Transcript thesis (ppt)

CAGING OF RIGID POLYTOPES VIA
DISPERSION CONTROL OF POINT FINGERS
Peam Pipattanasomporn
Advisor: Attawith Sudsang
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Motivation?
!
!
!
2
Better Approach?
3
Overview
Master Thesis
2-Squeeze
(2006)
Proposed
Ph.d. Thesis
Additional
Chapters
n-Squeeze
(2008)
Fix Cage
(2011)
2-Stretch
(2006)
Robust Cage
(2012)
n-Stretch
(2008)
Imperfect Shape
(2010)
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Overview
Master Thesis
2-Squeeze
(2006)
Proposed
Ph.d. Thesis
Additional
Chapters
n-Squeeze
(2008)
Fix Cage
(2011)
2-Stretch
(2006)
Robust Cage
(2012)
n-Stretch
(2008)
Imperfect Shape
(2010)
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2-Squeeze, How?
• Keep distance below a value
• Given object shape, solve:
– Where to place the fingers?
– The upperbound distance?
“Distance”
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2-Squeeze
• Possible escape path (object frame)
Distance
Along the path
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2-Squeeze
• “Better” escape path
Distance
Upperbound
“Better”
Initial
Along the path
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2-Squeeze
• Find an Optimal Escape Path in C-Free
Abstracted set of
escape configurations
(abstracted)
C-Obstacle
a
(a,b)
b
Configuration Space (4D)
Workspace (2D)
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2-Squeeze
• Find an Optimal Escape Path in C-Free
Abstract set of
escape configurations
(abstract)
C-Obstacle
(a,b)
Configuration Space (4D)
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C-Free Decomposition
C-Obstacle
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Paths connecting Terminals
C-Obstacle
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Finite Categorization of Paths
C-Obstacle
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Straight Path
Distance : |a-b|2
a
b
(linear interpolation)
Along the path
(a,b)
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Moving Across Convex Subsets
C-Obstacle
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Through Convex Intersections
C-Obstacle
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Requirements For The Algorithm
Distance(x)
x
Convex
Rigid Transformation Invariant
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Convex & RTI Examples
x1
d3
x3
d1
d2
x2
• d1 + d2 + d3
• d12 + d22 + d32
• max(d1, d2, d3)
• Larger  Loose cage
• Fingers at a point  Smallest
“Formation Size”
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Results (n-Squeeze)
Size: d12+d22+d32+d42
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Squeezing?
1
1
1
3
3
2
2
3
2
1-DOF Scaling ONLY
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“Size” & “Deformation”
1
1
3
3
2
Smaller size
Slightly Deformed
2
Reference Formation
Same size
No deformation
Larger size
Deformed
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“Size” & “Deformation”
1
1
3
3
2
Smaller size
Slightly Deformed
2
Reference Formation
Same size
No deformation
Same Formation
Larger size
Deformed
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“Size” & “Deformation”
1
3
Reference Formation
2
Smaller size
Slightly Deformed
Same size
No deformation
Same Formation
Larger size
Deformed
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“Size” & “Deformation”
1
3
Reference Formation
2
Smaller size
Slightly Deformed
Same size
No deformation
Same Formation
Larger size
Deformed
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“Size” & “Deformation”
• |r|22(x) = |A†x|22
– “Scale” or “Size” (w.r.t. reference)
• D(x) = |A(r; t) – x|22
– “Deformation upto Scale”
(w.r.t. reference)
A stores information of the reference.
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Squeezing ?
Convex
& RTI
1
Size = |r|22 < ???
1
1
3
3
2
D≤0
2
3
2
1-DOF Scaling ONLY
Convex
& RTI
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Squeezing
1
2
|r|
2
Size* =

;D≤0
;D>0
D≤0
D>0
Size = |r|22 < ???
1
1
D>0
3
3
2
D≤0
2
3
2
x
1-DOF Scaling ONLY
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Fix Formation Cage
Convex
& RTI
Convex
Constraint
Size* = 1
1
Size* ≤ 1
3
2
“Squeeze”
Size*  1
“Stretch”
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Robust Caging
• Keep error (deformation)
below a value
• Given object shape, find:
– Where to place the fingers
– The upperbound error
Independent Capture Regions
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n-Squeeze vs Fix Formation
KEEP
SIZE
BELOW UPPERBOUND
ERROR (DEFORMATION)
BELOW UPPERBOUND
OPTIMAL ESCAPE PATH
SIZE
MINIMIZE
UPPERBOUND DISTANCE ERROR (DEFORMATION)
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Error Tolerance
1
D2
2
3
=
inf
r,tϵR
2
1 1
2 2
3 3


