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Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2011, Article ID 684034, 8 pagesdoi:10.1155/2011/684034

Research Article

A Mathematical and Numerically Integrable Modelingof 3D Object Grasping under Rolling Contacts betweenSmooth Surfaces

Suguru Arimoto and Morio Yoshida

RIKEN-TRI Collaboration Center for Human-Interactive Robot Research, Nagoya, Aichi 463-0003, Japan

Correspondence should be addressed to Morio Yoshida, [email protected]

Received 31 March 2011; Accepted 25 July 2011

Academic Editor: Antonio Munjiza

Copyright © 2011 S. Arimoto and M. Yoshida. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

A computable model of grasping and manipulation of a 3D rigid object with arbitrary smooth surfaces by multiple robot fingerswith smooth fingertip surfaces is derived under rolling contact constraints between surfaces. Geometrical conditions of pure rollingcontacts are described through the moving-frame coordinates at each rolling contact point under the postulates: (1) two surfacesshare a common single contact point without any mutual penetration and a common tangent plane at the contact point and(2) each path length of running of the contact point on the robot fingertip surface and the object surface is equal. It is shownthat a set of Euler-Lagrange equations of motion of the fingers-object system can be derived by introducing Lagrange multiplierscorresponding to geometric conditions of contacts. A set of 1st-order differential equations governing rotational motions of eachfingertip and the object and updating arc-length parameters should be accompanied with the Euler-Lagrange equations. Furthermore, nonholonomic constraints arising from twisting between the two normal axes to each tangent plane are rewritten into a setof Frenet-Serre equations with a geometrically given normal curvature and a motion-induced geodesic curvature.

1. Introduction

In relation to the recent development of robotics researchand neurophysiology, there arises an important question on astudy of the functionality of the human hand in grasping andobject manipulation interacting physically with environmentunder arbitrary geometries of objects and fingertips. Anotherquestion also arises as to whether a complete mathematicalmodel of grasping a 3D rigid object with an arbitraryshape can be developed and used in numerical simulationto validate control models of prehensile functions of aset of multiple fingers. In particular, is it possible todevelop a mathematical model as a set of Euler-Lagrangeequations that govern a whole motion of the fingers-objectsystem under rolling contact constraints between each robotfingertip and a rigid object with an arbitrary smooth surface.

Traditionally in robotics research, a rolling contactconstraint between two rigid-body surfaces is defined as thezero velocity of one translational motion of the common

contact point on the fingertip surface relative to another onthe object surface [1]. Therefore, rolling contact constraintsare expressed in terms of velocity relations called a Pfaffianform. Montana [2] presented a complete set of all velocityrelations of a rolling contact by using the normalized gaussframe for expressing given smooth surfaces of fingertips anda 3D object. Based on Montana’s set of Pfaffian forms, Paljuget al. [3] formulated a dynamic model for the control ofrolling contacts in multiarm manipulation. However, it isuncertain whether the derived model of equations of motioncan be computationally integrable in time in case of rollingcontacts between general smooth surfaces, since in [3] only alimited case of ball-plate contacts was numerically simulated.Another work by Cole et al. [4] tried to simulate a 3Dgrasping, but it is uncertain whether it could overcome theproblem of arise of a nonholonomic constraint pointed outby Montana [2].

Even in case of 2D grasping by means of dual robotfingers with smooth fingertip surfaces, the integrability of

2 Modelling and Simulation in Engineering

Pfaffian forms of rolling contact constraints was shown veryrecently in our previous paper [5], where a complete setof computational models of Euler-Lagrange equations ofmotion of the whole fingers-object system and a pair of first-order differential equations expressing update laws of arc-length parameters along smooth contour curves of the objectwere given. Instead of the zero relative-velocity assumptionof rolling contact, the following set of postulates for purerolling contact is introduced:

(1) two contact points on each contour curve mustcoincide at a single common point without mutualpenetration,

(2) the two contours must have the same tangent at thecommon contact.

