Reference software for finding Chebyshev best-fit geometric elements

This paper summarizes work carried out in a project supported by the European Communities' Bureau of Reference (BCR) to develop reference software for finding best-fit geometric elements, under specified criteria, to coordinate data. The geometric elements considered are the line, plane, circle, sphere, cylinder, and cone. The criteria for determining the elements are, generally, minimum zone (MZ) and, where appropriate, minimum circumscribed (MC) and maximum inscribed (Ml). The software developed implements methods founded on optimization theory. Two approaches are described. the first implements mathematical programming methods that exploit the particular structure of the problems considered and provide a unified approach to their solution; the second, applicable to the MZ and, where appropriate, MC problems for the line (in two-dimensions (2-D)], plane, circle (in 2-D) and sphere, implements a combinatorial method that returns all global solutions.     

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.     

Predictive framework for cutting force induced cylindricity error estimation in end milling of thin-walled components

The static deflections of cutting tool and workpiece are the primary source for the deviation of machined components from the design specifications during end milling of thin-walled geometries. The deviations are expressed as per the Geometric Dimensioning and Tolerancing (GD&T) principles using size, form, and orientation of the features. This paper proposes a computational framework to estimate cutting force induced cylindricity error during end milling of thin-walled circular components. The framework combines computational elements such as Mechanistic force model, Finite Element Analysis (FEA) based workpiece deflection model, Cantilever beam formulation based tool deflection model, and Particle Swarm Optimization (PSO) based cylindricity estimation algorithm. It has been observed that the static deflections of a cutting tool and thin-walled component influence the cylindricity error considerably. The inevitable aspects associated with the end milling of thin-walled circular components such as concave-convex side machining and workpiece rigidity are investigated subsequently. It was observed that the cylindricity error during concave side machining is considerably smaller due to geometric configuration imparting adequate stiffness to thin-walled components. The study also demonstrated that an appropriate combination of productive cutting conditions and the component thickness could reduce cylindricity error considerably. The outcomes of the present study are substantiated by conducting a set of computational simulations and end milling experiments over a wide range of cutting conditions. The computational framework proposed in the present study can assist process planners in selecting appropriate cutting conditions to manufacture thin-walled circular components within tolerance limits specified by the designer.     

Evaluation of circularity from coordinate and form data using computational geometric techniques

Data for evaluating circularity error can be obtained from coordinate measuring machines or form measuring instruments. In this article, appropriate methods based on computational geometric techniques have been developed to deal with coordinate measurement data and form data. The computational geometric concepts of convex hulls are used, and a new heuristic algorithm is suggested to arrive at the inner hull. Equi-Distant (Voronoi) and newly proposed Equi-Angular diagrams are employed for establishing the assessment features under different conditions. The algorithms developed in this article are implemented and validated with the simulated data and the data available in the literature.     

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.     

A comparison of different algorithms for circularity evaluation

The measurement and evaluation of circularity of cylindrical components is very important for the majority of cylindrical workpieces used in precision engineering. Further, since the measurement and evaluation of cylindricity is more complex and time consuming, only circularity is evaluated for most applications.The evaluation of circularity from a circularity graph and/or from digital data requires a suitable algorithm. The most commonly used criterion for this has been the least squares criterion, though it is known that this does not necessarily give the best solution. Hence, an attempt has been made in this paper to propose different algorithms and compare them. The algorithms considered are based on the methods of least squares, intuition, general second-degree equation for a circle, best-fit ellipse and simplex search. A comparison of these methods is presented.     

Evaluating the geometric characteristics of cylindrical features

This paper presents mathematical models and efficient methodologies for the evaluation of geometric characteristics that define form and function of cylindrical features; namely cylindricity and straightness of median line. These two problems have similar structures and can be solved by comparable procedures. Based on the proposed methodologies, the cylindricity error evaluation can be performed using any of the following criteria: the least squares cylinders, minimum circumscribed cylinders, maximum inscribed cylinders or minimum zone cylinders. The procedures have been tested for accuracy and efficiency. The results indicate that they provide accurate results quickly.     

Method for cylindricity error evaluation using Geometry Optimization Searching Algorithm

According to the geometrical characteristics of cylindricity error, a method for cylindricity error evaluation using Geometry Optimization Searching Algorithm (GOSA) has been presented. The optimization method and linearization method and uniform sampling could not adopt in the algorithm. The principle of the algorithm is that a hexagon are collocated based on the reference points in the starting and the end measured section respectively, the radius value of all the measured points are calculated by the line between the vertexes of the hexagon in the starting and the end measured section as the ideal axes, the cylindricity error value of corresponding evaluation method (include minimum zone cylinder method (MZC), minimum circumscribed cylinder method (MCC) and maximum inscribed cylinder method (MIC)) are obtained according to compare, judgment and arranged hexagon repeatedly. The principle and step of using the algorithm to solve the cylindricity error is detailed described and the mathematical formula and program flowchart are given. The experimental results show that the cylindricity error can be evaluated effectively and exactly using this algorithm.     

