Proceedings of the 2011 IEEE
International Conference on Robotics and Biomimetics
December 7-11, 2011, Phuket, Thailand

A Grasp Performance Criterion for Robot Hands Considering Multiple
Aspects of Tasks and Hand Conﬁgurations
Masahiro Sato and Tsuneo Yoshikawa
Abstract— In this paper, a task-oriented grasp performance
criterion for robot hands is discussed. The objectives of grasp
criteria proposed in the past are stable grasping and the tasks
that require adequate forces and moments on the grasped
object. These criteria do not mainly consider the other physicalquantities such as positions and velocities on the grasped object
and the mechanical property of robot hands. We proposea grasp criterion for various kinds of tasks that evaluates
the feasibility of the task with considering hand mechanisms
and hand conﬁgurations. Our proposed method determine
the grasping position by considering the efﬁciency of grasped
object’s position, velocity, and force. As a result, this criterion
decides the grasping position and the hand conﬁguration.
Additionally, we have shown that our proposed criterion canbe applied to the design problem of robot hands.

I. INTRODUCTION
Grasping and manipulation of various objects are fundamental functions of robot hands. Recently, many multiﬁngered robot hands have been developed[1][2][3][4]. In
addition, robot hands whose speciﬁcation is like human’s
hands are also developed[5][6]. In the future, robot hands
will be expected to deal with a large varieties of objects
and different kinds of tasks. One issue for these robot hands
to realize dexterous grasping and manipulation like humans
do is to decide the position to grasp. Selecting the grasping
position affects “grasping and manipulation”. One approachto decide the grasping position for a robot is a grasp criterion.
Accomplishing dexterous tasks needs a precise decision of
the grasping position because the grasping position affects
the feasible region of positions, velocities, and forces on thegrasped object. Consequently, a grasp criterion will have an
important role to deal with complicated tasks.
Various kinds of grasp criteria have been studied. One
approach is using analysis based on geometry and dynamics.
Many grasp criteria of this type have been proposed[7][8].
Chinellato et al. proposed the grasp criterion using thetriangle which consists of ﬁnger tip positions[9]. Liegeois
suggested an index using joint angles of ﬁnger to decide
the best grasp[10]. Ferrari et al. proposed a grasp criterion whose index is the wrench space consisting of forces
and torque which are applied to the grasped object[11].
This idea is applied to a grasp simulator of robot hands
called GraspIt![14][16]. Some previous researches focusedon tasks of robot hands. Li and Sastry developed a grasp
criterion using the forces and torque which are applied tothe object[15]. Li et al. proposed a grasp criterion for tasks
with the requirement of forces on the object such as holding
a pitcher to pour water from the bottle to the cup[16].
However, these criteria do not simultaneously consider otherphysical quantities such as positions and velocities on thegrasped object, which are required to achieve dexterous
tasks. However, these criteria do not consider the mechanicalproperty of robot hands.
The other approach is using experimental results of subjects. Sugiyama et al. measured the grasping positions of subjects and discussed the relationship between the object shapeand the grasping positions[17]. Ikeda et al. used the different
length of the cylinder and evaluated the best length to be
grasped by human[18]. Balasubramanian et al. investigatedthe method of grasping by measuring human’s grasping andimproved the robustness of grasping[19]. Additionally, theobjective of these previous grasp criteria is stable grasping.
Therefore, the possibility about applying these criteria toskillful tasks is not mainly discussed.
In a previous paper[20], we have proposed a grasp criterion which considers multiple aspects of tasks: positions,velocities, and forces on the grasped object. Considering
these physical quantities makes it possible to deal withskillful tasks that requires a ﬁnite region of positions and/or
velocities. This criterion evaluates the feasibility of the taskwith comparing the desired region and the feasible regionof these physical quantities at certain grasping positions. In
addition, we have calculated the feasible region of physicalquantities by considering the mechanism of robot hands.
However, this criterion only evaluates grasping positions
and hand conﬁgurations are not mainly considered as anevaluation index.
In this work, we consider hand conﬁgurations in addition
to grasping positions and apply our proposed method to thedesign problem of robot hands. Furthermore, we improve thedeﬁnition of the desired region of these physical quantities
more intuitive and simple and made it possible to reduce
calculation cost. We give some numerical examples of tasks
in order to examine the validity of our proposed method.
Additionally, we compare our proposed method with two
criteria based on the wrench space.

