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

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.     

Adaptive Monte Carlo and GUM methods for the evaluation of measurement uncertainty of cylindricity error

Measurement uncertainty is one of the most important concepts in geometrical product specification (GPS). The “Guide to the expression of uncertainty in measurement (GUM)” is the internationally accepted master document for the evaluation of uncertainty. The GUM method (GUMM) requires the use of a first-order Taylor series expansion for propagating uncertainties. However, when the mathematical model of measurand is strongly non-linear the use of this linear approximation may be inadequate. Supplement 1 to GUM (GUM S1) has recently been proposed based on the basis of probability density functions (PDFs) using the Monte Carlo method (MCM). In order to solve the problem that the number of Monte Carlo trials needs to be selected priori, adaptive Monte Carlo method (AMCM) described in GUM S1 is recommended to control over the quality of the numerical results provided by MCM.The measurement and evaluation of cylindricity errors are essential to ensure proper assembly and good performance. In this paper, the mathematical model of cylindricity error based on the minimum zone condition is established and a quasi particle swarm optimization algorithm (QPSO) is proposed for searching the cylindricity error. Because the model is non-linear, it is necessary to verify whether GUMM is valid for the evaluation of measurement uncertainty of cylindricity error. Then, AMCM and GUMM are developed to estimate the uncertainty. The procedure of AMCM scheme and the validation of GUMM using AMCM are given in detail. Practical example is illustrated and the result shows that GUMM is not completely valid for high-precision evaluation of the measurement uncertainty of cylindricity error if only the first-order terms in the Taylor series approximation are taken into account. Compared with conventional methods, not only the proposed QPSO method can search the minimum zone cylindricity error precisely and rapidly, but also the Monte Carlo simulation is adaptive and AMCM can provide control variables (i.e. expected value, standard uncertainty and lower and higher coverage interval endpoints) with an expected numerical tolerance. The methods can be extended to the evaluation of measurement uncertainty of other form errors such as roundness and sphericity errors.     

圆柱度误差评定及其不确定度估计

基于新一代产品几何技术规范(Geometrical Product Specification,GPS)的操作及操作算子技术给出了圆柱度误差的最小二乘数学模型;根据GUM(Guide to the Expression of Uncertainty in Measurement)建议的方法,导出了该模型的不确定度估计公式;通过蒙特卡罗模拟法和实际测量计算结果的比较,验证了所导出的不确定度估计公式的有效性。     

Measurement uncertainty evaluation of conicity error inspected on CMM

The cone is widely used in mechanical design for rotation, centering and fixing. Whether the conicity error can be measured and evaluated accurately will directly influence its assembly accuracy and working performance. According to the new generation geometrical product specification(GPS), the error and its measurement uncertainty should be evaluated together. The mathematical model of the minimum zone conicity error is established and an improved immune evolutionary algorithm(IIEA) is proposed to search for the conicity error. In the IIEA, initial antibodies are firstly generated by using quasi-random sequences and two kinds of affinities are calculated. Then, each antibody clone is generated and they are self-adaptively mutated so as to maintain diversity. Similar antibody is suppressed and new random antibody is generated. Because the mathematical model of conicity error is strongly nonlinear and the input quantities are not independent, it is difficult to use Guide to the expression of uncertainty in the measurement(GUM) method to evaluate measurement uncertainty. Adaptive Monte Carlo method(AMCM) is proposed to estimate measurement uncertainty in which the number of Monte Carlo trials is selected adaptively and the quality of the numerical results is directly controlled. The cone parts was machined on lathe CK6140 and measured on Miracle NC 454 Coordinate Measuring Machine(CMM). The experiment results confirm that the proposed method not only can search for the approximate solution of the minimum zone conicity error(MZCE) rapidly and precisely, but also can evaluate measurement uncertainty and give control variables with an expected numerical tolerance. The conicity errors computed by the proposed method are 20%-40% less than those computed by NC454 CMM software and the evaluation accuracy improves significantly.     

新一代GPS操作技术在平面度误差评定中的应用

现代产品几何技术规范(GPS)中提出的操作及算子技术,为实现几何产品检验/认证过程的数字化和规范化提供了必要的技术基础。文章在阐述GPS操作及操作算子技术的基础上,进一步以平面度误差的评定为例,详细分析了传统的平面度误差评定方法中存在的不足,研究并分析了基于GPS的平面度误差数字化评定的实现思路及关键技术。     

平面度坐标测量的不确定度计算

目前的坐标测量只是给出平面度最小二乘检验的结果,并没有给出检验结果的不确定度.根据平面度最小二乘检验的基本原理和ISO/TS 14253-2给出的不确定度传递公式,提出了一种平面度坐标测量的不确定度的计算方法.该方法的特点是将平面方程的系数看作一个随机向量,通过计算该随机向量的均值和协方差矩阵来确定平面方程和平面度的检验结果及其不确定度.这不仅保证了平面度检验结果的完整性,而且符合新一代产品几何规范(GPS)标准的要求,从而可以提高工件检验的准确性.实验结果表明,根据平面度最小二乘检验的结果及其不确定度,可以根据ISO 14253-1给出的判定原则定量地判定平面是否合格.     

