A method for the selection of algorithms for form characterization of nominally spherical surfaces

This paper presents a method for the selection of algorithms for the form characterization of nominally spherical surfaces. The form characterization is applied to discrete data obtained from a three-dimensional (3-D) high-precision measurement system. In this paper, five sphere-fitting algorithms are outlined and compared. The fitting algorithms under investigation are: linear and nonlinear least-squares sphere fit, minimum zone sphere fit, four-point sphere fit, and sphere fit by error curve analysis. A method for selection of the best-fit algorithm, based on a spectral analysis of the surface irregularity is suggested and applied to the surface measurement of worn electrical switching contacts.     

高精度光斑中心定位算法

光斑中心定位是光学测量中的关键技术之一,检测算法的精度和速度直接影响了测量的精度及速度,传统的检测算法如灰度质心法、Hough变换法等在检测精度或速度上存在不足。鉴于此,提出了一种高精度光斑中心定位算法,该算法不仅能定位光斑中心还能拟合出圆半径。用计算机生成的光斑和实验生成的光斑对该算法进行验证,并与其他传统算法进行比较,结果表明,该算法的误差小于0.5像素且比其他经典算法更精确。     

Method for the evaluation of optical encoders performance under vibration

Optical encoders are the preferred choice for position measurement, both linear and angular, when high accuracy is required. Their performance is widely affected by deformation, temperature and vibration. This last aspect is analysed, showing the limitations found in conventional methods of performance evaluation. The determining parameters, such as sweep type, sampling rate, trigger etc. are examined and a final diagram of measuring accuracy versus frequency is presented, where aspects as resonance effects and measuring uncertainty are identified and discussed. Experimental analyses have been made in a commercial linear optical encoder and show that its measuring error is three times its precision, and wide areas of large measuring uncertainty are present in the frequency span. The information obtained through this method can be used to improve encoder design and mounting conditions in the machine, reducing the total error.     

Characterization of instrument drift using a spherical artifact

Drift is a common and inevitable error source in measurements. Currently there are two main approaches to address instrument drift in image or area-based measurements, drift calibration with target tracking and active feedback correction. We propose an alternative approach to drift calibration for profilometers, particularly high speed instruments such as confocal microscopes or scanning white light interferometers. The method is based on sequential measurements of a spherical artifact whose diameter is larger than the field of view. A best fit sphere algorithm is used to determine the movement of the spherical artifact's center over time. This reduces drift measurement uncertainty because it uses height data over the full field of view, compared to target tracking strategies that involve tracking small features. Simulation results show that under practical conditions, e.g. with typical noise levels and typical drift rates, this method is quite effective and can yield measurements with low uncertainty. The measurement is demonstrated on a commercial confocal microscope to determine drift rate magnitude and direction.     

微小球面曲率半径的测量研究

验证了测量平面面形误差的重叠四步平均算法（OAF）应用于测量微小球面曲率半径的可行性。采用Linnik显微移相干涉装置，测量了光纤连接器端面的曲率半径，给出了测量数据及处理结果；通过对已知曲率半径的标准微小凸球面样板的测量，得到相对误差为0．093％。同时，从理论上对OAF和其他算法对PZT位移误差的抑制能力作了分析比较。结果表明，OAF算法可用于微小球面曲率半径的测量，且相对于其他算法而言，其对PZT位移误差具有更强的抑制能力。     

Concepts and Limitations of Using Optical Profilometry for Air Bearing Simulations of Hard Disk Sliders

The topography of hard disk sliders is commonly measured with optical profilometers that produce a regularly spaced matrix of topography data corresponding to the pixel array of the CCD camera. This paper explores a new concept that combines optical profilometry and air bearing simulations to achieve an accurate method for predicting fly height early in the manufacturing process. Various sources of error in the topography measurement are analyzed with regards to their impact on the accuracy of air bearing simulations. The findings are illustrated by a feasibility study based on numerical simulations of a slider with a negative pressure air bearing design.     