Owing to these postulates, the path length of one contactpoint running on each fingertip contour curve and that ofanother contact point running on the object contour mustcoincide, that is, the constraint can be expressed eventuallyin the level of position variable. Hence, it is shown in [6]that Pfaffian constraints are integrable, and their integralforms are derived explicitly by using the moving framecoordinates. It is further shown [6] that the quantities of thesecond fundamental form of concerned contour curves donot appear in the Euler-Lagrange equations but play a keyrole in the update laws of arc-length parameters of the curves.

This paper aims at extending such a moving framecoordinates approach for mathematical modelling of 2Dgrasping to computable mathematical modelling of 3Dgrasping of a rigid object with arbitrary smooth surfacesunder the following set of 3D rolling contact constraints:

(a1) two contact points on each curved surface mustcoincide at a single common point without mutualpenetration,

(a2) the two curved surfaces have the same tangent at thecommon contact point, that is, each surface has thesame unit normal with mutually opposite directionat the common contact point,

(a3) the two path lengths running on their correspondingsurfaces must be coincident.

In the previous paper [7], a set of Euler-Lagrangeequations of motion of the fingers-object system have beenderived by using the moving frame coordinates, but anyexplicit set of update laws of moving frame coordinateshave been not given yet. In particular, any mathematicalrole of the quantities of the second fundamental form of acontact curve running on a concerned surface has not yetbeen studied in a mathematically explicit way. Therefore,it still remains unsolved to construct a complete set ofequations of 3D grasping under rolling contact constraintsin the situation of arbitrary given geometry of surfaces. Thispaper shows that nonholonomic constraints arising fromrelative twisting among the two normal axes at the contactpoint can be naturally resolved into determination of eachof geodesic curvatures of the curves of the contact point onthe fingertip surface and the object surface. Another second

fundamental form of normal curvature on each surface isassumed to be extracted from a data structure of a givenrigid body object, together with that of unit normal at eachspecified point on its surface. Thus, a set of 3D Frenet-Serretequations with normal and geodesic curvatures that updatethe moving frame coordinates are determined and shown tobe computationally integrable together with the set of Euler-Lagrange equations of motion of the whole system.

2. Preliminary Results on Derivation ofEuler-Lagrange Equations

Consider firstly a physical situation that a pair of multijointrobot fingers is grasping a 3D rigid body as seen in Figure 1.In this figure, the inertial frame denoted by O-xyz is fixedin the Euclidean space E3, and local coordinates systemsdenoted by Oi-XiYiZi for i = 1, 2 are introduced at eachrobot fingerend. The local coordinates system of the object isdenoted by Om-XYZ as shown in Figure 1, where Om standsfor the object mass center. Next, denote each locus of pointsof contact between the two surfaces by a curve γi(si) (3-dimensional vector in E3) with length parameter si on itscorresponding surface Si (i = 0, 1), where i = 0 signifies theobject, and i = 1 does the left hand fingerend. It is possibleto assume that, given a curve γ1(s1) as a locus of points ofcontact on S1 and another curve γ0(s0) as a locus of contactpoints on S0, the two curves coincide at contact point P1 andshare the same tangent plane T1 at P1 (see Figure 2). Further,during continuation of rolling contact, the two curves γ0(s0)and γ1(s1) can be described in terms of the same lengthparameter s in such a way that s0 = s + c0 and s1 = s + c1,where c0 and c1 are constant.

Second, suppose that at some s of the length parameterthe two curves γ0(s0) and γ1(s1) coincide at P1(s). Since γ0(s0)is described in local coordinates Om-XYZ, its expression inthe frame coordinates is given by

γ0(s0) = Π0γ0(s0), (1)

where Π0 is a 3× 3 rotational matrix composed of three unitorthogonal vectors rX , rY , and rZ that are expressed in theinertial frame coordinates O-xyz as shown in Figure 1, thatis,

Π0 = (rX , rY , rZ). (2)

Since γ0(s0) is parametrized by length parameter,γ′0(s0) = dγ0(s0)/d(s0) must be expressed by the unittangent vector b0(s0) at P1(s) lying on the tangent plane T1.According to (a1) and (a2), it is possible to suppose that thereexist the two unit normals n0(s0) and n1(s1) expressed incorresponding local coordinates Om-XYZ and O1-X1Y1Z1,respectively (see Figure 2). Then, it is possible to certify that

n0 = −n1, (3)

at s0 = s1 = s, where

n0 = Π0n0, n1 = Π1n1, (4)

Modelling and Simulation in Engineering 3

O

y

x

x

z

q11

J0

J1

q12

J2 = O1

Y1

q22

q21

X1

ϕ1

η1

P1S(u, v)

−n(u, v)rY

rZ Om rX

P2

• O2

q23

yy

Tangent plane at P1

∂S∂u

∂S∂v

Figure 1: A pair of robot fingers grasping a 3D rigid object withsmooth surfaces.

