1 dissertation workshop: algorithms, models and metrics for the design of workholding using part...
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Dissertation Workshop:
Algorithms, Models and Metricsfor the Design of Workholding Using
Part ConcavitiesK. Gopalakrishnan
IEOR, U.C. Berkeley.
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• Review
• Unilateral Fixtures - Experiments
• Deformation Space
• Two Point Deform Closure Grasps
• Assembly Line Simulation
• Conclusion
Outline
3
• 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,- /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
8
3D vg-grips
• Use plane-cone contacts:– Jaws with conical grooves: Edge contacts.– Support Jaws with Surface Contacts.
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va
I
IIIII
IV
3D vg-grips: Phase I
z
• Fast geometric tests.
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3D vg-grips: Phase II
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Examples
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• Review
• Unilateral Fixtures
• Deformation Space
• Two Point Deform Closure Grasps
• Assembly Line Simulation
• Conclusion
Outline
13
Ford Motor Co.
++
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Ford D219 Door model
• Datum points.
• Spot welding access.
• Variation in tolerances.
• Multiple parts.
• Clamping mechanism.
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Ford D219 Door model
WELDING
A4C
A1C
A2C
A3R
A5R
A6C
A7C
A8RA9R
B1CB2C
B3C
B4R B5R
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• Complete algorithm. BFS.
• Scale independent quality metric.
• New Experiments.
• Stay-in and stay-out regions (for datum points).
• Rigorous algorithm and clarification of concepts.
Unilateral Fixtures: Improvements
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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
Mirror
Dial Gauge
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Experimental Apparatus
A1 A2A3
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0.0250.020.0150.010.005
A1-A3
77.43
A1-A2
31.74
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 relaxation (inches)
Experiment Results
"Unilateral Fixtures for Sheet Metal Parts with Holes" K. Gopalakrishnan, Ken Goldberg, Gary M. Bone, Matthew, Zaluzec, Rama Koganti, Rich Pearson, Patricia Deneszczuk. Accepted to the IEEE Transactions on Automation Sciences and Engineering.
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• Review
• Unilateral Fixtures - Experiments
• Deformation Space
• Two Point 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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C-Space
• C-Space: describes position and orientation.
• Each DOF is a coordinate axis.
y
x
/3
(5,4)
y
x
q
4
5
/3(5,4,- /3)
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Obstacles• Obstacles prevent parts from moving freely.
• Images in C-space are called C-obstacles.• Rest is Free Space.
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Mesh M
Part E
Deformable parts: FEM• Part represented as Mesh.• Stiffness properties assigned. F = K X.• X = nodal displacement vector.
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Topology violating
configuration
Undeformed part Allowed deformation
Avoiding mesh collisions: DT
Example for for system of parts
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Avoiding collisions: D-obstacles
No collision Collision
Collision No collision
(with obstacle)
1 3
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Slice of complement of D-obstacle.
Nodes 1,2,3 fixed.
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Free Space: Dfree
Slice with nodes 1-4 fixedPart and mesh
1
2 3
5
4
x
y
x5
y5
x5
y5
x5
y5
Slice with nodes 1,2,4,5 fixed
x3
y3
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Nominal configuration
Deformed configuration
D-Space and Potential Energy
• Nodal displacement:
Distance preserving transformation.
X = T (q - q0)
q0
q
• For FEM with linear elasticity and linear interpolation,
U = (1/2) XT K X
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Deform Closure
qA
qB
• Equilibrium configuration:
Local minimum of U.
• Increase in potential energy UA needed to release part.
• Deform Closure if UA > 0.q
U(q)
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• Frame invariance.
• Form-closure Deform-closure of
equivalent deformable part.
TheoremsM
E
x1
y1
x 1
y 1
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Numerical Example
4 Joules 547 Joules
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• D-Obstacle symmetry
- Prismatic extrusion of identical shape along multiple axes.
• Symmetry of Topology preserving space (DT).
