Gripping Sheet Metal Parts at Vertices

K. Gopalakrishnan

A Project for CS 287

Outline

• Introduction and Motivation

• Related Work

• Gripping Sheet Metal Parts

• Quality Metric

• Extensions

• Analysis of Results

• Conclusions & Future Work

Introduction

• Grooves in cylindrical jaws used to grip sheet-metal parts

Jaw

Part

Motivation

• Simple reliable grips.

• Form-Closure achieved in 3D.

• Self-aligning grips.

• Very small Footprint.

Outline

• Introduction and Motivation

• Related Work

• Gripping Sheet Metal Parts

• Quality Metric

• Extensions

• Analysis of Results

• Conclusions & Future Work

Related Work

• V-grips (Form-closure only in 2D).

[Gopalakrishnan, Goldberg, 2002]

• Multi DOF grippers in Robotic Fixtureless

Assembly.[Plut, Bone, 1997]

Related Work

• Form-Closure & Force Closure

– [Mason, 2001]

– [Rimon, Burdick, 1995 & 1996]

• Necessary & Sufficient Conditions (number of

contacts)

– [Realeaux, 1963]

– [Somoff, 1900]

– [Mishra, Schwarz, Sharir, 1987]

– [Markenscoff, 1990]

Related Work

Caging Grasps [Rimon, Blake, 1999]

Efficient Computation of Nguyen regions [Van der Stappen, Wentink, Overmars, 1999]

Multi-DOF Grips for Robotic Fixtureless Assembly [Plut, Bone, 1996 & 1997]

Outline

• Introduction and Motivation

• Related Work

• Gripping Sheet Metal Parts

• Quality Metric

• Extensions

• Analysis of Results

• Conclusions & Future Work

Problem Definition

We first analyze piecewise planar sheet metal parts.

Assumptions:

• Frictionless contacts.

• Linear part edges (perimeter & holes).

• First order form-closure only.

• Jaws are vertical parallel cylinders with horizontal V-shaped grooves.

• Part thickness negligible.

V-shaped Grooves

• Consists of intersection of 2 frustums as shown:

• Forces exerted normal to each frustum at point of contact.

Jaw

Part

Analysis of Free Motion

• Any motion can be broken down to component

motions:

• Translations: ex, ey, ez

• Rotations: rx, ry, rz

• Any infinitesimal motion = sequence of

components.

Coordinate System

• x-z plane contains axes of jaws.

• Jaws close along x.

• Jaw axes parallel to z.

x

z y

Results from V-grips

Step1: We consider a pair of 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 strictly in the other’s Region IV

means it is a contracting v-grip.

Form-Closure in the x-y plane

• Test of 2D v-grip is applied.

• Ensures that distance decreases for ex, ey, rz.

• Next, we consider ex, ez, ry.

x

z y

Form-Closure in the x-z plane

• Test of 2D v-grip is applied in x-z plane.

• However, consider the jaws as the part and part as

contacts.

• Ensures that distance decreases for ex, ez, ry.

• For infinitesimal motions, vertices still lie strictly

in corresponding region.

x

z y

Rotation about x axis

• Rotation about x axis always increases.

• Reason: Only parts which are horizontal at

contacts are considered.

Theorem

• Any sheet-metal part which is horizontal at the

points of contact is held in Form-Closure if

– Part is in expanding/contracting 2D v-grip

for horizontal projection.

– Jaws are held in contracting/expanding 2D

v-grip by the part in the plane containing

axes of jaws.

• These conditions are sufficient but not necessary.

Algorithm

• For every face of part, generate all pairs of

concave vertices on faces parallel to it.

• Test each pair for Form-Closure.

• If in Form-Closure, add to list of grips.

• Sort list by quality metric.

Outline

• Introduction and Motivation

• Related Work

• Gripping Sheet Metal Parts

• Quality Metric

• Extensions

• Analysis of Results

• Conclusions & Future Work

Quality Metric for v-grips

• Based on sensitivity to relaxing of jaws.

Maximum change in orientation with one jaw still at a vertex.

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

l

l-l

v a v b

Suggested Metric

• Sensitivity of orientation to relaxing of jaws.

• Consider all values of |tan()/l| for v-grips in x-y

and x-z planes.

• Take maximum value for worst change in

orientation.

• Intuitive but not rigorous.

Outline

• Introduction and Motivation

• Related Work

• Gripping Sheet Metal Parts

• Quality Metric

• Extensions

• Analysis of Results

• Conclusions & Future Work

Friction

• Is it possible to extend by just adding friction cone

to the regions?I

II

IV

III

Diagonal Planes at Contacts

• Horizontality is used only when analyzing rx.

• If distance can be shown to reduce otherwise,

horizontal assumption can be removed.

• E.g. Plane contains a line parallel to y-axis.

Outline

• Introduction and Motivation

• Related Work

• Gripping Sheet Metal Parts

• Quality Metric

• Extensions

• Analysis of Results

• Conclusions & Future Work

Strength of Conditions

• Compare results generated by sufficient conditions

with those generated by brute force.

lh

Benchmarking Part

1

Comparison

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0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25

Min

imum

hal

f-v

angl

e of

fru

stum

l

Sufficient Conditions

Brute Force

ComparisonM

inim

um h

alf-

v an

gle

of f

rust

um

h

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

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Brute Force

Sufficient Conditions

Outline

• Introduction and Motivation

• Related Work

• Gripping Sheet Metal Parts

• Quality Metric

• Extensions

• Analysis of Results

• Conclusions & Future Work

Future Work

• Necessary & Sufficient Conditions.

• Acquisition.

• Trajectory Prediction.

• Friction.

• Second-order form-closure.

• Design of jaws for part.