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Chapter8 Workpiecelocalization

Motivation

Geometricalgorithms forworkpiecelocalization

Performanceand reliabilityanalysis

Sampling andProbe RadiusCompensation

Implementationandapplications

Conclusion

Reference

Chapter 8 Workpiece localization

1

Lecture Notes for

A Geometrical Introduction to

Robotics and Manipulation

Richard Murray and Zexiang Li and Shankar S. SastryCRC Press

Zexiang Li1 and Yuanqing Wu1

1ECE, Hong Kong University of Science & Technology

July 28, 2010


Motivation





Conclusion

Reference


2


1 Motivation

2 Geometric algorithms for workpiece localization

3 Performance and reliability analysis

4 Sampling and Probe Radius Compensation

5 Implementation and applications

6 Conclusion

7 Reference


Motivation





Conclusion

Reference

8.1 MotivationChapter 8 Workpiece localization

3

In 1952, MIT Servo Lab(G.S.Brown) developed, incollaboration with Parsons, the first CNC milling machine.

The MIT numerically controlled milling machine

Manual machine with an operator

Giddings & Lewis 5-axis Skin Miller (1957)

Kearney & Trecker NC Turning Fujitsu & Fraice NC Mill(1958)


Motivation





Conclusion

Reference


4

In 1959, APT was developed, followed by extensiveactivities in CAD

CAD CAM CNC

Setup& Fixturing

(AutoCAD⋯) (UGII, MasterCam⋯) (Fanuc, Siemens⋯)


Motivation





Conclusion

Reference


5

◻ Conventional Approaches:

Jigs, �xtures and hard gauges:⇒ Expensive!

Manual Setup Time:⇒ consuming & expensive


Motivation





Conclusion

Reference

8.1 A Computer-aided Setup SystemChapter 8 Workpiece localization

6

CAD/CAMdata

Arbitrarily place &fixture workpiece withgeneral purpose fixtures

(Robots and/orprogrammable fixtures)

Probe and measure pointdata from the

workpiece surfaces

Compute the location& orientation ofthe workpiece

Modify & optimizetool path with computed

transformation

CNC Machine


Motivation





Conclusion

Reference

8.2 The ProblemChapter 8 Workpiece localization

7

◻ Possible Geometries:

x yz

xy

z

(a)Regular Workpiece Regular localization(b)Symmetry Symmetric localization(c)Partially machined Hybrid localization/envelopment(d)Raw stock Envelopment


Motivation





Conclusion

Reference


8

(a) Regular workpiece and the Euclidean group SE(3):

◻ Rotational Motion:

SO(3) = {R ∈ R3×3∣RTR = I, detR = 1}so(3) = {ω ∈ R3×3∣ωT = −ω} ≅ R3

ω = [ 0 −ω3 ω2ω3 0 −ω1−ω2 ω1 0

] LieAlgebra

ofSO(3)

x

y

z

Exp:

so(3)→ SO(3) ∶ ω → eω = Rω ∈ R3 Expomential coordinates of R x

y

z ω

eω


Motivation





Conclusion

Reference


9

◻ General rigid motion :

SE(3) = {(p,R)∣p ∈ R3,R ∈ SO(3)}

g = [ R p0 1 ] ∈ SE(3) ∶ Euclidean group of R3

se(3) = {[ ω v0 0 ] ∈ R4×4∣ω, v ∈ R3} ∶ Lie Alegebra ofSE(3)

xy

z

xy

z

ξ = [ ω v0 0 ] ξ = [ v

ω ]Exp:

se(3) → SE(3) ∶ ξ → eξ

:Screw motion


Motivation





Conclusion

Reference


10

(b) Symmetric workpiece and homogeneous space :

◻ Symmetry Subgroups:

A Cylinder:

G0 = {[ eλ1 e3 λ2e30 1

]∣ λ1, λ2 ∈ R}x y

z

x y

z

A Plane:

