Applying particle swarm optimization algorithm to roundness error evaluation based on minimum zone circle

Minimum zone circle (MZC) method and least square circle (LSC) method are two most commonly used methods to evaluate roundness, but only the MZC method complies with the standard definition and can obtain the minimum roundness error value. The determination of the center of MZC is a nonlinear optimization problem which is suitable to be solved by particle swarm optimization (PSO) algorithms. In this paper, the standard PSO algorithm was introduced and theory analysis about the impact of value selection of some important parameters, such as inertia weight ω, on the algorithm’s stability and convergence was carried on so as to provide basis for giving these parameters better values. Furthermore, the superiority of making ω decrease linearly with iterations was verified through a computation experiment in terms of stability and accuracy, compared with the other three cases of ω = 1, 0.5, 0. Based on the analysis, the novel PSO algorithm, with ω decreasing linearly from 0.9 to 0.4 and the LSC center as the initial positions of the particles, is implemented to obtain MZC-based roundness errors of sampling points collected from circular section profiles by a coordinate measuring machine (CMM). By comparing the novel PSO–MZC results with the LSC-based results, it is concluded that the former are a little smaller than the latter, which verifies that the novel PSO algorithm is feasible to calculate roundness error and the fact that a LSC-based one is generally larger than a MZC-based result; the values of the two roundness errors are both related to sample size and increase with an increase in the sample size with a decreasing increment.     

圆度误差评定方法的研究

圆度误差评定是否准确,将直接影响到机械产品的性能和寿命.介绍了4个简单而有效的算法来评定圆度误差,即最小外接圆、最大内接圆、最小区域法和最小二乘圆法.利用MATLAB对上述算法进行了验证,说明文中算法有效.     

Uncertainty Calculation of Roundness Assessment by Automatic Differentiation in Coordinate Metrology

Recently, Coordinate Measuring Machines (CMMs) are widely used to measure roundness errors. Roundness is calculated from a large number of points collected from the profiles of the parts. According to the Guide to the Expression of Uncertainty in Measurement (GUM), all measurement results must have a stated uncertainty associated the them. However, no CMMs give the uncertainty value of the roundness, because no suitable measurement uncertainty calculation procedure exists. In the case of roundness measurement in coordinate metrology, this paper suggests the algorithms for the calculation of the measurement uncertainty of the roundness deviation based on the two mainly used association criteria, LSC and MZC. The calculation of the sensitivity coefficients for the uncertainty calculation can be done by automatic differentiation, in order to avoid introducing additional errors by the traditional difference quotient approximations. The proposed methods are exact and need input data only as the measured coordinates of the data points and their associated uncertainties.     

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.     

一种基于MATLAB的圆度评定方法

圆度的评定和计算 ,实质上是根据圆度的定义构造函数模型、再进行函数优化求解的过程。提出一种基于MATLAB的圆度评定方法 ,利用MATLAB优化工具箱 ,为采用最小区域圆法、最小二乘圆法、最小外接圆法和最大内接圆法实现圆度的评定提供了新的选择。     

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.     

Simple and efficient algorithms for roundness evaluation from the coordinate measurement data

A common problem of quality control and inspection of circular parts is the measurement of their roundness. Recently, the coordinate measuring machines (CMMs) have been used to measure roundness errors by collecting a large number of points from the profile of the rounded parts to meet the international standards. Direct evaluation of roundness from this large number of pointes is complex and time consuming. Therefore, efficient algorithms should be designed to meet the standards and to simplify and accelerate the computation process. This paper introduces simple and efficient algorithms to evaluate the roundness error from the large number of points obtained by CMMs using three internationally defined methods: Minimum Circumscribed Circle (MCC), Maximum Inscribed Circle (MIC) and Minimum Zone Circles (MZC). A software has been developed using C++ to apply these algorithms on the data obtained by CMMs. The developed algorithms were verified by comparing their results with the results obtained by a commercial instrument and the maximum variation between the two results did not exceed than ±2.27%. The efficiency of the introduced algorithms was verified in terms of computation time and the results proved the efficiency of the developed algorithms.     

