Softness Effects on Manipulability and Grasp Stability
Tetsuyou Watanabe

Abstract— This paper presents a novel analysis for the effects
of softness at the fingertip on the manipulability and stabilityof grasping. The stability for grasping can be regarded thathow much magnitude of external wrench we can balance.
We formulate manipulability and the set of generable object
wrenches for grasping system, taking deformation of thefingertips into consideration, and show that the increase of
the softness decreases the manipulability while it increases
generable object wrench. The validity of our analysis is shownby numerical examples.

I. I NTRODUCTION
Recently, there is much attention on robots executing
everyday manipulation in human environment [1]. Multifingered hand manipulation is necessary to execute such kindof tasks. If developing such robot hand systems, we shouldtake the affinity to human into consideration. Therefore,
grasping with deformable fingertips or contact areas havebeen researched [2]. They developed controller for grasping
and manipulating the object with deformable fingertips.
However, there are still many unclear issues about the effectsof the softness around contact area (deformation of thefingertips). This paper focuses on and analyzes the effect of
the softness at the contact area on manipulability and grasp
stability: how much easily we can manipulate the object andhow much magnitude of external wrench we can compensate.
Manipulability is a well-known concept to evaluate theperformance of robotic manipulator [3]. For a single-arm
manipulator, it is defined as the set of generable end-effector
velocities in the task space when the set of generable joint
velocities is given. When the given set of joint velocities
is a unit ball, the set of end-effector velocities becomes an
ellipsoid. The ellipsoid is called manipulability ellipsoid. The
volume of the ellipsoid can be regarded as a quality measure
to evaluate the performance in velocity domain. It is called
manipulability measure. Based on the manipulability, many
quality measures such as condition number are proposed[3]. This concept can be extended to the general constraining system such as robotic hands [4]–[10]. In a general
constraining system, object velocity is evaluated instead ofendeffector velocity. For a dual-arm system, Chiacchio etal. [4] discussed manipulability. Bicchi et al. [5] analyzed
manipulability for general grasping system including wholearm manipulation system. After that, Bicchi et al. [6], Wen etal. [7], and Park et al. [8] analyzed manipulability for general
constraining systems with underactuated joints. T. Watanabe
This work was not supported by any organization
T. Watanabe are with College of Science and Engineering,
Kanazawa
University,
Kanazawa,
920-1192,
Japan
and

te-watanabe@ieee.org[9], [10] presented a way for manipulability analysis taking joint torques required for grasping into consideration.
However no-researchers concerned the softness effects on
manipulability for grasping systems.
The magnitude of generable resultant object wrench has
been regarded as a criterion index representing graspingstability since it corresponds to how much magnitude ofexternal wrench we can compensate. By linearizing thefrictional cone [11], the set of generable resultant object
wrench can be expressed by a convex polyhedron. Then, thevolume of the set can be calculated by, for example, qhull
algorithm [12], and it can be a criterion index. The radius of
the sphere inscribed the set can also be a criterion index since
it represents the maximum magnitude of external wrench
which can be balanced in any direction. These criterion
indexes are embedded in GRASPIT [13] which is simulation
software for grasp planning. Based on these kinds of criterion
indexes, grasp planning problems such as investigations ofgrasping points and posture have been researched. On theother hand, if the contact area is deformable, the moment
around normal direction can be applied. Such a contact iscalled soft-finger contact model, and corresponding frictional
condition was researched [14], [15]. However, it is not clear
how the change of the softness around contact area affects
friction and the generable resultant object wrench.
Concerning these, this paper gives the following contributions.
Criterion index for manipulability including softness
effect: We formulate kinematical relationship between fingertip, contact and object velocities, taking the softness effect(deformation of contact area) into consideration. Then, weconstruct the set of generable object velocities under thebound on the magnitude of fingertip velocities. From thederived set of generable object velocities, we present new
criterion index for manipulability which can take the softness
effect into consideration.
Criterion index for grasp stability including softness effect: We formulate frictional conditions based on the contact
area associated with the deformation of fingertip, which can
explicitly represent how the change of the softness affects
friction. Linearizing the formulated frictional conditions, weconstruct the set of generable object wrenches, expressedby a convex polyhedron. From the set, we derive the newcriterion index for grasp stability which can take the softness
effect into consideration.
Softness effect on manipulability and grasp stability:
Based on the derived criterion index, we analytically show
the decrease of softness (increase of stiffness) causes theincrease of manipulability. It will be also shown by numericalObject

