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.     

Application of Monte Carlo simulation for estimation of uncertainty of four-point roundness measurements of rolls

Large-scale rotors in the paper and steel industry are called rolls. Rolls are reground at regular intervals and roundness measurements are made throughout the machining process. Measurement systems for roundness and diameter variation of large rolls (diameter <2000 mm) are available on the market, and generally use two to four sensors and a roundness measurement algorithm. These methods are intended to separate roundness of the rotor from its movement. The hybrid four-point method has improved accuracy, even for harmonic component amplitudes. For reliable measurement results, every measurement should be traceable with an estimation of measurement uncertainty. In this paper, the Monte-Carlo method is used for uncertainty evaluation of the harmonic components of the measured roundness profile under typical industrial conditions. According to the evaluation, the standard uncertainties for the harmonic amplitudes with the hybrid method are below 0.5 μm for the even harmonics and from 1.5 μm to 2.5 μm for the odd harmonics, when the standard uncertainty for the four probes is 0.3 μm each. The standard uncertainty for roundness deviation is 3.3 μm.     

Laser tracker performance quantification for the measurement of involute profile and helix measurements

Standard-conforming measurements for a large involute gear were performed with a manually operated laser tracker system and the corresponding task-specific measurement uncertainties were estimated. Especially, readers using laser trackers for inspecting large involute gears will get information of a taskspecific measurement performance for the first time, which significantly differs from the laser tracker machine specification. To ensure unambiguous and repeatable measurement results, user-friendly auxiliary tools are used, which allows the operator to probe the measurement points according to existing guidelines and standards. Measurements were taken on a robust and highly accurate large involute gear measurement standard of the Physikalisch-Technische Bundesanstalt (PTB) under laboratory conditions. The size of this gear measurement standard complies with those gears used in wind power plants. The external gear materializes a left and a right hand gear as well as a spur gear. The obtained results of profile, helix and surface measurements are presented. These research activities were carried out at the PTB in the department of coordinate metrology.     

Use of coordinate measuring machine to measure angles by geometric characterization of perpendicular planes. Estimating uncertainty

The purpose of this paper is to estimate the uncertainty of angle measurements using a Coordinate Measuring Machine (CMM), through the geometrical characterization of two concurrent sides of a steel angle gauge block with four perpendicular sides.For the calculation of the uncertainty associated with the measurement and investigating the errors of the CMM associated with orientation and length in the work volume, two models: linear statistical model behavior of CMM and the Mitutoyo model behavior are used and compared. After having established two behavioral models for the CMM we have determined the values of the angles and their uncertainty by using Monte Carlo Method. The results show that the proposed methods are suitable to investigate CMM hardware performance and determine the contribution of machine variables to measurement uncertainty. We can affirm that the statistical model behavior is more immediate and less laborious in terms of calculation and implementation time than the Mitutoyo model.     

三坐标机测量齿轮齿廓的不确定度评价

介绍了坐标测量中几种常用的不确定度评价方法.指出传统的三坐标测量机的测量不确定度评价方法大都不适用于评价坐标测量中面向对象的测量不确定度,并对使用蒙特卡洛方法评价测量不确定度进行了研究.首先,根据三坐标测量机详细标定文件及补偿策略说明建立测量模型.然后,将测量中的采样点通过测量模型生成大量测量结果,并以此评价测量不确定度.在齿廓评价实验中,评定齿廓误差的测量不确定度为0.96 μm时,多次评价结果之间的最大差值不超过0.03 μm,具有可靠的理论依据和较稳定的评定结果.文章指出,目前商用三坐标测量机大都不能为特定的测量对象提供测量不确定度报告,使用蒙特卡洛方法有希望改变此现状.     

