Advanced Robotics 24 (2010) 1345–1364

brill.nl/ar

Full paper
A Riemannian-Geometric Approach for Intelligent Control
and Fingertip Design of Multi-fingered Hands
Suguru Arimoto a,b,∗ , Morio Yoshida b , Masahiro Sekimoto a and Ji-Hun Bae ca

Research Organization of Science and Engineering, Ritsumeikan University, 1-1-1 Nojihigashi,
Kusatsu, Shiga 525-8577, Japanb
RIKEN-TRI Collaboration Center, RIKEN, Nagoya, Aichi 463-0003, Japanc
Center for Cognitive Robotics Research, KIST, Seoul, South Korea
Received 7 August 2009; accepted 13 October 2009

Abstract
Based upon the analysis of modeling and control of two-dimensional (2-D) grasping and manipulation ofarbitrary rigid objects under rolling contact constraints, geometrical conditions for the design of desired
fingertip shapes of robot fingers are discussed from the Riemannian-geometric standpoint. A required condition is given as an inequality expressed in terms of quantities of the second fundamental form of fingertip
contour curves, although the quantities do not enter into the Euler–Lagrange equation of motion of theoverall fingers/object system. Satisfaction of the inequality is necessary for stabilization of grasping by using fingers–thumb opposable control signals without use of external sensings or knowledge of an object tobe grasped. At the same time, asymptotic convergence of a solution to the closed-loop dynamics of 2-D
precision prehension by a pair of multi-joint robot fingers is proved under rolling contact constraints andthe existence of redundancy in the system’s degrees of freedom. This is regarded as an extension of the
Dirichlet–Lagrange stability theorem to a dynamical system with redundancy and geometric constraints.
© Koninklijke Brill NV, Leiden and The Robotics Society of Japan, 2010
Keywords
Multi-fingered hands, Riemannian geometry, fingers–thumb opposability, precision prehension, stablegrasping

1. Introduction
In Napier’s famous book [1], entitled Hands, he said that ‘opposition of the thumbis one of the hallmarks by which humans can be authenticated as it were’. As tothe fingers–thumb opposability, Napier also defined: ‘Opposition is a movement* To whom correspondence should be addressed. E-mail: arimoto@fc.ritsumei.ac.jp© Koninklijke Brill NV, Leiden and The Robotics Society of Japan, 2010

DOI:10.1163/016918610X501462

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by which the pulp surface of the thumb is placed squarely in contact with — ordiametrically opposite to — the terminal pads of one or all of the remaining digits’.
This paper tackles not only a control problem for stabilization of motion of
precision prehension (two-dimensional (2-D) object grasping) based on the fingers–
thumb opposability, but also a design problem of fingertip shapes that must be
fit to the most important function of hands, i.e., fingers–thumb opposition. When
fingertips are a rigid body, different from the softness of human digits, mathematical modeling of 2-D grasping becomes relatively easier even if rolling contact
constraints are taken into account [2, 3]. However, if grasping of a 2-D object
with arbitrary smooth contour curves is to be treated, modeling of rolling contact
constraints becomes difficult because the constraints are regarded originally to be
non-holonomic, as discussed in a representative text book [4]. In a typical ‘ball–plate’ problem of grasping when fingertips are hemispheric and an object to be
grasped is a parallelepiped, it has been shown that the Pfaffian form expressing arolling contact constraint is integrable and its integral is described in an explicit
form. Very recently, in 2009, it was shown that Pfaffian forms of rolling contact
constraints are integrable in the sense of Frobenius for a general class of 2-D graspings even if both the fingertips and object have arbitrary shapes [5]. In Ref. [5], theEuler–Lagrange equation of motion of the fingers/object system is derived, which
are characterized by arclength parameters of contour curves along which rolling
contact motions arise. At the same time, an update law of arclength parameters
is derived as a first-order differential equation in which quantities of the second
fundamental form of contour curves are involved. This follows from a differentialgeometric assumption on rolling contact proposed by Nomizu [6].
This paper first summarizes the previous results of 2-D grasping under the circumstances of the arbitrary geometry of fingertip and object shapes, and shows that,in a special case when the object is a parallelopiped, stable grasping is realized by acontrol signal based upon the fingers–thumb opposability. It is then shown that thecontrol signal can be regarded as a gradient of a potential function whose minimum
is attained at an equilibrium posture. Then, necessary conditions for an adequate
shape of fingertip contours are given by characterizing a relation of the curvaturesof contour curves with changeable lengths between the contact point and the fingertip center. Compared with typical fingertips like a hemisphere, a type of the moon
13 days old, a cresent moon or an ellipse, a contour curve of the human thumb ismostly fit to minimization of the artificial potential function based on the fingers–
thumb opposability. In Section 5, the necessity condition is enlarged by treating thearbitrary geometry of rigid objects based upon a testbed problem of stabilization ofrotational motion by using a single robot finger. In the final section, a proof of convergence of motions of the closed-loop dynamics to an equilibrium posture of stablegrasping is given in the case that the object is a parallelopiped. A general problemof finding a stable control signal for a class of 2-D objects with an arbitrary shapeis still open and remains unsolved.

