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

Dynamic Force/Torque Equilibrium for Stable Graspingby a Triple Robotic Fingers System
Kenji Tahara, Suguru Arimoto and Morio Yoshida
Abstract— This paper proposes a stable object grasping
method to realize dynamic force/torque equilibrium by using a
triple robotic fingers system with soft and deformable hemispherical fingertips. In the authors’ previous works, “Blind
Grasping” control scheme, which realizes stable object grasping
without use of any external sensing such as vision, force, or
tactile sensing in the case of using a pair of robot fingers,
has been proposed. This control methodology is based on aunique configuration of human hand, called “Fingers-Thumb
Opposability”. In this paper, a ternary finger in addition to apair of fingers is introduced not only to expand a stable regionof grasping, but also to enhance dexterity and versatility ofthe multi-fingered robotic hand system. To this end, a “Blind
Grasping” manner is modified in order to install it in the triple
fingers system. First, dynamics of the triple robotic fingers
system and a grasped object with considering rolling constraints
is modeled, and a control input based on the blind graspingmanner is designed. Next, the closed-loop dynamics is derived
and a stability analysis is shown briefly. Finally, its usefulness
is discussed through numerical simulation results.

I. I NTRODUCTION
A multi-fingered robotic hand system to perform dexterous
manipulation tasks like human has been one of the most
attractive topics in the field of developing anthropomorphic
robotic devices so far. However, it is still now hard to
realize a human-like robotic hand system in the functional
sense. So far, many works related to the object grasping
have been reported [1–5]. Especially, a model for object
grasping with rolling constraints has been treated in several
works [7–9] since Montana had presented one of the rigid
rolling constraint models [6]. In these previous works, modeling and analysis of object grasping have been investigatedmainly from the kinematic viewpoint. Also object grasping
controllers have mostly been designed based on the inverse
dynamics. Therefore, they need the knowledge of an object
mass center’s location in advance, and the exact informationof contact points’ locations between the grasped object andeach fingertip in realtime. In other words, they need nottreat the problem of stability and convergence for closedloop dynamics explicitly under such a strong assumpition
This work was partially supported by Japan Society for the Promotion of
Science (JSPS), Grant-in-Aid for Scientific Research (B) (20360117) and“the Kyushu University Research Superstar Program (SSP)”, based on thebudget of Kyushu University allocated under President’s initiative.
K. Tahara is with the Organization for the Promotion of Advenced Research, Kyushu University, Fukuoka, 819-0395 JAPANtahara@ieee.org
S. Arimoto is with the Research Organization of Scinece andEngineering, Ritsumeikan University, Kusatsu, 525-8577 JAPANarimoto@fc.ritsumei.ac.jp
K. Tahara, S. Arimoto and M. Yoshida are with the RIKEN-TRI Collaboration Center for Human-Interactive Robot Research, RIKEN, Nagoya,463-0003 JAPAN yoshida@nagoya.riken.jp

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

y

qt3 O

qm1
qm2
qm3

qm4

x
qi2

qi1
qi3

x0t
x0m xm x0i xt
xi
Oc.m.
qi4
Y rY
rZ rX
X
ω Z
Fig. 1.

qt1
qt2

qt4
qt5

z

Triple robotic fingers systemthat all the information about the object mass and exact
contact point positions is available in realtime. On the other
hand, Prattichizzo and Bicchi [10] have treated dynamic
analysis of a general manipulation system. Nevertheless,nonholonomic rolling constraints have not yet been taken
into consideration explicitly, and it has not mentioned about
control design. In recent years, Arimoto et al. have proposed
one of the dynamic object grasping methods, which need notuse any contact point information or object mass information,using a pair of soft robotic fingers with considering rolling
constraints [11–13]. Subsequently the authors have improved
it and developed a dynamic object manipulation method,
called “Blind Grasping” based on their works. This control
method is inspired by the unique configuration of human
hand, called “Fingers-Thumb Opposability” [14]. Using thiscontroller, stable object grasping is realized not only without
use of any external sensing, but also without paying highcomputational costs such as calculation of inverse dynamics,
motion planning according to optimization of certain performance indices, or puzzling about illposedness of inverse
kinematics. However, we have assumed that a grasped object
does not spin around the opposition axis arising between
each contact point on the fingertip during movement so far,because the spinning motion is uncontrollable for a pair ofrobotic fingers with opposability. In this paper, we introduce
a ternary finger in addition to a pair of robotic fingers totake away this strong assupmtion. It is easy to expect thatby introducing a surplus finger, a stable region to satisfy thecondition of force-torque equlibrium to immobilize the object
is larger than the case of a pair of robotic fingers because thenumber of wrench vectors is increasing. Moreover, it may
enhance the dexterity and versatility intuitively. However

