Joint 48th IEEE Conference on Decision and Control and28th Chinese Control Conference
Shanghai, P.R. China, December 16-18, 2009

FrC06.5

Stability Analysis of Grasped Object by Soft-Fingers
with 3-Dimensional Deformation based on Moment Stability
Akira Nakashima and Yoshikazu Hayakawa
Abstract— This paper analyze stability of an object grasped
by soft-ﬁngers in 3-dimensional space based on moment stability. We ﬁrstly deﬁne the moment stability as a criterion forstability of a grasped object when the object is perturbed forthe orientation. In detail, the moment stability means that thegrasping force can be the restoring force when the object isperturbed. Based on the moment stability, the stability conditionof an object grasped by hard-ﬁngers is derived. We indicate that
contact points to satisfy the condition are restricted to upper
locations of the center of mass and the class of the object shapeis only a kind of hollow objects. Next, the condition of an object
grasped by semispherical soft-ﬁngers thirdly is considered. Twoﬁngered grasp by soft-ﬁngers can satisfy the force-closure which
is traditional major criterion for static grasp. The considerationshows a novel result that two-ﬁngered grasp can satisfy the
moment stability with only two ﬁngers while the hard-ﬁngers
need at least three numbers for the force-closure and can notsatisfy the moment stability. On the other hand, it is indicatedthat the torsion moments do not effect on the moment stability
well and it is necessary to add the third ﬁnger in a conditionof the contact points. Numerical examples ﬁnally are shown.

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
Authors are with Mechanical Science and Engineering, Graduate Schoolof Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Japanakira@haya.nuem.nagoya-u.ac.jp, and Y. Hayakawa is also withRIKEN-TRI Collaboration Center, RIKEN, 2271-103, Anagahora, Shimoshidami, Moriyama-ku, Nagoya, Japan.

978-1-4244-3872-3/09/$25.00 ©2009 IEEEin 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
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 have beenstudied. Compared with hard-ﬁngers made from hard materials, the soft-ﬁngers have a lot of advantages [8]: the contact
friction is larger than that of the hard-ﬁnger since the contact
is surface; the soft-ﬁnger can reduce the impact force incontact establishment; it can ﬁt on various shapes of theobject; there exists moment friction about the contact normal.
Inoue et al. [9], [10], [11], [12] proposed a parallel distributed
deformation model of a semispherical soft ﬁngertip andconsidered stability of two-ﬁngered grasp based on the local
minimum elastic energy with the rolling contact in twodimensional space. On the other hand, Nakashima et al.
[13] proposed a radial distributed deformation model of asemispherical soft ﬁngertip and showed a stability conditionof two-ﬁngered grasp to satisfy the moment stability basedon the stiffness effect. These studies are restricted in 2dimensional space and can not consider the moment friction
about the contact normal because of 2-dimensional deformation model. This moment is new feature in 3-dimensional
deformation of a soft-ﬁnger and one of the advantages ofa soft-ﬁnger as mentioned previously. Two-ﬁngered graspby soft-ﬁngers can satisfy the force-closure [14], which isa traditional major criterion for static grasp. However, it is
not evident how the moment effects on restoring force whena grasped object is perturbed.
In this paper, we study stability of an object grasped bysoft-ﬁngers in 3-dimensional space based on the stiffness
effect. The study [4] showed that the stiffness effect due
to the object rotation is much larger than the one due tothe object translation and the stiffness effect due to therolling contact stabilize the object. Therefore, we consider
the object rotation and the rolling contact. We ﬁrstly deﬁne
the moment stability as a criterion for stability of a grasped
object when the object is perturbed for the orientation. Basedon the moment stability, the stability condition of an object
grasped by hard-ﬁngers is derived. The problem setting area speciﬁed case of the analysis in Ref. [4]. We indicatethat contact points to satisfy the condition are restricted toupper locations from the center of mass and the class ofthe object shape is only a kind of hollow objects. Next, thecondition of an object grasped by semispherical soft-ﬁngers

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k ∈ R3 is the rotation axis and ζ ∈ R is the amount of therotation. It is assumed that the object is only rotated aroundthe center of mass. It is also assumed that the contact force
f i and the contact point on the object pi are invariant. This
assumption can be realized by an appropriate control method
if a joint structure satisﬁes the manipulable condition [14].
We deﬁne the moment stability as(ζk)T M (ζ, k) < 0 for any k, ζ,TFig. 1.

