Chinese Journal of Mechanical Engineering

, Volume 25, Issue 5, pp 875–881 | Cite as

Monte carlo method for the uncertainty evaluation of spatial straightness error based on new generation geometrical product specification

  • Xiulan Wen
  • Youxiong Xu
  • Hongsheng Li
  • Fenglin Wang
  • Danghong Sheng


Straightness error is an important parameter in measuring high-precision shafts. New generation geometrical product specification(GPS) requires the measurement uncertainty characterizing the reliability of the results should be given together when the measurement result is given. Nowadays most researches on straightness focus on error calculation and only several research projects evaluate the measurement uncertainty based on “The Guide to the Expression of Uncertainty in Measurement(GUM)”. In order to compute spatial straightness error(SSE) accurately and rapidly and overcome the limitations of GUM, a quasi particle swarm optimization(QPSO) is proposed to solve the minimum zone SSE and Monte Carlo Method(MCM) is developed to estimate the measurement uncertainty. The mathematical model of minimum zone SSE is formulated. In QPSO quasi-random sequences are applied to the generation of the initial position and velocity of particles and their velocities are modified by the constriction factor approach. The flow of measurement uncertainty evaluation based on MCM is proposed, where the heart is repeatedly sampling from the probability density function(PDF) for every input quantity and evaluating the model in each case. The minimum zone SSE of a shaft measured on a Coordinate Measuring Machine(CMM) is calculated by QPSO and the measurement uncertainty is evaluated by MCM on the basis of analyzing the uncertainty contributors. The results show that the uncertainty directly influences the product judgment result. Therefore it is scientific and reasonable to consider the influence of the uncertainty in judging whether the parts are accepted or rejected, especially for those located in the uncertainty zone. The proposed method is especially suitable when the PDF of the measurand cannot adequately be approximated by a Gaussian distribution or a scaled and shifted t-distribution and the measurement model is non-linear.