Masahiro Sato is with College of Information Science and Engineering, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan,

sato.masahiro.0214@gmail.com

Tsuneo Yoshikawa is with College of Information Science and Engineering, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan, yoshi

kawa@ci.ritsumei.ac.jp,(((

1547

ΣH

Z

Z

X6

X

ΣH

lf
Y

X

Y
lr
Z

ΣO
Object

O

Z

X2

X

Y

ΣO

Feasible set

Y

X1

Required set(a) Required set and feasible set(b) Calculation method of safety margin

Fig. 1.

X

r O = (xO , yO , zO , φO , ϕO , ψO )(c) Analytic model

Principles of our proposed criterion

II. P ROPOSED G RASP C RITERION
Y

ΣO

φ

X

Y

ΣO

X

We consider a robot hand having with multi-artuculated
ﬁngers that grasps an object by its ﬁngertips as shown in
Fig. 1 (a). To determine a grasping position, we proposea grasp criterion considering multiple aspects of tasks andhand conﬁgurations. We deﬁne multiple aspects of tasks as
the feasibility for position, velocity, and force on the grasped
object.
A. Required Set and Feasible Set

Task requirement: position

Required position set

Ẏ

In order to evaluate the feasibility of the task at certain
grasping position, we use two kinds of set (Fig. 1 (a)). One isa required set which means the desired region of one physicalquantity (position, velocity, or force) to achieve the task. The
other set is a feasible set which indicates the realizable regionof this physical quantity at certain grasping position andhand conﬁguration. Inclusive relation between these two setsshows whether the task can be achieved at certain grasping
position and hand conﬁguration or not.

Ẋ

B. Evaluation Function(a) Example of required position set

Y
φ̇

X

ΣO

ΣO

Task requirement: velocity

Required velocity set(b) Example of required velocity set

Y
ΣO

M

X

Fy
ΣO

Task requirement: force

Required force set(c) Example of required force set

Fig. 2.

Designing method for required set

Fx

We propose a evaluation function that calculates an inclusive relationship between the required set and the feasible set.
It means that this evaluation function shows the possibility
for achieving the task. We deﬁne the evaluation function asa safety margin. The larger safety margin indicates the lower
risk to fail the task. In other words, the optimum grasp has
the largest safety margin among all the candidates and hand
conﬁguration.
Assume that lf is the distance from the origin to the edgeof the feasible set in the direction of cartesian coordinate in
N dimension space (N = 3 for the 2 dimensional plane andN = 6 for the 3 dimensional space). and lr is the distancefrom the origin to the edge of the required set in the same
direction. Then, a safety margin a is deﬁned byl
min lfr if lr = 0,(1)a=0 otherwise.
The safety margin is the minimum value of lf /lr with respect
to all possible directions. Let the safety margin of positions,velocities, and forces be ap , av , and af at certain grasping1548

position, respectively. We use agm as an index with thefollowing deﬁnition:
agm = min {ap , av , af } .

(2)