Frame coordinates

y

O

z O1

x

ω1

T1

Object coordinates

Yω0

Z

Om Xb1 b0

n1

e1

P1

S1

S0n0

e0

Figure 2: Definition of the moving frame coordinates systemcentering at the rolling contact point.

and Π1 = (rX1, rY1, rZ1), rX1 denotes the unit vectorof X1-axis of O1-X1Y1Z1 expressed in the inertial framecoordinates, and rY1 and rZ1 have a similar meaning.

In what follows, we denote vectors ni and bi for i = 0, 1with upper bar when they are expressed in the inertial framecoordinates as seen in Figure 2. We also denote the derivativeof Πi in time t by Πi and similarly the derivatives of ni and bi

in t by ni, and bi. If we assume that the instantaneous axis ofangular velocity of the object through the mass center Om isdenoted by ω = (ωx,ωy ,ωz)

T in the frame coordinates, then

the angular velocity vectorω attached to the local coordinatesOm-XYZ can be defined in such a way that

ω = Π0(ωX ,ωY ,ωZ)T = Π0ω, (5)

where we define ω = (ωX ,ωY ,ωZ)T. It is well known in thetext books [8–10] that

Π0 = Π0Ω0, (6)

where

Ω0 =

⎛⎜⎜⎝

0 −ωZ ωY

ωZ 0 −ωX

−ωY ωX 0

⎞⎟⎟⎠. (7)

It is easy to check that, in the illustrative case of aspherical left hand fingertip shown in Figure 1, we have

Π1 = Π1Ω1, Ω1 =

⎛⎜⎜⎝

0 p1 0

− p1 0 0

0 0 0

⎞⎟⎟⎠, (8)

where p1 = q11 + q12 because both the rotational axes ofjoints J1 and J2 have the same direction in z-axis of the framecoordinates O-xyz.

Let us now interpret the first postulate (a1) in amathematical form described by

r1(q1)

+ Π1γ1(s) = r0(x) + Π0γ0(s), (9)

where r1(q1) denotes the position vector of O1 (the centerof the left hand fingerend) expressed in terms of theframe coordinates and q1 = (q11, q12)T, and r0(= x) doesthe position vector of Om (the object mass center) alsoexpressed in the frame coordinates and x = (x, y, z)T. Then,differentiation of (9) in t yields

(r1 − r0) + Π1γ1 + Π1b1

(ds1

dt

)= Π0γ0 + Π0b0

(ds0

dt

).

(10)

If during rolling of the contact point the tangent vector b1

of the fingerend has the same direction as that of b0 (of theobject), that is, if b1 = b0, then on account of (a3), (10)reduces to

(r1 − r0) + Π1Ω1γ1 −Π0Ω0γ0 = 0. (11)

According to the previous paper, multiplication of therotation matrix of the moving frame coordinates defined byΠ0Ψ0 from the right yields

(Rn1,Rb1,Re1) � (r1 − r0)TΠ0Ψ0

− ωT(γ0 ×Ψ0)

+ ωT1

(γ1 ×Ψ∗1

)

= 0,

(12)

where we define e0 = n0 × b0, e1 = n1 × b1, and

Ψ0 = (n0, b0, e0), Ψ∗1 = (−n1, b1,−e1) (13)


