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Computer-Aided
  Inspection

   D. ElKott


BACKGROUND

SAMPLING
methodology
quadrilateral
surfs.
extension
case study
   ...
Computer-Aided
  Inspection

   D. ElKott


BACKGROUND

SAMPLING
methodology
quadrilateral
surfs.
extension
case study
ASS...
Computer-Aided
  Inspection

   D. ElKott
                                                             Background
        ...
Computer-Aided
  Inspection

   D. ElKott
                                                                                ...
approach                                                                      
                                           ...
Computer-Aided
  Inspection

   D. ElKott
                                                                                ...
ASSESSMENT            
                                                                                                   ...
Computer-Aided
  Inspection

   D. ElKott
                                                                          Backgr...
Computer-Aided
  Inspection

   D. ElKott


BACKGROUND

SAMPLING
methodology
quadrilateral
surfs.
extension
case study
ASS...
Computer-Aided
  Inspection

   D. ElKott
                                                                    The Methodol...
Computer-Aided
  Inspection

   D. ElKott
                                    The Problem, and Approach to Solution
BACKGR...
Computer-Aided
  Inspection

   D. ElKott
                                         Sampling Quadrilateral Surfaces
       ...
Computer-Aided
  Inspection

   D. ElKott
                                        Sampling Quadrilateral Surfaces
        ...
Computer-Aided
  Inspection

   D. ElKott
                                         Sampling Quadrilateral Surfaces
       ...
Computer-Aided
  Inspection

   D. ElKott
                                      Sampling Quadrilateral Surfaces
          ...
Computer-Aided
  Inspection

   D. ElKott
                                                      Extending the Work
       ...
Computer-Aided
  Inspection

   D. ElKott
                                                                 Extending the W...
Computer-Aided
  Inspection

   D. ElKott
                                                        Extending the Work
     ...
Computer-Aided
  Inspection

   D. ElKott
                                                        Extending the Work
     ...
Computer-Aided
  Inspection

   D. ElKott
                                                      Extending the Work
       ...
Computer-Aided
  Inspection

   D. ElKott
                                                     Extending the Work
        ...
Computer-Aided
  Inspection

   D. ElKott
                                                                                ...
Computer-Aided
  Inspection

   D. ElKott
                                                                                ...
Computer-Aided
  Inspection

   D. ElKott


BACKGROUND

SAMPLING
methodology
quadrilateral
surfs.
extension
case study
ASS...
Computer-Aided
  Inspection

   D. ElKott
                                The Problem and Approach to Solution
BACKGROUND
...
Computer-Aided
  Inspection

   D. ElKott
                                     Sampling Strategy Implementation
          ...
Computer-Aided
  Inspection

   D. ElKott
                                      Localization of CMM Measurement
          ...
Computer-Aided
  Inspection

   D. ElKott
                                   Localization of CMM Measurement
             ...
Computer-Aided
  Inspection

   D. ElKott
                                Localization of CMM Measurement
                ...
Computer-Aided
  Inspection

   D. ElKott
                                   Localization of CMM Measurement
             ...
Computer-Aided
  Inspection

   D. ElKott
                                    Constructing the Substitute Geometry
       ...
Computer-Aided
  Inspection

   D. ElKott
                                 Sampling Plan Implementation–An
BACKGROUND     ...
Computer-Aided
  Inspection

   D. ElKott
                                 Sampling Plan Implementation–An
BACKGROUND     ...
Computer-Aided
  Inspection

   D. ElKott
                                 Sampling Plan Implementation–An
BACKGROUND     ...
Computer-Aided
  Inspection

   D. ElKott
                                Sampling Plan Implementation–An
BACKGROUND      ...
Computer-Aided
  Inspection

   D. ElKott


BACKGROUND

SAMPLING
methodology
quadrilateral
surfs.
extension
case study
ASS...
Computer-Aided
  Inspection

   D. ElKott
                                                                     Conclusions...
Computer-Aided
  Inspection

   D. ElKott
                                                                     Conclusions...
Computer-Aided
  Inspection

   D. ElKott
                                                                                ...
Computer-Aided
  Inspection

   D. ElKott


CURVATURE
manufacture
measures
UNCERTAINTY

ERRORS

GEOMETRY
diff.geometry
BRep...
Computer-Aided
  Inspection

   D. ElKott
                                           Curvature effect on manufacturing
CURV...
Computer-Aided
  Inspection

