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2nd International Summer School on Geometric Methods in Robotics, Mechanism Design and Manufacturing Research-Lecture 08 Workpiece Localization

2nd International Summer School on Geometric Methods in Robotics, Mechanism Design and Manufacturing Research-Lecture 08 Workpiece Localization

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  • 1. Chapter 8 Workpiece localization 1 Chapter Lecture Notes for Workpiece localization A Geometrical Introduction to Motivation Geometric Robotics and Manipulation algorithms for workpiece localization Richard Murray and Zexiang Li and Shankar S. Sastry Performance CRC Press and reliability analysis Sampling and Probe Radius Compensation Zexiang Li and Yuanqing Wu Implementation and ECE, Hong Kong University of Science & Technology applications Conclusion Reference July ,
  • 2. Chapter 8 Workpiece localization 2 Chapter 8 Workpiece localization Chapter Workpiece Motivation localization Motivation Geometric algorithms for workpiece localization Geometric algorithms for workpiece localization Performance and reliability analysis Performance and reliability Sampling and Probe Radius Compensation analysis Sampling and Probe Radius Implementation and applications Compensation Implementation and Conclusion applications Conclusion Reference Reference
  • 3. Chapter 8 Workpiece localization 8.1 Motivation 3 In 1952, MIT Servo Lab(G.S.Brown) developed, in collaboration with Parsons, the first CNC milling machine. Chapter Workpiece localization Motivation Geometric algorithms for workpiece localization Manual machine with an operator Performance and reliability analysis The MIT numerically controlled milling machine Sampling and Probe Radius Compensation Implementation and applications Giddings & Lewis 5-axis Skin Miller (1957) Conclusion Reference Kearney & Trecker NC Turning Fujitsu & Fraice NC Mill(1958)
  • 4. Chapter 8 Workpiece localization 8.1 Motivation 4 In 1959, APT was developed, followed by extensive Chapter activities in CAD Workpiece localization Motivation Geometric algorithms for workpiece localization Performance and reliability analysis Sampling and Probe Radius Compensation Implementation and applications Setup& Fixturing Conclusion Reference CAD CAM CNC (AutoCAD⋯) (UGII, MasterCam⋯) (Fanuc, Siemens⋯)
  • 5. Chapter 8 Workpiece localization 8.1 Motivation 5 ◻ Conventional Approaches: Chapter Workpiece localization Motivation Geometric algorithms for workpiece Jigs, xtures and hard gauges: ⇒ Expensive! localization Performance and reliability Manual Setup Time: ⇒ consuming & expensive analysis Sampling and Probe Radius Compensation Implementation and applications Conclusion Reference
  • 6. Chapter 8 Workpiece localization 8.1 A Computer-aided Setup System 6 CAD/CAM data Chapter Workpiece localization Motivation Arbitrarily place & fixture workpiece with Modify & optimize Geometric general purpose fixtures tool path with computed algorithms for workpiece (Robots and/or transformation localization programmable fixtures) Performance and reliability analysis Probe and measure point Sampling and data from the CNC Machine Probe Radius Compensation workpiece surfaces Implementation and applications Conclusion Compute the location & orientation of Reference the workpiece
  • 7. Chapter 8 Workpiece localization 8.2 The Problem 7 ◻ Possible Geometries: Chapter Workpiece localization z Motivation y x Geometric algorithms for z workpiece x y localization Performance and reliability analysis Sampling and Probe Radius Compensation (a)Regular Workpiece Regular localization Implementation (b)Symmetry Symmetric localization and applications (c)Partially machined Hybrid localization/envelopment Conclusion (d)Raw stock Envelopment Reference
  • 8. Chapter 8 Workpiece localization 8.2 The Problem 8 (a) Regular workpiece and the Euclidean group SE(3): Chapter Workpiece ◻ Rotational Motion: × SO( ) = R ∈ R RT R = I, detR = localization Motivation Geometric × so( ) = ω ∈ R ωT = −ω ≅ R algorithms for workpiece ˆ ˆ ˆ localization z Performance −ω and reliability ω ω= −ω analysis ˆ ω −ω Lie Algebra Sampling and Probe Radius ω of y SO( ) Compensation x z ˆ ω Implementation Exp: and so( ) → SO( ) ∶ ω → eω = R applications ˆ Conclusion ˆ ˆ eω y ω ∈ R Expomential coordinates of R Reference x
