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.     

A design advisor for the optimal inspection of circularity tolerance

In the wake of growing importance for quality and the need to reduce inspection costs simultaneously, the need for a scientific method of selecting an optimum inspection strategy for coordinate measuring machine (CMM) based inspection has become very important. The inspection error resulting from CMM inspection is greatly affected by the profile irregularities and the sampling strategy, which includes sample size, sampling methods, and algorithms used for form evaluation. This paper describes a system that can recommend an optimal inspection plan based on the needs of the user. A design of experiments (DOE) based approach is used to relate the inspection error with sampling strategies. Surface irregularities are included in the form of lobes formed on the profile. A new two-way model is proposed that works in both directions between the sampling strategy and the performance metrics. The results indicate that the number of lobes and the sampling method used have little impact on the inspection error, while the sample size and form evaluation algorithms have a significant influence. An inspection plan advisor is presented, which provides an inspection plan based on the estimated shape and acceptable measurement error.     

A method for axis straightness error evaluation based on improved artificial bee colony algorithm

An improved artificial bee colony (IABC) optimization algorithm for the accurate evaluation of minimum zone axis straightness error from a set of coordinate measurement data points was proposed. In the proposed algorithm, the opposition-based learning method was employed to produce initial population and scouts, the employed bees used greedy selection mechanism to update the best food source achieved so far one by one, and a new search mechanism inspired by differential evaluation was used for onlookers. The nonlinear mathematical model for axis straightness error evaluation and the fitness function of IABC were introduced in detail. Four classical test functions were selected in the experiments; the simulation results verified the feasibility of IABC algorithm. According to two practical examples, the results obtained by the IABC algorithm are more accurate and efficient than other conventional methods. It is a unified approach for other form and position error evaluations and is well suited for high-precision measuring equipments such as the CMM.     

一种复杂二次曲面轮廓度评定方法

为了建立复杂二次曲面轮廓度评定中计算机数据处理的理论模型和实现方法,提出了一种不要求满足小误差假设、不使用微分线形化评定二次曲面轮廓度的方法。该方法首先通过最小二乘法得到一个初始二次曲面,然后用模式搜索对初始二次曲面系数进行调整,直到找到满足最小区域原则的理想二次曲面,其目标函数值作为被测曲面的轮廓度。在计算过程中使用坐标变换将一般型的二次曲面化为标准型,既简化了轮廓度的计算,而且被测曲面可以在测量范围内任意放置。对抛物面轮廓度评定表明目标函数随着模式搜索的进行逐渐减小,模式搜索得到的抛物面轮廓度值比用最小二乘法得到的轮廓度值小得多,因此该方法更好地反映了被测二次曲面表面形状误差。     

评定直线度误差的最小二乘法与最小包容区域法精度之比较

介绍了直线度误差评定的最小二乘法和最小包容区域法的算法模型与实现方法。在三坐标测量机上对八种不同被测直线进行了采样点坐标数据提取,分别用最小二乘法和最小包容区域法的基于搜索逼近-逐次旋转逼近法进行了给定平面内直线度误差的评定。结果表明:最小二乘法的评定结果与最小包容区域法的基于搜索逼近-逐次旋转逼近法的评定结果完全一致,即直线度误差的最小二乘法评定结果符合最小条件。     

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.     

空间直线度误差评定的新算法

针对在坐标测量机CMM上测得的空间直线度误差测点集(测点序列),为提高空间直线度误差的评定精度,运用误差理论和最小二乘原理,提出了基于测点集中心的最小二乘算法(LSABC算法),推导出了LSABC算法的数学模型。并根据其基于测点集中心的特点,在按LSABC算法对全部测点进行高低排序后,以其中两最高点的中点为基点,设计出了LSABC改进算法。最后,利用UG/GRIP语言对以上新算法编制了计算机程序,以进行数字实验验证。数字实验结果表明:新算法比传统的最小二乘算法和其他多种算法具有更高的正确度。     

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.     