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In Napier’s book [1], John Hunter’s principle is quoted at p. 9: ‘structure was theintimate expression of function’ and ‘function was conditioned by environment’. By
invoking three heros (Hunter, Charles Bell and Charles Darwin), Napier continued:‘John Hunter turned our attention from the structure of the hand to its function; Bellrelated the function of the hand to the environment; and Darwin demonstrated thatthe environment, by process of natural selection, gave birth to structure’.
2. Euler–Lagrange Equations of Motion
Very recently, a complete model of 2-D grasping of a rigid object with an arbitrary
shape by a pair of robot fingers with arbitrarily given fingertip shapes (see Fig. 1)
was presented based upon the differential-geometric assumptions of rolling contacts[2]. The assumptions are summarized as:(i) Two contact points on the contour curves must coincide at a single common
point without mutual penetration.
(ii) The two contours must have the same tangent at the common contact point.
As pointed out in previous papers [5, 7], these two conditions as a whole are
equivalent to Nomizu’s relation [6] concerning tangent vectors at the contact point
and normals to the common tangent. As a result, a set of Euler–Lagrange equationsof motion of the overall fingers/object system is presented in the following forms:M ẍ −(fi n̄0i + λi b̄0i ) = 0(1)
I θ̈ +

i=1,2(−1)i {fi (bT0i γ0i ) − λi (nT0i γ0i )} = 0

(2)

i=1,2

Figure 1. Pair of 2-D robot fingers with a curved fingertip makes rolling contact with a rigid object
with a curved contour.

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S. Arimoto et al. / Advanced Robotics 24 (2010) 1345–1364
1
Gi (qi )q̈i +Ġi (qi ) + Si (qi , q̇i ) q̇i + fi {JiT (qi )n̄0i − (−1)i (bTi γi )ei }2+ λi {JiT (qi )b̄0i − (−1)i (nTi γi )ei } = ui ,i = 1, 2,(3)where qi denotes the joint vector as q1 = (q11 , q12 , q13 )T and q2 = (q21 , q22 )T , θ̇
denotes the angular velocity of rotation of the object around the object mass centerOm expressed by position vector x = (x, y)T in terms of the inertial frame coordinates O-xy and Ji (qi ) = ∂ri /∂qiT for i = 1, 2, where ri denotes the position of Oi
in O-xy coordinates. It is implicitly assumed that all axes of rotation of finger joints
are in the z-axis orthogonal to the horizontal xy-plane and, therefore, the effect ofgravity is ignored. In general, there is no reason to equip each robot finger withthree joints but the two joints must be necessary for further realization of dexterity
like control of the orientation angle of an object if the object geometry is known.
Equation (1) expresses the translational motion of the object with mass M and (2)its rotational motion with inertia moment I around the mass center Om . At the contact point Pi , bi denotes the unit tangent vector expressed in local coordinates of
Oi -Xi Yi fixed to the fingertip of finger i (i = 1, 2) as shown in Figs 1 and 2, and ni
denotes the unit normal to the tangent expressed in terms of Oi -Xi Yi . Similarly, b0i
and n0i are the unit tangent and normal at Pi expressed in terms of local coordinates
Om -XY fixed to the object as shown in Figs 1 and 2. All these unit vectors are determined uniquely from the assumptions (i) and (ii) on the rolling contact constraints
at each contact point Pi dependently on each corresponding value si of arclengthparameter for i = 1, 2 as shown in Fig. 2. In accordance with the assumptions (i)and (ii), a single length parameter s1 is used commonly for expressing the positionof contact point P1 along the left fingertip contour and that of P1 along the object
contour at the left side. Similarly, s2 is commonly used for expressing the locationof contact point P2 along b02 and b2 . Equation (3) expresses joint motions of finger
i with the inertia matrix Gi (qi ) for i = 1, 2 and e1 = (1, 1, 1)T and e2 = (1, 1)T . All
position vectors γi and γ0i for i = 1, 2 are defined in Fig. 2, and expressed in their

Figure 2. Definitions of tangent vectors bi and b0i and normals ni and n0i at contact points Pi for
i = 1, 2.

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corresponding local coordinates, respectively. Both the unit vectors b̄0i and n̄0i areexpressed in the inertial frame coordinates as:b̄0i = 0 b0i ,n̄0i = 0 n0i ,0 = (rX , rY ),(4)

where 0 ∈ SO(2), and rX and rY denote the unit vectors of X- and Y -axes of theobject in terms of the frame coordinates O-xy. In (1)–(3), fi and λi are Lagrange
multipliers that correspond to the following rolling contact constraints, respectively:(5)Qbi = (ri − rm )T b̄0i + bTi γi − bT0i γ0i = 0, i = 1, 2
T
TT
Qni = (ri − rm ) n̄0i − ni γi − n0i γ0i = 0, i = 1, 2,(6)
where ri denotes the position vector of the fingertip center Oi expressed in terms of
the frame coordinates O-xy and rm the position vector of Om in terms of O-xy. In
parallel with Euler–Lagrange equations (1)–(3), arclength parameters si (i = 1, 2)
should be governed by the following formulae of the first-order differential equation[5]:dsi{κ0i (si ) + κi (si )}(7)= (−1)i (θ̇ − ṗi ), i = 1, 2,
dt
where κi (si ) denotes the curvature of the fingertip contour for i = 1, 2, κ0i (si ) the
curvature of the object contour at contact point Pi corresponding to length parameter si for i = 1, 2, p1 = q11 + q12 + q13 , and p2 = q21 + q22 . Throughout the paper
we use (˙) to denote the differentiation of the content of the parentheses ( ) in time tas θ̇ = dθ/dt in (7) and ( ) for that of ( ) in length parameter si as illustrated byγi (si ) = dγi (si )/dsi . As discussed in the previous papers, we have:dγi (si ),b0i (si ) = γ0i (si ), i = 1, 2,=(8)
bi (si ) = γi (si )dsiand:ni (si ) = κi (si )bi (si ),n0i (si ) = κ0i (si )b0i (si ),i = 1, 2,(9)

and further:bi (si ) = −κi (si )ni (si ),b0i (si ) = −κ0i (si )n0i (si ),i = 1, 2.

