Algorithms, Models and Metricsfor Workholding using

Part Concavities.

K. Gopalakrishnan

IEOR, U.C. Berkeley.

Workholding

Grasping Fixturing

Conventional Fixtures• Bulky• Complex• Multilateral• Dedicated, Expensive• Long Lead time,

Designed by

human intuition

Ideal Fixtures• Compact• Simplified• Unilateral• Modular, Amortizable• Rapid Setup,

Designed by

CAD/CAM software

Inspiration

• GBL (Global Body Line) (Toyota, 1998-)– Multiple models.– Fewer Jigs/Fixtures.

Workholding: Basic concepts

• Immobility– Any part motion causes

collision

• Force Closure– Any external Wrench

resisted by applying suitable forces

C-Space

C-Space (Configuration Space):

• [Lozano-Perez, 1983]

• Dual representation of part position and orientation.

• Each degree of part freedom is one C-space

dimension.

y

x

/3

(5,4)

y

x

4

5

/3(5,4,- /3)

Phy

sica

l spa

ceC

-Spa

ce

Avoiding Collisions: C-obstacles

• Obstacles prevent parts from moving freely.• Images in C-space are called C-obstacles.

• Rest is Cfree.

Phy

sica

l spa

ceC

-Spa

ce

x

y

Workholding and C-space

• Multiple contacts.

• 1 Contact = 1 C-obstacle.

• Cfree = Collision with no

obstacle.

• Surface of C-obstacle: Contact, not collision.

Phy

sica

l spa

ceC

-Spa

ce

x

y

Form Closure

• A part is grasped in Form Closure if any

infinitesimal motion results in collision.

• Form Closure = an isolated point in C-free.

• Force Closure = ability to resist any wrench. Phy

sica

l spa

ceC

-Spa

ce

x

y

First order Immobility

• Consider escape path.

• Distance to C-obstacles.

• Truncate to First order.

First order Immobility

Phy

sica

l spa

ceC

-Spa

ce

First order ImmobilityIn n dimensions there are n(n+1)/2 DOF:

n translations n(n-1)/2 rotations

For first order immobility, n(n+1)/2+1 are necessary and sufficient

Fast Test for First OrderImmobility

• Any infinitesimal motion

on the plane is a rotation.

• No center of rotation possible for a part in Form-Closure.

• Try to identify possible centers.

+ -

+ -

+-

++

-

-

Workholding: Rigid parts

• Number of contacts– [Reuleaux, 1876], [Somoff, 1900]

– [Mishra, Schwarz, Sharir, 1987], [Markenscoff, 1990]

• Nguyen regions– [Nguyen, 1988]

• Form and Force Closure– [Rimon, Burdick, 1995]

• Immobilizing three finger grasps– [Ponce, Burdick, Rimon, 1995]

[Mason, 2001]

Workholding: Rigid parts

+ -

+-

++-

-

• Caging Grasps– [Rimon, Blake, 1999]

• Summaries of results– [Bicchi, Kumar, 2000]– [Mason, 2001]

• C-Spaces for closed chains– [Milgram, Trinkle, 2002]

• Fixturing hinged parts– [Cheong, Goldberg, Overmars,

van der Stappen, 2002]• Contact force prediction

– [Wang, Pelinescu, 2003]

2D v-grips

Expanding.

Contracting.

AlgorithmStep1: We list all concave vertices.

Step2: At these vertices, we draw normalsto the edges through the jaw’s center.

Step3: We label the 4 regions as shown:

I

II

IV

III

Theorem:

Both jaws lie strictly in the other’s Region I means it is an expanding v-grip

orBoth jaws lie in the other’s Region IV, at least

one strictly, means it is a contracting v-grip

• Maximum change in orientation occurs with one jaw at a vertex.

• The metric is given by |d/dl|.• Using sine rule and neglecting 2nd order terms,

|d/dl| = |tan()/l|

l

l-l

v a v b

Ranking Grips

3D v-grips

3D v-grip:

– Start from a stable initial orientation.

– Close jaws monotonically.

– Deterministic Quasi-static process.

– Final configuration is a 3D v-grip if only vertical translation is possible.

• Input: A CAD model of the part and the position of its center of mass.

• Output: A list (possibly empty) of all 3D v-grips.

Phase I

A candidate 2D v-grip occurs at end of phase I

This is because a minimum height of COM occurs at minimum jaw distance

Phase II

All configurations in Phase II are

candidate 2D v-grips.

