Study of detailed deviation zone considering coordinate metrology uncertainty

Abstract

The detailed Deviation Zone Evaluation (DZE) based on the measurement of the discrete points is a crucial task in coordinate metrology. The knowledge of detailed deviation zone is necessary for any form of intelligent dynamic sampling approach in coordinate metrology or any downstream manufacturing process. Developing the desired knowledge of the deviation zone using only a finite set of the data points always needs a set of efficient interpolation and extrapolation techniques. These methods are selected based on the nature of the perusing pattern of the geometric deviation. The objective of this work is to study the efficiency of a DZE approach for the various combinations of the manufacturing errors and coordinate metrology accuracies. The first employed DZE method is governed by a Laplace equation to estimate the geometric deviations and a Finite Difference scheme is used to iteratively solve the problem. The other DZE method utilizes a metaheuristic approach based on Genetic Programming. Several cases of surfaces manufactured by various levels of fabrication errors and also different types of metrology systems are studied and the convergence of the employed methodologies are analyzed. It is shown how efficient the DZE solutions are to reduce the uncertainty of the resulting deviation zone based on the number of points acquired during the measurement process. The DZE solutions are successful to minimize the number of the required inspected points which directly reduces the cost and the time of inspection. The results show a great improvement in reliability of deviation zone evaluation process.

Introduction

Development of computational algorithms to improve efficiency and reliability of coordinate metrology process has been a challenging research task during the last three decades. Concurrent to rapid technological changes of the Coordinate Measurement Machines (CMMs), the demand for more advanced computational algorithms to plan, control, and processing results of coordinate metrology are increased. However, the reliability and efficiency of these computational algorithms are always limited by inherent sources of uncertainties due to the nature of the coordinate metrology process.

Skilled operators are generally required to oversee the measurements and interpret measurement results. It is desirable, therefore, to improve the efficiency of these machines by developing measurement algorithms that produce valid results in minimum time. Because the total measurement time is proportional to the number of sample points required for the measurement cycle, minimizing the number of collected sample points is an important factor to consider when developing an optimal measurement plan. As measurement costs are directly influenced by the number of acquired points, it is desirable that the acquired points are chosen such that with a smaller number of points the surface estimation has the best discrepancy when compared with the ideal geometry.

A surface reconstruction always consists of getting a computational model that resembles more accurately to the real object. The coordinate metrology is based on a scattered cloud of points obtained from the measured object. There is a vast literature on geometry extraction from point clouds, which includes local polynomial fits, global smooth approximations, Voronoi/Delaunay-based methods, and level set methods. Earlier heuristic approaches gave no guarantees, but new powerful methods provide more accurate and better precision to the real shape. These methods mostly have an acquisition of infinitely dense data, and they also undergo some restrictive assumptions on object smoothness and regards the evenness of the sampling [1].

The research is conducted on three main computational tasks of coordinate metrology i.e. Point Measurement Planning (PMP), Substitute Geometry Estimation (SGE), and Deviation Zone Evaluation (DZE) [2], [3], [4]. Researches on PMP focus on size and location of the measurement process. The PMP objective is to achieve a more precise representation of the measured geometries using fewer points. This goal is usually approached by proposing an algorithm that analyzes the trend of errors for the acquired points. The objective of SGE is to find the best substitute geometry based on the fitting criteria in the inspection process. An objective function is defined which also represents the fitting error. The typical objective functions in SGE process are developed based on total least square of Euclidian distances of the measured points from the substitute geometry, or maximum of Euclidian distances of the measured points from the substitute geometry. An optimization process needs to be employed to minimize this objective function when a set of geometric parameters is given [2].

The last computational task which is also the main focus of this research work, is DZE. Any manufacturing process has a limited level of accuracy which results in geometric deviation of the final manufactured surfaces from the originally designed model. In order to minimize these deviations, it is important to fully address the behaviour and severity of the deviations. DZE is the effort to model the deviation zone of the entire manufactured surfaces using the available finite number of sample points.

Early stage research in all three areas were limited to inspection of the geometries with only primitive features such as plane, circle, rectangle, cylinder, cone, and sphere. In the rest of this section, a few prominent works on PMP and SGE which also reflect on DZE are briefly reviewed following by a more detailed review of the works focused on the DZE problem.

As in the realm of SGE, Shuanmugam [5] governed the median approach and least square assessment method to estimate the flatness and circularity of measured data. He realized that the results of least square are affected by all measurements while median method is only affected by maximum and minimum of readings. On the other hand, for the same set of data, the median approach had better convergence rather than the least square method.