Key words

uncertainty evaluation Monte Carlo method spatial straightness error quasi particle swarm optimization minimum zone solution geometrical product specification 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    International Standard Organization. ISO/TS 14253-3-2002 Geometrical product specifications (GPS)-Inspection by measurement of workpieces and measuring equipment-Part 3: Guidelines for achieving agreements on measurement uncertainty statements[S]. Switzerland: ISO copyright office, 2002.Google Scholar
  2. [2]
    International Standard Organization. ISO/TS 12780-1-2003 Geometrical product specifications (GPS)—Straightness—Part 1: Vocabulary and parameters of straightness[S]. Switzerland: ISO copyright office, 2002.Google Scholar
  3. [3]
    International Standard Organization. ISO 1101-1996 Technical drawings—Geometrical tolerancing[S]. Switzerland: ISO copyright office, 1996.Google Scholar
  4. [4]
    LIAO Ping, YU Shouyi. A method of calculating 3-D line error using genetic algorithm[J]. Journal of Central South University of Technology, 1998, 29(6): 586–588. (in Chinese)Google Scholar
  5. [5]
    HUANG J. An exact minimum zone solution for three-dimensional straightness evaluation problems[J]. Precision Engineering, 1999, 23(3): 204–208.CrossRefGoogle Scholar
  6. [6]
    ZHANG Q, FAN K C, LI Z. Evaluation method for spatial straightness errors based on minimum zone condition[J]. Precision Engineering, 1999, 23(3): 264–272.CrossRefGoogle Scholar
  7. [7]
    MAO Jian, CAO Yanlong. Evaluation method for spatial straightness errors based on particle swarm optimization[J]. Journal of Engineering Design, 2006, 13(5): 291–294. (in Chinese)Google Scholar
  8. [8]
    HERMANN G. Robust convex hull-based algorithm for straightness and flatness determination in coordinate measuring[J]. Acta Polytechnica Hungarica, 2007, 4(4): 111–120.MathSciNetGoogle Scholar
  9. [9]
    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.CrossRefGoogle Scholar
  10. [10]
    HUANG Fugui, CUI Changcai. New method for evaluation arbitrary spatial straightness error[J]. Chinese Journal of Mechanical Engineering, 2008, 44(7): 221–224. (in Chinese)MathSciNetCrossRefGoogle Scholar
  11. [11]
    WANG Jinxin, JIANG Xiangqian, XU Zhengao, et al. Uncertainty calculation of spatial straightness in three dimensional measurement[J]. Journal Huazhong University of Science & Technology (Natural Science Edition), 2005, 33(12): 1–3. (in Chinese)Google Scholar
  12. [12]
    ZHAO Fengxia, ZHANG Linna, ZHENG Yuhua, et al. Spatial straightness error evaluation and its uncertainty estimation based on GPS[J]. Journal of Mechanical Strength, 2008, 30(3): 441–444. (in Chinese)Google Scholar
  13. [13]
    BIPM Joint Committee for Guides in Metrology. JCGM 101, Evaluation of measurement data-Supplement 1 to the “Guide to the expression of uncertainty in measurement”-Propagation of distributions using a Monte Carlo method[S]. Sevres: Joint Committee for Guides in Metrology, 2008.Google Scholar
  14. [14]
    COX M G, SIEBERT B R L. The use of a Monte Carlo method for evaluating uncertainty and expanded uncertainty[J]. Metrologia, 2006, 43(4): S178–S188.CrossRefGoogle Scholar
  15. [15]
    KRUTH J P, GESTEL N V, BLEYS P. Uncertainty determination for CMMs by Monte Carlo simulation integrating feature form deviations[J]. Manufacturing Technology, 2009, 58(1): 463–466.Google Scholar
  16. [16]
    LIAN Huifen, CHEN Xiaohuai. Uncertainty evaluation in roundness measurement based on Monte Carlo method[J]. Tool Technology, 2010, 44(6): 82–84. (in Chinese)Google Scholar
  17. [17]
    WUBBELER G, KRYSTEK M, ELSTER C. Evaluation of measurement uncertainty and its numerical calculation by a Monte Carlo method[J]. Measurement Science Technology, 2008, 19(1): 1–4.Google Scholar
  18. [18]
    International Standard Organization. ISO/TS 15530-3-2004(E), Geometrical Product Specifications (GPS)-Coordinate measuring machines (CMM): Technique for determining the uncertainty of measurement-Part 3: Use of calibrated workpieces or standards[S]. Switzerland: ISO copyright office, 2004.Google Scholar
  19. [19]
    WEN X L, SONG A G. An improved genetic algorithm for planar and spatial straightness error evaluation[J]. International Journal Machine Tools & Manufacture, 2003, 24(1): 88–91.MathSciNetGoogle Scholar
  20. [20]
    KENNEDY J, EBERHART R. Particle swarm optimization[C]// Proceedings of the IEEE International Conference on Neural Networks (ICNN), Perth, Australia, November 27–December 1, 1995: 1 942–1 948.Google Scholar
  21. [21]
    WEN X L, HUANG J C, SHENG D H, et al. Conicity and cylindricity error evaluation using particle swarm optimization[J]. Precision Engineering, 2010, 34(2): 338–346.CrossRefGoogle Scholar
  22. [22]
    RAFAJLOWICZA E, SCHWABE R. Halton and hammersley sequences in multivariate nonparametric regression[J]. Statistics & Probability Letters, 2006, 76(8): 803–812.MathSciNetCrossRefGoogle Scholar
  23. [23]
    LEI G. Adaptive random search in quasi-Monte Carlo methods for global optimization[J]. Computers & Mathematics with Applications, 2002, 43(6): 747–754.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    BBHM W, GEYER-SCHULTS A. Exact uniform initialization for genetic programming, foundations of genetic algorithms IV[M]. California: Morgan Kaufmann, 1997.Google Scholar
  25. [25]
    MAARANEN H, MIETTINEN K, MAKELA M. Quasi-random initial population for genetic algorithms[J]. Computers and Mathematics with Applications, 2004, 47(12): 1 885–1 895MathSciNetCrossRefGoogle Scholar
  26. [26]
    International Standard Organization. ISO/TS 15530-4-2008, Geometrical Product Specifications(GPS)-Coordinate measuring machines (CMM): Technique for determining the uncertainty of measurement-Part 4: Evaluating task-specific measurement uncertainty using simulation[S]. Switzerland: ISO copyright office, 2008.Google Scholar
  27. [27]
    International Standard Organization. ISO/TS 14253-2-1999, Geometrical Product Specification (GPS)-Inspection by measurement of workpieces and measuring equipment-Part 2: Guide to the estimation of uncertainty in GPS measurement, in calibration of measuring equipment and in product verification[S]. Switzerland: ISO copyright office, 1999.Google Scholar

Copyright information

© Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Xiulan Wen
    • 1
  • Youxiong Xu
    • 1
  • Hongsheng Li
    • 1
  • Fenglin Wang
    • 1
  • Danghong Sheng
    • 1
  1. 1.Automation DepartmentNanjing Institute of TechnologyNanjingChina

Personalised recommendations