2257

x0i
Index
Fingerpair of robotic fingers, and we will show the reason why it
can be realized through modeling and analyzing an overall
dynamics which includes some physical interaction between
each fingertip and a grasped object, and some numericalsimulations in this paper.

λYi
fi

x0t λXt

MiddleFingerxM

λXm

II. T RIPLE ROBOTIC F INGERS S YSTEM

x0m

ft
Oc.m. fm

Thumb

x
O

y

Fig. 2.
Centroid xM of the triangle made by the center of eachhemispherical fingertip

in the case of a triple robotic fingers system, especiallylike our proposed fingers configuration as shown in Fig. 1,
each center of contact area between fingertips and the object
surfaces makes a triangle, because the ternary finger (now,it is the index finger) is not at the opposite position fromany other fingers. This geometrical difference is one of themost remarkable points in the case of a triple robotic fingers
system, and thereby the force-torque equiriblium condition
to immobilize the object is completely different. Therefore
in this paper, we modify the blind grasping control signal
to be applicable for our triple robotic fingers system in thefollowing way:
usj = −fd
J0 T (x0j − xM ) − C j q̇ j ,ri + rm + rt j(1)

where x0 j denotes each center of hemispherical fingertips,
rj the radius of each fingertip, and xM the centroid of thetriangle made by each center of the hemispherical fingertip
as shown in Fig. 2. Apparently, the centroid is given in thefollowing way:1
(x0i + x0m + x0t ) ,(2)
3
where q̇ j denotes each joint angular velocity for each finger,
C j > 0 each diagonal positive definite matrix, fd the nominal desired grasping force, and J0 j each Jacobian matrixof each center of the hemispherical fingertips with respect
to each joint angular velocity. The subscript of j = i, m, t
stands for the index finger, middle finger, and thumb in all
equations. Now, let us consider an x-y cross section of ourmodel as shown in Fig. 2. It is well known that in the caseof immobilizing a 2-dimensional rectangular parallelepiped
object, it needs four frictionless fingers [15]. However inour model, each finger can generate not only a normal
grasping force expressed as fi , fm , ft , but also tangential
rolling constraint forces expressed as λYi , λXm , λXt andthereby a force/torque equilibrium to immobilize an object
can be established by using these rolling constraint forceseven though only three fingers are utilized. It is one of
the critically different points compared with grasping by axM =Let us consider a triple robotic fingers system as shownin Fig. 1. This system is composed of a 4 d.o.f. middle
finger and a 5 d.o.f. thumb whose root positions are fixedin an inertial frame and located oppositely to each other,and a 4 d.o.f. index finger whose root position forms
an isosceles triangle together with the root positions ofother fingers. In other words, the index finger is not at
the opposite side from any other fingers. Assume that thegrasped object is a rectangular solid, and the shape of eachfingertip is hemispheric and made of some soft material
such as silicon rubber, and thereby each soft fingertip canmake area contact with the object surfaces. In addition,
each fingertip can perform rolling without slipping on theobject surfaces during movements. Note in this paper thatthe gravity effect is ignored to have a physical insight into
analyzing physical interaction and stability of the system,and to indicate explicitly the difference of the force/torqueequilibrium condition between the case of using a pair ofrobotic fingers and that of using triple robotic fingers. As
shown in Fig. 1, symbol O denotes the origin of Cartesian
coordinates which commonly regarded as the first, secondon third joint center of the thumb. Symbol Oc.m. denotes thecenter of the object mass, and also the origin of object local
coordinates. The position of Oc.m. in Cartesian coordinatesis expressed by x = (x, y, z)T . An instantaneous rotational
axis of the object, which is expressed by the object local
coodinates Oc.m. , is ω = (ωx , ωy , ωz )T and it means thatorientation angular velocities around each axis of Oc.m.
respectively. Also the position of the center of each contact
area in Cartesian coordinates is expressed by xj .
A. Constraints
When each fingertip is rolling on the object surfaces, there
arise nonholonomic rolling constraints during movements.
These constraint models are derived here. The object orientation in Cartesian coordinates is expressed by the rotational
matrix R such that(3)
R = rX rY rZ ∈ SO(3),where rX , rY , rZ ∈ R3 are mutually orthonormal vectors
on the object frame. It is known that this rotational matrix R
is one of the members of the group SO(3), and it satisfies
the folloing relation


