The 15th International Conference on Advanced Robotics
Tallinn University of Technology
Tallinn, Estonia, June 20-23, 2011

Dynamic Object Manipulation using a Multi-Fingered Hand-Arm System:
Enhancement of a Grasping Capability
using Relative Attitude Constraints of Fingers
Akihiro Kawamura, Kenji Tahara, Ryo Kurazume and Tsutomu Hasegawa

Abstract— In this paper, an enhancement of a dynamic object
grasping and manipulation method, which has been proposedby us previously, is presented. This enhancement makes it
possible to grasp more various shaped objects which could nothave been grasped by our previous method. In our previous
method, a force/torque equilibrium condition to satisfy stableobject grasping has been realized by only using grasping forces
normal to an object surfaces. It is because each fingertip is
soft and hemispheric and then rolling constraints arise during
movement, and these phenomena cannot be stopped while
existing the rolling constraint forces. Therefore, satisfying the
dynamic force/torque equilibrium condition depends not only
on a configuration of the multi-fingered hand system, but also
on the shape of the grasped object. In this paper, a class ofsatisfying the force/torque equilibrium condition is expanded
by generating counter tangential forces to suspend the rolling
motion of each fingertip. In order to generate the counter tangential force, a relative attitude constraint between each finger
is introduced. Firstly, a rolling constraint between each fingertip
and object surface is given. Then, a relative attitude constraint
control signal to generate constantly-produced tangential forces
is designed. Finally, it is demonstrated through numericalsimulations that the proposed control method accomplishes
to grasp arbitrary shaped polyhedral objects and regulate its
position and attitude, simultaneously.

qa5
qa3
qa4

q21
q23
q24

q11

q32

q13
q14

q15

q31

ωy

q33
q34

qa2

x02

z
x03

x01
Fig. 1.

x

ωx

O

qa1

ωz

Multi-fingered hand-arm systemseveral shaped objects are unable to grasp in stable using thismethod. In this method, a dynamic force/torque equilibrium
condition to immobilize the object must be satisfied by only
normal grasping forces component on each fingertip. It is because a tangential grasping force component induces a rolling
motion of each fingertip. Namely, the tangential grasping
force component on each fingertip must be vanished in orderto satisfy a dynamic force/torque equilibrium condition in a
final state. However, this requirement is quite strict, and hard
to be satisfied in case of grasping an arbitrary polyhedral
object. In order to relax the requirement, we add an anothercontrol signal which makes a counter tangential force tosuspend a rolling motion of each fingertip and satisfy adesired force/torque equilibrium condition in a final state.
Tahara et al. have proposed a stable object grasping andmanipulation method that can generate a counter tangential
force at the final state [10], [11]. However, this method
cannot regulate a position and attitude of the grasped object
to each desired value precisely. It is because their method
uses a time integration of each joint angular velocity, andthe integral term generates some deviations from a desired position and attitude of a grasped object. This paper
proposes a novel object manipulation method to enhance
our previously proposed one [8], [9]. A relative attitude
constraint between each finger is introduced in addition to
our previous controller. This artificial constraint makes acounter tangential force to cancel the tangential reaction
force induced by the rolling motion of each fingertip. Using
the artificial constraint forces makes it possible to grasp morevarious shaped objects. In addition, the dynamic stability of

I. I NTRODUCTION
Humans perform dexterous manipulation tasks using theirarms and hands. This ability is one of the most fundamental
and important skills for robots working around humans.
Aiming at this target, many robotic systems and control
strategies for grasping and manipulation of an object havebeen proposed [1–4]. Among them, several methods related
to a multi-fingered hand-arm system has been proposed toaccomplish a dexterous grasping and manipulation tasks likea human hand [5–7]. We also have proposed the dynamic
object grasping and its attitude control method for an arbitrary polyhedral shaped object using a multi-fingered handarm system with soft hemispherical fingertips, so far [8], [9].
This method can realize stable grasping and a desired object
attitude by using a sensory feedback methodology. However,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 Faculty of Engineering, Kyushu University,
Fukuoka, 819-0395, JAPAN tahara@ieee.org
R. Kurazume and T. Hasegawa are with the Faculty of Information
Science and Electrical Engineering, Kyushu University, Fukuoka, 819-0395,
JAPAN {kurazume, hasegawa}@ait.kyushu-u.ac.jp