   D. ElKott
                                                                     Curvature M...
Computer-Aided
  Inspection

   D. ElKott
                                                  Curvature Measures
CURVATURE
m...
Computer-Aided
  Inspection

   D. ElKott
                                               Measurement Uncertainty
CURVATURE...
Computer-Aided
  Inspection

   D. ElKott
                                                 Sources of Measurement Errors
C...
Computer-Aided
  Inspection

   D. ElKott
                                              Sources of Measurement Errors
CURV...
Computer-Aided
  Inspection

   D. ElKott
                                              Geometry of Curves and Surfaces
CU...
Computer-Aided
  Inspection

   D. ElKott
                               Nonmanifold Boundary Representation
CURVATURE
man...
Computer-Aided
  Inspection

   D. ElKott
                                          n-Sided surface patches
CURVATURE
manu...
Computer-Aided
  Inspection

   D. ElKott
                                          What’s With n-Sided Surfaces!
CURVATUR...
Computer-Aided
  Inspection

   D. ElKott
                                                Visual Continuity Measures
CURVA...
Computer-Aided
  Inspection

   D. ElKott
                  Trimmed Surface Substitute Geometry
CURVATURE
manufacture
meas...
Computer-Aided
  Inspection

   D. ElKott
                                            CMM Probe Scanning
CURVATURE
manufac...
Computer-Aided
  Inspection

   D. ElKott
                                   Initial Correspondence Points
CURVATURE
manuf...
Computer-Aided
  Inspection

   D. ElKott
                                  Localization of CMM Measurement
              ...
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Ph D Thesis Defense Presentation

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PhD thesis defence presentation.

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Ph D Thesis Defense Presentation