工件圆度误差测量不确定度评定

为了实现工件圆度误差的不确定度评定,对基于三坐标测量机的工件圆度轮廓数据的采样策略、圆度评定方法及不确定度评定方法进行研究。首先,根据工件圆度轮廓特征进行实验测量,获取不同工件的多个样本。接着,基于最小二乘法和微分进化优化算法对样本的圆度误差进行了误差评定。然后,在分析比较误差大小的基础上,说明了采用的采样策略和微分进化评定算法。最后,基于圆度误差评定结果运用了测量不确定度表示指南(GUM)和蒙特卡洛方法(MCM)进行不确定度评定。实验结果表明:微分进化算法与最小二乘法相比均值差最大达到1.1μm, MCM方法比GUM方法得到的标准不确定度均值小0.02μm。合理的采样点数、微分进化算法及MCM不确定度评定方法可以得到更稳定可靠、精度高的评定结果。     

激光跟踪仪多边测量的不确定度评定

激光跟踪仪多边测量是大型高端装备制造现场溯源的重要手段,正确评定其不确定度是确保制造过程量值统一、结果可靠的关键。本文提出了一种准确、快速的激光跟踪仪多边测量的不确定度评定方法。从仪器误差、环境干扰及靶球制造误差等方面分析激光跟踪仪多边测量的不确定度来源。针对多边测量的输出量为多维向量的特点,重点研究基于多维不确定度传播律(GUM法)的不确定度合成方法,同步评定目标点坐标和跟踪仪站位的不确定度。最后,介绍了点到点长度的不确定度计算方法。实验表明:GUM法评定的不确定度结果与蒙特卡洛法(MCM法)的结果相比,坐标不确定度偏差小于0.000 2 mm,相关系数偏差小于0.01,满足数值容差,且GUM法用时仅为MCM法的0.08%;点到点长度测试的En值均小于1。因此,基于GUM法评定激光跟踪仪多边测量的不确定度具有可行性及高效性,且评定结果正确、可靠。     

极大熵法在几何量测量不确定度评定中的应用

为提高新一代GPS标准体系中几何量测量不确定度的评定精度,提出了一种基于极大熵检验的曲线拟合方法.该方法根据极大熵原理,将极大熵函数法应用到几何量测量要素操作的拟合中,把曲线拟合转化为光滑函数优化问题进行求解.最后以圆度为例,利用极大熵法和最小二乘法对测量数据进行了处理.对比结果表明,该方法简单易行,精度高,符合新一代GPS标准的要求.     

现代产品几何技术规范(GPS) 的不确定度理论及应用技术研究

新一代GPS（generation geometrical product specification）在测量不确定度的基础上扩展了不确定度的概念,将不确定度应用到几何产品的“功能、规范、认证”的全过程。文中着重分析新一代GPS不确定度的构成、相互关系及GPS过程量化统一的内在规律性;明确不确定度与操作算子之间的关系,揭示GPS不确定度理论的概念基础、应用规律及技术经济性;研究新一代GPS基本原则与不确定度之间的关系规律,给出减小GPS不确定度的对策和措施;并在此基础上,以产品质量认证为例,进一步分析研究GPS测量不确定度规范及其应用技术。     

新一代GPS中平面度公差域的研究

在现代制造行业中,技术标准的发展成为影响制造业的重要因素,目前推行的为新一代产品几何技术规范(geometrical product specification and verification,简称GPS)。如何根据新一代GPS的要求,制定一系列标准规范已成为很多国家的重要课题。本文根据新一代GPS的要求,利用尺寸公差和几何公差的独立原则,通过对平面度相关自由度变量的确定来规范所要求的平面,并分析平面度公差域的范围;通过实例分析求解平面度公差域的变化范围,得出一种求平面度公差域范围的方法。     

改进蜂群算法在平面度误差评定中的应用

为了准确快速评定平面度误差,提出将改进人工蜂群( MABC)算法用于平面度误差最小区域的评定.介绍了评定平面度误差的最小包容区域法及判别准则,并给出符合最小区域条件的平面度误差评定数学模型.叙述了MABC算法,该算法在基本人工蜂群算法( ABC)模型的基础上引入两个牵引蜂和禁忌搜索策略.阐述了算法的实现步骤,通过分析选用两个经典测试函数验证了MABC算法的有效性.最后,应用MABC算法对平面度误差进行评定,其计算结果符合最小条件.对一组测量数据的评定显示,MABC算法经过0.436 s可找到最优平面,比ABC算法节省0.411 s,其计算结果比最小二乘法和遗传算法的评定结果分别小18.03μm和6.13 μm.对由三坐标机测得的5组实例同样显示,MABC算法的计算精度比遗传算法和粒子群算法更有优势,最大相差0.9 μm.实验结果表明,MABC算法在优化效率、求解质量和稳定性上优于ABC算法,计算精度优于最小二乘法、遗传算法和粒子群算法,适用于形位误差测量仪器及三坐标测量机.     

