Robot Grasping of Deformable Objects

Department of Computer Science, Iowa State University

Robot Grasping of Deformable Objects

Yan-Bin Jia

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

Department of Computer ScienceIowa State UniversityAmes, IA 50010, USA

June 5, 2014


Rigid Body Grasping – Form Closure

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

frictionless contacts

𝑥

𝑦


Rigid Body Grasping – Force Closure

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

contact friction cones

Not form closure.

Form closure does not imply force closure.


Barrett Hand Grasping a Foam Object


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 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.


Displacement-Based Scheme

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.


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)


Linear Plane Elasticity

vu

yx

yx

Displacement field:

𝑓 1

𝑓 2


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


Finite Element Method (FEM)

KU T

21

:1

n

displacements at nodal points

K: stiffness matrix (symmetric & positive semidefinite)

Strain energy:


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


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


Deformation from Contact Displacements

Boundary nodes in contact with grasping fingers:

mi

i

1

Forces at nodes not in contact:

0kf miik ,,1

known

𝑝𝑖

𝑝 𝑗

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


Reduced Stiffness Matrix

Forces at m contact nodes:

CFmm 22

Strain energy:

CT

21

KT21

VDbasis matrixof . finger

placement.

Deformation:

reduced stiffness matrix


Squeeze with Two Point Fingers

ip

jp

j

i

Minimizing potential energy is equivalent to maximizing strain energy.

CT

21max

1

Solution:

ji

ij

jipppp

ppu

21ˆ

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


Pure Squeeze

Issues with a stable squeeze

object translation or rotation during deformation.

namely, not necessarily orthogonal to .

Pure squeeze :

)(null KVD v̂

squeeze depth


Example for Comparison

(stable squeeze)

Deformation under (pure squeeze)

Deformation under

¿(0.91,0 .35) ¿ (0.55,0 .21)


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.


Positional Constraints & Contact Analysis

Deformation update during a grasp needs positional constraints.

Resort to varying finger contacts

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?


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 sliding on a finger

Maintain two sets:


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( ll

Total deformation when event +1 happens:


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


More Contact Events

Event C – Stick to Slip

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

Event D – Slip to Stick

The polar angle stops changing at squeeze depth.

𝑝𝑘

𝜃

𝑓 𝑘


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.


Experiment

Young’s modulus PaPoisson’s ration Contact cof

slip

stick


Stick to Slip


Stick to Slip back to Stick

Second (convex) shape


Experiment with Ring-like Objects

(𝐸 ,𝜇 ,h ,𝜌 )Degenerate shells.


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.


Acknowledgement

US National Science Foundation

IIS-0915876

Documents

Robot Grasping of Deformable Objects