(10)

It is well known from text books on differential geometry of curves and surfaces(see, e.g., Ref. [5]) that (9) and (10) constitute Frenet–Serre’s formulae for thefingertip contour curves and object contours. Note that all equations (1)–(3) arecharacterized by length parameters si for i = 1, 2 through unit vectors n0i , b0i , bi ,and ni , and vectors γ0i and γi expressed in each local coordinates, but quantities ofthe second fundamental form of contour curves, i.e., κi (si ) and κ0 (si ) for i = 1, 2,
do not enter into (1)–(3). It is shown that the set of Euler–Lagrange equations of
motion from (1) to (3) can be derived by applying the variational principle to the
Lagrangian of the system(fi Qni + λi Qbi ),(11)
L(X; s1 , s2 ) = K(X, Ẋ) −
i=1,2

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where X denotes the position state vector defined as:
X = (x, y, θ, q1T , q2T )T ,and:
K(X, Ẋ) =

(12)

 1
M 2
I
(ẋ + ẏ 2 ) + θ̇ 2 +q̇ T Gi (qi )q̇i .
2
2
2 i

(13)

i=1,2

Note that K(X, Ẋ) is independent of the shape parameters s1 and s2 , but Qni and
Qbi defined by (5) and (6) are dependent on si for i = 1, 2, respectively. The variational principle is written in the following form:
 t1{δL + uT1 δq1 + uT2 δq2 } dt = 0.
(14)
t0

From this it follows that:



 ∂
∂
1
Ġ(X) + S(X, Ẋ) Ẋ +Qni + λi
Qbi
fi
G(X)Ẍ +
2
∂X
∂X
i=1,2
 
u1
,
(15)
=B
u2
where G(X) = diag(M, M, I, G1 (q1 ), G2 (q2 )), S(X, Ẋ) is a skew-symmetric matrix and B denotes the 8 × 5 constant matrix defined as B T = (03×5 , I5 ), where
03×5 signifies the 3 × 5 zero matrix, and I5 the 5 × 5 identity matrix.
3. Fingers–Thumb Opposable Control Signals
In order to design adequate control signals for a pair of multi-joint fingers as shownin Fig. 1, we suppose that the kinematics of both the robot fingers are known, andmeasurement data of joint angles and angular velocities are available in real-time,
but the geometry of an object to be grasped is unknown and the location of its masscenter together with its inclination angle cannot be measured or sensed. This supposition is reasonable because the structure of robot fingers is fixed for any object,
but the object to be grasped is changeable from time to time. This standpoint is coincident to the start point of Riemannian geometry that, if the robot (both the robot
fingers) has its own internal world, then the robot kinematics based upon quantitiesof the first fundamental form like γi (si ) and bi (si ) together with qi and q̇i must beaccessible because these data are intrinsic to the robot’s internal world. However,any quantities of the second fundamental form like κi (si ) (i = 1, 2) cannot be determined from the robot’s intrinsic world. By the same season, we assume that thepositions of finger centers O1 and O2 denoted by r1 and r2 are accessible from theintrinsic robot world, and further the Jacobian matrices defined by Ji (qi ) = ∂ri /∂qiT
for i = 1, 2 are also assumed to be intrinsic, i.e., real-time computable. Thus, let us
now consider a class of control signals defined by the form:ui = −ci q̇i + (−1)i βJiT (qi )(r1 − r2 ) − αi N̂i ei ,i = 1, 2,(16)

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where β stands for a position feedback gain common for i = 1, 2 with physical unit
N/m, α is also a positive constant common for i = 1, 2, N̂i is defined as:N̂i = eTi {qi (t) − qi (0)} = pi (t) − pi (0),i = 1, 2,(17)

and ci denotes a positive constant for joint damping for i = 1, 2. The first term on
the right-hand side of (16) stands for damping shaping, the second term plays a roleof fingers–thumb opposition and the last term avoids possibly some excess motionof rotation of the object through contacts. Note that the sum of inner products of ui
and q̇i for i = 1, 2 is given by: αi  d βT2r1 − r2  +N̂ 2 −q̇i ui = −ci q̇i 2 .
(18)
dt 2
2 i
i=1,2

i=1,2

i=1,2

Substitution of control signals of (16) into (3) yields:1
Gi q̈i +Ġi + Si q̇i + ci q̇i − (−1)i βJiT (r1 − r2 ) + αi N̂i ei
2+ fi {JiT n̄0i − (−1)i (bTi γi )ei }+ λi {JiT b̄0i − (−1)i (nTi γi )ei } = 0,i = 1, 2.

(19)

Hence, the overall closed-loop dynamics is composed of the set of Euler–Lagrange
equations of (1), (2) and (19) that are subject to four algebraic constraints of (5)and (6), and the pair of the first-order differential equations of (7) that governs
the update law of arclength parameters s1 and s2 . It should be also remarked that,
according to (18), the sum of inner products of (1) and ẋ, (2) and θ̇ , and (19) and q̇i
for i = 1, 2 yields the energy relation:
dE(X, Ẋ) = −ci q̇i 2 ,
(20)
dt
i=1,2

where:
E(X, Ẋ) = K(X, Ẋ) + P (X) αiβP (X) = r1 − r2 2 +N̂ 2 ,
2
2 i

(21)
(22)

i=1,2

and K(X, Ẋ) is the total kinetic energy defined by (13), and P (X) is called theartificial potential energy that is a scalar function depending on only q1 and q2 . It is
important to note that the closed-loop dynamics of (1), (2) and (19) can be written
into the general form, correspondingly to (15):1∂P (X)
G(X)Ẍ +Ġ(X) + S(X, Ẋ) + C Ẋ +
2
∂X