An object grasped by hard-ﬁngers.

thirdly is considered. A 3-dimensional deformation model ofa soft-ﬁnger and the contact kinematics with the deformation
are utilized which has been proposed in [15]. Two-ﬁngered
grasp by soft-ﬁngers can satisfy the force-closure which istraditional major criterion for static grasp. The considerationshows a novel result that two-ﬁngered grasp can satisfy the
moment stability with only two ﬁngers while the hard-ﬁngers
need at least three numbers for the force-closure and can
not satisfy the moment stability. On the other hand, it isindicated that the torsion moments do not effect on the
moment stability very well and it is necessary to add thethird ﬁnger in a condition of the contact points. Numericalexamples ﬁnally are shown for the consideration.
II. S TABILITY A NALYSIS OF H ARD -F INGER G RASP
We ﬁrstly analyze stability of an object grasped by apair of hard-ﬁngers in 2D space as shown in Fig. 1. The
ﬁngers and object are rigid bodies and the ﬁngers are pointy.
ΣO is the reference frame attached at the center of massof the object. Note that vectors without left superscripts
are expressed in ΣO . f i , pi ∈ R3 (i = 1, · · · , N ) are
the contact force and the position vector at the ith contact
point respectively. The grasp of the object is deﬁned as thefollowing equilibrium equations of force and moment:Nwhere M (ζ, k) := [Mx My Mz ] ∈ R is the moment
caused by the object rotation ω = ζk. Eq. (5) is the extensionof the 2-dimensional moment stability in [13] and meansthat M (ζ, k) is the restoring force against the rotational
displacement ω = ζk as shown in Fig. 1.
The resultant moment M (ζ, k) of the hard-ﬁnger grasp isgiven byN
M (ζ, k) =(Rk (ζ)pi ) × fi ,(6)i=1

where Rk (ζ) ∈ R3×3 is the rotation matrix about k-axis
through ζ deﬁned as [14]⎡ 2⎤kx vζ + cζkx ky vζ − kz sζ kx kz vζ + ky sζ
ky2 vζ + cζ
ky kz vζ − kx sζ⎦,Rk (ζ)=⎣kx ky vζ + kz sζ
kx kz vζ − ky sζ ky kz vζ + kx sζ
kz2 vζ + cζ
(7)where cζ := cos ζ, sζ := sin ζ and vζ := 1 − cos ζ. It is here
assumed that the perturbed rotation amount ζ is small. From
the assumption and (3), (7) is linearized as⎤⎡1−kz ζ ky ζ
1−kx ζ ⎦ = I3 + S(ζk). (8)Rk (ζ) = ⎣ kz ζ
−ky ζ kx ζ1
Substituting (8) into (6), we get
M (ζ, k) =
=

N

i=1
N

i=1



fi + mg = 0,

i=1
N


(1)
=−

S(pi )fi = 0,
T

where m is the mass of the object, g := [0 0 g] , g =−9.8 [m/s2 ] is the gravitational vector and S(pi ) ∈ R3×3 isthe skew-symmetric matrix deﬁned as
⎤
⎡
0
−pzi
p yi
0
−pxi ⎦ ,(3)
S(pi ) := ⎣ pzi−pyi pxi
0

S(pi )fi +N

S(S(ζk)pi )fi

i=1

0

N


T
pT
ζk,
i fi I3 − pi fi(9)where the ﬁrst term of the second row equals 0 from (2)and the second term is simpliﬁed by easy calculation [14].
Substituting (9) into (5) leads to(ζk)T M (ζ, k) = −(ζk)T Kh (ζk),

(10)

where
Kh :=

by which a cross product pi × fi is expressed aspi × fi = S(pi )fi .