G0 = {[ eλ1 e3 λ2e1 + λ3e20 1

]∣ λi ∈ R}x y

z

x y

z


Motivation





Conclusion

Reference


11

◻ Configuration Space:

SE(3)/G0 = {gG0∣g ∈ SE(3)}g1 ∼ g2 iff g1 ⋅ g

−12 ∈ G0

Elements ofSE(3)/G0 ∶ [g], gG0 or g

G0

e gG0

g

Proposition 1SE(3)/G0is a differentiable manifold of dimension

dim(SE(3)) − dim(G0), with a transitive action

µ ∶ SE(3) × SE(3)/G0 → SE(3)/G0

(h, gG0)→ hgG0


Motivation





Conclusion

Reference


12

◻ Canonical Coordinates:

g0 ∶ Lie algebra ofG0

M0 ⊕ g0 = se(3)Define:Exp:

M0 ⊕ g0 → SE(3)(m, h)→ emeh

gG0

g

Let(ξ1,⋯, ξr) be a basis of M0,and m = y1 ξ1 +⋯+ yr ξr

Ψ ∶ SE(3)/G0 → Rr , g ↦ (y1 ,⋯, yr)

is well defined, and provide a canonical coor-dinate system for SE(3)/G0

I

g0m0

gG0

0

SO(3)

exp log


Motivation





Conclusion

Reference

8.2 Problem formulationChapter 8 Workpiece localization

13

1 Regular Localization: Data:{yi}i=1⋯nFind g ∈ SE(3), xi ∈ Si

min ε(g , x1, . . . , xn) = n

∑i=1

∥yi − gxi∥2

2 Symmetric Localization:

Find g ∈ SE(3)/G0 s.t.

min ε(g , x1, . . . , xn) = n

∑i=1

∥yi − gxi∥2

3 Hybrid Localization/EnvelopmentProblem:

{yi}i=1⋯n ∶ Finished surface with G0

{zi}i=1⋯m ∶ Unmachined surface

x y

z

x y

z

x y

z

x y

z

x y

z

yi

δi


Motivation





Conclusion

Reference

8.2 Problem formulationChapter 8 Workpiece localization

14

Find g0 ∈ SE(3)/G0 , xi ∈ Si s.tmin εl(g) = n

∑i=1< g−1yi − xi, ni >2

x y

z

yi

δi

ni

ωi

g−1zjzj

Letg(λ) = g0G0(λ)

Find g(λ) ∈ SE(3),ωj ∈ Sj,s.t.min εl = n

∑j=1< g−1(λ)zj − ωj, nj >2

and < g−1(λ)zj − ωj , nj >≥ δj , j = 1,⋯,m


Motivation





Conclusion

Reference

8.2 Analytic resultsChapter 8 Workpiece localization

15

ε(g , x1 ,⋯, xn) = ∑ni=1 ∥yi − gxi∥2

§ Define:

x = 1n ∑n

i=1 xi; y = 1n ∑n

i=1 yi; x′i = xi − x; y′i = yi − yW = ∑i y

′ix′Ti = UΣVT (SVD)

Σ = [ σ1 0 00 σ2 00 0 σ3

]Theorem 1 ():

If Rank(W) = 3 (i.e.n ≥ 4),∃!(R∗ , p∗)minimize ε(⋅, x1 ,⋯, xn) and { R∗ = VUT

p∗ = x − R∗yε∗ =∑

i

∥x′i∥2 +∑i

∥y′i∥2 − 2∑i

σi


Motivation





Conclusion

Reference


16

Proof :

ε(R, p, ⋅) = ∑ ∥gyi − xi∥2= n∥Ry + p − x∥2 +∑i ∥Ry′i − x′i∥2⇒ p∗ = x − R∗y

ε(R) = ∑i ∥Ry′i − x′i∥2= ∑i(∥y′i∥2 + ∥x′i∥2´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶a

) − 2∑i < Ry′i , x′i >= a − 2tr(RW)

