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

Pinching 2D Object with Arbitrary Shape by Two Robot Fingers
under Rolling Constraints
Morio Yoshida, Suguru Arimoto, and Kenji Tahara

O

Abstract— Modeling of pinching an object with arbitrary
shape by a pair of robot fingers with hemispherical ends
in a horizontal plane is proposed in a mathematical andcomputational manner. Since the curvature of an object contourwith an arbitrary curve is variable according to the changeof the contact point between the object surface and the rigid
finger tip, the arclength paremeter “s” explicitly appears in
the overall fingers-object dynamics. It is shown that the overall
fingers-object system should be accompanied with the firstorder differential equation of the parameter “s” that includes
the curvatures of both the object contour and finger-tip curve.
A control input, which is of the same category as the control
input called “blind grasping” appeared in our former papers,
is utilized for the realization of stable grasp. The control input
does neither need to use the kinematic information of the object
nor use any external sensing. Finally, numerical simulations arecarried out in order to confirm the effectiveness of our proposedmodel and control input.

O

,

x
q21

q11
y
P1 Om

q12
q13

Q1

Fig. 1.

ϕ1 ϕ2

Q2

X

q22

P2
Y

A pair of robot fingers pinching an object with arbitrary shape

I. I NTRODUCTION
The fact that humans easily and smoothly manipulate
many kinds of objects has attracted many people since themotion of this movement includes the intriguing properties;
finger-thumb opposition, redundancy of finger joints, arbitrary shape of a finger tip or a pinched object, and soft androlling contact between a finger tip and an object surface.
These properties are beneficial and effective to realize
the robots used in daily life, and robotic researchers havedevoted their time to analyze, simulate, and create thefinger robots with these properties [1] [2] [3]. Most of
explorations in robotics, however, remain in kinematics or
motion planning for the realization of force-torque closure
to accomplish stable grasp. The rolling geometry between
two surfaces of arbitrary shape was discussed in detail [4][5]. This research focused on semi-dynamic or kinematic
approaches and did not show any explicit forms representing
the physical interaction between two bodies in the wrench
space. Researchers in multibody dynamics propose manymodels in a dynamic sense but do not consider rolling contact
in our sense [6] [7]. They have missed an important aspectof physical interaction between two bodies under rolling
constraints in a dynamic sense, despite the strong demandThis work was partially supported by Japan Society for the Promotionof Science (JSPS), Grant-in-Aid for Scientific Research (B) (20360117).
M. Yoshida, S. Arimoto, and K. Tahara are with the RIKEN-TRICollaboration Center for Human-Interactive Robot Research, RIKEN,
Nagoya, Aichi, 463-0003, Japan yoshida@nagoya.riken.jp
S. Arimoto is with the Department of Robotics, Ritsumeikan University,
Kusatsu, Shiga, 525-8577, Japan arimoto@fc.ritsumei.ac.jp
K. Tahara is with the Organization for the Promotion ofAdvanced Research, Kyushu University, Fukuoka, 819-0395, Japantahara@ieee.org

978-1-4244-3804-4/09/$25.00 ©2009 IEEEfor unveiling the correlated mechanism to realize interactive
robots with human movements.
Arimoto et al. [8], around 2000, proposed the 2-D dynamic
pinching model by a pair of robot fingers with hemisphericalends under rolling constraints, by the aid of the assumptionthat the velocity of the finger tip in the tangent direction must
be equal to the velocity on the object surface when the shapeof the pinched object is limited to flat surfaces. This model
was extended in 2006 to a three dimensional model [9]. A
mathematical model was derived, as a set of equations of
motion of the fingers-object system under Pfaffian constraints
due to rolling constraints, and the equations representing
infinitesimal rotations of the object. The control input called“blind grasping”, which need not use object information or
external sensing, comfirms stable grasping. Because of thelack of the total number of wrench vectors on the object,spinning motion arises around the opposing axis by two
contact points between finger tips and object surfaces. This
proposed model has six wrench vectors on the object but
seven wrench vectors are needed to satisfy force/torquebalance in a three dimensional case. A physically faithfulviscosity model [10] was introduced to blunt the spinningmotion. However, all models proposed by Arimoto et al. [9]were restricted to the flat surfaces.
In this paper, a new model of pinching an object with
arbitrary shape by a pair of robot fingers with hemisphericalends in a horizontal plane is proposed in a mathematicaland computational manner (see Fig.1). A control input forrealizing stable grasping, which is of the same category of
the control input called “blind grasping”, is proposed. In

