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On the Existence of Form-Closure Configurations on a Grid

A. Frank van der Stappen

Presented by K. Gopalakrishnan

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• Inspiration

• Introduction

• Form Closure on a Grid

• Extensions

• Conclusion

Outline

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• Modular fixturing

• Existence of Fixtures– [Zhuang, Goldberg, 96]

• Purely geometric approach

Inspiration

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• Inspiration

• Introduction & Related Work

• Form Closure on a Grid

• Extensions

• Conclusion

Outline

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Ways to hold parts

• Form Closure– Any part motion causes

collision

• Force Closure– Any external Wrench resisted

by applying suitable forces

[Mason, 2001]

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C-Space

C-Space (Configuration Space)Describes position and orientation.

We represent each degree of freedom of a part as a C-space coordinate.

y

x

/3

(5,4)

y

x

4

5

/3(5,4,-/3)

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ObstaclesObstacles prevent parts from moving freely.

These images of obstacles in C-space are called C-obstacles.

The rest is Free Space.

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Form Closure

Form Closure:

All adjacent points are collisions

Difficult to Determine.Hence, distance from C-obstacles is used. [Rimon & Burdick, 96]

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Form Closure in C-space

“Active” obstacles only

Small motion cannot result in reduction of distance.

Distance not easy to compute either.

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First order Form-Closure• Consider Infinitesimal motion.

• Taylor Expansion for distance.

• Truncate to First order.

• Equivalent to replacing surfaces by tangents.

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First order Form-Closure

In n dimensions there are n(n+1)/2 DOF. n translations n(n-1)/2 rotations

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

– [Somoff, 1900]

– [Mishra, Schwarz, Sharir, 1987]

– [Markenscoff, 1990]

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Fast Test for First OrderForm-Closure

• 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.

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Fast Test

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Fast Test

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Fast Test

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Fast Test

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Failure of First order Form-Closure

Higher order terms neglected.

Uncertainty when first order term is 0.

We need to look at Second order terms.

For generic parts in 2 or 3 dimensions, 3 or 4 point contacts are sufficient. [Rimon, Burdick, 1995]

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Second Order Form-Closure

Example: 1st order approximation:

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Second order Form-Closure

First order approximation allows a pure rotation about the centroid.

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• Inspiration

• Introduction

• Form Closure on a Grid

• Extensions

• Conclusion

Outline

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Problem Definition

• Rigid polygonal part.

• Frictionless point contacts.

• No parallel edges.

To prove that:

The part can be held in form-closure by 4 point-contacts that lie on 2 perpendicular lines.

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Notation

• P Polygonal Part

• e any edge of P

• a any point on e

• le(a) normal to e at a

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Proof

• Consider the largest inscribed circle C in P.

• C has center m

• Let this touch the part at a1, a2, a3.

a1 a2

a3

m

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Proof

• 3 radial vectors positively span R2.

• m only possible center of rotation.

• Contacts obtained in neighborhood of ai.

a1 a2

a3

m

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Case I

• At least 1 ai is a vertex.

• Has to be Concave.

a1 a2

a3

m

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Case I

• If no normal at vertex passes thru m, we are done

a1 a2

a3

m

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Case I

• If a normal at the vertex passes through m,

a1 a2

a3

m

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Case I

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Case II

• All 3 contacts at edges.

• Again, m is the only possible

• Let the least angle between radii be between ma1 and ma3.

• We choose a4 to make a1a4 and a2a3 gridlines.

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Case II-1

• m is to the left of normal at a4.

• Move the horizontal line slightly upward.

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Case II-2

• m is to the right of normal at a4.

• Rotate both axes by equal angles.

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Case II-3

• m is to on the normal at a4.

• Coordinated translation of horizontal and vertical gridlines.

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• Inspiration

• Introduction

• Form Closure on a Grid

• Extensions

• Conclusion

Outline

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Parallel Edges

• Counterexamples.

• Topology not maintained.

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Contact at Grid Vertex

• Not feasible (e.g. convex polygons).

• Additional contact on another line required.

• Feasibility not guaranteed.

• Requires large enough part.

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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.

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Jaws with non-zero radii

• Contact at Convex vertex not desired.

• As radius increases, Minkowsky sum becomes disc.

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• Inspiration

• Introduction

• Form Closure on a Grid

• Extensions

• Conclusion

Outline

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Conclusion

• Existence of solution proved under assumptions.

• Validity of assumptions shown.

• Non zero radius incorporated.

• Extension to non-polygonal parts open.

• Accessibility to clamps ignored.

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