An Improved Measurement Uncertainty Calculation Method of Profile Error for Sculptured Surfaces

The current researches mainly adopt "Guide to the expression of uncertainty in measurement(GUM)" to calculate the profile error. However, GUM can only be applied in the linear models. The standard GUM is not appropriate to calculate the uncertainty of profile error because the mathematical model of profile error is strongly non-linear. An improved second-order GUM method(GUMM) is proposed to calculate the uncertainty. At the same time, the uncertainties in different coordinate axes directions are calculated as the measuring points uncertainties. In addition, the correlations between variables could not be ignored while calculating the uncertainty. A k-factor conversion method is proposed to calculate the converge factor due to the unknown and asymmetrical distribution of the output quantity. Subsequently, the adaptive Monte Carlo method(AMCM) is used to evaluate whether the second-order GUMM is better. Two practical examples are listed and the conclusion is drawn by comparing and discussing the second-order GUMM and AMCM. The results show that the difference between the improved second-order GUM and the AMCM is smaller than the difference between the standard GUM and the AMCM. The improved second-order GUMM is more precise in consideration of the nonlinear mathematical model of profile error.     

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.     

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。     

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.     

Flatness error evaluation and verification based on new generation geometrical product specification (GPS)

New generation geometrical product specification (GPS) links the whole course of a geometrical product from the research, development, design, manufacturing and verification to its release, utilization, and maintenance. Measurement process is one of the most important part of verification/inspection in the new generation GPS. With the knowledge-intensive and globalization trend of the economy, unifying the evaluation and verification of form errors will play a vital role in international trade and technical communication. Considering the plane feature is one of the most basic geometric primitives which contribute significantly to fundamental mechanical products such as guide way of machine tool to achieve intended functionalities, the mathematical model of flatness error minimum zone solution is formulated and an improved genetic algorithm (IGA) is proposed to implement flatness error minimum zone evaluation. Then, two evaluation methods of flatness error uncertainty are proposed, which are based on the Guide to the Expression of Uncertainty in Measurement (GUM) and a Monte Carlo Method (MCM). The calculating formula and the propagation coefficients of each element and correlation coefficients based on GUM and the procedures based on MCM are developed. Finally, two examples are listed to prove the effectiveness of the proposed method. An investigation into the source and effects of different uncertainty contributors for practical measurement on CMM is carried out and the uncertainty contributors significant are analyzed for flatness error verification. Compared with conventional methods, the proposed method not only has the advantages of simple algorithm, good flexibility, more efficiency and accuracy, but also guarantees the minimum zone solution specified in the ISO/1101 standard. Furthermore, it accords with the requirement of the new generation GPS standard which the measurement uncertainty characterizing the reliability of the results is given together. And it is also extended to other form errors evaluation and verification.     

Implementation of unscented transform to estimate the uncertainty of a liquid flow standard system

First-order partial derivatives of a mathematical model are an essential part of evaluating the measurement uncertainty of a liquid flow standard system according to the Guide to the expression of uncertainty in measurement (GUM). Although the GUM provides a straight-forward method to evaluate the measurement uncertainty of volume flow rate, the first-order partial derivatives can be complicated. The mathematical model of volume flow rate in a liquid flow standard system has a cross-correlation between liquid density and buoyancy correction factor. This cross-correlation can make derivation of the first-order partial derivatives difficult. Monte Carlo simulation can be used as an alternative method to circumvent the difficulty in partial derivation. However, the Monte Carlo simulation requires large computational resources for a correct simulation because it considers the completeness issue whether an ideal or a real operator conducts an experiment to evaluate the measurement uncertainty. Thus, the Monte Carlo simulation needs a large number of samples to ensure that the uncertainty evaluation is as close to the GUM as possible. Unscented transform can alleviate this problem because unscented transform can be regarded as a Monte Carlo simulation with an infinite number of samples. This idea means that unscented transform considers the uncertainty evaluation with respect to the ideal operator. Thus, unscented transform can evaluate the measurement uncertainty the same as the uncertainty that the GUM provides.     

空间直线度最小二乘评定结果的不确定度估计

评定模型的非线性和不可微给空间直线度最小二乘评定结果不确定度的估计带来困难,由空间直线度最小二乘评定模型出发,参照ISO国际测量不确定度表示指南,对影响不确定度估计的一些因素加以简化,推导了不确定估计公式。根据测量数据将推导公式估计结果与蒙特卡罗法运算的结果进行比较,结果表明采用本方法与蒙特卡罗方法获得的结果相符合,且本方法所用时间较少。     

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。     

基于蒙特卡罗法与GUM法的直线度测量不确定度评定

由于形位误差测量的复杂性和测量结果评定的多样性,导致在实际测量结果中形位误差的不确定度评定成了难题。通过GUM法和蒙特卡罗法对直线度的测量不确定度进行评定。首先,根据最小二乘法得到直线度的误差模型;然后采用GUM方法对测量结果进行不确定度评定,采用蒙特卡罗仿真方法对测量值进行模拟仿真,从而得到直线度误差的不确定度;设置实验对比,通过数据分析验证了蒙特卡罗方法评定的可行性,为形位误差测量结果不确定度评定提供了更加简便的方法。     

