*Profile Coordinate Metrology Based on Maximum Conformance to TolerancesFaculty of Engineering and Applied Science University of Ontario Institute of Technology Oshawa, Ontario, CanadaDr. Ahmad Barariahmad.barari@uoit.ca

CMSC - Coordinate Metrology Systems Conference (CMSC 2011), Phoenix, Arizona, 25 28 July 2011

Three Basic Computation Tasks in Coordinate Metrology

*Integration Inspection System: Current Research and Final Goal

G Nominal geometryG Actual geometryG Substitute geometryG* Optimum substitute geometrypi'G*Fitting ProcessGG'piiXZYSix-DOF rigid body transformationGeometric Deviations

Substitute GeometryObjective FunctionGeometric DeviationThe Best Substituted GeometryeiDesired Geometry in the Reference coordinate Systempipi*pi*: Corresponding pointpi: i th measured pointni: normal vector at the corresponding pointDG: Desired geometryT: Transformation matrixDeviation Zone Evaluation of a Single Geometric Feature

*Tolerance Zone & Residual Deviations Tolerance Zone Definition(ASME Y14.5)Upper tolerance limitLower tolerance limitNormal vectorGuGlG'pi'XZYGpi(u* ,v*)(pi, G) Pi(u* ,v*)

*A Drawback of Common Fitting Methods A Measured Point that Complies to the Tolerance ZoneAcceptReject

*Application: Over-Cut & Under-Cut in Closed Loop Machining G'pi'XZYG*pi(pi, G*) GCommon fitting methods are not suitable for closed-loop machining & inspection.

* Required Properties for the Fitting FunctionFitting to the tolerance zone and not to the nominal geometryFitting to eliminate the over-cut situation Fitting to minimize the under-cut by minimizing the residual deviations Residual Deviation FunctionMinimization ObjectiveFitting Function Maximum Conformance to Tolerance (MCT) function (p)

*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.

*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:

*General Form of Penalty Function (Juliff, 1993; Patton et al., 1995; Back et al., 1997)Penalty Function Distance metric function Penalty factorFor any point with the over cut condition the (pi',Gl) 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

*Adaptive Penalty FunctionError Model-Local two directional sinusoid waves

*Distribution of Geometric DeviationsError Model-Local two directional sinusoid waves MDZMCT=0 : standard normal density: half-normal density

*Penalty factor, C, controls the velocity of transition from the state of the standard normal to the target state of extremely skewedViolation of the Feasible Solution Area selecting a relatively small penalty factor (C=10).compensate the violation of the feasible solution by adoption of lower tolerance limitSolutions:

*Distribution of the Geometric Deviations (DGD)Problem 2: is the region with maximum deviation sampled?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 Surfaces Geometric CharacteristicsAssumption: Gradient of the deviations is a direct function of the proximity of the Surface points with a high confidence level.

*Example: Effect of Systematic Machining ErrorsKinematic Modeling of Generic Orthogonal Machine Tools Using Homogeneous Transformation Matrices

*NURBS Presentation of Machined SurfaceQuasistatic Linear OperatorJacobian Matrix of the Actual Machining Point

*Behavior of Systematic ErrorsA 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

*Geometric Deviations Resulting from Systematic ErrorsGeometric DeviationsHorizontal Machining Center Systematic Error VectorsVertical Machining Center

*Search-Guided Sampling Monitoring Continuity in Probability Density Function (PDF) of Geometric Deviations

*Iterative Search Hessian FunctionPositive Maximum Absolute HessianNegative Maximum Absolute Hessian

*Fitting Uncertainty Using the Search Method Error Models (magnification: 100):1-Quasistatic errors of a vertical machine tool2-One directional sinusoid wave 3-Two directional sinusoid waves 4-Local two directional sinusoid waves

*Stratified SamplingError Model-Local two directional sinusoid waves 144 Stratified Points64 Stratified Points64 Random Points

*Result of Search-Guided SamplingError Model-Local two directional sinusoid waves

*Estimation of Uncertainty-Results100 Times MiniMax Inspection Using Five Different Data Capturing Method (2000 Experiments)

*Plug-In Uncertainty Bootstrap EstimationPlug-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 functionThe 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 Plug-In Uncertainty-Bootstrap Results100 Bootstrap Replications of Inspection of Five Different Data Capturing Method (2000 Experiments)

*Distribution of the Geometric Deviations (DGD)Pragmatic Space

*Interpolation of Geometric DeviationsRecall:the variations of non-rigid transformation vectors of the machined point has a direct relationship with the distance of the nominal points.A Proximity ProblemVoronoi Diagram

Delaunay Triangle (ORourke, 1998; Okabe at el., 2000)

*Interpolation ProcedurePosition in the Parametric Space:A Location between Sites any location on the uv parametric plane belongs to an individual Delaunay triangle

*Case Study: DGD of a NURBS surfaceStamping Die of front door of a vehicle with the general dimensions of 1150mm1080mm35mm (Forth order uniform, non-periodic NURBS surface with 16 control points)

*Simulation of Machining (Vertical Machine Tool)PDF of Residual DeviationsInspection (Search procedure captures in 163 data points)Step 1Step 2

Mean of geometric deviation (mm)Standard deviation of geometricdeviationMaximum geometric deviation Minimum geometric deviationDeviation (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]

*Interpolation of DeviationsDevelopment of DRD Step 3

*InspectionMachiningApplication: Closed-Loop of Machining and Inspection

*Setup & Machining PhaseAlignmentSetupRoughingReference PatchFinishing CC-LinesFinished Part

*Inspection PhasePhysical MeasurementCylindrical FitDGDVirtual Data CapturingFinal Inspection

*Experiment #1:Flexible Knot LocationsBeforeAfter

*Experiment #2- First Degree NURB SurfaceControl NetSecond Degree NURBS SurfaceFirst Degree NURBS SurfaceCC-LinesUpper Tolerance=0.006 mmLower Tolerance=-0.007 mm

*Setup & Machining PhaseAlignmentSetupRoughingReference PatchFinishing CC-LinesFinished Part

*Inspection PhasePhysical MeasurementCylindrical FitDGDVirtual Data CapturingFinal Inspection

*Experiment #ResultsBeforeAfter

*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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