ICROS-SICE International Joint Conference 2009
August 18-21, 2009, Fukuoka International Congress Center, Japan

Manipulation of 2D Object with Arbitrary Shape by Robot Finger
under Rolling Constraint
Morio Yoshida1 , Suguru Arimoto1,2 and Kenji Tahara3,1
1

RIKEN-TRI Collaboration Center for Human-Interactive Robot Research, RIKEN, Nagoya, Japan(Tel: +81-52-736-5866; E-mail: yoshida@nagoya.riken.jp)
2
Department of Robotics, Ritsumeikan University, Shiga, Japan(E-mail: arimoto@fc.ritsumei.ac.jp)
3Organization for the Promotion of Advanced Research, Kyushu University, Fukuoka, Japan(E-mail: tahara@ieee.org)Abstract: Contact of two contours of a pinched object and a robot ﬁnger tip, with arbitrary shape, is expressed interms of differential geometry. The overall ﬁnger-object dynamics with rolling and contact constraints is derived as EulerLagrange’ equation of motion and the ﬁrst differential equation with curvatures of the contours is derived for updating
the length parameters. Despite the complicated mathematical structure, a control input signal is proposed, which can beconstructed without using object geometrical information or external sensing, and it is shown that it is effective to stabilize
rotation of the object. Consequently, numerical simulations are carried out in order to demonstrate the practicality of theproposed model and control signal.
Keywords: Robot Finger, Rolling Constraint, Arbitrary Shape, Numerical Simulation

1. INTRODUCTION

ferential geometry and in a computational manner (seeFig.1). Despite the complicated mathematical structure,a simple control input, which does not need to use object information or external sensing, is proposed to realize stable grasping. Finally numerical simulations arecarried out to testify the validity of the derived model andproposed control signal.

Object manipulation with various shapes such as a coffee cup, pencil, and plate has close ties with our daily
lives. This is easy for humans by virtue of multiple joints,
sensory feedback, and accumulation of learned data. But
it is difﬁcult for robots since the scientiﬁc explorationof such interactive tasks of manipulation is not yet advanced. The understanding of physical interaction between an object and robot needs to be advanced in orderto realize a human-friendly robot.
This has led many robotics researchers to examine
robotic hands [1]. Most of the explorations in robot hands
have focused on kinematics or motion planning for therealization of force-torque closure to accomplish stablegrasp. Rolling geometry between two surfaces of objects,
an important phenomenon expressing a physical interaction of rigid bodies, was investigated [2]. In the ﬁeldof multibody dynamics, a lot of useful models expressing contact situations are presented [3]. However, theymissed an aspect of the physical interaction between twobodies under rolling constraints in a dynamical sense, despite the demand for unveiling the mechanism of suchmutual action among bodies to realize interactive robots
with human movements.
Arimoto et al. formulated the 2-D and 3-D dynamicpinching models by a pair of robot ﬁngers with hemispherical ends under rolling constraints [4]. This was restricted to the ﬂat surfaces. The group extended the dynamics of pinching to the case of treating objects witharbitrary shape in a horizontal plane [5]. Very recently,
this agument is extended to cover the case of 2-D pinching under the arbitrary geometry [6].
In this paper, a 2-D dynamic pinching model with thetwo contours on arbitrary shapes of a pinched object anda robot ﬁngertip is rederived from the standpoint of dif-xOq1yθ
n1

q2
OXOm

b1 Y

Fig. 1 Robot ﬁnger pinching an object pivoted at a ﬁxedpoint Om

2. DYNAMICS
In order to understand a rolling phenomenon between
two contours with arbitrary shape, the simplest mechanical setup is considered (see Fig.1). The robot ﬁnger witha ﬁnger end of arbitrary shape has two degrees of freedom (2.D.O.Fs). The object pivots around the ﬁxed point
Om = (x, y). It is assumed that motion of the overallﬁnger-object system is restricted to a horizontal plane andthat the effect of gravity can be ignored. q1 and q2 express- 695 -

