Proceedings of the
47th IEEE Conference on Decision and Control
Cancun, Mexico, Dec. 9-11, 2008

ThB05.4

Stability Analysis of Grasped Object by Soft-Fingers
based on Moment Stability
Akira Nakashima, Li Jingtai and Yoshikazu Hayakawa
Abstract— This paper analyze stability of an object grasped
by a pair of soft-ﬁngers in two-dimensional space based on
moment stability. We ﬁrstly deﬁne the moment stability as acriterion for stability of a grasped object when the object isperturbed for the orientation. Based on the moment stability,
the stability condition of an object grasped by a pair of hardﬁngers is derived. We indicate that contact points to satisfy
the condition are restricted to upper locations of the centerof mass. Next, the condition of an object grasped by a pairof semispherical soft-ﬁngers thirdly is considered. We derive a
sufﬁcient condition for the moment stability and indicate thatthe contact points to satisfy the condition can be in both upperand lower locations of the center of mass. Numerical examples
ﬁnally are shown. An example shows the grasp situation not tosatisfy the condition is established by a slender object with far
lower contact locations and its grasp is stable for the faults.

I. I NTRODUCTION
Many researchers have tried to introduce robots into
human’s daily environments. Since the robots are aimed to dovarious tasks instead of human, multi-ﬁngered robot hands
are effective as end-effectors. Multi-ﬁngered robot hands
have capability to grasp variously-shaped objects because thehands can grasp with multi contacts and can control grasping
force via multi joint inputs.
There are many research for grasp stability of an object
grasped by balanced contact forces when the object isperturbed from its equilibrium point. When the object isdisplaced and the balanced forces are invariant, the forces
generate the resultant force and moment to the object. Thisis referred to as the stiffness effect and is a direct measureof quasi-static grasp stability [1]. Cutkosky and Kao [2]derived the stiffness matrix between resultant force/momentand small displacement of a grasped object as a functionof geometry of the grasped object and contact condition.
Montana [3] analyzed the stiffness effect of an grasped object
with rolling contact in 2D space concerned with the curvatureof the object. Maekawa et al. [4] analyzed the stiffness effectwith rolling contact in 3D space and derived the stiffness matrix to evaluate the stiffness effect of the translation/rotationof the object and the contact movement due to the rolling
contact. Since it is assumed that ﬁnger-tips are rigid bodies
in the all studies, the stiffness effect destabilize the grasped
object except for the stiffness effect due to rolling contact[4]. While these studies do not control contact forces, there
are studies where the stiffness effect is controlled by contact
All authors are with Mechanical Science and Engineering, Graduate
School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya,Japan akira@haya.nuem.nagoya-u.ac.jp, and Y. Hayakawa is alsowith RIKEN-TRI Collaboration Center, RIKEN, 2271-103, Anagahora,
Shimoshidami, Moriyama-ku, Nagoya, Japan.

978-1-4244-3124-3/08/$25.00 ©2008 IEEE

forces to stabilize a grasped object around its equilibrium[5], [6], [7]. In this paper, we concentrate on the stiffness
effect without controlling contact forces.
Recently, soft-ﬁngers made from soft materials has beenstudied. Compared with hard-ﬁngers made from hard materials, the soft-ﬁngers have a lot of advantages [8]: thecontact friction is larger than that of the hard-ﬁnger since
the contact is surface; the soft-ﬁnger can reduce the impact
force in contact establishment; it can ﬁt on varrious shapesof the object; there exists moment friction about the contact
normal. Inoue et al. [9], [10] considered an one-dimensional
deformation model of a semispherical soft ﬁngertip with thegeometric and material nonlinearities and proposed a quasistatic manipulation method based on the local minimum
elastic energy with the rolling contact in two-dimensional
space. Furtheremore, they proposed a two-dimensional deformation model and studied dynamic stability [11], [12].
However, they showed the stability of the soft-ﬁnger grasp
by only numerical simulations. Furthermore, since they did
not decouple the contact movements due to the rolling anddeformation, the stability effect of the rolling was not shown.
In this paper, we analytically study stability of an object
grasped by a pair of soft-ﬁngers in two-dimensional space
based on the stiffness effect. The study [4] showed that the
stiffness effect due to the object rotation is much larger thanthe one due to the object translation and the stiffness effectdue to the rolling contact stabilize the object. Therefore,
we consider the object rotation and the rolling contact. We
ﬁrstly deﬁne the moment stability as a criterion for stabilityof a grasped object when the object is perturbed for theorientation. Based on the moment stability, the stability
condition of an object grasped by a pair of hard-ﬁngers
is derived. We indicate that contact points to satisfy thecondition are restricted to upper locations of the center ofmass. Next, the condition of an object grasped by a pair ofsemispherical soft-ﬁngers thirdly is considered. We derivea sufﬁcient condition for the moment stability and indicatethat the contact points to satisfy the condition can be in both
upper and lower locations of the center of mass. Numericalexamples ﬁnally are shown. An example shows the grasp
situation not to satisfy the condition is established by aslender object with far lower contact locations and its grasp
is stable for the faults.
II. S TABILITY A NALYSIS OF H ARD -F INGER G RASP
We ﬁrstly analyze stability of an object grasped by a pairof hard-ﬁngers in 2D space as shown in Fig. 1. The object
is assumed to be a convex polyhedron. ΣO is the reference4582

