The 2009 IEEE/RSJ International Conference onIntelligent Robots and Systems
October 11-15, 2009 St. Louis, USA

Dynamic Grasping for an Arbitrary Polyhedral Object
by a Multi-Fingered Hand-Arm System
Akihiro Kawamura, Kenji Tahara, Ryo Kurazume and Tsutomu Hasegawaqa 3

Abstract— This paper proposes a novel control method forstable grasping using a multi-fingered hand-arm system with
soft hemispherical finger tips. The proposed method is simple
but easily achieves stable grasping of an arbitrary polyhedral object using an arbitrary number of fingers. Firstly, weformulate nonholonomic constraints between a multi-fingered
hand-arm system and an object constrained by rolling contact
with finger tips, and derive a condition for stable grasping bystability analysis. A new index for evaluating the possibility ofstable grasping is proposed and efficient initial relative positions
between finger tips and the object for realizing stable grasping
are analyzed. The stability of the proposed system and thevalidity of the index are verified through numerical simulations.

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I. I NTRODUCTION
A multi-fingered hand-arm system has been expected torealize dexterous grasping like a human hand. Robots withthis system will be able to perform various manipulation
tasks even in an unstructured environment safely. Many
robotics systems and control methods for grasping an object
with an arbitrary shape have been proposed [1–5]. Especially,
dynamic grasping controllers using rolling constraints havebeen reported [6–9]. However, these controllers are based on
inverse dynamics calculation, and object information suchas the mass and the shape of the object is required. In
contrast, there are several researches for dynamic grasping ofan unknown object using external sensing devices such as avision sensor and a tactile sensor [10–12]. In these methods,
sensing devices are costly and sensing error must be taken
into consideration though object information is not requiredin advance.
On the other hand, Wimböck et al. [13] proposed a dynamic grasping method for an arbitrary object which requires
neither object information nor external sensors. However,the stability of the system and the convergence performanceof the closed-loop dynamics were not discussed. Moreover,rolling constraints between fingertips and the object surfaces
are not considered in their method. Similarly, Arimoto et al.
[14–17] have proposed a dynamic object grasping method
without object information and external sensors. The object
This work was partially supported by ”the Kyushu University Research
Superstar Program (SSP)”, based on the budget of Kyushu University
allocated under President’s initiative.
A. Kawamura is with the Graduate School of Information Science andElectrical Engineering, Kyushu University, Fukuoka, 819-0395, JAPAN

kawamura@irvs.is.kyushu-u.ac.jp
K. Tahara is with the Organization for the Promotion of Advanced Research, Kyushu University, Fukuoka, 819-0395, JAPANtahara@ieee.org
R. Kurazume and T. Hasegawa are with the Faculty of Information
Science and Electrical Engineering, Kyushu University, Fukuoka, JAPAN{kurazume, hasegawa}@ait.kyushu-u.ac.jp

978-1-4244-3804-4/09/$25.00 ©2009 IEEE

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x02
x01

x03
Fig. 1.

&z

z

O

qa 2

qa1

&x

x

Multi-fingered hand-arm system

is limited to the one which consists of two flat and parallel
surfaces, though they verified the stability of the system andthe convergent performance of the closed-loop dynamics.
This paper proposes a novel control method for stablegrasping of an arbitrary polyhedral object using a multifingered hand-arm system with soft hemispherical finger
tips. The proposed method is simple but easily achieves
stable grasping of an object without use of preliminary
information for a grasped object and any external sensors
using an arbitrary number of fingers. Firstly, we formulate
nonholonomic constraints between a multi-fingered handarm system and an object constrained by rolling contact with
finger tips. The nonholonomic constraints for rolling contact
were proposed by Arimoto et al. [14–17] for an object with
two flat and parallel surfaces. We expand these constraints
for an object with an arbitrary polyhedral shape. The numberof fingers is also variable in our formulation. Secondly,
we derive Lagrange’s equation of motion for a hand-arm
system, and propose a new control signal which achieves
stable grasping. A new index for evaluating the possibilityof stable grasping is proposed and efficient initial relative
positions between finger tips and an object for realizing
stable grasping are analyzed. Using this index, it is clarified
that appropriate initial positions of finger tips depend on the
shape of the object and the ratio of the size of the object
and the radius of the finger tip. The stability of the proposedsystem and the validity of the index are verified throughnumerical simulations.

