B659: Principles of Intelligent Robot Motion
Spring 2013
David Tidd
Grasp Quality
• Given two different grasps, how can they be
compared?
– Are they stable? -> Force closure
– How stable are they? -> Grasp quality metrics
c1
c1
c3
c3
c2
Agenda
•
•
•
•
Point force generalization
Wrench space
Grasp quality metrics
Simulation method
Contact Types
• Type of contact determined by colliding geometries
– Point: point on plane (stable), point on point or line
(unstable)
– Line: line on plane or nonparallel line (stable), line on
parallel line (unstable)
– Plane: plane on plane
• Unstable contacts ignored in analysis
Point-Plane
Point-Line
Point-Point
Everything as a Point Contact
• Line contact -> 2 points
• Plane contact -> convex hull of points
• Any distribution of normal forces across a region can
be represented as a weighted sum of point forces
along that region’s convex hull
Point Contacts with Coulomb Friction
• A point contact with friction is able to apply more
than just a normal force
• “Friction cone” is the vector space of all possible
forces a point can apply due to friction
• f = fn+ ft where |ft| ≤ |μs*fn|
n
n
Approximating Friction Cones
• Pyramidal approximation converts vector
space to finite set of vectors
– 8-sided approximation used in simulation
Wrenches
• Each point force also applies torque
– τ=dxf
• Wrench is a force-torque pair
– The i-th point contact has m wrenches, one for each force
in the pyramidal approximation
– d is the vector from the point contact to the torque origin
– λ is a constant relating force to torque for analysis
• λ = 1/r was chosen to make torque size invariant
Wrench Space
• For 3D objects, wrench space is 6D
– 3D for force, 3D for torque
– For 2D objects, it’s 3D
fy
fx
τz
Wrench Hulls
• Set of wrenches from ONE point contact = boundary
of what wrenches can be applied from that one point
• Set of wrenches from ALL point contacts = convex
hull in wrench space, total possible range of
wrenches that can be applied
2D Example
Is this grasp stable?
c1
d1
f1,1
f1,2
COM
f2,1
f2,2
d2
c2
2D Example
• 2 point contacts
• 4 wrenches
• Force closure?
c1
d1
– Yes
• What about torque?
• Direction of d x f
– All torque is in same
direction, out of page
f1,1
f1,2
COM
f2,1
f2,2
d2
c2
2D Example
• Ignore fx for now
c1
w1,1
τout
w2,2
d1
Wrench hull
w1,2
w2,1
-fy
+fy
Does not
contain origin,
not stable
τin
f1,1
f1,2
COM
f2,1
f2,2
d2
c2
2D Example
• What if there was a 3rd point?
c1
τout
w1,1
w2,2
d1
Wrench hull
w1,2
w2,1
-fy
+fy
f1,1
f3,1
f1,2
f3,2
w3,2
d3
Does contain
origin, stable
w3,1
τin
c3
COM
f2,1
f2,2
d2
c2
Grasp Quality
• Both of these grasps are stable
– But how stable are they?
c1
c1
d1
f1,1
f3,1
f1,2
f3,2
d1
COM
f1,1
f3,1
f1,2
f3,2
d3
c3
d3
c3
COM
f2,1
f2,2
d2
c2
Grasp Quality Metrics
• Quality is how well a grip can resist disturbances
• Worst case scenario
– How efficiently can a grip resist disturbance wrenches at
its weakest point?
• Weakest means the direction (in wrench space) at
which the sum normal force is converted to the
desired wrench least efficiently
– Grip a pencil at the end and try to resist torque
– Now try it while gripping the center
– The center requires much more normal force to get the
same wrench
Worst Case Scenario
• The point on the wrench hull
that is closest to the origin is
the weakest point
• Disturbances in the opposite
direction are hardest to -f
y
resist
• Metric ε = The radius of the
largest ball that can be
enclosed in the wrench hull
– Varies from 0 to 1 due to
normalization of wrenches
τout
w1,1
w2,2
Hard to resist
w2,1
w1,2
+fy
ε
w3,2
w3,1
τin
Physical Meaning of ε
• In the worst case, the sum magnitude of the
contact wrenches would need to be 1/ε times
the disturbance wrench
Grasp Quality Metrics
• So are these equal?
τout
w1,1
w1,2
w2,2
w1,1
w2,1
w1,2
-fy
τout
+fy -fy
ε
w3,2
+fy
ε
w3,2
w3,1
τin
w3,1
τin
Average Case Scenario
• How efficiently can a grip resist a disturbance
wrench on average?
• Metric ν = Volume of the convex hull in
wrench space
• The three point contact has more volume, so
it is more stable on average
Grasp Simulation Method
• Set hand configuration except for distal links
• Iterate configuration of distal links and check for
collisions with object
• Continue until all links have collided
Only one solution
found. There could
be better solutions.
How to determine
initial configuration?
Grasp Analysis Method
•
•
•
•
Decompose the collisions into point contacts
Covert point contacts into sets of wrenches
Construct wrench hull
Compute quality metrics
Grasp Search
• Each hand configuration maps to one grasp via
simulation
• The total possible grasp space is equivalent to the
initial configuration space of the hand
• Explore a subset of C-space using finite steps
• Other methods?
Discussion