]. 计算机集成制造系统，2024，30(7)：2283-2295. HUANG Hua，ZHI Xiaobo，LI Jiaran，et al. Dynamic prediction method for comprehensive errors of CNC machine tools based on data-model drive[J]. Computer Integrated Manufacturing System，2024，30(7)：2283-2295.
[58] 魏新园，钱自强，吴秋源，等. 数据机理驱动的机床主轴热精度建模方法研究[J]. 仪器仪表学报，2024，44(12)：111-119. WEI Xinyuan，QIAN Ziqiang，WU Qiuyuan，et al. Research on data-driven modeling method for thermal accuracy of machine tool spindle[J]. Journal of Instrumentation，2024，44(12)：111-119.
[59] 李波，马帅，刘强，等. 基于深度神经网络的立式机床热误差建模研究[J]. 组合机床与自动化加工技术，2023(5)：160-163. LI Bo，MA Shuai，LIU Qiang，et al. Research on thermal error modeling of vertical machine tool based on depth neural network[J]. Modular Machine Tool & Automatic Manufacturing Technique，2023(5)：160-163.
[60] ZHANG L，LIU J，ZHUANG C. Digital twin modeling enabled machine tool intelligence:a review[J]. Chinese Journal of Mechanical Engineering，2024，37(2)：46-71.
[61] 张敏. 基于激光干涉仪的数控机床几何误差检测与辨识[J]. 机械工程师，2006(9)：76-78. ZHANG Min. The geometric error detection and distinguishing of laser interfere eater-based CNC machine tool[J]. Mechanical Engineer，2006(9)：76-78.
[62] 周汉辉. 机床三维空间关键几何误差的来源解析——XM-60多光束激光干涉仪的应用案例分享[J]. 世界制造技术与装备市场，2023(2)：80-83. ZHOU Hanhui. Source analysis of key geometric errors in three-dimensional space of machine tools-application case sharing of XM-60 multi beam laser interferometer[J]. World Manufacturing Technology and Equipment Market，2023(2)：80-83.
[63] 黎坚，周艳红，龚海，等. 多坐标数控机床通用运动学模型的建立[J]. 中国机械工程，1999(1)：12-14+3. LI Jian，ZHOU Yanhong，GONG Hai，et al. Establishment of a universal kinematic model for multi coordinate CNC machine tools[J]. China Mechanical Engineering，1999(1)：12-14+3.
[64] SARTORI S，ZHANG G X. Geometric error measurement and compensation of machines[J]. CIRP Annals，1995，44(2)：599-609.
[65] SCHWENKE H，KNAPP W，HAITJEMA H，et al. Geometric error measurement and compensation of machines-an update[J]. CIRP Annals，2008，57(2)：660-675.
[66] GUAN L，MO J，FU M，et al. An improved positioning method for flank milling of S-shaped test piece[J]. The International Journal of Advanced Manufacturing Technology，2017，92：1349-1364.
[67] PAN F Y，LI M，YIN J. Error identification for 3-axis machine tool based on laser interferometer[J]. Advanced Materials Research，2012，490：309-314.
[68] 曹永洁，万军，傅建中，等. 激光干涉仪在机床定位精度测量中的误差分析[J]. 机床与液压，2007(4)：163-164，183. CAO Yongjie，WAN Jun，FU Jianzhong，et al. Errors analysis of laser interfere eater during the linear positioning accuracy measure end in machine tools[J]. Machine Tool & Hydraulics，2007(4)：163-164，183.
[69] 刘志松，王永青，刘阔. 基于球杆仪的立式加工中心圆度的测试与分析[J]. 组合机床与自动化加工技术，2018(5)：104-107. LIU Zhisong，WANG Yongqing，LIU Kuo. Roundness error measurement and analysis of vertical machining center with double ball-bar[J]. Modular Machine Tool & Automatic Manufacturing Technique，2018(5)：104-107.
[70] 周汉辉. 数控机床三维空间误差补偿技术的应用[J]. 航空制造技术，2011，54(6)：48-51. ZHOU Hanhui. Application of 3D spatial error compensation technology for CNC machine tools[J]. Aeronautical Manufacturing Technology，2011，54(6)：48-51.
