1

Dissertation Workshop:

Algorithms, Models and Metricsfor the Design of Workholding Using

Part ConcavitiesK. Gopalakrishnan

IEOR, U.C. Berkeley.

2

• 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

4

Modular Fixturing

• Existence and algorithm: Brost and Goldberg, 1996.

5

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.

6

2D v-grips

Expanding.

Contracting.

7

• 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.

9

va

I

IIIII

IV

3D vg-grips: Phase I

z

• Fast geometric tests.

10

3D vg-grips: Phase II

11

Examples

12

• Review

• Unilateral Fixtures

• Deformation Space

• Two Point Deform Closure Grasps

• Assembly Line Simulation

• Conclusion

Outline

13

Ford Motor Co.

++

14

Ford D219 Door model

• Datum points.

• Spot welding access.

• Variation in tolerances.

• Multiple parts.

• Clamping mechanism.

15

Ford D219 Door model

WELDING

A4C

A1C

A2C

A3R

A5R

A6C

A7C

A8RA9R

B1CB2C

B3C

B4R B5R

16

• 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

17

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

18

Apparatus: Schematic

BaseplateTrack

Slider Pitch-Screw

Mirror

Dial Gauge

19

Experimental Apparatus

A1 A2A3

20

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.

21

• Review

• Unilateral Fixtures - Experiments

• Deformation Space

• Two Point Deform Closure Grasps

• Assembly Line Simulation

• Conclusion

Outline

22

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

• Generalization of C-Space.

• Based on FEM model.

D-Space

23

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)

24

Obstacles• Obstacles prevent parts from moving freely.

• Images in C-space are called C-obstacles.• Rest is Free Space.

25

Mesh M

Part E

Deformable parts: FEM• Part represented as Mesh.• Stiffness properties assigned. F = K X.• X = nodal displacement vector.

26

Topology violating

configuration

Undeformed part Allowed deformation

Avoiding mesh collisions: DT

Example for for system of parts

27

Avoiding collisions: D-obstacles

No collision Collision

Collision No collision

(with obstacle)

1 3

42

Slice of complement of D-obstacle.

Nodes 1,2,3 fixed.

28

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

29

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

30

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)

31

• Frame invariance.

• Form-closure Deform-closure of

equivalent deformable part.

TheoremsM

E

x1

y1

x 1

y 1

32

Numerical Example

4 Joules 547 Joules

33

• 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

34

• Review

• Unilateral Fixtures - Experiments

• Deformation Space

• Two Point Deform Closure Grasps

• Assembly Line Simulation

• Conclusion

Outline

35

• Given:

Deformable polygonal part.

FEM model.

Pair of contact nodes.

• Determine:

Optimal jaw separation.

Optimal?

Problem Description

M

E

n0

n1

36

• If Q = UA:

Quality metric

37Q = min { UA, UL }

Stress

Strain

Plastic Deformation

L

Quality metric

38

• Given:

Deformable polygonal part.

FEM model.

Pair of contact mesh nodes.

• Assume:

Sufficiently dense mesh.

Linear Elasticity.

Problem Description

M, K

E

n0

n1

39

• 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

40

A

B

C

E

F

G

D

Contact Graph: Edges

Adjacent mesh nodes:

A

B

C

D

E

F

G

H

H

41

Potential Energy vs. ni

nj

kij

Pot

entia

l Ene

rgy

(U)

Distance between FEM nodes

Undeformed distance

Expanding

Contracting

42

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

43

Contact Graph: Edges

Non-adjacent mesh nodes:

44

• Traversal with minimum increase in energy.

• FEM solution with two mesh nodes fixed.

ni

nj

Deformation at Points of Interest

45

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

46

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* ,

)

47

U (

vo,

)

, U (

v*,

)

Threshold Potential Energy

U ( v*, )

U ( vo, )

UA ( )

UA ( ) = U ( v*, ) - U ( vo, )

48

UA (

)

, UL (

)

Quality Metric

UA ( )UL ( )

Q ( )

49

• Possibly exponential

number of pieces.

• Sample in intervals of .

• Error bound on max. Q =

* max { 0(ni, nj) *

kij }

Numerical Sampling

Q

(

)

50

• 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

51

V -

Algorithm for UA(i)

• List of known least-work nodes: .

• List of estimated least work for vertices adjacent to .

52

V -

Algorithm for UA(i)

53

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.

54

• Review

• Unilateral Fixtures - Experiments

• Deformation Space

• Two Point Deform Closure Grasps

• Assembly Line Simulation

• Conclusion

Outline

55

• Review

• Unilateral Fixtures - Experiments

• Deformation Space

• Two Point Deform Closure Grasps

• Assembly Line Simulation

• Conclusion

Outline

56

• 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

57

• 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

58

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.

59

• 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

60

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/