2264

II. A M ULTI F INGERED H AND A RM S YSTEM
In this section, we define a model of a hand-arm systemcomposed of an arm and a multi-fingered hand. An example
of a multi-fingered hand-arm system treated here is illustrated in Fig. 1. An object to be grasped is an arbitrary
polyhedral object whose surfaces touched by finger tips are
flat. All finger tips maintain rolling contact with the object
surfaces, and do not slip and detach from the surfaces during
movement of the tips. Assume that fingertips roll within theranges of hemisphere surfaces, and they don’t deviate fromeach contact surface. Note that the gravity effect is ignored inthis paper in order to have a physical insight into analyzing
physical interaction and stability of the system. As shownin Fig. 1, O denotes the origin of Cartesian coordinates.
x0i ∈ R3 is the center of each contact area. Hereafter, thesubscript of i refers to the ith finger in all equations. The
degrees of freedom of the arm and the ith finger are Na
and Ni , respectively. The joint angle of the arm is expressedby q a ∈ RNa . Similarly, the joint angle of the ith finger
is expressed by q 0i ∈ RNi. q denotes the joint angles
 ofT
the arm and all the fingers = (q a , q 01 , q 02 , ..., q 0N ) . N
is the number of the fingers. As shown in Fig. 2, Oc.m.
denotes the center of the object mass and the origin of local
coordinates. Its position in Cartesian coordinates is expressedT
as x = (x, y, z) ∈ R3 . Instantaneous rotational axis of theobject at Oc.m. is expressed by ω. The orientation angularvelocities around each axis of Cartesian coordinates x, y, z
are expressed as ωx , ωy , ωz respectively.
A. Constraints
3-dimentional rolling constraints with area contacts aremodeled here. The orientation of the object in Cartesian
coordinates can be expressed by the rotational matrix R such
that
Rob

= (rX , r Y , r Z ) ∈ SO (3) ,= (C iX , C iY , C iZ ) ,

vi

=



∆ri Ċ iY − ωi × C iY ,

C iY

& rY
Oc.m.
Yi

rX

xi

vi

rZ

∆ ri

Zi

Finger tip

Center of
finger tip

x0i

ri

Xi

Center of the
Object mass

Fig. 2.

Contact model at the center of the contact area

are expressed such that the velocity of the center of thecontact area on the finger tip, given as (3), should equal
to the velocity of the center of the contact area on the object
surface∆ri C T= Ẋi
(4)iX Ċ iY − ω i × C iY
= Żi ,
(5)
∆ri C TiZ Ċ iY − ω i × C iY
where

Xi

= −C ⊤iX (x − x0i )

(6)

Zi

−C ⊤
iZ (x − x0i ).

(7)=Equations (4) and (5) denote nonholonomic rolling constraints on the object surfaces. These constrains are of linearwith respect to each velocity vector, and thereby they can be
expressed as Pfaffian constraints as follows:
X iq q̇ + X ix ẋ + X iω ωZ iq q̇ + Z ix ẋ + Z iω ω

= 0
= 0,

(8)
(9)

where


(3)

where ω i ∈ R3 is the orientation angular velocity vectors foreach robotic finger on the contact frames, ri is the radius ofeach finger tip, and ∆ri are the distance between the centerof the finger tips and the contact surfaces (see Fig. 2). v iis on the tangential plane at the center of the contact area
(now, they are surfaces of the object). The rolling constraints

C iZ
C iX

(2)

where RCi is the rotational matrix between the object frameto the contact frames. Now, we consider the velocity of thecenter of the contact area v i must satisfy&iObject(1)

where rX , r Y , r Z ∈ R3 are mutually orthonormal vectors
on the object frame. It is known that this rotational matrix
is one of the members of the group SO(3). In addition tothis, we define contact frames at the center of each contact
area as follows:
Rob RCi

Center ofcontact area

Contact area

T
X iq = ∆ri C T
iZ J Ωi − C iX J 0i
 X = CT
ix

iX

T
 X iω = {C iX × (x − x0i )} − ∆ri C T
iZ
TT Z iq = −∆ri C iX J Ωi − C iZ J 0i
 Z ix = C TiZT
Z iω = {C iZ × (x − x0i )} + ∆ri C T
iX ,
T