[71] 王超. 三轴数控机床空间误差测量、建模与补偿关键技术研究[D]. 武汉：华中科技大学，2019. WANG Chao. Research on key technologies of volumetric error measurement，modeling and compensation for three-axis CNC machine tools[D]. Wuhan：Huazhong University of Science & Technology，2019.
[72] 李小力. 数控机床综合几何误差的建模及补偿研究[D]. 武汉：华中科技大学，2008. LI Xiaoli. Modelling and Compensating for the comprehensive geometric errors of CNC machine tools[D]. Wuhan：Huazhong University of Science and Technology，2008.
[73] 杨帆，杜正春，杨建国，等. 数控机床误差检测技术新进展[J]. 制造技术与机床，2012(3)：19-23. YANG Fan，DU Zhengchun，YANG Jianguo，et al. Review on geometric error measurement of machine tools[J]. Manufacturing Technology and Machine Tools，2012(3)：19-23.
[74] 刘又午，刘丽冰，赵小松，等. 数控机床误差补偿技术研究[J]. 中国机械工程，1998(12)：54-58. LIU Youwu，LIU Libing，ZHAO Xiaosong，et al. Research on error compensation technology for CNC machine tools[J]. China Mechanical Engineering，1998(12)：54-58.
[75] ZHANG G，OUYANG R，LU B，et al. A displacement method for machine geometry calibration[J]. CIRP Annals，1988，37(1)：515-518.
[76] 范晋伟，田越，宋国荣，等. 基于14条位移线测量法的数控机床误差参数辨识技术[J]. 北京工业大学学报，2000(2)：11-15. FAN Jinwei，TIAN Yue，SONG Guorong，et al. Technology of CNC machine error parameter identification based on fourteen displacement measurement line[J]. Journal of Beijing Polytechnic University，2000(2)：11-15.
[77] CHEN G，YUAN J，NI J. A displacement measurement approach for machine geometric error assessment[J]. International Journal of Machine Tools & Manufacture，2001，41(41)：149-161.
[78] 粟时平，李圣怡，王贵林. 三轴机械几何误差辨识新方法的研究[J]. 中国机械工程，2002，13(21)：1818-1820. SU Shiping，LI Shengyi，WANG Guilin. Research on a new method for identifying geometric errors in three-axis machinery[J]. China Mechanical Engineering，2002，13(21)：1818-1820.
[79] 黄天喜. 10线法辨识三轴数控机床几何误差及误差补偿研究[D]. 长沙：中南大学，2010. HUANG Tianxi. Research on geometric error identification and error compensation of three axis CNC machine tool using the 10 line method[D]. Changsha：Central South University，2010.
[80] LI J，XIE F，LIU X，et al. Geometric error identification and compensation of linear axes based on a novel 13-line method[J]. The International Journal of Advanced Manufacturing Technology，2016，87：2269-2283.
[81] 李杰，谢福贵，刘辛军，等. 五轴数控机床空间定位精度改善方法研究现状[J]. 机械工程学报，2017，53(7)：113-128. LI Jie，XIE Fugui，LIU Xinjun，et al. Analysis on the research status of volumetric positioning accuracy improvement methods for five-axis NC machine tools[J]. Journal of Mechanical Engineering，2017，53(7)：113-128.
[82] 李传东. 摇篮式五轴数控机床几何误差辨识方法研究[D]. 哈尔滨：哈尔滨理工大学，2021. LI Chuandong. Research on the identification method of geometric error of cradle five axis NC machine tool[D]. Harbin：Harbin University of Science and Technology，2021.
[83] WANG C，LIOTTO G. A theoretical analysis of 4 body diagonal displacement measurement and sequential step diagonal measurement[J]. WIT Transactions on Engineering Sciences，2003，44.
[84] CHAPMAN M A V. Limitations of laser diagonal measurements[J]. Precision Engineering，2003，27(4)：401-406.
[85] SVOBODA O. Testing the diagonal measuring technique[J]. Precision Engineering，2006，30(2)：132-144.
[86] IBARAKI S，HATA T. A new formulation of laser step diagonal measurement—Three-dimensional case[J]. Precision Engineering，2010，34(3)：516-525.