978-1-4577-1157-2/11/$26.00 ©2011 IEEE

y

q22
q12

8

Center ofContact area contact area

the overall system is ensured.
In what follows, firstly a nonholonomic rolling constraint
between each fingertip and object surface is given. Thisrolling constraint is based on the Arimoto’s model [10–13],
and in this paper we expand Arimoto’s model to the case thatthe object has an arbitrary polyhedral shape [8]. An arbitrary
number of fingers are also allowed in our formulation.
Secondly, Lagrange’s equation of motion for the overallsystem is derived. After that, a control signal for stablegrasping and manipulation of a grasped object with a new
additional term to make a relative attitude constraint between
each finger is designed. Finally, it is demonstrated througha numerical simulation that our control method makes it
possible to grasp the arbitrary shaped polyhedral object withregulating a position and attitude of it. In addition, a detailed
analysis of the stability for the closed-loop dynamics isshown in Appendix.

Center of
finger tip
ObjectωxrZCenter of theobject mass

Fig. 2.

Contact model at the center of each contact area

where R is a rotational matrix which expresses an attitude of
the object in Cartesian coordinates, and RCi is a rotational
matrix which expresses a relative attitude between the object
frame to the contact frames. The y-axis of RCi is signified
by a unit vector C iY that is taken normal to the contact
surface. Now, the rolling constraint can be expressed as
Pfaffian constraints in the following form [8]:






X ix
X iω
X iq
q̇ +
ẋ +
ω = 0,
(2)
Z iq
Z ix
Z iω
where
⎡

T
X iq = Δri C T
iZ J Ωi − C iX J 0i
⎢ X = CT
ix
⎢
iX
⎢
T
⎢ X iω = {C iX × (x − x0i )} − Δri C T
iZ⎢⎢ Z iq = −Δri C T
J Ωi − C T
J 0i
iX
iZ
⎢
⎣ Z ix = C TiZT
Z iω = {C iZ × (x − x0i )} + Δri C T
iX ,

(3)

and Δri is the perpendicular distance between the center of
each fingertip andcontact surface (see Fig. 2). A symbolNJ Ωi ∈ R3×(Na + i=1 Ni ) is the Jacobian matrix for theattitude angular velocity of each fingertip
N with respect tothe joint angular velocity q̇ ∈ RNa + i=1 Ni , and J 0i ∈N
R3×(Na + i=1 Ni ) is the Jacobian matrix for the velocityof the center of each fingertip ẋ0i with respect to the joint
angular velocity, respectively.
B. Contact Model of Soft Finger-Tip
The physical relationship between a deformation of thefingertip at the center of each contact area and its reproducing
force is given on the basis of the lumped-parameterized
model which have been proposed by Arimoto et al [12]. The
reproducing force fi which is normal to the object surface
at the center of each contact area is given as follows:d(ri − Δri )fi = f¯i + ξi dt(4)¯
fi = k(ri − Δri )2 ,A. Constraints
A 3-dimensional rolling constraint with area contact ismodeled here. Now, each contact frame at the center of eachcontact area is introduced. It is given as follows:(C iX , C iY , C iZ ) ,

