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1 Profile Coordinate Metrology Based on Maximum Conformance to Tolerances Faculty of Engineering and Applied Science University of Ontario Institute of Technology Oshawa, Ontario, Canada Dr. Ahmad Barari ahmad.barari@uoit.ca CMSC - Coordinate Metrology Systems Conference (CMSC 2011), Phoenix, Arizona, 25 – 28 July 2011

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II.Substitute Geometry Estimation (SGE) The Best Substituted Geometry εCiεCi eiei εTiεTi Desired Geometry in the Reference coordinate System pipi I.Point Measurement Planning (PMP) Substituted Geometry in the Reference coordinate System p i : i th Measured point riri Tolerance envelope III.Deviation Zone Evaluation (DZE) Probability Density Function of Geometric Deviations f(e) e Three Basic Computation Tasks in Coordinate Metrology

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3 Integration Inspection System: Current Research and Final Goal SGEPMP DZE c) Integrated computational tasks SGEPMP DZE b) Recent research on integration of tasks PMPSGEDZE a) Sequential tasks in traditional coordinate metrology SGEPMP DZE d) Integrated Inspection System Design & Manufact. Data

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G Nominal geometry G′ Actual geometry G″ Substitute geometry G* Optimum substitute geometry pi'pi' G*G* Fitting Process G″ p i ″ Δ(p i ′,G″) Δ G′ (G″) G G'G' pipi εiεi X Z Y Six-DOF rigid body transformation Geometric Deviations

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Substitute Geometry Objective Function Geometric Deviation The Best Substituted Geometry eiei Desired Geometry in the Reference coordinate System pipi pi*pi* p i *: Corresponding point p i : i th measured point n i : normal vector at the corresponding point D G : Desired geometry T: Transformation matrix Deviation Zone Evaluation of a Single Geometric Feature

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6 Tolerance Zone & Residual Deviations Tolerance Zone Definition (ASME Y14.5) Upper tolerance limit Lower tolerance limit Normal vector Gu”Gu” Gl”Gl” G'G' pi'pi' X Z Y G”G” p i ″(u*,v*) Δ(p i ′, G ” ) Residual Deviation P i (u *,v * )

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7 A Drawback of Common Fitting Methods A Measured Point that Complies to the Tolerance Zone p1'p1' p 1 ″ Gu″Gu″ Gl″Gl″ G″ Δ(p 5 ′,G″) p2'p2' p3'p3' p4'p4' p5'p5' p 2 ″ X Z Y Measured point Corresponding point p1'p1' p 1 ″ Gu″Gu″ Gl″Gl″ G″ Δ(p 4 ′,G″) p2'p2' p3'p3' p4'p4' p5'p5' p 2 ″ X Z Y Measured point Corresponding point Corresponding point in the previous tolerance zone Accept Reject

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8 Application: Over-Cut & Under-Cut in Closed Loop Machining G'G' pi'pi' X Z Y G*G* pi″pi″ Δ(p i ′, G * ) Over-Cut Under-Cut G Closed-Loop Machining Strategy: Correction of Under-Cut Regions Elimination of Over-Cut Regions Common fitting methods are not suitable for closed-loop machining & inspection.

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9 Required Properties for the Fitting Function Fitting to the tolerance zone and not to the nominal geometry Fitting to eliminate the over-cut situation Fitting to minimize the under-cut by minimizing the residual deviations Residual Deviation Function Minimization Objective Fitting Function G G'G' pi'pi' p i ″ Δ(p i ′,G l ″) Δ G′ (G″) GuGu GlGl G″G″ Gu″Gu″ Gl″Gl″ Δ(p i ′,G u ″) X Z Y G G'G' pi'pi' X Z Y GuGu GlGl Gu*Gu* Gl*Gl* G*G* Λ G′ (Gu * ) pi″pi″ Maximum Conformance to Tolerance (MCT) function ( p→∞ )

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10 Objectives in Closed-Loop Inspection and Machining Fitting criteria for Closed-Loop Machining & Inspection: Inspection Based on Machining: Machining Based on Inspection: To develop a fitting methodology to construct the substitute geometry that minimizes the required compensation operations but maximizes the compensation capability of the geometric deviations. To develop a method to determine number and location of the measured points based on the characteristics and properties of the actual machined surface, to reduce uncertainty of the process. To develop a compensation procedure based on the inspection results. The procedure should be capable to interpolate the compensation requirements between the measured points for the entire machined surface.