  1. 1. Computer-Aided Inspection D. ElKott BACKGROUND SAMPLING methodology quadrilateral surfs. extension case study CAD-Based Inspection of Sculptured Surfaces ASSESSMENT approach probe path localization sub. geometry Diaa F. ElKott demo CONTRIBUTIONS Department of Mechanical Engineering BIBLIOGRAPHY McMaster University November 3, 2006
  2. 2. Computer-Aided Inspection D. ElKott BACKGROUND SAMPLING methodology quadrilateral surfs. extension case study ASSESSMENT approach probe path localization I. Background sub. geometry demo CONTRIBUTIONS BIBLIOGRAPHY 2
  3. 3. Computer-Aided Inspection D. ElKott Background Sculptured vs Primitive Surfaces BACKGROUND SAMPLING methodology quadrilateral surfs. extension case study ASSESSMENT Part with primitive surfaces Part with sculptured surfaces approach probe path localization sub. geometry demo CONTRIBUTIONS BIBLIOGRAPHY 3
  4. 4. Computer-Aided Inspection D. ElKott Background The Manufacture of Sculptured Surfaces BACKGROUND SAMPLING methodology quadrilateral surfs. CAD/CAM/CAE extension case study ASSESSMENT
  5. 5. approach probe path localization sub. geometry demo CONTRIBUTIONS BIBLIOGRAPHY ! Challenge Algorithm errors (CAD, CAM, inspection, analysis.) Measurement uncertainty. 4 Hardware errors (CMM, machining centre.)
  6. 6. Computer-Aided Inspection D. ElKott Background The Manufacture of Sculptured Surfaces BACKGROUND SAMPLING methodology quadrilateral surfs. CAD/CAM/CAE extension case study clamp CMM probe
  7. 7. ASSESSMENT approach probe probe path (3) scanning path localization (1) sub. geometry clamp demo (2) CONTRIBUTIONS physical BIBLIOGRAPHY ! model Research Issues Limitations in the reported literature Sampling of sculptured surfaces. Assessment of the physical object geometry. Sampling strategy implementation. 5
  8. 8. Computer-Aided Inspection D. ElKott Background Objectives, and Scope of Research BACKGROUND SAMPLING methodology quadrilateral surfs. Objectives extension case study 1 Develop a CAD-based system to generate the sampling strategy of ASSESSMENT sculptured surfaces. approach probe path localization 2 Address the issues associated with the implementation of the sub. geometry sampling plan. demo CONTRIBUTIONS 3 Develop methods for the physical object assessment using the CMM BIBLIOGRAPHY measurement data. Scope of research Continuous probe scanning clamp CMM Answer the following questions: probe 1 Where to measure a model (3) probe scanning path with sculptured features? (1) clamp 2 How to assess the quality of (2) the physical object? physical model 6
  9. 9. Computer-Aided Inspection D. ElKott BACKGROUND SAMPLING methodology quadrilateral surfs. extension case study ASSESSMENT approach probe path localization II. Sampling of Sculptured Surfaces sub. geometry demo CONTRIBUTIONS BIBLIOGRAPHY 7
  10. 10. Computer-Aided Inspection D. ElKott The Methodology BACKGROUND SAMPLING Criterion methodology Isoparametric line sampling quadrilateral upper sampling plan surfs. tolerance bound extension x u , v case study isoparametric ASSESSMENT u=const. curves substitute approach geometry probe path localization CAD v =const. sub. geometry model demo sampling CONTRIBUTIONS X3 plan error BIBLIOGRAPHY CAD model X2 lower sampling plan X1 tolerance bound Challenges Shape complexity. Substitute geometry construction. Tools 8 C++, SMLibTM geometric modelling kernel.
  11. 11. Computer-Aided Inspection D. ElKott The Problem, and Approach to Solution BACKGROUND SAMPLING methodology quadrilateral The problem surfs. extension Find the sample line distribution that results in an acceptable error case study between the substitute geometry and the CAD model. ASSESSMENT approach probe path localization Approach to sampling sub. geometry demo NURBS parameters sample curves curvature analyses sample size CONTRIBUTIONS Geometry point-based rep. Preliminary substitute geom. Sampling Plan CAD model BIBLIOGRAPHY Processing Sampling Assessment max. no. of scanning curves sample curves sampling plan deviations sample size error tolerance matrix substitute geom. Modify NO Terminate Sampling Plan ? sample curves sample size Output YES substitute geometry deviations matrices Sampling Plan continuity checks 9