PR0002/09/0000-0695 ¥400 © 2009 SICE

the joint angles of the robot ﬁnger, and θ the angle of theobject. In the coordinate system, numerical values of allangles are positive in counterclockwise direction. Local
coordinate Om -XY is attached to the pinched object, andlocal coordinate O01 -X1 Y1 is attached to the ﬁnger tip ofthe robot. O-xy expresses the inertial frame coordinates.
The contour of the left side of the object is expressed bya curve attached to the local coordinates (X(s), Y (s)) by
using arclength parameter s. Similarly the contour of theﬁnger tip is expressed by a curve attached to the local coordinates (X0 (s), Y0 (s)). P1A and P1B are the contact

l n1
ln0

P1
P1

O01
l b0

X0
x n0

P1A

P1BP0b0

lb1

n1
Omb1

b1

Y0
b0
s=0

θ1

X0

s=0

P1O01x n0P0
q1 +q2

ψ0ϕ1

P1AP1Bb0Y0b0n1θθ1yθThe length P0� P1� is expressed in the inertial coordinates
as follows:

xOmb1

Fig. 3 Geometrical relationship of ln0 , ln1 , lb0 , and lb1

X

P1

P0� P1� = (x − x01 ) cos(θ + θ1 )−(y − y01 sin(θ + θ1 ))Yb1This length is equal to ln0 + ln1 and then we deﬁne Q1
as follows:

Fig. 2 Geometrical relationship between local coordinates Om -XY and O01 -X0 Y0−Q1

point between the robot ﬁngertip and the object surface
from the robot and object views respectively. Similarly
n0 and n1 are unit normal vectors orthogonal to the tangent plane at the contact from the robot and object views
respectively. The angle between the vector n1 and X axis
is expressed by θ1 as follows:θ1 (s) =arctan {X � (s)/Y � (s)}arctan {X0� (s)/Y0� (s)}ϕ1 = π − (q1 + q2 ) + (θ + θ1 (s))ϕ1 − ψ0 = π/2(1)(2)(13)

whereQ = Q1 − lb0 {π + θ + θ1 (s) − q1 − q2 }(15)

dQ=0dt(16)

andand also we derive the equation:

It is noted that n0 and b0 must be equal to −n1 and b1
respectively in order to satisfy the rolling constraint. Wedeﬁne ln0 , ln1 , lb0 , and lb1 (see Fig.3) as follows:
X0 (s) sin ψ0 + Y0 (s) cos ψ0−X(s) sin θ1 − Y (s) cos θ1X0 (s) cos ψ0 − Y0 (s) sin ψ0−X(s) cos θ1 + Y (s) sin θ1(12)We deﬁne the contact constraint as a holonomic constraint in the form:�π�+ (ln0 + ln1 )Q = Q + lb0 ψ0 +2
�π�= Q1 + (ln0 + ln1 ) − lb0 ϕ1 − ψ0 −2= 0(14)vectors n0 and n1 and the unit tangent vectors b0 and b1
are expressed as follows:��sin(θ + θ1 )(3)
b1 =cos(θ + θ1 )��cos(θ + θ1 )n1 =(4)
− sin(θ + θ1 )��− cos(q1 + q2 + ψ0 )(5)
b0 =sin(q1 + q2 + ψ0 )��sin(q1 + q2 + ψ0 )n0 = −(6)cos(q1 + q2 + ψ0 )====(x − x01 ) cos(θ + θ1 )−(y − y01 sin(θ + θ1 ))ln0 (s) + ln1 (s)We deﬁne the relative angle ϕ1 by the deﬁnitions of θ1and ψ0 as follows:

where X0� (s) = dX0� /ds and Y0� (s)/ds. The unit normal

lb0 (s)lb1 (s)ln0 (s)ln1 (s)==where X � (s) = dX(s)/ds and Y � (s) = dY (s)/ds.
Similarly the angle between the vector n0 and X0 is expressed by ψ0 as follows:ψ0 (s) =(11)

�(7)
(8)
(9)
(10)

⎞dθ∂Q/∂θ, ∂Q/∂q1 , ∂Q/∂q2 ⎝ dq1 ⎠ = 0dq2
�⎛which is reduced to�- 696 -lb1 (s), nT
1 (s)J01 (q) + lb0 (s)e

�
T(17)

⎞dθ⎝ dq1 ⎠ = 0 (18)dq2⎛

where J01 (q) = ∂(x01 , y01 )/∂(q1 , q2 ). The lengthOm P1� + P0� O01 (see Fig.3) is expreesed in the inertial
frame as follows:
Om P1� + P0� O01=

where κ0 (s) expresses the curvature of the ﬁngertip’scontour and κ1 (s) the curvature of the object’s contour.
κ0 (s) and κ1 (s) are deﬁned as follows (see [6]):−(x − x01 ) sin(θ + θ1 )−(y − y01 ) cos(θ + θ1 ) (19)κ0κ1This length is equal to lb0 (s) + lb1 (s) and we deﬁne R1
as follows:
R1= −(x − x01 ) sin(θ + θ1 )−(y − y01 ) cos(θ + θ1 )= lb0 (s) + lb1 (s)�Though the contours of the ﬁnger tip and object are ofcomplicated geometry, we propose a simple control input
which does not need to use the information of the object
and the data of external sensing, in order to stabilize theobject. This is described as��fd T
x01 − xu = −cq̇ − J01 (q)(32)
y01 − yr(20)