0
−ωz
ωy
0
−ωx ,
(4)
Ṙ = R [ω×] , [ω×] = ωz
−ωy

ωx

0

where ω = (ωx , ωy , ωz ) is the angular velocity vector for
the instantaneous rotational axis of the object expressed by
the object local coordinates. Obviously, (4) is not integrable

2258

T

analytically, and it causes one of the nonholonomic constraints with respect to the angular velocities of the object.
On the other hand, the translational velocity of the center of
each contact area on the fingertip vj can be expressed in thefollowing way:⎡vi = (ri − Δri ) (−ṙX + Ωi × rX )⎣ vm = (rm − Δrm ) (−ṙY + Ωm × rY )(5)
vt = (rt − Δrt ) (ṙ Y − Ωt × rY ) ,where Ωj is the orientation angular velocity vector for eachrobotic finger at the center of each contact area, and Δrj
is the fingertip’s displacement at the center of each contact
area such that⎡Δri = (ri + Di ) + rX T (x − x0i )⎣ Δrm = (rm + Wm ) + rY T (x − x0 m )(6)
Δrt = (rt + Wt ) − rY T (x − x0 t ),where Di is the object depth from the object mass center tothe object surface at the index finger’s side, and Wm + Wt
is the object width. Obviously, v j is on the tangential planeat the center of each contact area which is the object surface
itself in this case. The fingertip’s orientation angular velocity
vectors Ωj can be expressed as a linear homogeneous form
with respect to each joint angular velocity such thatΩj = JΩj q̇ j .

(7)

The rolling constraints can be expressed in such a waythat the velocity of the center of each contact area on thefingertips vj equals to that on the object surface such that(ri − Δri )rY T (−ṙX + Ωi × rX ) = Ẏi(8)(ri − Δri )rZ T (−ṙ X + Ωi × rX ) = Żi(rm − Δrm )rX T (−ṙY + Ωm × rY ) = Ẋm(9)(rm − Δrm )rZ T (−ṙY + Ωm × rY ) = Żm(rt − Δrt )rX T (ṙ Y − Ωt × rY ) = Ẋt(10)(rt − Δrt )rZ T (ṙY − Ωt × rY ) = Żt ,

Obj

ect

sur

face

rt -Δrt
λXt
Ωt

rX
rY

ft
X
Y

vt
Z

x0t

xt
rZ
λZt
Δrt
Contact area

Fig. 3.

Contact model at the center of the contact area for the thumb

Di

Wm

-Xm

Y

X

Zi

x0i

Oc.m.

Zm

Z

x0m

-Yi
xt

xm

x

O y

z
Fig. 4. Projected distances for the index finger (Yi , Zi ) and the middle
finger (Xm , Zm ) on each object surface

where Yi , Zi , Xm , Zm , Xt , Zt are the projected distances
between the object mass center Oc.m. and the center of eachcontact area to X, Y and Z axes of the local frame on theobject mass center as shown in Fig. 4. These can be given
as follows:⎡
Zi = −rZT(x−x0 i )Yi = −rY T(x−x0i ),⎣ Xm = −rXT(x−x0m ),Zm = −rZT(x−x0 m ) (11)T
Xt = −rX (x−x0t ),Zt = −rZT(x−x0t ).
Eventually, from (7)∼(11) we obtain(ri − Δri ) −ωz + rZ T JΩ i q̇i = Ẏi(ri − Δri ) ωy − rY T JΩi q̇ i = Żi(rm − Δrm ) ωz − rZ T JΩ m q̇ m = Ẋm(rm − Δrm ) −ωx + rX T JΩ m q̇ m = Żm(rt − Δrt ) −ωz + rZ T JΩt q̇t = Ẋt(rt − Δrt ) ωx − rX T JΩt q̇ t = Żt ,