(see Figure 2), and we use Π0 = Ψ0 = Π1Ψ∗ and

γ0 ×Ψ0 =(γ0 × n0, γ0 × b0, γ0 × e0

), (14)

and γ1×Ψ∗1 has a similar meaning. Equation (13) means thethree equalities Rn1 = 0, Rb1 = 0, and Re1 = 0 that constitutethe set of three Pfaffian forms expressing the rolling contactconstraint of zero-relative velocity. In the previous paper [7],it is shown that the Pfaffian forms of (13) are integrable withthe integral forms

ddt

(Qn1,Qb1,Qe1) = (Rn1,Rb1,Re1) = 0, (15)

where

(Qn1,Qb1,Qe1) = (r1 − r0)TΠ0Ψ0 + γT1Ψ

∗1 − γT

0Ψ0,

Ψ1 = (n1, b1, e1)(16)

provided that b0 = b1.By virtue of the integrability of each Pfaffian form of

rolling contact constraints, the Lagrangian of the system iswritten into

L = K −Mgy −∑

i=1,2

Pi(qi)

−∑

i=1,2

(fiQni + λiQbi + ηiQei

),

(17)

where

K =∑

i=1,2

12qTi Gi(qi)

+M

2‖x‖2 +

12ωTHω. (18)

In these equations, M denotes the mass of the object, H , theinertia matrix of the object around its mass center, Gi(qi), theinertia matrix of finger i, Pi, the potential energy of finger i,g, the gravity constant, and fi, λi, and ηi express Lagrangemultipliers corresponding to constraints

(Qn1,Qb1,Qe1) = (0, 0, 0) (19)

and x = dr(x)/dt. Then, by applying the variational principleto L described as

∫ t1

t0

⎧⎨⎩δL +

∑

i=1,2

uTi δqi

⎫⎬⎭dt = 0, (20)

with control input ui at finger joints, it is possible to obtainthe following set of Euler-Lagrange equations:

Mx −∑

i=1,2

(fin0i + λib0i + ηie0i

)+ Mgey = 0, (21)

Hω + ω ×Hω −∑

i=1,2

γ0i ×(fin0i + λib0i + ηie0i

) = 0,

(22)

Gi(qi)qi +

{12Gi(qi)

+ Si(q, q)}

qi +∂Pi(qi)

∂qi

− JTi

(qi){

fini − λibi + ηiei}

−WTi

{γi ×

(fini − λibi + ηiei

)} = ui, i = 1, 2,(23)

where ey = (0, 1, 0)T, and the meaning of Wi will beexplained later. It should be noted that the sum of innerproducts of x and (21), ω and (22), and qi and (23) fori = 1, 2 yields the energy relation

∑

i=1,2

qTi ui =

ddt

⎧⎨⎩K + Mgy +

∑

i=1,2

Pi(qi)⎫⎬⎭. (24)

3. Necessary Conditions for Updating MovingFrame Coordinates

In order to always keep the tangent vector b1 of the fingerendat the contact point P1 to coincide with the tangent vectorb0 of the object surface at the same common contact point,we first show that the following two equations should besatisfied necessarily:

ωT0 b0 = ωT

1 b1, (25)

(κn0 + κn1)ds0

dt= ωT

0 e0 + ωT1 e1, (26)

as shown in detail in (A1) of Appendix A, where κn0 denotesthe normal curvature of the object surface at the contactpoint P1, and κn1 does that of the fingerend surface S1

at P1. Both the normal curvatures κn0 and κn1 should bedetermined in accordance with the geometric structure oftheir corresponding surfaces, once the direction of each locusof contact points, that is, b1 or b0, is given. Similarly, asshown in (A2) of Appendix A, the conditions b0 = b1 and

b0 = b1 imply

(ωT

0 n0 + ωT1 n1

)+ (κe0 + κe1)

ds0

dt= 0, (27)

where κe0 denotes the geodesic curvature of the object surfaceat P1, and κe1 has a similar meaning. This equation does notdetermine each κe0 or κe1 individually. Therefore, let us tryto differentiate e0 and e1 in t. However, as shown in (A3) ofAppendix A, we rederive only (25) and (26).