The minimum safety margins which have the highest risk
for the task are evaluated. Note that, intuitively speaking, theposition is related to the workspace evaluation, the velocity isrelated to the manipulability analysis, and the force is related
to the grasping force analysis.
III. C ALCULATION M ETHOD FOR R EQUIRED S ET AND
F EASIBLE S ET
A. Analytic Model
We consider the model that the robot hand with n ﬁngers
grasp the object (Fig. 1 (c)). Additionally, we deﬁne r O =(xO , yO , zO , φO , ϕO , ψO ) as the position vector consisting of
position at midpoint (xO , yO , zO ) and posture of the object(φO , ϕO , ψO ), θ i = [θi1 , θi2 , · · · , θim ]T ∈ m as the joint
angle vector on ith ﬁnger, θ = [θ 1 , θ 2 , · · · , θ n ]T ∈ m×n
as the joint angle velocity vector of all ﬁngers, τ i =[τi1 , τi2 , · · · , τim ]T ∈ m as the joint torque vector on ith
ﬁnger, τ = [τ 1 , τ 2 , · · · , τ n ]T ∈ m×n as the joint torquevector on all ﬁngers.
The followings are the conditions which are also considered.
• The ﬁngers are solid and make frictional point contacts
with the object. Therefore, the contact points between
ﬁngertips and the object can be regarded as free revolutejoints.
• All ﬁngers do not intersect.
• Gravitational force does not work on them.
B. Required Set
The required set is speciﬁed by the task requirement. An
example of the required set is shown in Fig. 2. If the object
is required to move within a limited range of x, y, φ axis,
the required position set is given by a cube (Fig. 2 (a)). If
the task requires unilateral velocity on φ axis, the required
velocity set is designed to be a line along φ axis (Fig. 2 (b)).
In order to resist the forces which has a ﬁxed magnitude but
might be applied from all the directions, the required force
set should be the circle on Fx Fy plane (Fig. 2 (c)).
C. Feasible Set
1) Feasible Position Set: The feasible position set is therealizable region of the position vector r O and is calculated
by using inverse kinematics. The inverse kinematic of thismodel is obtained byθ = f inv (r O )(3)Note that f inv (·) denotes an inverse kinematics.
The condition is a range of joint angles calculated in order
to avoid singular conﬁgurations. Hence, the range of joint
angles is set to be 0 ≤ θij ≤ π. Therefore the feasible
position set is deﬁned as follows:2) Feasible Velocity Set: A feasible velocity set denotes the realizable region of the velocity vector ṙ O =(ẋO , ẏO , żO , φ̇O , ϕ̇O , ψ̇O ).
Let us solve ṙ O from joint angular velocities of ﬁngers.
Assume that θ̇ i is the angular velocity at ﬁnger i, we
can solve the equation of ﬁngertips by using two types of
expression. One route is using forward kinematics of ﬁnger iand the other route is using r O . Given that pf i is the positionof the ﬁngertip i, the velocity of pf i is obtained bydgki (θ i ) = J Ei θ̇ i ,dt
dṗf i = goi (r O ) = J Gi ṙ
dtṗf i =

(6)

where gki (θ i ) is the kinematics of ﬁnger i, goi (r O ) is
position vector pf i in terms of r O , J Ei is the Jacobianmatrix at ﬁnger i, J Gi is the grasp matrix at ﬁnger i. From(5) and (6), we obtain
J Gi ṙ = J Ei θ̇ i .

(7)

J G ṙ = J E θ̇

(8)Therefore, we have

where

⎡
⎢
⎢
JE = ⎢
⎣

J E1

O
J E2
..

.
O
J En
⎡
⎤
J G1
⎢ J G2 ⎥⎢⎥J G = ⎢ . ⎥ ∈ 2n×6 ,⎣ .. ⎦⎤⎥⎥⎥ ∈ 2n×2n ,
⎦

J Gn
⎤
θ̇ 1
⎢ θ̇ 2 ⎥⎥⎢θ̇ = ⎢ . ⎥ ∈ mn .
⎣ .. ⎦⎡

θ̇ nAside from singular conﬁgurations, J E has an inverse matrix. Therefore, joint angular velocity vector θ̇ is representedbyθ̇ = J −1
E J G ṙ.

(9)Note that we assume that robotic hand does not have redundancy.
The condition of the feasible velocity set is a range of joint
angular velocities. We deﬁne θmax as the maximum absolutevalue of a joint angular velocity. In this paper, θmax is set
to be 1. Hence, the feasible velocity set is obtained by
SF v = ṙ | θ̇ = J −1
E J G ṙ, | θ̇ij |≤ 1 ,i ∈ [0, n], j ∈ [0, m].

SF p = {r | θ = f inv (r O ), 0 ≤ θij ≤ π} ,i ∈ [0, n], j ∈ [0, m].

(5)

(4)

1549

(10)

in order to utilize the friction, Assumed that μ is coefﬁcientof static friction, the condition is given by

Contact point idif i · ai1.
>| fi |
1 + μ2ai

Therefore, feasible force set is deﬁned as
1
f i · ai>,SF f = F O | F O = Gf , ai · f i ≥ 0,
| fi |
1 + μ2(19)| τ̇ij |≤ 1 , i ∈ [0, n], j ∈ [0, m].

Friction conefiFig. 3.

(18)

Friction cone and vectors for force constraints on ﬁnger i.

D. Calculation Algorithm
3) Feasible Force Set: The feasible force set is the realizable region of the force vector F O = (f TO , nTO )T consistingof force at the center of the mass f O = (fOx , fOy , fOz )Tand moment of the object nO = (nOφ , nOϕ , nOψ )T .
Here, we describe how to calculate the force vector F O .
Assume that f i is the force at ﬁngertip i, f i can be
represented as
f i = J TEi

−1

τ i.

