a geometric approach to designing a programmable force field with a unique stable equilibrium for...
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A Geometric Approach to Designing a Programmable Force Field with a
Unique Stable Equilibrium for Parts in the Plane
Attawith Sudsang and Lydia E. Kavraki
Department of Computer Science
Rice University
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Part Manipulation
A very important operation in automation Specific problems:
position, orient, sort, and separate parts
Stage 1
Stage 2
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“Traditional” Parts Feeders
Commercial Vibratory Bowl Feeder
[McKnight Automation]
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Flexible Part Feeders
New challenges: Parts are smaller and more fragile Parts are manufactured in large quantities Efficiency, flexibility and robustness of assembly
Parallel jaw gripper: [Goldberg 93],[Rao and Goldberg 95]
1 JOC and 2 JOC systems: [Erdman 95],[Erdman and Mason 96],
[Lynch 96], [Akella, Huang, Lynch Mason 97]
Fences: [Wiegley,Goldberg,Peshkin, Brokowski 96]
Examples:
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Recent Paradigm: Programmable Force Fields
Field realized on a plane
Force and torque exerted on
the contact surface of the part
Advantages Little or no sensing No damage to the parts A wide variety of parts
[ Similation by Bohringer and Donald]
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Programmable Force Fields: Implementation
MEMS: Böhringer, Donald, McDonald 94-99 Arrays of directed air jets: Berlin et al. 98-99 Arrays of small motors, Messner et al. 97-00 Vibrating plates, Reznik and Canny 98-01
and others
[Pictures by Bohringer, Donald et al]
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Programmable Force Fields: Analysis
Abstaction:
Part is subjected to force F and torque M Equilibrium configuration: position and orientation fixed Part is at an equilibrium: F = 0 and M = 0
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Equilibrium Configurations: A Brief History
Task: Define fields in which a given part has few equilibrium
configurations
Overview:1994: O(kn2) and O(kn)
and conjecture for unique equilibrium
1997: 2
2000: 1 existential proof
2001: 1 CONSTRUCTIVE PROOF
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1994: Squeeze and Unit Radial Fields
Squeeze Fields:
O(kn2) equilibria for polygons
Unit Radial + Squeeze: O(kn) equilibria for polygons
[Böhringer, Donald, McDonald]
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1997: Elliptic Field
[Kavraki]
0,,
),(
baba
withbxax
2 stable equilibria
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2000: One Equilibrium [Böhringer, Donald, Kavraki and Lamiraux] [Lamiraux and Kavraki]
Unit radial + Constant
One stable equilibrium for parts
whose center of mass pivot point
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Pivot Point
A fixed point on the part’s coordinate frame situated
at the center of the radial field at equilibrium
In general: pivot point center of mass
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2000: One Equilibrium
Unit Radial + Constant
Problem: proof is not constructive
[Böhringer, Donald, Kavraki and Lamiraux] [Lamiraux and Kavraki]
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2001: One Equilibrium + Constructive Proof
Linear Radial + Constant
[Sudsang and Kavraki]
Combined in a way that depends
on the GEOMERTY of the part
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Notation: Radial Field
)(, rfO
is the unit radial field
1,
0
0Ex.
rO kh, is a linear radial field (h,k constant) centered at O
Pivot point defined in a similar way: p
Pivot
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Notation: Constant Field
0),(
yxf
Important: A constant field can be written as
the sum of two linear radial fields
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Constant Field as Sum of Linear Radial Fields
0),(
yxf is a combination of
kd where,k,
f
deandk,
f
err
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Constant Field as Sum of Linear Radial Fields
0),(
yxf
Proof:
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Unique Equilibirum: Main Result
d, h, k, c > 0 d is the distance of the center of mass and the pivot
point under:
If d>0 there is a unique equilibrium
and at equilibrium the pivot point under K at
and center of mass at
0
kd-
c)(k h ,0
0 K
r
0
0
0
d -
0
0
c)(2k h ,0
0
r+
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Unique Equilibrium
0
kd- k-,
0
0
r
k,0
d-
r
c)(2k h ,0
0
r
c)(k h ,0
0
r
+
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Unique Equilibrium
k,0
d-F 1
r c)(k h ,
0
0F 2
r
d : distance of p and p’ p’: center of mass
p : pivot point under F2
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Why a Unique Equilibrium?
Step 1: Identifying possible equilibria
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Step2: Stability Using Potential Fields
Plane: Potential
Configuration Space: Lifted Potential
)(2
2),( 2222 yx
ckyxhkdxyxu
qS
drruqU )()( Sq= surface occupied by the
part at configuration q
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Stability Using Potential Fields
Equilibria Configurations = Local Minima of U We show U has a local minimum (and a local maximum)
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Unique Equilibrium
k,0
d-F 1
r c)(k h ,
0
0F 2
r
p’: center of mass
p : pivot point under F2
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Future Work and Open Problems
Computation of the pivot point Implementation issues: Friction, Discretization,
Convergence Can programmable force fields be used to assemble
and separate parts?
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Flexible Feeders Parallel jaw gripper:
[Goldberg 93][Rao and Goldberg 95] 1 JOC and 2 JOC systems:
[Erdman 95][Erdman and Mason 96][Lynch 96][Akella, Huang, Lynch, Mason 95]
Fences:
[Wiegley,Goldberg,Peshkin, Brokowski 96] Programmable force fields and related devices:
[Bohringer, Donald, McDonald 94-99][Bohringer, Bhatt, Goldberg 95], [Bohringer, Bhatt, Donald, Goldberg 96][Bohringer, Cohn et al 97][Biegelsen, Jackson, Berlin, Cheng 97][Coutinho and Will 97][Frei et al 99][Luntz, Messner, Choset 97-98][Resnick and Canny 98-01][Suh et al 99][Yim 99-01] and others.
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Notation: Pivot Point
A fixed point p on the part’s
coordinate frame situated
at the center of the field when
an equilibrium is achieved
Example: rO k,
0 A k drF when the center of mass is at O
In general: pivot point center of mass
p
Pivot point