[87] 鲁志政，陈志俊，沈颖华，等. 激光矢量分步对角线法的精度分析与研究[J]. 机械设计与研究，2008(3)：93-96. LU Zhizheng，CHEN Zhijun，SHEN Yinghua，et al. The error analysis and research of the laser vector step diagonal measure meant[J]. Machine Design and Research，2008(3)：93-96.
[88] 黄奕乔，杨建国. 数控机床体积误差激光分步体对角线测量的优化方法[J]. 组合机床与自动化加工技术，2017(2)：54-59. HUANG Yiqiao，YANG Jianguo. Optimized method of laser step body diagonal measurement of volumetric errors of CNC machine tools[J]. Modular Machine Tool & Automatic Manufacturing Technique，2017(2)：54-59.
[89] 任永强，杨建国，沈金华，等. 基于体对角线的机床垂直度误差的高效测量分析[J]. 中国机械工程，2005(16)：1435-1438. REN Yongqiang，YANG Jianguo，SHEN Jinhua，et al. Analysis on efficient measurement of squareness errors along body diagonals for machine tools[J]. China Mechanical Engineering，2005(16)：1435-1438.
[90] 鲁志政. 数控机床误差的辨识新方法研究及补偿应用[D]. 上海：上海交通大学，2008. LU Zhizheng. Research in the new method of identification and complement application for CNC errors[D]. Shanghai：Shanghai Jiaotong University，2008.
[91] 沈金华. 数控机床误差补偿关键技术及其应用[D]. 上海：上海交通大学，2009. SHEN Jinhua. Key technique and application in error compensation for CNC machine tools[D]. Shanghai：Shanghai Jiao Tong University，2009.
[92] 张舒洁. 数控机床运动误差分步辨识法理论与实验研究[D]. 上海：上海交通大学，2010. ZHANG Shujie. Theoretical and experiment research on motion error identification of CNC machine tools based on multi-step identification method[D]. Shanghai：Shanghai Jiao Tong University，2010.
[93] SUN G，HE G，ZHANG D，et al. Body diagonal error measurement and evaluation of multi-axis machine tool using multi-beam laser interferometer[J]. International Journal of Advanced Manufacturing Technology，2020(107)：4545-4559.
[94] WANG S，HE G，ZHANG D，et al. Innovative design methods for the geometric accuracy of machine tool guide rail oriented to spatial accuracy[J]. Journal of Manufacturing Processes，2024，119：483-498.
[95] 姜辉. 五轴数控机床几何与热误差实时补偿关键技术及其试验研究[D]. 上海：上海交通大学，2014. JIANG Hui. Key technologies and experimental research on real-time compensation for geometrical and thermal errors of five-axis CNC machine tool[D]. Shanghai：Shanghai Jiao Tong University，2014.
[96] 任永强，刘国良，叶飞帆，等. 基于体对角线机床位置误差的激光矢量测量分析[J]. 上海交通大学学报，2005(9)：1413-1417. REN Yongqiang，LIU GuoLiang，YE Feifang，et al. Analysis of laser vector measurement based on body diagonals for volumetric position errors on machine tools[J]. Journal of Shanghai Jiao Tong University，2005(9)：1413-1417.
[97] IBARAKI S，SATO G，TAKEUCHI K. ‘Open-loop’ tracking interferometer for machine tool volumetric error measurement—two-dimensional case[J]. Precision Engineering，2014，38(3)：666-672.
[98] Leica AT₄0₂绝对激光跟踪仪[EB/OL]. [2025-01-03]. http://www.hexagonmetrology.com.cn/ProductList_7_7.aspx.
[99] FARO Laser Tracker: Vantage[EB/OL]. [2025-01-03]. http://www.faro.com/zh-cn/products/ metrology/faro-laser- tracker/.
[100] API-Radian激光跟踪仪[EB/OL]. [2025-01-03]. http://www.apisensor.com/cn/.
[101] Laser tracker Polaris [EB/OL]. [2025-01-03]. https://www.pimicos.com/.
[102] 沈睿. 激光跟踪仪多站测量的精度提升方法研究[D]. 成都：电子科技大学，2020. SHEN Rui. Research on the method of improving the accuracy of multi station measurement of laser tracker[D]. Chengdu：University of Electronic Science and technology，2020.