47th IEEE CDC, Cancun, Mexico, Dec. 9-11, 2008

ThB05.4

Finger 2

Finger 1

f2

f1
l1

f1

γm

l2

y

− α1

Friction
Corn

f2

p1
O

f 1 p1
mg

α2

x

γf

p2

p1

p2

h1
Fig. 1.

frame attached at the center of mass of the object. Notethat vectors without left superscripts are expressed in ΣO .
f i , pi ∈ R2 (i = 1, 2) are the contact force and the position
vector at the ith contact point respectively. li and hi are thedistances of the ith contact point from the center of mass
along the normal and tangent, respectively. αi is the angle ofthe side grasped by the ith ﬁnger from the y-axis of ΣO . The
two red thick dashed lines at the ith contact point represent
the boundaries of the friction cone, which is deﬁned as

Grasp situations to satisfy the sufﬁcient condition (10).

where M (θ) is the moment caused by the object rotation θ.
(5) means that M (θ) is the restoring force for the rotational
displacement θ as shown in Fig. 2.
The moment M (θ) in the hard-ﬁnger grasp is given by
M (θ) = [(R(θ)p1 )×]f 1 + [(R(θ)p2 )×]f 2 ,= [pi ×]RT (θ).

(1)

where m is the mass of the object, g is the gravitational
constant and [pi ×] ∈ R1×2 is the 2D cross product deﬁned
as[pi ×] := [−pyi pxi ], pi := [pxi pyi ]T .
(4)
For the system of the hard-ﬁnger grasp, we consider the
stiffness effect due to the object rotation θ. It is assumed thatthe object is only rotated around the center of mass and thecontact force f i is invariant. We deﬁne the moment stabilityasM (θ)θ < 0,
(5)

 
cos θ
sin θ fix[pi ×]
M (θ) =− sin θ cos θ fiy
i=1



 
2

fiy
f
=
[pi ×] cos θ ix + sin θ
fiy
−fix
=

0

f1

y
p2

x

f2

Finger 1y

R (θ ) p2

R (θ ) p1
O

cos θ −


2


pT
i fi

sin θ

i=1

T
= −(pT1 f 1 + p2 f 2 ) sin θ,(8)where the ﬁrst term of the third row equals 0 from (3). The
following theorem holds:
Theorem 1: The necessary and sufﬁcient condition of the
moment stability for the hard-ﬁnger grasp isT
pT
1 f 1 + p2 f 2 > 0.

pTi f i > 0, i = 1 and 2.

Finger 2

f2[pi ×]f i(9)

Let us substitute the condition (9) of Theorem 1 by thefollowing sufﬁcient condition

M (θ )

Finger 2

i=1
2

i=1

θ

f1

(7)



2


where eni , eti ∈ R are the normal and tangent vectors andμi is the static friction coefﬁcient.The grasp of the object isdeﬁned as the following equilibrium equations:
0f1 + f2 +
= 0,
(2)
−mg
[p1 ×] f 1 + [p2 ×] f 2 = 0,(3)

Finger 1(6)

where R(θ) ∈ R2×2 is the rotation matrix. Calculating[(R(θ)pi )×] with (4) leads to[(R(θ)pi )×]= −pix sin θ − piy cos θ pix cos θ − piy sin θ

2Fig. 2.

(c)

Substituting (7) into (6), we getTfT
i eti ≤ μi |f i eni |,

O

(b)

h2

An object grasped by a pair of two hard-ﬁngers.

p1

pi

mg

(a)
Fig. 3.

p2 f 2x

Moment stability for rotation perturbation.