(10)

and ω = (ωx , ωy , ωz ) ∈ R3 is the angular velocity
vector for the instantaneousrotational axis of the object.
NJ Ωi ∈ R3×(Na + i=1 Ni ) is the Jacobian matrix for theorientation angular velocities of the finger tips with respect

2265

N

to the joint
angular velocity q̇ ∈ RNa + i=1 Ni . J 0i ∈3×(Na + N
i=1 Ni ) is the Jacobian matrix for the center ofR
the finger tip x0i with respect to the joint angle, respectively.

where S (q, q̇) is skew-symmetric matrix, u is a vector of
the input torque. In addition, λiX and λiZ denote Lagrange’s
multipliers.
III. C ONTROL I NPUT

B. Contact Model of Soft Finger-Tip
In this paper, the physical relationship between the deformation of the finger tip at the center of the contact area
and its reproducing force is given on the basis of lumpedparameterized model proposed by Arimoto et al [14]. The
reproducing force f (∆r) in the normal direction to theobject surface at the center of the contact area is given as
follows:dfi = f¯i + ξi dt(ri − ∆ri )(11)f¯i = k(ri − ∆ri )2 ,where k is a positive stiffness constant which depends on thematerial of the finger tip. In the second term of the righthand side of the upper equation of (11), ξi (∆ri ) is a positive
scalar function with respect to ∆ri , and thereby the viscous
force increases according to the expansion of the contact
area.
C. Overall Dynamics
The total kinetic energy for the overall system can be
described as follows:1
1
1
K = q̇ T H q̇ + ẋT M ẋ + ω T Iω,
(12)
2
2
2
N
N
where H ∈ R(Na + i=1 Ni )×(Na + i=1 Ni ) is the inertia

Tahara et al. [17] proposed a simple control method of a
triple robotic fingers system for stable grasping. We propose
a new control method based on Tahara’s method for a multifingered hand-arm system. This control signal is designed sothat the center of each finger tip approaches each other. The
control signal is given as follows:NP (∆ri ) =

i=1

N

i=1

H (q) q̈ +
+

T

N

d
E = −q̇TC q̇ −ξ∆ṙi2 ≤ 0
dt
i=1

M ẍ +N 
i=1

I ω̇ +

E = K + V + ∆P ≥ 0
1
1
1
K = q̇ T H q̇ + ẋTM ẋ + ωTIω
2
2
2
A
2
2
(x01 − x02 ) + (x02 − x03 )V=22+... + (x0N − x01 )

∆P =

N



T
−fi C iY + X T
λ
+
Z
λ
ix iX
ix iZ = 0 (15)N1İ + S ω −{C iY × (x − x0i )}fi
2
i=1

i=1

where

i=1

0

δri

(20)
(21)
(22)

(23)


f¯i (∆rdi + φ) − f¯i (∆rdi ) dφ, (24)=∆rdi − ∆ri
f
.
 d
N
N
i=1 ri(25)(26)∆rdi is an initial value of ∆ri . V plays a roll of an artificial
potential energy arisen from the control input. Equation (21)is evident since K, V and ∆P are positive as far as 0 ≤∆rdi − δri < ri . In addition, Ė ≤ 0 during movement.
Equations (20) and (21) yield


N
∞

T
2
q̇ C q̇ +ξ∆ṙi dt0i=1

≤ E (0) − E (t) ≤ E (0) ,

N 


T
XT
λ
+
Z
λ
+
iω iX
iω iZ = 0, (16)

(19)

δr
i

A =

For the object:(18)

By substituting (17) into (14) and taking a sum of innerproduct of (19) and closed loop dynamics expressed by (14),(15) and (16), we obtain

1Ḣ (q) + S (q, q̇) q̇
2

i=1

N

1 
x0i ,
N i=1

Λ̇ = (q̇, ẋ, ω) .

