Form Error Evaluation Research Articles

Evaluation and Measurement of 3D Form Errors of Ground Curve Surface

Evaluation and measurement of form errors distributed on 3D ground curved surface are proposed due to the difficulty of processing measured points of 3D ground curve surface in comparison with axisymmetric curved surface. First form curved surface grinding is conducted by arc-shaped diamond grinding wheel, second ground curved surface is measured by contact measurement to obtain 3D measuring data, next transfer mode from CNC grinding to measuring reference frames is established, then effective and applicable 3D compensation arithmetic for probe sphere error is introduced, finally 3D form errors are investigated in connection with reference frame transfer and probe sphere compensation. It is confirmed that form error PV of is improved form 203 m / 8 cm2 is obtained by using reference frame transfer and probe sphere compensation, enhancing measuring accuracy by about 29 %.

Evaluation of form error at semi-spherical tools by use of image processing

In this study, a method is proposed for dimensional measurements of semi-spherical parts for the standard tool electrodes used in Electrical Discharge Machining (EDM). The tool electrode to be measured is magnified by a profile projector. Pictures of some equatorial plane projections of the electrodes are captured by a digital camera before and after machining. The pictures were calibrated with the geometric camera calibration. Then, the profiles are extracted from their background using the adaptive threshold, which is an image processing method. 2D coordinates are first obtained and then converted to 3D coordinates. Using the obtained data, sphericity and radius are calculated by means of the minimum zone method. It is seen that the proposed method can be used for determining the EDM tool electrodes’ form error.

Genetic algorithm-based form error evaluation

Form error evaluation of geometrical products is a nonlinear optimization problem, for which a solution has been attempted by different methods with some complexity. A genetic algorithm (GA) was developed to deal with the problem, which was proved simple to understand and realize, and its key techniques have been investigated in detail. Firstly, the fitness function of GA was discussed emphatically as a bridge between GA and the concrete problems to be solved. Secondly, the real numbers-based representation of the desired solutions in the continual space optimization problem was discussed. Thirdly, many improved evolutionary strategies of GA were described on emphasis. These evolutionary strategies were the selection operation of ‘odd number selection plus roulette wheel selection’, the crossover operation of ‘arithmetic crossover between near relatives and far relatives’ and the mutation operation of ‘adaptive Gaussian’ mutation. After evolutions from generation to generation with the evolutionary strategies, the initial population produced stochastically around the least-squared solutions of the problem would be updated and improved iteratively till the best chromosome or individual of GA appeared. Finally, some examples were given to verify the evolutionary method. Experimental results show that the GA-based method can find desired solutions that are superior to the least-squared solutions except for a few examples in which the GA-based method can obtain similar results to those by the least-squared method. Compared with other optimization techniques, the GA-based method can obtain almost equal results but with less complicated models and computation time.

Evaluation of form data using computational geometric techniques—Part I: Circularity error

The present work deals with evaluation of form error from the measured profiles obtained using a form tester, namely roundness/cylindricity measuring instrument. In Part I, details of circularity evaluation are presented. Due to eccentricity in component setting and radius-suppression inherent in the measurement, circularity error has to be evaluated with reference to a limacon. A computational geometry-based algorithm is proposed for establishing minimum circumscribed, maximum inscribed and minimum zone limacons. A new type of control hull for directly constructing equi-angular diagrams and a new procedure for updating are introduced. Validation has been done with bench-mark data set and corresponding results available in the literature. Being geometry-based algorithm, it is simple to follow and each iteration can be visualized and interpreted geometrically. On comparison with simplex search method, the proposed algorithm is found to be computationally efficient in terms of accuracy and time taken. The proposed methods can be easily implemented in computer-aided roundness measuring instruments. Extension of this work for evaluation of cylindricity error has been dealt in Part II.