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θ

the case of rolling contacts with an object with arbitrary
shape, its contour curvature varies according to the changeof the contact and then the arclength parameter “s” must
be taken into account to describe movement of the contact
point along the object’s contour curve. The overall fingersobject system is shown to be governed by Euler-Lagrange’s
equations parametrized by the arclength parameter and at
the same time the first order differential equation of theparameter “s” is derived. It is noted that Lagrange’s equations representing the overall fingers-object system and the
update equation of “s” should be solved simultaneously sothat they satisfy the principle of causality. Finally, numericalsimulator is constructed based upon our proposed model,
and by applying our proposed control signals, numericalsimulations are carried out for validating of the model andthe effectiveness of the control inputs.
II. DYNAMICS
Modeling of pinching a rigid object with arbitrary shape by
two robot fingers with 3 DOFs and 2 DOFs is schematicallyshown in Fig.1. In the coordinate system, numerical valuesof all angles are positive in counterclockwise direction. The
finger tips of two robot fingers are of hemispherical shapeand rigid. We introduce the local coordinate Om -XY fixedat the object frame, and define unit vectors rX on the X axis
and rY on the Y axis (see Fig.2). The left-side contour of theobject is expressed by a curve attached to the local coordinate (X(s1 ), Y (s1 )) (see Fig.3), and similarly the right-side
contour by a curve with the local coordinate (X(s2 ), Y (s2 ))(see Fig.4), with the aid of arclength parameters si (i = 1, 2).
P1 is the contact point between the finger tip and the object
surface, b1 the tangent unit vector of both the finger tip
and the contour of the object at contact point P1 , and n1
the normal unit vector to the tangent vector b1 . The angle
between the normal unit vector n1 and the X axis is denotedby θ1 , and is determined as follows:θ1 (s1 ) = arct (X  (s1 )/Y  (s1 ))



X
rX

Om =(xm ,ym)n1

1Q1 s=0Y (s)1θ1rYYP1 =(x1 ,y1) P1
O01(x01,y01)

Fig. 2.

θ1 b1
YRelationship between local coordinate O-XY and fingertip O01

O

x

θ

y

P1

1
Q s=0
1

n1

r1 P1

(1)

O01

Fig. 3.

(2)X
lb1(s)1
Om =(x, y)ln1(s)1 Y1

where X  (s1 ) = dX(s1 )/ds1 and Y  (s1 ) = dY (s1 )/ds1 .
Similarly θ2 (see Fig.4) is determined as follows:θ2 (s2 ) = arct (X  (s2 )/Y  (s2 ))y

Y (s)1

θ1
X (s)1
θ

P1

θ1 b1
Y

y

b1

Y

Geometrical relationship based on length parameter “s1 ”

where X (s2 ) = dX(s2 )/ds1 and Y (s2 ) = dY (s2 )/ds2 .
P1 P1 is expressed in the local coordinate (X(s1 ), Y (s1 )) as
follows:
P1 P1 = ln1 (s1 ) = −X(s1 ) cos θ1 (s1 ) + Y (s1 ) sin θ1 (s1 )(3)Similarly the contact constraint of the right side of the object
is derived as follows:
Q2

On the other hand, P1 P1 is represented in the inertia frame
O-xy as follows:
P1 P1