We are now in a position to find a necessary condition formaintaining the equality of (25) for the time being. To thisend, it is important to see that the time rate of the equality(25) reduces to

{(ωT

0 e0

)κe0 −

(ωT

1 e1

)κe1

}ds0

dt

= ωT1 b1 − ωT

0 b0 +(−κn0ω

T0 n0 + κn1ω

T1 n1

),

(28)

as shown in (A4) of Appendix A. Then, this equation togetherwith (27) leads to

⎛⎝ 1 1

ωT0 e0 −ωT

1 e1

⎞⎠⎛⎝κe0

κe1

⎞⎠ds0

dt=⎛⎝−(ωT

0 n0 + ωT1 n1

)

ξ1

⎞⎠,

(29)


where

ξ1 �(−κn0ω

T0 n0 + κn1ω

T1 n1

)ds0

dt+ ωT

1 b1 − ωT0 b0. (30)

Thus, it is possible to determine each geodesic curvatureindividually by inverting the coefficient 2× 2 matrix of (29)in the following way:

κe0ds0

dt=−ωT

1 e1

(ωT

0 n0 + ωT1 n1

)+ ξ1

ωT1 e1 + ωT

0 e0, (31)

κe1ds1

dt=−ωT

0 e0

(ωT

0 n0 + ωT1 n1

)− ξ1

ωT1 e1 + ωT

0 e0

. (32)

Finally, it is possible to see that the moving framesdenoted by Ψ0 and Ψ1 should satisfy the Frenet-Serreequations

ddtΨ0 = Ψ0

⎛⎜⎜⎝

0 κn0 0

−κn0 0 −κe0

0 κe0 0

⎞⎟⎟⎠

ds0

dt,

ddtΨ1 = Ψ1

⎛⎜⎜⎝

0 κn1 0

−κn1 0 −κe1

0 κe1 0

⎞⎟⎟⎠

ds0

dt.

(33)

4. Sufficient Conditions for Updating theMoving Frame Coordinates

In the Frenet-Serre equation of (31), the coefficient κn0 calledthe normal curvature is determined by the geometric shapeof the object surface at point P1 denoted by γ0(s0) in the localcoordinates Om-XYZ, and the other normal curvature κn1

in (32) is also determined similarly. The geodesic curvaturesκe0 and κe1 are determined via the instantaneous motion ofrolling contact, so that they satisfy (27) and (28). We nowshow under the postulates (a1) to (a3) that if κe0 and κe1 aredetermined by (31) and (32), respectively, and the tangentvectors b0(0) and b1(0) of the moving frame coordinates atthe initial time are chosen to coincide with each other and atthe same time to satisfy (25) at t = 0, then for any t > 0, itfollows that

ddt

b1 = ddt

b0,ddt

e1 = − ddt

e0,

b1 = b0, e1 = e0.

(34)

To prove this, first note again that the postulates (a1) and (a2)imply

n0 = −n1,ddt

n0 = − ddt

n1. (35)

Next, note that the sum of (31) and (32) implies (27). At thisstage, suppose that b1 is not coincident with b0 at some t > 0though both b1 and b0 are lying on the same tangent plane.

In other words, suppose that bT1 b0 = cos θ and θ(t) stands

for θ(t, s1(t), s0(t)), then, as shown in Appendix B, we have

θ = ωT1 n1 + ωT

0 n0 + (κe0 + κe1)ds0

dt. (36)

Further, as discussed in (B3) of Appendix B, geodesic curva-tures κe0 and κe1 should be defined as

κe0 = ∂θ

∂s0, κe1 = ∂θ

∂s1. (37)

It is important to remark that (36) was first derived byMontana [2] as a nonholonomic constraint of rolling.Equation (36) can be interpreted by Murray et al. [1]as the nonholonomic constraint that governs the rotatingmotion of one tangent plane to the fingerend relative toanother tangent plane of the object surface caused by relative“twisting” between the axis of normal n1 and that of n0.Nevertheless, it is further important to note that if κe0 andκe1 are set as shown in (31) and (32), respectively, then theright hand side of (36) becomes zero due to (27). That is,(31) and (32) imply θ = 0. Therefore, if at the initial timeb1(0) = b0(0), then the setting of (31) and (32) for geodesiccurvatures κe1 and κe0 leads to b1(t) = b0(t) for t > 0 asfar as ωT

1 e1 +ωT0 e0 /= 0. Since n1 = −n0 as far as the contact is

maintained, the equality b1(t) = b0(t) implies e1(t) = −e0(t)for t ≥ 0, then, from (A1) of Appendix A, (25) and (26)follow. At the same time, from (26) and (27), it follows thatdb1/dt = db0/dt in view of the first two equations of (A2) ofAppendix A. Thus, it is concluded that all equalities in (34)follow.