ω = [ 0 −ω3 ω2ω3 0 −ω1−ω2 ω1 0

]ωR ∈ TRSO(3)

eR

ωωR


Motivation





Conclusion

Reference


17

< dεR, ωR >= d

dt∣t=0ε(etωR) = −2tr(ωRW) = 0,∀ω⇒ RW Symmetric

LetRW = S

⇒ S2 =WT⋅W

S = (WT⋅W) 1

2 = V [ ±σ1 ±σ2±σ3]VT

ε∗(⋅) =∑i

(∥y′i∥2 + ∥x′i∥2) − 2tr(WTW) 12 ⇒ { R∗ = VUT

p∗ = x − R∗y◻


Motivation





Conclusion

Reference


18

Proposition 2A necessary condition for x∗i , i = 1,⋯, n, to minimize

ε(R, p, ⋅)is that (xi = Ψi(ui , vi))(B){ < x′i − (p + Ryi), Ψui >= 0< x′i − (p + Ryi), Ψvi >= 0 i = 1,⋯, n

where Ψi ∶ R2 → R3, parametric equation of Si, and

ε(R, p) =∑i

∥Ryi + p − x∗i ∥2 =∑i

< Ryi + p − x∗i , n >2ni

gyi − xi < gyi − xi , n >gyi

xi

λni = gyi − xi


Motivation





Conclusion

Reference


19

◻ Localization Algorithms:

(g0 , x0i ) → (g′, x′i) →⋯ (g∗, x∗i )(B) (B)

(1) Variational Algorithm:

{ R = VUT

p = x − Ry(3) Hong-Tan Algorithm:

gk+1 = e ξ ⋅ gkFind ξ = [ v

ω ]by minim.

ε(ξ) =∑i

< (I + ξ)gkyi − xi , ni >2

A ⋅ ξ = b

(2) Tangent Algorithm:

gk+1 = e ξ ⋅ gk ξ = [ ω v0 0 ]

≈ (I + ξ)gkω, v ∈ R3

Find ξ = [ vω ]by minim.

ε(ξ) =∑i

∥(I + ξ)gkyi − xi∥2

A ⋅ ξ = b


Motivation





Conclusion

Reference


20

Algorithm 1: Algorithm( Alternating Variable Method )

Input Y = {yi}ni , yi ∈ SiStep 0 (a)Set k=0

(b)Initialize g0

(c)Compute y0i = (g0)−1yi(d)Compute x0i(e)Compute ε0 = ε(g0 , x0)(f)k=k+1

Step 1 (a)Newton’s algorithm for xki(b)Compute gk using (xki , gk−1)(c)Compute yki = (gk)−1y(d)Compute εk = ε(gk , xk)(e)If(1 − εk/εk−1) < δ1exit,else(f)Set k = k + 1, return to Step1 (a)


Motivation





Conclusion

Reference

8.2 Performance evaluationChapter 8 Workpiece localization

21

Algorithms

1 Variational Algorithm

2 ICP Algorithm

3 Tangent Algorithm

4 Meng’s Algorithm

5 Hong-Tan Algorithm

Performance Criteria

1 Robustness

2 Accuracy

3 E�ciency


Motivation





Conclusion

Reference


22

Regions of convergence in terms of the maximal orientation errors foreach of the algorithms


Motivation





Conclusion

Reference


23

Accuracy of estimation achieved by each of the algorithms as afunction of the number of measurement points


Motivation





Conclusion

Reference


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Computational efficiency by each of the algorithms as a function ofthe number of measurement points

Summary:

Algorithms Robustness Accuracy EfficiencyHong-Tan Good(−20 ∼ 20○) highest highestVariational Better(−30 ∼ 30○) high highTangent Best(−60 ∼ 60○) high high


Motivation





Conclusion

Reference

8.2 Symmetric localizationChapter 8 Workpiece localization

25

Find g ∈ SE(3)/G0 , xi ∈ F to

minimize ε(g , x1 ,⋯, xn) = n

∑i=1∥yi−gxi∥2

Choose

M0 ⊕ g0 = se(3)M0 = span{ξ1 ,⋯, ξk}

x y

z

x y

z

Algorithm 2: Algorithm:(Symmetric Localization)Input: (a)Measurement data{yi}i=1,⋯,n