Gear & Shaft

We assume that the gear is a cylinder (no teeth)

This part is symmetric about the axis (one redundant degree of freedom). Search is thus reduced to 0 dimensions!

l1 l2

2r

2R

rR

rRlt

2 to allow gripping.

Gear & Shaft: Solution

Work-surface

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

10.6 10.8 11 11.2 11.4 11.6 11.8 12 12.2 P

art O

rien

tati

on

Final position Jaw separation

Grasp progress

Part OrientationShaft Trajectory

Example without Symmetry

Orthogonal views:

Initial 3Dpart orientation Final 3D v-grip

x

y

z

y

Unilateral Fixtures

• “Unilateral” loading of body panels.

• Fixture lies on interior of assembled body.

• Reconfigurable fixtures.

Proposed Modular Components

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

– Support Jaws with Surface Contacts.

Definition: Vg-grips

• Rigid approximation.

• <va, vb> is a vg-grip if:

– Jaws engage part at va, vb.

– Achieves form closure.

• Not easy to check.

Notation: Coordinate Axes

z

x: line joining vertices Projection perpendicular to x

x

Sufficient Test

• Form-closure is achieved if:1. 2D v-grip in x-y plane.

2. 2D v-grip in x-z plane (same nature as 1)3. qij, i=a,b; j=1,2; penetrate cone (angle with axis less

than half-cone angle)

qij

rij

ex qij = ex x rij.

Proof: Outline

• Any displacement of part guarantees jaw

displacement.

• Jaws are rigid.

• Thus Form-closure is achieved.

Quality Metric

• Maximum sensitivity 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

Mirror

Dial Gauge

Experimental Apparatus

A1 A2A3

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 in March 2004 to the IEEE Transactions on Automation Sciences and Engineering.

Secondary Jaws

• Grasp planning: Combining Geometric and Physical models- [Joukhadar, Bard, Laugier,

1994]

• Bounded force-closure- [Wakamatsu, Hirai, Iwata,

1996]

• Minimum Lifting Force- [Howard, Bekey, 1999]

Holding Deformable Parts

Holding Deformable Parts

• Manipulation of flexible sheets- [Kavraki et al, 1998]

• Quasi-static path planning.- [Anshelevich et al, 2000]

• Robust manipulation- [Wada, Hirai, Mori,

Kawamura, 2001]

Deformable parts

• “Form closure” does not apply:

Can always avoid collisions by deforming the part.

• Deformation Space: A Generalization of Configuration Space.

• Based on Finite Element Mesh.

D-Space

Deformable Polygonal parts: Mesh• Planar Part represented as Planar Mesh.• Mesh = nodes + edges + Triangular elements.• N nodes• Polygonal boundary.

D-Space• A Deformation: Position of each mesh node.• D-space: Space of all mesh deformations.• Each node has 2 DOF.• D-Space: 2N-dimensional Euclidean Space.

30-dimensional D-space

Nominal mesh configuration

Deformed mesh configuration

Deformations

• Deformations (mesh configurations) specified as list of translational DOFs of each mesh node.

• Mesh rotation also represented by node displacements.

• Nominal mesh configuration (undeformed mesh): q0.

• General mesh configuration: q.

q0

q

D-Space: Example• Simple example:

3-noded mesh, 2 fixed.• D-Space: 2-dimensional Euclidean Space.• D-Space shows moving node’s position.

x

y

Phy

sica

l spa

ceD

-Spa

ce

q0

Topological Constraints: DT

x

y

Phy

sica

l spa

ceD

-Spa

ce

• Mesh topology maintained.• Non-degenerate triangles only.

DT

Topology violating

deformation

Undeformed part

Allowed deformation

Self Collisions and DT

TDq

TDq

D-Obstacles

x

y

Phy

sica

l spa

ceD

-Spa

ce

• Collision of any mesh triangle with an object.

• Physical obstacle Ai has an image DAi in D-Space.

A1

DA1

D-Space: Example

Phy

sica

l spa

ce

x

y

D-S

pace

• Dfree = DT [ (DAiC)]

Free Space: Dfree

Slice with only node 5 moving.

Part and mesh

1

2 3

5

4

x

y

Slice with only node 3 moving.

x3

y3x5

y5

x5

y5

x5

y5

c

ii

Tfree DADD

Phy

sica

l spa

ceD

-Spa

ce

Nodal displacement

• X = q - q0: vector of

nodal translations.

• Equivalent to moving origin in D-Space to q0.

D- space

q0

q

Potential Energy

• Linear Elasticity.

• K = FEM stiffness matrix. (2N 2N matrix for N nodes.)

• Forces at nodes:

F = K X.