(11)

Let di = (dix , diy , diz )T be the distance vector from thecenter of the mass of the object to the ﬁngertip i. Then, f O
and nO can be obtained by
n

fO =

f i,

(12)

i=1
n

(di × f i ).

nO =

(13)

i=1

Therefore, F O is deﬁned as:



fO
nO
≡ Gf


G=
f=



I3
I3
(d1 ×) (d2 ×)f1

f2

Step1 Calculate each safety margin (ap , av , af ) of grasping position candidates and the hand conﬁguration
Step2 Select minimum safety margin agm
Step3 Select maximum agm from all grasping position
candidates and hand conﬁgurations
If the grasping positions and the hand conﬁguration are notdecided because there are multiple minimum safety margins,repeat Step2 – Step3 by using the second smallest value andthe third smallest value.
IV. N UMERICAL E XAMPLE
A. Task speciﬁcations
In this paper, we examine tasks on 2 dimensional planewith two-ﬁngered robotic hand. We examine the validity ofour proposed method with the following tasks:
Pulling the object (Fig. 4)
Drawing lines (Fig. 5)• Opening/Closing a bottle cap (Fig. 6)
•
•



FO =

where

The best grasping positions and the hand conﬁguration fortasks are decided by using following steps:

···

···
I3
· · · (dn ×)
T
fn
.

(14)

,

The model and grasping positions candidate is shown in
Fig. 4 - Fig. 6. Task requirements and parameters are listed
in TABLE I. In our previous work[20], we only evaluatedgrasping positions. Here, we also evaluate distance d andposture φ as it is shown in Fig. 7.

(15)
(16)

Conditions for the feasible force set is the grasping force
and the friction cone. In order to keep grasping the object,forces on ﬁngertips should be applied to the object. Let ai =(aix , aiy , aiz )T be the inward unit normal vector of the object
surface on ﬁnger i, the condition of grasping force is obtainedby

B. Calculation results
Calculation results of each task are shown in TABLE
II, Fig. 8. If the object and the hand intersects, the safetymargins are set to be 0. According to our proposed method,
the best grasp of these tasks are the followings:(17)

Pulling the object: Position A, d = 1.0, φ = 0.0 [rad]Drawing lines with a pen: Position E, d = 1.0, φ = 0.0[rad]• Opening/Closing a bottle cap: Position B, d = 1.0

The friction cone is also essential condition to keep
grasping the object. Forces should be inside the friction cone

As it is shown in in the safety margins agm in TABLE
II, our proposed method can be used to decide the hand
conﬁguration and the posture.

ai · f i ≥ 0.

•
•

1550

F G H

Y

ΣO
1.0

E
D
C
B
A

X
1.0

Y

d
1.0

ΣH

(a) ImageFig. 4.

1.0(b) Model

0.2
0.2

1.0

0.2
0.2 0.2

0.5

1.0 X

(c) Grasping position candidate

Task speciﬁcations, analytic model, and grasping position candidate for numerical examples (Example 1: Pulling the object)

1.0

Y

d

1.0

(a) Image

E
D
C
B
A

1.0

ΣO

Fig. 5.

0.2

X

Y

1.0

ΣH

1.0X

(b) Model

0.2
0.2
0.2

1.0

0.2

0.5

(c) Grasping position candidate

Task speciﬁcations, analytic model, and grasping position candidate for numerical examples (Example 2: Drawing lines with a pen)

Y
1.0
d
1.0
(a) Imageπ/4 [rad]

1.0

C

R1.0

ΣH

π/2 [rad]

0.25

B
1.0
X

0.25

A

F
E

D

(c) Grasping position candidate(b) Model

Task speciﬁcations, analytic model, and grasping position candidate for numerical examples (Example 3: Opening/Closing a bottle cap)
1.5

1.5

1.0

1.0

1.0

0.5

0.5

0.5

0.0
-0.5

y [−]

1.5

y [−]

y [−]

Fig. 6.

ΣO
Y

G

X

0.0
-0.5

0.0
-0.5

-1.0

-1.0

-1.0

-1.5

-1.5

-1.5

-2.0
-2.0-1.5-1.0-0.5 0.0 0.5 1.0 1.5 2.0-2.0-2.0-1.5-1.0-0.5 0.0 0.5 1.0 1.5 2.0-2.0-2.0-1.5-1.0-0.5 0.0 0.5 1.0 1.5 2.0x [−](a) d = 0.5 [−], φ = 0.0 [rad]Fig. 7.

x [−](b) d = 1.0 [−], φ = 0.0 [rad]x [−](c) d = 0.5 [−], φ = π/4 [rad]Example of candidates of distance d and posture φ.