Invited review article: measurement uncertainty of linear phase-stepping algorithms

Phase retrieval techniques are widely used in optics, imaging and electronics. Originating in signal theory, they were introduced to interferometry around 1970. Over the years, many robust phase-stepping techniques have been developed that minimize specific experimental influence quantities such as phase step errors or higher harmonic components of the signal. However, optimizing a technique for a specific influence quantity can compromise its performance with regard to others. We present a consistent quantitative analysis of phase measurement uncertainty for the generalized linear phase stepping algorithm with nominally equal phase stepping angles thereby reviewing and generalizing several results that have been reported in literature. All influence quantities are treated on equal footing, and correlations between them are described in a consistent way. For the special case of classical N-bucket algorithms, we present analytical formulae that describe the combined variance as a function of the phase angle values. For the general Arctan algorithms, we derive expressions for the measurement uncertainty averaged over the full 2π-range of phase angles. We also give an upper bound for the measurement uncertainty which can be expressed as being proportional to an algorithm specific factor. Tabular compilations help the reader to quickly assess the uncertainties that are involved with his or her technique.     

误差补偿技术在相位偏移干涉测量中的应用

在研究泰曼—格林相位偏移干涉仪测量原理基础上，分析了位移驱动器移相误差对五幅移相计算结果的影响，一阶线性误差和二阶非线性误差是相位偏移干涉测量技术中产生相位误差的主要因素；提出了五幅算法移相误差补偿技术，该方法直接从相位偏移干涉图中计算移相过程中存在的一阶及二阶移相误差，对五幅算法结果进行误差修正；采用玻璃平晶为测试对象，建立了泰曼一格林干涉仪移相误差补偿原理试验系统。试验结果表明在同时存在一阶移相误差及二阶移相误差情况下，采用提出的移相误差补偿方法可以将位移测量精度提高6倍，相当于采用氦氖激光器的倍程干涉仪中位移精度达到1．0nm。     

An investigation of the robustness of the nonlinear least-squares sphere fitting method to small segment angle surfaces

This paper presents an investigation of the nonlinear least-squares sphere fitting algorithm (TLS_A). The work concentrates on investigating the reliability of the TLS_A algorithm when applied to a small segment angle of a sphere. The definition of small segment angle is discussed in the paper and taken to be below 1° (in both x and y directions) of the spherical surface. This application of the TLS_A method is important when it is used on data from optical scanning systems where the measurements are limited by the gauge range and the angular tolerance of the sensor. The TLS_A algorithm has been first compared with the TLS_D algorithm suggested by Forbes for this application. The results show that the TLS_A algorithm can be used in small surface segment angles. The main study is focused on testing the algorithm on a sphere superimposed with surface irregularities (sensor/measurement noise or roughness). Two properties of the TLS_A algorithm are covered: the bias and the uncertainty of the estimated radius. Both simulation and theoretical approaches have been attempted. A new algorithm to estimate the bias of the TLS_A algorithm has been derived in this paper based on Box's method. Together with uncertainty estimation, which can be produced by using either a conventional method or Zhang's error propagation function (EPF), a comprehensive understanding of the TLS_A algorithm in this application is thus achieved, and used to develop a number of recommendations for the precision metrology of spherical and near spherical surfaces.     

FILTERED REFERENCE AND FILTERED ERROR LMS ALGORITHMS FOR ADAPTIVE FEEDFORWARD CONTROL

A unified and consistent formulation is developed for both filtered reference and filtered error forms of the instantaneous steepest decent, or LMS, algorithm when used to adapt FIR feedforward controllers. Both algorithms minimise the mean-square value of the same, output, error function. The two algorithms are first formulated for single-input single-output linear plants. It is argued that since the behaviour of the two algorithms is equivalent in the case of slow adaption, the conditions on the accuracy of the plant model for stability should also be the same in both cases. This is expressed as a bound on the unstructured multiplicative uncertainty of the plant. Filtered reference and filtered error algorithms are also derived for multiple-input multiple-output (MIMO) linear systems, although the filtered reference algorithm is found not to have a simple block diagram interpretation. In the MIMO case, the filtered error form of the algorithm can have considerable computational advantages over the filtered reference form. Finally the two algorithms are extended to the case of non-linear plants and/or controllers which are modelled as feedforward neural networks. In the non-linear case the two formulations of the LMS algorithm reduce to two forms of the widely used backpropagation algorithm.     