If differentiating this relationship with respect to time, we
get

t1i
ωi ri ni b1i ωo
pcii
Σo po
Σi pi
Δri
Δ
ai
Σ

ṗci = ṗi + (ri − Δri )(ω i × ni ) + v i − Δṙi ni

Fingertip

Fig. 1.

where
v i = (ri − Δri )(ṅi − ω i × ni )expresses the surface velocity on the ith fingertip.
On the other hand, the relationship between pci and po isgiven byModel for analysis

pci = po − ai ni + b1i t1i + b2i t2iexamples. We also show the decrease of softness (increaseof stiffness) causes the decrease of grasp stability: we canbalance larger external object wenches with softer fingertips.
The rest of this paper is organized as follows. First, weformulate the criterion indexes for manipulability and grasp
stability. After that, we show the numerical examples toconfirm the analysis.

ṗci = ṗo + ω o × (−ai ni + b1i t1i + b2i t2i )(5)

From (3) and (5), the following equation is obtained.

A. Effect on manipulability
Here, we discuss manipulability [3] to understand how
easily the hand can manipulate the object. Fig.1 shows themodel of the fingertip and the object. This model is based on
the models of Xydas et. al. [15] and Arimoto et. al. [16]. For
the convenient, we discuss only fingertip motion, and assume
that we can move the fingertip in arbitrary directions. Everyfingertip has a shape of deformable hemisphere. The object
has flat surface at the contact point and is rigid body. We
consider to grasp the object with n fingertips. Let Σo and Σibe the coordinates located at the gravity center of the object
and at the center of hemisphere of ith fingertip. Let be po andpi are the positions of the origin of Σo and Σi , and pci be theposition of the center of the contact surface between the ith
fingertip and the object. ω o and ω i are angular velocities ofΣo and Σi . ni is unit contact normal directing inside of theobject. t1i and t2i are unit contact tangential vectors. ri is thehemisphere radius of the fingertip and Δri is the deformation
amount by contact. Let fni be the component of contact force
in normal direction ni . We assume the following relation at
the steady state.
(1)

where ki and ζ are material and geometric parameters
associated with softness. Note that if according to the modelsof Arimoto [16] and Inoue et. al. [17], ζ = 2. If supposing
grasping the object with certain constant contact forces, we
can regard Δri as a parameter expressing skin softness. If
Δri is large, it means the skin is soft and vice versa.
The relationship between pci and pi is given by
pci = pi + (ri − Δri )ni(4)

where ai , b1i and b2i denote the distances between Σo andpci along the directions of ni , t1i and t2i , respectively. Notethat ai is constant since it represents the distance between Σoand the side of the object. If differentiating this relationshipwith respect to time, we get− ȧi ni + ḃ1i t1i + ḃ2i t2i
II. S OFTNESS EFFECTS

fni = ki Δriζ

(3)

(2)

ṗi + (ri − Δri )(ω i × ni ) − Δṙi ni= ṗo + ω o × (−ai ni + b1i t1i + b2i t2i )(6)Here we used the following relationships:ȧi = 0, v i = ḃ1i t1i + ḃ2i t2iwhich expresses there is no slippage between the fingertip
and the object (the surface velocity on the fingertip is equal
to the one on the object surface) and is non-holonomic
constraint.
On the other hand, if fingertip is soft, then the angularvelocity around the contact normal direction can be applied
to the object.
nTi ω i = nTi ω o .
From (6) and (7), we get




ṗi
ṗo
GTfi
= ni Δṙi + GToi
ωi
ωo

(7)

(8)

where
GTfi = GTfri + Δri GTfdi ,I 3×3 −ri [ni ×]T,
Gf ri =nTi 
 O 1×3
O 3×3 [ni ×],
GTfdi =
 O 1×3 O 1×3
I3−[(−ai ni + b1i t1i + b2i t2i )×]
T
Goi =
nTi
O 1×3

(9)

If aggregating (8) for all n contact points, we get
GTf ẋf = N Δṙ + GTo ẋo := p̃˙ c(10)

where⎡term including Δri .