[103] ZHANG Z，HU H. Measuring geometrical errors of linear axis of machine tools based on the laser tracker[C]//2011 IEEE International Conference on Robotics and Bionics. IEEE，2011：2276-2281.
[104] 王金栋，郭俊杰，费致根，等. 基于激光跟踪仪的数控机床几何误差辨识方法[J]. 机械工程学报，2011，47(14)：13-19. WANG Jindong，GUO Junjie，FEI Zhigen，et al. Method of geometric error identification for numerical control machine tool based on laser tracker[J]. Journal of Mechanical Engineering，2011，47(14)：13-19.
[105] WANG J，WANG Q，LI H. The method of geometric error measurement of NC machine tool based on the principle of space vector’s direction measurement[J]. International Journal of Precision Engineering and Manufacturing，2019，20(4)：511-524.
[106] WANG J，GUO J. The identification method of the relative position relationship between the rotary and linear axis of multi-axis numerical control machine tool by laser tracker[J]. Measurement，2019，132：369-376.
[107] AGUADO S，SAMPER D，SANTOLARIA J，et al. Identification strategy of error parameter in volumetric error compensation of machine tool based on laser tracker measurements[J]. International Journal of Machine Tools & Manufacture，2012，53(1)：160-169.
[108] AGUADO S，SANTOLARIA J，SAMPER D，et al. Influence of measurement noise and laser arrangement on measurement uncertainty of laser tracker multilateration in machine tool volumetric verification[J]. Precision Engineering，2013，37(4)：929-943.
[109] 殷建，李明. 基于激光跟踪仪的五轴机床旋转轴误差测量[J]. 中国激光，2015，42(4)：244-251. YIN Jian，LI Ming. Errors measurement for rotation axis of five-axis machine tool based on laser tracker[J]. Chinese Journal of Lasers，2015，42(4)：244-251.
[110] SOICHI I，GOH S，KUNITAKA T. ‘Open-loop’ tracking interferometer for machine tool volumetric error measurement—two-dimensional case[J]. Precision Engineering，2014，38(3)：666-672.
[111] 刘少朋. 基于体对角线误差检测的数控机床几何误差辨识方法研究[D]. 天津：天津大学，2018. LIU Shaopeng. An identification strategy research of CNC machine tool geometric errors based on body diagonal errors measurement[D]. Tianjin：Tianjin University，2018.
[112] WANG J，CHEN P，DENG Y，et al. New algorithms for motion error detection of numerical control machine tool by laser tracking measurement on the basis of GPS principle[J]. Review of scientific instruments，2018，89(1).
[113] 李晴朝. 五轴数控机床空间误差检测、补偿与动态误差控制方法研究[D]. 成都：电子科技大学，2022. LI Qingchao. Study on volumetric error measurement，compensation and dynamic error control of five-axis CNC machine tools[D]. Chengdu：University of Electronic Science and Technology of China，2022.
[114] 马守东，高栋，路勇. 基于最小PDOP的跟踪仪顺次多站测量站位优化[J]. 计量学报，2023，44(2)：157-164. MA Shoudong，GAO Dong，LU Yong. Optimization the position of the laser tracker based on the minimum PDOP[J]. Acta Metrologica Sinica，2023，44(2)：157-164.
[115] ZHA J，ZHANG H. Geometric error identification of gantry-type CNC machine tool based on multi-station synchronization laser tracers[J]. Chinese Journal of Mechanical Engineering，2024，37(1)：46.
[116] 李万红. 激光跟踪仪高精度测角误差补偿技术研究[D]. 合肥：合肥工业大学，2014. LI Wanhong. Research on high accuracy angular error compensation technology for laser tracker[D]. Hefei：Hefei University of Technology，2014.
[117] 陈章位，祖洪飞，洪伟，等. 基于多基站激光跟踪仪的机器人位姿精度测试方法[J]. 计测技术，2021，41(1)：10-16. CHEN Zhangwei，ZU Hongfei，HONG Wei，et al. The accurate robot pose measurement based on multi base station laser tracker[J]. Measurement Technology，2021，41(1)：10-16.