TABLE IP HYSICAL QUANTITIES FOR REQUIRED SET AND CANDIDATE FOR THE BEST GRASP
Physical quantity xr
yr
φr
Position
Posture φDistance d
Position
0.0 [-] 0.25 [-] 0.0 [rad]Pulling the object
Velocity
0.0 [-] 0.5 [-] 0.0 [rad/s] A - H (8 pattern) 0 [rad], π/4 [rad], π/2 [rad] 0 [-], 0.5 [-], 1.5 [-]Force
0.0 [-] 0.5 [-] 0.0 [-]Position
0.5 [-] 0.0 [-] 0.0 [rad]Drawing lines with a pen Velocity
0.5 [-] 0.0 [-] 0.0 [rad/s] A - E (5 pattern) 0 [rad], π/4 [rad], π/2 [rad] 0 [-], 0.5 [-], 1.5 [-]Force
0.5 [-] 0.0 [-] π/12 [-]Position
0.0 [-] 0.0 [-] π/6 [rad]Opening/Closing a bottle cap Velocity
0.0 [-] 0.0 [-] π/6 [rad/s] A - G (7 pattern) 0 [rad], π/4 [rad], π/2 [rad] 0 [-], 0.5 [-], 1.5 [-]Force
0.0 [-] 0.0 [-] π/6 [-]Task

C. Discussion
According to the result of Example 1 (Fig. 8 (a)), ourproposed method selects the grasping position A which
can guarantee comparatively larger workspace than othergrasping positions B – H. Thus, this grasp criterion could
decide the grasping position for the task with the requirementof the region of positions. This is because our proposed
method consider the positions, velocities, and forces on thegrasped object. This aspect is one of the advantages ofour criterion compared with the previous grasping criterion
which focuses on stable grasping and the tasks with therequirement of forces on the grasped object.
The result of Example 2 (Fig. 8 (c)) indicates that our
proposed method can deal with tasks with the requirementof forces on the grasped object similar to the grasp criteria
proposed in the past. We found that the evaluation function

1551

TABLE II
N UMERICAL CALCULATION RESULTS OF TASKSDistance d
Posture φ
Safety margin agm Position candidates Selected safety margin Selected positiond = 0.5
φ = 0 [rad]ap (1.60 [-])
Ad = 0.5 φ = π/4 [rad]ap (0.62 [-])
A–B
d = 0.5 φ = π/2 [rad]d = 1.0
φ = 0 [rad]av (1.71 [-])A–C
ap (3.60 [-], 3rd lowest)
A
Pulling the object
d = 1.0 φ = π/4 [rad]av (1.59 [-])
A–D
d = 1.0 φ = π/2 [rad]d = 1.5
φ = 0 [rad]av (1.29 [-])
A–E
d = 1.5 φ = π/4 [rad]d = 1.5 φ = π/2 [rad]d = 0.5
φ = 0 [rad]af (0.53 [-])
Cd = 0.5 φ = π/4 [rad]af (0.08 [-])
Cd = 0.5 φ = π/2 [rad]d = 1.0
φ = 0 [rad]ap (1.20 [-])
E
ap (3.60 [-], 1st lowest)
E
Drawing lines with a pen
d = 1.0 φ = π/4 [rad]af (0.12 [-])
Dd = 1.0 φ = π/2 [rad]d = 1.5
φ = 0 [rad]ap (0.93 [-])
Ed = 1.5 φ = π/4 [rad]ap (0.29 [-])
Ed = 1.5 φ = π/2 [rad]d = 0.5
ap (0.52 [-])A
Opening/Closing a bottle cap d = 1.0
ap (1.04 [-])B
ap (1.04 [-], 1st lowest)Bd = 1.5
ap (0.60 [-])
B
Safety margin [-]

Task

5.00
4.50
4.00
3.50
3.00
2.50
2.00
1.50
1.00
0.50
0.00

Safety Margin (Position)
Safety Margin (Velocity)

Safety Margin (Force)
Minimum Safety Margind = 1.0

A

B

C

D

E

F

G

Position A
φ = 0.0 [rad]H

Position candidates(a) Example 1: Pulling the object, d = 1.0, φ = 0 [rad](b) Best grasp (Example 1: Pulling the object)

Safety margin [-]

