algorithms, models and metrics for the design of workholding using part concavities k....
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![Page 1: Algorithms, Models and Metrics for the Design of Workholding Using Part Concavities K. Gopalakrishnan, Ken Goldberg, IEOR, U.C. Berkeley](https://reader036.vdocuments.net/reader036/viewer/2022081514/56649edc5503460f94bec557/html5/thumbnails/1.jpg)
Algorithms, Models and Metricsfor the Design of Workholding Using
Part Concavities
K. Gopalakrishnan, Ken Goldberg,
IEOR, U.C. Berkeley.
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• Review
• Unilateral Fixtures - Experiments
• Deformation Space
• Optimal Deform Closure Grasps
• Assembly Line Simulation
• Conclusion
Outline
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• Bulky• Complex• Multilateral• Dedicated, • Expensive• Long Lead time• Designed by
human intuition
Conventional Fixtures
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Modular Fixturing
• Existence and algorithm: Brost and Goldberg, 1996.
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C-Space and Form Closure
y
x
/3
(5,4)
y
x
q
4
5
/3(5,4,- p/3)
C-Space (Configuration Space):• Describes position and orientation.• Each degree of freedom of a part is a C-space axis.• Form Closure occurs when all adjacent
configurations represent collisions.
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2D v-grips
Expanding.
Contracting.
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• N-2-1 approachCai et al, 1996.
• Decoupling beam elementsShiu et al, 1997.
• Manipulation of sheet metal partKavraki et al, 1998.
Deformable parts
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3D vg-grips
• Use plane-cone contacts:– Jaws with conical grooves: Edge contacts.– Support Jaws with Surface Contacts.
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Examples
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• Review
• Unilateral Fixtures - Experiments
• Deformation Space
• Optimal Deform Closure Grasps
• Assembly Line Simulation
• Conclusion
Outline
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Quality Metric
• Sensitivity of orientation to infinitesimal jaw relaxation.
• Maximum of Rx, Ry, Rz.
• Ry, Rz: Approximated to v-grip.
• Rx: Derived from grip of jaws by part.
Jaw Jaw
Part
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Apparatus: Schematic
BaseplateTrack
Slider Pitch-Screw
Dial Gauge
Mirror
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Experimental Apparatus
A1 A2A3
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-0.025 -0.02 -0.015 -0.01 -0.005
A1-A3
31.74
A1-A2
77.43
0.3
0.2
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.1
0
Orie
ntat
ion
erro
r (d
egre
es)
Jaw separation (inches)
Experiment Results
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Ford D219 Door model
++
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A4C
A1C
A2C
A3R
A5R
A6C
A7C
A8RA9R
B1CB2C
B3C
B4R B5R
D219 Door: Contact set
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• Review
• Unilateral Fixtures - Experiments
• Deformation Space
• Optimal Deform Closure Grasps
• Assembly Line Simulation
• Conclusion
Outline
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• Lack of definition of fixtures/grasps for deformable parts.
• Generalization of C-Space.
• Based on FEM model.
D-Space
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Topology violating
configuration
Undeformed part Allowed deformation
Topology Preservation
Example for for system of parts
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D-free: Examples
Slice with 1-4 fixedPart and mesh
1
2 3
5
4
x
y
x5
y5
x5
y5
x5
y5
Slice with 1,2,4,5 fixed
x3
y3
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• For FEM with linear elasticity and linear interpolation,
P.E. = (1/2) XT K X
D-Space and Potential Energy
qA
qB
UT
• Increase in potential energy UA needed to release part.
• Deform Closure if UA > 0.
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• Frame invariance.
• Form-closure Deform-closure of
equivalent deformable part.
Results
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Numerical Example
1
2
3
4
3
4
1
2
4
2
3
11
4
3
2
4 Joules 547 Joules
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• D-Obstacle symmetry
- Prismatic extrusion of identical shape along multiple axes.
- Point obstacles are identical but displaced.
• Symmetry of Topology preserving space (DT).
- Superset: Non-degenerate meshes.
Symmetry in D-Space
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• Review
• Unilateral Fixtures - Experiments
• Deformation Space
• Optimal Deform Closure Grasps
• Assembly Line Simulation
• Conclusion
Outline
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• Given:
Deformable polygonal part.
FEM model.
Pair of contact nodes.
• Determine:
Optimal jaw separation.
Optimal?
Problem Description
M
E
n0
n1
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• Consider:
- Threshold P.E. UA.
- Additional P.E. needed for plastic deformation UL.
• Q = min { UA, UL }
Quality metric
n0
n1
L
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• Assume sufficiently dense mesh.
• Points of interest: contact at mesh nodes.
• Construct a graph:
Each graph vertex = 1 pair of perimeter mesh nodes.
O(p2) graph vertices.
Contact Graph
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• Traversal with minimum increase in energy.
• FEM solution with two mesh nodes fixed.
ni
nj
Deformation at Points of Interest
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Potential Energy vs. ni
nj
kij
Pot
entia
l Ene
rgy
(U)
Distance between FEM nodes
Undeformed distance
Expanding
Contracting
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A
B
C
E
F
G
A
B
C
D
E
F
D
G
H
Contact Graph: Edges• Traversal with minimum increase in energy.
