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REDUCING UNCERTAINTY OF COORDINAL METROLOGY BY EMPLOYING AN INTEGRATED COMPUTATIONAL SYSTEM Ahmad Barari 1 1 Faculty of Engineering and Applied Science, University of Ontario Institute of Technology, Oshawa, Canada, [email protected] Abstract: Integrated Inspection System (IIS) is an efficient solution to combat the inherent plug-in uncertainty in coordinate metrology. Using IIS, the computational tasks are performed concurrently or as the elements of a closed-loop structure. It is shown that implementing this concept reduces inspection uncertainty by developing opportunities for communication and adaptability between the three basic computational tasks in the typical coordinate metrology processes, i.e., planning for measurement of discrete point, estimating the best substitute geometry, and evaluation of the deviation zone. Key words: Coordinate Metrology, Minimum Deviation Zone, Point Measurement Planning, Inspection Uncertainty, Integrated Inspection System. 1. INTRODUCTION The purpose of this research is to develop a robust and reliable platform to perform coordinate metrology of surfaces. Although, surface coordinate metrology is a key process in many different engineering and scientific applications, unfortunately it is still influenced by several concerning sources of uncertainties which reduce the reliability of this assumed very precise process. The traditional coordinate metrology is completed by sequential performing of three analytical tasks of Point Measurement Planning (PMP), Substitute Geometry Estimation (SGE) and Deviation Zone Evaluation (DZE). In such systems, the inspection accuracy is subject to sorts of “plug-in uncertainty”, i.e. the uncertainty due to estimating some aspects of a probability distribution on the basis of sampling. Such sources of uncertainties can adversely affect the accuracy and reliability of the inspection. Recently, Integrated Inspection System (IIS) has been studied as an efficient solution to combat the inherent plug-in uncertainty in coordinate metrology. In an IIS, the computational tasks can be performed concurrently or as the elements of a closed-loop structure. In this way, the inspection uncertainty can be reduced using the developed opportunities for communication and adaptability. To benefit from potential advantages of IIS, sophisticated techniques for on-line generating the uncertainty information are required. Also, at each computational task, advanced methods are required to efficiently adapt the process based on the uncertainty information. In order to develop an efficient IIS, an estimation of geometric deviations by DZE needs to be dynamically utilized by PMP to acquire the most useful data set from the measuring part. Hence, gradual progress of DZE plays an important role in IIS implementation. Moreover as the number of measured points increases, estimation of the optimum substitute geometry by SGE becomes a challenging task due to highly nonlinearity of the resulting optimization problem. Reliable methodologies to gradually estimate the geometric deviation zone and adaptively change the PMP and SGE for both primitive geometric features and sculptured surfaces needs to be developed. This paper first presents the methodologies to establish a loop between SGE and PMP. Using a statistical pattern recognition technique the distribution of work-piece’s geometric errors is studied and new sample points are captured accordingly. The results show that implementation of this method can reduce the uncertainty in inspection of a sculptured surface up to 60%. Then it explains a method to model the distribution of geometric deviations. The method is developed by utilizing a geometric map of the measured deviations to a parametric surface, and uses the results to tessellate the work-piece surface based on the proximities to measurement locations and a Veronoi diagram. 2. BACKGROUND Recent studies show the inherent uncertainty resulting by computational tasks in coordinate metrology is significant. The results, which appear in Reference [1], show that the computational uncertainty for measuring and Auto-body profile can be as high as 300 µm. Comparing this result with the effect of other sources of uncertainty in coordinate metrology such as equipment, environment, work-piece and operator [2], it can be seen that computational uncertainty in the most cases can be even higher than the expanded uncertainty of all other sources. Under these conditions, the total expanded uncertainty of CMM inspection can be unfeasible either for quality or for process control purposes. The concept of integration of three basic computation tasks, i.e. Point Measurement Planning (PMP), Substitute Geometry Estimation (SGE) and Deviation Zone Evaluation (DZE) is discussed in Reference [3]. An integrated computational model can significantly solve the problem by providing an online share of information between the computational tasks. Figure 1 schematically shows the sequential system utilized in the common practices versus the recent research achievements toward developing a robust