5. A Numerically Integrable Set of DifferentialEquations Under Rolling Constraints

It is now possible to show a set of all the differential equationsof motion of the fingers-object system under rolling contactconstraints. In what follows, we use the suffix “0i” forexpressing variables on quantities of the object at contactpoint Pi of the i-th finger and the suffix i for those ofthe fingerend surface of finger i at Pi. For convenience,we use ω instead of ω0. In the following, we give theset of Euler-Lagrange equations, first-order equations ofrotation matrices Π0 and Πi, update equations of lengthparameters, and Frenet-Serre equations for updating movingframe coordinates at contact points

Mx−∑

i

Π0Ψ0iλi −Mgey = 0, (Ex)

Hω + ω ×Hω −∑

i

γ0i ×Ψ0iλi = 0, (Eω)

Giqi +{

12Gi + Si

}qi +

∂Pi∂qi

+{JTi Πi + WT

i γi×}Ψ∗i λi = ui,

(Ei)


ddtΠ0 = Π0Ω0,

ddtΠi = ΠiΩi, (Er)

ddtsi = ωTe0i + ωT

i eiκn0i + κni

, (Es)

ddtΨ0i = Ψ0iK0i

dsidt

,ddtΨi = ΨiKi

dsidt

, (Ef s)

where

K0i =

⎛⎜⎜⎝

0 κn0i 0

−κn0i 0 −κe0i

0 κe0i 0

⎞⎟⎟⎠,

Ki =

⎛⎜⎜⎝

0 κni 0

−κni 0 −κei0 κei 0

⎞⎟⎟⎠,

Ψ0i = (n0i, b0i, e0i), Ψi = (ni, bi, ei),

λi =(fi, λi,ηi

)T, Ji(qi) = ∂ri

(qi)

∂qTi

,

(38)

and γ0i denotes the contact point Pi on the object surfaceexpressed by the object local coordinates Om-XYZ, and γidoes that of Pi on the fingerend surface of finger i expressedby the fingerend local coordinates Oi-XiYiZi. In (Ei), Wi is anm× 3 matrix depending on qi, where m denotes the degreesof freedom. In the case of a pair of robot fingers depicted inFigure 1, it is obvious to see that

W1 =

⎛⎜⎜⎝

0 0

0 0

1 1

⎞⎟⎟⎠, W2 =

⎛⎜⎜⎝

1 0 0

0 sin q21 sin q21

0 cos q21 cos q21

⎞⎟⎟⎠. (39)

It should be remarked again that (Ef s) expresses a set ofFrenet-Serre equations for determining each moving framecoordinates at contact point Pi, and then the geodesiccurvatures κe0i and κei are determined in the same manneras shown in (31) and (32). Further, computation of ωi and ωappearing in (31) and (32) through ξi defined by (30) can beexecuted simultaneously via numerical integration of (Eω)and (Ei). In practice, it is possible to compute ωi by

ω = H−1

⎧⎨⎩−ω ×Hω +

∑

i

γ0i ×Ψ0iλi

⎫⎬⎭. (40)

Analogously, it is possible to compute ωi since ωi must beexpressed by a function form of V(qi)qi, and qi can becalculated by multiplying (Ei) by G−1

i (qi) from the left.

6. Conclusions

A computational model of dynamics of 3D object graspingand manipulation under rolling contact constraints bymeans of multiple multijoint robot fingers with smoothfingerend surfaces is derived on the basis of the postulates

of pure rolling contact constraint. The postulates are sum-marized: (1) at the contact point, the fingerend and objectsurfaces share a common tangent plane with each normalwith opposite direction and (2) the path length of contactpoints running on the fingerend is coincident with thatrunning on the object surface. The postulates are adoptedby referring to Nomizu’s work [11] in which it is assumedthat any relative twist motion does not arise. The proposedmodel is composed of a set of 2nd-order Euler-Lagrangeequations derived by using the moving frame coordinatesand 1st-order Frenet-Serre equations together with 1st-order differential equations governing update laws of lengthparameters and rotational motions of the local coordinates.The nonholonomic constraint arising from possible relativetwist of the two normal axes at the contact point isresolved into determination of the geodesic curvatures ofthe fingerend and object surfaces. This leads to a conclusionthat the whole set of simultaneous differential equationswith constraints are numerically integrable (as a preliminaryresult of numerical simulation, see [12]).