Applications of integrated motion controllers for precise CNC machines

The error resources of precise motion control systems are basically categorized into linear and nonlinear effects. To realize the precise motion of industrial computer numerical control (CNC) machines, this paper presents an integrated motion control structure with modular algorithms, including both the linear control and the nonlinear compensation. In the linear control design, this study applies three algorithms: (1) feedforward control to address the tracking errors, (2) cross-coupled control to reduce the contouring errors, and (3) digital disturbance observer to lessen the effects of modeling errors and disturbances in real applications. The results indicate that the linear motion controller achieves greatly improved accuracy in both tracking and contouring by reducing the servo lags and mismatched dynamics of the different axes. However, the adverse effect due to friction still exists and cannot be eliminated by applying the linear motion controller only. This study further integrates the nonlinear compensator and develops friction estimation and compensation rules for CNC machines. The digital signal processors are suitable to implement all the developed linear and nonlinear algorithms, and the present controllers have been successfully applied to industrial CNC machines. Experimental results on a vertical machining center indicate that, under different feed rates, the CNC machine with the integrated motion controller significantly reduces the maximum contouring error by 135% on average.     

Establishing circle and circular-cylinder references using computational geometric techniques

Form deviations of cylindrical features present in the manufactured parts are measured using coordinate measuring machine (CMM) and expressed as circularity or cylindricity errors evaluated using appropriate reference features. In the present work, computational geometric techniques are used to establish a circle as reference feature, and a heuristic algorithm is proposed to get a unique convex inner hull. Using the concept of equi-distant lines and diagrams, minimum circumscribed (MC), maximum inscribed (MI), and minimum zone (MZ) circles are established. For the first time, algorithms purely based on computational geometric concepts have been developed in the present work to arrive at MC, MI, and MZ circular cylinders. As the algorithms and the implementation details are explained with simple data sets, the practitioners can easily understand these methods and implement them in CMMs for the evaluation of circularity and cylindricity errors. The algorithms are also tested on larger datasets, and in all cases, accurate results are obtained in less than a second.     

遗传算法在流体机械优化设计中的应用

作为一种全新的随机优化方法 ,遗传算法已广泛地应用于透平机械设计的多个方面 ,例如遗传算法直接用于气动元件的形状优化设计 ,遗传算法与一定的计算流体动力学方法结合参与流场的分析和计算等等。本文是对遗传算法在流体机械优化设计方面应用的一个综述。     

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.     

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...     

The Application of Grey Theory to the Prediction of Measurement Points for Circularity Geometric Tolerance

The objective of this study is to improve the efficiency of achieving accurate and speedy inspection when a coordinate measuring machine (CMM) is used to measure circularity geometric tolerance. Grey prediction using grey theory was applied in developing the heuristic algorithm for predicting the number of measuring points required for measuring circularity geometric tolerance. The heuristic algorithm was used to plan the number of measuring points of the next workpiece and to predict the circularity geometric tolerance dimensions. This step provides a better foundation for on-line inspection to determine the number of measuring points required for measurement inspection of the next workpiece. It can also predict whether the circularity geometric tolerance of the next workpiece will conform with the circularity geometric tolerance dimension on the design drawing. This heuristic algorithm could also be used to determine whether the manufacturing process requires modification, in order to save human and material resources and reduce failure rate. In this paper, a circularity heuristic algorithm for deciding the number of measuring points for the circularity geometric dimension is proposed to reduce the angle between measuring points. In this heuristic algorithm, the addition of measuring points is determined by whether the predicted circularity value exceeded the scope of the grey interval. When more points are to be inspected, equiangular inspection is again performed. Once the predicted circularity value exceeds the limitation of the design drawing circularity geometric dimensions, the manufacturing process is examined immediately. The concept of adding measuring points for on-line inspection is combined with the grey prediction and the standard deviation concept to establish the circularity heuristic algorithm for deciding the number of the measuring points. Experimental data are also introduced to verify the measuring point algorithm for the circularity geometric dimension of a workpiece. The objective is to reach a balance between the smallest number of measuring points required for inspection and inspection accuracy. Consequently, it is possible to avoid using too many measuring points, which increases the inspection time, while achieving the required measurement accuracy. using too many measuring points, which increases the inspection time, while achieving the required measurement accuracy.     