x0i r
i

Finger tip

In this section, a model of a hand-arm system is given.
An example of the system treated here is illustrated inFig. 1. This system has soft and hemispherical fingertips
and enough number of DOFs to regulate a position andattitude of the grasped object. All fingertips maintain rolling
contact with the object surfaces, and do not slip and detachfrom the surfaces during manipulation. It is assumed thateach fingertip rolls within the ranges of its hemisphericsurfaces, and they do not transfer to another surface during manipulation. Additionally, assume that object attitude
information is available in real time by using some sortof external sensors (e.g. vision sensor, etc.). Note that thegravity effect is ignored in this paper in order to clarify
a physical insight into analyzing physical interaction andstability of the system. As shown in Fig. 1, a symbol O
denotes the origin of Cartesian coordinates, and x0i ∈ R3 isthe center position of each fingertip. Hereafter, a subscripti refers to the ith finger in all equations. The number of
DOFs of the arm and the if inger are Na and Ni , respectively.
A joint angle vector of the arm is q a ∈ RNa . Similarly,
a joint angle vector of the ith finger is q 0i ∈ RNi . The
joint angle vector of the overall system including the armT
and all fingers is q = (q a , q 01 , q 02 , ..., q 0N ) , where N is
the number of the fingers. As shown in Fig. 2, the center
position of each contact area is xi ∈ R3 , and a symbol
Oc.m. denotes the center position of an object mass and theorigin of local coordinates. A position of Oc.m. in CartesianTcoordinates is x = (x, y, z) ∈ R3 . An instantaneousrotational axis of the object at Oc.m. in Cartesian coordinatesTis ω = (ωx , ωy , ωz ) ∈ R3 .

=

xi

rX

II. A M ULTI F INGERED H AND A RM S YSTEM

R·RCi

CiZ
CiY

rY

Δri

CiX

ωi

where ri is the radius of each fingertip, k is a positive
stiffness constant and it depends on the material of thefingertip. Also, ξi is a positive scalar function with respect(1)
9

to Δri , and it indicates that the viscous force depends onthe contact area.
Additionally, we introduce another viscosity force between
each fingertip and the object surfaces that affects toward a
torsional motion around the normal vector C iY . The energy
dissipation function to generate the torsional viscosity force
is given as follows:

rx f i

bi
2
||C T
(5)
iY (ω − ω i ) || ,2
where bi is a positive scalar function which depends on thematerial of fingertip and the contact area, and ω i ∈ R3 is anattitude angular velocity vector of each fingertip expressedby each contact frame.

Rf i

fN

rx f (i+1)
Rf (i+1)

Final State
Relative State

Fig. 3.

Relative state and final state

N

N

(6)

i=1

For the object:NN

T−fi C iY + X T
ix λiX + Z ix λiZ = 0 (7)up = K p

i=1

∂Ti T
T
XT
+
iω λiX + Z iω λiZ = 0, (8)
∂ω
i=1
i=1

N

uo = K owhere S (q, q̇) signifies a skew-symmetric matrix, u stands
for an input torque vector. In addition, λiX and λiZ denote
Lagrange’s multipliers.

+ (r y × r yd ) + (r z × r zd )} ,NN
i=1 ri j=1

JT
0j (xc − x0j ) − C q̇,(11)where Ko > 0 is a positive scalar constant. A desired
attitude of the object is expressed by a rotational matrix
Rd = (r xd , r yd , r zd ). Also r x × r xd is an object attitude
error vector with respect to the x-axis between a present
attitude r x and the desired attitude r xd . The attitude error
vector with respect to other axes y and z can be defined in thesimilar way, and they are mutually orthonormal.Namely, thesummation of these attitude error vectors indicates a desired
instantaneous rotational axis of the object, and the desired
object attitude can be realized if a contact force and momentfrom each finger is added to the object according to thisdesired axis.
Finally, a new control signal ust to constrain a relative attitude of each finger is designed. This control signal generates
a relative attitude constraint force between each finger. This
artificial constraint force plays a role of a counter tangential
force to cancel the rolling constraint force generated by therolling motion (the rolling constraint force is expressed by

In this section, a control signal for stable grasping andmanipulation of an arbitrary polyhedral object is designed.
The proposed controller consists of four parts, the first one isfor stable grasping, the second one is for a position control,
the third one is for an attitude control, and the last one isa newly added term to constrain a relative attitude between
each finger to suspend the rolling motion of each fingertip.
The first, second and third one are the same as previously
proposed controller [8], [9]. The last one is the most different
and remarkable point compared with the previous controller.
Firstly, a control signal for stable grasping us is designed
so that the center of each fingertip approaches each other[8]. It is given as follows:fdJTΩj {(r x × r xd )j=1