  • 9. Chapter 8 Workpiece localization 8.2 The Problem 9 ◻ General rigid motion : Chapter SE( ) = (p, R) p ∈ R , R ∈ SO( ) Workpiece localization z Motivation Geometric g= R p ∈ SE( ) ∶ Euclidean group of R y z algorithms for workpiece x localization y Performance x × se( ) = ∈R ω, v ∈ R ∶ Lie Alegebra ofSE( ) and reliability analysis ˆ ω v Sampling and Probe Radius ξ= ξ= Compensation ˆ ˆ ω v v Implementation ω and applications Exp: se( ) → SE( ) ∶ ξ → eξ Conclusion ˆ ˆ Reference :Screw motion
  • 10. Chapter 8 Workpiece localization 8.2 The Problem 10 (b) Symmetric workpiece and homogeneous space : Chapter Workpiece localization ◻ Symmetry Subgroups: Motivation A Cylinder: Geometric algorithms for G = eλ e λ ,λ ∈R workpiece ˆ z localization λe z Performance and reliability y x y analysis x Sampling and Probe Radius A Plane: Compensation Implementation and G = eλ ˆ e λ e +λ e λi ∈ R z z applications Conclusion y x y Reference x
  • 11. Chapter 8 Workpiece localization 8.2 The Problem 11 ◻ Configuration Space: Chapter SE( ) G = {gG g ∈ SE( )} G Workpiece localization e gG Motivation − g g ∼ g iff g ⋅ g ∈G Geometric algorithms for workpiece localization Performance and reliability Elements ofSE( ) G ∶ [g], gG or g analysis Sampling and Proposition 1 SE( ) G is a differentiable manifold of dimension Probe Radius Compensation Implementation dim(SE( )) − dim(G ), with a transitive action and applications µ ∶ SE( ) × SE( ) G → SE( ) G Conclusion Reference (h, gG ) → hgG
  • 12. Chapter 8 Workpiece localization 8.2 The Problem 12 ◻ Canonical Coordinates: g ∶ Lie algebra ofG Chapter M ⊕ g = se( ) Workpiece gG localization Motivation g Geometric Define: algorithms for Exp: M ⊕ g → SE( ) workpiece localization g (m, h) → em ˙h Performance m ˆ ˆ and reliability analysis ˆ ˆ e Sampling and Let( ξ , ⋯, ξr ) be a basis of M , ˆ ˆ and m = y ξ + ⋯ + yr ξr Probe Radius exp log Compensation ˆ ˆ I SO( ) Implementation Ψ ∶ SE( ) G → Rr , g ↦ (y , ⋯, yr ) and applications ˜ gG Conclusion Reference is well defined, and provide a canonical coor- dinate system for SE( ) G
  • 13. Chapter 8 Workpiece localization 8.2 Problem formulation 13 Regular Localization: Data:{yi }i= ⋯n Find g ∈ SE( ), xi ∈ Si Chapter Workpiece z z n min ε(g, x , . . . , xn ) = localization yi − gxi y Motivation i= x y x Geometric algorithms for workpiece Symmetric Localization: Find g ∈ SE( ) G s.t. localization Performance and reliability z z n min ε(g, x , . . . , xn ) = analysis Sampling and yi − gxi x y x y Probe Radius i= Compensation Implementation and Hybrid Localization/Envelopment z applications Problem: δi {yi }i= ⋯n ∶ Finished surface with G Conclusion x y yi {zi }i= ⋯m ∶ Unmachined surface Reference
  • 14. Chapter 8 Workpiece localization 8.2 Problem formulation 14 Find g ∈ SE( ) G , xi ∈ Si s.t g − zj ni zj Chapter z δi n min εl (g) = < g − yi − xi , ni > Workpiece ωi localization x y yi Motivation i= Geometric Let g(λ) = g G (λ) algorithms for workpiece localization Performance and reliability Find g(λ) ∈ SE( ), ωj ∈ Sj ,s.t. ¯ analysis n min εl = < g − (λ)zj − ωj , nj > Sampling and Probe Radius Compensation j= Implementation and applications and Conclusion < g − (λ)zj − ωj , nj >≥ δj , j = , ⋯, m Reference
  • 15. Chapter 8 Workpiece localization 8.2 Analytic results 15 ε(g, x , ⋯, xn ) = ∑n yi − gxi i= Chapter Workpiece § Define: x = n ∑n xi ; y = n ∑n yi ; xi′ = xi − x; yi′ = yi − y localization Motivation ¯ i= ¯ i= ¯ ¯ W = ∑i yi′ xi T = UΣV T (SVD) Geometric ′ algorithms for workpiece σ Σ= localization σ Performance and reliability σ analysis Theorem 1 (): If Rank(W) = (i.e.n ≥ ), Sampling and Probe Radius R∗ = VU T Compensation ∃!(R∗ , p∗ ) minimize ε(⋅, x , ⋯, xn ) and p∗ = x − R∗ y Implementation and ¯ ¯ applications Conclusion ε∗ = xi′ + yi′ − σi Reference i i i
  • 16. Chapter 8 Workpiece localization 8.2 Analytic results 16 Proof : ε(R, p, ⋅) = ∑ gyi − xi Chapter Workpiece = n R¯ + p − x + ∑i Ryi′ − xi′ localization Motivation y ¯ Geometric ⇒ p∗ = x − R∗ y ¯ ¯ algorithms for ε(R) = ∑i Ryi′ − xi′ workpiece = ∑i ( yi′ + xi′ ) − ∑i < Ryi′ , xi′ > localization Performance and reliability analysis = a − tr(RW) a Sampling and Probe Radius ˆ ω Compensation ˆ ωR Implementation ω ω= and −ω e applications ˆ ω −ω ωR ∈ TR SO( ) −ω ω R Conclusion ˆ Reference
  • 17. Chapter 8 Workpiece localization 8.2 Analytic results 17 < dεR , ωR >= t= ε(e R) = − tr(ωRW) = , ∀ω d ˆ tω ˆ ˆ Chapter dt ⇒ RW Symmetric Workpiece localization Motivation Let RW = S Geometric algorithms for workpiece ⇒ S = WT ⋅ W localization Performance S = (W T ⋅ W) = V and reliability ±σ analysis ±σ VT Sampling and ±σ Probe Radius Compensation Implementation ε∗ (⋅) = ( yi′ + xi′ ) − tr(W T W) ⇒ R∗ = VU T p∗ = x − R∗ y and applications i ¯ ¯ Conclusion Reference ◻
