Straightness and flatness tolerance evaluation: an optimization approach

This paper presents an optimization approach that could be used to calculate exact values of straightness and flatness errors as defined by the ANSI Y14.5M standards on geometric dimensioning and tolerancing. The straightness and flatness error evaluation problems are formulated as nonlinear optimization problems with linear objective function and nonlinear constraints. Because of the special structure of the problem, a linear search method is developed that reduces the nonlinear problem to a linear programming problem with only two constraints. Examples are presented to compare the optimization approach with the least-squares method and some exact methods. The results show that the optimization procedures presented in this paper provide exact values of straightness and flatness errors and are superior to the existing methods in terms of computation time.     

Tolerance-based localization algorithm: form tolerance verification application

Current localization techniques have been successfully used for aligning sculptured surfaces with CAD models in inspection applications. However, tolerance specifications are not considered as an integral part of the localization process. The tolerance verification and comparison with measured surfaces occur at a later step to accept or reject the manufactured part. This two-step process prolongs the inspection time. For the first time, this paper presents a novel localization algorithm for inspection that integrates the tolerance specifications as an optimality criterion. A closed-form solution algorithm that applies 3D rigid body transformation using quaternion and uses a minimum acceptable deviation zone approach was developed. The formulation is based on the mathematical definitions from ANSI Y14.5.1 standards (American National Standard Institute) for form tolerances. The new iterative minimum acceptable deviation zone localization algorithm is formulated using four types of form tolerances: straightness of a median line, straightness of a surface line, flatness and cylindricity. It is applied and compared to several benchmark examples for validation. The results demonstrated the ability of the new localization approach to achieve comparable results but with less computation effort due to using a constraint satisfaction problem and a closed-form solution algorithm in the formulation. The merit of the new approach stems from its ability to increase the efficiency of tolerance verification during the inspection process. The applicability of the proposed algorithm to various types of tolerance is highlighted.     

Evaluation of minimum zone flatness by means of nonlinear optimization techniques and its verification

This article deals with the application of some nonlinear optimization techniques for minimum zone flatness. The convergence criteria of the techniques, namely the downhill simplex method and the repetitive bracketing method, are considered. The least-squares method is also applied, and subsequently the three methods are compared from the viewpoint of computational accuracy. A surface profile measuring system and a noncontact sensor are used to obtain three-dimensional data. The measured data are expressed by means of perspective mapping. Subsequently, the relationship among the above three methods is clarified according to accuracy and efficiency of the computation. Furthermore, some examples of the relationship between the manufacturing method and the flatness value, and the technique of a skilled hand are described.     

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.     

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.     

Geometry of the minimum zone flatness functional: planar and spatial case

The zone straightness and flatness functional is constructed from the definition of the measure. The geometry of the straightness functional is illustrated in the plane, and in three dimensions, a novel means to visually represent flatness is described using the zone separation body. The zone separation body is a new construction that is uniquely associated with every measurement dataset and can be used to represent the flatness functional visually.     

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。     

A new minimum zone method for evaluating straightness errors

A new minimum zone method for straightness error analysis 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 line rotation scheme, is developed for the straightness analysis of planar lines. Extended works on the error analysis of spatial lines by the least parallelepiped enclosures are also described. Some examples are given in terms of the minimum zone and least-squares. Finally, this easy-to-use method is illustrated by an example that demonstrates that, for a planar line, the minimum zone solution can even be found without the use of a computer.     

基于多目标优化的空间直线度误差评定

根据最小区域定义及数学规划理论，建立了空间直线度评定的非线性规划模型，指出了这模型实质上是多目标优化的问题并将该优化问题转化成单目标优化问题。由于该非线性规划模型还是凸的、二次的，因此提出了用逐次二次规划的解法（SQP法）来实施。SQP法在评定过程中保留了模型中的非线性信息，对初始参数的要求低，且稳定、可靠、效率高。几个算例的结果均满足凸规划全局最优判别准则，这就有力的验证了上述结论。     

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.     

Evaluation of spherical form errors—Computation of sphericity by means of minimum zone method and some examinations with using simulated data

Because of the practical difficulties of measuring whole spherical surface form errors, no concrete three-dimensional (3-D) verification has yet been developed. This article deals with the calculation of the value of spherical form errors; that is, sphericity. The iterative least-squares method in which the problem is linearized and the minimum zone method in which the downhill simplex method, one of the nonlinear optimization techniques, is applied are considered. The data to be analyzed are not obtained by actual measurement of a spherical surface, because there is no such a measuring system in my laboratory, but simulated by applying surface harmonics (Laplace's spherical function) with a computer. Then, the application conditions for downhill simplex method are investigated. Furthermore, the roundness values of the spherical surface are compared with the sphericity by means of the minimum-zone method.     

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.     

Straightness and flatness evaluation using data envelopment analysis

In today's world of precision engineering, robustness and accuracy in the evaluation of the form tolerances are considered as competitive advantages for manufacturing enterprises. Amongst various methods for accurate and robust evaluation, which have been studied, nonlinear optimization methods based on operational research have proved to be successful as far as they can ensure unique and global convergence in practical applications. However, it is well known that ensuring the convergence is the most difficult thing to deal with for a nonlinear optimization technique because the performance is in general highly sensitive to parameter setting. Therefore, this paper introduces a robust linear programming formulation-based algorithm in which the performance is not dependent on the quality of parameters. Interestingly, in this algorithm, the data envelopment analysis technique is used to form a convex hull that decides the minimum enclosed zone in a robust manner. From the computational experiments, it is shown that the proposed algorithm can be a promising alternative to the traditional nonlinear optimization method for straightness and flatness evaluation.     

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.     

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.     

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。     

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。     

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.     

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。     

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.