(10)

For simplicity, the side angles are set to αi = 0. The areasof f 1 and f 2 to satisfy (10) are represented by the shaded
areas in (a) and (b) of Fig. 3. It is obvious that there do notexist f 1 and f 2 when the contact positions are under thecenter of mass. In order to construct the grasp of the object,
the contact force f i also has to satisfy the friction condition(1), which is illustrated in (c) of Fig. 3. γfi and γmi are the4583

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ThB05.4

angles of the boundaries of the friction cone and (10) fromthe contact normal, respectively. These angles are deﬁned asγfi := tan−1 μi , γmi := tan−1

hi
.
li

(11)

Therefore, it is necessary for the distances li and hi of thecontact point to satisfyγmi ≤ γfi

hi
≤ μi
li

⇔

xF2

L2

l1

y

h1

− α1
Fφ1

(12)

β2
O

yF1

as shown in (c) of Fig. 3.
Remark 1: From (12), it is easy to grasp a slender object
with the upper contact points while it is difﬁcult to grasp awide object with central contact points. In other words, thecontact locations for the stability are restricted to far upperfrom the center of mass.

F2

yF 2
F r2

x

h2

l2
F φ2

α2

β1

F1

xF1
Fig. 5.

L1
An object grasped by a pair of soft-ﬁngers.

III. STABILITY A NALYSIS OF S OFT-F INGER G RASP
A. System Conﬁguration
In this section, we secondly analyze stability of an object
grasped by a pair of soft-ﬁngers in 2D space. The deformation model of the soft-ﬁnger is 2D reduction model of the
3D model proposed in [13]. Fig. 4 shows the semi-spherical
soft ﬁnger with the radius ri . ΣFi is ﬁxed at the center ofthe ith ﬁnger and the yFi -axis is perpendicular to the base.
The deformation area of the soft-ﬁnger is deﬁned as thedashed area, which is the overlapped area between the ﬁnger
and the object. The contact point before the deformation isrepresented by the contact angle φi . Since the contact surface
is the circle from the geometry of the deformation deﬁnition,
the contact point after the deformation is deﬁned as thecenter of the contact surface. In the latter, the contact point
after the deformation is simply called the contact point. We
deﬁne the deformation displacements δri and δφi as the polar
coordinates of the contact point. δri is the radial displacementand δφi is the angle displacement in the inverse directionof φi . ΣCFi is the contact frame attached to the contact
point and the xFi -axis is in the contact normal. The forces
produced by the deformation are assumed to be generated
based on Hook’s law with respect to the small elements of the

O bject

yF i

C ontact Point
befor e Defor.

xC Fi
F inger i

yC F

i

i

φi
Fi

Fig. 4.

C Fi

Fri := kri πδr2i
1
δφi
2
×
,
Fφi := kφi πri33φi (π − φi ) ri − δri

δr
xF i

Deformation model of a semi-spherical soft ﬁnger.

i

(13)
(14)

where kri and kφi are the stiffness coefﬁcients. The directions of Fri and Fφi are in xCFi and yCFi -axes, respectively.
The derivation and experimental validation are shown inAppendices A and B.
Fig. 5 shows the object grasped by the soft-ﬁngers. Notethat Fri and Fφi are normal and tangential to the object
surface. Fri and Fφi satisfy the following equilibrium equations:
ROCF1 F1 + ROCF2 F2 = 0,

[p1 ×] ROCF1 F1 + [p2 ×] ROCF2 F2 = 0,

(15)
(16)

where Fi := [Fri Fφi ]T ∈ R2 , ROCFi := R(θFi )R(φi −δφi ), R(θFi ) is the rotation matrix from ΣFi to ΣO , andR(φi − δφi ) is the rotation matrix from ΣCFi to ΣFi . Li isthe length of the line segment between the centers of the ith
ﬁnger and the object. βi is the angle of this line segmentfrom the normal to the grasped edge. These are deﬁned asli,ri − δri (0) + hi
ri − δri (0) + hi,
Li :=cos βiβi := tan−1

C ontact Point
after Defor.
Defor med
A rea

δφ

deformation displacements. From Hook’s law, the generated
forces by the deformation δri and δφi are derived as

(17)
(18)

where δ ri (0) and δφi (0) are the deformation displacementsin the initial equilibrium grasp to satisfy (15) and (16). The
other parameters and vectors are the same as in Fig. 1.
B. Rolling Contact with Deformation
The contact point Fi pCFi ∈ R2 expressed in ΣFi is givenby
(ri − δri ) cos(φi − δφi )
Fi
.
(19)
pCFi =(ri − δri ) sin(φi − δφi )4584

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ThB05.4

LOi

ObjectSurfacewhere
Mr (θ) :=dδ riContact Move.
due to Defor.

d ( Fi pCd F )Contact Move.
on Surface
Fi

i

pC′ F

Fi

i

dφi
dδ φi
Fig. 6.

iri − δ ri

Differentiating (19) leads toṗCFi = Fi ṗrCF − Fi ṗdCF ,
i

where

i


0
φ̇ ,
ri − δri i
 

δ̇ri
1
0
Fi d
ṗCF = R(φi − δφi ).
i0 ri − δri δ̇φiṗrCF = R(φi − δφi )
i

(20)

(26)

−hi Fφi (θ).

