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.     

A new model for general polygon matching problems

Polygon matching is widely used in inspection and measurement applications. The profile of a polygon is a set of edges. Unlike other geometric features, a polygon does not have a regular shape and its number of edges can be any integer greater than two, which causes problems in finding an optimum matching solution. This paper proposes a new model that guarantees the optimum solution for the matching of any polygon shape. The essence of the proposed model is to transform the matching problem into an evaluation of the straightness error of the edges, which simplifies the computation and gives an exact solution in the matching of any shape of polygon.     

基于改进遗传算法的圆度误差评定

提出了一种应用遗传算法计算满足最小区域法的圆度误差的新思路,并对传统的遗传算法提出了一些改进,理论上可以获得全局最优解。仿真结果表明,该方法可以在变量的全局范围内有效、正确的评价圆度误差。     

Roundness: A closed form upper bound for the centroid to minimum zone center distance by worst-case analysis

The minimum zone tolerance (MZT) meets the ISO 1101 definition of roundness error: it determines two concentric circles that contain the roundness profile and such that the difference in radii is the least possible value.     

An exact minimum zone solution for three-dimensional straightness evaluation problems

A mathematical model giving the exact solution for three-dimensional (3-D) straightness evaluation based on the minimum zone criterion is presented in this paper. This model builds upon the convex hull of the measured point set to show that the 3-D straightness is determined simply by some measured points on the vertices of the convex hull. The axis of the critical cylinder giving the minimum objective value is proved to be parallel to one of the edges of the convex hull. This model is effective and easy for implementation.     

Circularity error evaluation: Theory and algorithm

Many procedures for the evaluation of circularity error based on different criteria have been developed. The procedures that are based on the minimum radial separation criterion are either too complex or lack an algorithmic approach to find optimal solution. This paper presents an optimization-based technique to find the value of circularity error based on the minimum radial separation criterion. The problem is formulated as a nonlinear optimization problem. Based on the developed necessary and sufficient conditions a generalized nonlinear optimization procedure is presented. The performance of the developed procedure is analyzed for different size problems generated using a simulation program. Results indicate that the procedure is accurate and very efficient in solving large size real life problems.     

An assessment of “variation conscious” precision fixturing methodologies for the control of circularity within large multi-segment annular assemblies

The fixturing of large segmented-ring assemblies is of importance to a number of key high value industries such as the aerospace and power generation sectors. This study examines methods of optimising the circularity of segmented-ring assemblies, and how the manufacturing variation within each element (i.e. segment wedge) contributes to overall assembly variability. This has lead to the definition of two original assembly methodologies that aim to optimise an assembly, so that circularity errors are minimised for a given set of components. The assembly methods considered during this study include a radial Translation Build (TB) and a Circumscribed Geometric (CG) approach, both of which are compared to a traditional Fixed Datum (FD) build method. The effects of angular, radial, parallelism/flatness and chord length variability within the component geometry, and their effect on the circularity of the final annular assembly are examined mathematically and experimentally. Furthermore, the inherent loss of assembly circularity due to differences between component and assembly sagitta is also considered, along with the stepping caused by dissimilar adjacent component radii as a result of manufacturing variation. Experimental results show that the CG build method offers a significant improvement in circularity in most situations over the benchmark FD build method. This contrasts the TB results that proved to be the least consistent in terms of circularity, but better in the control of angular breaking errors within the assembly.     

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.     

A study on analyzing the problem of the spherical form error

Many methods to evaluate the form error of a sphere have been studied over the years. Most of these, such as the optimum methods, employed the approximate local solution to obtain the desired results. In this paper, three mathematical models are constructed to evaluate the solutions of the minimum circumscribed sphere, the maximum inscribed sphere, and the minimum zone sphere by directly resolving the simultaneous linear algebraic equations. Examples are given to verify that the model is admissible and reliable. These simple mathematical methods are verified to be useful for determining the exact solution.     

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 new strategy for inspecting roundness features

This article addresses the problem of inspection of roundness error (in contrast to the measurement of roundness error) and the idea of GO-NOGO inspection software. A new approach is presented in which the formalism of linear programming is applied through reexpressing the initial geometrical problem and adding further constraints. The ultimate position is to provide a set of constraints from which the required inspection information can be obtained by examining only the feasibility of the mathematical progrma, with no need to consider its iteration toward optimality. Feasibility may be explored using an existing, although not widely known, algorithm to guide the GO-NOG decision. The basis of the algorithm is given along with practical examples of how it works, Tests indicate that on real data it usually involves a computational effort only modestly higher than using an exchange algorithm to find an optimal fit. Thus, it appears quite practical for applications to the relatively small computer of a standard instrument.     

评定直线度误差的精确算法与程序实现

本文讨论了评定直线度误差的计算机精确算法，给出相应的程序框图，并介绍了程序的实现方法。通过在实验室和企业中的实际运用，证明了该方法的有效性和程序的可靠性。     

评定直线度误差的精确算法与程序实现

本文讨论了评定直线度误差的计算机精确算法,给出相应的程序框图,并介绍了程序的实现方法。通过在实验室和企业中的实际运用,证明了该方法的有效性和程序的可靠性。     

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.     

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.     

直线度平面度公差的快速评定测点分类法

在直线度、平面度公差判定的最小包容区域法中提出一个新的、快速的实施方法。新方法将所有测量点分成“高点”、“低点”和“鞍点”三种类型。并指出最小包容区域法中的最高点只出现在“高点”中,最低点只出现在“低点”中,最高(低)点不会出现在“鞍点”中。这样极大的减少了搜索的范围,提高了软件的效率,而且测量点越多,效果越显著。通过70个测点平面度评定的典型算例,表明此算法比传统的最小区域法要快几十倍。     

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.     

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.     

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.