(pi + S(ζk)pi ) × fi

i=1

(2)

i=1

(5)

3

N


T
Khi , Khi := pTi fi I3 − pi fi .

i=1

(4)

For the system of the hard-ﬁnger grasp, we consider the
stiffness effect due to the object rotation ω = ζk, whereSince (10) is the quadratic form, the condition to hold themoment stability (5) is given by [16]8193K̄h > 0,(11)

FrC06.5
relations hold:pxi fxi + pyi fyi < 0 (i = 1, 2, 3),1
px1 = px2 = −2px3 = px < 0, fx1 = fx2 = − fx3 = fx > 0,2py1 = −py2 = py < 0, py3 = 0, fy1 = −fy2 > 0, fy3 = 0,(18)pz1 = pz2 = pz3 = pz > 0, fz1 = fz2 = fz3 = fz .

The gravity direction is in the z-axis. This example is atypical grasp situation where the contact points surroundthe center of mass (CM) and the grasping forces are in the
direction to the CM. From (18), it is easily obtained thatk̄h33 < 0, k̄h12 = 0 and k̄h13 . By these facts and (17),
detK̄h3 = k̄h11 k̄h22 k̄h33 − (k̄h23 )2 < 0.
(19)

Fig. 2.

Therefore, it is necessary to hold k̄h33 > 0 in order to satisfy
detK̄h3 > 0. This condition is satisﬁed in the strange grasp
situation of a hollow object of Fig. 2 (b), where the grasping
forces are in the inverse direct to the CM. Furthermore, itis necessary to satisfy (17). A sufﬁcient condition of (17) is
given byGrasp situations not to and to satisfy (12).

whereK̄h :=

NK̄hi , K̄hi :=i=1

1(Khi + KhTi ).
2

pyi fyi +pzi fzi > 0, pxi fxi +pzi fzi > 0 (i = 1, 2, 3). (20)

From Sylvester’s criterion [16], The condition to satisfy (11)is(12)
detK̄hj > 0, j = 1, 2, 3,
where K̄hj ∈ Rj×j is the jth principal minors of K̄ h .
Here, we consider a condition to satisfy the moment
stability. The principal minors K̄hj are calculated from (10)and (11) as
detK̄h1 = k̄h11
detK̄h2 = k̄h11 k̄h22 − (k̄h12 )2

(13)
(14)

detK̄h3 = k̄h11 k̄h22 k̄h33 + 2k̄h12 k̄h13 k̄h23−k̄h11 (k̄h23 )2 − k̄h22 (k̄h12 )2 − k̄h33 (k̄h13 )2 , (15)For space limitation, the condition pyi fyi + pzi fzi > 0
is considered. The areas of f 1 and f 2 to satisfy (20) arerepresented by the shaded areas in (c) and (d) of Fig. 2. It is
obvious that there do not exist f 1 and f 2 when the contact
positions are under the center of mass. In order to construct
the grasp of the object, the contact force f i also has to satisfy
the friction condition illustrated in (3) of Fig. 2 (e):TfT
i eti ≤ μi |f i eni |,where eni , eti ∈ R3 are the normal and tangent vectors
and μi is the static friction coefﬁcient. Fig. 2 (f) shows thefriction cone and the area of (20), where γfi and γmi are theangles of the boundaries of the friction cone and (20) fromthe contact normal respectively. These angles are deﬁned aswhere k̄hij is the component of K̄h at the ith row and jth
column given byk̄h11 :=N
N
X
1X
(pyi fyi + pzi fzi ), k̄h12 := −(py fx + pxi fyi )2 i=1 i ii=1k̄h13 := −

N
N
X
1X
(pzi fxi + pxi fzi ), k̄h22 :=(pxi fxi + pzi fzi )
2 i=1
i=1

k̄h23 := −

N
N
X
1X
(pzi fyi + pyi fzi ), k̄h33 :=(pxi fxi + pyi fyi )
2 i=1
i=1

(16)

From (13), (14) and (16), the condition to satisfy detK̄hj >0 (j = 1, 2) is easily obtained ask̄h11 > 0, k̄h22 > 0.

(17)

Since the condition of (15) depends on a grasp situation, it isconsidered in an example of Fig. 2 (a), where the following(21)

γfi := tan−1 μi , γmi := tan−1

hi
,
li

(22)

where li and hi are the distances of the ith contact point fromthe CM along the normal and tangent respectively. Therefore,
it is necessary for the distances li and hi of the contact point
to satisfy
hi
≤ μi
(23)
γmi ≤ γfi ⇔li
as shown in (f) of Fig. 2.
Remark 1: From (23), 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 CM. Furthermore, Fig. 2 (b) implies that kinds ofobject shape to satisfy the moment stability are restricted tohollow objects. These results indicate that it is impossible torealize the moment stability by hard-ﬁngers.