  12. 12. Computer-Aided Inspection D. ElKott Sampling Quadrilateral Surfaces Sampling Algorithms BACKGROUND SAMPLING methodology quadrilateral surfs. Automatic sampling extension case study 1 Preliminary sampling plan. ASSESSMENT approach 2 Assess plan → ∆. probe path localization 3 IF (!TERM) GOTO 4 ELSE END sub. geometry demo 4 Add sample at ∆max , GOTO 2. CONTRIBUTIONS BIBLIOGRAPHY Example: 15 lines, ∆max = 5µm 10
  13. 13. Computer-Aided Inspection D. ElKott Sampling Quadrilateral Surfaces Sampling Algorithms BACKGROUND SAMPLING methodology quadrilateral surfs. Curvature-based sampling extension case study 1 Assess surface → κ. ASSESSMENT approach 2 Preliminary sampling plan. probe path localization 3 WHILE (!TERM) GOTO ?? sub. geometry demo 4 Add sample at κmax . DO WHILE CONTRIBUTIONS BIBLIOGRAPHY Example: 23 lines, ∆max = 50µm 11
  14. 14. Computer-Aided Inspection D. ElKott Sampling Quadrilateral Surfaces Sampling Algorithms BACKGROUND SAMPLING methodology quadrilateral surfs. Isoplanar sampling extension case study 1 Assess surface. ASSESSMENT approach 2 Construct N isoplanar sections. probe path localization 3 Construct substitute geometry. sub. geometry demo 4 END. CONTRIBUTIONS BIBLIOGRAPHY Example: 23 lines, ∆max = 70µm 12
  15. 15. Computer-Aided Inspection D. ElKott Sampling Quadrilateral Surfaces Limitations BACKGROUND SAMPLING methodology quadrilateral surfs. extension Algorithm limitations case study ASSESSMENT Quadrilateral surface sampling algorithms cannot generate sampling plans approach for the following features: probe path localization Trimmed surfaces. sub. geometry demo n-sided surface patches. CONTRIBUTIONS BIBLIOGRAPHY Models composed of multiple surfaces. Research issues CAD model representation. Sampling algorithm. Substitute geometry construction algorithm. Sampling plan assessment. 13
  16. 16. Computer-Aided Inspection D. ElKott Extending the Work Trimmed Surfaces – The Building Block BACKGROUND SAMPLING methodology quadrilateral surfs. Point-based representation extension case study ASSESSMENT approach probe path localization sub. geometry demo CONTRIBUTIONS BIBLIOGRAPHY 14
  17. 17. Computer-Aided Inspection D. ElKott Extending the Work Trimmed Surfaces – The Building Block BACKGROUND SAMPLING methodology quadrilateral surfs. Point-based representation extension case study ASSESSMENT approach probe path localization sub. geometry demo CONTRIBUTIONS BIBLIOGRAPHY   ¡T Pi,j = ui vj Oij ∆κu ij ∆κv ij , i = 1, . . . , N u , j = 1, . . . , N v @ 1 for Si,j ∈ ST Oij = 15 0 for Si,j ∈ ST /
  18. 18. Computer-Aided Inspection D. ElKott Extending the Work Trimmed Surfaces – The Building Block BACKGROUND SAMPLING methodology quadrilateral surfs. Hybrid sampling extension case study 1 Criteria for ranking of isoparametric lines (dynamic!): ASSESSMENT The point occurrence, R1. approach The surface mean curvature, R2. probe path localization The deviation between the substitute geometry and CAD, R3. sub. geometry demo 2 A combined weight is calculated for each isoparametric curve: CONTRIBUTIONS @ BIBLIOGRAPHY (αR1 + βR2) / (α + β) → preliminary. R= (αR1 + βR2 + γR3) / (α + β + γ) → update. Demonstration 16
  19. 19. Computer-Aided Inspection D. ElKott Extending the Work Trimmed Surfaces – The Building Block BACKGROUND SAMPLING methodology quadrilateral surfs. Substitute geometry algorithm extension case study Requirement: Fit adjacent surface patches such that cross-boundary ASSESSMENT derivative continuity is maintained. approach probe path Algorithm: skinning of cross sectional curves with boundary localization sub. geometry constraints [1]. demo CONTRIBUTIONS BIBLIOGRAPHY Sampling plan assessment The sampling plan is assessed according to two criteria: 1 Deviation between the substitute geometry and the CAD model (automatic). 2 Visual continuity between adjacent surface patches (semi-automatic). 17
  20. 20. Computer-Aided Inspection D. ElKott Extending the Work Example – The Human Ear Model BACKGROUND SAMPLING methodology quadrilateral surfs. The model extension case study Model composed of two surfaces, sewed along their partition line. ASSESSMENT approach probe path localization sub. geometry demo CONTRIBUTIONS BIBLIOGRAPHY 18