An experimental method for assessing the contribution of the production process variations to the task-specific uncertainty of coordinate measurements

The present paper deals with an important source of variation often neglected in uncertainty estimates of coordinate measurements: the between-sample variation. This source of variation becomes relevant to the measurement uncertainty when significant interactions occur between variations in the production process and specific limitations in the measurement process. This paper discusses some particular types of interaction which usually lead to between-sample variation in coordinate measurements. For analysing the statistical significance of the between-sample variation and for assessing its contribution to the measurement uncertainty, an experimental method using multiple calibrated workpieces is described. A case study involving the uncertainty estimation of a roundness measurement process using a coordinate measuring machine is presented. The results show the importance of considering the between-sample variation in uncertainty evaluations of coordinate measurements.     

基于蒙特卡罗仿真方法的大尺寸测量不确定度分析

为评价大尺寸测量不确定度,特别是多站融合测量不确定度,引入蒙特卡罗仿真方法.该方法根据各传感器单元的概率特性重复采样,生成测量结果的样本,统计样本得到坐标测量不确定度,并利用计算机可视化样本的三维散点图.通过不同样本数对运算时问和准确性影响的实验,确定样本数为500.以激光跟踪仪为例进行实验,比较蒙特卡罗法、统计法和解析法三种方法得到的不确定度结果,吻合情况较好,其中与解析法比较最大偏差仅为2.7 um实验结果表明,蒙特卡罗仿真方法可以准确评价大尺寸测量仪器及多站融合测量不确定度,融合精度优于各局部精度.     

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.     

Monte Carlo simulation of scanning electron microscope signals for lithographic metrology

Two computer codes for simulating the backscattered, transmitted, and secondary-electron signals from targets in a scanning electron microscope are described. The first code, MONSEL-II, has a model target consisting of three parallel lines on a three-layer substrate, while the second, MONSEL-III, has a model target consisting of a two-by-two array of finite lines on a three-layer substrate. Elastic electron scattering is determined by published fits to the Mott cross section. Both plasmon-generated electrons and ionized valence electrons are included in the secondary production. An adjustable quantity, called the residual energy loss rate, is added to the formula of Joy and Luo to obtain the measured secondary yield. The codes show the effects of signal enhancement due to edge transmission, known as blooming, as well as signal reduction due to neighboring lines, known as the “black-hole” effect.     

Evaluation of coordinate measurement uncertainty with use of virtual machine model based on Monte Carlo method

Advances in modern manufacturing techniques implies more efficient production but also new tasks for coordinate metrology. The main of them is evaluation of accuracy of the measurement, because according to technological requirements, results of measurements are useful only when they are given with their accuracy. Currently used methods for uncertainty assessment are difficult and require knowledge and measuring experience. It is therefore important to implement correct and validated methods that will also be easy to implement and will not require broad metrological knowledge from the personnel. Presented here simulation method, based on Monte Carlo method is one of them. The article presents the conception, implementation and validation of this method.     

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.     

Design and analysis of experiments in CMM measurement uncertainty study

In applying a coordinate measuring machine to measure a mechanical object, many factors affect the measurement uncertainty. Although a number of studies have been reported in evaluating measurement uncertainty, few have applied the factorial design of experiments (DOE) to examine the measurement uncertainty, as defined in the ISO Guide to the Expression of Uncertainty in Measurement (GUM). This research applies the DOE approach to investigate the impact of the factors and their interactions on the uncertainty while following the fundamental rules of the GUM. The measurement uncertainty of the location of a hole measured by a coordinate measuring machine is used to demonstrate our methodology.     

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.     

Application of Monte Carlo modelling in the measurement of photomask linewidths at the national physical laboratory

A scanning electron microscope (SEM) fitted with a helium-neon laser interferometer is used to measure the widths of features on photomasks. In this way the magnification of the SEM can be known very precisely. Algorithms which use the backscattered electron (BSE) signal, yielding good measurement repeatability, have been developed, but in order to be able to relate measurements made on the image to the physical dimensions of the artefact, it has been necessary to model the image formation process. The modelled predicted offsets to be applied to the experimental measurements are governed largely by the angle of the sloping sides of the chromium edges of photomask lines; knowing this angle, it is possible to use a simple geometric relation to calculate what offset should be applied.     