8194

FrC06.5

Fig. 3.

Soft ﬁnger and Contact situation.

III. S TABILITY A NALYSIS OF S OFT-F INGER G RASPIn this section, we secondly analyze stability of an object
grasped by a pair of soft-ﬁngers in 3D space. A deformation
model of a soft-ﬁnger and a contact kinematics are necessary for the analysis. We use a 3-dimensional deformation
model and the rolling contact with deformation as a contact
kinematics, which are proposed in [15].
A. 3-Dimensional Deformation ModelIn this paper, a soft-ﬁnger means a semispherical soft
material, the ﬂat part of which is attached to a rigid baseas in the left ﬁgure of Fig. 3. We make a deformation modelof a soft-ﬁnger with a contact to a plane of a polyhedral
object as shown in the right of Fig. 3.
The frame and coordinates of the conﬁguration of the contact point is deﬁned here. The ﬁnger frame ΣFi is attachedto the center of the soft-ﬁnger as in the left of Fig. 4 (a).
The yFi - and zFi -axes are in the ﬁnger base and the xFi -axis
goes through the ﬁnger. Since the soft-ﬁnger is a semisphere,
the contact point on the soft-ﬁngertip before deformation

is expressed by the polar coordinates (θi , φi , ri ), where θi ,φi and ri are the angles and radius. When the ﬁnger isdeformed with the contact on the object, the contact surface
is the circle from the geometry of the ﬁnger and object asshown in the left ﬁgure of Fig. 4 (a). Therefore, the contact
point after deformation is deﬁned as the center of the contact
surface. The right of Fig. 4 (a) shows the cross section along
the longitude through the contact point of the left ﬁgure.
The deformed area overlapping the object is expressed by
the compressive deformation displacement δri ∈ R+ in theradial direction. As shown in the left of Fig. 4 (b), thedeformations along the ﬁnger surface are expressed by theshearing and compressive deformation angles δθi , δφi ∈ R
in the inverse directions of (θi , φi ). In the right of Fig. 4 (b),the contact frame ΣCFi is attached at the contact point,
which is deﬁned such that the xCFi - and yCFi -axes are in
the longitude and latitude through the contact point and thezCFi -axis is in the normal to the ﬁnger surface. The torsion
deformation with respect to the contact surface is expressedby the shearing deformation angle δφi about the zCFi -axis
in the inverse direction. Deﬁne the deformation coordinatesas δi := [δri δθi δφi δψi ]T ∈ R4 .
From Hook’s law, the forces produced by the deformations
are derived as the functions of the deformations δ i :Fri (δri ) = kri πδr2i1Mθi (δθi , φi ) = kθi πri4 1 + cφi (1 + cφi )(1 + c2φi ) δθi
10


2
1
1
3
Mφi (δφi , φi ) = kφi ri
δφi+3
π − φiφi21(24)Mψi (δri , δψi ) = kψi π ri2 − (ri − δri )2 δψi .
10
Fri is the force in the radial direction and Mθi and Mφi are
the moments about the xFi - and zF i - axes in the left ﬁgureof Fig. 4 (b) where cφi := cos φi . Mψi is the moment about
the zCFi -axis in the right ﬁgure of Fig. 4 (b).
B. Rolling Contact with Deformation(a) Polar coordinates and radius coordinate δri

Let us denote the variables to express the contact conﬁguration without the deformation in order to consider the
rolling contact. Fig. 5 represents the neighborhood of the ith
contact. ΣCFi and ΣCOi are the coordinate frames attachedon the surfaces of the ith ﬁnger and the object with theorigins at the ith contact point. The zFi - and zi -axes ofthe frames are outward and normal to the surfaces of the
ObjectF inger i

ψi
xC F

xC F

i

Fi

pC F (
i

fi

xC Oi
yC Oi

i

C Fi

)

O

zC Fi

yC Fi

zC O

i

pC O ( oi )
i

C Oi

Fi

O

(b) Spherical and torsion coordinates (δθi , δφi ), δψiFig. 4.