 ∂
∂
Qni + λi
Qbi = 0,
fi
(23)
+
∂X
∂X
i=1,2

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S. Arimoto et al. / Advanced Robotics 24 (2010) 1345–1364

where C = diag(02 , 0, c1 I3 , c2 I2 ). This can be also obtained by applying the principle of variation to the Lagrangian:(fi Qni + λi Qbi ).
(24)
L = K(X, Ẋ) − P (X) −i=1,2

4. Necessary Conditions for the Design of Fingertip Shape
It is known [8] that, in a simple case of ‘ball–plate’ pinching, a solution to theclosed-loop dynamics corresponding to (23) under some holonomic constraints ofrolling contacts converges to a steady (equilibrium) state that minimizes the potential P (X) under the constraints. However, a stabilization problem of control signals
like (16) still remains unsolved or rather has not yet been tackled not only in a
general setup of arbitrary geometry like the situation shown in Fig. 1, but also in amore simple case where the object to be grasped is a parallelepiped but the fingertip
shapes are arbitrary. In this paper, we will tackle this simple problem and show thatminimization of such an artifically introduced potential can lead to stable graspingunder some good design of fingertip shapes (Fig. 3).
First, we remark that, since the first term of P (X) in (22) is the squared norm of−−−−→
the vector O2 O1 times β/2, it must be a function only dependent on length parameters s1 and s2 . Then, it will be that minimization of the squared norm r1 − r2 2over rolling contact motions is attained when the straight line P1 P2 connecting
the two contact points becomes parallel to the vector (r1 − r2 ). That is, U (X)(=(β/2)r1 − r2 2 ) is minimized when O1 O2 becomes parallel to P1 P2 . To show
this directly from the set of Euler–Lagrange equations (1), (2) and (19) seems difficult even in this case. Instead, we remark that (r1 − r2 ) can be expressed in termsof length parameters si for i = 1, 2 as:r1 − r2 = −1 γ1 + 2 γ2 + 0 (γ01 − γ02 ),(25)

Figure 3. Minimization of the squared norm r1 − r2 2 over rolling motions is attained when thestraight line P1 P2 connecting the two contact points becomes parallel to the vector (r1 − r2 ), i.e.,
O1 O2 becomes parallel to P1 P2 .

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1353

where i ∈ SO(2) denotes the rotational matrix of Oi -Xi Yi to be expressed in theframe coordinates O-xy. Since the object is rectangular, all b0i and n0i for i = 1, 2
are invariant under the change of si for i = 1, 2. Therefore, as seen from Fig. 3, if
the object width is denoted by lw and zero points of s1 and s2 are set as shown in
Fig. 3, then it is possible to write (25) as:
r1 − r2 = (s1 − s2 )b̄01 + (−bT1 γ1 + bT2 γ2 )b̄01 − lw n̄01 + (nT1 γ1 + nT2 γ2 )n̄01 . (26)Since b̄01 ⊥ n̄01 , U (X) can be expressed as:ββr1 − r2 2 = {d 2 (s1 , s2 ) + l 2 (s1 , s2 )}22= U (s1 , s2 ),(27)

d(s1 , s2 ) = s1 − s2 − bT1 γ1 + bT2 γ2(28)l(s1 , s2 ) = −lw + (nT1 γ1 + nT2 γ2 ).

(29)

U (X) =
where:

Note that the artificial potential U (X) can be regarded as a scalar function definedin terms of length parameters s1 and s2 . When minimization of U (s1 , s2 ) over someparameter intervals si ∈ Ii = (ai , bi ) is considered for i = 1, 2, it is important tonote that the vector (r1 − r2 ) is originally subject to the constraint:
V (s1 , s2 ) = (r1 − r2 )T n̄01 − nT1 γ1 − nT2 γ2 − nT01 γ01 − nT02 γ02 = 0,(30)which is obtained by subtraction of Qn2 from Qn1 defined in (5) and (6). Hence, byintroducing a Lagrange multiplier η, minimization of the function:
W (s1 , s2 ; η) = U (s1 , s2 ) + ηV (s1 , s2 ),must be equivalent to that of U (X). Then it follows that:∂W= (−1)i βκi (r1 − r2 )T ξ̄ i + ηκi bTi γi ,∂siwhere we define, for abbreviation:
ξ̄ i = (nTi γi )b̄0i + (bTi γi )n̄0i ,i = 1, 2,

i = 1, 2.

(31)

(32)

(33)

The derivation of this equation is discussed in Appendix A. At this stage, we remarkthat the vectors ξ̄ i for i = 1, 2 appear at contact points P1 and P2 as indicated inFig. 3. Evidently from the right-hand side of (32), if we set:
η = β(r1 − r2 )T n̄01(=−β(r1 − r2 )T n̄02 ),(34)

and at the same time:(r1 − r2 )T b̄0i = 0,i = 1, 2,(35)

then (32) implies:
∂W
= 0,
∂si

i = 1, 2.

(36)

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S. Arimoto et al. / Advanced Robotics 24 (2010) 1345–1364−−−−→

In view of the geometrical meaning of (35) that the vector O2 O1 ⊥ b̄0i , whenr1 − r2 becomes perpendicular to b0i from some starting posture by rolling contact motion, si for i = 1, 2 must have the same value s ∗ and bT1 γ1 = bT2 γ2 . That is,satisfaction of the conditions:bT1 γ1 = bT2 γ2 ,(37)
s1 = s2 = s ∗ ,
−−−−→
−−−−→
reduces to that O2 O1 becomes parallel to P2 P1 as shown in Fig. 3. Under (37):∂W= 0,∂si si =s ∗i = 1, 2.