Symmetry in D-Space
1
32
4
4
21
3
5
5
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• Review
• Unilateral Fixtures - Experiments
• Deformation Space
• Two Point 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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• If Q = UA:
Quality metric
37Q = min { UA, UL }
Stress
Strain
Plastic Deformation
L
Quality metric
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• Given:
Deformable polygonal part.
FEM model.
Pair of contact mesh nodes.
• Assume:
Sufficiently dense mesh.
Linear Elasticity.
Problem Description
M, K
E
n0
n1
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• Points of interest: contact at mesh nodes.
• Construct a graph:
Each graph vertex = 1 pair of perimeter mesh nodes.
p perimeter mesh nodes.
O(p2) graph vertices.
Contact Graph
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A
B
C
E
F
G
D
Contact Graph: Edges
Adjacent mesh nodes:
A
B
C
D
E
F
G
H
H
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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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Contact Graph
0.1
0.4
0.71
1.3
1.6
1.9
2.2
2.5
2.8
3.1
3.4
3.74
4.3
4.6
4.9
5.2
5.5
5.8
0.1
0.8
1.5
2.2
2.9
3.6
4.3
5
5.7
0
0.1
0.2
0.3
0.4
0.5
0.6
REDO
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Contact Graph: Edges
Non-adjacent mesh nodes:
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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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U (
v(n
i, n j),
)
Peak Potential Energy Given release path
0.1
0.4
0.71
1.3
1.6
1.9
2.2
2.5
2.8
3.1
3.4
3.74
4.3
4.6
4.9
5.2
5.5
5.8
0.1
0.8
1.5
2.2
2.9
3.6
4.3
5
5.7
0
0.1
0.2
0.3
0.4
0.5
0.6
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0.1
0.4
0.71
1.3
1.6
1.9
2.2
2.5
2.8
3.1
3.4
3.74
4.3
4.6
4.9
5.2
5.5
5.8
0.1
0.8
1.5
2.2
2.9
3.6
4.3
5
5.7
0
0.1
0.2
0.3
0.4
0.5
0.6
Peak Potential Energy: All release paths
U (
v* ,
)
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U (
vo,
)
, U (
v*,
)
Threshold Potential Energy
U ( v*, )
U ( vo, )
UA ( )
UA ( ) = U ( v*, ) - U ( vo, )
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UA (
)
, UL (
)
Quality Metric
UA ( )UL ( )
Q ( )
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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
0.1
0.4
0.71
1.3
1.6
1.9
2.2
2.5
2.8
3.1
3.4
3.74
4.3
4.6
4.9
5.2
5.5
5.8
0.1
0.8
1.5
2.2
2.9
3.6
4.3
5
5.7
0
0.1
0.2
0.3
0.4
0.5
0.6
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V -
Algorithm for UA(i)
• List of known least-work nodes: .
• List of estimated least work for vertices adjacent to .
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V -
Algorithm for UA(i)
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Numerical Example
Undeformed
= 10 mm.
Optimal
= 5.6 mm.
Rubber foam.
FEM performed using ANSYS.
Computing Deform Closure Grasps, K. "Gopal" Gopalakrishnan and Ken Goldberg, submitted to Workshop on Algorithmic Foundations of Robotics (WAFR), Oct. 2004.
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• Review
• Unilateral Fixtures - Experiments
• Deformation Space
• Two Point Deform Closure Grasps
• Assembly Line Simulation
• Conclusion
Outline
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• Review
• Unilateral Fixtures - Experiments
• Deformation Space
• Two Point Deform Closure Grasps
• Assembly Line Simulation
• Conclusion
Outline
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• 2D v-grips: Fast necessary and sufficient algorithm.
• 3D v-grips: Fast path planning.
• Unilateral Fixtures:
- Combination of fast geometric and numeric approaches.
- Quality metric.
Contributions
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• D-Space and Deform-Closure:
- Defined workholding for deformable parts.
- Frame invariance.
- Symmetry in D-Space.
• Two Jaw Deform-Closure grasps:
- Fast algorithm for given jaw separation.
- Error bounded optimal separation.
• Assembly line simulation: Cost analysis for modular tooling.
Contributions
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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
61
http://ford.ieor.berkeley.edu/vggrip/