  • 18. Chapter 8 Workpiece localization 8.2 Analytic results 18 Proposition 2 A necessary condition for xi∗ , i = , ⋯, n, to minimize Chapter ε(R, p, ⋅)is that (xi = Ψi (ui , vi )) Workpiece localization < xi′ − (p + Ryi ), Ψui >= Motivation (B) i = , ⋯, n < xi′ − (p + Ryi ), Ψvi >= Geometric algorithms for workpiece where Ψi ∶ R → R , parametric equation of Si , and localization Performance and reliability ε(R, p) = Ryi + p − xi∗ = < Ryi + p − xi∗ , n > analysis Sampling and ¯ Probe Radius gyi i i Compensation ni < gyi − xi , n > Implementation gyi − xi and λni = gyi − xi applications Conclusion xi Reference
  • 19. Chapter 8 Workpiece localization 8.2 Analytic results 19 ◻ Localization Algorithms: Chapter (g , xi ) → (g ′ , xi′ ) →⋯ (g ∗ , x∗ ) i Workpiece localization (B) (B) Motivation (1) Variational Algorithm: Geometric (2) Tangent Algorithm: R = VU algorithms for T p = x − R¯ gk+ = e ξ ⋅ gk ξ= workpiece ˆ localization ¯ y ˆ ˆ ω v ≈ (I + ξ)gk Performance and reliability (3) Hong-Tan Algorithm: ˆ analysis Sampling and Probe Radius gk+ = e ξ ⋅ gk ˆ ω, v ∈ R Find ξ = Compensation v by minim. Implementation and Find ξ = v ω by minim. ω ε(ξ) = (I + ξ)gk yi − xi applications ˆ Conclusion ε(ξ) = < (I + ξ)gk yi − xi , ni > ˆ i i A⋅ ξ =b Reference A⋅ ξ =b ¯ ¯
  • 20. Chapter 8 Workpiece localization 8.2 Analytic results 20 Algorithm 1: Algorithm( Alternating Variable Method ) Chapter Workpiece Input Y = {yi }n , yi ∈ Si i localization Step 0 (a)Set k=0 Motivation (b)Initialize g Geometric (c)Compute yi = (g )− yi algorithms for (d)Compute xi (e)Compute ε = ε(g , x ) workpiece localization Performance and reliability (f)k=k+1 analysis Step 1 (a)Newton’s algorithm for xik Sampling and (b)Compute g k using (xik , g k− ) (c)Compute yik = (g k )− y Probe Radius Compensation Implementation (d)Compute εk = ε(g k , xk ) (e)If( − εk εk− ) < δ exit,else and (f)Set k = k + , return to Step1 (a) applications Conclusion Reference
  • 21. Chapter 8 Workpiece localization 8.2 Performance evaluation 21 Algorithms Chapter Variational Algorithm Workpiece localization ICP Algorithm Motivation Geometric Tangent Algorithm algorithms for workpiece Meng’s Algorithm localization Performance Hong-Tan Algorithm and reliability analysis Sampling and Performance Criteria Probe Radius Compensation Robustness Implementation and Accuracy applications Conclusion E ciency Reference
  • 22. Chapter 8 Workpiece localization 8.2 Performance evaluation 22 Chapter Workpiece localization Motivation Geometric algorithms for workpiece localization Performance and reliability analysis Sampling and Probe Radius Compensation Implementation Regions of convergence in terms of the maximal orientation errors for and applications each of the algorithms Conclusion Reference
  • 23. Chapter 8 Workpiece localization 8.2 Performance evaluation 23 Chapter Workpiece localization Motivation Geometric algorithms for workpiece localization Performance and reliability analysis Sampling and Probe Radius Compensation Implementation Accuracy of estimation achieved by each of the algorithms as a and applications function of the number of measurement points Conclusion Reference
  • 24. Chapter 8 Workpiece localization 8.2 Performance evaluation 24 Chapter Workpiece localization Motivation Geometric algorithms for workpiece localization Performance and reliability analysis Sampling and Computational efficiency by each of the algorithms as a function of Probe Radius Compensation the number of measurement points Implementation and Summary: applications Algorithms Robustness Accuracy Efficiency Conclusion Hong-Tan Good(− ∼ ○ ) highest highest Reference Variational Better(− ∼ ○ ) high high Tangent Best(− ∼ ○ ) high high
  • 25. Chapter 8 Workpiece localization 8.2 Symmetric localization 25 Find g ∈ SE( ) G , xi ∈ F to Chapter n minimize ε(g, x , ⋯, xn ) = yi −gxi Workpiece localization Motivation i= Geometric algorithms for workpiece Choose z z localization Performance and reliability M ⊕ g = se( ) x y x y M = span{ξ , ⋯, ξk } analysis Sampling and Probe Radius Compensation Algorithm 2: Algorithm:(Symmetric Localization) Implementation and Input: (a)Measurement data{yi }i= ,⋯,n applications (b)CAD description of F Conclusion Output Optimal solution g ∗ ∈ SE( ) G , xi ∈ F Reference