(38)

Finally, it is important to check the positivity of the Hessian matrix H =(∂ 2 U/∂si ∂sj ). Bearing in mind the form of (32) together with (34), we obtain(see the details in Appendix A):∂ 2U= κi (−1)i β(r1 − r2 )T n̄0i (κi nTi γi + bTi γi )∂si ∂si si =s ∗


2 1
T
= −βl(s1 , s2 )κi+ ni γi , i = 1, 2,
κi

(39)

and:
∂ 2U
= 0,
∂s1 ∂s2 si =s ∗(40)

where l(s1 , s2 ) is defined by (29). Since l(s1 , s2 ) < 0 from the geometrical meaningof the situation shown in Fig. 3, it is possible to conclude that the potential function
U (s1 , s2 ) is minimized at the posture satisfying (37) provided that:1> −nTi (si )γi (si ), i = 1, 2,(41)
κi (si )for all si belonging to (s ∗ − δi , s ∗ + δi ) with some δi > 0 for i = 1, 2.
Geometric and physical meanings of (41) are well illustrated in Fig. 4. Even if
the fingertip is spherical and its curvature κ(s) is constant (i.e., equal to 1/r, wherer denotes the radius), a design of fingertip shape like a moon 13 days old does notsatisfy the inequality of (41), since the center of the sphere is located inside thefingertip (Fig. 4a). In the case of a fingertip like a crescent moon shown in Fig. 4b,
however, the inequality of (41) is satisfied. It is interesting to note that, in the caseof an elliptic fingertip, the inequality of (41) is satisfied in the vicinity of contact
position P1 as shown in Fig. 5a when the line O1 P1 is perpendicular to the Y1 axis, but there arise possible contact points denoted by P  that may not satisfy (41),when P  moves further toward the endpoint P0 along the contour of the ellipsoid(Fig. 5b).
A mathematical proof of a solution trajectory to the closed-loop dynamics of(1), (2) and (19) under holonomic constraints of (5) and (6) together with the setof first-order differential equations of (7) starting from a posture in a neighborhood
the equilibrium state shown in Fig. 3 will be discussed in Section 6.

S. Arimoto et al. / Advanced Robotics 24 (2010) 1345–1364

(a)

1355

(b)

Figure 4. Design of a fingertip geometry is sensitive to the necessary condition for minimization of
the artificial potential U (s1 , s2 ). (a) Fingertip like a moon 13 days old. (b) Fingertip like a crescent
moon.

(a)

(b)

Figure 5. Fingertip design using a half of the ellipse. (a) Smallest curvature arises when
O1 P1 ⊥ Y1 -axis. (b) Curvature increases as P  runs along the contour toward the Y1 -axis.

5. Further Necessity of Fingertip Curvatures
In order to obtain a more robust condition on the design of the shape of a robot finger
against more generality of the geometry of objects to be grasped stably, consider atestbed control problem of grasping a rigid object with an arbitrary geometry by aplanar multi-joint robot finger as depicted in Fig. 6. In this problem, the object ispivoted around the point Om fixed to the inertial frame O-xy. The object contour
curve is denoted by γ0 (s) with the aid of the arclength parameter expressed in local
coordinates Om -XY fixed to the object and the fingertip contour curve by γ1 (s)expressed in local coordinates O1 -X1 Y1 fixed to the fingertip as shown in Fig. 6.

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S. Arimoto et al. / Advanced Robotics 24 (2010) 1345–1364

Figure 6. Testbed control problem of grasping a rigid object with arbitrary geometry.

Also, unit tangent vector b1 and normal n1 at contact point P1 along the fingertip
contour and similarly b0 and n0 at P1 along the object contour are defined as shownin Fig. 6. It is shown in the previous paper [5] that the Euler–Lagrange equations of
motion of this finger/object system are described in the forms:
I θ̈ − f (γ0T b0 ) + λ(γ0T n0 ) = 01Ġ(q) + S(q, q̇) q̇ + f {J1T (q)n̄0 + (γ1T b1 )e}
G(q)q̈ +2+ λ{J1T (q)b̄0 + (γ1T n1 )e} = u,

(42)

(43)

where n̄0 = 0 n0 , b̄0 = 0 b0 , 0 ∈ SO(2) with respect to the counterclockwise
rotation angle θ from x-axis in the O-xy frame coordinates, I and G(q) denote theobject inertia moment around Om and the finger inertia matrix, respectively, andq = (q11 , q12 , q13 )T and e1 = (1, 1, 1)T . If we use a control signal:
u = −cq̇ − βJ1T (q)(r1 − r0 ),

(44)

−−−→
−−−−→
where r1 and r0 denote position vectors OO1 and OOm , respectively, and β andc are an appropriate positive constant, then the closed-loop dynamics obtained bysubstituting (44) into (43) can be written as:1Ġ + S q̇ + cq̇ + βJ1T (q)(r1 − r0 )G(q)q̈ +2+ f {J1T (q)n̄0 + (γ1T b1 )e} + λ{J1T (q)b̄0 + (γ1T n1 )e} = 0.

(45)

Evidently the set of this equation and (42) can be regarded as the Euler–Lagrange
equations of motion derived from applying the variational principle to the Lagrangian:L = K(X, Ẋ) − U (X) − f Qn − λQb ,(46)

S. Arimoto et al. / Advanced Robotics 24 (2010) 1345–1364

1357

Figure 7. Geometric meaning of the length of O1 Om .

where X = (θ, q T )T and:U (X) =
β
r1 − r0 22

Qn = (r1 − r0 )T n̄0 − (γ0T n0 + γ1T n1 )Qb = (r1 − r0 )T b̄0 − (γ0T b0 − γ1T b1 ).