(b)CAD description of FOutput Optimal solution g∗ ∈ SE(3)/G0 , xi ∈ F


Motivation





Conclusion

Reference


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Algorithm 3:Input Y = {yi}ni , yi ∈ SiStep 0 (a)Set k=0

(b)Initialize g0(c)Solve for x0i , i = 1,⋯, n(d)Calculate ε0 = ∑i ∥yi − gix0i ∥2

Step 1 (a)

Let gk+1 = emgk, m ∈ Adgk(M0)Solve for m by minimizaingε(m) = ∑i ∥yi − gk+1xki ∥2or ε(m) = ∑i < g−1k+1yi − xki , nki >2

(b)Solve for xk+1i(c)Calculate εk+1(d) If(1 − εk+1/εk) > ε,Set k = k + 1,

go to step1(a).Else exit.

Adgk(g0)Adgk (M0)


Motivation





Conclusion

Reference


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Example: A plane in R3

x(u, v) = ue1 + ve2G0 = {[ eλ1 e3 λ2e1 + λ3e2

0 1] ∣λi ∈ R}

g0 = span{ξ1, ξ2, ξ6} M0 = span{ξ3, ξ4, ξ5}gwm

computed solution R1 = [ 0.9878 0.0104 0.15510.0104 0.9912 −0.1322−0.1551 0.1322 0.9790

]q1 = [ −0.0073 0.0061 3.8215 ]T

SVD R2 = [ 0.5913 0.7914 0.15510.0104 0.9912 −0.1322−0.1551 0.1322 0.9790

]solution q2 = [ 1.6483 0.2603 3.5934 ]TExact R0 = [ 0.9776 −0.1372 0.1596

0.1596 0.97759 −0.1372−0.1372 0.15963 0.9776]

transform q0 = [ 1.0 2.0 4.0 ]TSimulation results for a plane inR3


Motivation





Conclusion

Reference


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◻ Performance Evaluation:

Robustness with re-spect to initial condi-tions

Efficiency compari-son


Motivation





Conclusion

Reference

8.2 Discrete symmetryChapter 8 Workpiece localization

29

Composite Feature:

AB

C xy

z

GA = {em1 ξ1+m2ξ3+m3 ξ5 ∣m1,m2,m3 ∈ R}GB = {em1 ξ2+m2 ξ3+m3 ξ4 ∣m1 ,m2,m3 ∈ R}GC = {em1 ξ1+m2 ξ2+m3 ξ6 ∣m1,m2,m3 ∈ R}

⇒ GABC = GA ∩GB ∩Gc = IQ:SE(3)/GABC = SE(3) a unique solution?

Correct Solution

xy

z

xy

zxy

z

xy

z


Motivation





Conclusion

Reference


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Plane:G0 = SE(2) ×D D = {1,−1}Identify configurations differing byG0

Cube:GABC = GA ∩GB ∩GC = {I, eπξ4 , eπξ5 , eπξ6}● Solution:Filter out solutions with deviating home point

● Remark: Aξ = b, ξ = [ vω ]

Rank(A) = 6⇒ Regular localization

Ker(A) = Lie algebra ofG0

Ker(A)� = m0⇒ ξ = A+b

x

yz

AB

C xy

z


Motivation





Conclusion

Reference

8.2 The hybrid algorithmChapter 8 Workpiece localization

31

{yi}ni=1 FSLÐÐ→ g0 ∈ SE(3)/G0

Algorithm 4: Algorithm:(The Envelopment Algorithm)

Input (a)Meas. data {zi}mi=1(b)CAD model and a basis (η1 ,⋯, ηr)for g0(c)g0 ∈ SE(3)/G0from the FSL algorithm

Output Optimal Solution g∗ ∈ SE(3)Step 0 (a)Set k=0 and g0 = g0

(b)Compute ω0i and n0i ,i = 1,⋯,m

(c)Calculate ε0e = ∑i < (gi)−1zi − ω0i , n

0i >2


Motivation





Conclusion

Reference


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Algorithm 5:

