Evaluation method for spatial straightness errors based on minimum zone condition

A nonlinear mathematical model for spatial straightness error evaluation based on the minimum zone condition is established in this paper. According to the error analysis, it is proved that the mathematical model for spatial straightness error evaluation cannot be linearized. A criterion for verification of the existence and uniqueness of the minimum zone solution is proposed. A new computational method is also proposed, and practical examples are given. Finally, the correctness of this method is demonstrated using a geometrical solution. This new method is convenient for computation of uniqueness and exactness of the minimum zone solution.     

A fast and simple algorithm for evaluation of minimum zone straightness error from coordinate data

Various methods have been suggested in the past to determine the minimum zone straightness error, but suffer from various drawbacks. A new, fast and simple algorithm is proposed to calculate the straightness error from planar coordinate data. It guarantees the minimum zone solution. An example and test data are provided. Results of simulation experiments to establish the time computational complexity of the algorithm are also presented.     

平面内直线度误差最小区域法的完备性研究

针对国家标准GB/T 11336-2004中用于平面内直线度误差最小包容区域评定的极点计算法对有些数据的处理结果不符合最小包容区域的高-低-高或低-高-低原则,即存在不完备性,因此提出定向极点搜索与迭代的评定方法。理论分析和平面内直线度误差数据的处理实例证明本方法完全符合平面内直线度误差评定的最小包容区域的高-低-高或低-高-低原则。数据处理的结果还进一步显示新方法完备性好,评定结果精准、可靠,计算过程绝对收敛、速度也更快,可以克服国家标准GB/T 11336-2004中平面内直线度误差评估的极点计算法存在的缺陷。     

Evaluation of angular error between two lines

The error caused by the fitting of a set of data points to two lines having a specified angle is defined as the angular error. The evaluation of angular error needs to find two pairs of parallel lines that follow the angular constraint and bound the data points under the minimum zone criterion. The evaluation of angular error is difficult in mathematics, so it is usually simplified by treating one line as the datum and the form error of the other line calculated from the datum as the solution. The datum is assumed to be perfect that doesn’t tell the real properties of the actual line but gives incorrect solutions in general. To give an exact solution to the angular error problem based on the minimum zone criterion, a new mathematical model is proposed in this paper. The basic idea of this model is to rotate the data points of one line to the same direction as that of the other line so that the evaluation of angular error is simplified as solving a straightness problem. The angular error obtained from the proposed model is proved to be minimum. The proposed model offers a simple approach to solve the tough angular error problem, and it also provides a feasible tool to explore more complicated problems, like 3D angular error and polygon error.     

评定直线度误差数据处理方法的分析与比较

用基础的测量方法和数据处理方法对直线度误差进行评定。测量方法采用水平仪的节距法。数据处理方法采用两端点连线法、最小二乘法、最小区域法。每一种方法都分别用计算法和作图法评定直线度误差。采用不同的方法对同一组数据评定直线度误差,其结果不尽相同,并对其精度进行比较,从而得出应用这三种方法评定直线度误差的差别。     

基于凸多边形的直线度误差的评定

通过对两平行直线包容测量数据点的分析,得到了两平行直线具有最小距离时必经过凸多边形的3个顶点的条件。根据上述条件,提出了基于凸多边形的直线度误差评定的方法,该方法满足最小包容区域。为求解直线度误差,采用矢量积构造凸多边形。通过实例验证了该方法的正确性和有效性。     

直线度误差评定方法简述

本文简要叙述了用两端点法、最小二乘法和最小区域法来评定平面内直线度误差的求解方法和步骤。     

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.     

A new minimum zone method for evaluating flatness errors

A new minimum zone method for flatnes error analyis is proposed in this article. Based on the criteria for the minimum zone solution and strict rules for data exchange, a simple and rapid algorithm, called the control plane rotation scheme, is developed for the flatness analysis of a flat surface. Experimental work was performed, and some examples are given in terms of the minimum zone and least-squares solutions.     

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。     

Application of neural network interval regression method for minimum zone straightness and flatness

The goal of this paper is to develop an accurate, efficient, and robust algorithm for the minimum zone (MZ) straightness and flatness. In this paper, we use an interval bias adaptive linear neural network (NN) structure together with least mean squares (LMS) learning algorithm, and an appropriate cost function to carry out the interval regression analysis. From the results, we can see that both the straightness and flatness results from the interval regression method by NN can converge closer to the definition of the MZ straightness and flatness, respectively, than that of the least-squares (LSQ) method. The interval regression method by NN developed in this paper is applicable in the linear regression analysis that has a complicated constraint, and where the LSQ method cannot be used.     

基于新一代GPS的空间直线度误差快速几何逼近算法

基于新一代产品几何技术规范(Geometrical Product Specification and Verification,GPS)的操作及操作算子技术提出了一种新的空间直线度评定方法--快速几何逼近算法.按照最小区域的原则,给出了空间直线度快速几何逼近算法的数学模型及优化目标.该算法原理简单,没有复杂的数学运算...     