r+t-

1
2
3
2
“Placement Error upto
Scale”
“Placement Error”
1
Ep 2
3
=
inf
|r|2=1
tϵR2
1 1
2 2
3 3



r+t-
1
2
3
p
NOT CONVEX!
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Approximation

inf g(r)
|r|2=1

min
i ϵ{1,…, m}
inf g(r)
r ϵ Ri
R2
R3
R1
R4
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Approximation

inf g(r)
|r|2=1

min
i ϵ{1,…, m}
inf g(r)
r ϵ Ri
R2
Min of Convex Functions
(not convex)
R3
R1
R4
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Optimal Path
Min of a Convex Function is Convex
f = f1 = min(f1)
Optimal Path
f1 = f
f1 = f = f2
f = f2
Min of Two Convex Functions
f = min(f1, f2)
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Optimal Path
f1 = f
f1 = f = f2
f = f2
???
Min of Two Convex Functions
f = min(f1, f2)
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Optimal Path
What is the optimal path, starting from the minimal points?
f1=f=f2
f(x)
f=f1
f=f2
1
2
x
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Critical Point
Consider…
f1=f=f2
f(x)
f=f1
f=f2
1,2
1
2
x
Only the points under the water level are reachable
when the maximum deformation is limited to below the water level.
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Optimal Path
: minimizer for a CONVEX optimization problem:
minimize L s.t.
f1(x) < L
f1=f=f2
f2(x) < L
f(x)
f=f1
f=f2
1,2
Critical Value
1
2
x
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Critical Point
f1=f=f2
f(x)
f=f1
f=f2
1,2
Critical Value
1
2
x
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Min of Multiple Convex Functions
f= f2
f= f1
f= f3
Min of Multiple Convex Functions
f = min(f1, f2 , f3)
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Min of Multiple Convex Functions
2,3
f= f2
1,2
2
3
f= f1
1,3
f= f3
1
Min of Multiple Convex Functions
f = min(f1, f2 , f3)
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Search Space
2
1,2
1,3
1
2,3
3
Min of Multiple Convex Functions
f = min(f1, f2 , f3)
Include all possible
between any two regions: f=fi , f=fj
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Optimal Path
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Results
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Results
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Shape Uncertainty
Exact Object
(Unknown)
sensor
Scanned Object
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Idea
• Cage subobject  Cage object ?
• Fingers must not penetrate the object.
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Idea
• Find placements that cage subobject, outside
superobject.
Exact Object
(Unknown)
Exact boundary (unknown)
but inbetween the bounds.
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Applications
•
•
•
•
•
Simplification
Curved Surface, Spherical Fingers
Shape Uncertainty
Slightly Deformable Object
Partial Observation
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Results
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Results
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Conclusion
• O(c log c) exact algorithms
– Squeeze, Stretch, Squeeze & Stretch
– c : # decomposed convex features
• O(cm2 log( cm2 ) ) approximate algorithm
– m : # approximation facets
• Extension to three dimension.
• Trade error tolerance with uncertainty.
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Q&A
:o
:V
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Publications
•
Journal Papers
– Peam Pipattanasomporn, Attawith Sudsang: Two-Finger Caging of Nonconvex Polytopes. IEEE
Transactions on Robotics 27 (2011)
– Thanathorn Phoka, Pawin Vongmasa, Chaichana Nilwatchararang, Peam Pipattanasomporn
and Attawith Sudsang: Optimal independent contact regions for two-fingered grasping of
polygon. Robotica (2011)
•
Conference Papers
– Peam Pipattanasomporn, Attawith Sudsang: Object caging under imperfect shape
knowledge. ICRA 2010
– Thanathorn Phoka, Pawin Vongmasa, Chichana Nilwatchararang, Peam Pipattanasomporn,
Attawith Sudsang: Planning optimal independent contact regions for two-fingered forceclosure grasp of a polygon. ICRA 2008
– Peam Pipattanasomporn, Pawin Vongmasa, Attawith Sudsang: Caging rigid polytopes via
finger dispersion control. ICRA 2008
– Peam Pipattanasomporn, Pawin Vongmasa, Attawith Sudsang: Two-Finger Squeezing Caging of
Polygonal and Polyhedral Object. ICRA 2007
– Peam Pipattanasomporn, Attawith Sudsang: Two-finger Caging of Concave Polygon. ICRA
2006
– Thanathorn Phoka, Peam Pipattanasomporn, Nattee Niparnan, Attawith Sudsang: Regrasp
Planning of Four-Fingered Hand for Parallel Grasp of a Polygonal Object. ICRA 2005
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