⎤

ẋf 1


⎢ ẋf 2 ⎥
ṗi
ṗo
⎢
⎥
ẋf i =
, ẋf = ⎢ . ⎥ , ẋo =ωiωo⎣ .. ⎦
ẋf n
⎡ T ⎤
Go1
⎢ GTo2 ⎥⎢⎥GTf = diag[GTfi ], N = diag[ni ], GTo = ⎢ . ⎥⎣ .. ⎦
GTonwhere diag expresses a block diagonal matrix.
Then, we consider manipulability; how much magnitudeof the object velocity (ẋo ) we can generate with the fingertip
velocity (ẋf ) whose magnitude is bounded. We consider thegenerable object velocity underẋTf W f ẋf ≤ 1(11)where W f denotes the weight matrix which is a positive
definite symmetric matrix.
GTf is a full row rank matrix. Then, from (10), we get
T +T˙ẋf = (GTf )+(wf ) p̃c + (I − (Gf )(wf ) Gf )kf

(12)

−1
T
−1
−1
denotes thewhere (GTf )+(wf ) = W f Gf (Gf W f Gf )weighted psedo-inverse matrix of Gf and kf denotes anarbitrary vector. Similarly to the analyses of Yoshikawa [3]and Bicchi [18], we condier the generable p̃˙ c with miminumcost. Then we get the following relationship:

1 ≥ ẋTf W f ẋf= p̃˙ Tc (GTf W −1 Gf )−1 p̃˙ c + kTf (I − (GTf )+f−1 ˙≥ p̃˙ Tc (GTf W −1p̃c .
f Gf )

T 2(wf ) Gf ) kf

(13)

Here, we consider N Δṙ + GTo ẋo = p̃˙ c shown in (10).
This relationship includes Δṙ, but does not include Δr.
Hence, if concerning the effect of Δr (softness) on thegenerable object velocity ẋo , we only have to consider
T−1
GTf W −1
f Gf . What (the norm of) Gf W f Gf is largemeans that we can generate large object velocity and viceversa. Then, we focus on GTf W −1
f Gf . For the convenient,
we assume
T−1
GTf W −1
f Gf = diag[Gf i W f i Gf i ]which means W f = diag[W f i ] can be held. From (9), we
getT−1
GTfi W −1
f i Gf i = Gf ri W f i Gf ri
T−1+ Δri (GTfri W −1
f i Gf di + Gf di W f i Gf ri )+ Δri2 GTfdi W −1
f i Gf diHere, we define the following two matrices, respectively,
expressing the term which does not include Δri and theGTfi W −1
f i Gf i = G1f i + G2f i(14)

G1f i = GTfri W −1
f i Gf ri
TT−1
G2f i = Δr i (Gf ri W −1
f i Gf di + Gf di W f i Gf ri )+ Δr 2i GTfdi W −1
f i Gf diHere, we suppose that

W f i11O 3×3
W fi =, W f i11 ∈ R3×3 . (15)
O 3×3 wf i22 I 3×3
Then, from (9), G2f i can be rewritten by
G2f i =

(Δri2 −2ri Δri )[ni ×][ni ×]T
wf i22

O 3×1

O 1×3
0

 O,
(16)

since Δri ≤ ri . Here O  A means A is positive semideninite. From this relationship and (14), we can get thefollowing relationshipGTfi W −1
f i Gf i = G1f i + G2f i  G1f i .