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 new biogeography-based optimization (BBO) algorithm for the flexible job shop scheduling problem

Biogeography-based optimization (BBO) algorithm is a new kind of optimization technique based on biogeography concept. This population-based algorithm uses the idea of the migration strategy of animals or other species for solving optimization problems. In this paper, the BBO algorithm is developed for flexible job shop scheduling problem (FJSP). It means that migration operators of BBO are developed for searching a solution area of FJSP and finding the optimum or near-optimum solution to this problem. In fact, the main aim of this paper was to provide a new way for BBO to solve scheduling problems. To assess the performance of BBO, it is also compared with a genetic algorithm that has the most similarity with the proposed BBO. This similarity causes the impact of different neighborhood structures being minimized and the differences among the algorithms being just due to their search quality. Finally, to evaluate the distinctions of the two algorithms much more elaborately, they are implemented on three different objective functions named makespan, critical machine work load, and total work load of machines. BBO is also compared with some famous algorithms in the literature.     

数控机床进给速度对圆度的影响测试与分析

对球杆仪的测试应用及研究现状进行了阐述,并介绍了球杆仪的测试原理。针对某型立式加工中心的X-Y平面进行了圆度误差测试,圆度测试在500 mm/min,1 000 mm/min,3 000 mm/min的进给速度下进行。给出了不同进给速度下的圆度图和圆度误差数据,得到了伺服轴的进给速度越大,圆度误差越大的结论。对球杆仪测试的圆度误差进行了误差分离,得到了不同进给速度下的误差数据及排序,并对各项误差的原因及影响进行了分析。     

试验验证圆度测量中圆心偏对球杆仪测试结果的影响

基于Renishaw QC20-W球杆仪对某型卧式加工中心进行了圆度测试,给出了圆度测试的结果并对结果进行了分析.试验结果表明,在对数控机床进行圆度测试时,圆心偏移对于轨迹误差测量的影响可以忽略.最后分析了球杆仪解算器的误差分离功能,得出了在对数控机床进行圆轨迹误差测试时可以不考虑圆心偏干扰的结论.     

Evaluation of conditional multiple concentric circular arcs

Measurement and evaluation of circularity of cylindrical components is well established using roundness measuring instruments, but when circular track

Problems of this and more complex nature have been faced while evaluating the tool-changer arm-holder tracks of NC machines. Since no methods are avail     

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.     

A queuing approach for a tri-objective manufacturing problem with defects: a tuned Pareto-based genetic algorithm

In this research, a manufacturing facility with independent workstations to remanufacture nonconforming products is investigated. Each workstation is first modeled as an M/M/m queuing system with m being a decision variable. Then, a tri-objective integer nonlinear programming models is developed to formulate the problem. The first objective tries to minimize the waiting times of products, while the second one tries to maximize the minimum reliability of machines at the workstations. Since minimization of the waiting times results in using a large number of machines with higher idle times, the third objective is considered to minimize the mean idle time of the machines. The aim is to determine optimal number of machines at each workstation. Since the problem belongs to the class of NP-hard problems, the non-dominated sorting genetic algorithm-II (NSGA-II) is utilized to find Pareto fronts. Because there is no benchmark available in the literature to validate the results obtained, the non-dominated ranked genetic algorithm (NRGA) is used as well. In both algorithms, not only the best operators are selected but also all of their important parameters are calibrated using statistical analysis. The performances of the algorithms are statistically compared using the t test. Besides, the multiple attribute decision-making method of TOPSIS is used to determine the better algorithm. The applicability of the proposed model and the solution algorithms is demonstrated via some illustrative examples.