Tlb1 (s), bT
1 (s)J01 (q) − ln0 (s)e(21)�⎞dθ⎝ dq1 ⎠ = 0 (22)dq2
⎛Equation (22) can be integrated in the sense of Frobenius.
We deﬁne the rolling constraint as follows:(23)

R = ln0 (s) {π + θ + θ1 (s) − q1 − q2 } + R1

(24)

where=

where c and r are positive constants. The ﬁrst term of theright hand side of eq.(32) works as damping. The second
term is introduced to cease the rotational motion of theobject.
By applying the proposed control input (eq.(32)) to theoverall ﬁnger-object dynamics (eqs (27) and (28)), we obtain the closed-loop dynamics as follows:��
1G(q)q̈ +Ġ + S + cq̇
2∂R fd∂Q+ Δλ− N1 e = 0∂q
∂qr
∂R fd
∂Q+ Δλ+ N1 = 0I θ̈ + Δf∂θ∂θrwherefd
fd
Δf = f + Q1 , Δλ = λ + R1r
r
N1 = lb0 Q1 − ln0 R1 = −lb0 ln1 + ln0 lb1+Δf

R + {lb0 (s) + lb1 (s)}�π�=0−ln0 (s) ψ0 (s) +2R(30)(31)

3. CONTROL SIGNAL & CLOSEDDYNAMICS

Rolling constraint expresses the condition of rolling on
the surface without slipping. The tangent velocity of theobject on the tangent plane of the contact point P1 must
be equal to the tangent velocity of the ﬁnger tip on thesame tangent plane, at instant t, as follows:∂∂ln0 ϕ1 + R1 = 0∂t∂twihch is reduced to= −X0�� (s)Y0 (s) + X0� (s)Y0�� (s)= X �� (s)Y � (s) − X � (s)Y �� (s)and alsodR=0(25)
dt
The total kinetic energy of the ﬁnger robot and object isobtained as follows:11(26)
K = q̇ T G(q)q̇ + I θ̇222
where G(q) stands for the inertia matrix of the robot ﬁnger and I for the inertia moment of the object around Om .
By deﬁning the Lagrangian L = K and installing tangent
vectors (∂θ, ∂q 1 ,∂q 2 ) satisfying eqs (18) and (22), thedynamic equations of motion of the overall ﬁnger-object
system are derived:(33)(34)

(35)
(36)

4. NUMERICAL SIMULATIONI θ̈ + f lb1 − λln1 = 0(27)����
1T(q)n1 + lb0 e
G(q)q̈ +Ġ + S q̇ + f J01
2
� T
�
λ J01
(q)b1 − ln0 e = u (28)where q = (q1 , q2 )T . From the diﬁnition of directional
derivatives of b0 and b1 (see[6]), the arclength parameters should be updated as follows
�ds �= q̇1 + q̇2 − θ̇{κ0 (s) + κ1 (s)}(29)
dt

We carry out numerical simulations in order to verify the effectiveness of our proposed model and control input. The physical parameters of the overall system shown in Table 1 and the parameters of control input shown in Table 2 are used in the simulations. As
an example of an object with arbitrary shape, the curvec0 (s) = (X0 (s), Y0 (s)) of the ﬁngerend’s contour, attached to the local coordinate O01 -X0 Y0 , is used as follows (see Fig.4-a):√
1 + 4 × 202 × s2
X0 (s) = 0.035 −(37)2 × 20Asinh(2 × 20 × s)(38)Y0 (s) =2 × 20

Similarly the curve c(s) = (X(s), Y (s)) of the object’s
contour, attached to the local coordinate Om -XY , is used
as follows (see Fig.4-b):X(s) = −0.03
�
1 + 4 × 502 × (s − 3.363 × 10−3 )2+(39)
2 × 50- 697 -Table 1 Physical parameters of the ﬁngers and object.

-0.05

-0.05

0.500 [N]0.006 [Nms]0.010 [m]1500
3000
225.0 × 104
900.0 × 104-0.1-0.1ΔfThe value of zero

-0.15
012time[s]0300fr NThe value of zero-0.003

Y [m]

Y0[m]

0Om

01time[s]2-0.005

301Fig. 8 frd N1

00.01

X0[m]

0.02

0.03

-0.03.
-0.02

-0.01

X [m]

00.01

1.