(27)

rai (θ) = Li cos(βi − θ) − hi .
(21)
(22)

Fi r
ṗCF
i

represents the contact movement which is causedby the rolling of the ﬁngertip of the radius ri − δri withthe rotation velocity φ̇i , while Fi ṗdCF represents the contactimovement which is caused by the deformation movementof the ﬁngertip with the deformation velocity δ̇ri and δ̇φi .
This is illustrated in Fig. 6, where Fi pCFi and Fi pCF areithe contact points before and after the contact movement,
and dφi , dδri and dδφi are the increments of φi , δri and δφirespectively. The contact point Fi pCFi moves to Fi pCF byithe movements d(Fi prCF ) and d(Fi pdCF ) due to the rollingiiand deformation. Since Fi ṗrCF is the contact movement oni
the ﬁngertip surface while Fi ṗdCF is not the movement on theisurface but rather the deformation movement, only Fi ṗrCFiequals the velocity of the contact point on the object surface:


0
Fi r
ṗCF = R(φi − δφi )
,
(23)
i
L̇Oi
where the right term is the contact velocity on the object
and LOi is the increment of the contact point on the object.
Substituting (21) into (23), we get(ri − δri )φ̇i = L̇Oi .

Li sin(βi − θ)Fri (θ),

Mr (θ) and Mφ (θ) are the moments generated by Fri (i =1, 2) and Fφi (i = 1, 2) respectively. Since Fri (θ) andFφi (θ) are composed of δri , δφi and φi from (13) and (14),
we derive δri , δφi and φi as functions of θ in the following.
Fig. 7 shows the geometric relation at ith contact point
when the object is rotated through θ. The left and rightﬁgures show the relations before and after the rotation
respectively. In the right ﬁgure, rai (θ) is the distance of thecontact point from the center of the ﬁnger. From the similar
triangles with respect to the point a, rai (θ) is obtained as



Fi

i=1
2

i=1

pC F

Rolling motion with the deformation.

FiMφ (θ) :=2


From (28), we get the radius deformation δri (θ):δri (θ) = ri − rai (θ) = ri + hi − Li cos(βi − θ).

(29)

Next, we derive the rotation deformation δφi (θ) and thecontact angle φi (θ):δφi (θ) := δφi (0)+Δδφi (θ), φi (θ) := φi (0)+Δφi (θ), (30)where Δδφi (θ) and Δφi (θ) are the increments from thegrasping equilibrium due to the rotation θ. From Fig. 7, theincrement of the relative angle δφi (θ) − φi (θ) equals to θ:
δφi (θ) − φi (θ) − (δφi (0) − φi (0)) = Δφi (θ) − Δδφi (θ) = θ.
(31)
In addition to the equation (31), the following rolling contact
equation holds from (24):(ri − δri (θ))d(Δφi (θ)) = d(LOi (θ)).

(32)

The contact displacement LOi (θ) on the object due to therotation θ is obtained as
LOi (θ) = Li sin βi − Li sin(βi − θ),(33)

where the ﬁrst and second terms are the distances of thecontact points from the center of mass along the object
surface before and after the object rotation θ respectively.

θ

(24)
hi

This equation is signiﬁcant to derive the deformation due tothe object rotation θ in the next subsection.

hi
C ontact
Pointrai (θ )Liβi − θLia

C. Derivation of Moment due to Object Rotation
The moment M (θ) caused by the rotation displacement
θ is derived here. From the geometric relations between thecontact points and the center of mass shown in Fig. 5, M (θ)
is easily given by
M (θ) = Mr (θ) + Mφ (θ),

(28)

(25)

4585

θ
Fi

Fig. 7.

δφ ( 0) − φi ( 0)iFiδφ (θ ) − φi (θ )iGeometry at the ith contact point with object rotation θ.

47th IEEE CDC, Cancun, Mexico, Dec. 9-11, 2008

ThB05.4

From the two equations (31) and (32), we can derive δφi (θ)
and φi (θ). From (32), (33) and (28), d(Δφi ) is expressed as
Li cos(βi − θ)ra (θ) + hidθ = idθrai (θ)rai (θ)


hi
dθ.
= 1+
rai (θ)d(Δφi ) =(34)

Integrating (34) from 0 to θ leads tohi
Δφi (θ) = θ +  2(f (0) − f (θ)),
Li − h2i

(35)




 tan βi −θ + Li −hi 2Li +hi .