III. C ONTROL I NPUT

us =

(10)

where Kp is a positive scalar constant and xd is a desired
position of the object. In addition, we also introduce a control
signal uo to regulate an attitude of the grasped object. It is
given as follows [9]:{C iY × (x − x0i )}fi
i=1
N

N

+

JT
0j (xd − xc ),j=1

NI ω̇ + ω × Iω −

N
where C ∈ R(Na + i=1 Ni )×(Na + i=1 Ni ) > 0 is a positive
definite diagonal matrix that expresses the damping gain foreach joint. Also fd denotes a nominal desired grasping force,Nand xc = N1 i=1 x0i .
Secondly, an object position control signal is designed.
A position of the center of mass for the grasped object iscommonly used as the object position. However, even using
some external sensing methods, it is quite difficult to know
an actual position of the center of mass during manipulation
with no previous knowledge of the grasped object. In thisstudy, we use xc as a virtual object position instead of theactual position of the center of mass. The object position
control signal up is given in the following:

Lagrange’s equation of motion for the overall system considering the rolling constraint explicitly is given as follows:
For the multi-fingered hand-arm system:


N
1
∂Ti T
H (q) q̈ +Ḣ (q) + S (q, q̇) q̇ +
2
∂ q̇
i=1

M ẍ +fu

fst

x0(i+1)ry f i
Rf i,(i+1)rel

C. Overall Dynamics

TT
JT
0i C iY fi + X iq λiX + Z iq λiZ = u,fR

rz f (i+1)

x0i

Ti =

+

ry f (i+1)

rz f i(9)10TABLE I
P HYSICAL PARAMETERS
Triple-fingered hand-arm system

TABLE IID ESIRED GRASPING FORCE AND GAINS
fd
Kp
Ko
Kst
Ca
C1
C2
C3
xd

1st link length
la11.300[m]li10.300[m]la2
1.000[m]li20.200[m]2nd link lengthla30.175[m]li30.140[m]3rd link length
1st mass center
lga1
0.650[m]lgi10.150[m]ndlga20.500[m]lgi20.100[m]2 mass center
lga3
0.0875[m]lgi30.070[m]3rd mass center
1st mass
ma1
1.300[kg]mi1
0.250[kg]ma2
1.000[kg]mi2
0.150[kg]2nd mass
ma3
0.400[kg]mi3
0.100[kg]3rd massst−1
1 Inertia I a1
diag(7.453, 7.453, 0.260)×10 [kg·m2 ]2nd Inertia I a2
diag(3.397, 3.397, 0.128)×10−1 [kg·m2 ]diag(0.291, 0.291, 0.500)×10−1 [kg·m2 ]3rd Inertia I a3
1st Inertia I i1
diag(7.725, 7.725, 0.450)×10−3 [kg·m2 ]diag(2.060, 2.060, 0.120)×10−3 [kg·m2 ]2nd Inertia I i2
3rd Inertia I i3
diag(0.538, 0.538, 0.031)×10−3 [kg·m2 ]Radius of fingertip ri
0.070[m]52
Stiffness coefficient ki
1.000×10[N/m]

Damping function ξi
1.000× ri2 − Δri2 π[Ns/m2 ]Rd

TABLE IIII NITIAL CONDITION
q̇
qa
q 01
q 02
q 03
ẋ
x
ω

ObjectMass mLength of each side
Inertia I

0.061[kg]0.42[m]diag (4.589, 2.592, 4.589) × 10−2 [kg · m2 ]R

f R in Fig. 3). Namely, the dynamic force/torque equilibrium
condition to immobilize a grasped object can be realized
by the artificial tangential constraint force f st because therolling motion of each finger can be suspended by thisartificial constraint forces. The control signal for the relative
attitude constraint between each finger ust is given as
follows:NJT
ust = Kst
Ωi r xf i × r xf i,(i−1)d + r xf i,(i+1)d