(12)
(13)
(14)

These (12)∼(14) express nonholonomic rolling constraints
on the object surfaces, and they are in linear homogeneouswith respect to each velocity vector. Therefore, they can be
rewritten as Pfaffian constraints which are given as:
Yq i q̇ i + Yxi ẋ + Yω i ω = 0(15)
Zq i q̇ i + Zx i ẋ + Zω i ω = 0

Xq m q̇ m + Xx m ẋ + Xω m ω = 0(16)
Zq m q̇ m + Zxm ẋ + Zωm ω = 0

Xq t q̇ t + Xx t ẋ + Xω t ω = 0(17)
Zq t q̇ t + Zxt ẋ + Zω t ω = 0,
where⎡
Yq i = −(ri − Δri )rZ T JΩ i + rY T J0 i ,⎢ Yx i = −rY T , Yω i = (Zi , 0, − Di )T⎢⎣ Zq = (ri − Δyi )rY T JΩi + rZ T J0i ,i
Zx i = −rZ T , Zω i = (−Yi , Di , 0)T⎡
Xq m = (rm − Δrm )rZ T JΩ m + rX T J0 m ,⎢ Xxm = −rX T , Xω i = (0, − Zm , Wm )T⎢⎣ Zq = −(rm − Δrm )rX T JΩ m + rZ T J0 m ,m
Zx m = −rZ T , Zω i = (−Wm , Xm , 0)T⎡
Xq t = (rt − Δrt )rX T JΩt + rZ T J0t ,⎢ Xxt = −rY T , Xωt = (0, − Zt , − Wt )T⎢⎣ Zq = (rt − Δrt )rX T JΩ t + rZ T J0 t ,t
Zx t = −rZ T , Zω t = (Wt , Xt , 0)T ,

2259

(18)

(19)

(20)

and J0i , J0m ∈ R3×4 , J0t ∈ R3×5 are the Jacobianmatrices for each x0 j with respect to each joint angle
respectively.
B. Soft Fingertip Model
The lumped-parametrized contact model for the deformation of the fingertip at the center of each contact area which
has been proposed by Arimoto et al. [11] is used here. Thismodel considers only the deformation of normal direction,and doesn’t include that of tangential direction. Namely, we
assume that the deformation of tangential direction doesn’t
occur during manipulation for the sake of simplifying themodel construction. The reproducing force fj (Δrj ) at thecenter of each contact area can be expressed as follows:
fj = f¯j + ξj Δṙj ,f¯j = kΔrj2 ,

T
T T
T
T
T T
X m = (Xq T
m , Xx m , Xω m ) , X t = (Xq t , Xx t , Xω t ) ,T
T
T T
and Z j = (Zq j , Zx j , Zω j ) . In addition, λ =
T
T T
T
T
(λT
i , λm , λt ) and λi = (λYi , λZi ) , λm = (λXm , λZm ) ,T
λt = (λXt , λZt ) denote Lagrange’s multipliers. Eventually,
we obtain Lagrange’s equation of motion such that
For the triple fingers:

where H j denotes the inertia matrix for each finger, M =diag(m, m, m) the mass of the object, and I is the inertia
tensor for the object represented by the principal axes ofinertia. On the other hand, the total potential energy for theoverall system is given as follows:
Δrj

Pfj =

P =
j=i,m,tj=i,m,tf¯j (Δrj )dζ,(24)

M ẍ+(fi −λXm −λXt )rX +(fm −ft −λYi )rY−(λZi +λZm +λZt )rZ = 0∂TjT
I ω̇ + ω × Iω +∂ω
j=i,m,t
 