f (θ) := ln i −hi  tan βi2−θ − L
Li +hi

(36)

where

From (31), δφi (θ) is obtained ashi

δφi (θ) =  2(f (0) − f (θ)).
Li − h2i(37)Remark 2: The deformation displacements δri (θ) andδφi (θ) are derived as the functions of θ since the contact
movement due to the deformation Fi ṗdCF is not included ini(32). If Fi ṗdCF is not decoupled from (32), δri (θ) and δφi (θ)
ican not be driven analytically.
D. Sufﬁcient Condition of Moment Stability
We consider a condition for the moment (25) to satisfy the
moment stability condition (5). The condition (5) is rewrittenasM (θ)θ = (Mr (θ)−Mr (0))θ+(Mφ (θ)−Mφ (0))θ < 0, (38)where Mr (0) = −Mφ (0) from M (0) = 0.
First, consider the moment Mr (θ) of (26). Since rai (θ)
of (28) is the distance of the contact point, the following
inequality has to be satisﬁed:
0 < rai (θ) < ri .

(39)

Substituting (28) into (39) and solving the resultant equation
with respect to cos(βi − θ) lead tohiri + hi< cos(βi − θ) <
.
(40)
Li
Li
Under the condition (40), the following lemma holds:
Lemma 1: Suppose that (40) holds. Then, the following
relations hold:d(41)
Mr (θ) ≤ 0, (Mr (θ) − Mr (0))θ < 0.
dθ
Proof: Differentiating (26) results indM (θ)
kr πδri
= i
g(θ),
dθ
3
where



r i + hig(θ) := cos(βi − θ) −
6Li

(42)

2 
2
r i + hi2−
− . (43)
6Li
3

Since dMdθ(θ) of (42) is convex downward with respect tocos(βi − θ), the maximum of dMdθ(θ) is given by the left ori≤ 1.
right boundaries of (40). Firstly consider the case riL+h
i
Then,

2 
2
ri + h iri + h i2−−g(θu ) = 5
6L
6Li
3
 i

2
ri + h i2− 1 ≤ 0,
(44)
=
3
Li
2 
2

ri + h iriri + hi2g(θl ) = 5
−
−
−
6Li
Li
6Li
3


2
ri + h i2− 1 ≤ 0,
(45)
<
3
Li

where θu and θl are deﬁned asri + hicos(βi − θu ) =,Licos(βi − θl ) =

hi
.
Li

i
Next consider the case riL+h> 1. Then, g(θu ) takes theidifferent value given byg(θu ) =Li − (ri + hi )
< 0,
3Li

(46)

where θu is deﬁned ascos(βi − θu ) = 1.
The maximum of g(θ) is one of (44), (45) and (46). Therefore, from (42), (43), (41) holds.
Remark 3: Lemma 1 indicates that the moment producedby the radius forces Fri (θ) (i = 1, 2) always effects on therestoring force to stabilize the object against the rotational
perturbation.
Second, consider the moment Mφ (θ) of (27). Substituting
the following formula of trigonometric functionscos(βi − θ) =1 − tan2 βi2−θ

1 + tan2 βi2−θinto (40) and solving the resultant inequality with respect totan βi2−θ yield

 


tan βi − θ  < Li − hi .
(47)

2 
Li + hi
Under the condition (47), the following lemma holds:
Lemma 2: Suppose that (47) holds. Then, Δφi (θ) andΔδφi (θ) of (35) and (37) are monotone increasing.
Proof: From (35) and (37), it is sufﬁcient to check theproperty of f (θ) of (36). f (θ) is rewritten asf (θ) = ln f1 (f2 (θ)),

(48)

where




 x + ai  , ai := Li − hi
f1 (x) := x − ai 
Li + h iβi − θ
f2 (x) := tan.
2
As for the function f1 (x),


x+a
−2ai
=
< 0, f (−ai ) = 0.
x−a(x − ai )2

4586

(49)
(50)

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ThB05.4

Therefore, f1 (x) > 0 and f1 (x) > 0 in the case |x| < aifrom (47). On the other hand, the function f2 (x) is monotone
decreasing because
1βi − x
=−
< 0.
tan
2
2 cos2 βi2−x
Furthermore, |f2 (x)| < ai from (47). Since f1 (f2 (θ)) ismonotone decreasing and ln x is monotone increasing, f (θ)
is monotone decreasing. Therefore, Δφi (θ) and Δδφi (θ) of(35) and (37) are monotone increasing because Δφi (θ) andΔδφi (θ) have −f (θ).
A sufﬁcient condition for the moment (25) to satisfy the
moment stability condition (5) is derived:
Theorem 2: Suppose thatδφi {(2φi − π)(rai + hi ) − Li sin(βi − θ)φi (π − φ)}+φi (π − φi )hi > 0.

(51)

Then, (5) holds.
Proof: From (14), differentiating Fφi (θ) leads to
2
1
d×Fφ (θ) = kφ πri3 ×dθ i3{rai φi (π − φi )}2
δφi (2φ − π)(rai + hi ) +
φi (π − φi ) {hi − Li δφi sin(βi − θ)} . (52)

From (52) and (51), the following holds:dFφ (θ) > 0.
dθ
From (27), (53) yields to(Mφ (θ) − Mφ (0))θ < 0.

