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 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.     

The relationship between the minimum zone circle and the maximum inscribed circle and the minimum circumscribed circle

Following the minimum zone criterion set forth in the current ANSI and ISO standards, evaluation of roundness error is formulated as a non-differentiable unconstrained optimization problem and hard to handle. The maximum inscribed circle and minimum circumscribed circle are all easily solved by iterative comparisons, so the relationship between the minimum zone circle and maximum inscribed circle, minimum circumscribed circle is proposed to solve efficiently the minimum zone problem. Based on the known minimum zone circle, the maximum inscribed circle and minimum circumscribed circle can be easily determined. The relationship is implemented and validated with the data available in the literature.     

A new strategy for circularity problems

The evaluation of circularity based on the minimum zone criterion as a non-linear and non-convex problem, which requires a substantial amount of computational effort in general, is investigated. A new strategy for improving the computational efficiency by collecting data points at the farthest and nearest locations from current minimum radial separation center until all collected data points meet the optimum criterion is proposed. A number of mathematical models developed in this paper indicate that the minimum circularity can be determined by using a small number of critical data points. The validated results show that the proposed strategy offers an effective way to identify the critical data points at the early stage of computation and gives an efficient approach to solve the circularity problems, especially when the number of data point is large.     

Minimum zone evaluation of sphericity deviation based on the intersecting chord method in Cartesian coordinate system

Along with the developments of manufacturing and machining technology, spherical parts with high-precision are widely applied to many industrial fields. The high-quality spherical parts depend not only on the design and machining techniques but also on the adopted measurement and evaluation approaches. This paper focuses on the minimum zone evaluation model of sphericity deviation in Cartesian coordinate system. A new method, i.e. intersecting chord method, is proposed to solve the problem of constructing 3 + 2 and 2 + 3 models of the minimum zone reference spheres (MZSP). The modelling method employs intersecting chords rather than characteristic points to construct the geometrical structure of evaluation model. Hence, the efficiency of processing data is improved without compromising the accuracy of deviation evaluation. In the modelling process, the two concentric spheres of minimum zone model are simplified as an intersecting chords structure, the virtual centre generated by the intersecting chords can be used to judge whether the searched object is the maximum object or not, which decrease the positioning error of the minimum zone centre and reduce the difficulty of constructing models. To test and verify the performances of intersecting chord method, two experiments are performed to confirm the effectiveness of the proposed method, and the results indicate that the proposed method is more trustworthy against accuracy and computation time than other methods required to achieve the same results.     

Conicity error evaluation using sequential quadratic programming algorithm

Form error evaluation plays an important role in processing quality evaluation. Conicity error is evaluated as a typical example in this paper based on sequential quadratic programming (SQP) algorithm. The evaluation is carried out in three stages. Signed distance function from the measured points to conical surface is defined and the cone is located roughly by the method of traditional least-squares (LS) firstly; the fitted cone and the measured point coordinates are transformed to simplify the optimal mathematical model of conicity error evaluation secondly; and then optimization problem on conicity error evaluation satisfying the minimum zone criterion is solved by means of SQP algorithm and kinematic geometry, where approximate linear differential movement model of signed distance function is deduced in order to reduce the computational complexity. Experimental results show that the conicity error evaluation algorithm is more accurate, and has good robustness and high efficiency. The obtained conicity error is effective.     

Intersecting chord method for minimum zone evaluation of roundness deviation using Cartesian coordinate data

Minimum zone evaluation of roundness deviation is a very important and complex problem in precision measurement. Along with the continuous development of precision machining technology, it has become an increasingly prominent issue of how to quickly and accurately evaluate the minimum zone roundness deviation from a large number of coordinate data. In this paper, an intersecting chord method is first proposed to realize the minimum zone model of roundness deviation with coordinate data. The new modelling method uses the crossing relationship of chords to construct the intersecting structure and the 2 + 2 evaluation model of the minimum zone roundness deviation, which can not only accurately determine the position of minimum zone centre but also greatly improve the computational efficiency of modelling process. Using the related chords and their extreme points to generate a virtual centre, this may reduce the deviation between the intersecting chords structure and the centre of the minimum zone evaluation. The proposed method makes use of the geometric relationship of chords, so the minimum zone roundness deviation can be obtained without the optimal method or the point-by-point method. The validation test of the proposed method is designed to analyze a coordinate dataset published in other literature. Comparing the proposed method with the published method, it is easy to show that the relative error between two results is less than 0.4%. Finally, an experiment is also given to indicate that the calculation accuracy and the evaluation efficiency of the proposed method achieve a satisfactory conclusion.     

