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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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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
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
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.
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)
Department of Computer Science, Iowa State University
Barrett Hand Grasping a Foam Object
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.
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.
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?
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).
Department of Computer Science, Iowa State University
Linear Plane Elasticity
vu
yx
yx
Displacement field:
𝑓 1
𝑓 2
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
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
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)
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
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
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
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
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
Department of Computer Science, Iowa State University
Matrix for Solution of Deformation
0TBBA
M
if (two or more contacts)
C
M 1
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
Department of Computer Science, Iowa State University
Reduced Stiffness Matrix
Forces at m contact nodes:
Cf mm 22
Strain energy:
CT21
KT21
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
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
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)
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.
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:
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:
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
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
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.
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
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
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.
Department of Computer Science, Iowa State University
Experiment
Young’s modulus PaPoisson’s ration Contact cof
slip
stick
Department of Computer Science, Iowa State University
Stick to Slip
Department of Computer Science, Iowa State University
Stick to Slip back to Stick
Second (convex) shape
Department of Computer Science, Iowa State University
Experiment with Ring-like Objects
(𝐸 ,𝜇 ,h ,𝜌 )Degenerate shells.
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.
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.
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:
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.
Department of Computer Science, Iowa State University
An Example
Resistance by a stable squeeze.
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.
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.
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.
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 .
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
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.
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
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
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.
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.
Department of Computer Science, Iowa State University
Acknowledgement
US National Science Foundation
IIS-0915876
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