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Department of Computer Science, Iowa State University Robot Grasping of Deformable Planar Objects Yan-Bin Jia (with Ph.D. students Feng Guo and Huan Lin) Department of Computer Science Iowa State University Ames, IA 50010, USA

Robot Grasping of Deformable Planar Objects

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Yan-Bin Jia (with Ph.D. students Feng Guo and Huan Lin ) Department of Computer Science Iowa State University Ames, IA 50010, USA. Robot Grasping of Deformable Planar Objects. Rigid Body Grasping – Form Closure. The object has no degree of freedom ( Reuleaux , 1875). . - PowerPoint PPT Presentation

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Page 1: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Robot Grasping of Deformable Planar Objects

Yan-Bin Jia

(with Ph.D. students Feng Guo and Huan Lin)

Department of Computer ScienceIowa State UniversityAmes, IA 50010, USA

Page 2: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Rigid Body Grasping – Form Closure

The object has no degree of freedom (Reuleaux, 1875).

frictionless contacts

What cannot be generated by the contact force?

𝑥

𝑦

forces in the -direction

torques about the -direction

Page 3: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Rigid Body Grasping – Force Closure

The contacts can apply an arbitrary wrench (force + torque)to the object (Nguyen 1988).

contact friction cones

These wrench vectors positively span the 3D wrench space W.

Equivalently, their convex hull contains the origin in the interior.

Each force (normal or tangential) at a contact generates a vector in the 3D wrench space W (6D for a 3D object).

Not form closure.

Form closure does not imply force closure.

They can resist an arbitrary external wrench.

Page 4: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Related Work on Rigid Body Grasping Form closure grasps

Bounds on # contact points: Mishra et al. (1986); Markenscoff et al. (1987) Synthesis: Brost & Goldberg (1994); van der Stapper et al. (2000)

Force closure grasps

Testing & synthesis: Nguyen (1988); Trinkle (1988); Ponce et al. (1993); Ponce et al. (1997)

Caging: Rimon & Blake (1999); Rodriguez et al. (2012)

Grasp metrics: Kerr & Roth (1986); Li & Sastry (1988); Markenscoff & Papadimitriou (1989); Mirtich & Canny (1994); Mishra (1995); Buss et al. (1988); Boyd & Wegbreit (2007)

Page 5: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Barrett Hand Grasping a Foam Object

Page 6: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Deformable Body Grasping Is Difficult

Form closure impossible (infinite degrees of freedom)

Force closure inapplicable (changing geometry, growing contacts)

High computation cost of deformable modeling using the finite element methods (FEM)

Very little research done in robotics (most limited to linear objects)

Wakamatsu et al. (1996); Hirai et al. (2001); Gopalakrishnan & Goldberg (2005);Wakamatsu & Hirai (2004); Saha & Isto (2006); Ladd & Kavraki (2004)

Contact constraints needed for modeling do not exist at the start of a grasp operation.

Page 7: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Displacement-Based Deformable Grasping

A change of paradigm from rigid body grasping.

Specified forces cannot guarantee equilibrium after deformation.

Deformation computed under geometric constraints ensures force and torque equilibrium.

Easier to command a finger to move to a place than to exert a prescribed grasping force.

Specify finger displacements rather than forces.

Page 8: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Positional Constraints & Contact Analysis

Deformation update during a grasp needs positional constraints.

Resort to varying finger contacts

They are maintained by friction.

Contact regions grow or shrink.

Individual contact points slide or stick.

Incrementally track contact configuration!

Instantaneous deformation is assumed in classical elasticity theory.

How can we predict the final contact configuration from the start of a grasp operation?

Page 9: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Assumptions Deformable, isotropic, planar or thin 2-1/2 D object.

Two rigid grasping fingers coplanar with the object.

Frictional point or area contacts.

Gravity ignored.

Small deformation (linear elasticity).

Page 10: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Linear Plane Elasticity

vu

yx

yx

Displacement field:

𝑓 1

𝑓 2

Page 11: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Strains

Extensional strain – relative change in length

before

after

x uxu

xu

xx

0

limyv

y

y 'y

x

'xyu

xv

Shear strain – rotation of perpendicular lines toward (or away) from each other.

xv

yu

xy

Page 12: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Strain Energy

dxdyEEhUs xyyyxx

2222 12

212

Theorem 1 Any displacement field that yields zero strain energy is linearly spanned by three fields:

, , and

translation rotation

Page 13: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Finite Element Method (FEM)

a) Discretize the object into a triangular mesh.

b) Obtain the strain energy of each triangular element in terms of the displacements of its three vertices.

c) Sum up the strain energies of all elements.

