﻿ 基于三维最小二乘方法的空间直线度误差评定
 文章快速检索 高级检索

1. 北京航空航天大学 机械工程及自动化学院, 北京 100191;
2. 北京航空航天大学 虚拟现实技术与系统国家重点实验室, 北京 100191

Spatial straightness error evaluation based on three-dimensional least squares method
Wang Bingjie1, Zhao Junpeng1, Wang Chunjie1,2
1. School of Mechanical Engineering and Automation, Beijing University of Aeronautics and Astronautics, Beijing 100191, China;
2. State Key Laboratory of Virtual Reality Technology and Systems, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract:Spatial straightness error is very important for the assessment of mechanical product precision. The spatial straightness error evaluation algorithm with high precision is needed in real project. In order to evaluate the spatial straightness error more accurately, a mathematical model of spatial straight line fitting was established based on the national standard (GB/T 11336—2004) and three-dimensional least squares method, the exact solution to the model was deduced. The diameter of the minimum cylindrical surface of the least squares was obtained by using the method of spatial projection, coordinate transformation and lattice method. The proposed method was validated by numerical experiments. It is not only more accurate and robust, but also easy to be implemented.
Key words: spatial straightness error     spatial straight line fitting     method of spatial projection     coordinate transformation     three-dimensional least squares method

 图 1 最小二乘法评定直线误差度示意图Fig. 1 Diagram of spatial straightness error evaluation with LSM

δxyz=0时,即

 图 2 搜索区间节点Fig. 2 Nodes in search interval
2 数值实验与结果分析

 测量点数据来源 测点与包容圆柱面接触方式 评定直线度误差值/μm LSM算法 3DLSA算法 本文算法 文献[4] 2点接触 771845.6700 7.2448 6.2399 文献[3] 3点接触 18.1000 13.5000 9.9764 文献[15] 2点接触 16.6671 17.1252 16.0472 文献[6] 2点接触 30.0000 36.3000 28.4760 文献[5] 2点接触 171324.3700 17.1252 16.0472

1) 文献[4]测量点数值数量级不同,LSM算法结果为771845.6700,3DLSA算法得出的结果为7.2448,本文方法结果为6.2399;

2) 文献[3]与文献[6]测量点数值数量级相同,LSM算法结果分别为18.1000和30.0000,3DLSA算法结果分别为13.5000和36.3000,本文方法结果分别为9.9764和28.4760;

3) 文献[15]与文献[5]中测量点坐标相对差值相同,只是调换了坐标轴次序,它们的空间直线度误差值应该相同,但是LSM算法的评定结果相差较大,3DLSA算法满足此要求,本文方法亦满足此要求,而且本文方法评定的直线度误差值更小.

1) 本文采用三维最小二乘方法建立了空间直线拟合的数学模型,并给出了该数学模型的精确解,完善了空间直线拟合的理论基础.

2) 本文方法相较于算例中提及的算法具有更好的稳定度和准确度.

3) 本文方法计算效率较高可应用于精密测量以及数据处理中.