Adjacent mesh nodes:
Non-adjacent mesh nodes:
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U (
v(n
i, n j),
)
Peak P.E.: Given release path
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Peak P.E.: All release paths
U (
v* ,
)
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U (
vo,
)
, U (
v*,
)
Threshold P.E.
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UA (
)
, UL (
)
Quality Metric
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• Possibly exponential
number of pieces.
• Sample in intervals of .
• Error bound on max. Q =
* max { 0(ni, nj) *
kij }
Numerical Sampling
Q
(
)
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• Calculate UL.
• To determine UA:
Algorithm inspired by Dijkstra’s algorithm for sparse graphs.
Fixed i
Insert pic of contact graph drawn on 2D P.E. graph
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V -
Algorithm for UA(i)
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V -
Algorithm for UA(i)
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Numerical Example
Undeformed
= 10 mm.
Optimal
= 5.6 mm.
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• Review
• Unilateral Fixtures - Experiments
• Deformation Space
• Optimal Deform Closure Grasps
• Assembly Line Simulation
• Conclusion
Outline
![Page 42: Algorithms, Models and Metrics for the Design of Workholding Using Part Concavities K. Gopalakrishnan, Ken Goldberg, IEOR, U.C. Berkeley](https://reader036.vdocuments.net/reader036/viewer/2022081514/56649edc5503460f94bec557/html5/thumbnails/42.jpg)
• Review
• Unilateral Fixtures - Experiments
• Deformation Space
• Optimal Deform Closure Grasps
• Assembly Line Simulation
• Conclusion
Outline
![Page 43: Algorithms, Models and Metrics for the Design of Workholding Using Part Concavities K. Gopalakrishnan, Ken Goldberg, IEOR, U.C. Berkeley](https://reader036.vdocuments.net/reader036/viewer/2022081514/56649edc5503460f94bec557/html5/thumbnails/43.jpg)
• 2D v-grips.
• 3D v-grips.
• 3D vg-grips.
• Unilateral Fixtures.
• D-Space and Deform-Closure.
• Optimal Deform-Closure grasps.
• Assembly line simulation.
Topics completed
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Publications• Computing Deform Closure GraspsK. "Gopal" Gopalakrishnan and Ken Goldberg, submitted to Workshop on Algorithmic Foundations of Robotics (WAFR), Oct. 2004.
• D-Space and Deform ClosureA Framework for Holding Deformable Parts, K. "Gopal" Gopalakrishnan and Ken Goldberg. IEEE International Conference on Robotics and Automation, May 2004.
• Unilateral Fixtures for Sheet Metal Parts with HolesK. "Gopal" Gopalakrishnan, Ken Goldberg, Gary M. Bone, Matthew Zaluzec, Rama Koganti, Rich Pearson, Patricia Deneszczuk, tentatively accepted for IEEE Transactions on Automation Science and Engineering. Revised version December 2003.
• “Unilateral” Fixturing of Sheet Metal Parts Using Modular Jaws with Plane-Cone ContactsK. “Gopal” Gopalakrishnan, Matthew Zaluzec, Rama Koganti, Patricia Deneszczuk and Ken Goldberg, IEEE International Conference on Robotics and Automation, September 2003.
• Gripping Parts at Concave VerticesK. "Gopal" Gopalakrishnan and K. Goldberg, IEEE International Conference on Robotics and Automation, May 2002.
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• Optimal node selection.
- Given a deformable part and FEM model.
- Determine optimal position of a pair of jaws.
- Optimal: Minimize deformation-based metric over all FEM nodes.
Future work
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1 “Unilateral” Fixturing of Sheet Metal Parts Using Modular Jaws with Plane-Cone Contacts, K. “Gopal” Gopalakrishnan, Matthew Zaluzec, Rama Koganti, Patricia Deneszczuk and Ken Goldberg, IEEE International Conference on Robotics and Automation (ICRA), Sep. 2003.
2 D-Space and Deform Closure: A Framework for Holding Deformable Parts, K. "Gopal" Gopalakrishnan and Ken Goldberg, IEEE International Conference on Robotics and Automation (ICRA), May 2004.
3 Computing Deform Closure Grasps, K. "Gopal" Gopalakrishnan and Ken Goldberg, submitted to Workshop on Algorithmic Foundations of Robotics (WAFR), Oct. 2004.
Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr MayMay
QualifyingExam
Ford Research Laboratory:
Designed fixture prototype.
D-Space:Finalized definitions and derived initial results.
Submitted ICRA '04 paper2.
ICRA '03 paper presented1.
Revised T-ASE paper3 and
performed new experiments.
Optimizing deform closure
grasps.
Optimal node selection for
deform-closure.
Dissertation workshop.
Write Thesis.Submitted
WAFR’04 paper
Revise WAFR ’04 paper.
Ford Research Laboratory:Finish prototype and
experiments with new modules and mating parts.
D-Space:Formalize basic
definitions.
Submit ICRA '04 paper.
Improve locator optimization
algorithm
Complete mating parts algorithm.
Submit IROS’04 paper
Locator strategy for multiple
parts.
Cutting planes/heuristics for MIP formulation.
Pro
pose
d tim
elin
e (in
May
’03)
Cur
rent
Tim
elin
e (in
Mar
ch ’0
4)
Assembly line simulation for cost
effectiveness.
Timeline
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http://ford.ieor.berkeley.edu/vggrip/