(17)

From (16), it can be seen that (the norm of) G2f i decreases
with the increase of Δri , which indicates that (the norm of)
GTf W −1
f Gf decreases with the increase of Δri . Therefore,
we can say that generable object velocity decreases with
the increase of Δri . The magnitude of Δri depends on thecontact force and stiffness at the contact area, as it can beseen from (1). If comparing two grasps where the magnitude
of contact forces are the same but stiffness at the contact area
is different, the generable object velocity for larger stiffness
is larger than that for smaller stiffness.
On the other hand, from (17), it can be seen that generable
object velocity when the deformation at the contact area
causes is smaller than that when the deformation at thecontact area can be neglected (fingertip is rigid).
Summarizing, softness (stiffness) affects manipulability,
namely generable object velocity decreases with the increaseof softness (decrease of stiffness).
Now, from(10) and (13), we have1 ≥ ẋTf W f ẋf−1≥ (N Δṙ + GTo ẋo )T (GTf W −1
(N Δṙ + GTo ẋo ).
f Gf )(18)

This set can be regarded as an ellipsoid in [Δṙ T ẋTo ]T space.
Here, we will consider to map this ellipsoid onto ẋo space.
For this purpose, we consider the completing square withrespect to Δṙ:
(Δṙ − A2 ẋo )T A1 (Δṙ − A2 ẋo ) + ẋTo A3 ẋo ≤ 1.

(19)

where A1 , A2 and A3 are the matrices derived by thecompleting square. From this formulation, it can be seenthat the ellipsoid (18) is divided equally in Δṙ direction by
Δṙ = A2 ẋo (see Fig.2). If substituting Δṙ = A2 ẋo into theellipsoid (18), we get{Δṙ = A2 ẋo ,ẋTo A3 ẋo ≤ 1}.
Δr

are some hypotheses for principle of friction, but normally,
if two surfaces contacts each other, the total resistance force
for tangential motion can be expressed by [19].
.
Δr=A2xo

common area

Ftan = Acontact τshearing.
xo

Fig. 2.

Mapping of the ellipsoid onto ẋo space

which is common area of the ellipsoid (18) and Δṙ = A2 ẋo .
In this set, ẋTo A3 ẋo ≤ 1 expresses the feasible area/range ofẋo in the common area. This set is equivalent to the ellipsoid
mapped onto ẋo space.
Comparing the coefficients of (18) and (19), we have
−1N.
A1 = N T (GTf W −1
f Gf )

A1 is regular since N has full column rank. Then, A2 andA3 can be written by
T
T
−1
−1 T
Go
A2 = −A−1
1 N (Gf W f Gf )

= N+
GT .
((GT W −1 G )−1 ) off

A3 = Go (I − N N +
)T
−1
−1 )
((GT
f W f Gf )−1
(GTf W −1(I − N N +)GTof Gf )((GT W −1 G )−1 )fff

Now, we express the object velocity which we would liketo evaluate, with weighted matrix W o :˙ o = W o ẋx̃Using this relationship, the generable object velocity can be
expressed by the following ellipsoid:
˙ To M x̃˙ o ≤ 1.
x̃

(20)

−1
M = W −To A3 W o

From this, we will define the following criterion index:
Im =where Ftan is the total resistance force, Acontact is the truearea of contact (The two surfaces actually touch at a discrete
number of contact spots, and Acontact corresponds to thesum of the all areas) and τshearing is the resistance force
per unit area. If according to the model given by Persson[20]–[22], the real contact area corresponds to the apparent
contact area if the contact surface is enough smooth andcontact pressure is enough high. In other cases (which isusual), the real contact area is proportional to the load fn(contact normal force):
Acontact =1
λmax (M ),(21)

where λmax (M ) denotes the maximum eigenvalue of M .
This index expresses the maximum object velocity which is
generable in any direction.
Remark that M includes Gf i given in (9), and Gf i
includes Δri (see (1)). Then, we need to give not only
fingertip configuration but also grasping force to determine
M.
B. Effect on stability/friction
We suppose that grasp stability means that we can balancelarge external wrench. Here, we analyze the effect of softness
on how much magnitude of resultant wrench can be applied
to the object. For the purpose, we focus on friction. There

