Algorithms, Models and Metricsfor the Design of Workholding Using

Part Concavities

K. Gopalakrishnan, Ken Goldberg,

IEOR, U.C. Berkeley.

• Review

• Unilateral Fixtures - Experiments

• Deformation Space

• Optimal Deform Closure Grasps

• Assembly Line Simulation

• Conclusion

Outline

• Bulky• Complex• Multilateral• Dedicated, • Expensive• Long Lead time• Designed by

human intuition

Conventional Fixtures

Modular Fixturing

• Existence and algorithm: Brost and Goldberg, 1996.

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.

2D v-grips

Expanding.

Contracting.

• N-2-1 approachCai et al, 1996.

• Decoupling beam elementsShiu et al, 1997.

• Manipulation of sheet metal partKavraki et al, 1998.

Deformable parts

3D vg-grips

• Use plane-cone contacts:– Jaws with conical grooves: Edge contacts.– Support Jaws with Surface Contacts.

Examples

• Review

• Unilateral Fixtures - Experiments

• Deformation Space

• Optimal Deform Closure Grasps

• Assembly Line Simulation

• Conclusion

Outline

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

Apparatus: Schematic

BaseplateTrack

Slider Pitch-Screw

Dial Gauge

Mirror

Experimental Apparatus

A1 A2A3

-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

Ford D219 Door model

++

A4C

A1C

A2C

A3R

A5R

A6C

A7C

A8RA9R

B1CB2C

B3C

B4R B5R

D219 Door: Contact set

• Review

• Unilateral Fixtures - Experiments

• Deformation Space

• Optimal Deform Closure Grasps

• Assembly Line Simulation

• Conclusion

Outline

• Lack of definition of fixtures/grasps for deformable parts.

• Generalization of C-Space.

• Based on FEM model.

D-Space

Topology violating

configuration

Undeformed part Allowed deformation

Topology Preservation

Example for for system of parts

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

• 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.

• Frame invariance.

• Form-closure Deform-closure of

equivalent deformable part.

Results

Numerical Example

1

2

3

4

3

4

1

2

4

2

3

11

4

3

2

4 Joules 547 Joules

• 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

• Review

• Unilateral Fixtures - Experiments

• Deformation Space

• Optimal Deform Closure Grasps

• Assembly Line Simulation

• Conclusion

Outline

• Given:

Deformable polygonal part.

FEM model.

Pair of contact nodes.

• Determine:

Optimal jaw separation.

Optimal?

Problem Description

M

E

n0

n1

• Consider:

- Threshold P.E. UA.

- Additional P.E. needed for plastic deformation UL.

• Q = min { UA, UL }

Quality metric

n0

n1

L

• 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

• Traversal with minimum increase in energy.

• FEM solution with two mesh nodes fixed.

ni

nj

Deformation at Points of Interest

Potential Energy vs. ni

nj

kij

Pot

entia

l Ene

rgy

(U)

Distance between FEM nodes

Undeformed distance

Expanding

Contracting

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:

U (

v(n

i, n j),

)

Peak P.E.: Given release path

Peak P.E.: All release paths

U (

v* ,

)

U (

vo,

)

, U (

v*,

)

Threshold P.E.

UA (

)

, UL (

)

Quality Metric

• Possibly exponential

number of pieces.

• Sample in intervals of .

• Error bound on max. Q =

* max { 0(ni, nj) *

kij }

Numerical Sampling

Q

(

)

• 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

V -

Algorithm for UA(i)

V -

Algorithm for UA(i)

Numerical Example

Undeformed

= 10 mm.

Optimal

= 5.6 mm.

• Review

• Unilateral Fixtures - Experiments

• Deformation Space

• Optimal Deform Closure Grasps

• Assembly Line Simulation

• Conclusion

Outline

• Review

• Unilateral Fixtures - Experiments

• Deformation Space

• Optimal Deform Closure Grasps

• Assembly Line Simulation

• Conclusion

Outline

• 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

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.

• 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

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

http://ford.ieor.berkeley.edu/vggrip/