The coordinates of the deformation of the soft-ﬁnger.

Fig. 5.

8195

Contact coordinates before the deformation.

FrC06.5
ith ﬁnger and the object, respectively. The contact pointin the ﬁnger surface is represented by the position vectorFi
pCF (θi , φi ) ∈ R3 . Similarly, the contact point on theiobject surface is represented by O pCO (xoi , yoi ) ∈ R3 . Noteithat (θi , φi ), i.e., the some of the previous polar coordinatesof the soft-ﬁnger, and (xoi , yoi ) are the 2D local coordinates
on the ﬁnger and object surfaces in x- and y-axes of ΣCFiand ΣCOi . In addition, since x- and y-axes of ΣCFi and ΣCOi
are in the same contact tangent surface, let ψi be the angle
between the x- axes of ΣCFi and ΣCOi without the torsion
deformation δψi as shown in Fig. 5. Then, the conﬁgurationof the contact points is described by the contact coordinates
ηi := (θi , φi , xoi , yoi , ψi ) [14].
Considering the deformations δi into the contact coordinates ηi , the contact point Fi pCF after the deformation andithe contact frame RF CFi are expressed as




Fi

Fig. 6.

MCi ∈ R are the contact force and moment expressed inΣCFi and they satisfy the following equilibrium equations:NX

Ri sΦi0cΦisP hiipCFi = Ri cΘi cΦi , RFi CFi = −sΘi −sΦi cΘi cΘi cΦi
Ri sΘi cΦicΘi −sΦi sΘi sΘi cΦi

i=N

(25)
where cΘi = cos Θi , sΘi = sin Θi , cΦi = cos Φi , sΦi = sin Φi ,Ri := ri − δri , Θi := θi − δθi , Φi := φi − δφi .

(26)

In this paper, we consider the pure rolling contact [14].
The pure rolling contact means that the ﬁnger does not slipon the object without rotationally slipping around the zCFi axis. The relationship between the contact coordinates isgiven by [15]

 

 
dxoi
− sin Ψi Ri cos Φi 0 dθicos Ψi=(27)
0
Ri dφidyoi− sin Ψi − cos Ψi




Rψi

dψi = dθi sin Φi ,where (dθi , dφi ), (dxoi , dyoi ) and dψi are the small displacements of the contact coordinates andΨi := ψi + δψi .

(29)(27) means that the contact displacements of the ﬁnger andobject on the contact surface equal to each other. Rψi is therotation matrix between the (x, y) coordinates of ΣCFi andΣCOi . Mgfi translates the angle displacements to the linear
ones. (28) means that the angle displacements in the zCFi
due to ψi and θi around the xFi -axis equal to each otherbecause the angle between the zCFi - and xFi -axes is π − Φi .
Remark 2: Note that there do not exist the small displacements of the deformations δi in the rolling contact
equations of (27) and (28). This is because the deformations
δi do not include the displacements on the ﬁnger surface andcan denote only the shift of the contact point expressed inΣCFi due to the deformed soft-ﬁnger.
C. Derivation of Moment due to Object Rotation
The problem setting is mentioned for preliminary. The leftof Fig. 6 shows an object grasped by the soft-ﬁngers. The
object is assumed to be a convex polyhedron. FCi ∈ R3 andROCFi FCi = 0,

N “”X
S(pi )ROCFi FCi + MCi ROCFi ez = 0,

(30)

(31)

i=1

where ez := [0 0 1]T , ROCFi := R(θFi )RFi CFi , R(θFi )is the rotation matrix from ΣFi to ΣO . FCi and MCi areexpressed by the deformation forces and moment of (24) as
Mθi −Mψi sin Φi y
Fφ, FCi = i , FCz i = Fri , MCz i = Mψi .
Ri cos Φi
Ri(32)
In the right ﬁgure, pLi ∈ R3 is the center of the ﬁnger and
pFi Ci ∈ R3 is the vector from the center of the ﬁnger to thecontact point expressed in ΣO . Therefore, the contact point
pi is given by

FCxi =

pi = pLi + pFi CFi , pFi CFi := RFi Fi pCFi .

Mgfi

(28)

An object grasped by soft-ﬁngers.