Q(1 − ν 2 )fn
E(23)

where Q is the coefficient related with contact surface
roughness, ν and E are Poisson’s ratio and elastic modulus
2(1−ν 2 )(1−ν 2 ))where (1−ν= E1 1 + E2 2 (the subscript 1 and 2
E
corresponds to the two materials constructing the contact
surface) In this case, maximum static frictional force is given
byFtan = μsof t fn ,

f

(22)

μsof t =Q(1 − ν 2 )τshearingE(24)

μsof t can be regarded as maximum static frictional coefficient. From this equation, we can see the relationshipbetween frictional coefficient and elastic modulus, which
was not considered in conventional analyses of grasping. If
comparing the two fingertips whose softness (E) is different
but whose τshearing is the same (for example, because thesurface is made of the same material), μsof t for the softer
fingertip is larger than the other one, and then applicable
frictional forces are also larger. This might be one of thereasons why grasping with softer fingertip can achieve morestable grasping.
In general, if the fingertip is deformable, it is calledsoft-finger contact. In this model, not only frictional forces
but also contact moment around normal direction can beapplied to the object. This paper also considers the contact
moment. Similarly to the methodology of Xydas et. al. [15],we derive local frictional forces in tangential direction at
local infinitesimal area of the contact area (from the contactpressure on that area), and then derive frictional forces andcontact moment at the contact area by integrating the local
frictional forces with respect to the whole contact area. By
deriving the local frictional forces according to (24), thefrictional condition can be represented by
Fi = {f i , mi ||f ti |2
|mi |2+≤ 1, fni ≥ 0},2(μi fni )(αi aci μi fni )2
(25)
 T 
t
1i
fni = nTi f i , f ti = T i f i =
f i.
tT2i

Maximum magnitude of moment around normal [Nm]where f i is contact force vector, mi is contact moment, μi
is frictional coefficient corresponding to μsof t in (24), f ti istangential force vector, aci is the radius of the contact area,and αi is constant number.
Xydas et.al. [15] presented the relationship between radiusof contact area aci and contact normal force fni
γaci = αci fni(26)

2= 2ri /ki2γ and ζ = 1/(2γ) (0 ≤ γ ≤ 1/3). From
where αci(25) and (26), frictional condition can be rewritten by
2

2
1.8
1.6
1.4
e
in
dl
ate
m
ici
ox
pr
Ap

1.2
1
0.8
0.6
0.4
0.2
0
0

ξ0i

2

|f ti ||mi |Fi = {f i , mi |+1+γ 2 ≤ 1, fni ≥ 0},(μi fni )2(μmi fni)(27)
μmi = αi αci μi
μmi can be regarded as frictional coefficient for moment.
This condition equals to conventional frictional condition forsoft-finger contact, if assuming γ = 0 which corresponds tothe case when the radius is proportional to contact normal
force. Then it can be said that this condition is the extensionof conventional frictional condition.
From (10) and the principle of virtual work, we have
 


Gf
wf
(28)
=
wc
wo
Go
where wc = [wTc1 wTc2 · · · ]T , wci = [f Ti mi ]T , wf denotes
fingertip wrench corresponding to contact force f i andcontact moment mi , wo denotes resultant wrench applied
to the object.
Now, supposing the upper bound of contact normal forceis ξi , we represent the generable object wrench set as convex
polyhedron. If tangential contact forces are zero, maximum
magnitude of generable frictional moment around contact
normal direction is given by
1+γ.
μmmi fni

Fig. 3 shows the relationship between this maximum frictional moment and contact normal force. It can be seenfrom this figure that if contact normal force is very small,
generable moment is almost zero, and as the contact normal
force increases, the moment becomes to be generable. Then,
we approximate this relationship by the two lines, as shownin Fig. 3. The dot line in Fig. 3 is the tangent line of thecurve at the point where contact normal force is half of
its maximum. Let ξ0i be the contact normal force at theboundary (between the range where frictional moment canbe generated and the range where it cannot). Note that ξ0i
is supposed to be small compared to ξi . Then, the generable
maximum frictional moment can be represented by
μmmi (fni − ξ0i )where μmmi denotes the gradient of the approximated line.
Based on this approximation, firstly, we consider the casewhen contact normal force fni is its maximum ξi . If contact
normal force is fixed, the set of contact wrenches satisfying
frictional condition can be represented by ellipsoid shown in