Rf i,jrel = RTf jini Rf iini ,(16)

where Rf iini is a rotational matrix between an initial attitudeof ith fingers. A hand-arm system used in the simulation isthat it consists of an arm part which has 5 DOFs and a
triple-fingered hand part which has one 5 DOFs finger andtwo 4 DOFs fingers. The shape of the grasped object is ahexahedron that is made of combining two tetrahedrons, andit cannot satisfy the force/torque equilibrium condition by
only using normal grasping force components. Namely, thisgrasped object cannot be grasped by only using our previous
controller [8], [9]. The parameters of the triple-fingered handarm system and the object are shown in Table I. Table IIshows a desired nominal grasping force, each gain and adesired position and attitude of the grasped object. Table III
shows an initial condition. Figures 4 and 5 show the transient
responses of the object position and attitude. We see fromthese figures that both the position and attitude of the object
converge to each desired value, respectively. Additionally, we
also confirmed that the elements of Δλ∞ and the velocitiesof the overall system converge to zero. These results indicatethat our proposed controller realizes stable grasping and thedesired position and attitude of the object, even it cannot
satisfy the force/torque equilibrium condition by only using
normal grasping forces. In addition to the simulation, we
conduct another simulation without the new control signal
ust . The top figure in Fig. 6 shows an initial state of thesimulation, and the lower two figures show the state at 5
seconds later. We can see the effectiveness of ust to suspend
the rolling motion of the fingertips through Fig. 6.

+r yf i × r yf i,(i−1)d + r yf i,(i+1)d
+r zf i × r zf i,(i−1)d + r zf i,(i+1)d , (12)

where
(13)

Rf i,jd = (r xf i,jd , r yf i,jd , r zf i,jd )= Rf j Rf i,jrel , (j = (i − 1) , (i + 1)) (14)where Kst > 0 is a positive scalar constant. Both Rf i and
Rf i,jd are rotational matrices that express a present anddesired attitude of ith fingertip, respectively. Also Rf i,jrel
is a relative rotational matrix between an attitude of ith andj th fingers. In Fig. 3, f u is a grasping force generated by
the control signal of stable grasping and manipulation of theobject except that generated by the relative attitude constraint
input ust . Therefore, the total control signal u is given as
follows:u = us + up + uo + ust .

0[rad/s](−0.176, −1.700, 1.903, 1.360, 0.519)T [rad](0.014, 0.042, −0.997, 1.590, 0.306)T [rad](0.011, −0.921, 1.218, 1.010)T [rad](−0.056, −0.806, 1.074, 1.075)T [rad]0 [m/s](−0.120, 0.529, 0.775)T [m]0 [rad/s]
⎡
⎤
0.77
0.07
−0.67
⎣ −0.04
1.00
0.05 ⎦
0.67
−0.01
0.74

to keep an initial relative attitude of all fingertips. It is givenin the following:i=1

Rf i = (r xf i , r yf i , r zf i )10.0[N]
3.333
0.119
4.286
diag(1.003, 0.651, 0.735, 0.278, 0.177)×10−1 [Ns·m/rad]diag(0.606, 0.687, 0.786, 0.642, 0.198)×10−2 [Ns·m/rad]diag(0.468, 0.780, 0.318, 0.099)×10−2 [Ns·m/rad]diag(0.648, 0.780, 0.318, 0.099)×10−2 [Ns·m/rad]T⎡(0.000, 0.450, 0.900) [m]⎤0.82
0.21 −0.54⎣ −0.22 0.97
0.05 ⎦
0.53
0.08
0.84

(15)

The stability of the overall system can be verified. It canbe found in Appendix.
IV. N UMERICAL S IMULATION
Numerical simulations to show the effectiveness of proposed controller is conducted here. In the simulations, a desired relative attitude constraint Rf i,jrel in (14) is configured
11