  • 26. Chapter 8 Workpiece localization 8.2 Symmetric localization 26 Algorithm 3: Chapter Input Y = {yi }n , yi ∈ Si i Workpiece localization Step 0 (a)Set k=0 Motivation (b)Initialize g Geometric (c)Solve for xi , i = , ⋯, n (d)Calculate ε = ∑i yi − gi xi algorithms for workpiece localization Performance Let gk+ = em gk , m ∈ Adgk (M ) ˆ ˆ and reliability Solve for m by minimizaing ˆ (a) ε(m) = ∑i yi − gk+ xik analysis Step 1 ˆ or ε(m) = ∑i < gk+ yi − xik , nk > − Sampling and Probe Radius ˆ i Adg k (g ) Adg k (M ) Compensation k+ Implementation (b)Solve for xi (c)Calculate εk+ (d) go to εk+ εk ) > ε,Set k = k + , If( − and applications Conclusion step1(a).Else exit. Reference
  • 27. Chapter 8 Workpiece localization 8.2 Symmetric localization 27 Example: A plane in R Chapter x(u, v) = ue + ve Workpiece G = λ e +λ e λi ∈ R localization Motivation eλ ˆ e Geometric g = span{ ξ , ξ , ξ } M = span{ ξ , ξ , ξ } ˆ ˆ ˆ ˆ ˆ ˆ algorithms for workpiece localization gwm Performance . . − . . computed solution R = − . and reliability . . analysis . . Sampling and q =[ − . . . ]T . . − . . R = Probe Radius SVD − . Compensation . . . . Implementation solution q =[ . . . ]T and . − . . applications Exact R = . . − . − . . . q =[ . . ]T Conclusion transform . Reference Simulation results for a plane inR
  • 28. Chapter 8 Workpiece localization 8.2 Symmetric localization 28 ◻ Performance Evaluation: Chapter Workpiece localization Motivation Robustness with re- Geometric spect to initial condi- algorithms for workpiece tions localization Performance and reliability analysis Sampling and Probe Radius Compensation Implementation and Efficiency compari- applications son Conclusion Reference
  • 29. Chapter 8 Workpiece localization 8.2 Discrete symmetry 29 Composite Feature: z GA = {em ξ +m ξ +m ξ m , m , m ∈ R} Chapter Workpiece localization y x C +m ξ +m ξ Motivation GB = {em ξ m , m , m ∈ R} B A GC = {em ξ +m ξ +m ξ m , m , m ∈ R} Geometric algorithms for workpiece z ⇒ GABC = GA ∩ GB ∩ Gc = I localization Performance Q:SE( ) GABC = SE( ) a unique solution? and reliability analysis y x Sampling and z z Probe Radius Compensation z Implementation y x y x and applications y x Conclusion Reference Correct Solution
  • 30. Chapter 8 Workpiece localization 8.2 Discrete symmetry 30 Plane: G = SE( ) × D D = { , − } z Chapter Workpiece Identify configurations differing byG y localization Motivation Cube: GABC = GA ∩ GB ∩ GC = I, eπ ξ , eπ ξ , eπ ξ Geometric ˆ ˆ ˆ algorithms for workpiece x localization Performance Solution: z and reliability analysis Filter out solutions with deviating home point Sampling and y x C Remark: Aξ = b, ξ = ω Probe Radius Compensation v Rank(A) = ⇒ Regular localization Implementation B A Ker(A) = Lie algebra ofG and applications Ker(A) = m Conclusion ⇒ ξ = A+ b Reference
  • 31. Chapter 8 Workpiece localization 8.2 The hybrid algorithm 31 Chapter {yi }n → g ∈ SE( ) G Workpiece localization FSL i= Motivation Geometric Algorithm 4: Algorithm:(The Envelopment Algorithm) (a)Meas. data {zi }m algorithms for workpiece Input (b)CAD model and a basis (η , ⋯, ηr )for g localization i= ˆ ˆ Performance (c)g ∈ SE( ) G from the FSL algorithm Optimal Solution g ∗ ∈ SE( ) and reliability analysis Output Sampling and Step 0 (a)Set k=0 and g = g (b)Compute ωi and ni ,i = , ⋯, m Probe Radius Compensation Implementation (c)Calculate εe = ∑i < (gi )− zi − ωi , ni > and applications Conclusion Reference
  • 32. Chapter 8 Workpiece localization 8.2 The hybrid algorithm 32 Chapter Workpiece localization Algorithm 5: Step 1 (a)Let g k+ = g k eλ , λ ∈ g , and solve for λ ∈ Rr Motivation ˆ ˆ Geometric algorithms for (b)Solve for ωk+ and nk+ , i = , ⋯, m i i workpiece (c) Calculate εk+ (d) If ( − εk+ εe ) < ε localization e k and < (g ) zi − ωk+ , nk+ >≥ δi , k+ − Performance e and reliability then report the solution g ∗ = g k+ ; analysis i i Sampling and else set k=k+1 (e)If k ≤ K ,then go to step1(a); else, exit Probe Radius Compensation Implementation and applications Conclusion Reference
  • 33. Chapter 8 Workpiece localization 8.2 The hybrid algorithm 33 Chapter Example: 1 Workpiece localization Motivation S ∶ Finished surface Geometric algorithms for workpiece localization G = {e(λ ξ +λ ξ +λ ξ ) λi ∈ R} ˆ ˆ ˆ Performance and reliability M = span{ ξ , ξ , ξ } ˆ ˆ ˆ analysis Sampling and S , S , S ∶ Un nished surface Probe Radius Compensation ¯ ¯ ¯ Implementation and applications Conclusion Reference