KU T

21

:1

n

:K

displacements at nodal points

stiffness matrix (symmetric & positive semidefinite)

Page 14: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Energy Minimization

Total potential energy:

fK TT 21

load potential

:f vector of all nodal forces

0

Deformation is described by nodal displacements that minimize and satisfy the boundary conditions.

fK

U21

Page 15: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Stiffness Matrix

Null space is spanned by three -vectors:

,0,1,,0,1 T ,1,0,,1,0 T .,,,, 11T

nn xyxy

translations of all nodes rotation of all nodes

Spectral decomposition:

TVVK

)0,0,0,,,( 321 ndiag :V orthogonal matrix

Page 16: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Deformation from Contact Displacements

Boundary nodes in contact with grasping fingers:

mi

i

1

Forces at nodes not in contact:

0kf miik ,,1

Problem 1 Determine , , …, , and .

mi

i

f

ff

1

known

𝑝𝑖

𝑝 𝑗

Page 17: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Submatrices from Stiffness Matrix

K

12 1 i12i

mi212 mi

12 1 i 12i 12 mi mi2 n222 n

12 1iv

ki

ki

k

mv

vv

,2

,12 1

null space

contact node indices

𝐴=∑𝑘=1

2𝑛− 3 1𝜆𝑘

𝑣𝑘𝑣𝑘𝑇

𝐵

Page 18: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Solution Steps

fK fVV T

fVV TT k

n

nkkk

Tk

n

k k

vgvfv

2

22

32

1

)(1

ki

kin

nkk

ki

kiTk

n

k ki

j

j

j

j

j vv

gvv

fv,2

,122

22,2

,1232

1

)(1

0

gf

M

0,, 21222 fvvv Tnnn

0fBTmj ,...,1

n

n

n

ggg

g

2

12

22

projections of onto null space

Page 19: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Matrix for Solution of Deformation

0TBBA

M

if (two or more contacts)

C

M 1

Page 20: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Uniqueness of Deformation

Theorem 2 uniquely determines the displacement field (and thus the deformed shape) if .

Computational complexity

))(( 2 nmnO

a) Singular value decomposition (SVD) of .

)( 3nO

b) Deformed shape (i.e., )

)( 2/3nOm is small

Page 21: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Reduced Stiffness Matrix

Forces at m contact nodes:

Cf mm 22

Strain energy:

CT21

KT21

Page 22: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Squeeze by Two Point Fingers

ip

jp

j

i

Minimizing potential energy is equivalent to minimizing strain energy.

CT21max

1

Solution:

ji

ij

jipppp

ppu

21ˆ

Stable squeeze: the two point fingers move toward each other). squeeze depth

Page 23: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Pure Squeeze

Issues with a stable squeeze

object translation or rotation during deformation.

namely, not necessarily orthogonal to .

Pure squeeze :

)(null K v̂ uAuAvˆˆˆ where

squeeze depth

Page 24: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Example for Comparison

(stable squeeze)

Deformation under (pure squeeze)

Deformation under

¿(0.91,0 .35) ¿ (0.55,0 .21)

Page 25: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Squeeze Grasp with Rounded Fingers

Translate the fingers to squeeze the object.

Contact friction.

Initial point contacts and .

Contacts growing into segments.

To prevent rigid body motion, and must form force closure on an identical rigid object.

lies inside the two contact friction cones.

Page 26: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Contact Configuration

Which nodes are in contact.

Which of them are sticking and which are sliding.

sliding sticking

Sliding nodes position constraints.

Sticking nodes force constraints.

Deformation update based on FEM:

indices of nodes sticking on a finger

indices of nodes slipping on a finger

Maintain two sets:

Page 27: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Overview of Squeeze Algorithm

and change whenever a contact event happens:

Between events and +1, compute extra deformation based on the current values of and .

= 0, ,

Squeeze depth is sequenced by all such contact events:

)()()1( lll

Total deformation when event +1 happens:

Page 28: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Squeeze Grasp Algorithm

?

success

Compute reduced stiffness matrix

from

Contact Event Analysis

min extra squeeze Update

Either finger slips?

failure

yes

no

yesno

Page 29: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Movement of a Contact Node

A sticking node moves with its contacting finger.

tkˆ t̂

A sliding node also slides on its contacting finger.

)(

)(

sinsincoscosˆ

lkk

lkk

k rt

𝑝𝑘 𝑝𝑘

)(lk

𝑝𝑘

k

𝑝𝑘

Page 30: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Deformation under Extra Squeeze

mi

i

1

depending on for every sliding node

constraint equations

Every sliding node must receive a contact force on one edge of its friction cone.

variables

n

1

Cf

All s are solvable.

Page 31: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Contact Events

Check for all values of extra squeeze depth at which a eventcould happen, and select the minimum.

Event A – New Contact

Event B – Contact Break

𝑝𝑘

𝑝𝑘

0kf

𝑝𝑘

𝑂rOpk

𝑝𝑘

𝑂𝑟

Page 32: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

More Contact Events

Event C – Stick to Slip

Contact force is rotating out of the inward friction cone at .