(a) The curve of(b) The curve of
the ﬁngertip’s contour
the object’s contour
Fig. 4 The local coordinates of the ﬁngertip and the object

1.

00-1

-1
01time[s]2301Fig. 10 q̇122

30-1

1R [m]

θ [rad/s]time[s]Fig. 11 q̇2(×10-10).

3q2
The value of zero2q2[rad/s]-0.01

q1[rad/s]

-0.02

22.

q1
The value of zero

-0.01

time[s]Fig. 9 s

-0.01
-0.02

30.005-0.001

0.01

02

time[s]0.01-0.002O01

1Fig. 7 ΔλFig. 6 Δf

0.02
0.01

Δλ
The value of zero

-0.15

f N[Nm]rinternal force
damping coefﬁcientconstant value
CSM gain
CSM gain
CSM gain
CSM gain

fdcr
γf 1
γλ1
ωf 1
ωλ1

0Δ f[N]

Table 2 Parameters of control signals & CSM gains

0Δλ[N]

0.065 [m]0.065 [m]0.045 [kg]0.040 [kg]0.040 [kg]s[m]lengthlengthweightweightobject weight

l11
l12
m11
m12M

maintains the rolling and contact constraints during thesimulation. Consequently it is conﬁrmed through the numerical simulations that our proposed model, control input, and simulation method are effective.

-2-30.
01time[s]2R
The value of zero-4

θ
The value of zero

-1-5

301(×10-9)

23

Fig. 13 R

Fig. 12 θ̇(a) Initial pose(b) After 3 seconds
Fig. 5 Motion of pinching a 2-D object with arbitrary
shapetime[s]6Q
The value of zeroQ[m]4Asinh(2 × 50 × (s − 3.363 × 10−3 ))(40)
2 × 50Itis arclength parameter,� is noted that, because s �X0� (s)2 + Y0� (s)2 = 1 and X � (s)2 + Y � (s)2 = 1.
If a solution to the closed-loop dynamics (eqs. (33) and(34)) converges to an equilibrium point, it should satisfyq̈ → 0,θ̈ → 0,q̇ → 0,θ̇ → 0, Δf → 0, Δλ → 0,and (fd /r)N1 → 0. It is conﬁrmed, from Figs.6 ∼ 12,that key variables of the closed dynamics converge to theequilibrium point. These graphs and the pictures in themiddle of the simulation (see Fig.5) show that motion of
the robot ﬁnger achieves stable grasping by applying ourproposed control input. It is also shown, from Figs.13 and14, that the “Constraint Stabilization Method” (CSM) [7]

20Y (s) =01time[s]23Fig. 14 Q

5. CONCLUSION
A 2-D dynamic pinching model with two contours of aﬁngertip and object with arbitrary geometry is proposed.
It is important that our numerical simulation method becomes easy because rolling and contact constraints treating the contours of arbitrary shapes are fortunately expressed as holonomic constraints in a 2-D case. Our proposed control input is simple and practical though the ﬁn-- 698 -gertip and object models have complicated shapes. Finally it is conﬁrmed that our proposed model and control
input are effective, in terms of the graphs and visualization of the results of numerical simulation.

6. ACKNOWLEDGMENTS
This work was partially supported by Japan Society forthe Promotion of Science (JSPS), Grant-in-Aid for Scientiﬁc Research (B) (20360117).

REFERENCES[1] R. M. Murray, Z. Li, and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation, CRC
Press, Boca Raton, 1994.
[2] D.J. Montana. The kinematics of contact and grasp,
Int. J.of Robotics Research, Vol.7, No.3, pp.17-32,
1988.
[3] A.A. Shabana, Computational Dynamics, Second
Edition, Wiley-Interscience, New York, USA, 2001.
[4] S. Arimoto, Control Theory of Multi-ﬁngeredHands, Springer-Verlag, London, 2008.
[5] S. Arimoto, M. Yoshida, M. Sekimoto, and K.
Tahara. A Reimannian-Geometry Approach forControl of Robotic Systems under Constraints,
SICE J. of Control, Measurement , and System Integration, Vol.2, No.2, pp.107-116, 2009.
[6] S. Arimoto, M. Yoshida, M. Sekimoto, and K.
Tahara. Modeling and Control of 2-D grasping of an
object with arbitrary shape under rolling contact, accepted for publication in SICE JCMSI, Vol.2, 2009.
[7] 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.

- 699 -