Support vector regression for determining the minimum zone sphericity

Several methods have been investigated to determine the deviation of manufactured spherical parts from ideal geometry. One of the most popular is the least squares technique, which is still widely employed in coordinate measuring machines used by industries. The least squares algorithm is optimal under the assumption that the data set is very large and has the inherent disadvantage of overestimating the minimum tolerance zone, resulting sometimes in the rejection of good parts. In addition, it requires that the data be distributed normally. The support vector regression approach alleviates the necessity for these assumptions. While most fitting algorithms in practice today require that the sampled data accurately represent the surface being inspected, support vector regression provides a generalization over the surface. We describe how the concepts of support vector regression can be applied to the determination of tolerance zones of nonlinear surfaces; to demonstrate the unique potential of support vector machine algorithms in the area of coordinate metrology. In specific, we address part quality inspection of spherical geometries.     

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.     

最小包容区域法评定平面度误差的程序设计

介绍了最小包容区域法评定平面度误差的JAVA程序设计。采用三角形准则、交叉准则和直线准则分别求取最小包容区域,从三者中选取最小值作为平面度误差。得到的误差值具有唯一性。对数据采集的准备工作无特别要求,操作较简便。适用于三坐标测量机及其它仪器对平面度检测时数据的处理。     

多关节坐标测量机的数学模型和参数标定

根据Denavit-Hartenberg方法,运用齐次坐标变换等数学工具,提出了具有普适性的杆件单元模型,建立了6自由度多关节坐标测量机的数学模型.在此基础上研究了模型中系统参数及运动参数的标定问题,设计了一种全新的标定块,并使用该标定块根据最小二乘法和迭代法快速地实现了参数标定.     

用蚁群算法求解关节式坐标测量机的最佳测量区

本文提出了一种求解最佳测量区的方法,以进一步提高关节式坐标测量机的测量精度。首先,根据关节式坐标测量机的测量模型,建立了基于圆编码器测角误差的关节式坐标测量机误差模型。利用蒙特卡洛理论得到6个关节转角的随机数,采用数值法仿真分析测量机的测量空间。然后将包含测量空间的一立方体区域等间隔切割成343个小立方体区域,采用蚁群算法确定每个小区域由于圆编码器误差所引起的最大测量误差。最后,通过比较找到其中最大测量误差最小的区域,即为最佳测量区。研究结果表明,对于所研究的关节式坐标测量机,各个小区域的最大误差为0.069 9~0.189 6mm,其中最小值为0.069 9mm的区域为-100mm≤x≤100mm,-100mm≤y≤100mm,400mm≤z≤600mm。采用本文方法确定的最佳测量区在测量空间内为一个立方体区域,故在最佳测量区进行较高精度的测量具有实用性和可操作性。     

Measuring the errors of form of industrial surfaces: prediction and optimization

To acquire maximum information on the geometrical errors of industrially made surfaces at a minimum cost, a method for estimating conditional probabilities of a random signal (Bayesian prediction) is applied to three-dimensional metrology. First, a surface is interpolated between data acquired on a coordinate measuring machine (CMM). Then, for a given probability, limit surfaces are computed that bind a region of space containing the known data and the most probable interpolation of the missing data of the surface. These bounds can be treated as the surface; their points can be considered as if they were actual CMM data when fitting a tolerance zone or a datum feature to the data. For Bayesian prediction, the basic hypotheses on the signal are stationarity, ergodicity, and gaussian density. Deviations from these hypotheses and their consequences on the prediction are taken into account and corrections are proposed.     

Roundness measurement using limacons

Most roundness measuring systems which find automatically the ‘reference circles’ to a profile have used, for convenience, a limacon figure as an approximation to a circle. However this figure has important geometrical properties relevant to normal roundness measurement and can be regarded as the bais of an analytical system rather than an instrumental convenience. This paper examines some of the implications of the limacon method under practical conditions by comparison with circular references. It includes a comparison of roundness measurements taken using limacon references according to minimum zone, minimum circumscribing, maximum inscribing and least squares criteria     