Another aspect of SGE is to fit geometric features to scattered points. A very popular approach is to use least square fitting. Yau et al. [6] used a unified least square method for evaluation of various geometric features including the primitive features and sculptured surfaces. They minimized the sum of square of errors of the estimated feature regarding to the nominal ones. Therefore, a best fit of the nominal geometry was extracted. Since their algorithm has an iterative approach, it has the drawback of high time consumption for convergence. However, their results had a better accuracy over similar methods that uses linearization.

Kim and Kim [10] showed that geometric tolerances of circular features can be improved by a linear approximation using minimum variance. They used Chebyshev algorithm for linearization of least square method over sampled points. Menq et al. [11], modeled the error of measurement by minimizing the sum of the squared distances of the measurement data from the surface. They also considered the rigid body transformation matrix in the error calculation for the CMM path planning.

Samuel et al. [12] used a convex hull approach for evaluation of straightness and flatness of geometric features. After constructing the convex hull by divide and conquer algorithm. Their algorithm finds the minimum distance from two parallel lines and consider this amount as the straightness error of the manufactured surface. For flatness evaluation, they used the same approach but on a 3-Dimensional space and two parallel planes to find the error. Therefore, a minimum zone evaluation of the measured surface is provided. Their algorithm is quite robust and provides an imperative minimum zone; however, it might not be the best minimum zone possible for a set of measured data.

Another aspect of surface evaluation is the uncertainty analysis of the geometry. In one study, Yan et al. [13], provided the uncertainty analysis and variation reduction related to parts produced by end-milling. They divided their work to three stages: geometry errors decomposition, uncertainty analysis and variation reduction. The decomposition algorithm uses the superposition of waviness trend of the measured data to reconstruct a smooth surface close to the nominal one. This stage uses a bi-cubic B-Spline surface with the measured points as the control points in a regression model for best fit. However, the errors related to the fitted B-Spline patches are highly related to the number of acquired points. In order to overcome this drawback, they devised an iterative algorithm to increase the number of patches based on the randomness of the fitting error. In the second stage, they modeled the uncertainty parameters of the CMM using single decomposition on the transformation matrix of the CMM. In this approach, they could form the sensitivity matrix for the uncertainty parameter. By assuming that each uncertainty parameter is normally distributed, they fitted a normal distribution on the uncertainty parameters extracted from the decomposition of the transformation matrix. The fitted distribution is then tested for normality using skewness method where the bounds on the normal distribution are extracted for analysis. The method is verified through computational simulations and comparison of experimental data which showed an accuracy of relying in the bounds for at least 95% of the data [13], [14], [15].

Many researchers studied PMP including the discussions about the optimum number and location of the measured points. Narayanan et al. [16] studied the sample size effect on the evaluation of geometric errors. They established an algorithm that uses the asymptotic distribution of the sampled points to find the best sample size to model the geometric errors for a required accuracy. The accuracy is defined based on the range of errors that the CMM data has. They also assumed that the errors of the acquired data follow a normal distribution. Therefore, substituting the normal distribution of the errors in the probability range of errors lead to the required number of measurements for a specific range. They verified their method through actual measurement practice. The least squares of the error from the approximated range and actual range is between -1.5 and 3 which shows good convergence.

One of the very popular approaches in PMP is based on sequence algorithms. Kim and Raman [7] compared four different strategies for selecting points to evaluate flatness using CMMs. The strategies consisted of the Hammersley sequence sampling [8], the Halton–Zar-emba sequence sampling [9], the aligned systematic sampling, and the systematic random sampling. The results show that for high density point clouds, systematic random sampling has a lower discrepancy.

Ahn et al. [17] discussed the source of geometric deviations in machined parts. They modeled the volumetric errors caused by a 3-axis machine tool. They approached the problem by assessing the typical uncertainties caused by a 3 axis machine. Using rigid body kinematics model and homogeneous transformation, individual error components were analyzed. They considered spindle, tool, bed, head, saddle, work-piece, table and column as the sources of error. Therefore, by applying a beta distribution, the probability density function of positional error can be extracted. This model is used for approximating the uncertainty at an unknown position. They tested the algorithm for an arbitrary part but no comparison was provided.