=(x − x01 ) cos(θ + θ1 )−(y − y01 ) sin(θ + θ1 (s1 )) − r1

(4)=−(r2 + ln2 (s2 ))(6)

ln2 (s2 ) =X(s2 ) cos θ2 (s2 ) − Y (s2 ) sin θ2 (s2 ) (7)Om P1 is expressed in the local coordinate (X(s1 ), Y (s1 ))as follows:= −(x − x01 ) cos(θ + θ1 (s1 ))−(y − y01 ) sin(θ + θ1 (s1 ))= −(r1 + ln1 (s))(x − x02 ) cos(θ + θ2 (s2 ))−(y − y02 sin(θ + θ2 (s2 )))where

Then, the contact constraint of the left side of the object isderived as the holonomic constraint:
Q1

=

(5)

1806

Om P1 = lb1 (s1 ) = X(s1 ) sin θ1 + Y (s1 ) cos θ1

(8)

P2

θ
θ2

O02

θ+θ2
θ

Om
θ2

ln2

R2 (t)

ϕ2lb2 (s2 ) =

X
x

b2

= −(x − x02 ) sin(θ + θ2 (s2 ))−(y − y02 ) cos(θ + θ2 (s2 )) = lb2 (s2 ) (16)b2n2

θ + θ2

Similarly the rolling constraint at the right hand finger isderived as follows:∂∂φ2+ R2 (t) = 0
(15)
r2
∂t
∂t
where

ϕ2

θ2

X(s2 ) sin θ2 (s2 ) + Y (s2 ) cos θ2=−π + (θ + θ2 ) − (q21 + q22 )=−π + (θ + θ2 ) − p2

(18)

and p2 = q21 + q22 . Since eq.(15) can be integrated as
discussed in eq.(10) (see [11]), we similarly defineYR̃2 = R2 (t, s2 (t)) − R2 (0, s2 (0))θ
yFig. 4.

YR2 (t, s2 ) = −r2 {θ + θ2 (s2 ) − p2 } + s2 + R2 − lb2 (s2 ) (20)and ∂R2 (t, s2 )/∂t = 0 leads to eq.(15). The following
equation can be confirmed (see [11])On the other hand, Om P1 is represented in the inertia frame
as follows:= −(x − x01 ) sin(θ + θ1 )−(y − y01 ) cos(θ + θ1 ) = lb1 (s1 )(9)Rolling contact means that the robot finger tip rolls on theobject surface without slipping. In other words, the object’s
velocity along vector b1 at contact point P1 must be equal
to the finger tip’s velocity along vector b1 at the point, atinstant t, as follows:
r1

∂
∂φ1
+ R1 (t) = 0
∂t
∂t

=

π + (θ + θ1 ) − (q11 + q12 + q13 )=π + (θ + θ1 ) − p1T− cos(θ + θ1 ), −ln1 ) (dq 1 , dx, dy, dθ) = 0 (22)T

where b1 (θ)=(sin(θ + θ1 ) , cos(θ + θ1 )) ,dq 1 =(dq11 ,dq12 ,dq13 )T , J1 =∂(x01 , y01 )/∂(q11 , q12 , q13 ), and e1 =(1,1,1)T .
Similarly, eq.(12) is reduced to T
J2 (q2 )b2 (θ) + r2 e2 , − sin(θ + θ2 ),T− cos(θ + θ2 ), ln2 ) (dq 2 , dx, dy, dθ) = 0 (23)(11)

where b2 (θ)=(sin(θ + θ2 ) , cos(θ + θ2 )) ,dq 2 =(dq21 ,dq22 )T ,J2 =∂(x02 , y02 )/∂(q21 , q22 ), and e2 =(1,1)T .
By associating Lagrange’s multipliers fi with the constraints Qi =0 (i=1, 2)(eqs.(5) and (6)), we define a Lagrangian: 1
1q̇iT Gi (qi )q̇ i + M (ẋ2 + ẏ 2 )
L =
2
2
i=1,2

