Gripping Parts at Concave Vertices

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Gripping Parts at Concave Vertices

• K. Gopal Gopalakrishnan
• Ken Goldberg
• U.C. Berkeley

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Outline

• Inspiration
• Related Work
• 2D v-grips
• 3D v-grips
• Conclusion

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Inspiration
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Parts position and orientation are fixed.
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Advantages

• Inexpensive
• Lightweight
• Small footprint
• Self-Aligning
• Multiple grips

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Outline

• Inspiration
• Related Work
• 2D v-grips
• 3D v-grips
• Conclusion

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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
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Orders of Form-Closure

• First second order form-closure
• Rimon, Burdick, 1995
• For first order form-closure, n(n1)/21 contacts
  are necessary and sufficient
• Realeaux, 1963
• Somoff, 1900
• Mishra, Schwarz, Sharir, 1987
• Markenscoff, 1990

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Other 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
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Outline

• Inspiration
• Related Work
• 2D v-grips
• 3D v-grips
• Conclusion

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2D v-grips

Expanding.
Contracting.
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2D Problem Definition

• We 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.

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2D Problem Definition

• Let va and vb be two concave vertices.
• We call the unordered pair ltva, vbgt a v-grip if
  jaws placed at these vertices will provide
  frictionless form-closure of the part.

va
vb
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2D Problem Definition
Input Vertices of polygons representing the
parts boundary and/or holes, in
counter-clockwise order, and jaw radius. Output
A list (possibly empty) of all v-grips sorted by
quality measure.
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2D Algorithm
Step1 We list all concave vertices. Step2 At
these vertices, we draw normals to the edges
through the jaws center. Step3 We label the
4 regions as shown
Theorem Both jaws lie strictly in the others
Region I means it is an expanding v-grip or Both
jaws lie in the others Region IV, at least one
strictly, means it is a contracting v-grip
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Conditions for V-grip
Configurations like this are also
contracting v-grips
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The Distance Function s(s1,s2)

• Represents the distance between any 2 points
  on the parts perimeter.
• The points are represented by an arclength
  parameter s.
• Blake, Taylor, 1993 Rimon, Blake, 1998

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Proving the Theorem
The proof lies in proving the equivalence of
these 4 statements For any pair of concave
vertices ltv1,v2gt,

• A v1 and v2 both lie strictly in the others
  region I.

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

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

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

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

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

• B ? C

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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 s contour.

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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.
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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.

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Ranking Grips

• Maximum change in orientation occurs with one jaw
  at a vertex.
• The metric is given by dq/dl.
• Using sine rule and neglecting 2nd order terms,
• dq/dl tan(f)/l

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Ranking Grips Example
D
C
A
B
Metric evaluates grasp AC as better than BD
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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

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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.
• Apply 2D algorithm to transformed part.
• O(n log n) time required.

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Outline

• Inspiration
• Related Work
• 2D v-grips
• 3D v-grips
• Conclusion

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3D v-grips
Initial orientation
Final orientation after v-grip
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3D v-grips
Initial orientation
Final orientation after v-grip
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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.

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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.
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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
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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
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Phase II
All configurations in Phase II are candidate 2D
v-grips.
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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.

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3D Algorithm Termination.

1. 3D v-grip.

• 3D equilibrium grip.
• Part can move but Gripper cannot close.

• The part falls away.
• All termination conditions checked in
  wrench-space.

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Example Gear Shaft
Orthogonal views
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Gear Shaft
We assume that the gear is a cylinder (no teeth)
to allow gripping.
This part is symmetric about the axis (one
redundant degree of freedom). Search is thus
reduced to 0 dimensions!
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Gear Shaft Solution
Part Orientation
Shaft Trajectory
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3D Example without Symmetry
Orthogonal views
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3D Example Part Trajectory
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Outline

• Inspiration
• Related work
• 2D v-grips
• 3D v-grips
• Conclusion

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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.

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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

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