(47)
(48)
(49)

It should be remarked that Qn = 0 and Qb = 0 as seen from the geometric meaning
shown in Fig. 7. Note that Qb = 0 plays a role of holonomic constraint of rolling
contact in the tangent direction and Qn = 0 that of contact in the normal direction.
However, it is not obvious to see from the set of closed-loop dynamics of (42) and(45) what condition on the state of motion establishes stable grasping. To gain aphysical and geometric insight into the problem, note that the vector (r1 − r0 ) canbe written in the form:
r1 − r0 = (γ0T n0 + γ1T n1 )n̄0 + (γ0T b0 − γ1T b1 )b̄0 ,(50)

and the right-hand side depends on only θ in 0 and the length parameter s. Bearing
this in mind, define:
f = f + β(γ0T n0 + γ1T n1 )(51)
λ = λ + β(γ0T b0 − γ1T b1 )N1 = (γ1T b1 )(−γ0T n0 ) − (−γ1T n1 )(−γ0T b0 ),(52)

and see that (42) and (45) can be rewritten into the following formulae:I θ̈ − f (γ0T b0 ) + λ(γ0T n0 ) − βN1 = 01Ġ + S q̇ + cq̇ + f {J1T n̄0 + (γ1T b1 )e}
G(q)q̈ +2+ λ{J1T b̄0 + (γ1T n1 )e} + βN1 e = 0.

(53)

(54)

It should be remarked that both the (53) and (54) depend on length parameter s,
but not on quantities of the second fundamental form of contour curves. Instead,

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S. Arimoto et al. / Advanced Robotics 24 (2010) 1345–1364

the length parameter s must be governed by the following first-order differential
equation:ds= q̇ T e − θ̇ ,(55)
dtwhere κ1 denotes the curvature of the finger contour and κ0 that of the object
contour. Then, note that the artificial potential U (X) can be regarded as a scalarfunction of the length parameter in a geometric meaning shown in Fig. 7 such that:{κ1 (s) + κ0 (s)}ββr1 − r0 2 = {(γ0T n0 + γ1T n1 )2 + (γ0T b0 − γ1T b1 )2 }22
= U (s).

U (X) =

(56)

Thus, it is possible to see that the set of (53) and (54) can be regarded as theEuler–Lagrange equation obtained by applying the principle of variation to the Lagrangian:
L(X, Ẋ, s) = K(X, Ẋ) − U (s) − f Qn (X, s) − λQb (X, s).

(57)

The details are given in Appendix B. To show under what condition the function
U (s) is minimized, it is necessary to take differentiation of U (s) in s, that results
in:
β −1

dU
= (γ0T n0 + γ1T n1 )(bT0 n0 + bT1 n1 − κ0 γ0T b0 − κ1 γ1T b1 )ds
+ (−γ0T b0 + γ1T b1 )(−bT0 b0 + bT1 b1 − κ0 γ0T n0 + κ1 γ1T n1 )= −(κ0 + κ1 ){(γ1T n1 )(γ0T n0 ) + (γ1T b1 )(γ0T n0 )}.

(58)

From the geometric meaning shown in Fig. 8, this derivative vanishes if and only if
the area A0 equals to the area A1 , i.e.:
A0 = (γ1T b1 )(−γ0T n0 ) = (−γ0T b0 )(−γ1T n1 ) = A1 .

(59)

Figure 8. Minimization of U (s) is attained if and only if the straight line O1 Om meets the contact
point P1 or the area A0 (= −γ0T n0 × γ1T b1 ) is equal to A1 (= γ1T n1 × γ0 b0 ).

S. Arimoto et al. / Advanced Robotics 24 (2010) 1345–1364

1359

Hence, it follows from (52) that:β −1dU (s)= (κ0 + κ1 )N1 (s) = (κ0 + κ1 )(A0 − A1 ).
ds

(60)

Next, to take the second derivative of U (s) at N1 = 0, we calculate the derivative
N1 in s in such a way that:∂N1= (1 + κ1 γ1T n1 )(−γ0T n0 ) + (γ1T b1 )(−bT0 n0 − κ0 γ0T b0 )∂s
− (bT1 n1 − κ1 γ1T b1 )(γ0T b0 ) − (γ1T n1 )(1 + κ0 γ0T n0 )= −(γ0T n0 + γ1T n1 ) + (κ0 + κ1 ){(γ0T b0 )(γ1T b1 ) − (γ0T n0 )(γ1T n1 )}.

(61)

For the sake of simplifying notations, define:lni = −γiT ni ,lbi = γiT bi .

(62)Then, from (60) and (61) it follows that:β −1d2 U (s)= (κ0 + κ1 ){ln0 + ln1 + (κ0 + κ1 )(lb0 lb1 − ln0 ln1 )}.
ds 2 N1 =0(63)

Since N1 = 0 means lb1 = −(lb0 ln1 )/ ln0 , the Hessian (the second derivative ofU (s)) becomes positive when the following inequality is satisfied:κ0 + κ1 <ln0 + ln1
(1/ ln0 ) + (1/ ln1 )=,ln0 ln1 − lb0 lb1
1 + ε0(64)

where ε0 is defined as follows:
2
2
/ ln0
.
ε0 = −lb0 lb1 / ln0 ln1 = lb0(65)

Thus, it is possible to conclude that, for any object with a smooth contour satisfying:1> (1 + ε0 )ln0 = −(1 + ε0 )nT0 γ0 ,
κ0

(66)

if the curvature of the fingertip satisfies:1> (1 + ε0 )ln1 = −(1 + ε0 )nT1 γ1 ,
κ1 (s)(67)

then the Hessian becomes positive and minimization of U (s) is attained at s =s ∗ such that N1 (s ∗ ) = 0. Inequality (67) is a little more restrictive for κ1 (s) thaninequality (41).
Further comparison of (67) with (41) gives us an interesting observation that atypical human thumb has a contour curve shown in Fig. 9, which has a good margin
ε0 of (65) compared to other fingertip examples illustrated in Figs 4 and 5.