An important aspect of research is to estimate the contour’s profile of the measured object. Qiu et al. [18] developed a method for modeling the contours. The developed algorithm uses line and circle segments to interpolate the contour profile over the acquired points. Although the method is simple and robust, it suffers accuracy over freeform profiles and is not practical for 3-Dimentional cases. Weber et al. [19] modeled the form error of straightness, flatness, circularity and cylindericity. Their approach aimed to improve the robustness of non-linear method where least square was used. They applied Taylor expansion on non-linear approximation methods to find a unified linearization approach. The final formula is then solved to find the range bounds. The method provides a definitive accuracy of at least 88% in the worst case.

The sample size has an important effect on the efficiency of sampling process. Capello and Semeraro [20], studied the effect of sample size and the position of the measured points on the evaluation of the substitute geometry for circular features. They used Least Square Fitting to estimate the position of the feature over the sampled measurements and then studied the radius and position of the circles fitted. The results show that the error decreases by a factor of 4 if the number of samples points are increased from 5 to 30 for a circular hole. They also proposed an algorithm for substitute geometry fitting for 2-Dimensional and 3-Dimensional space called Harmonic Fitting Method (HFM) [21], [22]. The method considers two geometries, a true one which depends on the characteristics of the feature under the study and another one which is defined based on the sampled points using Least Square Method. Then the algorithm minimizes the distance between the two substitute geometry. They considered well-known features that are produced during a manufacturing process; however, the study of freeform surfaces is missing from the research.

The third direction of research which is also the tackled in this paper is DZE. Reconstruction of the measured surface is an important aspect in DZE. Most of the common methods for surface reconstruction are based on mesh generation where the dimensions are in macro scale. One of these methods is developed by Hoppe et al. [23] with an algorithm based on a set of distance functions. This algorithm has the ability to determine the topological type of the surfaces. It can also be used in higher dimensions to produce volumes.

Amenta et al. [24] also implemented a powerful algorithm based on the medial axis transform. This algorithm represents the surface of an object by governing the properties of Voronoi diagrams.

One of the recent representations of the surfaces and curves is NURBS (Non-Uniform Rational B-Splines). It is a Spline-Based representation of a parametric surface which is useful to model the surfaces with a thorough description of the object surface [25]. One of the most important aspects of NURBS is the standard representation of curves and surfaces [26] and the ability to be governed by modern standards like OpenGL and IGES. In order to fit a NURBS surface to a chaotic and scattered point cloud, several approaches have been studied. This representation can be used to model the deviation zone of the manufactured surfaces. Park et al. [27], proposed an algorithm to construct NURBS surface model from scattered point cloud. The algorithm estimates an initial model employing K-means clustering of NURBS patches using hierarchical graph representation. The model is then used to construct the G1 continuous NURBS patch network of the whole object. The overall estimation error is less than 3% in the case studies.

In another study, Gregorski et al. [28], proposed an algorithm to reconstruct B-Spline surfaces over scattered points. The algorithm separates the point cloud to a quad tree-like data structure and uses the sub-trees by least square fitting of quadratic surfaces to the point cloud. In the final stage, the quadratic surfaces are smoothened to bi-cubic surfaces and combined together as a uniform B-spline. In a similar work, Eck and Hoppe [29] developed an algorithm to fit a set of B-Spline patches over sampled points. They use tensor product B-spline patches proposed by Peters [30] using the surface splines. The algorithm requires meshing over the point cloud as the first stage before the B-Spline fitting is performed. The algorithm has also the ability to consider the design tolerances provided by the user.

Aquino et al. [31], proposed a method for three-dimensional surface reconstruction using NURBS. The algorithm consists of two steps. At first, they reduce the number of acquired points by dividing the point cloud into uniform region where each has a center of mass. Then each region’s boundary is estimated using a B-Spline curve to fully define the cross-section. Finally, they use the boundary curves to define number of knots and control points for the surface which covers that region.

It is shown in the previous research that modeling Laplace equation over the geometric deviations on a manufactured surface can be used for evaluation of surface deviations. This method is tested and implemented on Additive Manufactured surfaces [32]. The inherent waviness of such surfaces led to the assumption that the surface follows properties of a Harmonic function which in fact is the solution to the Laplace equation. On the other hand, dealing with dense point clouds and numerous sample data, the numerical solution of the so called function is the appropriate approach. However, the approach is not implemented and verified for different manufactured surfaces with various patterns of geometric deviations. This paper presents implementation and discussion on validating the use of a finite difference solution to the Laplace equation to model the geometric deviations by studying various manufactured parts with different levels of geometric accuracy. The manufactured surfaces are measured with various types of coordinate measurement systems with different levels of uncertainty. Also a second approach using a metaheuristic solution is presented and the convergence of the resulting DZE by two methods are compared.