  21. 21. Computer-Aided Inspection D. ElKott Extending the Work Example – The Human Ear Model BACKGROUND SAMPLING methodology quadrilateral surfs. u-Sampling plan and deviations extension case study Output of sampling algorithm when sampling both surfaces in the ASSESSMENT u-direction. approach probe path localization sub. geometry demo CONTRIBUTIONS BIBLIOGRAPHY 19
  22. 22. Computer-Aided Inspection D. ElKott Extending the Work Example – The Human Ear Model BACKGROUND SAMPLING methodology quadrilateral surfs. Visual continuity check extension case study VC 1 check to compare u−, and v − sampling plans. ASSESSMENT 1 approach probe path localization 0.9 sub. geometry VC1 continuity measure for substitute geometry demo 0.8 CONTRIBUTIONS 0.7 BIBLIOGRAPHY 0.6 0.5 0.4 0.3 0.2 0.1 20 u-sampling plan v-sampling plan 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 VC1 continuity measure for CAD model
  23. 23. Computer-Aided Inspection D. ElKott Extending the Work Example – The Human Ear Model BACKGROUND SAMPLING methodology quadrilateral surfs. Visual continuity check extension case study VC 2 check to compare u−, and v − sampling plans. ASSESSMENT 1 approach probe path localization 0.9 sub. geometry demo VC2 continuity measure for substitute geometry 0.8 CONTRIBUTIONS 0.7 BIBLIOGRAPHY 0.6 0.5 0.4 0.3 0.2 0.1 21 u-sampling plan v-sampling plan 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 VC2 continuity measure for CAD model
  24. 24. Computer-Aided Inspection D. ElKott BACKGROUND SAMPLING methodology quadrilateral surfs. extension case study ASSESSMENT approach probe path localization III. Physical Model Assessment sub. geometry demo CONTRIBUTIONS BIBLIOGRAPHY 22
  25. 25. Computer-Aided Inspection D. ElKott The Problem and Approach to Solution BACKGROUND SAMPLING methodology quadrilateral The problem surfs. extension case study Develop a framework for the implementation of the sampling strategy, and ASSESSMENT assessment of the physical object using the CMM measurement. approach probe path localization sub. geometry Approach to physical object assessment demo sampling hardware errors optimization algorithm CONTRIBUTIONS plan software errors objective function BIBLIOGRAPHY Sampling modified sampling plan CAD CMM measurement Data model Plan System data (filtered) Localization Refinement measurement physical data (filtered, model aligned) tolerances Deviations of Physical substitute Substitute fitting algorithm physical model Model Geometry fitting tolerances from CAD geometry Assessment Algorithm 23
  26. 26. Computer-Aided Inspection D. ElKott Sampling Strategy Implementation Probe Scanning Path BACKGROUND SAMPLING methodology quadrilateral The problem surfs. extension case study Calculate a nominal path for the continuous probe-scanning of the physical ASSESSMENT object. approach probe path localization sub. geometry Nominal probe scanning paths demo v CONTRIBUTIONS UV trimming valid UV BIBLIOGRAPHY curve probe path BRep model trimming curve edge avoidance distance modified probe intersection invalid UV scanning point Z probe path paths UV sample curve 24 u Y X
  27. 27. Computer-Aided Inspection D. ElKott Localization of CMM Measurement Problem and Solution Approach BACKGROUND SAMPLING methodology The problem quadrilateral surfs. extension Find the rigid body transformation that optimizes the alignment of the case study measurement data with the CAD model so as to minimize the influence of ASSESSMENT the CMM system, and machining errors on the accuracy of the substitute approach probe path geometry. localization sub. geometry demo CONTRIBUTIONS Why localization?! BIBLIOGRAPHY physical model nominal coordinate frame scanning Z point M Y M X M physical model actual ZCAD measurement CAD point CAD model 25 YCAD XCAD coordinate frame
  28. 28. Computer-Aided Inspection D. ElKott Localization of CMM Measurement Problem Formulation BACKGROUND SAMPLING methodology quadrilateral surfs. extension Problem formulation case study ASSESSMENT Align a cloud of scanned data points P with the BRep CAD model of the approach physical object such that: probe path localization sub. geometry P = {P0 , . . . , Pn−1 } demo Pi = {pi0 , . . . , pimi −1 } CONTRIBUTIONS BIBLIOGRAPHY using the objective function: H I € MinimizeFunction d n−1 1 Pm × dl2 e i ∀pl ∈P i=0 dl = GetShortestDistance(S, R × pl + T) H I H I H I xlnew xlold δx pnew l = d ylnew e = R(O, θx )×R(O, θy )×R(O, θz )× d ylold e + d δy e 26 zlnew zlold δz