(13)

N 


T
T
JT
= u, (14)
C
f
+
X
λ
+
Z
λ
iY
i
iX
iZ
0i
iq
iq

(17)





0

where P (∆ri ) is the elastic potential energy for each finger
generated by the deformation of the finger tip. Hence,
Lagrange’s equation of motion is expressed by applying thevariational principle as follows:
For the multi-fingered hand-arm system:

J 0j (xd − x0j ) − C q̇N
N
where C ∈ R(Na + i=1 Ni )×(Na + i=1 Ni ) > 0 is a diagonal
positive definite matrix that expresses the damping gain foreach finger, and fd is the nominal desired grasping force.
Now, an output vector of overall system is given as follows:ri −∆rif¯i (∆ri ) dζ,N
i=1 ri j=1

xd =matrix for the arm and the fingers, M = diag (m, m, m)
is the mass of the object, I = RĪRT and Ī ∈ R3×3 are
the inertia tensors for the object represented by the principal
axes of inertia. On the other hand, the total potential energy
for the overall system is given as follows:
P =

N


fd

u=

(27)

Equation (27) shows that the joint angular velocity q̇ (t) issquared integrable over time t ∈ [0, ∞). It shows that q̇(t) ∈2266

L2 (0, ∞). Considering the constraints shown by (4) and (5),it is clear that ẋ ∈ L2 (0, ∞) and ω ∈ L2 (0, ∞). Thereby,
the output of the overall system Λ̇(t) is uniformly continuous
since it is shown that Λ̇ → 0 and Λ̈ → 0 when t → ∞[15]. Therefore, it is obvious that the sum of the external
force applied to the finger system and the object ∆λ∞ isconverged to zero.
∆λ∞ = (∆λq , ∆λx , ∆λω ) → 0

For the multi-fingered hand-arm system:
1
Ḣ + S + C q̇
2
N 


T
ŃZi J T+
0i − ∆ri λiZ cos θsi J Ωi r X

H q̈ +

+

(28)

i=1

where

+

∆λx =

T
−fi C iY + X T
ix λiX + Z ix λiZ

i=1
N


∆λω = −

(30)

{C iY × (x − x0i )}fi
N 

i=1


T
XT
. (31)
λ
+
Z
λ
iX
iZ
iω
iω

As a consequence, it is shown the dynamic force/torqueequibilium condition for immobilization of the object issatisfied since each external force and each velocity become
zero. In other words, the stability of the overall systems isverified. However, it is necessary to consider the stability
from physical perspective since the stability is analyzed only
from a mathmatical viewpoint. There are some cases that oneof the centers of the contact areas is outside of the object
surfaces at the final state. Therefore, there is a requirement
to specify the shape of the object to argue the stability fromphysical perspective. Moreover we argue a final state of theoverall system mainly, because an overall movement depends
on each damping gain and initial contact position but it is
not depended on the final state of the system. Thereby inthis paper, we focus on a final state of the overall system
as the initial step of convergence analysis. Of course, it isimportant to consider the convergence of overall movement
and it is one of our next works.
In the next section, as an example, we introduce thecondition for an object with an arbitrary polygonal shape.
IV. S PECIFIC S TUDY
In this section, the condition for stable grasping of anarbitrary polyhedral object is shown. In addition, an exampleof the condition for a triangle pole is introduced.
The final positions of the finger tips are obtained when
E is minimized. On the other hand, the positions of thefinger tips when E is minimized (= Emin ) coincide with thepositions of the finger tips when V is minimized (= Vmin ).
Therefore, we derive the closed loop dynamics shown in (14)to (16) and V in (23) for an arbitrary polyhedral object.

T∆ri λiX J T
−
λ
J
iZ
Ωi
0i r YT−NZi J T
0i − ∆ri λiZ sin θsi J Ωi r Z




N
N  



−A
ŃW j − N ŃW i  J T
rX
0i


j=1
i=1



N
N  



Zj − N Zi  J T
−A
rY
0i


j=1
i=1



N
N  



NW j − N NW i  J T
−A
rZ
0i



i=1

+

N 

i=1

N 


T
T
∆λq =
JT
0i C iY fi + X iq λiX + Z iq λiZ − u (29)
i=1
N 


i=1
N 


i=1

=

j=1

0,

(32)

For the object:
M ẍ +N 
i=1

I ω̇ +++

−ŃZi rX + λiZ r Y + NZi r Z = 0

(33)