Utilization of voronoi diagrams for circularity algorithms

In the past many minimum zone center (MZC) algorithms have been developed. In opposition, many coordinate measuring machines (CMM) still use least-squares center (LSC) algorithms. A MZC algorithm that uses a computational geometry approach through the Voronoi diagrams to determine circularity can be performed with LSC. Both algorithms are compared by scanning the number of points of the set, the circularity value interval, and the workpiece radius. The differences between the results are compared to determine the relationship. The importance of the uncertainty of the machines is then compared with these differences.     

Modelling of dimensional and geometric error prediction in turning of thin-walled components

In die-mold manufacturing and aircraft industry, many components that have thin-walled features are produced by turning operation. The major problem encountered during internal or external turning is cutting force induced deflection of workpiece along the periphery as well as axial length of a component. The present research work aims to develop a mathematical model for estimating dimensional and geometric errors during turning of thin-walled hollow cylinder qualitatively and quantitatively. In the proposed model, a mechanistic approach which is semi-analytical in nature is followed to achieve accuracy of the predicting results. First of all, process geometry model for thin-wall turning is developed based on process geometry variables such as uncut chip thickness, actual feed per revolution, actual depth of cut, peripheral cutting speed, effective cutting area etc. Using these process geometry variables and mechanistic cutting constants, a force model of turning is developed to estimate the tangential and radial force components. Later on, based on the predicted forces, tool-workpiece combined deflection model is developed to estimate radial, diametric and various geometric errors of the turned surface. The developed models are able to predict radial, diametric and various geometric errors such as straightness, circularly and cylindricity errors without conducting expensive actual machining operation. Hence, the present study will be helpful to take care of precautionary measures for controlling of dimensional and geometric errors more efficiently and reliably. Therefore, an attempt has been made to provide a basic platform to machining practitioners and process planners for in-depth comprehension and characterization of dimensional and geometric errors of the entire turned surface for varying machining conditions.     

Experimental investigation,prediction and optimization of cylindricity and perpendicularity during drilling of WCB material using grey relational analysis

Manufacturing is always the heart of majority of industries. Drilling is an extremely important and an essential machining process which requires a lot of attention as in most of the cases it is required for assembly purposes. Majority of the holes produced during drilling are made with the help of Vertical Machining Centre (VMC) meant for pin- hole assembly. Though the tolerance is within limit, assembly problems arise due to the improper geometry of these holes. Various geometrical tolerances like cylindricity, circularity, perpendicularity and position errors are responsible for such assembly problems. This investigation is focussed on cylindricity and perpendicularity in the drilling of Wrought Cast Steel Grade B (WCB) material using SOMX 050204 DT insert. In this work, effect of machining variables like cutting speed, feed rate and depth of cut (canned cycle) are investigated and optimized using grey relational analysis (GRA). Reliable experiments are conducted based on a 3³ full factorial, replicated twice. Second order regression models are developed for predicting cylindricity and perpendicularity. The models’ adequacy has been checked by calculating correlation coefficient. It shows that the developed models are well fitted for the prediction of responses within the specific range of input variables.     

Verification of form tolerances part II: Cylindricity and straightness of a median line

Most inspectors measure form tolerances as the minimum zone solution, which minimizes the maximum error between the datapoints and a reference feature. Current coordinate measuring machines verification algorithms are based on the least-squares solution, which minimizes the sum of the squared errors, resulting in a possible overestimation of the form tolerance. Therefore, although coordinate measuring machines algorithms successfully reject bad parts, they may also reject some good parts. The verification algorithms developed in this set of papers compute the minimum zone solution of a set of datapoints sampled from a part. Computing the minimum zone solution is inherently a nonlinear optimization problem. This paper develops a single verification methodology that can be applied to the cylindricity and straightness of a median line problems. The final implementable formulation solves a sequence of linear programs that converge to a local optimal solution. Given adequate initial conditions, this solution will be the minimum zone solution. This methodology is also applied to the problems of computing the minimum circumscribed cylinder and the maximum inscribed cylinder. Experimental evidence that the formulations are both robust and efficient is provided.     

基于坐标测量机和拟粒子群进化算法的圆柱度误差检测与评定

建立了任意位置下基于坐标测量机检测的圆柱度误差最小区域解的数学模型,提出了采用拟粒子群进化算法求解最小区域圆柱度误差新方法。该算法使用实数编码,由拟随机Halton序列产生粒子的初始位置和速度,基于浓缩因子法修改粒子的速度。为了验证算法的有效性,对文献中测量数据采用提出的方法进行圆柱度误差计算并将结果与多种算法计算结果进行比较,同时在加工中心加工大量轴类零件,使用三坐标测量机对零件进行实测,应用该进化算法计算最小区域圆柱度误差并与三坐标测量机给出的结果进行比较。实验结果均证实了提出的方法不仅优化速度快、计算精度高,而且算法简单,需设置参数少,便于推广应用。     

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.     