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11 Barrier ensures that a feasible solution never becomes infeasible. However, this objective function can by highly non-linear with discontinuities. In practice, the optimization may have an infeasible initial condition and stuck there. It is likely to be stuck in a feasible pocket with a local minimum. Modification of MCT Function Two drawbacks of this objective function are: Adding a penalty condition instead of the barrier condition to avoid straying too far from the feasible solutions. Utilizing a method for iterative data capturing to escape from the local minima by increasing the energy level. Solutions:

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12 General Form of Penalty Function (Juliff, 1993; Patton et al., 1995; Back et al., 1997) Penalty Function Distance metric function Penalty factor For any point with the over cut condition the Δ(p i ',G l ″) is a negative value that monotonically decreases when the point moves further from the tolerance zone. Therefore it can be a good choice for the distance metric function. Modified Residual Deviation Function Gradient Transformation of Substitute Geometry Penalized zone Residual Deviation Function MCT Function

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13 Adaptive Penalty Function Error Model-Local two directional sinusoid waves

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14 Distribution of Geometric Deviations Error Model-Local two directional sinusoid waves MDZ MCT Half- Normal Distribution Shape factor α=0 : standard normal density α→∞ : half-normal density

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15 Penalty factor, C, controls the velocity of transition from the state of the standard normal to the target state of extremely skewed Violation of the Feasible Solution Area Penalty Factor Very Sever Penalty Function Convergence problem Violation of the Feasible Solution Large Small selecting a relatively small penalty factor ( C=10 ). compensate the violation of the feasible solution by adoption of lower tolerance limit Solutions:

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16 Distribution of the Geometric Deviations (DGD) Problem 2: is the region with maximum deviation sampled? p'p' Problem 1: how much is the deviation of an unmeasured point? Approach: Search-Guided Sampling (Adoptive) Assumption: Distribution of deviations on the manufactured surface has a continues Probability Density Function. Approach: Using Surface’s Geometric Characteristics Assumption: Gradient of the deviations is a direct function of the proximity of the Surface points with a high confidence level.

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17 Example: Effect of Systematic Machining Errors Kinematic Modeling of Generic Orthogonal Machine Tools Using Homogeneous Transformation Matrices Homogeneous Transformation of X-Axis misalignment homogeneous matrix Motion homogeneous matrix Identity matrix Homogeneous Transformation of Workpiece Homogeneous Transformation of Tool Actual Machined Point Zero offset Homogeno us vector

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18 NURBS Presentation of Machined Surface Quasistatic Linear Operator Jacobian Matrix of the Actual Machining Point Explicit Form of Machined Geometry NURBS Presentation of the Machined Surface: NURBS piecewise rational basis functions Matrix of the surface control points

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19 Behavior of Systematic Errors A Typical Vertical Machining Center (Calibration using laser interferometer, electronic levels, optical squares) A Typical Horizontal Machining Center (Calibration using laser interferometer, electronic levels, optical squares) Nominal Geometry

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20 Geometric Deviations Resulting from Systematic Errors Geometric Deviations Horizontal Machining Center Systematic Error Vectors Vertical Machining Center

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21 Search-Guided Sampling Monitoring Continuity in Probability Density Function (PDF) of Geometric Deviations On-Line Estimation of PDF In contrast, the only assumption in this works is the continuity of the true probability density function. Probability Pr that a given deviation e i will fall in a region Rg Window Function Range of Windowing Parzen Windows Method (Parzen, 1962 ) Window width

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22 Iterative Search Hessian Function Positive Maximum Absolute Hessian Negative Maximum Absolute Hessian

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23 Fitting Uncertainty Using the Search Method Error Models (magnification: 100×): 1-Quasistatic errors of a vertical machine tool 2-One directional sinusoid wave 3-Two directional sinusoid waves 4-Local two directional sinusoid waves

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24 Stratified Sampling Error Model-Local two directional sinusoid waves 144 Stratified Points 64 Stratified Points 64 Random Points

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25 Result of Search-Guided Sampling Error Model-Local two directional sinusoid waves

