Robot Grasping of Deformable Objects
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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
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Department of Computer Science, Iowa State University
Rigid Body Grasping – Form Closure
The object has no degree of freedom (Reuleaux, 1875).
frictionless contacts
𝑥
𝑦
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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
Not form closure.
Form closure does not imply force closure.
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Department of Computer Science, Iowa State University
Barrett Hand Grasping a Foam Object
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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 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.
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Department of Computer Science, Iowa State University
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.
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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)
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Department of Computer Science, Iowa State University
Linear Plane Elasticity
vu
yx
yx
Displacement field:
𝑓 1
𝑓 2
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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
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Department of Computer Science, Iowa State University
Finite Element Method (FEM)
KU T
21
:1
n
displacements at nodal points
K: stiffness matrix (symmetric & positive semidefinite)
Strain energy:
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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
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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
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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
known
𝑝𝑖
𝑝 𝑗
Theorem 2 uniquely determines the displacement field (and thus the deformed shape) if .
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Department of Computer Science, Iowa State University
Reduced Stiffness Matrix
Forces at m contact nodes:
CFmm 22
Strain energy:
CT
21
KT21
VDbasis matrixof . finger
placement.
Deformation:
reduced stiffness matrix
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Department of Computer Science, Iowa State University
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
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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 KVD v̂
squeeze depth
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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)
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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.
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Department of Computer Science, Iowa State University
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?
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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 sliding on a finger
Maintain two sets:
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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( ll
Total deformation when event +1 happens:
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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
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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 .
Event D – Slip to Stick
The polar angle stops changing at squeeze depth.
𝑝𝑘
𝜃
𝑓 𝑘
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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.
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Experiment
Young’s modulus PaPoisson’s ration Contact cof
slip
stick
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Department of Computer Science, Iowa State University
Stick to Slip
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Department of Computer Science, Iowa State University
Stick to Slip back to Stick
Second (convex) shape
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Department of Computer Science, Iowa State University
Experiment with Ring-like Objects
(𝐸 ,𝜇 ,h ,𝜌 )Degenerate shells.
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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.
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Department of Computer Science, Iowa State University
Acknowledgement
US National Science Foundation
IIS-0915876