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。     

基于误差转换的汽车曲轴圆度及圆柱度误差评价数学模型构建研究

针对国内汽车曲轴轴颈圆度误差、圆柱度误差检测普遍存在的效率低、精度低等问题,建立基于误差转换的平面曲线和空间曲线误差数学模型,结合圆和圆柱的数学表达建立满足最小包容条件的圆度和圆柱度误差评定数学模型,并采用遗传优化算法计算出符合最小评定要求的曲轴轴颈形位误差,解决了理想包容要素位姿参数不精确的问题。同时,建立基于图像域的汽车曲轴轴颈形状误差检测试验台,针对测量过程中连杆轴颈沿主轴颈公转运动,从而导致连杆轴颈图像域检测数据存在坐标不归一问题,以曲轴法兰端特征孔为基准,通过模板匹配特征与孔边缘提取实现了连杆轴颈圆度和圆柱度测量数据空间坐标归一化处理。以某型号发动机曲轴为例进行大样本误差检测试验,并与三坐标测量机测得的结果进行对比,数据分析表明提出的曲轴轴颈形状误差检测方法的精度为1μm,且重复检测误差在0.1μm以内,证明了其理论上的正确性及实践操作的可行性。     

A reliable method of minimum zone evaluation of cylindricity and conicity from coordinate measurement data

The form error evaluation of cylinders and cones is very important in precision coordinate metrology. The solution of the traditional least squares technique is prone to over-estimation, as a result unnecessary rejections may be caused. This paper proposes a reliable algorithm to calculate the minimum zone form errors of cylinders and cones, called a hybrid particle swarm optimization-differential evolution algorithm. The optimization is conducted in two stages, so that the program can hold a fast convergence rate, while effectively avoiding local minima. Experimental results demonstrate that the proposed algorithm can obtain very accurate and stable results for the calculation of cylindricity and conicity.     

Circle fitting from the polarity transformation regression

Geometrical fitting is useful in different fields of science and technology, in particular least squares minimum (LSM) methods are widespread in contact probing for coordinate measuring machines, as well as a reference shape for surface metrology. We present a new intuitive and simple LSM algorithm for circle fitting, the polarity transformation regression. It is a non-linear algebraic method from a generic geometric transformation. We derive the explicit expression of the model estimators from the data points. Then, the algorithm is compared with other methods based on simulation and some literature data sets. The proposed algorithm presents a comparable accuracy, low computational effort and good behavior with outliers based on the initial test, outperforming other well-known algebraic methods in some of the studied data sets. The basis of the algorithm is finally suggested for other potential uses.     

弧面凸轮廓面三坐标机测量与传动质量评价

在分析了弧面凸轮廓面数学模型的基础上,利用三坐标测量机对弧面凸轮廓面进行了等径点位测量,并提出了一种逐点比较法,将所有的测量数据用于确定凸轮坐标系与测量坐标系的位置关系。用非均匀B样条曲线拟合所有凸轮坐标系中的点坐标数据,由拟合曲线和滚子共轭运动的啮合方程求解实际共轭运动,从而评价弧面凸轮廓面的传动质量。最后,通过一个检测实例验证了该算法的正确性和可行性。     

基于三坐标测量机的白车身质量控制

介绍三坐标测量机的工作原理及在白车身检测中的应用。详细分析基于三坐标机测量白车身质量的各个环节,包括测量前准备、测量范围的确定、公差制定、测量、数据统计图表编制、偏差源分析、结论判断及问题整改等。将先进的三坐标测量技术应用于白车身质量检测,可以提高白车身的制造精度,从而提升整车的装配精度。     

三坐标测量机上形状误差评定的理论与方法

本文用线性化方法建立三坐标测量机上形状误差最小条件评定的统一数学模型，把形状误差的评定归结为线性规划问题，并采用有效集法求解。文章研究了有效集法在形状误差评定中的理论和特点，证明了有效集法的最优解条件包含了形状误差评定的最小条件和代数判别准则；阐述了该方法在处理形状误差时显得非常自然方便，简单易行，计算速度快等特点，是比单纯形法更优越的方法。文末给出一个圆柱度误差评定的实例和结果。     

Evaluating minimum zone flatness error using new method—Bundle of plains through one point

An attempt was made to create a new software solution for evaluating minimum zone (MZ) based flatness error using data acquired from a coordinate measuring machine (CMM). The authors tried to exploit one very useful characteristic of a reference plane for evaluating flatness error. Namely, it is shown that only one point located within a “cloud” of points can be used to generate reference planes for the purpose of evaluating flatness error. The method is named One Point Plane Bundle Method (OPPBM). A solution was created using exclusively basic analytic geometry relations/transformations and a general purpose worksheet tool. The results show that this solution can be used to determine very usable MZ flatness error values, which are significantly lower than values provided by the least square method. Execution times are also reasonable and acceptable. The method has been validated using the data from reference literature and experimental data measured by a CMM.     