Appendices

A. Necessary Conditions

(A1) Note that n1 = −n0 and (d/dt)n1 = −(d/dt)n0,

ddt

n0 = Π0n0 + Π0n0

= Π0Ω0n0 + Π0

(∂n0

∂s0

)ds0

dt

= Π0{ω0 × (b0 × e0)} − κn0Π0b0ds0

dt

=(ωT

0 e0 − κn0ds0

dt

)b0 −

(ωT

0 b0

)e0,

ddt

n1 =(ωT

1 e1 − κn1

ds1

dt

)b1 −

(ωT

1 b1

)e1.

(A.1)

If b0 = b1 and e0 = −e1, then it follows that

ωT0 b0 = ωT

1 b1, (∗)

(κn0 + κn1)ds0

dt= ωT

0 e0 + ωT1 e1. (∗∗)

(A2) Similarly, it follows that

ddt

b0 = Π0b0 + Π0b0

= Π0

{(ωT

0 n0

)e0 −

(ωT

0 e0

)n0

}

+ Π0(κn0n0 + κe0e0)ds0

dt

=(ωT

0 n0 + κe0ds0

dt

)e0

−(ωT

0 e0 + κn0ds0

dt

)n0,


ddt

b1 =(ωT

1 n1 + κe1ds1

dt

)e1

−(ωT

1 e1 + κn1ds1

dt

)n1.

(A.2)

These two equations imply (Ei) and

(ωT

0 n0 + ωT1 n1

)+ (κe0 + κe1)

ds0

dt= 0 (∗∗∗)

if b0 = b1 and (d/dt)b0 = (d/dt)b1.

(A3) Similarly, it follows that

ddt

e0 =(ωT

0 b0

)n0 −

(ωT

0 n0 + κe0ds0

dt

)b0,

ddt

e1 =(ωT

1 b1

)n1 −

(ωT

0 n1 + κe1ds1

dt

)b1.

(A.3)

These two equations imply (Eω) and (Es) if e0 = −e1,b0 = b1, and (d/dt)e0 = −(d/dt)e1.

(A4) Time rate of (Eω) reduces to

ωT0 b0 + ωT

0 b0 = ωT1 b1 + ωT

1 b1. (A.4)

Since bi = (κnini+κeiei)(ds0/dt) for i = 0, 1, the aboveequality reduces to

{(ωT

0 e0

)κe0 −

(ωT

1 e1

)κe1

}ds0

dt

=(−κn0ω

T0 n0 + κn1ω

T1 n1

)ds0

dt

+ ωT1 b1 − ωT

0 b0

� ξ1(ω1, ω0).

(A.5)

B. Preliminary Remarks on Geodesic Curvature

b0i = Π0b0i(tangent vectors on object surface i

),

n0i = Π0n0i(normal vectors on object surface i

),

bi = Πibi(tangent vectors on fingerend surface i

),

ni = Πini(normal vectors on fingerend surface i

),

(B.1)

both Ωi and Ω0 are skew symmetric.

(B1) Derivation of ∂θ/∂t where θ(t, si(t), si0(t)).

If bT1 b0 = cos θ, then

bT1Π

T1Π0e0 = cos

(π

2+ θ)= − sin θ,

−∂θ

∂tcos θ = θ

∂t

(bT

1ΠT1Π0e0

)

= bT1 Π

T1Π0e0 + bT

1ΠT1 Π0e0

= bT1Ω

T1Π

T1Π0e0 + bT

1ΠT1Π0Ω0e0

= {Ω1(e1 × n1)}TΠT1Π0e0

+ bT1Π

T1Π0Ω0(n0 × b0)

={(

ωT1 n1

)e1 −

(ωT

1 e1

)n1

}TΠT

1Π0e0

+ bT1Π

T1Π0

{(ωT

0 b0

)n0 −

(ωT

0 n0

)b0

}

=(ωT

1 n1

)eT

1ΠT1Π0e0 − bT

1Π1Π0b0

(ωT

0 n0

)

= −(ωT

1 n1 + ωT0 n0

)cos θ,

∂θ

∂t= ωT

1 n1 + ωT0 n0.