Form Error Evaluation: An Iterative Reweighted Least Squares Algorithm*

This paper establishes the equivalence between the solution to a linear Chebyshev approximation problem and that of a weighted least squares (WLS) problem with the weighting parameters being appropriately defined. On this basis, we present an algorithm for form error evaluation of geometric features. The algorithm is implemented as an iterative procedure. At each iteration, a WLS problem is solved and the weighting parameters are updated. The proposed algorithm is of general-purpose, it can be used to evaluate the exact minimum zone error of various geometric features including flatness, circularity, sphericity, cylindericity and spatial straightness. Numerical examples are presented to show the effectiveness and efficiency of the algorithm.

Form error evaluation of circles based on a finely-designed genetic algorithm

Form error evaluation of circles based on a finely-designed genetic algorithm

An improved genetic algorithm for planar and spatial straightness error evaluation

The measurement data obtained from the Coordinate Measuring Machines (CMMs) have to be further processed and analyzed to evaluate the form errors of manufactured components. An improved genetic algorithm (GA) is proposed to implement the minimum zone evaluation of planar and spatial straightness errors simultaneously. The algorithm employs the generation alternation model based Minimal Generation Gap (MGP) and blend crossover operators (BLX- α). Compared to traditional GAs, it is efficient and robust. Then, the objective function calculation approaches of planar and spatial straightness error are developed, which directly originate from the definition of minimum zone solution and conform to the ISO standard. Finally, the experimental results evaluated by different methods confirm the effectiveness of the proposed GA. Compared to conventional evaluation methods; it has the advantages of algorithm simplicity and good flexibility. Also it is a unified approach for other form error evaluations and is well suited for the form error evaluation on CMMs.

An overview on distance and pseudo distance functions and their applications

The fast and accurate collision detection and distance calculation between geometric models is fundamental in robotics, manufacturing and computer-simulated environments. This paper surveys the state of the art in the theory and algorithms of distance and pseudo distance functions between convex sets. An overview on the existing results, including fast algorithms for distance calculation, definitions and properties of various pseudo distance functions (the J-function, the growth distance, and the pseudo minimum translational distance), is presented. The distance and pseudo distance functions are versatile in a wide range of areas. Their applications in robot motion planning, grasp analysis/synthesis, form error evaluation, and engineering optimization is demonstrated.

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.

Analysis of Spherical Form Errors to Coordinate Measuring Machine Data.

Coordinates measuring machines (CMMs) are commonly utilized to take measurement data from manufactured surfaces for inspection purposes. The measurement data are then used to evaluate the geometric form errors associated with the surface. Traditionally, the evaluation of spherical form errors involves an optimization process of fitting a substitute sphere to the sampled points. This paper proposes the computational strategies for sphericity with respect to ASME Y14.5M-1994 standard. The proposed methods consider the trade-off between the accuracy of sphericity and the efficiency of inspection. Two approaches of computational metrology based on genetic algorithms (GAs) are proposed to explore the optimality of sphericity measurements and the sphericity feasibility analysis, respectively. The proposed algorithms are verified by using several CMM data sets. Observing from the computational results, the proposed algorithms are practical for on-line implementation to the sphericity evaluation. Using the GA-based computational techniques, the accuracy of sphericity assessment and the efficiency of sphericity feasibility analysis are agreeable.

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.

On determination of sample size in form error evaluation using coordinate metrology

During form error evaluation, as the accuracy requirement increases the measurement sample size should also be increased, thereby increasing the measurement time. In this paper, a new parameter is introduced based on the asymptotic distribution of the range of errors. This new parameter gives the probability that the form error (range of errors) is less than a predicted value based on the current sample size and the corresponding errors. Simulation studies are carried out and the results are presented.

Form error evaluation using L1-approximation

In general, systematic and accidental non-conformities contribute to the non-deterministic nature of the manufacturing processes. The measurement data generated during production and inspection should be further processed to get the desired information contained in it. The measurement data can be considered as the sum of the desired information and certain random errors and variations. Processing requires a method that reduces the effect of wild non-conformities giving due importance to the desired information present in the original measurement data. In this paper an efficient L 1-approximation is presented and used for form error evaluation of engineering surfaces to show its effectiveness when the measurement data contains wild points. The result is compared with the least-squares solution. The use of L 1-approximation for form error evaluation will also help to identify the wild points present on the surface of the part that can be compensated by further operation(machining) to produce a better quality part.