Comparison of GUM and Monte Carlo methods for evaluating measurement uncertainty of perspiration measurement systems

Measurement uncertainty is an important parameter to express measurement results including means and reliability. The uncertainty analysis of the biomedical measurement system needs to be established. A perspiration measurement system composed of several sensors was developed. We aim to estimate the measurement uncertainty of this system with several uncertainty sources, including airflow rate, air density, and inlet and outlet absolute humidity. Measurement uncertainty was evaluated and compared by the Guide to the expression of the uncertainty in measurement (GUM) method and Monte Carlo simulation. The standard uncertainty for the perspiration measurement system was 6.81 × 10⁻⁶ kg/s and the uncertainty percentage <10%. The major source of the uncertainty was airflow rate, and inlet and outlet absolute humidity. The Monte Carlo simulation could be executed easily with available spreadsheet software programs of the Microsoft Excel. GUM and Monte Carlo simulation did not differ in measurement uncertainty with precision to two decimal places. However, the sensitivity coefficient derived by GUM provided useful information to improve measurement performance, which was not evaluated with the Monte Carlo simulation method.     

Generalized expressions of second and third order for the evaluation of standard measurement uncertainty

The Guide to the Expression of Uncertainty in Measurement (GUM) requires the use of a first-order Taylor series expansion for propagating uncertainties. However, when the measurement function is strongly non-linear the use of this linear approximation may be inadequate and therefore higher order terms from the Taylor series cannot be neglected. The present paper aims to derive generalized expressions of second and third order for the evaluation of the estimate of a measurand and its associated standard uncertainty. A case study is given to illustrate an application of the proposed methods and the results obtained with the GUM method are compared to the corresponding ones when applying the method proposed in GUM Supplement 1.     

A computer simulation platform for the estimation of measurement uncertainties in dimensional X-ray computed tomography

The knowledge of measurement uncertainty is of great importance in conformance testing in production. The tolerance limit for production must be reduced by the amounts of measurement uncertainty to ensure that the parts are in fact within the tolerance. Over the last 5 years, industrial X-ray computed tomography (CT) has become an important technology for dimensional quality control. In this paper a computer simulation platform is presented which is able to investigate error sources in dimensional CT measurements. The typical workflow in industrial CT metrology is described and methods for estimating measurement uncertainties are briefly discussed. As we will show, the developed virtual CT (VCT) simulator can be adapted to various scanner systems, providing realistic CT data. Using the Monte Carlo method (MCM), measurement uncertainties for a given measuring task can be estimated, taking into account the main error sources for the measurement. This method has the potential to deal with all kinds of systematic and random errors that influence a dimensional CT measurement. A case study demonstrates the practical application of the VCT simulator using numerically generated CT data and statistical evaluation methods.     

Validation of the GUM using the Monte Carlo method when applied in the calculation of the measurement uncertainty of a compact prover calibration

This article presents the calibration of a compact prover using the weighing method. An evaluation of measurement uncertainty of the prover calibration has been developed using the GUM and Monte Carlo methodologies. A water draw kit was utilized to direct the liquid flow from the compact prover to a water container in order to weigh the transferred water mass on a balance. This amount of mass was used as reference for the calculation of the prover base volume. A modeling of the flow rate into the water draw kit as a function of time was conceived. This modeling was applied for calculating the error in the liquid volume of the water container due to the switching of two solenoid valves of the water draw kit. A mathematical model of the prover base volume has been developed. This model is non-linear and the two largest sources of uncertainty are related to the balance calibration certificate that together account for 31.84% of the uncertainty budget. This work showed that the GUM approach was validated by Monte Carlo method in the calculation of the measurement uncertainty of the calibration of a compact prover. The absolute differences of the respective endpoints of the coverage intervals of these two methods are less than 0.00023% of estimate of the prover base volume whose value is 151.427 dm³. This result was obtained for a coverage probability of 95% and ${10}^6$ Monte Carlo iterations. The density of the calibration water and its uncertainty have been calculated through an innovative approach.     

Can coverage factor 2 be interpreted as an equivalent to 95% coverage level in uncertainty estimation? Two case studies

The GUM modelling, its Bayesian modification and the Monte Carlo method (MCM) to estimate the uncertainty are compared in two practical measurement situations (finding reference value of relative humidity and a generic chemical instrumental analysis procedure). The results of the three approaches agree very well when there are no dominant input quantities with type A evaluated uncertainty estimated from small number of repeated measurements. In the opposite case the GUM gives underestimated expanded uncertainties (by up to 20–25%), compared to both other approaches. Analysis of the practical measurement situations reveals that even in the case of several dominating input quantities of similar uncertainty contributions, if one of them is distributed according to the t-distribution and has a low number (3–4) of degrees of freedom, the output quantity cannot be safely assumed Normally distributed and in such a case coverage factor 2 is not an equivalent to 95% coverage level.     

