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

Robot Grasping of Deformable Objects
Yan-Bin Jia
(joint work with Ph.D. students Feng Guo and Huan Lin)
Department of Computer Science
Iowa State University
Ames, IA 50010, USA
June 5, 2014
Department of Computer Science, Iowa State University
Rigid Body Grasping – Form Closure
The object has no degree of freedom (Reuleaux, 1875).
frictionless contacts
𝑦
𝑥
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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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Barrett Hand Grasping a Foam Object
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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)
 Contact constraints needed for modeling do not exist at the
start of a grasp operation.
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)
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Displacement-Based Scheme
A change of paradigm from rigid body grasping.
Specify finger displacements rather than forces.
 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.
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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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Linear Plane Elasticity
𝑓2
𝑓1
Displacement field:
 x   x  u
       
 y  y v
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Strains
Extensional strain – relative change in length
after
before
y
u
y
x
u u
 x  lim

x 0 x
x
u
y'

x'
v
x
v
y 
y
Shear strain – rotation of perpendicular lines
toward (or away) from each other.
x
 xy
u v


y x
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Finite Element Method (FEM)
Strain energy:
1 T
U   K
2
 1 
 
    :
 
 n
displacements at 𝑛 nodal points
K: stiffness matrix (symmetric & positive semidefinite)
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Energy Minimization
Total potential energy:
1 T
   K  T F
2
load potential
F:
vector of all nodal forces
Deformation is described by nodal displacements that minimize Π
and satisfy the boundary conditions.

0

K  F
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Stiffness Matrix
Null space is spanned by three 2𝑛-vectors:
1,0,,1,0 , 0,1,,0,1 ,  y1, x1,, yn , xn  .
T
T
T
translations of all nodes
rotation of all nodes
Spectral decomposition:
K  V V
V:
T
orthogonal matrix
  diag (1 ,, 2 n 3 ,0,0,0)
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Deformation from Contact Displacements
Boundary nodes 𝑝𝑖1 , … , 𝑝𝑖𝑚 in contact with grasping fingers:
𝑝𝑗
𝑝𝑖
  i1 
 
  
 
  im 
known
Forces at nodes not in contact:
fk  0
k  i1 , , im
Theorem 2  uniquely determines the displacement field 
(and thus the deformed shape) if 𝑚 ≥ 2.
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Reduced Stiffness Matrix
Forces at m contact nodes:
reduced stiffness matrix
F  C
2m  2m
Deformation:
  VD
basis matrix
of 𝐾.
finger
placement.
Strain energy:
1 T
 K 
2
1 T
 C
2
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Squeeze with Two Point Fingers
 i 
   
 j 
pj
Minimizing potential energy is
equivalent to maximizing strain
energy.
pi
1 T
max  C 
 1 2
 p j  pi 
1


Solution: uˆ 
2 pi  p j  pi  p j 
Stable squeeze: 𝜌𝑢 (the two point fingers move toward each other).
squeeze depth
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Pure Squeeze
Issues with a stable squeeze :
 object translation or rotation during deformation.
 namely,  not necessarily orthogonal to null(𝐾).
Pure squeeze :
  VD   null( K )
  v̂
squeeze depth
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Example for Comparison
(stable squeeze)
= (0.91,0.35)
(pure squeeze)
Deformation under 𝑢
= (0.55,0.21)
Deformation under 𝑣
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Squeeze Grasp with Rounded Fingers
Translate the fingers to squeeze the object.
 Initial point contacts 𝑝𝑖 and 𝑝𝑗 .
 Contact friction.
 Contacts growing into segments.
To prevent rigid body motion, 𝑝𝑖 and 𝑝𝑗 must form force closure on an identical
rigid object.
𝑝𝑖 𝑝𝑗 lies inside the two contact friction cones.
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Positional Constraints & Contact Analysis
Instantaneous deformation is assumed in classical elasticity theory.
Deformation update during a grasp needs positional constraints.
How can we predict the final contact configuration from the start?
Resort to varying finger contacts
 Maintained by friction.
 Contact regions grow or shrink.
 Individual contact points slide or stick.
Incrementally track contact configuration!
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Contact Configuration
 Which nodes are in contact.
 Which of them are sticking and which
are sliding.
Maintain two sets:
sliding
sticking
𝑇: indices of nodes sticking on a finger
𝑃: indices of nodes sliding on a finger
Deformation update based on FEM:
 Sliding nodes → position constraints.
 Sticking nodes → force constraints.
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Overview of Squeeze Algorithm
 𝑇 and 𝑃 change whenever a contact event happens:
 Squeeze depth 𝜌 is sequenced by all such contact events:
𝜌0 = 0, 𝜌1 , ⋯
 Between events 𝑙 and 𝑙+1, compute extra deformation ∆′ based on the
current values of 𝑇 and 𝑃.
 Total deformation when event 𝑙+1 happens:
(l 1)  (l )  '
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Contact Events
Check for all values of extra squeeze depth at which a event
could happen, and select the minimum.
 Event A – New Contact
𝑂
𝑝𝑘
𝑟 𝑂
𝑝𝑘
pk  O  r
 Event B – Contact Break
fk  0
𝑝𝑘
𝑝𝑘
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More Contact Events
 Event C – Stick to Slip
Contact force 𝑓𝑘 is rotating out of the inward friction cone at 𝑝𝑘 .
𝜃
𝑝𝑘
𝑓𝑘
 Event D – Slip to Stick
The polar angle 𝜃 stops changing at squeeze depth.
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Termination of Squeeze
At either one of the following situations:
 The object can be picked up against its weight vertically.
 A grasping finger starts to slip.
All contact nodes with the finger are slipping in the same direction.
 Strain at some node exceeds the material’s proportional limit.
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Experiment
slip
stick
Young’s modulus 𝐸 = 50000 Pa
Poisson’s ration 𝜗 = 0.3
Contact cof 𝜇 = 0.4
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Stick to Slip
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Stick to Slip back to Stick
Second (convex) shape
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Experiment with Ring-like Objects
(𝐸, 𝜇, ℎ, 𝜌) shells.
Degenerate
Type equation here.
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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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Acknowledgement
US National Science Foundation
IIS-0915876
Department of Computer Science, Iowa State University