  • 34. Chapter 8 Workpiece localization 8.2 The hybrid algorithm 34 Example: 2 Chapter Workpiece localization Motivation S , S : Finished surface G = {eλξ λ ∈ R} Geometric algorithms for M = {ξ , ξ , ξ , ξ , ξ } workpiece localization ˆ ˆ ˆ ˆ ˆ Performance and reliability ¯ S : Unfinished surface analysis Sampling and Probe Radius Compensation Implementation and applications Conclusion Reference
  • 35. Chapter 8 Workpiece localization 8.3 Reliability Analysis 35 Q:With measurement errors and a finite no. of sampling points, Chapter Workpiece how reliable is the computed solution? localization Motivation Example: Orientation: Geometric algorithms for workpiece localization ∆θ ∝ d Performance and reliability d d analysis Not too reliable More reliable Sampling and Translation: Probe Radius Compensation Implementation error ∝ sinΦ and applications d d Conclusion Not reliable in x-direction More reliable Reference
  • 36. Chapter 8 Workpiece localization 8.3 Reliability Analysis 36 {yi }n → g ∗ = (R∗ , p∗ ), xi∗ , ε∗ , → ga = (Ra , pa ) Estimate of i= Chapter n n ε∗ = < g ∗ yi − xi∗ , n∗ > ≤ εa = < ga yi − xi∗ , ni∗ > Workpiece localization i Motivation i= i= Geometric algorithms for Assume:εa = ε + ε∗ and workpiece < g ∗ yi − xi∗ , n∗ > , i = , ⋯, n localization Performance i and reliability < ga yi − xi∗ , n∗ > , i = , ⋯, n analysis Sampling and i Probe Radius Compensation are normally distributed, with variance ε∗ &εarespectively. Implementation ⇒F= ∶ F − distribution l = n − (dof) and εa applications ε∗ Let Fε (l, l)be critical value at the ε-level corresponding to Conclusion Reference dof (l, l)
  • 37. Chapter 8 Workpiece localization 8.3 Reliability Analysis 37 Chapter Workpiece P(F > Fε(l,l) ) = ε localization or P(F < Fε(l,l) ) = − ε Motivation Geometric The probability that F = ε∗ε+εl < Fε(l,l) is equal to − ε. algorithms for workpiece localization ∗ Performance Translational Reliability Letδp = (δpx , δpy , δpz ) and δ = δpx + δpx + δpx and reliability analysis ⎡ nT ⎤ Sampling and ⎢ ⎥ Probe Radius εp = δp [ n ⋯ nn ] ⎢ ⎥ ⎢ T ⎥ δp ∶= δp ⋅ Jp ⋅ δp Compensation T T ⎢ nn ⎥ ⎣ ⎦ Implementation and εa = ε∗ + εp applications Conclusion Reference
  • 38. Chapter 8 Workpiece localization 8.3 Reliability Analysis 38 ⇒ Chapter Workpiece localization e probability that < (Fε(l,l)− ) Motivation εp Geometric algorithms for ε∗ workpiece localization is equal to( − ε) Performance and reliability Proposition 3 analysis Sampling and Translational error d along any direction is bounded d ≤ ((Fε(l,l) − )ε∗ λp ) Probe Radius Compensation Implementation and applications where λp is the smallest eigenvalue of Jp Conclusion Reference
  • 39. Chapter 8 Workpiece localization 8.3 Reliability Analysis 39 Rotational Reliability: Assume ω = , and R∗ = eωθ Ra ≅ (I + ωθ)Ra Chapter Workpiece ˆ localization ˆ ⎡ (n × q )T ⎤ ⎢ ⎥ Motivation εr = ωT [ (n × q ) ⋯ (nn × qn ) ] ⎢ ⎢ ⎥ ω ⋅ θ ∶= ωT ⋅ Jr ωθ ⎥ Geometric ⎢ (nn × qn )T ⎥ algorithms for ⎣ ⎦ workpiece localization Performance and reliability analysis Proposition 4 Sampling and Rotational error θ along any direction is bounded by Probe Radius θ ≤ ((Fε(l,l) − l)ε∗ λr ) Compensation Implementation and applications where λp is the smallest eigenvalue of Jr . Conclusion Reference
  • 40. Chapter 8 Workpiece localization 8.4 Discrete symmetry 40 Chapter Workpiece localization Motivation touch probe Geometric Laser sensor algorithms for non-touch probe Optical sensor workpiece CCD sensor localization Performance Touch probe is a de facto choice. and reliability analysis High accuracy Sampling and Probe Radius Easy of use Compensation Implementation less calibration and applications Conclusion Reference
  • 41. Chapter 8 Workpiece localization 8.4 Discrete symmetry 41 ◻ Compensation–Probe Radius Error: Chapter Workpiece localization What we record is center point set {yi′ }. Motivation We need contact point set{yi }for local- Geometric algorithms for workpiece localization ization algorithms. Performance and reliability analysis Significant errors will be introduced if Sampling and Probe Radius not compensated since r is of several Compensation mms. Implementation and applications r: probe radius yi : contact point yi′ : probe cneter point Conclusion Reference ni : normal in Cw