Measurement error in prism coupling method and refractive index profile reconstruction in ion-exchanged waveguides

Measurement error in prism coupling method is studied and an analytic expression for the uncertainty of mode effective index measurement in an ion exchanged surface waveguide is derived. A novel algorithm to determine the profile of refractive index for ion-exchange waveguides is proposed. This method takes the values calculated by anti-WKB and linear segment interpolation as approximation and then uses the least squares fitting to eliminate the measurement error. Numerical results show that the proposed algorithm can effectively reconstruct the transverse refractive index profile of graded ion-exchange planar waveguides.     

Influence of linear-axis error motions on simultaneous three-axis controlled motion accuracy defined in ISO 10791-6

Five-axis machine tools, which combine three linear axes and two rotary axes, are required for accuracy in machining complex shapes. In this paper, to clarify the influence of simultaneous three-axis control motion measurements as specified in ISO 10791-6, the measured results using a ball bar and R-test are compared. As the motion trajectories of the linear axes are not identical in both measurement devices, it is expected that the error motions of the linear axes cause different measurement results depending on the measurement devices. Thus, the squareness errors between the linear axes and the error motions of the linear axes are assumed as error factors that influence the measured results in this study. A mathematical model of a five-axis control machine tool that considers the error motions and squareness errors of the linear axis is constructed, and the influence of those error factors on motion accuracy is examined using an experiment and a simulation. As a result, the squareness errors and error motions of the linear axis are observed to greatly affect simultaneous three-axis controlled motion accuracy.     

Step height measurement of microscale thermoplastic polymer specimens using contact stylus profilometry

Quantitative characterisation of the dimensions of micro-scale polymer specimens imposes higher demands on stylus profilometry. Due to the weak and time-dependent mechanical properties of polymers, one of the main error sources within tactile dimensional measurement for polymeric structures comes from the surface deformation of specimens under stylus scanning with a certain probing force. In this paper, a practical elastic–viscoplastic model is proposed for modelling the (steady-state) polymer deformations during tactile surface scanning. To experimentally determine the viscous material properties of polymers necessary for the proposed analytical model, a stylus-profiling based methodology has been developed. The feasibility of the proposed methodology and a measurement error compensation model has been experimentally verified. Preliminary experimental investigation on measurement of polymeric step height specimens with currently available stylus profilometers demonstrates that the specimen deformation induced tactile measurement deviation can be evidently reduced.     

液压油路气液两相传质的光学测量

为研究液压系统工作压力下管路中自由气体的气液传质过程,解决油液中气泡质量变化的光学测量问题,推导了高压油液环境下气泡直径测量的恒温及零温度梯度条件,设计了高压气液传质光学测量系统,并对该系统的气泡识别与跟踪的关键算法进行研究。根据毕渥数及牛顿冷却定律推导了测量气泡的极限尺寸,提出以气泡圆度为判定阈值的气泡图元识别算法。最后,利用相邻帧间气泡中心最小向量的方法实现对气泡圆心的跟踪。实验结果表明:该系统实现了14MPa下气液传质的光学测量,气泡半径测量误差小于4%,提供了高压油气传质的试验手段,基本满足液压油路气液两相传质的光学测量要求。     

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.     

Profile error evaluation of free-form surface using sequential quadratic programming algorithm

Profile error of free-form surface is evaluated in this paper based on sequential quadratic programming (SQP) algorithm. The optimal localization model is established with the minimum zone criterion firstly. Subsequently, the surface subdivision method or STL (STeror Lithography) model is used to compute the point-to-surface distance and the approximate linear differential movement model of signed distance is deduced to simplify the updating process of alignment parameters. Finally, the optimization model on profile error evaluation of free-form surface is solved with SQP algorithm. Simulation examples indicate that the results acquired by SQP method are closer to the ideal results than the other algorithms in the problem of solving transformation parameters. In addition, real part experiments show that the maximum distance between the measurement points and their corresponding closest points on the design model is shorter by using SQP-based algorithm. Lastly, the results obtained in the experiment of the workpiece with S form illustrate that the SQP-based profile error evaluation algorithm can dramatically reduce the iterations and keep the precision of result simultaneously. Furthermore, a simulation is conducted to test the robustness of the proposed method. In a word, this study purposes a new algorithm which is of high accuracy and less time-consuming.     