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S. Arimoto et al. / Advanced Robotics 24 (2010) 1345–1364

Figure 9. Sketch of a human-like thumb, where the fingertip center O1 is located beneath the base of
the fingernail.

6. Extension of the Dirichlet–Lagrange Theorem for Pinching a Rectangular
ObjectIn this section, we show that, when the line connecting the contact points P1 andP2 becomes parallel to the line O1 O2 as shown in Fig. 3, U (s1 , s2 ) is minimized
under the constraint of V = 0 [see (30)], and at the same time any solution to theset of closed-loop dynamics (1), (2) and (19) under rolling constraints (5) and (6)converges asymptotically to such an equilibrium posture, provided that the solution
trajectory starts in a neighborhood of the equilibrium state. To do this, define:
fi = fi + βl(s1 , s2 )
(68)
(69)
λi = λi − (−1)i βd(s1 , s2 ),and note that:−(−1)i βJiT (r1 − r2 ) = βJiT {l n̄0i − (−1)i d b̄0i },i = 1, 2.

(70)

Substituting (70) into (19) and referring to (68) and (69) yields:1Ġi + Si q̇i + ci q̇i + fi {JiT n̄0i − (−1)i (bTi γi )ei }Gi q̈i +2+ λi {JiT b̄0i − (−1)i (nTi γi )ei } + Ni ei = 0,i = 1, 2,(71)

where:
Ni = β{(−1)i l(bTi γi ) − d(nTi γi )} + αi {pi − pi (0)},i = 1, 2.

(72)

On the other hand, (1) and (2) can be rewritten into the forms:
I θ̈ −(73)
M ẍ − f1 n̄01 − f2 n̄02 − λ1 b̄01 − λ2 b̄02 = 0
TTT
f2 (b02 γ02 ) + λ1 (n01 γ01 ) − λ2 (n02 γ02 ) + SN = 0, (74)f1 (bT01 γ01 ) +1361

S. Arimoto et al. / Advanced Robotics 24 (2010) 1345–1364

where:
SN = βl(bT01 γ01 − bT02 γ02 ) − βd(nT01 γ01 + nT02 γ02 ) = β{(s1 − s2 )l + lw d}.

(75)

Now it is possible to show that the set of equations (73)–(75) together with (7)can be regarded as a set of Euler–Lagrange equations obtained by applying thevariational principle to the Lagrangian:( fi Qni + λi Qbi ),(76)
L = K(X, Ẋ) − U (s1 , s2 ) +i=1,2in which the external forces of damping ci q̇i for i = 1, 2 through finger joints aretaken into account. In fact, from (27)–(29) and (A.1) in Appendix A it follows that: dUi dsi
dU (s1 , s2 )dsi==(−1)i β{(nTi γi )d − (−1)i (bTi γi )l}κidt
dsi dt
dt
i=1,2
i=1,2

β{(nTi γi )d − (−1)i (bTi γi )l}(θ̇ − ṗi )
(77)
=
i=1,2

= β{(nT1 γ1 + nT2 γ2 )d + (bT1 γ1 − bT2 γ2 )l}θ̇ +

βNi ṗi ,

(78)

i=1,2

where Ni is defined as:
Ni = (−1)i (bTi γi )l − (nTi γi )d.

(79)

By using (28) and (29), (78) can be written as:dU (s1 , s2 )βNi ṗi= β{(l + lw )d + (s1 − s2 − d)l}θ̇ +
dt
i=1,2

T
= SN θ̇ +βNi ei q̇i .

(80)

i=1,2

Thus, we conclude that from (72) the variation of P takes the form:
 αi{pi − pi (0)}2
dP = d U +2[βNi + αi {pi − pi (0)}] dpi= SN dθ +i=1,2= SN dθ +

Ni eTi dqi .

(81)

i=1,2

The last term of the left-hand side of (74) comes directly from the first term of theright-hand side of (81) in the variational form of the potential P (X, s1 , s2 ). The lastterm Ni ei also comes directly from the last term of (81). Thus, it is possible toprove that, if the posture of the fingers–object system satisfying the condition thatO1 O2 is parallel to P1 P2 as shown in Fig. 3 is an isolated equilibrium state, thenthe posture must be asymptotically stable because the system is fully dissipated [9],1362