  • 42. Chapter 8 Workpiece localization 8.4 Discrete symmetry 42 ◻ Compensation–Our Proposed Method: Chapter Workpiece Note: yi′ = yi + rn′ localization Motivation i Geometric xi′ = xi + rni n′ = gni algorithms for workpiece localization i Performance and reliability analysis Sampling and Probe Radius Compensation Implementation yi′ : probe center point Cw and yi : contact point Cw xi′ : probe center point CM applications xi : contact point CM n′ :normal in CW Conclusion Reference r: probe radius ni : normal in CM i
  • 43. Chapter 8 Workpiece localization 8.4 Discrete symmetry 43 Chapter Workpiece The objective function becomes localization n n ε(g, x , ⋯, xn ) = = (yi + rn′ ) − (gxi + rn′ ) Motivation Geometric yi − gxi i i algorithms for i= i= workpiece n = yi′ − gxi′ localization Performance and reliability i= analysis Sampling and Probe Radius {yi′ } and {xi′ } lie on offset surfaces of the original ones ⇒ existing algorithms can be used to solve for g using {yi′ } Compensation Implementation and applications Conclusion Reference
  • 44. Chapter 8 Workpiece localization 8.4 Sampling 44 Chapter Workpiece localization Motivation Q:How many points should be probed? Geometric For a given number of points, where to probe? algorithms for workpiece Our computer-aided probing strategy uses: localization Reliability analysis to determine if the probed points are Performance and reliability adequate analysis Sampling and Sequential optimal planning to determine the locations Probe Radius Compensation where probing are to take place Implementation and applications Conclusion Reference
  • 45. Chapter 8 Workpiece localization 8.4 Sampling 45 Computer-Aided Probing Strategy–Reliability Analysis Recall the objective function Chapter n n ε(g, x , ⋯, xn ) = = g − yi − xi Workpiece localization yi − gxi Motivation i= i= g ∗ be the optimal solution of localization algorithms Geometric algorithms for Let xi∗ be the optimal solution of home points workpiece localization Performance ga be the actual transformation between CM and CW and reliability analysis It is easy to see Sampling and n n εa = ga yi − xi∗ − ≥ ε∗ = g ∗− yi − xi∗ Probe Radius Compensation Implementation i= i= and applications If we assume that sampling errors are normally distributed, Conclusion Reference ga yi − xi∗ and g ∗− yi − xi∗ are normally distributed. −
  • 46. Chapter 8 Workpiece localization 8.4 Sampling 46 Theorem 2 (): variance σ and X , ⋯, Xn is a random sample of size n of X, then Chapter the random variable U = ∑n (Xi − µ) σ will possess a chi-square i= Workpiece localization distribution with n dof. Motivation Theorem 3 (): Geometric algorithms for If U and V possess independent chi-square distributions with v workpiece localization and v degrees of freedom, respectively, then has the F distribution Performance with v and v degrees of freedom given by and reliability F= analysis U v Sampling and Probe Radius Vv Compensation Implementation where c is a constant related with v and v only. and f (F) = cF (v + v F) applications (v − ) (v +v ) Conclusion ⇒F= Reference εq is a F distribution ε∗
  • 47. Chapter 8 Workpiece localization 8.4 Sampling 47 By previous research, we define(assume n points are probed.) ⎡ nT ⎤ ⎡ (n × q )T ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ nT ⎥ ⎢ (n × q )T ⎥ J = N T N J = N T N ⎢ Np = ⎢ ⎥ Nr = − ⎢ ⎥ p Chapter ⎥ ⎢ ⎥ Workpiece ⎢ T ⎥ ⎢ ⎥ localization p p r r r ⎢ n ⎥ ⎢ T ⎥ (nn × qn ) ⎦ ⎣ n ⎦ ⎣ Motivation Geometric algorithms for workpiece where qi is the ith home point and ni is the corresponding localization normal vector Performance and reliability Translation error d along any direction is bounded by d ≤ ((Fε(l,l) )ε∗ λp ) analysis Sampling and smallest eigenvalues ofJp Probe Radius Compensation Rotation error along any direction is bounded by Implementation θ ≤ ((Fε(l,l) )ε∗ λr ) and applications smallest eigenvalues ofJr Conclusion Fε(l,l) : the critical value at the ε-level of the dof(l,l) Reference ε: the confidence limit l=n− : the degree of freedom
  • 48. Chapter 8 Workpiece localization 8.4 Sampling 48 ◻ Computer-Aided Probing Chapter Strategy–Optimal Planning: Workpiece localization Why the locations of measurement points are important? Motivation Geometric algorithms for workpiece localization Performance and reliability analysis Possible region of the object Sampling and Probe Radius Compensation the real object Implementation and Probe with error ball applications Conclusion Reference For 3D sculptured object, human intuition does not work well!