0.2

0.4

0.6

0.8

1

Normal contact force [N]Fig. 3. Relation between maximum moment around contact normal andnormal force (tangential force is zero and μmi = 2, γ = 0.25)

Fig. 4. Extending the formulation of Yu and Wenhan [23],we approximate this ellipsoid by convex polyhedron.
L K−1

wci =

λikκ wvikκ + λi10 wvi10 + λi1K wvi1K ,k=1 κ=1
L K−1

k=1 κ=1

λikκ + λi10 + λi1K = 1,

λikκ ≥ 0 (29)





πκ
2πk
) sin( )
L
K


πκ
2πk
t2i
) sin( )+ μi ξi sin(
0
L
K


πκ
o3×1
+ μmmi (ξi − ξ0i ) cos( )1Kwvikκ = ξi

ni
0

+ μi ξi cos(

t1i
0



where K is number of the segments in moment direction and
L is number of the segments in tangential direction. Note thatk = 1 only when κ = 0, K, and k = 1, 2, · · · , L in the othercases. By reallocating the number, (29) is rewritten by
l̂

wci =

l̂

λik wvik ,k=1

λik = 1,

λik ≥ 0k=1

(k = 1, 2, · · · , ˆl = L(K − 1) + 2)

(30)

By the convex polyhedron whose vertexes are this wvik andthe point where fni = ξ0i and the other components are all
zero (see Fig. 5), we approximate generable contact wrench
set:
l

l

λik wvik ,

wci =
k=1

λik = 1,

λik ≥ 0k=1

(k = 1, 2, · · · , l = L(K − 1) + 3),ni
wvil = ξ0i
0(31)

Now we will construct grasp wrench space (the set ofgenerable object wrenches) similarly to the way of Ferrari
and Canny [24]. From (28), every wvik in (31) is translated
at the object frame
wvoik = M f o Goi wvik

m

y
pC1
pC2

po

x
pC3

t1

Fig. 4. Linearized frictional tangential forces and moment around thecontact normal direction

Fig. 6.

Target system in numerical examples
TABLE IPARAMETER VALUES FOR NUMERICAL EXAMPLES

fn

ft
m

Fig. 5. Set of contact wrench satisfying frictional condition and approxicimated convex polyhedron (dashed line)

where M f o is weight matrix. Then, bounding the sum
magnitude of normal contact force, the grasp wrench spaceis given by
CH(

n


(wvoi1 , wvoi2 , · · · , wvoil ))(32)

Object base
Object heightri (i = 1, 2, 3)
W fi
Wf
Wo
W fo
po
pc1
pc2
pc3
ξi
αi
L
K
Q(1 − ν 2 )τshearing
grasping force whencalculating Im

regular hexagon (inradius: 0.025 (m))
0.1 (m)0.01(m)diag[1 1 1 ri2 ri2 ri2 ]diag[W f i ]diag[1 1 1 0.025 0.025 0.025]
diag[1 1 1 1/0.025 1/0.025 1/0.025][0 0 0]T (m)0.025[cos(60◦ ) sin(60◦ ) 0]T (m)0.025[cos(120◦ ) sin(120◦ ) 0]T (m)0.025[cos(−60◦ ) sin(−60◦ ) 0]T (m)
1 (N)
0.6
8
9
30000/π (N/m2 )[−0.433 − 0.75 0 0.866 0 0−0.433 0.75 0]Tiwhere CH denotes the convex hull, and the maximum
summation
n of normalized normal contact forces by ξi isunit: i=1 |fni /ξi | ≤ 1. If bounding the magnitude of every
normal contact force, the grasp wrench space isIII. N UMERICAL EXAMPLES

where A and b are the matrix and vector resulted from thetransformation.
Let A= col[aTi ] and b= col[bi ]. Generally the distance
between the origin and a hyperplane aTi wo = bi in thespace of w̃o is given by bi /|ai |. Therefore, if force closure
is satisfied, the maximum resultant object wrench which is
generable in any direction can be obtained by