Application of several computing techniques for minimum zone straightness

This research deals with the application of several algorithms to calculate the minimum zone straightness. Generally, in the evaluation of the minimum zone value of form errors such as straightness, flatness, roundness, and cylindricity, nonlinear optimization techniques are usually applied. The problem with the nonlinear technique that depends on computing algorithm is that the computing time may be prolonged. Therefore, some linear search techniques that are relatively easy to program are applied for straightness in this article. Furthermore, the problem can also be linearized by considering the characteristics of the measured profile for straightness; thus, the reduction of computing time will be achieved. Then, the problems caused by the above consideration are clarified. Consequently, the convergence criteria and comparison of results by means of several computing methods are investigated. The relationship between flatness and straightness values in some machining conditions and the comparison with the least-squares values are studied.     

Monte carlo method for the uncertainty evaluation of spatial straightness error based on new generation geometrical product specification

Straightness error is an important parameter in measuring high-precision shafts. New generation geometrical product specification(GPS) requires the measurement uncertainty characterizing the reliability of the results should be given together when the measurement result is given. Nowadays most researches on straightness focus on error calculation and only several research projects evaluate the measurement uncertainty based on "The Guide to the Expression of Uncertainty in Measurement(GUM)". In order to compute spatial straightness error(SSE) accurately and rapidly and overcome the limitations of GUM, a quasi particle swarm optimization(QPSO) is proposed to solve the minimum zone SSE and Monte Carlo Method(MCM) is developed to estimate the measurement uncertainty. The mathematical model of minimum zone SSE is formulated. In QPSO quasi-random sequences are applied to the generation of the initial position and velocity of particles and their velocities are modified by the constriction factor approach. The flow of measurement uncertainty evaluation based on MCM is proposed, where the heart is repeatedly sampling from the probability density function(PDF) for every input quantity and evaluating the model in each case. The minimum zone SSE of a shaft measured on a Coordinate Measuring Machine(CMM) is calculated by QPSO and the measurement uncertainty is evaluated by MCM on the basis of analyzing the uncertainty contributors. The results show that the uncertainty directly influences the product judgment result. Therefore it is scientific and reasonable to consider the influence of the uncertainty in judging whether the parts are accepted or rejected, especially for those located in the uncertainty zone. The proposed method is especially suitable when the PDF of the measurand cannot adequately be approximated by a Gaussian distribution or a scaled and shifted t-distribution and the measurement model is non-linear.     

A combinatorial optimization approach for evaluating minimum-zone spatial straightness errors

This paper presents a new and robust approach for the accurate evaluation of minimum-zone spatial straightness error from a set of coordinate measurement data points. The algorithm iteratively searches for the specific data points that define the minimum bound of the spatial straightness zone using combinatorial optimization. It is based on the fact that the minimum circumscribed cylinder of a point set, which is equivalent to the minimum spatial straightness zone of the measurement data, will pass through three, four, or five of the data points that constitute the convex hull vertices of the entire data set. Computed results have shown that although the presented approach may lead to increased computational time, it is robust and able to construct the exact minimum circumscribed cylinder for a given point set. The minimum-zone spatial straightness error can thus be evaluated with the best possible accuracy. The advantage of the presented algorithm is demonstrated via comparison with published computed results of existing algorithms.     

Precision measurement of cylinder straightness using a scanning multi-probe system

This paper describes a scanning multi-probe system for measuring straightness profiles of cylinder workpieces. The system consists of two probe-units, each having three displacement probes. The two probe-units, which are placed on the two sides of the test cylinder, are moved by a scanning stage to scan the two opposed straightness profiles of the cylinder simultaneously. A differential output calculated from the probe outputs in each probe-unit cancels the influence of error motions of the scanning stage, and a double integration of the differential output gives the straightness profile. It is verified that the difference between the unknown zero-values of the probes in each probe-unit (zero-difference) will introduce a parabolic error term in the profile evaluation result, which is the largest error source for straightness measurement of long cylinders. To make zero-adjustment accurately, the cylinder is rotated 180° and scanned by the probe-units again after the first scanning. The zero-differences of the probe-units, as well as the straightness profiles of the cylinder, can be accurately evaluated from the output data of the two measurements. The effectiveness of this method is confirmed by theoretical analysis and experimental results. An improved method, which can measure the variation of the zero-difference during the scanning, is also presented.     

噪声环境下直线度误差分离的最小均方误差方法

研究了噪声环境下直线度误差分离的最小均方误差方法。相对于常用的直线度误差分离中的最小二乘方法,最小均方误差解可更好抑制测量噪声对误差分离结果的影响。同时结合最小均方误差解和最小二乘解,本文还给出一种更好抑制噪声影响的最小范数混合解。     

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。     

传感器调零误差对直线误差分离的影响规律及其确定方法分析

分析了多点法直线 EST中传感器初始调零误差对直线误差分离结果的影响 ,指出不同方法的影响规律不同 :时域三点法中调零误差使直线形状误差按抛物线规律呈非线性增大 ,对直线误差评定产生很大影响 ;时域二点法中调零误差使直线形状误差线性增大 ,但对评定无影响 ;频域三点法的直线误差则不受调零误差的影响。提出了通过构造软基准来确定调零误差的两种方法 :时域频域结合法和对称反转配置法 ,可以克服常规方法的局限性     

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.