N 


1
−NY i λiZ − ŃXi Zi r Xİ + S ω +
2
i=1

N


(fi Xi + λiX Yi ) r Y

i=1
N 

i=1


−ŃY i λiZ + NXi Zi rZ = 0,

where
θ1 = 0T(i = 2, 3, ..., N ) θi = cos−1 C (i−1)Y C iY θsi = i θhh=1
 R = R−jθsi R−i π2
 Ci D = Y + ∆rii
 i
 N = X sin θ + D cos θ
i
si
i
si
 Wi

 ŃW i = Xi cos θsi − Di sin θsi NXi = fi sin θsi + λiX cos θsi ŃXi = fi cos θsi − λiX sin θsi NY i = Yi cos θsi + Xi sin θsi ŃY i = Yi sin θsi − Xi cos θsi NZi = fi cos θsi + λiX sin θsi
ŃZi = fi sin θsi − λiX cos θsi ,

(34)

(35)

where Yi is the distance from the center of the object mass
Oc.m. to the surface. From (32), (33) and (34), the scalarfunction V is given as follows:

2267

TABLE I

θ t1
A
B
C
D
E

x

X3

x

R ELATIONSHIP BETWEEN θti AND Xi

Y3

y

x

y

X1

θt1 [rad] θt2 [rad] θt3 [rad] X1 [m]
2.84
2.84
0.60
0.00
2.54
2.54
1.20
0.00
2.24
2.24
1.80
0.00
1.94
1.94
2.40
0.00
1.64
1.64
3.00
0.00

X3 [m]
0.0601IN
0.0122IN
0.0094IN-0.0190IN-0.1137 OUT (rj = 0.03 [m] , ∆rmin = 0.02 [m] , S = 6.38 × 103 m2 )TABLE II

z
Y2

θt 3

R ELATIONSHIP BETWEEN ri AND Xi

Y1

y

F
G
H

r[m]
0.030
0.050
0.070

∆rmin [m]
0.020
0.040
0.060

X1 [m]
0.00
0.00
0.00

X2 [m]
0.05040.06050.0705X3 [m]-0.0504-0.0605-0.0705

IN
IN
OUT

 2 
3(θt1 , θt2 , θt3 ) = (1.79, 1.79, 2.70) [rad], S = 6.38 × 10 m

X2x

X2 [m]-0.0601-0.0122-0.0094
0.01900.1137

θt 2

AFig. 3.

F

Cross-section view of polygonal column

C


  2

2
2
V = A X12 + X22 + ... + XN
+ D1 + D22 + ... + DN

− {(D1 X2 − X1 D2 ) sin θt2 + (D2 X3 − X2 D3 ) sin θt3+... + (DN X1 − XN D1 ) sin θt1 }− {(X1 X2 + D1 D2 ) cos θt2 + (X2 X3 + D2 D3 ) cos θt3

Fig. 4. Sample case of “IN” as shown in Table I (A, C) and II (F). IN :The case of success in which all the centers of the contact areas are insideof the object surfaces at the final state (V = Vmin )+... + (XN X1 + DN D1 ) cos θt1 }]
+

where

A
2
2
(Z1 − Z2 ) + (Z2 − Z3 )2+... + (ZN − Z1 )2 ,θtiθti

= θsi − θs(i−1)= θsi − θsN(i = 1)(i = 1).

(36)

E

H

(37)
(38)

θti is the external angle of the polygon parallel to the baseof the object (see Fig. 3). One of the prerequisite for theminimum value of V isZ1 = Z2 = ... = ZN .

(39)