(33)

3

phi ∈ R is the point with the minimum distance hi of theobject surface from the CM and n0hi ∈ R3 is the normal tothe object surface in the initial situation. nxoi , nyoi ∈ R3
are the unit vectors to express the x- and y-axes of ΣCOi .
We consider the moment produced by the soft-ﬁngers
when the object is rotated about k-axis through ζ from theequilibrium conditions (30) and (31). Because the contact
force and moment (FCi , MCi ) and the contact point pi
obviously depend on them from (24), (32), (33) and (25), it
is necessary to obtain the changed contact coordinates ηi (ζ)
and the deformation δi (ζ) due to the rotation (ζ, k):θi (ζ) := θi0 + Δθi (ζ), φi (ζ) := φ0i + Δφi (ζ)
xoi (ζ) := x0oi + Δxoi (ζ), yoi (ζ) := yo0i + Δyoi (ζ)ψi (ζ) := ψi0 + Δψi (ζ)δθi (ζ) := δθ0i + Δδθi (ζ), δφi (ζ) := δφ0 i + Δδφi (ζ)

0δψi (ζ) := δψ+ Δδψi (ζ),i(34)

where the variables with the superscript 0 are the initialvariables satisfying the initial equilibrium equations of (30)and (31), and the ones with the Δ is the increments from thegrasping equilibrium due to the rotation (ζ, k). The scheme
to obtain ηi and δi is shown in the following:1) To obtain δri (ζ) and (Δxoi (ζ), Δyoi (ζ)) from thegeometric viewpoint.

8196

FrC06.5
2) To obtain (Θi (ζ), Φi (ζ)) and Ψi (ζ) from the geometricviewpoint.
3) To obtain (Δθi (ζ), Δφi (ζ)) and Δψi (ζ) by solving
the rolling contact equations (27) and (28) with theobtained Φi (ζ) and Ψi (ζ).
4) To obtain (Δδθi (ζ), Δδφi (ζ)) and Δδψi (ζ) by (26) and(29) with the obtained (Θi (ζ), Φi (ζ)) and Ψi (ζ) and(Δθi (ζ), Δφi (ζ)).
1) Derivation of δri (ζ) and (Δxoi (ζ), Δyoi (ζ)): Since thecontact surface equals to the object surface as seen in Fig. 6,
the vector pFi CFi is in the same direction of phi given byphi (ζ) = Rk (ζ)(hi n0hi ) = hi nhi (ζ), nhi (ζ) := Rk (ζ)n0hi .
(35)
Therefore, by considering the distance between the center ofthe ﬁnger and the CM in the direction of phi , the magnitude
Ri (ζ) of pFi CFi is given by
Ri (ζ) = Lhi (ζ) − hi , Lhi (ζ) := nT
hi (ζ)pLi ,(36)

where Lhi (ζ) ∈ R is the distance between the center of the
ﬁnger and the CM in the direction of phi . From (26), δri (ζ)
is obtained as
δri (ζ) = ri + hi − Lhi (ζ).

(37)

Next, from pFi CFi = −Ri (ζ)nhi (ζ), (33) and (35), thecontact point pi expressed on the object surface with ζ = 0
is given by
p0i (ζ) = RkT (ζ)pLi − Ri (ζ)n0hi .

(38)

Therefore, (Δxoi (ζ), Δyoi (ζ)) are obtained as
Δxoi (ζ) = (n0xi )T Δp0i (ζ), Δyoi (ζ) = (n0yi )T Δp0i (ζ), (39)where Δp0i (ζ) := p0i (ζ) − p0i and p0i is the initial state.
2) Derivation of (Θi (ζ), Φi (ζ)) and Ψi (ζ): Substituting
pFi CFi = −Ri (ζ)nhi (ζ) and (25) into (33) yield⎡⎤ ⎡⎤sin Φi (ζ)αi (ζ)
T⎣cos Θi (ζ) cos Φi (ζ)⎦=⎣ βi (ζ) ⎦:= RF(θFi )nhi (ζ). (40)isin Θi (ζ) cos Φi (ζ)γi (ζ)
From (40), (Θi (ζ), Φi (ζ)) are obtained as
γi (ζ). (41)
Φi (ζ) = sin−1 αi (ζ), Θi (ζ) = sin−1cos Φi (ζ)
From Fig. 5, Ψi (ζ) is the angle between the xCFi and xCOi -axes deﬁned by the following corresponding unit
vectors:⎡⎤0
nxCF := ⎣− sin Θi (ζ)⎦ , nxCO := Rk (ζ)n0xo , (42)iiicos Θi (ζ)
where nxCF is the ﬁrst column of RFi CFi of (25). Considi
ering the sign of Ψi (ζ) which equals the sign of ψi (ζ), we
obtain the following Ψi (ζ):
Ψi (ζ) = aΨi cos−1 (nxCF )T nxCO ,
(43)
i