 [1] 张新宝,谢江平. 空间直线度误差评定的逼近最小包容圆柱法[J].华中科技大学学报:自然科学版,2011,39(12):6-9 Zhang Xinbao,Xie Jiangping.Evaluating spatial straightness errorby approaching minimum enclosure cylinder[J].Journal of Huazhong University of Science and Technology:Natural Science Edition,2011,39(12):6-9(in Chinese) Cited By in Cnki (5) [2] GB/T 11336—2004 直线度误差检测[S] GB/T 11336—2004 Measurement of departures from straightness[S](in Chinese) [3] 廖平,喻寿益. 基于遗传算法的空间直线度误差的求解[J].中南大学学报:自然科学版,1998,29(6):586-588 Liao Ping,Yu Shouyi.A method of calculating 3-D line error using genetic algorithms[J].Journal of Central South University:Science and Technology,1998,29(6):586-588(in Chinese) Cited By in Cnki (24) [4] 李淑娟,刘云霞. 基于坐标变换原理的最小区域法评定空间直线度误差[J].计测技术,2006,26(1):24-25 Li Shujuan,Liu Yunxia.A method for minimum zone evaluation of space linearity error based on principle of coordinate transformation[J].Metrology & Measurement Technology,2006, 26(1):24-25(in Chinese) Cited By in Cnki (12) [5] Ding Y,Zhu L M, Ding H.Semidefinite programming for chebyshev fitting of spatial straight line with applications to cutter location planning and tolerance evaluation[J].Precision Engineering,2007,31(4):364-368 Click to display the text [6] 黄富贵,崔长彩. 任意方向上直线度误差的评定新方法[J].机械工程学报,2008,44(7):221-224 Huang Fugui,Cui Changcai.New method for evaluating arbitrary spatial straightness error[J].Chinese Journal of Mechanical Engineering,2008,44(7):221-224(in Chinese) Cited By in Cnki (19) | Click to display the text [7] Wen X L, Xu Y X,Li H S,et al.Monte Carlo method for the uncertainty evaluation of spatial straightness error based on new generation geometrical product specification[J].Chinese Journal of Mechanical Engineering,2012,25(5):875-881 Click to display the text [8] Endrias D H, Feng H Y,Ma J,et al.A combinational optimization approach for evaluating minimum-zone spatial straightness errors[J].Measurement,2012,45(5):1170-1179 Click to display the text [9] 胡仲勋,杨旭静, 金湘中.LSM算法评定空间直线度误差的分析与改进[J].湖南大学学报:自然科学版,2010,37(2):27-31 Hu Zhongxun,Yang Xujing,Jin Xiangzhong.Analysis and improvement of the LSM algorithm for assessing spatial straightness error[J].Journal of Hunan University:Natural Sciences,2010,37(2):27-31(in Chinese) Cited By in Cnki (3) | Click to display the text [10] 郭际明,向巍, 尹洪斌.空间直线拟合的无迭代算法[J].测绘通报,2011(2):24-25 Guo Jiming,Xiang Wei,Yin Hongbin.Three-dimensional line fitting without iteration[J].Bulletin of Surveying and Mapping,2011(2):24-25(in Chinese) Cited By in Cnki (5) [11] 胡仲勋,杨旭静, 王伏林.空间直线度误差评定的LSABC算法研究[J].工程设计学报,2008,15(3):187-190 Hu Zhongxun,Yang Xujing,Wang Fulin.Research on LSABC algorithm for evalution of spatial straightness error[J].Journal of Engineering Design,2008,15(3):187-190(in Chinese) Cited By in Cnki (7) [12] 胡仲勋, 王伏林,周海萍.空间直线度误差评定的新算法[J].机械科学与技术,2008,27(7):879-882 Hu Zhongxun,Wang Fulin,Zhou Haiping.A new algorithm for evaluation of spatial straightness error[J].Mechanical Science and Technology for Aerospace Engineering,2008,27(7): 879882(in Chinese) Cited By in Cnki (8) [13] 胡仲勋,杨旭静, 金湘中.评定空间直线度误差的3DLSA算法研究[J].中国机械工程,2010,21(3):325-329 Hu Zhongxun,Yang Xujing,Jin Xiangzhong.Research on 3DLSA for evaluating spatial straightness errors[J].China Mechanical Engineering,2010,21(3):325-329(in Chinese) Cited By in Cnki (5) [14] 张筑生. 数学分析新讲(第二册)[M].北京:北京大学出版社,1990:294-295 Zhang Zhusheng.New lecture of mathematical analysis(book two)[M].Beijing:Peking University Press,1990:294-295(in Chinese) [15] Wen X L,, Song A G.An improved genetic algorithm for planar and spatial straightness error evaluation[J].International Journal of Machine Tools & Manufacture,2003,43(11): 1157-1162 Click to display the text

#### 文章信息

Wang Bingjie, Zhao Junpeng, Wang Chunjie

Spatial straightness error evaluation based on three-dimensional least squares method

Journal of Beijing University of Aeronautics and Astronsutics, 2014, 40(10): 1477-1480.
http://dx.doi.org/10.13700/j.bh.1001-5965.2013.0644