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。     

Precision modeling of form errors for cylindricity evaluation using genetic algorithms

A heuristic approach is proposed in this paper to model form errors for cylindricity evaluation using genetic algorithms (GAs). The proposed GAs method shows good flexibility and excellent performance in evaluating the engineering surfaces via measurement data involved with randomness and uncertainty. The numerical-oriented genetic operator is used as a basic representation for error modeling in the paper. The theoretical basis for the proposed Gas-based cylindricity evaluation algorithms is first presented. The performance of the method under various combinations of parameters and the precision improvement on the evaluation of cylindricity are carefully analyzed. One numerical example is presented to illustrate the effectiveness of the proposed method and to compare the Gas-based modeling results with those obtained by the least-squares method. Numerical results indicate that the proposed GAs method does provide better accuracy on cylindricity evaluation. The method can also be extended for solving difficult form error minimization and profile evaluation problems of various geometric parts in engineering metrology.     

Use of design of experiments and Monte Carlo method for instruments optimal design

A common issue in the design of measurement instruments is the comparison between different solutions in terms of components of the measurement chain, data processing or even measurement principles; the predicted instrumental uncertainty is the driving parameter for such a comparison. While in many situations the linearization of the measuring model allows using the standard ISO GUM procedure, in complex cases it might be necessary to proceed with Monte Carlo simulations as per ISO GUM supplement 1. This paper describes a method that combines the factorial design of experiments (DOE) and the ISO GUM supplement 1 uncertainty evaluation method to guide the instrument designer in the instrument configuration optimization. The proposed approach allows estimating, in the design phase, the overall instrumental uncertainty for different configurations, the instrument sensitivity to the accuracy in the measurements of its inputs and the effects on systematic and random measurement errors deriving from the choice of all instrumental variables. The use of data populations selected with the DOE criteria allows recovering valuable parameters equivalent to the sensitivity factors of the GUM linearized approach. The data analysis allows separating the critical factors that must be accurately controlled from those only weakly affecting the measurement uncertainty. The method has been applied to a case study where the metrological performances of a system devoted to the measurement of the acoustic radiation emitted by a vibrating panel in a reverberant enclosure had to be assessed.     

Evaluation of measurement uncertainty using mixed distribution for conducted emission measurements

For evaluation of measurement uncertainty for conducted emission measurements, we propose a new model which uses mixed distribution. Namely, evaluation of probability density function (PDF) for the measurand has been done using a Monte Carlo method and a modified least-squares method. As the Monte Carlo method required numerical calculation of approximate PDF values, pseudorandom number generator (PRNG) was developed for these requirements. For illustration, this work presents mixed distributions of two normal distributions, normal and rectangular distributions, and normal and also triangular distributions in case of conducted emission measurements. The results obtained by the Monte Carlo method and the modified least-squares method are compared to the corresponding results when applying the standard Guide to the Expression of Uncertainty in Measurement (GUM) procedure.     

Cylindricity modeling and tolerance analysis for cylindrical components

The circular and cylindrical features are fundamental geometric features in machines. Cylindricity error affects the fitting conditions of cylindrical components and impacts the performance of the precision products. In this paper, the cylindricity error was modeled using L-F functions and evaluated by particle swarm optimization algorithm. Then the contact method is developed to determine the position accuracy through the virtual assembling of the bore and shaft using Monte Carlo simulation. The effects of the cylindricity error and the number of lobes on the position error were analyzed in detail. The results indicate that the cylindricity error has more significant influence on the position accuracy between the cylindrical parts than the roundness error. Using the suggested method in the paper, the position accuracy can be rapidly predicted after the design tolerances are allocated or the geometrical errors are measured on manufactured parts.     

寄生式时栅传感器测量不确定度的分析与评定

为了全面分析寄生式时栅误差和不确定度来源,提高寄生式时栅的测量精度,建立符合国际GUM规范检测结果的不确定度评定模型,以84对级的寄生式时栅为研究对象,根据其测量原理分析所测量角度的计算公式,进而将不确定度来源分为插补脉冲个数的误差、插补脉冲信号的量化误差、行波信号的周期误差和环境误差四大类,从理论上建立各不确定度分量之间的理论传递关系,应用现代不确定度理论,推导出合成测量不确定度计算公式。搭建实验平台,利用示波器等仪器的测量结果评定各不确定分量具体数值大小,计算被测角度的合成测量不确定度值。通过与寄生式时栅整圆周的实际测量误差相比较,可以看出利用该评定方法评定的传感器角度测量不确定度与实际误差相符,因此可以用于寄生式时栅传感器的实际评定。