自动影像测量系统关键算法

为了实现对工件的自动影像测量,建立了自动影像测量系统。对该系统所采用的图元识别、图元测量、基于自动聚焦原理的高度测量等算法进行研究。根据圆形和矩形图元的面积和真圆度等特征参数介绍了图元识别算法。以直线和圆形图元为例分析了典型图元的测量算法,即在提取图元边缘点的基础上进行图元拟合。在分析比较能量谱等聚焦评价方法的性能的基础上,说明了采用自动聚焦原理进行高度测量的算法。最后,介绍了系统比例尺的设定方法以及光栅尺读数的误差补偿方法。实验结果表明:比例尺的标定精度为0.5 μm;图元的测量精度在微米级;高度测量精度为0.035 mm,基本满足自动影像测量的稳定可靠、精度高、抗干扰能力强等要求。     

Investigating the influence of selected factors on results of V-block cylindricity measurements

The article deals with the method of cylindricity measurements using V-blocks. It was assumed, as was in the case of roundness profile measurements before, that the V-block method can be applied to accurate cylindricity measurements of large machine parts directly on a machine tool or a work stand. Results of the statistical verification of the method show that its maximum error equals about 19% in relation to accurate radial method (for the confidence level P = 0.95). The next stage of the research work on the V-block cylindricity measurements was an analysis of potential sources of errors. These sources were divided into two groups: sources independent of the method parameters and sources strictly related to the method parameters. Authors investigated potential sources of errors and developed procedures allowing compensation of influence of most significant ones. The paper provides results of the theoretical and experimental research on an analysis and compensation of main sources of errors of the V-block method of cylindricity measurements.     

A unified approach to form error evaluation

Evaluation of form error is a critical aspect of many manufacturing processes. Machines such as the coordinate measuring machine (CMM) often employ the technique of the least squares form fitting algorithms. While based on sound mathematical principles, it is well known that the method of least squares often overestimates the tolerance zone, causing good parts to be rejected. Many methods have been proposed in efforts to improve upon results obtained via least squares, including those, which result in the minimum zone tolerance value. However, these methods are mathematically complex and often computationally slow for cases where a large number of data points are to be evaluated. Extensive amount of data is generated where measurement equipment such as laser scanners are used for inspection, as well as in reverse engineering applications.In this report, a unified linear approximation technique is introduced for use in evaluating the forms of straightness, flatness, circularity, and cylindricity. Non-linear equation for each form is linearized using Taylor expansion, then solved as a linear program using software written in C++ language. Examples are taken from the literature as well as from data collected on a coordinate measuring machine for comparison with least squares and minimum zone results. For all examples, the new formulations are found to equal or better than the least squares results and provide a good approximation to the minimum zone tolerance.     

Form Error Characterisation by an Optical Profiler

Form errors are deviations of the machined surface from the geometrical surface excluding position errors, waviness and roughness. From a functional point of view, as for surface roughness, form error characterisation is also important. In the present work, an optical profiler is used to measure and numerically characterise form errors such as roundness and cylindricity of cylindrical surfaces. A double orientation method using mean value analysis has been applied to separate the workpiece error from the spindle error during roundness measurement. Software is developed for data generation, fitting the reference data for assessing form errors in terms of statistical and functional parameters including new parameters. An optical profiler measures all the surface irregularities and hence can be used to study both micro and macro errors of the profile measured. A study of both roughness and roundness parameters along the circumferential direction is made for the unfiltered signal using different filter cut-off values. It is known that filtering greatly affects the value of the form error parameters measured. The form measurements obtained by the optical profiler are compared with the stylus profiler and the results are presented.     

A new strategy for circularity problems

The evaluation of circularity based on the minimum zone criterion as a non-linear and non-convex problem, which requires a substantial amount of computational effort in general, is investigated. A new strategy for improving the computational efficiency by collecting data points at the farthest and nearest locations from current minimum radial separation center until all collected data points meet the optimum criterion is proposed. A number of mathematical models developed in this paper indicate that the minimum circularity can be determined by using a small number of critical data points. The validated results show that the proposed strategy offers an effective way to identify the critical data points at the early stage of computation and gives an efficient approach to solve the circularity problems, especially when the number of data point is large.     

Computational methodologies for evaluating form and positional tolerances in a computer integrated manufacturing system

In any manufacturing process, the manufactured parts always show some kind of variation in their geometric characteristics (e.g. form, size, orientation, and position). To achieve the desired part quality, these geometric characteristics need to be maintained within predetermined design limits as defined by tolerances. After a part is manufactured, it is then inspected to tolerance specifications. In industries, coordinate measuring machines (CMMs) and vision systems are widely used to accomplish the verification tasks. Owing to the lack of proper definitions and implementational details of the correct assessment criteria for different geometric characteristics, it is difficult to develop appropriate algorithms for the inspection software of the CMMs. Though the standards have used the concepts of the tolerance zones (TZs) [1] and minimum zones (MZs) [2] for other geometric (e.g. form, location, etc.) characteristics, the incompleteness in their definitions creates ambiguities in implementing the inspection procedures. It is necessary to develop rigorous definitions of the geometric characteristics (of manufactured parts) that are to be established from discrete, manufactured data points in terms of either TZs or MZs. It is also important to establish the required mathematical criteria for assessing those geometrical characteristics of the manufactured features. In this paper the concepts of TZ and MZ have been thoroughly investigated for establishing practical methodologies for part inspection.