  • 49. Chapter 8 Workpiece localization 8.4 Sampling 49 ◻ Computer-Aided Probing Strategy–Fixture Model : Chapter locator 6 From fixture planning, we have locator i δyi = − niT (ri × ni )T ω = hT δξ Workpiece localization v i locator 1 z Motivation yi :theith locator error. rb Geometric l CB algorithms for δ:the workpiece location error. rW l x y workpiece Workpiece localization Combine equations at all locators, z v ⎡ δy ⎤ Performance ⎢ ⎥ and reliability ⎢ δy ⎥ CW y δy = ⎢ ⎥ = [ h h ⋯ hn ] δξ = GT δξ analysis x ⎢ ⎥ ⎢ δyn ⎥ Sampling and ⎣ ⎦ Probe Radius Compensation Implementation and δy = δyT δy = δξ T GGT δξ = δξ T Mδξ applications Note: Conclusion Reference the matrix M relates locator errors with workpiece location errors
  • 50. Chapter 8 Workpiece localization 8.4 Sampling 50 ◻ Computer-Aided Probing Strategy–Optimal Planning: We follow the D-optimization (Wang 00) with the index Chapter Workpiece localization max det(M) (in a point set domain) Motivation n Geometric Notice that M = GGT = hi hT i algorithms for i= workpiece localization If M contains n locators, we delete one from the Performance n locators, then and reliability ≤ pjj ≤ Mj = M − hj hT analysis det(Mj ) j Sampling and furthermore det(Mj ) = ( −pjj )det(M) pjj = hT M− hj Probe Radius Compensation j non-increasing! =M + (M hj )(M hj ) ( − pjj ) Implementation and and Mj− − − − T applications Conclusion By minimizing pjj (using sequential deletion method), we can sequentially optimize the index. Reference
  • 51. Chapter 8 Workpiece localization 8.4 Sampling 51 Chapter Workpiece localization ◻ Computer-Aided Probing Strategy–The Strategy: Motivation suppose the model has N’discretized points. d Geometric Let αr acceptable translation error bound algorithms for d workpiece αp acceptable rotation error bound localization ε the confidence limit d d Performance and reliability Input: CAD model of the workpiece, αr ,αp ,ε analysis Output: estimated transformation g within error bounds Sampling and Probe Radius Compensation Implementation and applications Conclusion Reference
  • 52. Chapter 8 Workpiece localization 8.4 Sampling 52 Manually probe 7 points (n=7) Two error Chapter bounds are Workpiece satisfied localization Reliability analysis Motivation Not Geometric satisfied algorithms for Set n=n+k workpiece N ≤ N′ localization n > N? Candidate Performance and reliability Error Probing analysis Yes points set No Sampling and Probe Radius Sequential planning Compensation Implementation and applications Probing Conclusion Reference success
  • 53. Chapter 8 Workpiece localization 8.4 Sampling 53 ◻ Computer-Aided Probing Strategy–Simulation: Chapter Workpiece localization Motivation Geometric algorithms for workpiece localization Performance and reliability Simulation Modle N’=1559 N=991 analysis Sampling and simulation setup: αr = . deg,αp = . mm,ε = Probe Radius d d Compensation % Implementation and given gd , normally distributed noise introduced applications Conclusion PII PC Reference two sequential optimal planning algorithms
  • 54. Chapter 8 Workpiece localization 8.4 Sampling 54 Chapter Sequential optimal planning: Workpiece Sequential deletion algorithm: we get final 6 localization points planning with 69.26s in MATLAB.det(M) = Motivation . × Geometric Sequential addition algorithm: algorithms for 1.Get 6 points maxdet(M) planning workpiece localization Random generation of points (G full rank) Performance Improve by interchange: and reliability Interchange a current point j and a candidate analysis point k,det(Mjk ) = pjk det(M) Sampling and pjk = hT M− hk Maximize pjp , we maximize j Probe Radius Compensation det(M)with one interchange we get nal points planning with . s in MATLAB.det(M) = . × Implementation and applications 2.Add point one by one Need averagely 0.066s Conclusion in MATLAB. Reference Need averagely 0.066s in MATLAB.
  • 55. Chapter 8 Workpiece localization 8.4 Sampling 55 Chapter Workpiece localization Motivation Geometric algorithms for workpiece localization Performance and reliability Simulation with deletion sequence, with µ = . σ = . ε= % analysis Sampling and Probe Radius Compensation Point number n 85 90 95 Translation error bound(mm) 0.1003 0.1004 0.0983 Implementation and Rotation error bound(degree) 0.1004 0.0968 0.0980 applications Succeed with 95 points! Conclusion Reference
  • 56. Chapter 8 Workpiece localization 8.4 Sampling 56 Chapter Workpiece localization Motivation Geometric algorithms for workpiece localization Performance and reliability Simulation with deletion sequence, with µ = . σ = . ε= % analysis Sampling and Probe Radius Point number n 215 220 225 Compensation Translation error bound(mm) 0.0977 0.0964 0.0945 Implementation Rotation error bound(degree) 0.1014 0.1005 0.099 and applications Succeed with 225 points! Conclusion Reference
  • 57. Chapter 8 Workpiece localization 8.4 Sampling 57 Chapter Workpiece localization Motivation Geometric algorithms for Comparison of σ = . andσ = . workpiece localization Performance and reliability analysis Sampling and Probe Radius Compensation Implementation and applications Conclusion Comparison of ε = %andε = Reference %
  • 58. Chapter 8 Workpiece localization 8.5 CAS system 58 User Interface Chapter Workpiece Probing Control localization Common Motivation Host Computer Auto-Probing Planning Parts Geometric algorithms for Workpiece Localization workpiece localization Tool Path Modification Performance and reliability analysis Motion Control Two CAS systems Different Sampling and Open architecture Parts Probe Radius CNC machine CNC Machine Probe Signal Collection Compensation Conventional CNC machine Implementation Machining and applications Different Probe System Parts Conclusion Reference