Step 1 (a)Let gk+1 = gkeλ , λ ∈ g0 , and solve for λ ∈ Rr

(b)Solve for ωk+1i and nk+1i , i = 1,⋯,m

(c) Calculate εk+1e(d) If (1 − εk+1e /εke) < ε

and < (gk+1)−1zi − ωk+1i , nk+1i >≥ δi,

then report the solution g∗ = gk+1;else set k=k+1

(e)If k ≤ K0,then go to step1(a); else, exit


Motivation





Conclusion

Reference


33

◇ Example: 1

S1 ∶ Finished surface

G0 = {e(λ1 ξ1+λ2 ξ2+λ3 ξ6)∣λi ∈ R}M0 = span{ξ3, ξ4, ξ5}S1, S2, S3 ∶ Un�nished surface


Motivation





Conclusion

Reference


34

◇ Example: 2

S1, S2: Finished surface

G0 = {eλξ3 ∣λ ∈ R}M0 = {ξ1, ξ2, ξ4, ξ5, ξ6}S1: Unfinished surface


Motivation





Conclusion

Reference

8.3 Reliability AnalysisChapter 8 Workpiece localization

35

Q:With measurement errors and a finite no. of sampling points,how reliable is the computed solution?●Example:Orientation:

d1 2

Not too reliable

∆θ ∝ 1d

d1 3

More reliableTranslation:

d1 3

4

Not reliable in x-direction

error ∝ 1sinΦ

d1 3

4

5

More reliable


Motivation





Conclusion

Reference


36

{yi}ni=1 → g∗ = (R∗ , p∗), x∗i , ε∗, Estimate ofÐÐÐÐÐÐ→ ga = (Ra , pa)ε∗ = n

∑i=1< g∗yi − x∗i , n∗i >2≤ εa =

n

∑i=1< gayi − x∗i , n∗i >2

Assume:εa = ε1 + ε∗ and< g∗yi − x∗i , n∗i >2, i = 1,⋯, n< gayi − x∗i , n∗i >2, i = 1,⋯, n

are normally distributed, with variance ε∗&εarespectively.

⇒ F = εaε∗ ∶ F − distribution l = n − 6(dof)

Let Fε(l, l)be critical value at the ε-level corresponding todof (l, l)


Motivation





Conclusion

Reference


37

P(F > Fε(l,l)) = εor

P(F < Fε(l,l)) = 1 − εThe probability that F = ε∗+εl

ε∗< Fε(l,l)is equal to 1 − ε.●Translational Reliability

Letδp = (δpx , δpy , δpz) and δ =√δ2px + δ2px + δ2pxεp = δTp [ n1 ⋯ nn ] ⎡⎢⎢⎢⎢⎣

nT1⋮nTn

⎤⎥⎥⎥⎥⎦ δp ∶= δTp ⋅ Jp ⋅ δp

εa = ε∗ + εp


Motivation





Conclusion

Reference


38

⇒�e probability that

εp

ε∗ < (Fε(l,l)−1)is equal to(1 − ε)

Proposition 3Translational error d along any direction is bounded

d ≤ ((Fε(l,l) − 1)ε∗/λp)1/2where λpis the smallest eigenvalue of Jp


Motivation





Conclusion

Reference


39

● Rotational Reliability:Assume∥ω∥ = 1, and R∗ = eωθRa ≅ (I + ωθ)Ra

εr = ωT [ (n1 × q1) ⋯ (nn × qn) ] ⎡⎢⎢⎢⎢⎣(n1 × q1)T⋮(nn × qn)T

⎤⎥⎥⎥⎥⎦ω ⋅ θ2 ∶= ωT

⋅ Jrωθ2

Proposition 4Rotational error θ along any direction is bounded by

θ ≤ ((Fε(l,l) − l)ε∗/λr)1/2where λp is the smallest eigenvalue of Jr.