T

where p1 = q11 + q12 + q13 . Eq.(10) can fortunately be
integrated in the sense of Frobenius (see [11]). In fact, we
defineR1 (t, s1 ) = r1 {θ + θ1 (s1 ) − p1 } + s1 + R1 − lb1 (s1 ) (12)and see that ∂R1 (t, s1 )/∂t = 0 is reduced to eq.(10). It isfound out that (see [11])∂R1∂R1 ds1
dR1
=
+
=0
dt
∂t
∂s1 dtdR2∂R2∂R2 ds2
=
+
=0
(21)
dt
∂t
∂s2 dtEq.(9) is reduced to the constraint form in terms of infinitesimally small variation:
 T
J1 (q1 )b1 (θ) − r1 e1 , − sin(θ + θ1 ),(10)

where ϕ1 is defined as follows (see Fig.1):
ϕ1

(13)

1
+ I θ̇2 − f1 Q1 − f2 Q2(24)2
where q 1 = (q11 , q12 , q13 )T ,q 2 = (q21 , q22 )T ,Gi (qi ) denotes
the inertia matrix for finger i (i = 1, 2), M and I denotethe mass and inertia moment of the object. Due to utilizing
the Lagrangian and associateing Lagrange’s multipliers λiwith eqs.(22) and (23), the dynamic equations of the overall
fingers-object system are derived as follows:

It is possible to defineR̃1 = R1 (t, s1 (t)) − R1 (0, s1 (0))

(19)

where

Geometrical relationship of the right side of an object

R1 (t)(17)I θ̈ − f1 lb1 (s1 ) + f2 lb2 (s2 )
(14)

1807

−λ1 ln1 (s1 ) + λ2 ln2 (s2 ) = 0

(25)

O

M ẍ − f1 cos(θ + θ1 ) + f2 cos(θ + θ2 )−λ1 sin(θ + θ1 ) − λ2 sin(θ + θ2 ) = 0(26)

M ÿ + f1 sin(θ + θ1 ) − f2 sin(θ + θ2 )−λ1 cos(θ + θ1 ) − λ2 cos(θ + θ2 ) = 0

(27)



y
Om =(x, y)

s=0
1



1
G1 (q1 )q̈1 +Ġ1 + S1 q̇1 + f1 J1T (q1 )n1 (θ)
2+λ1 J1T (q1 )b1 (θ) − r1 e1 = u11
G2 (q2 )q̈2 +Ġ2 + S2 q̇2 + f2 J2T (q2 )n2 (θ)
2+λ2 J2T (q2 )b2 (θ) + r2 e2 = u2

x

X
s=0
2

(X (s)1 ,Y (s)1 )(28)(X(s2 ),Y(s2))

(29)
Y

where



Fig. 5.
Local Coordinates (X(si ), Y (si )),(i = 1, 2) used in thecos(θ + θ1 )− cos(θ + θ2 )n1 (θ) =, n2 (θ) =, (30) simulation− sin(θ + θ1 )sin(θ + θ2 )and ui (i = 1, 2) stand for control inputs.
Since the arclength parameters si (i = 1, 2) depend on
the time parameter t, the parameters si (i = 1, 2) should be
updated as follows:ri
dsi=(ṗi − θ̇), i = 1, 2
dt
1 + ri κi (si )(31)

where κi (si )(i = 1, 2) denote the curvature of the object
contours at contacts as follows:








κi (si ) = X (si )Y (si ) − X (si )Y (si ),i = 1, 2(a) Initial pose

(32)

Fig. 6.