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。     

Profile error evaluation of free-form surface using sequential quadratic programming algorithm

Profile error of free-form surface is evaluated in this paper based on sequential quadratic programming (SQP) algorithm. The optimal localization model is established with the minimum zone criterion firstly. Subsequently, the surface subdivision method or STL (STeror Lithography) model is used to compute the point-to-surface distance and the approximate linear differential movement model of signed distance is deduced to simplify the updating process of alignment parameters. Finally, the optimization model on profile error evaluation of free-form surface is solved with SQP algorithm. Simulation examples indicate that the results acquired by SQP method are closer to the ideal results than the other algorithms in the problem of solving transformation parameters. In addition, real part experiments show that the maximum distance between the measurement points and their corresponding closest points on the design model is shorter by using SQP-based algorithm. Lastly, the results obtained in the experiment of the workpiece with S form illustrate that the SQP-based profile error evaluation algorithm can dramatically reduce the iterations and keep the precision of result simultaneously. Furthermore, a simulation is conducted to test the robustness of the proposed method. In a word, this study purposes a new algorithm which is of high accuracy and less time-consuming.     

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...     

Analysis of minimum zone sphericity error using minimum potential energy theory

Theoretical derivation of the minimum zone criteria of sphericity error based on the principle of minimum potential energy is proposed. All the measured data points are enclosed by two concentric spherical surfaces between which a fictitious spring is assumed to be placed. These two concentric spherical surfaces can be mathematically determined by five active data points. When the spring contracts, the potential energy of the simulated mechanical system tends to reduce which yields two new concentric spheres with smaller radial separation and new active data points. Finally, a stable state will be reached to the condition of minimum potential energy. The criteria conforming to such a state can be derived. A direct search scheme to the global minimum solution is also proposed. The clearance between such two concentric spherical surfaces is the minimum zone of spherical form error.     

Study on the Form Error Evaluation Model of a Conical-Shank Ball-End Milling Cutter

This paper presents a complete form error evaluation model of a conical-shank ball-end cutter. The complete conical-shank ball-end cutter comprises three major revolving portions including the conical shank, cylindrical section, and spherical head. By using the central line of the minimum circumscribed conical shank as a datum axis and employing profile data measured directly from cutter surfaces, the proposed evaluation method systematically computes the exact revolving envelopes of the conical shank, cylindrical part, and spherical part separately, and then combines them. One application example is presented to illustrate the effectiveness of the proposed modelling method. This paper provides a simple, logical, and precise method for the quality evaluation of a special rotating cutter and can be further extended for use in the quality assessment of various complicated cutters.     

单圈非重复性主轴回转精度评价

与工件圆度的误差评价相比,实现对不具有单圈重复性的主轴回转精度的评价较为困难.在分析主轴轴线定义及理想轴心可观测性的基础上,建立单圈非重复性主轴回转精度评价的数学模型,针对该数学模型,进行主轴回转误差的集合转换,使得转换后的集合能够适应于计算机处理的误差评价方法.然后利用极差极小化的原理,建立最小区域法的误差评定统一准则和作用表面的统一判别准则.利用这两个评判准则,可以顺利实现对主轴回转误差的最小区域法评价、最小外切圆法评价和最大内接圆法评价,从而提高单圈非重复性主轴回转误差的评价精度,同时也提高误差评价的效率.     

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.     

改进蜂群算法在平面度误差评定中的应用

为了准确快速评定平面度误差,提出将改进人工蜂群( MABC)算法用于平面度误差最小区域的评定.介绍了评定平面度误差的最小包容区域法及判别准则,并给出符合最小区域条件的平面度误差评定数学模型.叙述了MABC算法,该算法在基本人工蜂群算法( ABC)模型的基础上引入两个牵引蜂和禁忌搜索策略.阐述了算法的实现步骤,通过分析选用两个经典测试函数验证了MABC算法的有效性.最后,应用MABC算法对平面度误差进行评定,其计算结果符合最小条件.对一组测量数据的评定显示,MABC算法经过0.436 s可找到最优平面,比ABC算法节省0.411 s,其计算结果比最小二乘法和遗传算法的评定结果分别小18.03μm和6.13 μm.对由三坐标机测得的5组实例同样显示,MABC算法的计算精度比遗传算法和粒子群算法更有优势,最大相差0.9 μm.实验结果表明,MABC算法在优化效率、求解质量和稳定性上优于ABC算法,计算精度优于最小二乘法、遗传算法和粒子群算法,适用于形位误差测量仪器及三坐标测量机.     

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.     

基于遗传算法的球度误差评定

首先对球度公差评定问题进行了综述.然后根据圆度公差的数学定义,引申提出球度公差最小区域条件下的评定模型,并给出遗传算法的适应度函数.随后给出算法实现中的中的关键问题.最后用实例对算法进行了检验,计算结果表明基于遗传算法的球度误差优化算法不仅符合最小区域的条件,而且易于理解和实现,能够获得全局最优解,保证了高精度、高效率.     

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。     

基于最小包容区域的球度误差评定方法

针对球度误差评定方法存在原理误差或模型误差,提出了一种符合最小包容区域定义的球度误差评定方法,即将几何搜索逼近算法与基于最小包容区域法的球度误差评定的几何结构和定义结合起来的准确评定方法。对仿真数据和其他文献中的数据进行了评定。所提方法与其他方法的评定结果表明,所提方法可以准确地找到最小包容区域球的球心并给出球度误差的精确解。