Integration Inspection System (IIS). The architecture and general requirements for an IIS in form of a closed-loop between engineering tasks are discussed in Reference [4]. (a) parametric area. The point chosen for each element is randomly selected within the element. Figure 2 shows samples and the evaluated minimum deviation zone which is 0.06766 mm. There is, however, no guarantee that the extreme deviation in these samples is the same extreme deviation in simulated geometry. (b) Fig. 1. (a) Typical sequential computational system in coordinate metrology; (b) Current research achievements toward developing a robust Integration Inspection System (IIS) 3. INTEGRATION OF PLANNING FOR DATA SAMPLING AND ESTIMATION OF THE OPTIMUM SUBSTITUTE GEOMETRY In the traditional coordinate metrology, two sequential tasks are typically the collection of the measurement data based on a sampling plan, and then data analysis to derive the necessary geometric information from the data-point set in form of a substitute geometry. However, there is always an uncertainty about the correctness of the results of the whole process because the estimation of a substitute geometric is only based on a group of sampled points and due to the plug-in nature of this statistical estimation the uncertainty can be closely related to the location and number of the sampled points. A closed-loop established between the computational tasks of planning for point measurement and estimation of the substitute geometry can significantly reduce the inspection uncertainty [3]. An iterative sampling procedure based on online estimation of the substitute geometry is described here. In order to evaluate the geometric deviations a pattern recognition technique called Parzen Windows [3][5] is employed to estimate the Probability Density Function (PDF) of the geometric deviations between the actual and the substitute surfaces. By studying the discontinuities in the PDF, the possible errors in the previous sampling plan are being detected. Based on the obtained results, the respective compensation scheme is constructed. It serves to modify the sampling plan for a new sampling phase. An auto-body B-spline surface created by 16 control points with an overall dimension of 900 [mm] × 600 [mm] × 200 [mm] was used to verify the performance of this closedloop. An initial sampling of the surface based on a stratified strategy with 16 measured points was performed. A stratified strategy for initial sampling is applied. The surface is divided uniformly into 16 elements with the same Fig. 2. Inspection based on the 16 stratified samples In the next step the iterative sampling presented in Reference [3] was utilized. After 154 iterations of the closed-loop, only 24 points were selected as the critical points to evaluate the geometric deviation of the surface. Figure 3 and 4 show the total 154 measured points, and the final set of 24 points were selected as the most important points for DZE. Fig. 3. Evaluation based on all of the 154 points sampled in iterative sampling Using the 24 selected samples the evaluated minimum deviation zone is observed as high as 0.10778 mm which is almost 60% larger than what was evaluated before based on the 16 initial samples. This shows that the points obtained by iterative sampling are better representatives of the actual surface. The total 154 samples also yield the same evaluation of the deviation zone identical to what was obtained from the 24 selected samples. However, the computation time for 24 points was almost 4.8 times shorter than the evaluation time based on the 154 points. Although the estimation of the substitute geometry need to be performed several times in the iterative sampling method, but since the variables associated with the previous substitute geometry as the initial condition for the optimization, new substitute geometry is found quickly. The total iterative sampling computational time in this casestudy was almost 2.2 times longer than what was required for the initial 16 stratified points. Fig. 4. Evaluation based on the 24 points selected by iterative sampling Figure 5 shows how the density function changes by iterative sampling. It can be seen that gradually over each iteration, the shape of density function is extended, takes a more symmetric shape and the density of samples within the average deviations is increased. The final density function, when the algorithm is terminated shows a characteristic behavior similar to a Gaussian distribution function with a significant density around the mean and symmetric form. The mean value is also very close to zero which validates the high accuracy of both the sample planning as well as performed fitting. 