  29. 29. Computer-Aided Inspection D. ElKott Localization of CMM Measurement Problem Formulation BACKGROUND SAMPLING methodology quadrilateral surfs. extension Problem and solution case study ASSESSMENT approach probe path localization sub. geometry demo CONTRIBUTIONS BIBLIOGRAPHY 27
  30. 30. Computer-Aided Inspection D. ElKott Localization of CMM Measurement Problem Formulation BACKGROUND SAMPLING methodology quadrilateral surfs. extension Challenge case study ASSESSMENT 1 Shape complexity. approach probe path 2 Data localization without prior knowledge of correspondence. localization sub. geometry 3 Calculation of initial conditions for optimization. demo CONTRIBUTIONS BIBLIOGRAPHY Tools 1 C++. 2 Gnu Scientific Library (GSL). 3 SMLibTM . 28
  31. 31. Computer-Aided Inspection D. ElKott Constructing the Substitute Geometry Problem BACKGROUND SAMPLING methodology The problem quadrilateral surfs. extension Use the aligned CMM measurement to contruct a substitute geometry for case study the physical object using the algorithms for surface skinning of ASSESSMENT cross-section curves. approach probe path localization sub. geometry Challenge demo CONTRIBUTIONS Construct substite geometry for trimmed cross-section curves. BIBLIOGRAPHY Approach: localized measurement points valid model points extracted from CAD model space substitute geometry points at model edges points at model edges ZCAD 29 YCAD trimmed region XCAD
  32. 32. Computer-Aided Inspection D. ElKott Sampling Plan Implementation–An BACKGROUND Example SAMPLING Trimmed Surface Sampling methodology quadrilateral surfs. extension case study ASSESSMENT approach Surface Model probe path localization sub. geometry demo CONTRIBUTIONS BIBLIOGRAPHY 30
  33. 33. Computer-Aided Inspection D. ElKott Sampling Plan Implementation–An BACKGROUND Example SAMPLING Trimmed Surface Sampling methodology quadrilateral surfs. extension case study ASSESSMENT approach Solid Model, and Physical Object probe path localization sub. geometry demo CONTRIBUTIONS BIBLIOGRAPHY 31
  34. 34. Computer-Aided Inspection D. ElKott Sampling Plan Implementation–An BACKGROUND Example SAMPLING Trimmed Surface Sampling methodology quadrilateral surfs. extension case study ASSESSMENT approach Sampling Plan–virtual probe path localization sub. geometry demo CONTRIBUTIONS BIBLIOGRAPHY 32
  35. 35. Computer-Aided Inspection D. ElKott Sampling Plan Implementation–An BACKGROUND Example SAMPLING Trimmed Surface Sampling methodology quadrilateral surfs. extension case study ASSESSMENT approach Measurement Data, Object Assessment Results probe path localization sub. geometry demo CONTRIBUTIONS BIBLIOGRAPHY 0.080 mm 0.030 mm 0.005 mm 0.000 mm 33
  36. 36. Computer-Aided Inspection D. ElKott BACKGROUND SAMPLING methodology quadrilateral surfs. extension case study ASSESSMENT approach probe path localization IV. Conclusions sub. geometry demo CONTRIBUTIONS BIBLIOGRAPHY 34
  37. 37. Computer-Aided Inspection D. ElKott Conclusions Contributions, Limitations, Recommendations BACKGROUND SAMPLING methodology quadrilateral surfs. extension case study ASSESSMENT approach Contributions probe path localization Research sub. geometry demo 1 New and generalized methods for sampling sculptured surfaces. CONTRIBUTIONS 2 A framework for physical object assessment. BIBLIOGRAPHY Development 1 BRep-based approach. 2 Addressing several shape complexity issues 3 Linking the inspection planning and CMM measurement plan implementation. 4 Automation of several functions. 35
  38. 38. Computer-Aided Inspection D. ElKott Conclusions Contributions, Limitations, Recommendations BACKGROUND SAMPLING methodology quadrilateral surfs. extension case study ASSESSMENT approach Limitations probe path localization 1 Sampling is limited to isoparametric curves. sub. geometry demo 2 Accessibility assumption. CONTRIBUTIONS 3 Data localization is limited to cloud WRT CAD. BIBLIOGRAPHY Recommendations 1 Use part-specific shape features to guide sampling. 2 Consider CAD models which are based on data clouds. 3 Extend the work on the construction of substitute geometry. 36