It is interesting to note that the curvatures κi (si )(i = 1, 2) ofthe object contours appear in the update equations (eq.(31))
but not in the overall Lagrange’s equations (eqs.(25) ∼ (29)).
III. C ONTROL S IGNAL
Since a family of control inputs called “blind grasping” [9]neither need use the kinematic information of the object nor
any external sensing, that is, we do not need to consider the
shape of the object, we can apply the concept of our previous
control input [9] to this stabilization problem of grasping of
the object with arbitrary shape. Therefore a control input,
realizing stable grasping with an object with arbitrary shape,is proposed as follows:fd
x01 − x02
JiT (qi )ui = −ci q̇i + (−1)iy01 − y02
r1 + r2−ri N̂i ei ,i = 1, 2

Motion of pinching a 2-D object with arbitrary shape

IV. N UMERICAL S IMULATION
We construct a numerical simulator based on our proposedmodel and physical parameters of the fingers-object systemgiven in Table I. Numerical simulations are carried out by
applying our proposed control inputs (eq.(33)) using theparameters of control gains given in Table II and the initial

(33)

where
N̂i (t) = γi−1 ri (pi (t) − pi (0)) ,(b) After 5 seconds

i = 1, 2, (34)γi and ci (i = 1, 2) are positive constants, and pi (0) initialvalues of pi (t). The first term of the right hand side of eq.(33)
plays a role of damping. The second term is a signal based
upon the opposable force between O01 and O02 . The third
term compensates the difference among initial poses of theoverall system.

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TABLE IP HYSICAL PARAMETERS OF THE FINGERS AND OBJECT.
l11 = l21 = l22
l12
l13
m11
m12
m13
m21
m22
r0
ri (i = 1, 2)LM

lengthlengthlengthweightweightweightweightweightlink radius
radiusbase length
object weight

0.065 [m]0.039 [m]0.026 [m]0.045 [kg]0.025 [kg]0.015 [kg]0.045 [kg]0.040 [kg]0.005 [m]0.010 [m]0.063 [m]0.040 [kg]TABLE IIPARAMETERS OF CONTROL SIGNALS .
fdcγi (i = 1, 2)internal force
damping coefficient
regressor gain

1.000 [N]0.006 [Nms]0.001 [s2 /kg]

TABLE III
0θ

I NITIAL VALUES

f1

35.000 × π/18[rad]7.168 × π/18[rad]5.545 × π/18[rad]−4.008 × π/18[rad]−10.82 × π/18[rad]0.000 × π/18[rad]3.243 × 102 [m]7.500 × 102 [m]0.005 [m]0.001 [m]

-10

f1[N]

initial angle
initial angle
initial angle
initial angle
initial angle
initial angle
initial point
initial pointinternal arclength parameterinternal arclength parameterθ [deg]q11 (0)q12 (0)q13 (0)q21 (0)q22 (0)
θ(0)
x(0)
y(0)
s1 (0)
s2 (0)

2.5

-20

-30

2
0

1

2

3

4

0

5

1

2

3

4

5

time[s]time[s]Fig. 7. Trajectory of object angle Fig. 8. Trajectory of normal force
f1
θ
1.5

λ 1[N]
0.5

λ1

2
0

(35)

3

4

5

2

3

4

5

time[s]

1

1

0.5

<

0.5

<

N1

λ2

0
0

1

2

3

4

0

5

0

time [s]

1

2

3

4

5

time[s]Fig. 11. Trajectory of tangential
force λ2

Trajectory of N̂1

Fig. 12.

<

0.01

N2
0

<

-0.5

0.008

0.006

s1
0.004

-1

0

1

2

3

4

0

1

2

5

time[s]3

4

5time[s]Fig. 14. Trajectory of arclengthparameter s1

Trajectory of N̂2

V. C ONCLUSION

1809

1

Fig. 9. Trajectory of normal force Fig. 10. Trajectory of tangential
f2
force λ1

Fig. 13.
0.002

.