Sampling uncertainty in coordinate measurement data analysis

There are a number of important software related issues in coordinate metrology. After measurement data are collected in the form of position vectors, the data analysis software must derive the necessary geometric information from the point set, and uncertainty plays an important role in the analysis. When extreme fit approaches (L −∞ norm estimation approaches) are employed for form error evaluation, the uncertainty is closely related to the sampling process used to gather the data. The measurement points are a subset of the true surface, and, consequently, the extreme fit result differs from the true value. In this paper, we investigate the functional relationship between the uncertainty in an extreme fit and the number of points measured. Two major issues are addressed in this paper. The first addresses and identifies the parameters that affect the functional relationship. The second develops a methodology to apply this relationship to the sampling of measurement points.

シミュレーテッド・アニーリング法とＢｅｚｉｅｒテンプレートによる形状誤差特徴の抽出

Geometrical errors by manufacturing processes have been evaluated by the maximum width of tolerance zones which envelop the actual features. Details of error within such zones should however be investigated when the designer studies relation between errors and functionality of a product and compares and examines manufacturing methods to realize high accuracy for machined surfaces. A new method is proposed for the evaluation of form errors using a series of points such as measuring points on the product surfaces. This evaluation consists of two parts: 1. Form error features which consist of error models (Bezier Template) and error conditions, 2. Segmentation by fitting of form error features using the Simulated Annealing Method. Form error models are defined as a series of quadric or cubic Bezier curves. A set of Bezier control points can be used for classifying or characterizing typical geometrical errors. Segmentation and fitting of these form error features are realized by the Simulated Annealing Method, a statistical optimization method which can be used to search for the global minimum solution of a combinatrial problem. Our method allows the designer to estimate the effects of errors on functionality by using fitting Form Error Features as a substitute. This leads to functional tolerancing.

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.

An algorithm for form error evaluation — using the theory of discrete and linear Chebyshev approximation

Abstract ISO defines form error based on the concept of minimum zone, but does not describe the methods to obtain the same. In this paper, drawbacks of the available methods are discussed and a new algorithm is presented. This algorithm is guaranteed to give the minimum value of form error. The algorithm is well-suited for implementation on a computer which may be interfaced with a Coordinate Measuring Machine.

A study of in-process computer aided evaluation of size and form error for cylindrical workpieces

When the size and the form error of cylindrical workpiece are to be evaluated by in-process computer aided method, a procedure of in-process data conversion from geometrical quantities to digital quantities (G/D) has to be implemented. The paper presents a method of parameter selection for the procedure, which has been successfully used in a computer controlled automotive valve auto-classification machine. A criterion to choose maximum significant harmonic for correct evaluation of form error profile is proposed. The paper also presents a data smoothing approach to reduce the effect of noise when measuring a rotating workpiece so that good measurement accuracy can be achieved. The in-process size evaluation algorithms are also discussed.

Machine intelligence arbitration for geometrical form errors and geometrical adaptive control strategy

Evaluation of form errors is specified by the ISO in terms of the minimum-zone concept. A method is proposed for numerically determining the form errors based on the minimum-zone evaluation, and a general criterion is given for arbitrating the form errors. A general procedure for “machine-intelligent” arbitration is developed for geometrical form errors. A geometrical adaptive control strategy is presented on the basis of machine-intelligent arbitration. The use of the control strategy provides a convenient structure for the control system and produces an optimal control procedure for the cutting and grinding processes.

Statistical inferences of form error distribution function

The concept of statistical evaluation of form errors is illustrated and it is suggested that the form errors can be treated as stochastic variables which fit beta-distributions within arbitrary intervals. Results are in good agreement with experimental values.