Centroid Position of Fingertips xC [m]1

xd [2]

proposed previously in order to be able to grasp a morevarious shaped object using the relative attitude constraint
between fingers. Firstly, we formulated the nonholonomic
rolling constraint. Next, the new control signal to constrain
the relative attitude between fingers was designed in addition to our previously proposed grasping and manipulation
controller. After that, it was verified through the numericalsimulations that the proposed method realized the desired
position and attitude of the object stably, even in the casethat the force/torque equilibrium condition cannot be satisfied
by only using the normal grasping force component. In
addition, the stability of the overall system was verified byanalyzing the closed-loop dynamics in Appendix. In our
future works, we will conduct some experiments to verify
a practical usefulness of our proposed method. Furthermore,
we will expand the proposed control scheme for an object
with arbitrary smooth curved surfaces.

0.8

xC [2]
0.6

xd [1]
0.4

xC [1]

xC [0]

0.2

xd [0]

0
-0.2

0

5

15

10

20

Time[s]Transient responses of the centroid position of fingertips xC

Fig. 4.

Object Frame R and Desired Frame Rd

1.2

ry [1]

rz [2]

ryd [1]rzd [2]rxd [0]0.8

rx [2]
0.4

rx [0]

rxd [2]rz [1]

ry [0]0

ACKNOWLEDGMENT
This work was partially supported by ”the Kyushu University Research Superstar Program (SSP)”, based on the budgetof Kyushu University allocated under President’s initiativeand Grant-in-Aid for JSPS Fellows.

ryd [0]ryd [2]rzd [1]rx [1]R EFERENCES[1] M. R. Cutkosky, Grasping and Fine Manipulation, Kluwer Academic,
Dordrecht, Netherlands; 1985.
[2] R. M. Murray, Z. Li and S. S. Sastry, Mathematical Introduction to
Robotic Manipulation, CRC Press, Boca Raton; 1994.
[3] A. M. Okamura, N. Smaby and M. R. Cutkosky, “An overview ofdexterous manipulation,” Proc. of the 2000 IEEE Int. Conf. Robot.
Automat., pp. 255–262, SanFrancisco, CA, 2000.
[4] A. Bicchi, “Hands for dexterous manipulation and robust grasping: A
difficult road towards simplicity,” IEEE Trans. Robot. Automat., vol.
16, no. 6, pp. 652–662, 2000.
[5] K. Ohnishi, T. Tajima and Y. Saito, “Anthropomorphic hand control
for robotic hand-arm system,” IEEE Transactions on Systems, Man,
and Cybernetics, , vol.2, pp. 1255–1259, Nashville, TN, 2000.
[6] K. Nagai and T. Yoshikawa, “Grasping and manipulation byarm/multifingered-hand mechanisms,” Proc. of the 1995 IEEE Int.
Conf. on Robotics and Automation, pp. 1040–1047, May 21-27,
Nagoya, Japan, 1995.
[7] J.-H. Bae, S. Arimoto, R. Ozawa, M. Sekimoto and M. Yoshida,“A unified control scheme for a whole robotic arm-fingers system ingrasping and manipulation,” Proc. of the 2006 IEEE Int. Conf. Robot.
Automat., pp. 2131–2136, Orlando, Florida, 2006.
[8] A. Kawamura, K. Tahara, R. Kurazume and T. Hasegawa, “Dynamic
Grasping for an Arbitrary Polyhedral Object by a Multi-fingered HandArm System,” 2009 IEEE/RSJ Int. Conf. on Intelligent Robots andSystems, pp. 2264–2270, St. Louis, Missouri, 2009.
[9] A. Kawamura, K. Tahara, R. Kurazume and T. Hasegawa, “Dexterous
Object Manipulation by a Human-Like Hand-Arm System with Unified Controller,” 1st Int. Conf. on Applied Bionics and Biomechanics,
Venice, Italy, 2010 (to appear).
[10] K. Tahara, S. Arimoto and M. Yoshida, “Dynamic force/torque equilibrium for stable grasping by a triple robotic fingers system,” Proc.
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MO, 2009.
[11] K. Tahara, S. Arimoto and M. Yoshida, “Dynamic object manipulation
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AK, 2010.
[12] S. Arimoto, P.T.A. Nguyen, H.-Y. Han and Z. Doulgeri, “Dynamics
and control of a set of dual fingers with soft tips,” Robotica, vol. 18,
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grasping and manipulation,” Annual Review in Control, Vol. 31, pp.
189–209, 2007.