[118] 黄克，关立文，杨亮亮，等. 基于“S”形试件的五轴机床几何误差建模研究[J]. 机械设计与制造，2015(2)：189-193. HUANG Ke，GUAN Liwen，YANG Liangliang，et al. Geometric error modeling of five-axis CNC machine tools based on“s”shaped test piece[J]. Machinery Design & Manufacture，2015(2)：189-193.
[119] 任永强，杨建国，窦小龙，等. 五轴数控机床综合误差建模分析[J]. 上海交通大学学报，2003，37(1)：70-75. REN Yongqiang，YANG Jianguo，DOU Xiaolong，et al. Analysis on the error synthesis model of a five axis machine center[J]. Journal of Shanghai Jiao Tong University，2003，37(1)：70-75.
[120] 杨婧，冯其波. 数控机床空间几何误差测量研究进展[J]. 仪器仪表学报，2017，38(8)：1901-1911. YANG Jing，FENG Qibo. Research progress on spatial geometric error measurement of CNC machine tools[J]. Chinese Journal of Scientific Instrument，2017，38(8)：1901-1911.
[121] CHEN J，YUAN J，NI J，et al. Real-time compensation for time-variant volumetric errors on a machining center[J]. Transactions of the ASME，Journal of Engineering Industry，1993，(115)：472-479.
[122] 刘建军. 多轴数控机床几何误差辨识与补偿技术研究[D]. 成都：西南交通大学，2012. LIU Jianjun. Research on the identification and compensation technology of the geometric error for multi-axis CNC machine tools[D]. Chengdu：Southwest Jiaotong University，2012.
[123] FU J，FU Y，XU G，et al. Accuracy enhancement of five-axis machine tool based on differential motion matrix: Geometric error modeling，identification and compensation[J]. International Journal of Machine Tools & Manufacture，2015，89：170-181.
[124] 左维，李巍. 5轴数控机床空间误差模型及辨识方法研究[J]. 组合机床与自动化加工技术，2019(2)：45-48. ZUO Wei，LI Wei. Research on spatial error model and identification method of five axis CNC machine tool[J]. Modular Machine Tool & Automatic Manufacturing Technique，2019(2)：45-48.
[125] 侯家林. 基于多线测量方法的俯仰和偏摆误差研究[D]. 湛江：广东海洋大学，2022. HOU Jialin. Research on pitch and yaw errors based on multi line measurement method[D]. Zhanjiang：Guangdong Ocean University，2022.
[126] 焦安铃，陈光胜. 基于球杆仪的五轴数控机床误差快速检测方法[J]. 仪器仪表学报，2024，45(1)：138-148. JIAO Anling，CHEN Guangsheng. Fast error identification method for five-axis machine tools based on double ball-bar[J]. Chinese Journal of Scientific Instrument，2024，45(1)：138-148.
[127] WEIKERT S. R-Test，a new device for accuracy measurements on five axis machine tools[J]. CIRP Annals - Manufacturing Technology，2004，53(1)：429-432．
[128] 刘健. R-test五轴数控机床旋转轴误差测量系统设计[D]. 成都：西南交通大学，2018. LIU Jian. The design of rotation axis error measurement system for R-test of five-axis CNC machine tool[D]. Chengdu：Southwest Jiaotong University，2018.
[129] BRINGMANN B，Knapp W. Model-based ‘chase-the-ball’ calibration of a 5-axis machining center[J]. CIRP Annals，2006，55(1)：531-534.
[130] HONG C F，IBARAKI S. Non-contact R-test with laser displacement sensors for error calibration of five-axis machine tools[J]. Precision Engineering，2013，37(1)：159-171．
[131] LBARAKI C，OYAMA H，OTSUBO S. Construction of an error map of rotary axes on a five-axis machining center by static R-test[J]. International Journal of Machine Tools and Manufacture，2011，51(3)：190-200.
[132] LI F，XIE X，LIU J，et al. A geometric error identification method for the swiveling axes of five-axis machine tools by static R-test[J]. International Journal of Advanced Manufacturing Technology，2017，89(9-12)：3393-3405.
[133] ZHONG O，BI N，HUANG L，et al. Dynamic accuracy evaluation for five-axis machine tools using S trajectory deviation based on R-test measurement[J]. International Journal of Machine Tools and Manufacture，2018，125：20-33.