  39. 39. Computer-Aided Inspection D. ElKott Bibliography BACKGROUND SAMPLING L. Piegl and W. Tiller. methodology Cross-sectional design with boundary constraints. quadrilateral Engineering With Computers, 15:171–180, 1999. surfs. extension D. ElKott and S. Veldhuis. case study Isoparametric line sampling for the inspection planning of sculptured surfaces. Computer Aided Design, 37:189–200, 2005. ASSESSMENT approach D. ElKott and S. Veldhuis. probe path A CAD-based approach to CMM inspection of models with sculptured features. localization Accepted for Publication Engineering With Computers. sub. geometry demo J. Nelder and R. Mead. CONTRIBUTIONS A simplex method for function minimization. Computer Journal, 7:308–315, 1965. BIBLIOGRAPHY M. Galassi and J. Davies and J. Theiler and B. Gough and G. Jungman and M. Booth and F. Rossi. Gnu Scientific Library Reference Manual. December 2004. L. Piegl and W. Tiller. The N U RBSbook. Springer-Verlaag, 1997. 37
  40. 40. Computer-Aided Inspection D. ElKott CURVATURE manufacture measures UNCERTAINTY ERRORS GEOMETRY diff.geometry BRep n-sided surfs. continuity VII. Appendices vis.continuity sub.geometry SCANNING LOCALIZATION inititial corr. optimisation 38
  41. 41. Computer-Aided Inspection D. ElKott Curvature effect on manufacturing CURVATURE manufacture measures UNCERTAINTY ERRORS GEOMETRY Effect of surface curvature diff.geometry BRep Effect of Step Length Effect of Path Interval n-sided surfs. Overcut (τi ) continuity vis.continuity sub.geometry SCANNING LOCALIZATION λi inititial corr. CCi CCi+1 optimisation λi+1 ωi Cusp (η ) Undercut (τo ) CCi+2 39
  42. 42. Computer-Aided Inspection D. ElKott Curvature Measures CURVATURE manufacture measures UNCERTAINTY Surface vs. curve curvature ERRORS GEOMETRY Iso-curve diff.geometry Osculating Surface BRep Circle Normal n-sided surfs. continuity Iso-curve vis.continuity Principal sub.geometry Normal Isoparametric SCANNING Surface Curve LOCALIZATION inititial corr. optimisation Surface Minimum Principal Curvature Arc and Direction Surface Maximum Principal Curvature Arc and Direction Surface 40
  43. 43. Computer-Aided Inspection D. ElKott Curvature Measures CURVATURE manufacture measures UNCERTAINTY ERRORS GEOMETRY diff.geometry BRep Surface Curvature Measures ∆A, ∆K , ∆H n-sided surfs. continuity vis.continuity sub.geometry SCANNING LOCALIZATION inititial corr. optimisation 41
  44. 44. Computer-Aided Inspection D. ElKott Measurement Uncertainty CURVATURE manufacture measures UNCERTAINTY CMM measurement uncertainty ERRORS GEOMETRY A representation of all possible measurement errors in the CMM diff.geometry measurement. BRep n-sided surfs. continuity vis.continuity true point sub.geometry Z (unknown) SCANNING LOCALIZATION measured error inititial corr. point optimisation X Y 42
  45. 45. Computer-Aided Inspection D. ElKott Sources of Measurement Errors CURVATURE manufacture measures UNCERTAINTY ERRORS GEOMETRY Effect of probing direction diff.geometry BRep n-sided surfs. probing in surface probe probing in vertical probe normal direction direction continuity vis.continuity sub.geometry r r cp cp SCANNING n  n'  LOCALIZATION p p inititial corr. p' optimisation work work piece piece (a) Correct contact point, (b) Erroneous contact point, p = cp + r × n p = cp + r × n 43
  46. 46. Computer-Aided Inspection D. ElKott Sources of Measurement Errors CURVATURE manufacture measures UNCERTAINTY ERRORS GEOMETRY Effect of software diff.geometry BRep n-sided surfs. work work sample points piece continuity piece vis.continuity with measurement sub.geometry errors SCANNING C1 LOCALIZATION C2 inititial corr. optimisation work piece sampling C2 strategy #1 sampling strategy #2 C1 (g) Effect of fitting algorithm se- (h) Sampling strategy effect lection 44
  47. 47. Computer-Aided Inspection D. ElKott Geometry of Curves and Surfaces CURVATURE manufacture measures UNCERTAINTY ERRORS GEOMETRY osculating plane diff.geometry normal plane n BRep xu n-sided surfs. continuity vis.continuity b p xv sub.geometry x u , v SCANNING rectifying plane x ui , v j  tangent LOCALIZATION x s t plane inititial corr. optimisation X3 X3 X1 X2 X1 X2 (i) Moving trihedron of a space curve (j) Local frame, and tangent plane of a surface 45