Control by virtual gauge of skew surfaces for correction process

The automation of production has been considerably developed in recent years in all its phases from design and manufacturing to control. The design and manufacture of free form surfaces are a current practice in the industry; thus, the conformity problem of complex geometry parts is felt more and more. The co-ordinate measurement machine (CMM) has largely reduced the acquisition and processing time in the take up measurement operation, which explains the wide use of the CMM in the mechanical industry over the last decade. The modelling and controlling procedure proposed within this paper enables correction of complex surfaces during the parts manufacturing process. The method is based on real surfaces modeled by finite elements starting from cloud points obtained by a digitalizing procedure in the co-ordinate measurement machine. The procedure was applied on a tooth gear which equips the gear box manufactured in the Algeria Engines and Tractors Factory. A comparison between the real and ideal model has been established showing the defects of form in order to correct the grinding process. This comparison is carried out by associating a surface of perfect geometry to a group of palpated dots. The selected criterion of optimisation is the least squares method.     

面齿轮磨削齿面齿形误差修正

以面齿轮齿面的形状误差为研究对象,建立了面齿轮齿面数学方程,通过三坐标测量仪（CMM）对面齿轮齿面进行了误差测量。为提高面齿轮齿面的精度,提高其理论和工程的应用价值,提出了一种基于序列二次规划（SQP）的面齿轮齿面误差评定方法。建立了齿面误差的识别方程,应用序列二次规划算法对机床加工参数进行优化求解,并与最小二乘优化法进行对比,应用修正后的机床调整参数,再次通过三坐标测量仪对修正后的齿面误差进行检测,验证了该方法的可行性。     

Least squares evaluations for form and profile errors of ellipse using coordinate data

To improve the measurement and evaluation of form error of an elliptic section, an evaluation method based on least squares fitting is investigated to analyze the form and profile errors of an ellipse using coordinate data. Two error indicators for defining ellipticity are discussed, namely the form error and the profile error, and the difference between both is considered as the main parameter for evaluating machining quality of surface and profile. Because the form error and the profile error rely on different evaluation benchmarks, the major axis and the foci rather than the centre of an ellipse are used as the evaluation benchmarks and can accurately evaluate a tolerance range with the separated form error and profile error of workpiece. Additionally, an evaluation program based on the LS model is developed to extract the form error and the profile error of the elliptic section, which is well suited for separating the two errors by a standard program. Finally, the evaluation method about the form and profile errors of the ellipse is applied to the measurement of skirt line of the piston, and results indicate the effectiveness of the evaluation. This approach provides the new evaluation indicators for the measurement of form and profile errors of ellipse, which is found to have better accuracy and can thus be used to solve the difficult of the measurement and evaluation of the piston in industrial production.     

面向对象的大尺寸测量不确定度分析

测量结果必须说明不确定度。常规尺寸的解析法和标准比对法难以适用于评价大尺寸测量不确定度,尤其是特定拟合任务结果的不确定度。针对大尺寸测量的特殊性,研究基于蒙特卡罗的评价方法评价大尺寸特定测量对象的不确定度,并用计算机可视化直观表示离散点云。利用离线仿真评价测量不确定度,用于设计最优采样策略。以激光跟踪仪测量大型圆形截面工件分析为例,给出测点对称、均匀分布和半径约束等优化测量思想。最后应用于激光跟踪仪测量隧道构件的实例当中。仿真和实际实验表明,蒙特卡罗评价和离散点云表示法可准确、直观评价大尺寸测量特定测量对象的不确定度,制定的最优采样策略可提高测量精度。     

Investigation on sampling size optimisation in gear tooth surface measurement using a CMM

Coordinate measuring machines (CMMs) are widely used in the gear manufacturing industry. One of the main issues for contact inspection using a CMM is the sampling technique. In this paper the gear tooth surfaces are expressed by series of parameters and inspection error compensation and initial value optimisation method are presented. The minimum number of measurement points for 3D tooth surfaces are derived. If high precision is required, more points need to be inspected. The sampling size optimisation is obtained from the criterion equation. The surface form deviation and initial values are optimised using the minimum zone method and genetic algorithms. A feature-based inspection system for spur/helical gears is developed and trials and simulations demonstrate that the method developed is suitable and very effective. This revised version was published online in October 2004 with a correction to the issue number.     