Evaluation of measurement uncertainty of the high-speed variable-slit system based on the Monte Carlo method

Frontiers of Mechanical Engineering - This paper presents a dynamic and static error transfer model and uncertainty evaluation method for a high-speed variable-slit system based on a...     

Improvement of the measurement uncertainty of a high accuracy single sinker densimeter via setup modifications based on a state point uncertainty analysis

The single sinker densimeter with magnetic suspension coupling is one of the state of the art methods for the accurate measurement of fluid densities. The uncertainties of experimental pressure, density and temperature data, measured with a single sinker densimeter, were thoroughly evaluated following the uncertainty propagation law. The main uncertainty sources of each magnitude were determined. Based on this statistical study, several modifications were performed to reduce the uncertainty associated to each magnitude. Firstly two new PTR-25 probes were added. Secondly a new pressure transducer for the low pressure range was added. Finally the sinker of the densimeter was replaced by a bigger one to improve the balance reading. After these modifications the uncertainty of each magnitude was evaluated and validated with a Monte Carlo simulation. Results yielded a significant reduction of 44% in temperature uncertainty, more than 92% on pressure below 2 MPa, and more than 22% on density.     

Bremsstrahlung enhancement in electron probe microanalysis for homogeneous samples using Monte Carlo simulation

Fluorescence enhancement in samples irradiated in a scanning electron microscope or an electron microprobe should be appropriately assessed in order not to distort quantitative analyses. Several models have been proposed to take into account this effect and current quantification routines are based on them, many of which have been developed under the assumption that bremsstrahlung fluorescence correction is negligible when compared to characteristic enhancement; however, no concluding arguments have been provided in order to support this assumption. As detectors are unable to discriminate primary from secondary characteristic X‐rays, Monte Carlo simulation of radiation transport becomes a determinant tool in the study of this fluorescence enhancement. In this work, bremsstrahlung fluorescence enhancement in electron probe microanalysis has been studied by using the interaction forcing routine offered by penelope 2008 as a variance reduction alternative. The developed software allowed us to show that bremsstrahlung and characteristic fluorescence corrections are in fact comparable in the studied cases. As an extra result, the interaction forcing approach appears as a most efficient method, not only in the computation of the continuum enhancement but also for the assessment of the characteristic fluorescence correction.     

Evaluation of uncertainty in measurements based on digitized data

The paper deals with the uncertainty in measurement based on digital signal processing algorithms, like those achievable with the virtual instruments. The correct estimation of bias and uncertainty is discussed with reference to a simple case study. Three possible approaches to this question are examined and compared. It is shown how a Monte Carlo method, based on numerical simulations and implemented with commercial software packages, can allow virtual instruments to perform an auto-evaluation of both bias and uncertainty affecting their results. Some theoretical considerations, computer simulations and experimental tests are shown to support the proposed technique.     

Uncertainty analysis of a laser calibration system for evaluating the positioning accuracy of a numerically controlled axis of coordinate measuring machines and machine tools

This paper presents an uncertainty analysis of a Positional Error Calibrator based on a laser interferometer system. This laser calibration system is capable of evaluating the positioning accuracy of a numerically controlled axis of machine tools and coordinate measuring machines (CMM) under dynamic conditions. In order to assess the measurement uncertainty of this calibrator, an analysis of the uncertainty components that make up the uncertainty budget of this calibrator has been carried out. These uncertainty components can be classified into three categories as follows: (1) uncertainties intrinsic to the laser system; (2) uncertainties due to environmental effects; (3) measuring uncertainties due to the installation. The procedure for evaluating the uncertainty of this calibrator follows GUM (“Guide to the Expression of Uncertainty in Measurement”). This uncertainty analysis was carried out when this calibrator was used to assess the positional errors of the “X” axis of a moving bridge type CMM.