[134] TSUTSUMI M，IHARA Y，SAITO A，et al. Standardization of testing methods for kinematic motion of five-axis machining centers—draft proposal for ISO standard[C]// Proceedings of the Seventh Manufacturing and Machine Tool Conference，2008：95-96.
[135] ISO/CD 10791-6. Test conditions for machining centers—part 6：Accuracy of feeds，speeds and interpolations[S]. Switzerland：ISO，2009.
[136] 沈云波，刘建亭，李济顺，等. 平面正交光栅在数控机床误差测量中的应用[J]. 工具技术，2003，37(8)：48-51. SHEN Yunbo，LIU Jianting，LI Jishun，et al. Application of cross grid encoder in measuring kinematic error of NC machine tools[J]. Tool Engineering，2003，37(8)：48-51.
[137] 章婷. 多轴数控机床准静态空间误差建模及误差辨识方法研究[D]. 南京：南京航空航天大学，2016. ZHANG Ting. Research on quasi static volumetric error modeling and error identification of multi-axis NC machine tool[D]. Nanjing：Nanjing University of Aeronautics and Astronautics，2016.
[138] 张滔滔. 数控机床直线轴误差辨识与补偿[D]. 哈尔滨：哈尔滨工业大学，2016. ZHANG Taotao. Error identification and compensation of CNC machine tool’s linear axes[D]. Harbin：Harbin Institute of Technology，2016.
[139] 郭世杰，梅雪松，姜歌东. 基于平面光栅的机床几何误差测量与辨识[J]. 计算机集成制造系统，2020，26(8)：2037-2049. GUO Shijie，MEI Xuesong，JIANG Gedong. Geometric error measurement and identification of machine tool based on gross grid encoder[J]. Computer Integrated Manufacturing Systems，2020，26(8)：2037-2049.
[140] 姚思涵. 五轴数控机床几何误差建模与测量方法研究[D]. 天津：天津工业大学，2020. YAO Sihan.Research on geometric error modeling and measurement methods for five axis CNC machine tools[D]. Tianjin：Tiangong University，2020.
[141] 孔刘伟，王振忠，叶超，等. 五轴增减材混合加工中心集成开发技术研究[J]. 航空制造技术，2019，62(6)：53-59. KONG Liuwei，WANG Zhenzhong，YE Chao，et al. Research on integrated development technology of five axis additive and additive mixed machining center[J]. Aviation Manufacturing Technology，2019，62(6)：53-59
[142] AIA/NAS. NAS 979 uniform cutting TESTS—NAS series metal cutting equipment specifications[J]. AIA/NAS-NAS979，1969.
[143] 项四通，杜正春，杨建国. 数控机床几何与热误差测量和建模研究新进展[J]. 机械设计与研究，2019，35(6)：52–57. XIANG Sitong，DU Zhengchun，YANG Jianguo. Recent advances in measurement and modeling of geometric and thermal error of CNC machine tools[J]. Machine Design and Research，2019，35(6)：52-57.
[144] 赫巍巍，关立文，李大奇. 基于S试件的AC双转台五轴数控机床加工精度检测联合仿真[J]. 河北工程大学学报，2019，36(3)：95-100. HE Weiwei，GUAN Liwen，LI Daqi. Co-simulation for machining accuracy detection of five-axis CNC machine with ac dual rotary table based on the s-shaped test piece[J]. Journal of Hebei University of Engineering，2019，36(3)：95-100.
[145] 王伟，陶文坚，李晴朝. 五轴数控机床动态精度检验试件特性研究[J]. 机械工程学报，2017，53(1)：101-109. WANG Wei，TAO Wenjian，LI Qingzhao. Research on characteristic of test specimen for five-axis CNC machine tools[J]. Journal of Mechanical Engineering，2017，53(1)：101-109.
[146] MOU W P，SONG Z Y，GUO Z P，et al. A machining test to reflect dynamic machining accuracy of five axis machine tools[J]. Advanced Materials Research，2012，622-623：414-419.
[147] 陈耿祥，李迎光，郝小忠，等. S试件几何形状定义与特性分析[J]. 南京航空航天大学学报，2017，49(6)：793-797. CHEN Gengxiang，LI Yingguang，HAO Xiaozhong，et al. Geometry definition and properties analysis of S shape test piece[J]. Journal of Nanjing University of Aeronautics & Astronautics，2017，49(6)：793-797.