3

2

q11,q12,q13[rad/s]

0

q11
q12
q13
The value of zero

. .

s2

s2[m]

In this paper, a 2-D pinching model by two robot fingers ofan object is derived mathematically and tested numerically toensure whether the model works with physical reality. Since
the curvature of the object contour is variable according to
the change of the contact point between the object surface
and the rigid finger tip, the arclength paremeter “s” istaken into account to describe motion of the overall fingersobject system. The overall fingers-object system should beaccompanied with the first-order differential equation of thearclength parameter “s”. The dynamical equations that generate motion of the overall fingers-object system and the update
equations of arclength parameters should be numericallyintegrated simultaneously so that they satisfy the principle ofcausality. It is shown that the proposed control input called

0time[s](37)(38)

It is noted that, since the si (i = 1, 2) are arclengthX  (si )2 + Y  (si )2 = 1,(i = 1, 2). In theparameters,case, since the object contour in the right side is flat, thecurvature κ2 (s2 ) becomes zero. Their initial values must
be determined to satisfy the rolling and contact conditions(eqs.(5),(6),(12) and (20)), in order to carry out the simulations. The constraint stabilization method (CSM) [12] canbe used to maintain the rolling and contact constraints.
The motion obtained by the simulation is depicted in Fig.6.
Figs.8 ∼ 11 and 16 ∼ 20 show that all velocities of the
dynamic equations (eqs.(25) ∼ (29)) converge to zero, andthat all Lagrange’s multipliers converge to some constantvalues. These results show that motion of the overall fingersobject dynamics converges to some equilibrium state, andstable grasping is eventually realized from the viewpoint of
numerical simulation.

2

N1[N]

(36)

1

s1[m]

X(s2 ) =Y (s2 ) =2.5

λ 2[N]Y (s1 ) =

1

N2[N]

X(s1 ) =1 + 4 × 502 × s21[m]−0.03 +
2 × 50Asinh(2 × 50 × s1 )[m]
2 × 500.015 [m]s2 [m]

3

f2[N]values given in Table III in order to confirm the stabilityof motion of the overall fingers-object system (eqs.(25)∼ (29)). The curves c(si ),(i =1, 2) with local coordinates(X(si ), Y (si )) are used in the simulations as follows (seeFig.5):

f2

1.

-0.002.
.

0-1-0.004
-2

0

1

2

3

time[s]

4

5

0

1

2

3

4

5

time[s]Fig. 15. Trajectory of arclength Fig. 16. Trajectories of angularvelocities q̇11 ,q̇12 and q̇13
parameter s2

. .

q21
q22
The value of zeroq21,q22[rad/s]

1

.
.

.

0

[11] S. Arimoto, “Dexterity and Control of Robot Arms and Hands:
A Riemannian Geometry Approach”, IEICE Fundamentals Review,Vol.2, No.4, pp.1-11, 2009.
[12] J. Baumgarte, “Stabilization of constraint and integrals of motion in
dynamical systems”, Comput. Methods in Appl. Mech. and Eng., Vol.1,
No.1, pp.1–16, 1972.

0.02

x[m]

2

0

-1

-0.02

.

x
The value of zero

-2
0

1

2

3

4

5

0

1

time[s]

2

3

4

5

time[s]Fig. 17. Trajectories of angular Fig. 18.
Trajectory of translavelocities q̇21 and q̇22
tional velocity ẋ

0θ [rad/s]

.

y[m]

0.02

0

.

-1

-2

-3

.

.

-0.02

y
The value of zero

0

1

2

3

4

5

θ
The value of zero

-4
0

1

2

3

4

5

time[s]time[s]Fig. 19.
Trajectory of transla- Fig. 20. Trajectory of rotational
velocity θ̇tional velocity ẏ“blind grasping” is effective in realizing stable grasping as
far as numerical simulations are concerned. The problem ofstability of control inputs for our proposed pinching model
will be tackled more in detail from numerical simulation andat the same time from the mathematical analysis in a near
future.
VI. ACKNOWLEDGMENTS
This work was partially supported by Japan Society for
the Promotion of Science (JSPS), Grant-in-Aid for Scientific
Research (B) (20360117).
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