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

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