Motivation





Conclusion

Reference


40

touch probe

non-touch probe{ Laser sensorOptical sensorCCD sensor

Touch probe is a de facto choice.

High accuracy

Easy of use

less calibration


Motivation





Conclusion

Reference


41

◻ Compensation–Probe Radius Error:

What we record is center point set{y′i}.We need contact point set{yi}for local-ization algorithms.

Significant errors will be introduced ifnot compensated since r is of severalmms.

r: probe radius yi: contact pointni: normal in Cw y′i : probe cneter point


Motivation





Conclusion

Reference


42

◻ Compensation–Our Proposed Method:

Note:

y′i = yi + rn′ix′i = xi + rnin′i = gni

yi: contact point Cw y′i : probe center point Cwxi: contact point CM x′i : probe center point CMr: probe radius ni: normal in CM n′i :normal in CW


Motivation





Conclusion

Reference


43

The objective function becomes

ε(g , x1 ,⋯, xn) = n

∑i=1∥yi − gxi∥2 = n

∑i=1∥(yi + rn′i) − (gxi + rn′i)∥2

= n

∑i=1∥y′i − gx′i∥2

{y′i} and {x′i} lie on offset surfaces of the original ones

⇒ existing algorithms can be used to solve for g using {y′i}


Motivation





Conclusion

Reference

8.4 SamplingChapter 8 Workpiece localization

44

Q:How many points should be probed?For a given number of points, where to probe?Our computer-aided probing strategy uses:

Reliability analysis to determine if the probed points are

adequate

Sequential optimal planning to determine the locations

where probing are to take place


Motivation





Conclusion

Reference


45

Computer-Aided Probing Strategy–Reliability AnalysisRecall the objective function

ε(g , x1 ,⋯, xn) = n

∑i=1∥yi − gxi∥2 = n

∑i=1∥g−1yi − xi∥2

Let g∗be the optimal solution of localization algorithmsx∗i be the optimal solution of home pointsga be the actual transformation between CM and CW

It is easy to see

εa = n

∑i=1∥g−1a yi − x

∗i ∥2 ≥ ε∗ = n

∑i=1∥g∗−1yi − x∗i ∥2

If we assume that sampling errors are normally distributed,

g−1a yi − x∗i and g

∗−1yi − x∗i are normally distributed.


Motivation





Conclusion

Reference


46

Theorem 2 ():variance σ 2and X1 ,⋯,Xnis a random sample of size n of X, then

the random variable U = ∑ni=1(Xi − µ)2/σ 2will possess a chi-square

distribution with n dof.

Theorem 3 ():If U and V possess independent chi-square distributions with v1

and v2 degrees of freedom, respectively, then has the F distributionwith v1 and v2 degrees of freedom given by

F = U/v1Vv2

where c is a constant related with v1 and v2 only.

f (F) = cF 12 (v1−2)(v2 + v1F) 1

2 (v1+v2)

⇒ F = εq

ε∗is a F distribution


Motivation





Conclusion

Reference


47

By previous research, we define(assume n points are probed.)

Np =⎡⎢⎢⎢⎢⎢⎢⎣nT1nT2⋮

nTn

⎤⎥⎥⎥⎥⎥⎥⎦Nr = −

⎡⎢⎢⎢⎢⎢⎢⎣(n1 × q1)T(n2 × q2)T⋮(nn × qn)T

⎤⎥⎥⎥⎥⎥⎥⎦Jp = NT

p Np Jr = NTr Nr

where qi is the ith home point and ni is the correspondingnormal vectorTranslation error d along any direction is bounded by

d ≤ ((Fε(l,l))ε∗/λp)1/2 smallest eigenvalues ofJp

Rotation error along any direction is bounded by

θ ≤ ((Fε(l,l))ε∗/λr)1/2 smallest eigenvalues ofJr

Fε(l,l): the critical value at the ε-level of the dof(l,l)ε: the confidence limit

l = n − 6: the degree of freedom


Motivation





Conclusion

Reference


48

◻ Computer-Aided ProbingStrategy–Optimal Planning:Why the locations of measurement points are important?