0
−Zm
Zt
0
+ Zi fi +fm + 0 ft
−Y
X
−Xt
 i
m 


Zi
0
0
+ 0 λYi + −Zm λXm + −Zt λXt
W
−W
−D
 i
 m
 t
−Yi
−Wm
Wt
+ Di λZi + Xm λZm + Xt λZt = 0,

 t1 
t0

δ(K − P ) +uTj δq jdt =

j=i,m,tt1
t0

ξj Δṙjj=i,m,t

t1

+

0

∂Tj
∂Δṙj
+
∂ Λ̇
∂ Λ̇

δΛdt

T
T
T
T
T T
(Y T
i , Z i , X m , Z m , X t , Z t ) λδΛdt,

(25)

t0

0

(29)

(30)

0

where S j and S are skew-symmetric matrices.
Now, taking inner product of the input vector
T
T T
T
T
U = (uTwith the output vector
i , um , ut , 0 , 0 )
T
T
T
T
T T
Λ̇ = (q̇ i , q̇ m , q̇ t , ẋ , ω ) yeilds
TΛ̇ U =d(K + P ) = −(Tj + ξΔṙj2 ) ≤ 0.
dt
j=i,m,t(31)

This inequality expresses that the input-output pair satisfies
the passivity condition [17].

0where Pfj (Δrj ) is the potential energy for each fingertip
caused by the deformation of the fingertip. Eventually, Lagrange’s equation of motion can be obtained by applying thevariational principle in such a way(28)

For the object:

where ηj is a positive scalar function with resprect to Δrjwhich depend on material characteristics of the fingertips.
C. Overall Dynamics
The total kinetic energy of the overall system is given as:
1 T
11q̇ j H j q̇ j + ẋT M ẋ + ωT Iω,
(23)
K=
2
2
2
j=i,m,t1
Ḣm + Sm q̇m +− J0Tm rY fm
2
∂ q̇m
T
+ (Xq Tm , Zq m )λm = um (27)1∂TtT
Ḣ t + S t q̇ t ++ J0 Tt rY f t2∂ q̇ tT+ (Xq Tt , Zq t )λt = ut

H t q̈ t +

(26)

∂TmT

Hm q̈m +(21)

where k stands for a positive constant and ξj is a positive
scalar function regarding Δrj , and these are due to thefingertip’s material. In addition to this nonlinear springdamper model with respect to the normal of each contact
surface, we introduce a damping model with respect to twist
movements between the fingertips and the object surfaces as
the energy dissipation functions in the following way [16]:⎡Ti = ηi rX T (Rω − Ωi )2⎣ Tm = ηm rY T (Rω − Ωm )2(22)
Tt = ηt rY T (Rω − Ωt )2 ,1∂TiT
Ḣ i + S i q̇ i +− J0 Ti rX f i2∂ q̇ iT+ (Yq Ti , Zq i )λi = ui

H i q̈ i +III. D ESIGN OF E XTERNAL S ENSOR -L ESS G RASPINGC ONTROL S IGNAL
Let us design a control signal for stable grasping without
use of external sensing. We have already shown the basic
idea of our controller in (1) and (2) in Section I. In addition
to the control signal, we introduce an adaptive control signal
to compensate the grasping force from the surplus finger asfollows:
rj fd
J T N̂ j ,(32)
uc j = −3(ri + rm + rt ) ωj
where each N̂ j is updated correspoinding to the following
equation such that

T
T
T
T T
where Λ̇ = (q̇ T
i , q̇ m , q̇ t , ẋ , ω ) , uj is the input torqueT
T T
vectors for each finger, and Y i = (Yq T
i , Yx i , Yω i ) ,

2260

rj fd
d
N̂ j =
Γ−1 J ω q̇ ,dt
3(ri + rm + rt ) N j j j

(33)

−1
−1
−1
and ΓN−1
j = diag(γNjx , γNjy , γNjz ) > 0 is the adaptive
estimation gain for each N̂ j . These compensation control
terms play a important role to realize stable grasping in thecase of this triple fingers system. In the case of frictionlessfingers, it needs seven fingers to satisfy the form-closure toimmobilize a 3-dimensional rectangular solid object. However, this system has only three fingers. Furthermore, thegrasping force from the ternary finger is normal to thegrasping forces from other fingers. Therefore, in order torealize force/torque equilibrium to immobilize the object,
each finger should generate rolling constraint forces which
are partially shown in Fig. 2 as λYi , λXm , and λXt . These
rolling constraint forces at the final state cannot be generated
if only using the control signals given as (2), and thereby
these compensation control signals are necessary to generate
the rolling constraint forces at the final state. Eventually, we
design the total control signals to the triple robotic fingers
system according to the principle of linear superposition suchthatuj = us j + uc j .

