Analysis of
Multifingered Hands
Kerr and Roth 1986
Outline
 1. contribution of this paper
 1.1 Selection of Internal Grasp Forces
 1.2 Manipulating Objects within the Hand
 1.3 Hand Workspaces
 2. limitation of this paper
 3. future direction of this paper
1.1 Selection of Internal
Grasp Forces
 Determining how hard to squeeze an object in
order to ensure a secure grasp
 -f=WC（1）
 C=Cp+Ch （2）
 Cp = -WR+ f （3）
 Ch＝N l （4）
1.1 Selection of Internal
Grasp Forces
 Determining how hard to squeeze an object in
order to ensure a secure grasp
 Equation 1 shows that all forces must be balanced
 Then we consider friction at the contact points
 the friction cannot exceed the maximum available
force due to friction, determined by the coefficient
of static friction at each point
 and the tangential components of the friction must
balance out the contact forces so that the object
does not move
 Then we set up 2nd~8th equations
1.1 Selection of Internal
Grasp Forces
 Friction constraints
 For each contact,we have the constraint
cinormal ³ 0(5)
 We must also ensure that the tangential components of
the contact forces do not exceed the magnitude of the
equilibrating friction force. Using a Coulomb friction
model,this constraint is written as
(6)
2
2
cix + ciy £ mcinormal
1.1 Selection of Internal
Grasp Forces
 If we choose to use four planes, the linear
approximation to equation 6 is written as the set
of equations
 similarly for torsional friction...
cix + m cinormal ³ 0
 （7）
cix - m cinormal £ 0
ciy + m cinormal ³ 0 (8)
ciy - m cinormal £ 0
citorsion + mt cinormal ³ 0
citorsion - mt cinormal £ 0
1.1 Selection of Internal
Grasp Forces
 joint torque limits
 we also know the max torque that each finger
can exert… so the other constraints are that this
max torque cannot be exceeded and min
torque is required.
¶ T C £ t max

(9)
¶ T C ³ t min
1.1 Selection of Internal
Grasp Forces
 Through a series of computation and substituting


AC ³ P
(10)
AN l ³ P - ACp (11)
1.1 Selection of Internal
Grasp Forces
 Optimal selection of internal grasp force
 An optimal selection of l is one for which we are
furthest from violating any one of the
constraints.the distance of some point l0 from the
plane formed by the ith constraint of equation is
given by di = (AN)i l0 - (P - ACP )i (12)
 reformulating the formula to pose this problem
as a standard linear optimization problem ,we
get d £ (AN)i l -(P - ACP )i (13)
1.1 Selection of Internal
Grasp Forces
 These inequalities are written in more
conventional form if we create a vector ld which
is the vector l augmented with the distance
variable, d. Equation 13 then becomes

[(AN)i -1]ld ³ (P - ACp )i (14)
 the function we wish to maximize F(ld ) = d (15)
 Equations 14 and 15 form a linear programming
problem in its standard form, which is readily
solved by the Simplex method.
1.2. Manipulating Objects
within the Hand
 determining the motion of each finger joint
required to achieve a desired motion of the
object.
 The motion of the contact point on each
fingertip is identical to the motion of the
corresponding contact position on the object.If
the inverse kinematic solution for each finger is
known, the required finger-joint motions are easily
calculated.
1.2. Manipulating Objects
within the Hand
 Here,we will consider only the case of pure rolling
between the object and the fingertip.
 In the meantime We will consider the minimum
case of fingers with three joints.
Consider a simple case that
a finger in contact with an
object
1.2. Manipulating Objects
within the Hand
Going through the computation of math
.
vc = Ot + wt ´ (TR T c)
Tr T n = Br B n
.
vc = Ot - (Tr T c)Ä wt
.
T .
.
B .
Tr c = Br c
Ot = J p q
.
wt = Jo q
.
vc = [J p - (Tr T c)Ä Jo ]q
é.ù
é.ù
h
êa ú
Tr JTc ê . ú = Br J Bc ê . ú
ê ú
êëb úû
ëx û
.
vc = O b - (Br T c)Ä wb
Ot +Tr T c = Ob + Br B c
(16)-(23) for first row,(24)-(29) for
The second row
T .
B .
wt ´Tr T n + Tr n = wb ´ Br B n + Br n
é.ù
é.ù
h
êa ú
T
Ä
B
Ä
-(Tr n) J o q + Tr JTn ê . ú = -(Br n) w b + Br J Bn ê . ú
ê ú
êëb úû
ëx û
.
1.2. Manipulating Objects
within the Hand
 Finally we got
1.3. Hand Workspaces
 4 assumptions:
 The type of contact present between the
fingertips and the object is considered to be
point contact with friction,and the contact points
do not change during the motion.
 The hand workspace will be defined for a
particular configuration of contact points on the
object and for particular locations of the contact
points on the fingertips.
1.3. Hand Workspaces
 The workspace for each fingertip5 is known and is
considered fixed relative to the palm of the hand.
 Collisions or intersections between the object
and the finger links or between the finger links
them- selves will be ignored.
 with these four assumptions, determining the
workspace for a hand can be reduced to the
problem illustrated in figure 6 for a three-fingered
hand.
1.3. Hand Workspaces
1.3. Hand Workspaces
 Rather than deal with the complete sixdimensional configuration space of the object,
we will define the total workspace of the hand as
the volume swept out by one particular point q
fixed in the object as the object is moved
through all possible configurations. For our
purposes, we will choose q to be the spatial
center of the contact points on the object.
2. Limitations of this paper
 Assumptions
 1. for selection of internal grasp force, the paper
assume there are only friction constraint and joint
torque constraint.
 2. for manipulating objects within the hand, the
paper,Here,we will consider only the case of pure
rolling between the object and the fingertip.
 3. for hand workspaces, all the four assumptions
are too restrictive for practical purposes.
3. Future Direction of this
paper
 1.Adding other types of constraints,to make a real
case
 2. for hand workspaces,the treatment of this
problem here has not ad- dressed many of the
important issues relating to the workspace. The first
of these is the range of orientations possible at
each point in the total workspace.
 3. the analysis here assumes that the contact points
are fixed on both the fingertips and the object.
 4. The last important point not addressed above is
that of intersection between the object and the
fingers and between the fingers themselves.
3. Future Direction of this
paper
 There is no discussion about the influence of the
size and shape on manipulation. And it’s
important.
 there is no discussion about how to decide a
particular configuration and orientation of the
end-effector to pick an object from many
different possible configurations and orientations
 In the meantime,The issue of singular
configurations had not been covered.
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