(B.2)

(B2) With derivation of θ,

bT1Π

T1Π0e0 = − sin θ,

−θ cos θ = −(ωT

1 n1 + ωT0 n0

)cos θ

+ bT1Π

T1Π0(−κe0b0)

ds0

dt

+ (κn1n1 + κe1e1)TΠT1Π0e0

ds1

dt

= −(ωT

1 n1 + ωT0 n0

)cos θ

− κe0 cos θds0

dt− κe1 cos θ

ds1

dt,

θ = ωT1 n1 + ωT

0 n0 + (κe0 + κe1)ds1

dt.

(B.3)

(B3) With derivation of κe0 and κe1,

bT1Π

T1Π0e0 = − sin θ,

∂

∂s0bT

1ΠT1Π0e0 = − ∂θ

∂s0cos θ,

bT1Π

T1Π0(−κe0)b0 = − ∂θ

∂s0cos θ,

−κe0 cos θ = − ∂θ

∂s0cos θ,

κe0 = ∂θ

∂s0,

κe1 = ∂θ

∂s1.

(B.4)


References

[1] R. M. Murray, Z. Li, and S. S. Sastry, A MathematicalIntroduction to Robotic Manipulation, CRC Press, Boca Aton,Fla, USA, 1994.

[2] D. J. Montana, “The kinematics of contact and grasp,”International Journal of Robotics Research, vol. 7, no. 3, pp. 17–32, 1988.

[3] E. Paljug, X. Yun, and V. Kumar, “Control of rolling contacts inmulti-arm manipulation,” IEEE Transactions on Robotics andAutomation, vol. 10, no. 4, pp. 441–452, 1994.

[4] A. B. A. Cole, J. E. Hauser, and S. S. Sastry, “Kinematics andcontrol of multifingered hands with rolling contact,” IEEETransactions on Automatic Control, vol. 34, no. 4, pp. 398–404,1989.

[5] S. Arimoto, M. Yoshida, M. Sekimoto, and K. Tahara,“Modeling and control of 2-D grasping of an object witharbitrary shape under rolling contact,” SICE Journal of Control,Measurement, and System Integration, vol. 2, no. 6, pp. 379–386, 2009.

[6] S. Arimoto and M. Yoshida, “Modeling and control of 2Dgrasping under rolling contact constraints between arbitraryshapes: a Riemanniangeometry approach,” Journal of Robotics,vol. 2010, Article ID 926579, 13 pages, 2010.

[7] S. Arimoto, “Dynamics of grasping a rigid object with arbi-trary smooth surfaces under rolling contacts,” SICE Journal ofControl, Measurement, and System Integration, vol. 3, no. 3, pp.199–205, 2010.

[8] A. Gray, E. Abbena, and S. Salamon, Modern DifferentialGeometry of Curves and Surfaces with Mathematics, Chapman& Hall/CRC, Boca Aton, Fla, USA, 2006.

[9] S. Kobayashi, Differential Geometry of Curves and Surfaces,Shokabo, Tokyo, Japan, 1977, (revised in 1995 and inJapanese).

[10] M. Umehara and K. Yamada, Curves and Surfaces, Shokabo,Tokyo, Japan, 2001, (in Japanese).

[11] K. Nomizu, “Kinematics and differential geometry ofsubmanifolds—rolling a ball with a prescribed locus ofcontact,” Tohoku Mathematical Journal, vol. 30, pp. 623–637,1978.

[12] M. Yoshida and S. Arimoto, “A computational model of 3Dobject grasping with smoothgeometry under rolling contact,”in Proceedings of the 50th Annual Conference of the Societyof Instrument and Control Engineers (SICE ’11), pp. 919–923,2011.

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