where
fd
3(ri + rm + rt )ΔNj = J T
ω j(N̂ j − N j )⎡ri −Δri(rY IZ −rZ IY )⎢ Ni =ri⎢rm −Δrm⎢(rZ MX −rX MZ )⎢ Nm =⎢
rm⎣
rt −Δrt(rZ TX −rX TZ ) ,
Nt =
rt

Q=

Δfi = fi −Q·IX ,Δfm = fm −Q·MY⎢ Δft = ft −Q·TY ,ΔλYi = λYi +Q·IY⎢⎢ ΔλXm = λXm +Q·MX , ΔλXt = λXt +Q·TX(42)⎢⎣ ΔλZi = λZi +Q·IZ ,ΔλZm = λZm +Q·MZ
ΔλZt = λZt +Q·TZ ,⎡⎡IX = 2Ri −Xm −Xt
MX = 2Xm −Ri −Xt⎣ IY = 2Yi −Rm +Rt , ⎣ MY = 2Rm −Yi +Rt⎡ IZ = 2Zi −Zm −Zt⎡ MZ = 2Zm −Zi −Zt (43)
TX = 2Xt −Ri −Xm
Ri = ri −Δri +Di⎣ TY = 2Rt +Yi +Rm , ⎣ Rm = rm −Δrm +Wm
TZ = 2Zt −Zi −Zm
Rt = rt −Δrt +Wt⎡
Sωx = Q{−(rm −Δrm )MZ +(rt −Δrt )TZ }⎢ Sωy = Q(ri − Δri )IZ⎢⎢ Sωz = Q{−(ri −Δri )(2Yi −Wm +Wt )(44)⎢⎣+(rm −Δrm )(2Xm −Xt −Di )−(rt −Δrt )(2Xt −Di −Xm )} .

It is remarkable that the control inputs given as (34) arecomposed of only kinematic information, joint angles andangular velocities of the triple fingers system. Substituting
these control signals into (26)∼(30) yeilds the overall closedloop dynamics such that
For the triple fingers:

T
T
T
T T
Now, taking inner product of Λ̇ = (q̇ T
i , q̇ m , q̇ t , ẋ , ω )and (35)∼(39) yeilds

H i q̈ i +

(35)

1
∂TmT
−J0Tm rY Δfm
Hm q̈m + Ḣm +Sm +Cm q̇m +2∂ q̇m+(XqTm , ZqTm )Δλm − rmQΔNm = 0
(36)
∂TtT
1
Ḣ t +S t +C t q̇ t ++J0Tt rY Δft2∂ q̇t+(XqTt , ZqTt )Δλt +rt QΔN t = 0

(41)

and
⎡

(34)

1
∂TiT
Ḣ i +S i +C i q̇ i +−J0Ti rX Δfi
2∂ q̇ i+(YqTi , ZqTi )Δλi −ri QΔN i = 0

(40)

d
E
dt

=

E

=

V

=

ΔP

=

Ht q̈ t +(45)

j=i,m,t(37)

For the object:

2(q̇ T
j C j q̇ j + Tj + ξΔṙj ) ≤ 0−

K + ΔP + V ≥ 0
(46)

Q 2
2
2
2
2
2
IY +IZ +MX +MZ +TX +TZ
4
1 T
N̂ ΓNj N̂j ≥ 0
(47)
+
2 j
j=i,m,tδrjf¯j (Δrdj + ξ) − f¯j (Δrdj ) dξ, (48)j=i,m,t 0

M ẍ + (Δfi − ΔλXm − ΔλXt )rX+(Δfm − Δft − ΔλYi )rY− (ΔλZi + ΔλZm + ΔλZt )rZ = 0
S 
ωx
∂TjT
+ Sωy
I ω̇ + ω × Iω +∂ωSωz
j=i,m,t
 