rxd [1]ry [2]-0.4

rzd [0]rz [0]

-0.8

0

5

10

15

20

Time[s]Fig. 5.

Transient responses of the object frame R = (r x , r y , r z )Initial State

5s later without ust

5s later with ust

Fig. 6. Simulation for a position and attitude control of the grasped object

V. C ONCLUSION
This paper presented the new control signal to enhance thedynamic object grasping and manipulation method we have
12

A PPENDIX

loop dynamics expressed by (17), (18) and (19), we obtaindE = −q̇TC q̇ −Ti + ξΔṙi2 − D ≤ 0
dt
i=1
N

The stability of the system is proved here. The closed-loop
dynamics of the overall system is given by substituting (15)
into (6) such that
For the multi-fingered hand-arm system:H q̈ +

E = K + V + ΔP ≥ 0
1
1
1
K = q̇ T H q̇ + ẋTM ẋ + ωTIω22
2
V = Vs + V p + V o
⎫
⎧
N N
⎬
⎨
A
2
(x0i − x0j )
Vs =
⎭
4N ⎩


N
1
JT
Ḣ + S + C q̇ +0i C iY Δfi
2
i=1

N

+

N

XT
iq ΔλiX +
i=1
N

+

N

ZT
iq ΔλiZ +
i=1

∂Ti
∂ q̇
i=1

T

i=1
N 

+
i=1

1 T
J (B × C iY )Δri 0i



(17)

i=1

For the object:T−Δfi C iY + X T
ix ΔλiX + Z ix ΔλiZ
i=1
N 

xtr −

−
i=1

1
(B × C iY )
Δri


=0

(18)

N

I ω̇ + ω × Iω −{C iY × (x − x0i )}Δfi
i=1

N

N

∂Ti T
+
+
∂ω
i=1
i=1


N
1
+
(x − xi ) × xtr −B × C iY = 0, (19)
Δri
i=1
T
XT
iω ΔλiX + Z iω ΔλiZ0δri = Δrdi − Δri(30)

D = Dst + Do


N
d
T
Kst
Dst =
tr Rf i Rf (i+1) Rf i,(i+1)rel
dt
i=1


N
d
Kst (1 + 2 cos αi )
=
dt
i=1
T

N
Ko d
Do =
{C iY × (x − x0i )} B.
Δri dt
i=1

(31)

N

A = Nfd Δrjj=1⎢⎢ xtr = A (xc − x0i ) + Kp (xd − xc )⎢⎢ B = Kst B st + Ko B o⎢⎢ B st = (r xf j × r xf jd ) + (r yf j × r yf jd )⎢+ (r zf j × r zf jd )⎢⎢⎢ B o = (r xd × r x ) + (r yd × r y ) + (r zd × r z )⎢⎢ Δfi = fi − C T
iY xtr⎢T
1⎣ ΔλiX = λiX + C T
iX xtr + Δri C iZ BT
1ΔλiX = λiZ + C T
iZ xtr − Δri C iX B

(32)
(33)

In (30), Δrdi is Δri when fi equals to fd . In (32), αiT
is a rotational angle of RT
f i Rf (i+1) Rf (i+1)rel Rf irel . The
scalar function V plays a role of an artificial potential energy
arising from the control input. Also K, V and ΔP are
positive as long as 0 ≤ Δrdi − δri < ri , thereby (23) issatisfied. Equation (33) can be rearranged as follows [9]:

where
⎡

(26)

where

N

M ẍ +

(25)