2.50
2.00

Safety Margin (Position)
Safety Margin (Velocity)

Safety Margin (Force)
Minimum Safety Margind = 1.0

1.50
1.00

Position A

0.50

φ = 0.0 [rad]

0.00
A

B

C

D

E

Position candidates(c) Example 2: Drawing lines with a pen, d = 1.0, φ = 0 [rad](d) Best grasp (Example 2: Drawing lines
with a pen)

Safety margin [-]

3.00
2.50

Safety Margin (Position)
Safety Margin (Velocity)

Safety Margin (Force)
Minimum Safety Margind = 1.0

2.00

Position A
φ = 0.0 [rad]

1.50
1.00
0.50
0.00
A

B

C

D

E

F

G

Position candidates(e) Example 3: Opening/Closing a bottle cap, d = 1.0

Fig. 8.

(f) Best grasp (Example 3: Opening/Closing
a bottle cap)

Numerical calculation results of the taskshows the highest value at the nearest position from the pen
tip. This result is the similar result of the human for selecting
the grasping point where humans can easily imply forces
toward the pen tip.

the safety margin of force at candidate B is higher thanthat of candidates A and C. It is because of considering thefriction cones and this result coincided with the intuition of
the human.

Numerical result of Example 3 (Fig. 8 (e)) shows that thesafety margins of position at candidates A – C is higher thatthat of candidates D – F. This is because we consider theintersection between the object and the hand. Additionally,

Additionally, safety margins agm in TABLE II differ if dand φ is different. Especially, agm of φ = 0[rad] showshigher value than others. It is because we consider theintersection of the object with robotic hand.

1552

1.50
Safety Margin (Force)diSafety margin [-]Wrench space P

Contact point i

ε

wi

1.25

1.00

0.75

0.50

0.25

Friction cone

wi,j

0.00
A

B

The largest inscribed ball
1.00(b) Wrench space with inscribed ball

G

0.50

Polyhedron volume

Er [-]

V. C OMPARISON WITH THE G RASP C RITERION BASEDON W RENCH S PACE

H

Ball radius

0.50

0.30

0.20

0.25
0.10

0.00

A. Grasp Criterion Based on Wrench Space

0.00
A

B

C

D

E

F

G

H

A

Position candidates(b) Result (Volume of wrench space)y [−]We compare our proposed method with the grasp criterion
used in GraspIt![14]. The index of this criterion is the volumeof wrench space and the maximum radius of the inscribedball in it. We deﬁne the index of the volume, maximum
radius of the inscribed ball as Er , Ev , respectively. Assume
that the wrench at ﬁnger i is wi , the wrench space is P, wi
and P are deﬁned by


s
fi
=
αi,j wi,j ,wi =λr i × f i

1.5

1.0

1.0

0.5

0.5

0.0
-0.5

D

E

F

G

H

0.0

-1.0

-1.5

-1.5

(d) Best grasp (Proposed method)

C

-0.5

-1.0

x [−]

B

Position candidates(c) Result (Maximum radius)

1.5

-2.0
-2.0-1.5-1.0-0.5 0.0 0.5 1.0 1.5 2.0

j=1

-2.0
-2.0-1.5-1.0-0.5 0.0 0.5 1.0 1.5 2.0x [−](e) Best grasp (GraspIt!)sαi,j ≥ 0,

αi,j ≤ 1
i=1 j=1

P = ConvexHull

F

0.75

Wrench space and inscribed ball

n

E

0.40

Ev [-]

Fig. 9.

Dy [−](a) Approximated friction coneC

Position candidates(a) Result (Proposed method)

 n


(20)

Fig. 10. Comparison results with the grasp criterion based on wrench space(wi,1 , · · · , wi,m ) .

TABLE IIID ESIGN CANDIDATE FOR THE TASK

(21)