  48. 48. Computer-Aided Inspection D. ElKott Nonmanifold Boundary Representation CURVATURE manufacture measures UNCERTAINTY BRep ERRORS GEOMETRY diff.geometry Region BRep n-sided surfs. continuity vis.continuity Shell sub.geometry SCANNING FaceUse Face Surface LOCALIZATION inititial corr. optimisation LoopUse Loop EdgeUse Edge Curve VertexUse Vertex 46 Adapted from the SMLibTM online documentation.
  49. 49. Computer-Aided Inspection D. ElKott n-Sided surface patches CURVATURE manufacture measures UNCERTAINTY ERRORS 6-sided surface patch GEOMETRY diff.geometry BRep n-sided surfs. continuity vis.continuity sub.geometry SCANNING LOCALIZATION inititial corr. optimisation 47
  50. 50. Computer-Aided Inspection D. ElKott What’s With n-Sided Surfaces! CURVATURE manufacture measures UNCERTAINTY ERRORS Surface Unit Normal Vectors Cross-Boundary Derivatives GEOMETRY Along Boundary Curves Along Boundary Curves diff.geometry BRep n-sided surfs. continuity vis.continuity sub.geometry SCANNING LOCALIZATION inititial corr. optimisation 48
  51. 51. Computer-Aided Inspection D. ElKott Visual Continuity Measures CURVATURE manufacture measures UNCERTAINTY ERRORS curvature GEOMETRY circle for 1 diff.geometry BRep  1j n-sided surfs. E 0 continuity n  1 n  j j vis.continuity sub.geometry curvature SCANNING E 0 r1 circle for j 0 LOCALIZATION S1 pj  1j pj 1 r1 inititial corr. S1 j optimisation S0 S0 (k) VC 1 measure (l) VC 2 measure 49
  52. 52. Computer-Aided Inspection D. ElKott Trimmed Surface Substitute Geometry CURVATURE manufacture measures UNCERTAINTY ERRORS GEOMETRY diff.geometry BRep n-sided surfs. continuity vis.continuity sub.geometry SCANNING LOCALIZATION inititial corr. optimisation 50
  53. 53. Computer-Aided Inspection D. ElKott CMM Probe Scanning CURVATURE manufacture measures UNCERTAINTY ERRORS GEOMETRY diff.geometry clamp CMM BRep probe n-sided surfs. continuity vis.continuity sub.geometry probe (3) scanning path SCANNING (1) LOCALIZATION clamp inititial corr. optimisation (2) physical model 51
  54. 54. Computer-Aided Inspection D. ElKott Initial Correspondence Points CURVATURE manufacture measures UNCERTAINTY ERRORS GEOMETRY curvature CMM 2, j circle for probe v  diff.geometry BRep n-sided surfs. Rp continuity nj  vis.continuity sub.geometry SCANNING S LOCALIZATION inititial corr. optimisation pj curvature 1, j circle for 52
  55. 55. Computer-Aided Inspection D. ElKott Localization of CMM Measurement Data Post Procesing CURVATURE manufacture measures UNCERTAINTY ERRORS GEOMETRY Post processing of localized data diff.geometry BRep localized n-sided surfs. measurement continuity data points vis.continuity model sub.geometry boundary excluded SCANNING points LOCALIZATION inititial corr. optimisation CAD coordinate model frame boundary ZCAD CAD YCAD model XCAD 53
  56. 56. Computer-Aided Inspection D. ElKott Nelder-Mead Simplex Algorithm Minimization of function of n-variables CURVATURE manufacture START measures Initial Pi UNCERTAINTY y i = f (Pi ), i = 0 , , n → Get l , h, P P * = (1 + α )P − αPh ERRORS y * = f (P * ) GEOMETRY diff.geometry No Yes No BRep y * yl y * yi ,i ≠ h y * yh Ph = P * n-sided surfs. continuity Yes Yes vis.continuity P ** = (1 + γ )P * − γP P ** = β Ph + (1 − β )P sub.geometry y ** = f (P ** ) No y ** = f (P ** ) SCANNING LOCALIZATION No Yes y ** y l y ** y h inititial corr. optimisation Yes No Pi + Pl Pi = , i = 0, ,n Ph = P ** Ph = P * Ph = P ** 2 No Minimum Reached Yes EXIT Reproduced from Nelder and Mead, 1965 [4] 54
  57. 57. Computer-Aided Inspection D. ElKott Nelder-Mead Simplex Algorithm Minimization of function of n-variables CURVATURE START manufacture Initial Pi measures y i = f (Pi ), i = 0 , , n → Get l , h, P P * = (1 + α )P − αPh UNCERTAINTY y * = f (P * ) ERRORS No Yes No y * yl y * yi ,i ≠ h y * yh Ph = P * GEOMETRY Yes Yes diff.geometry P ** = (1 + γ )P * − γP P ** = β Ph + (1 − β )P BRep y ** = f (P ** ) No y ** = f (P ** ) n-sided surfs. continuity y ** y l No y ** y h Yes vis.continuity Yes No sub.geometry Pi = Pi + Pl , i = 0, ,n Ph = P ** Ph = P * Ph = P ** 2 SCANNING LOCALIZATION No Minimum Reached inititial corr. Yes optimisation EXIT The simplex [5] P0 = (x0 , x1 , ..., xn ) P1 = (x0 + δ0 , x1 , ..., xn ) P2 = (x0 , x1 + δ1 , ..., xn ) ... ... Pn = (x0 , x1 , ..., xn + δn ) 55

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