0)sin()cos(

k

kkf

Event D – Slip to Stick

The polar angle at stops changing at squeeze depth.

0dd k

Page 33: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Termination of Squeeze

A grasping finger starts to slip.

At either one of the following situations:

Strain at some node exceeds the material’s proportional limit.

The object can be picked up against its weight vertically.

All contact nodes with the finger are slipping in the same direction.

Page 34: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Experiment

Young’s modulus PaPoisson’s ration Contact cof

slip

stick

Page 35: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Stick to Slip

Page 36: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Stick to Slip back to Stick

Second (convex) shape

Page 37: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Experiment with Ring-like Objects

(𝐸 ,𝜇 ,h ,𝜌 )Degenerate shells.

Page 38: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Adversary Finger Resistance

Adversary finger tries to break a grasp via translation .

Grasping fingers and resist it it via translations and .

Initial contacts at ,, and .

Problem 2 What are the optimal and ?

Either and squeeze the object first or three fingers make contact with it simultaneously.

Page 39: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

What Optimality?

System potential energy can be made as large as possible.

Rigid body grasping

Total force/wrench to resist unit adversary force/wrench.

Deformable body grasping

Work to resist unit translation by adversary finger.

Optimality criterion should reflect the effort of resistance.

Page 40: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Work Minimization

Finger contact sets change during resistance:

: }

: }: }

1) Fixed point contacts ( ).

2) Fixed setment contacts ( ).

3) Change of contacts under Coulomb friction (general case).

Solution steps:

Page 41: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

The Case of Fixed Point Contacts

add

Cdd

W

T

F 2

1

2

1

021

Work done by grasping fingers:

Stable squeeze:

021 add021 apdpdp kji

Minimization is subject to

Pure squeeze:

)(col2

1

ACadd

Closed forms exist for optimal resisting translations and except in some degenerate cases.

Page 42: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

An Example

Resistance by a stable squeeze.

Page 43: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

The Case of Fixed Segment Contacts

Contact node is displaced by

=

if

if

if

Generally, closed forms exist for optimal resisting translations and .

Generalize over the case of fixed point contacts.

Page 44: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

General Case of Frictional Segment Contacts

Between two events the contact configuration does not change.

Sequence the translation by based on contact events.

Treat as fixed contact sets (only approximately for sliding nodes).

Directions of new translations and from minimizing extra work

JIt

tTt

lt

JIt

TtF ffW

21)(

under hypothesized extra unit translation by .

Distances of new translations are subject to the next contact event.

Page 45: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Outcomes of Resistance

Failure if either or slips before completes its translation.

Success otherwise (including slip of ).

Typically, and have squeezed the object for a grasp before makes contact with the object.

Page 46: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Simulation

𝐹 1

𝐹 2

𝐹 1

𝐹 2

𝐴

b) pushes the object via a translation under the resistance by and .

a) squeezes the object toward via a translation .

Page 47: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Resistance Trajectories

squeeze

resist

𝐹 2

𝐴

0019.00008.0

1d

0005.00007.0

2d

0044.00024.0

a

Page 48: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Resistance Experiment

force meter

Straightened trajectories for and for ease of control.

Work estimated as half of the product of translation with the sum of the initial and final force readings.

Page 49: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Simulation vs Experiment

Straightened trajectories for and .

2.12.1 0.828.0 1.53

0.00270.0086.57

0.018

Measurement errors.

Reasons for discrepancies:

Experiment (“Optimal”) Simulation

Page 50: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

“Optimal” vs “Arbitrary” Resistances

“Optimal” resistance as just presented.

“Arbitrary” resistance with a translation direction chosen arbitrarily.

894.0

447.0ˆ2d

0032.00012.00016.0004.0

, 21 dd

stable resistance

Page 51: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Summary

Displacement-based grasping strategy for deformable objects.

Stable and pure squeezes.

Event-driven algorithm combined with contact mode analysis.

Energy-based grasp optimality.

Computational efficiency from one-time matrix decomposition.

Page 52: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Future Work

Computationally efficient grasp outcome prediction.

Rigid body grasping vs. deformable body grasping.

Why deformable objects are often easier to grasp?

Soft fingers on a rigid body vs. Hard fingers on a deformable body

Design of grasping algorithms for 3D deformable objects.

Energy-based grasping metrics.

Page 53: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Acknowledgement

US National Science Foundation

IIS-0915876

Page 54: Robot Grasping of Deformable Planar Objects

Department of Computer Science, Iowa State University

Online Papers

IEEE International Conference on Robotics and Automation (2013, published)

http://www.cs.iastate.edu/~jia/papers/ICRA13.pdf

Extended version (submitted to International Journal of Robotics Research)

http://www.cs.iastate.edu/~jia/papers/IJRR13-submit.pdf

IEEE/RSJ International Conference on Intelligent Robots and Systems (2013, accepted)

http://www.cs.iastate.edu/~jia/papers/IROS13-submit.pdf