Possible region of the object

the real object

Probe with error ball

For 3D sculptured object, human intuition does not work well!


Motivation





Conclusion

Reference


49

◻ Computer-Aided ProbingStrategy–Fixture Model :From fixture planning, we haveδyi = − [ nTi (ri × ni)T ] [ vω ] = hTi δξyi:thei

thlocator error.δ:the workpiece location error.

v

rblrWl

x y

z

x y

zlocator 1

locator i locator 6

CW

CB

WorkpieceCombine equations at all locators,

δy =⎡⎢⎢⎢⎢⎢⎣δy1δy2⋮

δyn

⎤⎥⎥⎥⎥⎥⎦= [ h1 h2 ⋯ hn ] δξ = GTδξ

Note: ∥δy∥ = δyTδy = δξTGGTδξ = δξTMδξ

the matrix M relates locator errors with workpiece locationerrors


Motivation





Conclusion

Reference


50

◻ Computer-Aided Probing Strategy–Optimal Planning:

We follow the D-optimization (Wang 00) with the index

max det(M) (in a point set domain)

Notice that M = GGT = n∑i=1

hihTi

If M contains n locators, we delete one from then locators, then

Mj =M − hjhTjfurthermore det(Mj) = (1−pjj)det(M) pjj = hTj M−1hj

and M−1j =M−1 + (M−1hj)(M−1hj)T/(1 − pjj)

0 ≤ pjj ≤ 1det(Mj)

non-increasing!

By minimizing pjj (using sequential deletion method), we can sequentially optimizethe index.


Motivation





Conclusion

Reference


51

◻ Computer-Aided Probing Strategy–The Strategy:

suppose the model has N’discretized points.Let αd

r acceptable translation error boundαdp acceptable rotation error boundε the confidence limit

Input: CAD model of the workpiece, αdr ,α

dp ,ε

Output: estimated transformation g within error bounds


Motivation





Conclusion

Reference


52

Manually probe 7 points (n=7)

Reliability analysis

Set n=n+k

n > N?

Sequential planning

Probing

success

Error

Notsatisfied

No

Two errorbounds aresatisfied

Yes

N ≤ N′CandidateProbingpoints set


Motivation





Conclusion

Reference


53

◻ Computer-Aided Probing Strategy–Simulation:

Simulation Modle N’=1559 N=991

simulation setup:

αdr = 0.1deg,αd

p = 0.1mm,ε = 95%given gd, normally distributed noise introduced

PII 400 PC

two sequential optimal planning algorithms


Motivation





Conclusion

Reference


54

Sequential optimal planning:Sequential deletion algorithm: we get final 6points planning with 69.26s in MATLAB.det(M) =7.056 × 1011Sequential addition algorithm:1.Get 6 points maxdet(M) planning

Random generation of 6 points (G full rank)

Improve by interchange:Interchange a current point j and a candidatepoint k,det(Mjk) = p2jkdet(M)pjk = hTj M−1hkMaximize pjp , we maximize

det(M)with one interchangewe get �nal 6 points planning with 0.05s inMATLAB.det(M) = 9.180 × 1011

2.Add point one by one Need averagely 0.066sin MATLAB.Need averagely 0.066s in MATLAB.


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55

Simulation with deletion sequence,with µ = 0.01 σ2 = 0.01 ε = 95%Point number n 85 90 95Translation error bound(mm) 0.1003 0.1004 0.0983Rotation error bound(degree) 0.1004 0.0968 0.0980

Succeed with 95 points!


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Simulation with deletion sequence,with µ = 0.01 σ2 = 0.01 ε = 95%Point number n 215 220 225Translation error bound(mm) 0.0977 0.0964 0.0945Rotation error bound(degree) 0.1014 0.1005 0.099

Succeed with 225 points!