(53)

(54)

Therefore, from (38) and Lemma 1, (5) holds.
Remark 4: The next section shows an example not tohold the condition (51). However, the grasped object is stablesince the moment produced by the radius forces Fri (θ) (i =1, 2) are much larger than the moment produced by therotation forces Fφi (θ) (i = 1, 2). The grasp situation of theshown example is sinular. This can imply that the condition(51) can hold in ordinary cases of grasping.
IV. N UMERICAL E XAMPLES(a) Initial grasp.

(b) Moments when the object is rotated.

Fig. 8.

Numerical example 1.

initial situation of the grasping. The red, blue and blackarrows are the radial forces Fri , rotational forces Fφi andgravity force mg respectively. Fig. 8 (b) shows the moments
Mr (θ), Mφ (θ), M (θ) and Mh (θ), which are dashed blueline, red line, solid green line and black line, respectively.
Mh (θ) is the moment in the case of the hard-ﬁnger grasp,
which is deﬁned as (8). Note that the black line represents
Mh (θ) × 10. We conﬁrm that the all moments are therestoring forces and M (θ) is much larger than Mh (θ). Thisshows that the stability of the soft-ﬁnger grasp is much better
than that of the hard-ﬁnger grasp. Note that the range of θ
is given by the product set of the ranges of θ to satisfy (40),−π < δφi (θ) < π and 0 < φi (θ) < π with (30), (35)–(37).
Fig. 9 shows Fri and Fφi when the object is rotated in thepositive and negative directions. In the case θ > 0, the forces
Fr1 and Fφ1 to stabilize the object increase while the forces
Fr2 and Fφ2 to destabilize the object decrease. This indicatethat the forces produced by the soft-ﬁnger deformation effecton the stability well. The case θ < 0 is similar to the caseθ > 0.
Figs. 10-11 show the case where the object is grasped
at the lower contact points. The parameters and deﬁnitionsof the arrows and lines are the same as Figs. 8-9. In
Fig. 10 (a), we conﬁrm that the moments Mr , Mφ and M
are the restoring forces while the moment Mh destabilizes
the grasped object. This shows that the soft-ﬁnger grasp isstable even when the hard-ﬁnger grasp is unstable. In thecase θ > 0 of Fig. 11, the forces Fr2 and Fφ1 to stabilize the

Numerical examples are shown to conﬁrm the stability ofthe soft-ﬁnger grasp.
Fig. 8 (a) shows the case where the rectangular object isgrasped at the upper contact points. The radius of the ﬁnger
is ri = 10[mm] and the coefﬁcients are kri = 0.377, kφi =0.166[N/mm2 ] as shown in Appendix B. The height, widthand mass of the object are 120, 100[mm] and m = 0.3[kg]respectively. The distances of the ith contact point from thecenter of mass are li = 50 and hi = 50[mm]. The contact
angles are φ1 = φ2 = 90[deg]. The initial deformation
displacements are δr1 (0) = δr2 (0) = 2[mm] and δφ1 (0) =−δφ2 (0) = 10[deg]. From these parameters, (17) and (18),β1 = −β2 = 41[deg] and L1 = L2 = 76.5[mm]. The
parameters satisfy the condition (51). Fig. 8 (a) shows the

4587

θ

Fig. 9.

−θ

Fri and Fφi of the example 1 when the object is rotated.

47th IEEE CDC, Cancun, Mexico, Dec. 9-11, 2008(a) Initial grasp.

ThB05.4(b) Moments when the object is rotated.

Fig. 10.

Numerical example 2.

(a) Initial grasp.

(b) Moments when the object is rotated.

Fig. 12.

object increase while the forces Fr1 and Fφ2 to destabilize
the object decrease. This indicate that the forces producedby the soft-ﬁnger deformation effect on the stability although
the contact locations are changed. The case θ < 0 is similar
to the case θ > 0.
Let us consider a situation not to satisfy the sufﬁcientcondition (51). In the condition (51), it is obvious thatrai + hi > 0 and φi (π − φi ) > 0 from (39) and 0 < φi < π.
Therefore, the following conditions have to hold not tosatisfy (51):δφi > 0 : 2φi − π < 0, sin(βi − θ) > 0,
δφi < 0 : 2φi − π > 0, sin(βi − θ) < 0.