i

Fig. 7.

where

A rectangular object grasped by two soft-ﬁngers.

aΨi := sign (nxCF × nxCO )T pFi CFi .
ii3) Derivation of (Δθi (ζ), Δφi (ζ)) and Δψi (ζ): From thepure rolling equations (27) and (28), (Δθi (ζ), Δφi (ζ)) andΔψi (ζ) are given by⎤⎡d(Δxoi (ζ̃))
  ζΔθi (ζ)−1T=
Mgf(ζ̃)Rψ(ζ̃) ⎣ d(Δydζ̃ (ζ̃)) ⎦ dζ̃ (44)iioiΔφi (ζ)0Δψi (ζ) =

 ζ
0

dζ̃

d(Δθi (ζ̃))
sin Φi (ζ̃)dζ̃.
dζ̃

(45)

It is difﬁcult to obtain (Δθi (ζ), Δφi (ζ)) and Δψi (ζ) as
functions of ζ by integrating (44) and (45) analyticallybecause the obtained analytical functions of Ri (ζ), Θi (ζ)
and Ψi (ζ) are complex. Therefore, (Δθi (ζ), Δφi (ζ)) andΔψi (ζ) are obtained numerically in this paper.
4) Derivation of (Δδθi (ζ), Δδφi (ζ)) and Δδψi (ζ): From(26), (29) and (34), (Δδθi (ζ), Δδφi (ζ)) and Δδψi (ζ) are
obtained asΔδθi (ζ) = −Θi (ζ) + Δθi (ζ) + (θi0 − δθ0i )Δδφi (ζ) = −Φi (ζ) + Δφi (ζ) + (φ0i − δφ0 i )(46)Δδψi (ζ) = Ψi (ζ) − Δψi (ζ) − (ψi0 + δψ0 i ).

IV. N UMERICAL E XAMPLE AND D ISCUSSION
A Numerical example is shown to conﬁrm the moment
stability of the soft-ﬁnger grasp.
Fig. 7 shows the case where the rectangular object isgrasped at the lower contact points. The radius of the
ﬁnger is ri = 10[mm] and the stiffness coefﬁcients are
kri = 0.377, kφi = 0.166[N/mm2 ], kθi = 0.0266, kψi =0.0488[N/mm3 ] [15]. The height, width, depth and massof the object are 100[mm] and m = 1.0[kg] respectively.
The initial contact points are p01 = (0, −50, −25), p02 =(0, 50, −25)[mm]. The contact angles are θ10 = θ20 = 0[deg],
φ01 = 131, φ02 = 49[deg]. The initial deformations areδr01 = δr02 = 2.1[mm], δθ01 = −δθ02 = 0[deg], δφ0 1 = −δφ0 2 =11[deg], δψ0 1√= δψ0 2 = 0[deg]. The rotation axis and angles
are k = 1/ 3[1 1 1]T and ζ = ±2[deg], which generates
the rotation with respect to the all axes.

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FrC06.5

Fig. 8.

Simulation result.

Fig. 11.

Simulation result seen from the x-axis rotation.

ψFig. 8 shows the simulation result. The left ﬁgure shows
the inner product between the rotation axis ζk and theresultant moment M (ζ, k). It is conﬁrmed that the moment
stability is satisﬁed. Furthermore, the right ﬁgure shows themoments with respect to all the (x, y, z)-axes. It is conﬁrmedthat the signs of the moments are inverse with respect tothe rotation directions. When the two moments are not therestoring moments, the moment stability could hold if thelast moment is the restoring one and its effect is much bigger
than the effects of the others. However, the stability of its
case may reduce. Figs. 9–11 show the results seen from thex-, y- and z-axes respectively, where the the deformation
contact forces are illustrated as the solid sick black arrows.
In the each ﬁgure, it is conﬁrmed that the resultant moment
due to the deformation contact force with respect to thecorresponding axis is the restoring moment.