S. Arimoto et al. / Advanced Robotics 24 (2010) 1345–1364

no matter how the system is holonomically constrained. If both the fingers are of
1 d.o.f., then the total degrees of freedom of the system becomes single and therefore the equilibrium state is isolated. In the case of a redundant degrees-of-freedom
system like the setup illustrated in Fig. 1, it is necessary to extend the so-called
Dirichlet–Lagrange stability theorem [9, 10] to a system with redundancy in degreeof freedom together with holonomic constraints. Such an extension of the theorem
is possible as already discussed in a special class of robot control problems [11],but the details are too mathematically sophisticated and therefore will be discussed
in a future paper.
7. Conclusions
Characterization of a desired fingertip geometry of a robot hand for precision prehension of 2-D objects with arbitrary shapes is discussed on the basis of Euler–
Lagrange equations of motion of physical interactions between fingertip contour
curves and object surfaces. Based on the use of control signals from the fingers–
thumb opposability, a region of the desired fingertip contour is characterized byits curvature relative to the distance from the fingertip center to the tangent at thecontact point. From the stabilization point of view of grasping, it is shown that atypical human thumb has a wider region in the contour curve than other fingertip
geometries, such as an ellipse or a crescent moon type.
References
1. J. Napier, Hands. Princeton University Press, Princeton, NJ (1993).
2. S. Arimoto, Intelligent control of multi-fingered hands, Annu. Rev. Control 28, 75–85 (2004).
3. S. Arimoto, Control Theory of Multi-fingered Hands: A Modelling and Analytical-Mechanics Approach for Dexterity and Intelligence. Springer, London (2008).
4. R. Murray, Z. Li and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation. CRC
Press, Boca Raton, FL (1994).
5. 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, SICE J. Control Meas. Syst. Integrat. 2, 379–386(2009).
6. K. Nomizu, Kinematics and differential geometry of submanifolds — rolling a ball with a prescribed locus of contact, Tohoku Math. J. 30, 623–637 (1978).
7. S. Arimoto, M. Yoshida, M. Sekimoto, K. Tahara and J.-H. Bae, Modeling and control for 2-D
grasping of an object with arbitrary shape under rolling contact, in: Proc. SYROCO 2009, Gifu,
pp. 517–522 (2009).
8. S. Arimoto, A differential-geometric approach for 2D and 3D object grasping and manipulation,
Annu. Rev. Control 31, 189–209 (2007).
9. W. M. Oliva, Geometric Mechanics (Lecture Notes in Mathematics 1798). Springer, Berlin (2002).
10. F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems. Springer, New York (2005).
11. S. Arimoto, M. Sekimoto, H. Hashiguchi and R. Ozawa, Natural resolution of ill-posedness of
inverse kinematics for redundant robots: a challenge to Bernstein’s degrees-of-freedom problem,
Adv. Robotics 19, 401–434 (2005).

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1363

Appendix A
Partial differentiation of U (s1 , s2 ) of (27) in si results in:


∂U
∂d
∂l
=β d
+l
∂si
∂si
∂si
= (−1)i βκi {(nTi γi )d − (−1)i (bTi γi )l}= (−1)i βκi (r1 − r2 )T ξ̄ i ,i = 1, 2.

(A.1)

Further, since n̄0i and b̄0i are irrelevant to si and (r1 − r2 )T b0i = 0 at si = s ∗ , itfollows that (r1 − r2 )T ∂ ξ̄ i /∂si = l(s1 , s2 )∂(bTi γi )/∂si at si = s ∗ and:
∂ 2U
∂ ξ̄
= −βκi (r1 − r2 )T i
2
∂si si =s ∗∂si si =s ∗= −βκi l(bTi γi + κi nTi γi )= −βκi2 l(1/κi + nTi γi ),(A.2)

which shows (39).

Appendix B
From the definition of N1 in (52) and the derivative of U (s) in s as shown in (58),
it follows that:
ds
dU ds
dU
=
= β(κ0 + κ1 )N1 .
(B.1)
dt
ds dt
dt
Substitution of (55) into the right-hand side of (B.1) yields:dU= β(θ̇ − eT q̇)N1 ,
dt

(B.2)

dU (s) = −βN1 dθ + βN1 eT dq.

(B.3)

which shows:

The last term in the left-hand side of (53) comes from the first term of the right-handside of (B.3) and the last term in the right-hand side of (54) comes from the second
term of that of (B.3).

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S. Arimoto et al. / Advanced Robotics 24 (2010) 1345–1364

About the Authors
Suguru Arimoto received the BS from Kyoto University, in 1959, and DE fromthe University of Tokyo, in 1967. He was an Associate Professor of Osaka University during 1968–1973, and a Professor, during 1973–1990. In 1988, he joined
the Faculty of Engineering of the University of Tokyo as a Professor and, in 1997,
retired. Since 1997, he has been with Ritsumeikan University as a Professor. He isan IEEE Fellow (1983), an IEICE Fellow (2000), an RSJ Fellow (2003), a JSME
Fellow (2005) and an IEEE Life Fellow (2007). He was awarded a Royal Medalwith a Purple Ribbon from the Japanese Government in 2000, IEEE 3rd Millennium Memorial Medal in 2000, Pioneer Award from the IEEE R&A Society in 2006, and the 2007
Rufus Oldenburger Medal from the ASME.
Morio Yoshida received the BS, MS and DE degrees from Ritsumeikan University, Japan, in 2002, 2004 and 2007, respectively. He is currently a Research
Scientist in RIKEN-TRI Collaboration Center, RIKEN, Japan. His current research interests are robot hands, modeling and robot control.

Masahiro Sekimoto received the BS, MS and PhD degrees from Ritsumeikan
University, Japan, in 2003, 2005 and 2007, respectively. In 2007, he joined the
Research Organization of Science and Engineering of Ritsumeikan University,where he is currently a Postdoctoral Fellow. He was a Visiting Scholar at Georgia Institute of Technology, USA, in 2008. His research interests include control
design of redundant robots and mechanical analyses of human movements.

Ji-Hun Bae received the BS and MS degrees in Electrical Engineering from Myongji University, South Korea, in 1999 and 2001, respectively, and the PhD degreein Robotics from Ritsumeikan University, Japan, in 2004. He is currently a Research Scientist at DART (Division for Applied Robot Technology), KITECH(Korea Institute of Industrial Technology). From 2004 to 2006, he was a Postdoctoral Fellow at the 21st Century COE Program at Ritsumeikan University, Japan.
He worked as a Senior Researcher at PIRO, from 2006 to 2008, and as a Postdoctoral Fellow at KIST, from 2008 to 2009. His research interests include robot
manipulation, robot hands and arms, sensory feedback control, biological control and bio-mimetics.