(55)
(56)

conﬁrm that the moment M (θ) stabilizes the object while
the moment Mφ (θ) produced by Fφi destabilizes the object.
This is because the moment produced by Fri is much larger
than the moment Mφ . Note that the condition δφi > 0 (< 0)of (55) or (56) is broken when the rotation displacement θ
decreases (increases) becase δφi (θ) is monotone increasingfrom Lemma 2. Therefore, Fφ1 or Fφ2 can satisfy the condition (51). This result indicates that the grasping situation
not to satisfy (51) is composed of only particular the shapeof the object, contact points and contact angles and in thatsituation the soft-ﬁnger grasp also is stable.

(55) and (56) lead to

V. C ONCLUSIONS AND F UTURE W ORKS

δφi > 0 : φi  1, βi  1,
δφi < 0 : φi  1, βi  −1.

(57)
(58)

In the initial grasp, δφ1 > 0 and δφ2 < 0. Therefore, (57)and (58) correspond to i = 1 and i = 2 respectively. In
order to achieve (57) and (58), the object with the height
192[mm] and width 20[mm] is grasped. The distances of thelower ith contact point from the center of mass are li = 80
and hi = 10[mm]. The contact angles are φ1 = 30 and
φ2 = 150[deg]. The initial deformation displacements areδr1 (0) = δr2 (0) = 2[mm] and δφ1 (0) = −δφ2 (0) = 6[deg].
From these parameters, β1 = −β2 = 77[deg] and L1 =L2 = 81.2[mm]. Fig. 12 (a) shows the initial grasp andFig. 12 (b) shows the moments when the object is rotated.
Note that the dashed red line represents Mφ (θ) × 100. WeθNumerical example 3.

−θThis paper analyzed stability of an object grasped bya pair of soft-ﬁngers in two-dimensional space based on
moment stability. We ﬁrstly deﬁned the moment stability asa criterion for stability of a grasped object when the object isperturbed for the orientation. Based on the moment stability,
the stability condition of an object grasped by a pair of
hard-ﬁngers was derived. We indicated that contact points tosatisfy the condition are restricted to upper locations of thecenter of mass. Next, the condition of an object grasped by apair of semispherical soft-ﬁngers thirdly was considered. We
derived a sufﬁcient condition for the moment stability andindicated that the contact points to satisfy the condition canbe in both upper and lower locations of the center of mass.
Numerical examples ﬁnally were shown. An example shows
the grasp situation not to satisfy the condition is establishedby a slender object with far lower contact locations and its
grasp is stable for the faults.
In future works, it is necessary to consider the friction
condition and the limitation of ﬁnger joints. We will try to
optimize the soft-ﬁnger grasp with respect to the mentioned
conditions.
R EFERENCES

Fig. 11.

Fri and Fφi of the example 2 when the object is rotated.

[1] M. T. Mason and J. K. Salisbury, Robot Hands and the Mechanics ofManipulation, Cambridge, MA: MIT Press, 1985.
[2] M. Cutkosky and I. Kao: Computing and Controlling the Complianceof a Robot Hand, IEEE Trans. Robot. Automat., Vol. 5, No. 2, pp.
151-165, 1989.
[3] D. J. Montana: Contact Stability for Two-Fingered Grasps, IEEE
Trans. Robot. Automat., Vol. 8, No. 4, pp. 421-430, 1992.

4588

47th IEEE CDC, Cancun, Mexico, Dec. 9-11, 2008

δF

dθi

xF

d(δ Fr )
i
dfr

φi


yF

ri

(a) Compressive Force by δri .

Fig. 13.

d(δ Fφ )
i

i

dli
δφ

i

Contact
Surfacedθiφi − δφ
i

ΣF
i

i

zF
i

b
li

20

0.02

15

0.015

0.01

5

0.005

0

0

0.5

1

1.5

2

2.5

3

3.5

4

δ []

φi
q


yF
r 2 − x2ii(a) Compressive Force Fri .

Fig. 14.

i(b) Compressive Moment by δφi .

Finger-tip forces by δri and δφi .

A PPENDIX
A. Derivation of Forces by Deformation(1) Derivation of δ Fri :Consider the dashed area of the small element dθi rotatedabout the center axis of the contact surface through dψi in
Fig. 13 (a). dfri is the force generated by the deformation
at the point a. Since the force parallel to the contact
surface dfri sin θi is counteractive against −dfri sin θi at thesymmetric point b of a with respect to the center axis, it
is sufﬁcient to consider forces in the center axis dfri cos θi .
r −δSince the compression ratio at a is 1− riicosrθii , the generated
δ
force d( Fri ) of the small area at a is given by d(δ Fri ) =r −δ
2 × kri (1 − riicosrθii )ri dθi ri sin θi dψi × cos θi , where kri is
the stiffness coefﬁcient. Then, αi  πr −δδFr i =
2kri 1 − riicosrθii ri2 sin θi cos θi dψi dθi

4.5

5

0

1

2

3

4

5

6

7

8

9

10

δφ [δ](b) Compressive Moment Mφi .