Remark 3: The torsion moments of the soft-ﬁngers effecton the moment about the y-axis, which have been expected
to be the effective restoring moments, e.g., in [8], [14]. Since
this effect may be difﬁcult to be illustrated in Fig. 10, thecontact moments are shown in Fig. 12 in order to check its
effect. Note that the contact moments effect the y-axis. The
sum of the contact moments is not the restoring moment
because it is positive when ζ > 0. This fact imply that thetorsion moments may not effective to the moment stability
while its is effective to the force-closure.
Let us consider its reason. The change of the torsion
deformation Δδψi is effected on by Ψi and Δψi from (46),which are the terms to reduce each other. The changes areillustrated in Fig. 13, where the upper and lower ﬁgures show
the the changes on the ﬁrst ﬁnger-tip and the left side of theobject respectively when the object is rotated from ζ = 0
to 2 [deg]. The bule and red arrows represent the xCF1 and yCF1 - axes of ΣCF1 at the contact point on the ﬁnger.
The solid sick bule lines represent the shifts of the contact
point on the ﬁnger-tip surface due to the object rotation andthe rolling contact. The black and red circles represent theﬁrst and end of the shifts of ζ = 0, 2 [deg]. In the upper
ﬁgure, the changes ΔΨ1 and Δψ1 are shown as the changesof the xCF1 -axes. ΔΨ1 is directly caused by the relative
motion of the ﬁnger and object from (43) and very small
becuase of small object rotation ζ. On the other hand, Δψ1
is caused by the rolling contact from (45). Note that the shift
due to the rolling contact is equal to the shift on the object
surface in the lower ﬁgure, and is relatively bigger than theshift due the object rotation. This implies that Δψ1 has thedominant effect on the change of the torsion deformationΔδψ1 . Furthermore, Δψ1 is caused by the object rotation
about the zO -axis becuse it depends on the change Δθ1 whileψ

Fig. 9.

Simulation result seen from the z-axis rotation.

ζFig. 10.

Simulation result seen from the y-axis rotation.

Fig. 12.

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ζThe contact moments expressed in ΣO .

FrC06.5
with only two ﬁngers while the hard-ﬁngers needed at least
three numbers for the force-closure and could not satisfy the
moment stability. On the other hand, it was indicated that thetorsion moments did not effect on the moment stability welland it was necessary to add the third ﬁnger in a conditionof the contact points.
In the case of the dynamics, there are some different
points from the static analysis in this paper. Since the rolling
relationships of (44) and (45) does not hold becuase of its
kinematic forms, it is necessary to consider the acceralation
form. Furthremore, it is ncessary to consider an appropriate
control method to satisfy the moment stability, an example
of which is shown in [15]. A very simple method is the
PID control of the ﬁnger-tips rigidly. This idea is based on
the static analysis in this paper because the ﬁnger-tips are
assumed to be ﬁxed. The method would utilize the elastic
property to satisfy the moment stability.
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. 13. The changes of the contact conﬁgurations on the 1st ﬁnger andthe object.

Fig. 14.

The illustrations of the changes Δθ1 and ΔΨ1 .

ΔΨ1 is caused by the object rotaion about the yO -axis asshown in Fig. 14. Therefore, the moment about the yO -axis
due to the torsion deformation can not effect on the restoring
moment when the object is rotated about the zO -axis. In thatgrasp situation, it is necessary to add the third soft-ﬁnger
which has to produce the inverse moment against the sign ofζ about the y-axis. A example of its contact position of thethird ﬁnger is in the surfaces perpendicular to the x-axis.
V. C ONCLUSIONS AND F UTURE W ORKS
This paper analyzed stability of an object grasped by softﬁngers in 3-dimensional space based on moment stability.
In the case of the hard-ﬁnger grasp, we indicated that
contact points to satisfy the condition were restricted to upper
locations of the center of mass and the class of the object
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