Evaluation method for spatial straightness errors based on minimum zone condition

A nonlinear mathematical model for spatial straightness error evaluation based on the minimum zone condition is established in this paper. According to the error analysis, it is proved that the mathematical model for spatial straightness error evaluation cannot be linearized. A criterion for verification of the existence and uniqueness of the minimum zone solution is proposed. A new computational method is also proposed, and practical examples are given. Finally, the correctness of this method is demonstrated using a geometrical solution. This new method is convenient for computation of uniqueness and exactness of the minimum zone solution.     

基于新一代GPS的产品检验符合性不确定度评定*

新一代产品几何技术规范将测量不确定度的概念拓展至符合性不确定度,但并未给出相对应的评定方法。为全面估计产品检验中测量结果与产品规范所有可能的差异,基于新一代产品几何技术规范,研究产品检验符合性不确定度评定。基于产品几何技术规范定义,提出规范不确定度、方法不确定度、符合性不确定度的评定方法;借助不确定度的黑箱模型,通过测量结果统计学量值特性指标,评定执行不确定度。以产品圆度检验为例,研究符合性不确定度评定操作过程,基于符合性不确定度划分产品检验的合格区间。实例分析结果表明,规范不确定度和方法不确定度的量值与执行不确定度相当,不可忽略;由于符合性不确定度包含测量结果与图纸规范所有可能的不一致性,基于符合性不确定度进行产品合格判定更为可靠。     

新一代产品几何技术规范测量不确定度理论及应用技术

研究了测量不确定度的统一评定和表述方法，用新一代产品几何技术规范（GPS）基于GUM及系统量化统一的新思路，拓展了测量不确定度的概念，规范了测量不确定度的评定程序。在此基础上，将测量不确定度的概念拓展至GPS系统，利用拓展后不确定度的统计、量化特性，将产品的功能、规范与测量评定量化集成。通过不确定度的经济杠杆调节作用，实现过程资源配置的统筹优化，提高产品的综合效益。     

现代产品几何技术规范(GPS)体系及应用分析

现代产品几何技术规范GPS(dimensional geometrical product specification and verification)是ISO／TC213针对产品的设计与制造而规定的一系列宏观和微观的几何技术规范，是所有机电产品的技术标准与计量规范的基础。随着全球经济的发展和科学技术的进步，尤其是随着CAD／CAM／CAQ(computer aided design／computer aided manufacture／computer aided quality)的应用和发展，新工艺、新技术、新材料的应用以及加工精度从微米到纳米的提高，ISO／TC213 GPS也随之发生了巨大的变化，已经由以几何学为基础的第一代GPS，发展到以计量学为基础的第二代GPS。文中在阐述ISO／TC213 GPS标准体系的形成、特点及发展趋势的基础上，对其构成思路及矩阵模型进行深入分析和研究，进一步揭示出其新一代GPS标准体系的构造模式规律及本质特征。     

现代产品几何技术规范(GPS)中基于对偶性的操作技术及其应用

揭示现代产品几何技术规范（geometrical product specification and verification，GPS）中规范过程和检验／认证过程的物象对应关系，以及其表面模型存在着对偶性、操作技术存在着共性的内在规律性；井以几何误差的检验／认证过程为例，阐述利用操作及算子技术实现GPS计量过程数字化的可能性及应用规律；进一步明确基于对偶性的表面模型、操作及操作算子技术，对于实现GPS设计与计量的统一性、与CAX的集成性和数字化功能是至关重要的。     

Uncertainty Calculation of Roundness Assessment by Automatic Differentiation in Coordinate Metrology

Recently, Coordinate Measuring Machines (CMMs) are widely used to measure roundness errors. Roundness is calculated from a large number of points collected from the profiles of the parts. According to the Guide to the Expression of Uncertainty in Measurement (GUM), all measurement results must have a stated uncertainty associated the them. However, no CMMs give the uncertainty value of the roundness, because no suitable measurement uncertainty calculation procedure exists. In the case of roundness measurement in coordinate metrology, this paper suggests the algorithms for the calculation of the measurement uncertainty of the roundness deviation based on the two mainly used association criteria, LSC and MZC. The calculation of the sensitivity coefficients for the uncertainty calculation can be done by automatic differentiation, in order to avoid introducing additional errors by the traditional difference quotient approximations. The proposed methods are exact and need input data only as the measured coordinates of the data points and their associated uncertainties.     

新一代GPS中提取与滤波、拟合操作间关联特性的研究

基于对新一代GPS(geometrical product specification and verification)中关键操作技术的深入分析,揭示提取与滤波、拟合之间固有的内在规律性,给出操作间参数的选用原则,为统筹优化几何误差数字化评定中的操作策略提供技术基础;最终通过实例验证关键操作集成化思想对几何误差评定的高效稳定性,不仅有利于实现几何误差数字化计量精度和成本的优化,而且还推进了新一代GPS标准体系关键技术的应用研究。