In order to verify the validity of our approach, we show
numerical examples. Fig.6 shows the target system where
grasping the prism shaped object with 3 fingertips. Everyfingertip is a sphere with the diameter of ri =0.02 (m). Thebase is a regular hexagon whose inscribed circle is 0.025(m) in radius, and the height of the object is 0.1 (m). We
suppose that ki = πE, based on (1) and the model of Inoueand Hirai [17]. We approximate fictional condition by convex
polyhedron with 67 vertices where L = 8 and K = 9.
We approximated the relationship between generable contact
normal force and moment by the two lines, as shown in Fig.
3. The approximated line is the tangent line of the curve at
the point where contact normal force is half of its maximum.
The other setting parameters are listed at Table I.
Based on the setting, we calculated the criterion indexes
Im (in (21)) and If (in (35), derived using (32)). In order toinvestigate the influence of softness on manipulability index
and grasp stability, we calculated Im and If when changing
stiffness parameter ki (see (1)) from 10000 to 100000 (N/m2 )while ζ is 1.5, 1.5 + 1/6, 1.5 + 1/3, and 2. When calculating
Im , the grasping forces (internal forces for grasping) is set
as follows:

If = min bi /|ai |.

[−0.433 − 0.75 0 0.866 0 0 − 0.433 0.75 0]T

CH(

n


(wvoi1 , wvoi2 , · · · , wvoil ))

(33)

i


where
denotes the Minkowski sum, and the maximum
normal contact forces is ξi for ith contact point.
We will have convex hulls of nl points in (32) whileln points in (33). Therefore, from the viewpoint of computational effort, grasp wrench space given in (32) might
be useful. In both cases, the grasp wrench space can betransformed into the following form:{w̃o | Aw̃o ≤ b}

i

(34)

(35)

Manipulability index : Im (m/s)

0.7

R EFERENCES

0.68
0.66
0.64

γ=1.5

0.62

γ=1.67γ=1.83

0.6

γ=2

0.58
0.56
10000

Fig. 7.

30000

50000
70000
Stiffness parameter : ki

90000

Softness effect on manipulability: criterion index Im

Generable wrench index : If (N)

0.6
0.5
0.4
γ=1.67γ=1.83

0.2

γ=2

0.1
0
10000

Fig. 8.

γ=1.5

0.3

30000
50000
70000
Stiffness parameter : ki

90000

Softness effect on generable object wrench: criterion index If

The results are shown in Fig.7 and Fig.8. From Fig.7 andFig.8, it can be seen that with the increase of stiffness
parameter ki , manipulability increases while generable object
wrenches decrease. With the increase of γ, manipulability
decreases while generable object wrenches increase. Both ki
and γ are the parameters related with softness. It is seemedthat the effect of γ on manipulability is larger than thatof ki . On the other hand, the effect of ki on generable
object wrenches is larger than that of γ. Manipulability index
depends on Δri , while generable object wrenches largely
depend on ki . They might be one of the reasons.

IV. C ONCLUSION
In this paper, we analyzed the effects of softness at
the fingertip on the manipulability and generable object
wrenches. We formulated manipulability and the set of generable object wrench for grasping system, taking deformationof the fingertips into consideration. We showed that theincrease of the softness decreases the manipulability while it
increases generable object wrenches (stability from the view
point of grasping). The validity of our approach was also
shown by numerical examples. This analysis suggests thatthe softness of fingertip should change according to purposeof task, if we can. For example, when picking up (unknown)
object on table, softer fingertip would be preferable. Whenassembling parts, harder fingertip would be preferable. In
future, concerning this, we will develop a new system forfingertip, which can change stiffness of the fingertip.

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