4. INTEGRATION OF EVALUATION OF GEOMETRIC DEVIATIONS AND PLANNING FOR DATA SAMPLING Coordinate metrology provides deviation of the discrete points on a measured surface, but typically it is not capable to explore any information of the surface regions between these measured points. The developed methodology presents a continuous function to estimate the Distribution of Geometric Deviations (DGD) on the entire inspected surface [6]. The methodology is developed based on estimation of parametric proximity of the surface points to the actual measured points. Utilizing a Voronoi Diagram and its corresponding Delaunay Triangulation, the procedure to estimate DGD is developed [7]. The resulting DGD model can be employed to estimate the deviation values at any unmeasured point of the inspected surface when a detailed understanding of the surface geometric deviations is required. Implementation of the developed methodology is described and case study for a typical industrial part is presented. This methodology can be used for closed-loop of inspection and manufacturing processes when a compensation scheme is available to compensate the manufacturing errors based on the DGD model. Estimation of the geometric deviation for an unmeasured point of the actual surface is required when the geometric deviation values for the measured points are available as the attributes of the sites in the parametric space. It is shown in Reference [8] that the distribution of geometric deviations caused by quasi-static manufacturing errors is continuous because it is a direct function of existence continuity in the nominal geometry. Therefore, the geometric deviations can be considered as a continuous variable with known values at sites in the parametric space. When using a proper PMP a sufficient number of points from the appropriate locations are measured, the sites are available to represent the continuity of geometric deviation function. Geometric deviation of an arbitrary unmeasured location can be estimated by studying its proximity to the known geometric deviations of the measured sites. The measured sites are selected using a proper search method during PMP activity using the iterative search method explained in the Section 3. These sites are estimated as the representatives of critical deviation regions on the actual geometry, which are selected based on the continuity in distribution of the geometric deviation function. Let S be a two-dimensional parametric plane of uv in the parametric space with n sites of {s1,s2,s3,…,sn}. The parametric plane can be partitioned by assigning every point in the plane to its nearest site. This problem is known in computation graphics as the Voronoi Diagrams [7]. All those points assigned to si form the Voronoi Region, V(si). V(si) consists of all the locations at least as close to si as to any other site: V si x : si x s j x Fig. 5. Changes of the probability density function in iterative sampling j i ,x S (1) where, the bracket indicates the parametric distance of its inside terms. If a location happens to be equally close to two or more sites, the location belongs to the boundary between these sites. The set of all the locations that have more than one nearest neighbor forms the Voronoi diagram. Figure 6 shows the Voronoi diagram of 30 sites on the parametric uv plane. A planar triangulation, called Delaunay Triangulation, is produced by a straight line connection of the adjacent Voronoi sites if no four sites are co-circular and using the above properties it can be shown that the drawn straight lines avoid crossing [7]. where, er A B e i C and, si(u) and si(v) are the u and v coordinates of the site si. The required geometric deviation is found by inserting the parametric coordinates of the arbitrary location of r in Equation 1, and A r v si v s j u si u ek -ei e j -ei sk u si u B r u si u s j v si v ek -ei e j -ei sk v si v C s j u si u sk v si v s j v si v sk u si u Fig. 6. Voronoi Diagram resulted by 30 sites on the parametric plane Figure 8 illustrates the front driver’s door of a vehicle with the general dimensions of 1150mm×1080mm×35mm. An accurate stamping die needs to be manufactured for mass production of this part. The main body of the door is designed by a uniform, non-periodic Non-Uniform Rational B-Spline (NURBS) surface with control net of 16 control points. The control points are defined identically in the u and v directions. Coordinates of the controls points are listed in Appendix. The degrees of NURBS surface in the u and v directions are identical and equal to three (forth order polynomials). Figure 7 illustrates the Delaunay triangulation resulted by the Voronoi diagram of Figure 6. Fig. 8. Design tolerances of the stamping die to produce the car door Fig. 7. Superimposing the Delaunay triangles on the Voronoi diagram of 30 sites on the parametric plane Suppose r is an arbitrary location on the parametric plane and three nodes of its corresponding triangle are si, sj and sk. The associated deviations for three sites of si, sj and sk are three known values of ei, ej and ek respectively. Therefore, corresponding to the three sites of si, sj and sk, three discrete positions can be found respectively in the parametric space. The desired geometric deviation value er, is estimated by planar interpolation of three known positions of oi, oj and ok. The plane resulted by three positions of oi, oj and ok is as follows: u si u v si v e -ei (2) s u s u s v s v e -e 0 j i j i j i sk u si u sk v si v ek -ei In order to simulate manufactured surface of the die, machining