0
−Zm
Zt
0
0
Z
+ i Δfi +
Δfm +
Δft
−Y
X
−Xt
 i
m 


Zi
0
0
+ 0 ΔλYi + −Zm ΔλXm + −Zt ΔλXt
W
−W
−D
 i
 m
 t
−Yi
−Wm
Wt
+ Di ΔλZi + Xm ΔλZm+ Xt ΔλZt = 0,
0

0

(38)

where δrj = Δrj − Δrdj and it is positive definite in δrj
as long as 0 ≤ Δrdj + δrj < rj . By choosing adaptive gain
ΓN j adequately, the scalar function V ≥ 0 should have aminimum value Vmin in a neighborhood of the initial stateΛ(0) and thereby the scalar function E also has the minimum
value Emin in the following way:
Emin = K + ΔP + V − Vmin ≥ 0.

(49)

Therefore, from (45) and (49) we obtain the following
relation such that
∞
0

(39)

0

2261

2
(q̇ T
j C j q̇ j + Tj + ξΔṙj )dt

j=i,m,t≤ Emin (0) − Emin (t) ≤ Emin (0). (50)TABLE IThis means that each joint angular velcity q̇ j (t) is squared
integrable over time t ∈ (0, ∞), that is, q̇ j (t) ∈ L2 (0, ∞),and by considering six rolling constraints expressed by(12)∼(14), it is possible to see that ẋ ∈ L2 (0, ∞) andω ∈ L2 (0, ∞). Then the velocity vector Λ̇(t) becomes
uniformly continuous in time t, and finaly we can obtainthat Λ̇ → 0 at t → 0, and also we see Λ̈ → 0 as t → 0 [18].
These imply that

PARAMETERS FOR NUMERICAL SIMULATIONS
Triple robotic fingers
1st link length lj1
2nd link length lj2
3rd link length lj3
1st link mass center lgj1
2nd link mass center lgj2
3rd link mass center lgj3
1st link mass mj1
2nd link mass mj2
3rd link mass mj3
1st link inertia Ij1
2nd link inertia Ij2
3rd link inertia Ij3
Radius of fingertip rj
Stiffness coefficient kj
Damping function ξj
Damping function ηj

ΔD∞ = (Δf T , ΔλT , ΔN T )T → 0 as t → ∞, (51)⎡Δf = (Δfi , Δfm , Δft )TT
T T⎣ Δλ = (ΔλT
i , Δλm , Δλt )T
TT
ΔN = (ΔN i , ΔN m , ΔN Tt ) .
Equation (51) means that overall external forces which
affect both the fingers and the object converge to zero whentime t → ∞. This system is highly redundant and thereby
it should carefully treat the convergence of overall statevariables. Nevertheless, the convergence could be proven byintroducing the concept of “Stability on a manifold” [18] thatis omitted here. Eventually, the force/torque equilibrium forthe object is satisfied, and the object is thereby immobilized
in the dynamical sense.

ObjectMass mHeight h
Width Wm + Wt
Depth Di + d
Ineatia I

0.05 [kg]0.035 [m]0.02 [m] (Wm = Wt = 0.01 [m])0.035 [m] (Di = 0.0175 [m])diag(0.68, 1.02, 0.68)×10−5 [kg·m2 ]Desired grasping force and each gainfd
1.0 [N]
Ci, Cmdiag(1.5, 1.5, 1.0, 0.8)×10−3
Ct
diag(1.5, 1.5, 1.5, 1.0, 0.8)×10−3
ΓN i , ΓN m , ΓN mdiag(1.0, 1.0, 1.0)×10−4

IV. N UMERICAL S IMULATION
The parameters used in the simulations are given in Table
I. Figure 5 shows the simulation graphics for realizationof force/torque equilibrium to immobilize the object by the
triple robotic fingers system, and Fig. 6 shows the transient
responses of overall external force values ΔD∞ . We see
from Fig. 6 that all componets of ΔD∞ converge to zero. Itmeans that any external forces for each finger and the object
converge to the equilibrium state eventually. Moreover, Fig.
7 shows the transient responses of overall velocity vector
T
T
T
T T
Λ̇ = (q̇ T
j , q̇ m , q̇ t , ẋ , ω ) . We can confirm that from
Fig. 7, overall velocity vector Λ̇ converge to zero at the
final state. From these results, the force/torque equilibrium