基于网格模型的圆度在线检测系统开发

提出一种基于三角网格模型的圆度在线检测方法.根据圆柱的空间位置、分布的测点和测点的矢量来进行测量路径的设计.利用最小二乘法,计算圆柱测量截面的圆度误差.最后以CMM检测结果作为理想的标准,对圆度在线检测结果进行评估,结果得出该方法所带来的圆度误差非常小.     

轴承滚子凸度轮廓的最小二乘拟合与误差评定

为了实现对轴承滚子凸度轮廓误差的精确评定,依据圆弧修正型轴承滚子凸度素线轮廓的几何特征和形状误差的定义,基于最小二乘原理,研究了轴承凸度轮廓(两段圆弧和一段直线)的最小二乘拟合和误差评定方法。首先利用各测量点的曲率差值确定了圆弧段与直线段的相切参考点;其次分别选取两个参考点临近的测量点作为辅助相切参考点,并与对应的圆弧段测量点一起拟合出一系列的最小二乘圆弧并计算拟合误差;然后基于直线与两段圆弧相切的原则确定出一系列的直线方程并计算对应的直线度误差;通过比较判断最终确定出圆弧修正型轴承滚子凸度轮廓的最小二乘拟合及误差评定。实例结果表明:圆弧修正型凸度轮廓曲线的总误差0.020 9mm与文中设定标准凸度轮廓曲线引入的法向误差0.02mm相差4.5%。本方法可以有效的实现轴承凸度轮廓的拟合与误差评定,为平面多段曲线的最小二乘拟合提供了一种新的思路。     

Accurate evaluation of free-form surface profile error based on quasi particle swarm optimization algorithm and surface subdivision

Although significant progress has been made in precision machining of free-form surfaces recently, inspection of such surfaces remains a difficult problem. In order to solve the problem that no specific standards for the verification of free-form surface profile are available, the profile parameters of free-form surface are proposed by referring to ISO standards regarding form tolerances and considering its complexity and non-rotational symmetry. Non-uniform rational basis spline(NURBS) for describing free-form surface is formulated. Crucial issues in surface inspection and profile error verification are localization between the design coordinate system(DCS) and measurement coordinate system(MCS) for searching the closest points on the design model corresponding to measured points. A quasi particle swarm optimization(QPSO) is proposed to search the transformation parameters to implement localization between DCS and MCS. Surface subdivide method which does the searching in a recursively reduced range of the parameters u and v of the NURBS design model is developed to find the closest points. In order to verify the effectiveness of the proposed methods, the design model is generated by NURBS and the measurement data of simulation example are generated by transforming the design model to arbitrary position and orientation, and the parts are machined based on the design model and are measured on CMM. The profile errors of simulation example and actual parts are calculated by the proposed method. The results verify that the evaluation precision of freeform surface profile error by the proposed method is higher 10%-22% than that by CMM software. The proposed method deals with the hard problem that it has a lower precision in profile error evaluation of free-form surface.     

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.     

Measurement of form error of a probe tip ball for coordinate measuring machine (CMM) using a rotating reference sphere

This study presents a method of measurement of the form error of the tip ball in the tactile probing systems of a coordinate measuring machine (CMM) by using a rotating reference sphere. The measurement of the form error of the CMM probe tip was conducted without the use of additional external measuring instruments or sensors. The form errors of the probe tip ball and the reference sphere were separated from the probing coordinates of CMM by rotation of the reference sphere. The effectiveness of the proposed method was evaluated based on an uncertainty analysis. The uncertainty in measurement of diameter of the probe tip ball was estimated to be less than 0.5 μm.     

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.     

免形状测量机的误差分离技术研究

免形状(Form-free)测量机是基于零件"小偏差假设",不依赖被测对象的几何形状、无须知道几何参数、无须精密定位的情况下,也能进行几何尺寸和形状误差的测量与评定的测量机。针对此测量机,建立了误差补偿综合模型,为测量机误差测量和补偿提供了理论基础。提出基于激光追踪仪的几何误差分离方法,快速准确实现对测量机几何误差的检定。使用软件对分离算法进行仿真及实验,结果证明该分离算法是可行的,并将其与激光干涉仪测得的误差比较,两者非常吻合,证明此算法具有快速、精度高等优点,适用于坐标测量机的几何精度检测。