N Kp
2
(xd − xc )
(27)
2

N Ko
2
2
Vo =
(r x − r xd ) + (r y − r yd )
2

2
(28)
+ (r z − r zd )N  δriΔP =f¯i (Δrdi + φ) − f¯i (Δrdi ) dφ, (29)T
CT
iY BJ Ωi C iY

= 0,

(24)

Vp =

i=1

−

(23)

i=1 j=1NJT
Ωi (xi − x0i ) × xtr +

(22)

Do

=

2Ko cos β
i=1dlog Δri ,dt(34)

where β is a rotational angle from an initial attitude to adesired attitude. From (22) and |cos β| ≤ 1, we obtain



N
d
d


E + Dst − 
2Ko log Δri 


dt
dt

(20)

i=1

N

≤ −q̇ T C q̇ −Ti + ξΔṙi2 .

(35)

i=1

Now, we define a new scalar function a as follows:

N
d
a=1
if
i=1 2Ko dt log Δri ≥ 0
a = −1otherwise.

Now, an output vector of the overall system is given as
follows:
T
(21)Λ̇ = q̇ T , ẋT , ω T .

(36)

Then, (35) is rearranged using a as follows:dWdt

By taking a sum of the inner product of (21) and the closed
13

N

≤

−q̇ T C q̇ −
i=1




Ti + ξΔṙi2 ≤ 0,

(37)

N

where

+
N

N

W = E + Kst(1 + 2 cos αi )− a
i=1

i=1
N

2Ko log Δri .(38)1(x − xi )× xtr−
(B × C iY ) ,
Δri
i=1

+

i=1

Furthermore, we obtain(1 + 2 cos αi ) − a

2Ko log Δri

i=1
N




≥ E − N Kst − 
2Ko log Δri  .



i=1

(39)

i=1

The following equation is obtained if the right-hand side of(39) is positive.
W ≥ 0.

(40)

In other words, (40) is satisfied as long as all gains in thecontrol input are designed to satisfy the following equation

N




E − N Kst − (41)
2Ko log Δri  ≥ 0.

i=1

Equations (37) and (40) yield N∞q̇ C q̇ +0Ti + ξΔṙi2

T



dt

i=1

≤ W (0) − W (∞) ≤ W (0) .

(42)Equation (42) shows that the joint angular velocity q̇ (t) issquared integrable over time t ∈ [0, ∞). It indicates q̇(t) ∈L2 (0, ∞). Considering the rolling constraint shown by (2), it
is also clear that ẋ ∈ L2 (0, ∞) and ω ∈ L2 (0, ∞). Thereby,
the output of the overall system Λ̇(t) is uniformly continuous
and it is shown that Λ̇ → 0 and Λ̈ → 0 as t → ∞ [13].
Therefore, it is obvious that the sum of the nominal external
forces affecting on the hand-arm system and the object Δλ∞converge to zero.
Δλ∞ = (Δλq , Δλx , Δλω ) → 0,

(43)

where
N

Δλq =

T
T
JT
0i C iY Δfi + X iq ΔλiX + Z iq ΔλiZ
i=1
N

JT
Ωi (xi − x0i ) × xtr

+

i=1
N 
T
Ko C TiY BJ Ωi C iY −

+
i=1
N

Δλx =


1 T
J (B × C iY ) (44)Δri 0iT−Δfi C iY + X T
ix ΔλiX + Z ix ΔλiZ
i=1
N 

−
i=1
N

Δλω = −

1
(B × C iY )
xtr −
Δri

(46)

where Δλq denotes an external force applied to each jointof the hand-arm system. Also Δλx and Δλω are external
forces affecting on the object.
As a consequence, it is shown that the desired position
and attitude of the object is realized stably, because eachnominal external force and velocity becomes zero.

NN

W ≥ E + Kst

T
XTiω ΔλiX + Z iω ΔλiZ

(45)

{C iY × (x − x0i )}Δfi
i=114