[148] 张传景. 五轴数控机床几何误差建模方法研究[D]. 湛江：广东海洋大学，2023. ZHANG Chuanjing. Geometric Error modeling method research for five-axis CNC machine tools[D]. Zhanjiang：Guangdong Ocean University，2023.
[149] 黄伟. 基于激光跟踪仪的大型五轴数控机床空间精度检测及建模[D]. 成都：电子科技大学，2018. HUANG Wei. The measuring and modeling of the volumetric accuracy for large-scale five-axis machine tool based on laser tracker[D]. Chengdu：University of Electronic Science and Technology of China，2018.

面向误差补偿的数控机床空间误差建模与辨识方法研究进展

Research Progress on Spatial Error Modeling and Identification Methods for CNC Machine Tools Based on Error Compensation

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

编辑推荐

Metrics

本文评价

[1]	马帅, 冷杰武, 陈祝云, 李巍华, 李波, 刘强. 基于数字孪生和深度迁移学习的电主轴热误差建模方法[J]. 机械工程学报, 2025, 61(3): 52-66.
[2]	梁相龙, 姚志凯, 邓文翔, 姚建勇. 六自由度液压机械臂运动学标定和逆解研究[J]. 机械工程学报, 2025, 61(2): 346-357.
[3]	葛广言, 肖域坤, 吕军, 杜正春. 基于刚度缩聚和局部更新的薄壁件铣削力所致误差高效预测与实时补偿[J]. 机械工程学报, 2025, 61(17): 331-342.
[4]	刘志峰, 陈传海, 郭劲言, 黎志杰. 数控机床可靠制造新范式：由“功能上的可能”升级为“性能上的可靠”[J]. 机械工程学报, 2025, 61(12): 293-304.
[5]	冉琰, 田轲, 窦一凡, 金传喜, 张根保, 慕宗燚. 数控机床元动作链运动误差传递及综合精度建模[J]. 机械工程学报, 2023, 59(23): 211-220.
[6]	杨兆军, 何佳龙, 刘志峰, 李国发, 陈传海. 数控机床可靠性技术新进展[J]. 机械工程学报, 2023, 59(19): 152-163.
[7]	黄祖广, 王舒辉, 王金江, 张凤丽. 基于RAMS的数控机床综合评价方法研究[J]. 机械工程学报, 2022, 58(9): 218-230.
[8]	张建涛, 刘志峰, 李彦生, 姜凯, 杨聪彬, 张彩霞. 基于相似理论的重型数控机床-基础系统位移变形研究[J]. 机械工程学报, 2022, 58(7): 309-316.
[9]	夏纯, 张海峰, 李秦川, 柴馨雪. 基于等效运动链的并联机器人运动学标定方法[J]. 机械工程学报, 2022, 58(14): 71-84.
[10]	范晋伟, 谢本田, 叶倩, 陶浩浩. 基于区间理论的数控内圆复合磨床几何误差灵敏度分析方法研究[J]. 机械工程学报, 2022, 58(11): 220-230.
[11]	孟博洋, 李茂月, 刘献礼, WANG Lihui, LIANG S Y, 王志学. 机床智能控制系统体系架构及关键技术研究进展[J]. 机械工程学报, 2021, 57(9): 147-166.
[12]	李文龙, 谢核, 尹周平, 丁汉. 机器人加工几何误差建模研究：I空间运动链与误差传递[J]. 机械工程学报, 2021, 57(7): 154-168.
[13]	刘阔, 韩伟, 王永青, 刘海波, 宋磊. 数控机床进给轴热误差补偿技术研究综述[J]. 机械工程学报, 2021, 57(3): 156-173.
[14]	蓝益鹏, 姚婉婷, 杨文康, 雷城. 数控机床直线同步电动机磁悬浮系统的神经网络直接自适应控制[J]. 机械工程学报, 2021, 57(17): 236-242.
[15]	张俊, 蒋舒佳, 池长城. 2UPR&2RPS型冗余驱动并联机器人的运动学标定[J]. 机械工程学报, 2021, 57(15): 62-70.