A Study on an Evaluation Method for Form Deviations of 2D Contours from Coordinate Measurement

A practical approach towards evaluating form deviations of 2D complex contour profiles from the coordinate measurement data of discrete points is proposed. In this approach, a 2D complex contour is represented approximately by the combination of straight line and circular arc segments under the condition of satisfying the prespecified interpolation accuracy. The form deviation of a measured point is calculated relative to the approximate contour and the form deviation of the whole contour is evaluated using either the minimum zone criterion (MZC) or the least squares criterion (LSC). It has been demonstrated that the difference between the deviation definitions, one relative to the original contour and the other to the approximate contour, is not larger than the prespecified accuracy value, i.e. the evaluation precision of the form deviation can be sufficiently guaranteed. The effectiveness and the efficiency of the approach are verified with a practical example of a planar cam contour. Moreover, a weighted LSC technique is proposed in order to obtain a rational evaluation result for the measured points with non-uniform length interval.     

一种直角坐标系下圆度误差最小区域评价实现方法

针对直角坐标系下圆周截面形状误差评价,介绍了一种圆柱体截面圆度误差的测量与评定方法.建立了基于直角坐标系下的最小区域圆度误差三维评测模型,得到利用弦线截交关系快速评价圆度误差的方法.通过多次截交产生的虚拟中心定位,可以准确确定评价点的位置,达到了快速、精确利用最小外接球法评价球度误差的目的.通过分析表明,该方法计算效率...     

Evaluation of the form errors of axisymmetric components

The form errors of three-dimensional surfaces have an effect on the proper functioning of an assembly. Methods for evaluating form errors of surfaces of spherical and cylindrical components are available. However, in most machines and instruments the functional components are axisymmetric, e.g. spindles, and have varying geometry in the form of steps, tapers etc. Evaluation of the form errors at different zones on the component as well as a form error characteristic of the whole component is required in such cases. To do this coordinate data or circularity traces are obtained at different horizontal sections using a coordinate measuring machine or a roundness measuring instrument respectively and the required form error is then evaluated. Different algorithms developed for fitting the axis of the component which is the first step in the evaluation of form errors are described. The method of normal least-squares fitting was found to be superior to the general least-squares method.     

Research on the uncertainties from different form error evaluation methods by CMM sampling

As a part of the new measurement uncertainty system proposed in the new generation of Geometrical Product Specifications and Verification, the evaluation methods of uncertainties to form errors have been researched in mechanical engineering, which are calculated based on the error propagation principle and statistical concept under certain conditions. In this paper, the evaluation datum is obtained by using both the least squares method and the genetic optimization algorithm. Their computation uncertainties to flatness and roundness were compared with each other using the sample data from a coordinate measurement machine (CMM). The results show that the uncertainties obtained from the genetic algorithm-based method are similar to those from the least squares method according to their evaluation parameters. The evaluation uncertainties from different methods become a little smaller with more sample points. A more significant conclusion is that the evaluation uncertainties from two methods are so small that they almost do not affect the measurement uncertainties to form error, which, in fact, mainly comes from the CMM sampling. Therefore, for the efficiency and simplification of calculation, especially for the cylintricity with more parameters, the uncertainties from evaluation methods can be neglected where the precision is not so strict.     

高精度三维螺纹测量机接触式扫描测头动态特性研究

扫描测头是高精度三维螺纹综合测量机的核心部件,其动态特性严重影响了整机的精度。为了提高测量机的精度,对高精度螺纹三维尺寸测量线性扫描测头的动态特性进行了研究。首先分析了三维螺纹综合测量机用扫描测头的测量原理,然后建立了动态特性模型并提出了影响动态测量结果的因素,最后通过实验验证了这种测头结构的动态特性和测量处理方式的有效性。实验结果验证了影响因素的正确性,优化影响最大的采样间距因素可以过滤80%的无效数据点,从而使该段拟合平均残差平方和误差减小了91.0%,线性误差减小了67.4%,进而提高了三维螺纹测量机的测量精度。