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57

Comparison of σ2 = 0.01andσ2 = 0.02

Comparison of ε = 95%andε = 99%


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8.5 CAS systemChapter 8 Workpiece localization

58

Host Computer

User Interface

Probing Control

Auto-Probing Planning

Workpiece Localization

Tool Path Modification

CNC Machine

Motion Control

Probe Signal Collection

Machining

Probe System

CommonParts

DifferentParts

DifferentParts

Two CAS systems●Open architectureCNC machine●ConventionalCNC machine


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59

◻ Common Parts:Graphics User Interface

Model viewing control (Compatible

with other CAD so�ware)

Surface selection for surface probing

Visual manipulation of probed points

Probe SystemAlgorithms

Workpiece localization

Online compensation

Probing control

Optimal planning etc.


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CNC Machine

ProbeBody

Stylus

Workpiece

Machine Table

Servo Motor

Servo motor

Servo Motor

Moter Server

Moter Server

Moter Server

Motion Controller

Host Computer

Software Module

ProbeInterface

protected forConventional system programmable for

Open architecturesystem


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8.5 ExperimentsChapter 8 Workpiece localization

60

◻Auto Probing

◻Manual Probing

◻Computer Aided ProbingOptions

αdp = 0.1mm αd

r = 0.2deg ε = 95%


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61


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8.5 Video ShowChapter 8 Workpiece localization

62


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8.5 Video ShowChapter 8 Workpiece localization

63


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8.6 ConclusionChapter 8 Workpiece localization

64

Three important components of building a CAS system havebeen discussed.

Robust workpiece localization algorithms

Accurate probe radius compensation method

Computer-aided probing strategy

On the basis of these algorithms, two CAS systems have beenbuilt.

Simulation and experimental results show that the system issuitable for real-time implementation in manufacturing process.


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8.7 ReferenceChapter 8 Workpiece localization

65

Regular Localization :[1] P. Besl and N. McKay. A method for registration of 3-D shapes. IEEE Trans.on Pattern Analysis and Machine Intelligence, 14(2):239-256,1992[2] W.E. Grimson and T.Lozano-Perez. Model-based recognition and localizationfrom sparse range or tactile data. Int. J. of Robotics Research, 3(3):3-35, 1984[3] J. Hong and X. Tan. Method and apparatus for determine position andorientation of mechanical objects. U.S. Patent No.5208763, 1990[4] Z.X. Li, J.B. Gou and Y.X. Chu. Geometric algorithms for workpiecelocalization. IEEE Transactions on Robotics and Automation, 14(6):864-78, Dec.1998

[5] C.H. Meng, H. Yau, and G. Lai. Automated precision measurement of surface

profile in CAD-directed inspection. IEEE Trans.On Robotics and Automation,

8(2):268-278, 1992


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8.7 ReferenceChapter 8 Workpiece localization

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Symmetric Localization :[6] J.B. Gou, Y.X,Chu, and Z.X. Li. On the symmetric localization problem.IEEE Trans. on Robotics and Automation, 12(4): 553-540, 1998[7] Jianbo Gou. Theory and Algorithms for Coordinate Metrology , PhD thesis,HKUST, Nov 1998.Hybrid Localization :[8] Y.X,Chu, J.B. Gou, and Z.X. Li. On the hybrid localization/ envelopmentproblem. IEEE Trans. on Robotics and Automation, 12(4): 553-540, 1998[9] Yunxian Chu. Workpiece Localization:Theory Algorithms and Implementation.PhD thesis, HKUST, March 1999.Others :[10] Y.X. Chu, J.B. Gou, and Z.X. Li. Workpiece localization Algorithms:Performance evaluation and reliability analysis. Journal of manufacturing systems,18(2):113-126, Feb.1999[11] Michael Yu Wang. An optimum design for 3-D fixture synthesis in a pointset domain. IEEE Trans.On Robotics and Automation, 16(6):930-46, Dec. 2000[12] Michael Yu Wang. An optimum design for 3-D fixture synthesis in a pointset domain. IEEE Trans.On Robotics and Automation, 16(6):930-46, Dec. 2000

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