Statistical point cloud model to investigate measurement uncertainty in coordinate metrology

In this work an approach to investigate measurement uncertainty in coordinate metrology is presented, based on fitting Gaussian random fields to high-density point clouds produced by measurement repeats. The fitted field delivers a depiction of the spatial distribution of random measurement error over a part geometry, and can incorporate local bias information through further measurement or with the use of an external model to obtain a complete, spatial uncertainty map. The statistical model also allows the application of Monte Carlo simulation to investigate how error propagates through the data processing pipeline ultimately affecting the determination of features of size and the verification of conformance to specifications. The proposed approach is validated through application to simulated test cases involving known measurement error, and then applied to a real part, measured with optical and contact technologies. The results indicate the usefulness of the approach to estimate measurement uncertainty and to investigate performance and behaviour of measurement solutions applied to the inspection and verification of industrial parts. The approach paves the way for the implementation of automated measurement systems capable of self-assessment of measurement performance.     

Phase-measuring algorithm to suppress inner reflection of transparent parallel plate in wavelength tuning Fizeau interferometer

Optical thickness variation is a fundamental characteristic of transparent optical devices. When measuring this variation by using a wavelength-tuning Fizeau interferometer, the measurement accuracy depends on the phase-shifting algorithm being used. Therefore, the phase-shifting algorithm should compensate for any errors incurred during the measurement, including the phase-shift error, coupling errors, and bias modulation of intensity. Among these errors, however, the coupling errors between the higher harmonics resulting from the inner reflections of the transparent plate and phase-shift error have not previously been considered. This paper presents a derivation of a 19-sample phase-shifting algorithm that can compensate for the miscalibration and 1st-order nonlinearity of the phase shift, coupling errors, and bias modulation of intensity during wavelength tuning. The characteristics of the 19-sample algorithm were estimated with respect to the Fourier representation in the frequency domain. The phase error of measurements performed using the 19-sample algorithm was discussed and compared with that of measurements obtained using other conventional phase-shifting algorithms. Finally, the optical thickness variation of a fused silica parallel plate was obtained using a wavelength-tuning Fizeau interferometer and the 19-sample algorithm. The measurement accuracy was discussed by comparing the ripples in the crosstalk noise with those calculated using other phase-shifting algorithms. The experimental results indicated that the optical thickness variation measurement accuracy for the fused silica plate was approximately 2 nm.     

Two-machine flowshop scheduling to minimize mean flow time under simple linear deterioration

A real industrial production phenomenon, referred to as deteriorating jobs, has drawn increasing attention. However, most research on this issue considers only single-machine problems. Motivated by this limitation, this paper considers a simple linear deterioration model in a two-machine flowshop where the objective is to minimize the mean flow time. Several dominance rules and three lower bounds are proposed to speed up the search for an optimal solution, and several heuristic algorithms are provided to derive near-optimal solutions. In addition, a computational experiment is conducted to evaluate their performances. Results indicate that the algorithms perform well, and a combined heuristic algorithm is recommended for practitioners.     

Measurements of stylus radii

In stylus measurements of surface texture the measured results for roughness depend on the stylus radius. Therefore it is important to determine the stylus radius. Since stylus tips are not perfectly spherical, the local radius of curvature varies significantly over the surface which makes the determination of an effective radius difficult. Both the techniques used to generate stylus profiles and the subsequent algorithms used to derive an effective radius are discussed. Comparisons are made between three techniques: sharp-edge traces, optical microscopy and scanning electron microscopy. Several algorithms, including that prescribed by the American National Standard ANSI B46-1, are discussed. It is concluded that the radius scale method is accurate, unambiguous and easy to use for routine measurements in the laboratory.