In the case that the grasped object has arbitrary polyhedral
shape, Xi can be considered as an index for evaluating thestability of the system from the physical perspective. In fact,
stable grasping is realized if Xi is inside of each contact
surface when V is minimized to Vmin , that is to say, thefinal state of the system is stable in the physical viewpoint.
Equation (36) has three parameters Xi , Di and θti . As an
example, we show the relation between Xi and θti in Table I
and the relation between Xi and Di for Vmin in the case thatthe object is a triangle pole, where the size of the triangle
pole is normalized by the cross-section area S. Table II shows
the relation between Xi and the radius of each finger tip ri
for Vmin in order to show the relation between Xi and Di for
Vmin . In Table I and II, “IN” (see Fig. 4) means a case of
success in which all the centers of the contact areas are insideof the object surface at the final state, and “OUT” (see Fig.
5) means a case of failure in which at least one of the centersFig. 5. Sample case of “OUT” as shown in Table I (E) and II (H). OUT :The case of failure in which at least one of the centers of the contact areas
is outside of the object surfaces at the final state (V = Vmin )of the contact areas is outside of the object surfaces at the
final state. Table I and II show that the shape of the grasped
object and the radius of each finger tip determine whether astable grasping is realized or not at the final state. From these
considerations, we can say that initial positions of the finger
tips are valid for stable grasping if the initial positions areclose to the position obtained when V is minimized to Vmin .
Therefore, Vmin , the minimum value of the scalar function
V , can be used as an index for evaluating the possibility ofstable grasping.
V. N UMERICAL S IMULATION
We conduct numerical simulations for verifying the proposed approach. The parameters of the triple-fingered handarm system and the object in numerical simulation are shown

2268

TABLE III
P HYSICAL PARAMETERS

Xi [m]

Triple-fingered hand-arm system
1st link length la1
1.300[m]2nd link length la2
1.000[m]0.175[m]3rd link length la3
1st link length li1
0.300[m]2nd link length li20.200[m]1st mass ma1
1.300[kg]1st mass center lga1
0.650[m]2nd mass ma2
1.000[kg]2nd mass center lga20.500[m]0.400[kg]3rd mass center lga3
0.0875[m]3rd mass ma3
1st mass mi1
0.250[m]1st mass center lgi1
0.150[m]2nd mass mi2
0.150[m]2nd mass center lgi2
0.100[m]1st Inertia I a1
diag(7.453, 7.453, 0.260)×10−1 [kg·m2 ]2nd Inertia I a2 diag(3.397, 3.397, 0.128)×10−1 [kg·m2 ]3rd Inertia I a3 diag(0.291, 0.291, 0.500)×10−1 [kg·m2 ]1st Inertia I i1
diag(7.725, 7.725, 0.450)×10−3 [kg·m2 ]nd
2 Inertia I i2
diag(2.060, 2.060, 0.120)×10−3 [kg·m2 ]Radius of fingertip ri
0.070[m]5 [N/m2 ]Stiffness coefficient ki
1.000×10
 2
Damping function ξi 1.000× ri − ∆ri2 π[Ns/m2 ]

0.09
0.08
0.07
0.06

}É1X 2
fromV2X 2 for Vmin

0.01
0
-0.01

}É1

-0.06
-0.07
-0.08
-0.09

X3
}É1
X
3 for Vmin

X1
X 1 for Vmin

fromV3

0

5

10

15

Time[s]Zi [m]

0.05

Z1
ZZ1
2
Z2
Z3

0.04
0.03

Z3

0.02

0

5

10

15

Time[s]Object0.037[kg](0.092, 0.048, 0.048)[m](1.833, 1.833, 2.618)[rad]diag (1.273, 0.193, 1.148) × 10−3 [kg · m2 ]Fig. 6.
(39)

0.6

TABLE IV
D ESIRED GRASPING FORCE AND GAINS

0.3

0

5

Time[s]1015

1

TABLE V

0.5

ûq21
àqf
ûq22
àqf
û
q23

0

-0.5

àqf

-1

I NITIAL CONDITION

00[rad/s](−0.262, −1.571, 2.094, 0.785, 0.000)T [rad](0.000, −0.663, 1.934)T × 10−2 [rad](−0.262, −0.705, 2.185)T × 10−2 [rad](0.262, −0.705, 2.185)T × 10−2 [rad]0[m/s](0.000, 0.600, 0.800)T [m] 0[rad/s] 
1 0 0
 0 1 0 
0 0 1

5

10

15

Time[s]ûq3i [N]

1
0.5

ûq31
àqf
ûq32
àqf
û
q33

0

-0.5

àqf

-1

0

5

10

15

Time[s]
3
1.5
0
-1.5
-3

ûqa1
àqa
û
qa2
àqa
û
qa3
àqa
û
qa4
àqa
û
qa5

ûqa [N]