Experimental validation of the force model.



r −δ
where αi := arccos i ri ri .
(2) Derivation of δ Fφi :Fig. 13 (b) shows the cross section such that the crosssection of Fig. 13 (a) is moved along zF i -axis with the
distance zi . Consider the dashed area of the small element
dli with the width dzi . Since the compression ratios ofδδφi
the compressive and extensive directions are φφii and π−φ,i
the generated moment of the dashed area is given byδφiδ
d(Mφi ) = li × kφi ( π−φ+ φφii )dli dzi , where kφi is theistiffness coefﬁcient. Then, ri √ri2 −zi2
δφi
δφi
dli dziMφ i =
li kφi+π − φi φi
−ri 0


1
2
1
δφi .
= kφi ri3+(60)3
π − φi φi
Since the point of application of the moment Mφi is thecontact point with the length ri − δri from the center of the
ﬁnger, the force produced by rotational deformation δφi isgiven by
Mφ i
Fφi :=.
(61)r i − δ ri
B. Experimental validation of the force model
The material of the soft ﬁngers is an urethane resin withthe hardness C-5. The radius of the soft-ﬁnger is 10[mm].
Fig. 14 show the experimental results of Fri and Fφi of(13) and (14). The stiffness coefﬁcients in (13) and (14)are calculated by the least-squares method with the force
model of (13) and (14) and the experimental data. The
coefﬁcients are kri = 0.377, kφi = 0.166[N/mm2 ]. The lines
represent the force model of (13) and (14) with the estimatedcoefﬁcients. In Fig. 14 (a), the cross, square, triangle andcircle represent the forces of Fri with φi = 90, 80, 70
and 60[deg] respectively. In Fig. 14 (b), the solid, dashed,dashed and single-dotted and dotted lines correspond to theexperimental data with the cross, square, triangle and circle
which are the same as Fig. 14 (a). In Fig. 14 (a) and (b), we
conﬁrm that the lines are approximate to the experimental
data.

0

= πkri δr2i ,100[4] H. Maekawa, K. Tanie and K. Komoriya, Kinematics, Statics andStiffness Effect of 3D Grasp by Multiﬁngered Hand with Rolling
Contact at the Fingertip, Proc. of Int. Conf. on Robot. Automat., pp.
78-85, 1997.
[5] M. Kaneko, N. Imamura, K. Yokoi and K. Tanie, A Realization of
Stable Grasp Based on Virtual Stiffness Model by Robot Fingers, IEEE
Int. Workshop on Advanced Motion Control, pp.156–163, 1990.
[6] H. Maekawa, K. Tanie, K. Komoriya and M. Kaneko, Stable Grasp andManipulation of an Object by a Three-Fingered Hand Using Position
and Stiffness Control, Proc. of Int. Conf. on Robot. Automat., 1995.
[7] Y. Funahashi, T. Yamada, M. Tate and Y. Suzuki, Grasp stability
analysis considering the curvatures at contact points, Proc. Int. Conf.
Robot. Automat., pp. 3040–3046, 1996.
[8] K. B. Shimoga and A. A. Goldenberg: Soft Robotic Fingertips Part I:
A Comparison of Construction Materials, Int. J. Robot. Res., Vol. 15,
No. 4, 320-334, 1996.
[9] T. Inoue and S. Hirai, “Elastic Model of Deformable Fingertip forSoft-ﬁngered Manipulation,” IEEE Trans. on Robotics, vol. 22, no. 6,
pp. 1273–1279, 2006.
[10] T. Inoue and S. Hirai, Quasi-Static Manipulation with Hemispherical
Soft Fingertip via Two Rotational Fingers, Proc. Int. Conf. on Robot.
Automat., pp.1947–1952, 2005.
[11] T. Inoue and S. Hirai, Study on Hemispherical Soft-Fingered Handlingfor Fine Manipulation by Minimum D.O.F. Robotic Hand, Proc. Int.
Conf. on Robot. Automat., pp.2454–2459, 2006[12] T. Inoue and S. Hirai, Dynamic Stable Manipulation via Soft-Fingered
Hand, Proc. Int. Conf. on Robot. Automat., pp.586-591, 2007.
[13] A. Nakashima, T. Shibata and Y. Hayakawa, ”Grasp and manipulation
by soft ﬁnger with 3-dimensional deformation,” Int. Conf. on Mechatronics and Information Technologys, 2007.

0

0.025

[]

i

θi

i

δF

ri sin θi

a


zF
i

0.03

25

ri

αi

θi

i

φ

b δr

ΣF

30

Center Axis of Contact Surface
i

[]

xF

ThB05.4(59)4589