of the door is simulated by developing a model for quasi-static machining errors of a vertical machine tool. The methodology to develop this model is described at Reference [8]. The quasi-static machining errors are modeled as a linear operator that applies on points on nominal geometry to estimate the location of actual machined points considering interaction of all the machine tool quasi-static error sources. These points are used directly for the inspection process and estimation of DGD. The datum reference “A” identifies four constraints for the SGE task in inspection process, including: the translation along the X-axis and Y-axis, and rotation about the X-axis and Y-axis. As a result, the two degrees of freedom for SGE fitting are translation and rotation about the Z-axis. During the PMP activity 163 data points are selected using a search method [3]. Figure 9 shows the 163 measured points and the result of the SGE fitting. In Figure 9, points that conform to the tolerance zone are indicated by solid blue circles. Points that could not fit into the tolerance zone and show some residual deviations are indicated with two concentric red circles.. Maximum observed geometric deviation is 0.083133 mm. are illustrated by superimposing a color map on the NURBS geometry in Figure 11. Any dark blue segment has conformed to the tolerance zone with no residual deviation. Fig. 11. Color-map of residual deviations resulting from the machining of surface of die Fig. 9. Measured Point on the substitute geometry; measured point with deviations beyond the tolerance zone is indicated by two concentric red circles By transferring the corresponding parametric coordinates and the deviation of 163 measured points to the parametric space, first, the Voronoi diagram and Delaunay tessellation are calculated. Then, using the planar interpolation methods explained in the previous Section, the DGD is estimated. The resulting DGD is illustrated in Figure 10. Using the evaluated DGD, the geometric deviation for any point on the actual surface can be estimated. The presented color-map identifies all the regions of the geometry which do not comply with the desired tolerance zone. An additional manufacturing process such as compensating machining operation can be employed to correct these regions and compensate the effect of the machining quasi-static errors. 5. CONCLUSION Utilizing Integrated Inspection System (IIS) in coordinate metrology reduces the uncertainties of the inspection process which are inherent in the computational tasks. The customized IIS can be designed and developed based on the particular condition and requirements of the engineering applications that are supposed to utilize the inspection results. Developing comprehensive computational platforms for coordinate metrology that reduces the overall inspection uncertainty is the ultimate goal in this area. The results presented in this paper confirms possibility of developing IIS for variety of applications, for variety of parts with single-feature or multi-features and for variety of features including primitive geometric features and sculptured surfaces. AKNOWLEDMENTS The research support provided by the Natural Science and Engineering Research Council of Canada (NSERC) is greatly appreciated. Fig. 10. Distribution of Deviations on the parametric surface Using the described procedure the geometric deviations of 900 equi-parametric-points of the actual surface are calculated. These points are selected on a 30×30 parametric grid of the NURB surface. The resulting residual deviations REFERENCES [1] A. Barari, “Sources of Uncertainty in Coordinate Metrology of Automotive Body”, CD Proc. of 2nd CIRP International Conference on Assembly Tech. and Systems (CATS 2008), Toronto, Canada, 2008. [2] A. Weckenmann, M. Knauer, M., T. Killmaier, “Uncertainty of coordinate measurments on sheet-metal parts in the automotive industry”, Journal of Materials Processing Tech., v 115, pp. 9-13, 2001. [3] A. Barari, H. A. ElMaraghy, G. K. Knopf, “Search Guided sampling to reduce uncertainty of minimum zone estimation”, ASME Transaction- Journal of Computing and Information Science in Engineering, v 7:4, pp. 360-371, 2007. [4] A. Barari, R. Pop-Iliev, “Reducing Rigidity by Implementing Closed-Loop Engineering in Adaptable Design and Manufacturing Systems”, Journal of Manufacturing Systems, 28, pp. 47-54, 2009. [5] E. Parzen, “On estimation of a probability density function and mode”, The Annals of Mathematical Statistics, 33, pp. 065-76, 1962. [6] A. Barari, “Estimation of Detailed Deviation Zone in Coordinate Metrology”, 13th International Conference on Metrology and Properties of Engineering Surfaces, National Physical Laboratory, Middlesex, England, April 12-15, 2011. [7] A. Okabe, B. Boots, k. Sugihara, S.N. Chiu, “Spatial Tessellations; Concepts and Applications of Voronoi Diagram”, John Wiley & Sons Ltd, England, 2000. [8] A. Barari, H.A. ElMaraghy. W.A. ElMaraghy “Design For Machining of Sculptured Surfaces - A Computational platform”, ASME Transaction- Journal of Computing and Information Science in Engineering, 9-3, 13 pages, 2009.