R

àqf

-0.6

1.0[N]diag(3.080, 2.065, 2.483, 0.768, 0.487)×10−2 [Ns·m/rad]diag(0.164, 0.177, 0.064)×10−2 [Ns·m/rad]

q̇
qa
q 01
q 02
q 03
ẋ
x
ω

ûq11
àqf
ûq12
àqf
û
q13

0

-0.3

ûq2i [N]

fd
Ca
Ci

History of Xi reach to Xi for Vmin and history of Zi satisfiesûq1i [N]

Mass m
(Y1 , Y2 , Y3 )(θt1 , θt2 , θt3 )Inertia I

in Table III. Table IV and V show the desired grasping
force and gains, and the initial condition, respectively. The
results of the grasping simulation are depicted in Figs. 6–
10. From Fig. 6 which indicates Xi and Zi , we can seethat Xi converges to which V satisfies Vmin , and thus, thecondition of (39) is satisfied. In addition, Xi and Zi converge
to the values which satisfy the conditions for Vmin . Fromthese results, we can conclude that the analysis of Vmin
is useful and it can be regarded as one of the index forevaluating whether stable grasping is realized or not. This isalso effective for evaluating the initial positions of each finger
tips. Figures 7 and 8 show the elements of ∆λ∞ converges
to zero. It indicates that the sum of external force applied
to the system and the object converges to zero. The results
illustrate that the dynamic force/torque equibilium condition
for immobilization of the object is satisfied. Figures 9 and

àqa

0

5

10

15

Time[s]Fig. 7.

External force applied to the system convergence to zero

10 show q̇, ẋ and ω and we can confirm that the velocitiesof the overall system converge to zero.
VI. C ONCLUSION
This paper presented the novel stable grasping method
for an arbitrary polyhedral object by a multi-fingered handarm system. Firstly, the nonholonomic constraints of rolling
contact were formulated, and the conditions to realize thestable grasping was derived from the stability analysis of
the overall system. Additionally, a new index obtained by
the scalar function Vmin was proposed for evaluating thepossibility of stable grasping. This possibility depends only2269

û x [N]

4
2
0
-2

grasping.
Currently, the position and orientation of the object are notspecified explicitly. However, it would be easily possible tocontrol the position and orientation of the grasped object byreferring to the method proposed by Tahara et al. [17] andBae et al. [18]. In the future works, we would like to perform
some experiments to verify the usefulness of our proposed
method. Furthermore, we will expand the proposed control
scheme for an object with curved surfaces.

ûx1
àx3
ûx2
àx2
û
x3
àx1

0

5

Time[s]1015

û & [N]

0.2
0.1

û&1
à
û&2
à3
û
&3

0
-0.1

à2

-0.2

0

5

10

15

ACKNOWLEDGMENT
This work was partially supported by ”the Kyushu University Research Superstar Program (SSP)”, based on the budgetof Kyushu University allocated under President’s initiative.

Time[s]Angular Velocity [rad/s]

Fig. 8.

External force applied to the object convergence to zero

20

q&11
q&theta_12
q&theta_
13

10
0
-10

theta_-20

Angular Velocity [rad/s]

05Time[s]

10

15

20

q& 21
q&theta_22
q&theta_
23

10
0
-10

theta_

-20

0

5

10

15

Angular Velocity [rad/s]

Time[s]20

q&31
q&theta_32
q&theta_
33

10
0
-10

theta_-20

Angular Velocity [rad/s]

05Time[s]10

15

1

q&theta_a1
q&a 2
theta_
q& a 3
q&theta_a4
q&theta_
a5

0.5
0
-0.5
-1

0

5

10

15

theta_

Time[s]Fig. 9.

History of joint angler velocity of arm and each finger

Velocity [m/s]

0.1
0.05

x&
&
ypos_v
pos_v
z&

0
-0.05

pos_v

-0.1

0

5

10

15

Angular Velocity [rad/s]

Time[s]30

ωxωori_veyωori_ve
z

15
0
-15

ori_ve

-30

0

5

10

15

Time[s]Fig. 10.

History of translational and rotational velocity of grasped object

on the shape of potential function V , and it does notdepend on initial contact positions and soft finger contact
model. The usefulness of the proposed index was verified
through numerical simulations. This means that the proposed
method realizes stable grasping regardless of any externalsensing and that the preshaping of the hand-arm system
for approaching to the object is important to realize stable

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