• Potential Energy:

U(q) = (1/2) XT K X

Potential Energy “Surface”

• U : Dfree R0

• Equilibrium: q where U is at a local minimum.

• Stable Equilibrium: q where U is at a strict local minimum.

• Stable Equilibrium = “Deform Closure Grasp”

q

U(q)

Potential Energy Needed to Escape from a Stable Equilibrium• Consider:

Stable equilibrium qA, Equilibrium qB.

• Capture Region:

K(qA) Dfree, such that any configuration in K(qA) returns to qA.

qA

qB

q

U(q)

K( qA )

• UA (qA) = Increase in Potential Energy needed to escape from qA.

= minimum external work needed to escape from qA.

• UA: Measure of “Deform Closure Grasp

Quality”

qA

qB

q

U(q)

UA

Potential Energy Needed to Escape from a Stable Equilibrium

K( qA )

Deform Closure

• Stable equilibrium = Deform Closure where

• UA > 0.

qA

qB

q

U(q)

• Theorem: Definition of Deform closure grasp and UA is frame-invariant.

• Proof: Consider D-spaces D1 and D2.

- Consider q1 D1, q2 D2.

such that physical meshes are identical.

Theorem 1: Frame Invariance

xy

x

y

D1:

D2:

• There exists distance preserving linear transformation T such that

q2 = T q1.

• It can be shown that

UA2(q2) = UA1 (q1)

Theorem 1: Frame Invariance

xy

x

y

Form-closure of rigid part

Theorem 2: Form Closure and Deform Closure

Deform-closure of equivalent deformable part.

Numerical Example

4 Joules 547 Joules

• High Dimensional.

• Computing DT and DAi.

• Exploit symmetry.

Computing Dfree

DAi

Dfree

DTC

• Consider obstacle A and one triangular element.

• Consider the slice De of D, corresponding to the 6 DOF of this element.

• Along all other axes of D, De is constant.

• Extruded cross-section is a prism.

• The shape of DAe is same for all elements.

Computing DAi

1

32

4

5

1

32

4

5

• Thus, DA is the union of identical prisms with orthogonal axes.

• Center of DA is the deformation where the part has been shrunk to a point inside A.

• Similar approach for DT.

Computing DAi

1

32

4

5

1

32

4

5+

• Given:

Pair of contact nodes.

• Determine:

Optimal jaw separation.

Optimal?

Two Point Deform Closure Grasps

M

E

n0

n1

• If Quality metric Q = UA.

• Maximum UA trivially at = 0

Naïve Quality Metric

New Quality Metric

• Plastic deformation.

• Occurs when strain exceeds eL.

New Quality Metric

• Additional work UL done by jaws for plastic deformation.

• New Q = min { UA, UL }

Stress

Strain

Plastic Deformation

A

B

C

eL

A

B

C

UL

• Additional input:

eL : Elastic limit strain.

: allowed error in quality metric.

• Additional assumptions:

Sufficiently dense mesh.

Linear Elasticity.

No collisions

Problem Description

M, K

E

n0

n1

Potential Energy vs. ni

nj

kij

Pot

entia

l Ene

rgy

(U)

Distance between FEM nodes

Undeformed distance

Expanding

Contracting

• 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

A

B

C

E

F

G

D

Contact Graph: Edges

Adjacent mesh nodes:

A

B

C

D

E

F

G

H

H

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

Contact Graph: Edges

Non-adjacent mesh nodes:

• Traversal with minimum increase in energy.

• FEM solution with two mesh nodes fixed.

ni

nj

Deformation at Points of Interest

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

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* ,

)

U (

vo,

), U

( v

*,

)

Threshold Potential Energy

U ( v*, )

U ( vo, )

UA ( )

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

UA (

)

, UL (

)

Quality Metric

UA ( )UL ( )

Q ( )

• 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

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

Algorithm for UA(i)

Algorithm for UA(i)

U

Vertex v (traversed on path of minimum work)

U(v)

U(v*)

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.

• 2D v-grips

- Grasping at concavities.

- New Quality metric.

- Fast necessary and sufficient conditions.

• 3D v-grips:

- Gripping at projection concavities.

- Fast path planning.

Summary

• Unilateral Fixtures:

- New type of fixture: concavities at concavities

- New Quality metric.

- Combination of fast geometric and numeric approaches.

• D-Space and Deform-Closure:

- Defined workholding for deformable parts.

- Frame invariance.

- Symmetry in D-Space.

Summary

• Two Jaw Deform-Closure grasps:

- Quality metric.

- Fast algorithm for given jaw separation.

- Error bounded optimal separation.

Summary