  • 59. Chapter 8 Workpiece localization 8.5 CAS system 59 ◻ Common Parts: Chapter Graphics User Interface Workpiece localization Model viewing control (Compatible Motivation with other CAD so ware) Geometric algorithms for Surface selection for surface probing workpiece localization Visual manipulation of probed points Performance and reliability Probe System analysis Algorithms Sampling and Probe Radius Workpiece localization Compensation Implementation Online compensation and applications Probing control Conclusion Optimal planning etc. Reference
  • 60. Chapter Workpiece localization Motivation CNC Machine Software Module Geometric Probe Probe Host Computer algorithms for Body Interface workpiece localization Performance Stylus Motion Controller and reliability analysis protected for Sampling and Workpiece Conventional system programmable for Probe Radius Compensation Open architecture Servo Motor Moter Server system Implementation and Machine Table Servo motor Moter Server applications Servo Motor Moter Server Conclusion Reference
  • 61. Chapter 8 Workpiece localization 8.5 Experiments 60 Chapter Workpiece localization Motivation ◻Manual Probing Geometric algorithms for workpiece localization Performance and reliability analysis ◻Auto Probing Sampling and Probe Radius Compensation Implementation ◻Computer Aided Probing and applications Options αp = . mm αr = . deg ε= Conclusion d d Reference %
  • 62. Chapter 8 Workpiece localization 8.5 CAS system 61 Chapter Workpiece localization Motivation Geometric algorithms for workpiece localization Performance and reliability analysis Sampling and Probe Radius Compensation Implementation and applications Conclusion Reference
  • 63. Chapter 8 Workpiece localization 8.5 Video Show 62 Chapter Workpiece localization Motivation Geometric algorithms for workpiece localization Performance and reliability analysis Sampling and Probe Radius Compensation Implementation and applications Conclusion Reference
  • 64. Chapter 8 Workpiece localization 8.5 Video Show 63 Chapter Workpiece localization Motivation Geometric algorithms for workpiece localization Performance and reliability analysis Sampling and Probe Radius Compensation Implementation and applications Conclusion Reference
  • 65. Chapter 8 Workpiece localization 8.6 Conclusion 64 Chapter Three important components of building a CAS system have Workpiece localization been discussed. Motivation Robust workpiece localization algorithms Geometric algorithms for Accurate probe radius compensation method workpiece localization Computer-aided probing strategy Performance and reliability analysis On the basis of these algorithms, two CAS systems have been Sampling and Probe Radius built. Compensation Implementation and Simulation and experimental results show that the system is applications Conclusion suitable for real-time implementation in manufacturing process. Reference
  • 66. Chapter 8 Workpiece localization 8.7 Reference 65 Chapter Workpiece Regular Localization : localization [1] P. Besl and N. McKay. A method for registration of 3-D shapes. IEEE Trans. Motivation on Pattern Analysis and Machine Intelligence, 14(2):239-256,1992 [2] W.E. Grimson and T.Lozano-Perez. Model-based recognition and localization Geometric algorithms for from sparse range or tactile data. Int. J. of Robotics Research, 3(3):3-35, 1984 workpiece [3] J. Hong and X. Tan. Method and apparatus for determine position and localization orientation of mechanical objects. U.S. Patent No.5208763, 1990 Performance [4] Z.X. Li, J.B. Gou and Y.X. Chu. Geometric algorithms for workpiece and reliability localization. IEEE Transactions on Robotics and Automation, 14(6):864-78, Dec. analysis 1998 Sampling and Probe Radius [5] C.H. Meng, H. Yau, and G. Lai. Automated precision measurement of surface Compensation profile in CAD-directed inspection. IEEE Trans.On Robotics and Automation, Implementation and 8(2):268-278, 1992 applications Conclusion Reference
  • 67. Chapter 8 Workpiece localization 8.7 Reference 66 Symmetric Localization : Chapter [6] J.B. Gou, Y.X,Chu, and Z.X. Li. On the symmetric localization problem. Workpiece localization IEEE Trans. on Robotics and Automation, 12(4): 553-540, 1998 [7] Jianbo Gou. Theory and Algorithms for Coordinate Metrology , PhD thesis, Motivation HKUST, Nov 1998. Geometric Hybrid Localization : algorithms for [8] Y.X,Chu, J.B. Gou, and Z.X. Li. On the hybrid localization/ envelopment workpiece localization problem. IEEE Trans. on Robotics and Automation, 12(4): 553-540, 1998 [9] Yunxian Chu. Workpiece Localization:Theory Algorithms and Implementation. Performance and reliability PhD thesis, HKUST, March 1999. analysis Others : [10] Y.X. Chu, J.B. Gou, and Z.X. Li. Workpiece localization Algorithms: Sampling and Probe Radius Performance evaluation and reliability analysis. Journal of manufacturing systems Compensation ,18(2):113-126, Feb.1999 Implementation [11] Michael Yu Wang. An optimum design for 3-D fixture synthesis in a point and set domain. IEEE Trans.On Robotics and Automation, 16(6):930-46, Dec. 2000 applications [12] Michael Yu Wang. An optimum design for 3-D fixture synthesis in a point Conclusion set domain. IEEE Trans.On Robotics and Automation, 16(6):930-46, Dec. 2000 Reference