Gripping Parts at Concave Vertices

K. “Gopal” Gopalakrishnan

Ken Goldberg

U.C. Berkeley

• Inspiration

• Related Work

• 2D v-grips

• 3D v-grips

• Conclusion

Outline

Inspiration

Part’s position and orientation are fixed.

Advantages

• Inexpensive

• Lightweight

• Small footprint

• Self-Aligning

• Multiple grips

• Inspiration

• Related Work

• 2D v-grips

• 3D v-grips

• Conclusion

Outline

Basics of Grasping

• Summaries of results in grasping– [Mason, 2001]

– [Bicchi, Kumar, 2000]

• Rigorous definitions of Form and Force Closure– [Rimon, Burdick, 1996]

– [Mason, 2001]

[Mason, 2001]

Orders of Form-Closure• First & second order form-closure

– [Rimon, Burdick, 1995]

• For first order form-closure, n(n+1)/2+1 contacts are necessary and sufficient– [Realeaux, 1963]

– [Somoff, 1900]

– [Mishra, Schwarz, Sharir, 1987]

– [Markenscoff, 1990]

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]

Other Related Work

• Inspiration

• Related Work

• 2D v-grips

• 3D v-grips

• Conclusion

Outline

2D v-grips

Expanding.

Contracting.

2D Problem DefinitionWe first analyze two-dimensional parts on the

horizontal plane.

Assumptions:

• Rigid Part.

• No out-of-plane rotation.

• Polygonal perimeter and Polygonal holes.

• Frictionless contacts.

• Zero Jaw radii.

2D Problem Definition

Let va and vb be two

concave vertices.

We call the unordered pair <va, vb> a v-grip if

jaws placed at these vertices will provide frictionless form-closure of the part.

va vb

2D Problem Definition

Input: Vertices of polygons representing the part’s boundary and/or holes, in counter-clockwise order, and jaw radius.

Output: A list (possibly empty) of all v-grips sorted by quality measure.

2D Algorithm

Step1: 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

Conditions for V-grip

Configurations like this are also contractingv-grips:

The Distance Function: (s1,s2)

• Represents the distance between any 2 points on the part’s perimeter.

• The points are represented by an arclength parameter s.

• [Blake, Taylor, 1993] & [Rimon, Blake, 1998]

O

s

S1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

S1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

S10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Proving the Theorem

The proof lies in proving the equivalence of these 4 statements: For any pair of concave vertices <v1,v2>,

• A: v1 and v2 both lie strictly in the other’s region I.

• B: (v1,s2) and (s1,v2) are local maxima at (v1,v2).

• C: (v1,v2) is a strict local maximum of (s1,s2).

• D: The grasp at v1,v2 is an expanding v-grip.

And similarly for contracting grasps (with region IV and minima)

Proof: Sketch

• A B: The shortest distance from a point to a line is along the perpendicular.

• B C:

v2

v1 v'1

v'2

I

P

Q Rv

Proof: Sketch (Contd.)

• C D: Worst case analysis.

– Any motion results in a collision.

• D C: Assume form-closure but not C.

– Case I: Contracting v-grip.

– Form-Closure at non-extremum:

Slide part along constant contour.

v1

v2

2D Algorithm

Thus, If 2 vertices lie in region I of each other (A is true), an expanding v-grip is achieved (D is true).

We enumerate all pairs of Concave vertices and apply theorems 1 and 2 for each pair to check for v-grips to generate an unranked list of v-grips.

Ranking Grips

• Based on sensitivity to small disturbances.

• Relax the jaws slightly. (Change the distance between them.)

• Consider maximum error in orientation due to this.

l

l-l

v a v b

• 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

Metric evaluates grasp AC as better than BD

Ranking Grips: Example

D

A C

B

Computational Complexity

• O(n) to identify k concave vertices.

• O(k2) to list v-grips and evaluate metrics.

• O(k2 log k) to sort list.

• Total: O(n + k2 log k) for 0 radius

Jaws with non-zero radii

• Jaw has a radius r

• The part is transformed with a Minkowsky addition, offsetting the polygons with a disk of radius r.

• Apply 2D algorithm to transformed part.

• O(n log n) time required.

• Inspiration

• Related Work

• 2D v-grips

• 3D v-grips

• Conclusion

Outline

3D v-grips

Initial orientation Final orientation after v-grip

3D v-grips

Initial orientation Final orientation after v-grip

3D Problem Definition In 3D, v-grips can be achieved with a pair of frictionless

vertical cylinders and a planar work-surface.

Assumptions:

• Rigid part

• Part is defined by a polyhedron.

• Frictionless contacts

• Jaws have zero radii.

3D Problem Definition

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.

3D Algorithm

• We describe a numerical algorithm for computing all 3D v-grips.

• The grasp occurs in 2 phases:

– Rotation in plane

– Rotation out of plane

• We find part trajectory during the second phase.

We describe the algorithm for contracting 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.

3D Algorithm

• Enumerate starting positions.

• Identify 2D v-grips of projections.

• Compute Phase II trajectory:

– Incrementally close jaws.

– Find local minimum of COM height among candidate 2D v-grips.

– Check termination criteria.

1. 3D v-grip.

3D Algorithm: Termination.

3. The part falls away.

All termination conditions checked in wrench-space.

2. 3D equilibrium grip.

Part can move but Gripper cannot close.

Example: Gear & Shaft

Orthogonal views:

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

3D Example without Symmetry

Orthogonal views:

Initial 3Dpart orientation Final 3D v-grip

x

y

z

y

3D Example Part Trajectory

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

53 54 55 56 57 58 59

theta

phi

Part

Ori

enta

tion

Jaw separation (w)

Grasp Progress

Final orientation

• Inspiration

• Related work

• 2D v-grips

• 3D v-grips

• Conclusion

Outline

Conclusions: 2D

• Fast algorithm to find all 2D v-grips

• Quality Metric that is fast to compute and is consistent with intuition in most cases.

• Extended to non-zero jaw radii.

• Implemented in Java applet available online.

Conclusions: 3D

• 3D algorithm determines all 3D v-grips.

• The algorithm reduces a 6D search to a 1D search.

• Critical part parameters for Design for Mfg

http://alpha.ieor.berkeley.edu/v-grips