0.05 [m]0.03 [m]0.02 [m]0.025 [m]0.015 [m]0.010 [m]0.05 [kg]0.03 [kg]0.02 [kg]diag(1.04, 1.04, 0.06)×10−5 [kg·m2 ]diag(2.25, 2.25, 0.38)×10−6 [kg·m2 ]diag(0.67, 0.67, 0.25)×10−6 [kg·m2 ]0.010 [m]3.0 × 105 [N/m2 ]500×(2rj Δrj −Δrj2 )π [Ns/m2 ]0.1×(2rj Δrj −Δrj2 )π [Nms]Initial conditionq̇ i , q̇m , q̇t
qi
qm
qt
ẋ
x
ω
R
N̂ i , N̂ m , N̂ t

0 [rad/s](0.0, −0.91, −0.83, −0.25)T [rad](0.0, 1.09, 0.95, −0.35)T [rad](0.0, 0.0, 2.05, −0.95, −0.35)T [rad]0 [m/s](0.005,0.025,0.085) [m]
0 [rad/s]
I30

to immobilize the object is established in a dynamic mannerby using the proposed control scheme.
V. C ONCLUSION
This paper has proposed a stable object grasping methodby modifying the blind grasping control signal in order toutilize it for the triple robotic fingers system. The control
signal can be constructed only by using measurement dataof finger joint angles and angular velocities, and referring
to fingers’ kinematics. Its effectiveness in realization of
the force/torque equilibrium in object grasping is discussed
theoretically with the convergence analysis and confirmed
through numerical simulations. In this paper, we treated thespecific grasped object that is a rectangular solid. However,
our control scheme can be aplicable to other arbitrary polyhedral objects easily. The modeling and analysis for grasping ofan arbitrary polyhedral object will be treated in another paper[19]. In the next step of the work, we would consider anarbitrarily shaped 3-dimensional object which is constructedby only smooth curved surfaces.
ACKNOWLEDGMENT

Fig. 5. Simulation graphics for the realization of force/torque equilibriumof the object by the triple robotic fingers system

This work was partially supported by Japan Society for
the Promotion of Science (JSPS), Grant-in-Aid for Scien-2262

Angular Velocity [rad/s]

Δ f [N]

1

Δfi

0.5

Δfm

0

Δf

t

-0.5
-1
0

1

2

3

4

5

5

dqi1
dqi2
dq

i3

0

dqi4
-5
0

1

2

3

1

Δλ

Δ λ [N]

Yi

0.5

Δλ

0

ΔλXt

Xm

Δλ

Zi

-0.5
-1
0

ΔλZm
1

2

3

4

5

Δλ

Zt

dq

m1

dq

2

m2

dq

m3

0

dq

m4

-2
-4
0

1

2

3

Angular Velocity [rad/s]

ΔN i [Nm]

ΔNi1

0.1

ΔNi2

0

ΔN

-0.1

ΔNi4

i3

2

3

4

5

dq

t1

2

dqt2

0

dqt3

-2

dqt5

-4
0

dqt4

1

3

ΔN

0

ΔN

-0.5

ΔNm4

m2
m3

3

4

0.02
0
-0.02
0

5

1

2

Angular Velocity [rad/s]

ΔN t [Nm]

ΔNt1

0.5

ΔNt2

0

ΔNt3
ΔNt4

-0.5

ΔNt5
2

3

4

5

Time [s]

1

1

5

dx
dy
dz

Time [s]

-1
0

4

0.04

ΔNm1

Velocity [m/s]

ΔNm [Nm]

1

2

2Time [s]

0.5

1

5

4

Time [s]-104Time [s]

0.2

1

5

4

Time [s]

-0.2
0

4

Time [s]Angular Velocity [rad/s]

Time [s]

3

4

5

0.5

ωx

0

ωy

-0.5

ωz

-1
-1.5
0

Time [s]

1

2

3

4

5

Time [s]Fig. 6. Force/torque equilibrium realization to immobilize the object by
the convergence of external force/torque values ΔD∞ to zerotific Research (B) (20360117), and “the Kyushu University
Research Superstar Program (SSP)”, based on the budget of
Kyushu University allocated under President’s initiative.
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