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26 Estimation of Uncertainty-Results 100 Times MiniMax Inspection Using Five Different Data Capturing Method (2000 Experiments)

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27 Plug-In Uncertainty – Bootstrap Estimation Plug-in uncertainty comes from the fact that it is always unknown how much of the captured dataset is a good representation of the real distribution function The plug-in uncertainty is very much related to the probability of capturing critical points. A Bootstrap method is used to evaluate this probability. Estimation of Maximum Geometric Deviation P=(P1, P2, P3,…,Pn) and Θ =θ(P) Empirical probability density function of f̃=1/n on each of the observed values Bootstrap estimate of the standard error

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28 Estimation of Plug-In Uncertainty-Bootstrap Results 100 Bootstrap Replications of Inspection of Five Different Data Capturing Method (2000 Experiments)

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29 Distribution of the Geometric Deviations (DGD) Pragmatic Space X Z Y Cartesian Workspace Space p'p' Parametric Space u e v Mapping u*u* v*v* e s

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30 Interpolation of Geometric Deviations Recall: the variations of non-rigid transformation vectors of the machined point has a direct relationship with the distance of the nominal points. A Proximity Problem Voronoi Diagram Delaunay Triangle (O’Rourke, 1998; Okabe at el., 2000) Variation of Non-Rigid Body Transformations

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31 Interpolation Procedure Position in the Parametric Space: A Location between Sites any location on the uv parametric plane belongs to an individual Delaunay triangle u e v ekek sjsj sisi sksk oioi ojoj okok ejej eiei erer r

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32 Case Study: DGD of a NURBS surface Stamping Die of front door of a vehicle with the general dimensions of 1150mm×1080mm×35mm (Forth order uniform, non-periodic NURBS surface with 16 control points) Tolerance Specification

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33 Simulation of Machining (Vertical Machine Tool) PDF of Residual Deviations Inspection (Search procedure captures in 163 data points) Step 1 Step 2 Mean of geometric deviation (mm) Standard deviation of geometric deviation Maximum geometric deviationMinimum geometric deviation Deviation (mm) Parameters in the substitute surface [u v] Deviation (mm) Parameters in the substitute surface [u v] 0.0311260.0317160.083133[0.0201 0.0211]-0.040000[0.0229 0.9629]

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34 Interpolation of Deviations Development of DRD Step 3

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35 Inspection Machining Application: Closed-Loop of Machining and Inspection Brown & Sharpe CMM Renishaw PH9 Probe Head Horizontal Spindle Rotary Table Reported displacement accuracies: ±0.004mm Room Temperature: 20°C Whali CNC Machine Tool Five Axes Horizontal Spindle Rotary Table Reported Positional Accuracy: ±0.015mm Room Temperature: 20°C Material:Aluminum 6061 Coolant: Oil Tool: 12mm Ball-nose Depth of Cut: 0.25mm f t = 0.015(mm/min) and V=110 (m/min)

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36 Setup & Machining Phase Alignment Setup Roughing Reference Patch Finishing CC-Lines Finished Part

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37 Inspection Phase Physical Measurement Cylindrical Fit DGD Virtual Data Capturing Final Inspection

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38 Experiment #1:Flexible Knot Locations Before After

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39 Experiment #2- First Degree NURB Surface Control Net Second Degree NURBS Surface First Degree NURBS Surface CC-Lines Upper Tolerance=0.006 mm Lower Tolerance=-0.007 mm

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40 Setup & Machining Phase Alignment Setup Roughing Reference Patch Finishing CC-Lines Finished Part

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41 Inspection Phase Physical Measurement Cylindrical Fit DGD Virtual Data Capturing Final Inspection

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42 Experiment #Results Before After

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43 Conclusions A new fitting methodology for coordinate methodology is developed that maximizes conformance of the measured points to a given tolerance zone. Generating detailed information of the deviation zone on the measured surfaces should be based on the needs of the upstream processes such as compensating machining, finishing or reverse engineering. A methodology is developed to estimate distribution of the geometric deviations on a surface that is measured using discrete point sampling. Developed search method is an alternative approach in coordinate data capturing which significantly reduces plug-in uncertainty. Integration of computational tasks in coordinate metrology significantly reduces measurement uncertainties.

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