Speciﬁcation pattern joint i Angular velocity [-] Joint torque [-]
A
1
1.0
1.0
2
1.0
1.0
B
1
0.5
0.5
2
1.0
1.0
C
1
1.0
1.0
2
0.5
0.5
D
1
0.5
0.5
2
0.5
0.5

i=1

Note that f i is the ﬁnger tip force at the corner j and ﬁnger
i, r i is the vector from the center of mass to the ﬁnger
tip i, wi,j is a wrench at the corner j of ﬁnger i, αi,jis the weight coefﬁcient at the corner j of ﬁnger i, and“ConvexHull” denotes convex set. wi is the sum of wi,jcreated by approximating friction cone in Fig. 9. Cornerj is created by the friction cone which is approximated aspolyhedron.
B. Calculation results
We used the task: “Stable grasping” and conditions are set
to be d = 1.0, φ = 0.0 [rad]. Results are shown in Fig. 10
and the best grasp are determined as followings:• Proposed method: Position A• Wrench space volume (Ev ): Position F, G, and H• Maximum radius of inscribed ball (Er ): Position G
C. Discussion
The evaluation result of the wrench space shows that thebest grasp is grasping the longer part of the object because
these criteria are mainly considering the moments on theobject. It means that these criteria have possibility to selectthe grasping positions that is too wide for robot hands tograsp. On the other hand, the result of our proposed method

implies that the best positions are shorter part of the object
because we consider hand mechanisms. These results show
the validity of taking hand mechanisms into consideration.
VI. A D ESIGNING M ETHOD C ONSIDERING M ULTIPLE
A SPECTS OF TASKS AND H AND C ONFIGURATIONS
A. Design guide
In order to deal with the speciﬁc task, we applied our
proposed method to design the robot hand. We designed thestructure of the robot hand, actuator joint angular velocity,
and actuator joint torque.
B. Calculation results
We designed the hand mechanism of the hand for the task:“Pulling the object” by using the candidates in Fig. 11 (a)– (e) and TABLE III. Results are shown in Fig. 11 (f) – (j)and the best design is Design 1 and pattern A.

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1.0

1.0

O

0.8

X

1.0
(a) Model (Design 1)

0.8
X

1.00

0.50

0.00

2.00

1.50

1.00

0.50

B

C

D

Position candidates(f) Result (Design 1)

2.00

1.50

1.00

0.50

B

C

D

1.0
(d) Model (Design 4)

Position candidates(g) Result (Design 2)

Fig. 11.

B

C

D

Position candidates(h) Result (Design 3)

O

0.8
X

1.0
(e) Model (Design 5)
3.00

Safety Margin (Position)
Safety Margin (Velocity)
Safety Margin (Force)
Minimum Safety Margin

2.50

2.00

1.50

1.00

0.50

0.00
A

1.2

Y

1.0
X

3.00
Safety Margin (Position)
Safety Margin (Velocity)
Safety Margin (Force)
Minimum Safety Margin

2.50

0.00
A

1.0

O

1.0

3.00
Safety Margin (Position)
Safety Margin (Velocity)
Safety Margin (Force)
Minimum Safety Margin

2.50

0.00
A

0.8

Y

1.2

O

1.0
(c) Model (Design 3)

Safety margin [-]1.50

Safety margin [-]

2.00

1.0

X

3.00
Safety Margin (Position)
Safety Margin (Velocity)
Safety Margin (Force)
Minimum Safety Margin

2.50

0.8

Y

1.2

1.0
(b) Model (Design 2)

3.00

Safety margin [-]

0.8

1.2

1.2

1.0

O

Y

1.2

Safety margin [-]Y

Safety margin [-]1.0

Safety Margin (Position)
Safety Margin (Velocity)
Safety Margin (Force)
Minimum Safety Margin

2.50

2.00

1.50

1.00

0.50

0.00
A

B

C

D

Position candidates(i) Result (Design 4)

A

B

C

D

Position candidates(j) Result (Design 5)

Numerical calculation results of the design based on proposed method

C. Discussion
The safety margin of position is higher if the ratio of thelinks is 1:1. It seems that the condition of the intersection
between the link and the bottom of the hand affected theresult. In addition, the safety margin of velocity is higher if
the mechanism is symmetric. This is because the velocityof the object which is easier to apply is different if themechanism is not symmetric. From these result, our proposed
method can be applied to the designing method of the handbecause safety margins differ if the hand mechanism and thespeciﬁcation of actuator changes.
VII. C ONCLUSIONS AND F UTURE W ORKS
We proposed a grasp criterion which considers multiple
aspects of tasks such as positions, velocities, and forces on
the grasped object. Evaluation of the feasibility of the task
is done by comparing two kinds of set: a required set anda feasible set. Inclusive relationships of these sets shows
the feasibility of the task. The numerical calculation resultsof the example indicate that our grasp criterion determine
the grasping position and the hand conﬁguration. In the
future, we are planning to consider the dynamics to calculate
feasible velocity sets and feasible force sets. Constructing themethod of evaluating on continuous candidates is also future
work of our research.
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