IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 6, DECEMBER 2006 1273

[17] D. Vischer and O. Khatib, “Design and development of high-perfor-mance torque-controlled joints,” IEEE Trans. Robot. Autom., vol. 11,no. 4, pp. 537–544, Aug. 1995.

[18] J. Garcia de Jalón and E. Bayo, Kinematic and Dynamic Simulation ofMultibody Systems: The Real-Time Challenge. New York: Springer-Verlag, 1994.

[19] J. J. Craig, Introduction to Robotics: Mechanical and Control, 2nded. Reading, MA: Addison-Wesley, 1989.

[20] R. Stribeck, “The key qualities of sliding and roller bearings,” Zeitchriftdes Vereines Seutscher Ingenieure, vol. 46, no. 38, pp. 1342–1348,1902.

[21] B. Armstrong-Helouvry, P. Dupont, and C. C. de Wit, “A survey ofmodels, analysis tools and compensation methods for the control ofmachines with friction,” Automatica, vol. 30, no. 7, pp. 1083–1138,1994.

[22] C. C. de Wit, H. Olsson, K. J. Aström, and P. Lischinsky, “A new modelfor control of systems with friction,” IEEE Trans. Autom. Control, vol.40, no. 3, pp. 419–425, Mar. 1995.

[23] M. W. Spong, “Modeling and control of elastic joint robots,” Trans.ASME, J. Dyn. Syst., Meas., Control, vol. 109, pp. 310–319, Dec. 1987.

[24] E. D. Sontag, “Smooth stabilization implies coprime factorization,”IEEE Trans. Autom. Control, vol. 34, no. 4, pp. 435–443, Apr. 1989.

[25] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adap-tive Control Design. New York: Wiley-Interscience, 1995.

[26] M. C. Berg, N. Amit, and J. D. Powell, “Multirate digital control systemdesign,” IEEE Trans. Autom. Control, vol. 33, no. 12, pp. 1139–1150,Dec. 1988.

[27] Real-Time Workshop for Use with Simulink, 5th ed. Natick, MA:MathWorks Inc., 2002.

[28] R. L’Archevêque, M. Doyon, J.-C. Piedbœuf, and Y. Gonthier,“SYMOFROS: Software architecture and real time issues,” in Proc.DASIA, Data Syst. Aerosp., Montreal, QC, Canada, May 2000, vol.SP-457, pp. 41–46.

[29] DSP Blockset for Use with Simulink, User’s Guide. Natick, MA:MathWorks Inc., 1998.

[30] H. K. Khalil, Nonlinear Systems. New York: Macmillan, 1992.[31] R. K. Miller and A. N. Michel, Ordinary Differntial Equations. New

York: Academic, 1982.[32] F. C. Hoppensteadt, “Singular perturbations on the infinite interval,”

Trans. Amer. Math. Soc., vol. 123, no. 2, pp. 521–535, 1966.[33] M. J. Corless and G. Letmann, “Continous state feedback guaranteeing

uniform ultimate boundedness for uncertain dynmamic systems,” IEEETrans. Autom. Control, vol. AC-26, no. 5, pp. 1139–1144, Oct. 1981.

[34] W. Hahn, Stability of Motion. New York: Springer-Verlag, 1967.

Elastic Model of Deformable Fingertipfor Soft-Fingered Manipulation

Takahiro Inoue and Shinichi Hirai

Abstract—We propose a straightforward static elastic model of a hemi-spherical soft fingertip undergoing large contact deformation, as occurswhen robotic hands with the fingertips handle and manipulate objects,which is suitable for the analysis of soft-fingered manipulation because ofthe simple form of the model. We focus on formulating elastic force andpotential energy equations for the deformation of the fingers which arerepresented as an infinite number of virtual springs standing vertically.The equations are functions of two variables: the maximum displacementof the hemispherical fingertip and the orientation angle of a contactingplanar object. The elastic potential energy has a local minimum in ourmodel. The elastic model was validated by comparison with results of acompression test of the hemispherical soft fingertip.

Index Terms—Deformation model, elasticity, grasping, manipulation,robotic hand, soft fingertip.

I. INTRODUCTION

To date, various research has been done on manipulation of objectsby soft-fingered robotic hands. Most of the studies, particularly earlierstudies, focused only on contact mechanisms on various soft fingers.More recently, there has been an increase in studies on sensing mecha-nisms of human hand and designing control systems in robotic applica-tions to emulate the human capabilities which are applicable to robotichands. The conventional studies, however, have not been explicitly pro-viding any analytical exploration of the simplicity in grasping and ma-nipulating motions in terms of the soft-fingered handling. As a causeof the above-mentioned, it has been substantially difficult to derive afine elastic model of soft materials used in the fingertips.

Yokokohji et al. proposed a control scheme with visual sensorswhich can cancel the frictional twist/spin moment at the contactpoint of soft fingertips, and achieved stable grasping by spherical softfingertips [1], [2]. Maekawa et al. developed a finger-shaped tactilesensor covered with a soft and thin material, and proposed a controlalgorithm based on tactile feedback using the sensor, which needs noinformation about the geometry of the grasped object [3], [4]. Theymanaged to control the position of an object along a desired trajectory.In these papers, point-contacts were used to represent constraints ofrolling contact in their theoretical models, although the fingertips weremade from soft material such as rubber. Arimoto et al. verified thepassivity of equations of motion for a total handling system by using aLagrangian function incorporating the elastic potential energy due tothe deformation of soft fingertips [5], and compensated for the gravityeffect in three-dimensional space [6]. An elastic force model was alsoderived for the elastic potential energy of a system in which virtuallinear springs were arranged for simplicity in a radial pattern insidehemispherical soft fingertips. Doulgeri et al. discussed the problem ofstable grasping with deformable fingertips on which rolling constraintswere described as non-holonomic because of change in the effective

Manuscript received February 6, 2006; revised June 28, 2006. This paper wasrecommended for publication by Associate Editor H. R. Choi and Editor H. Araiupon evaluation of the reviewers’ comments. This work was supported by theIEEE. This paper was presented in part at the IEEE International Conference onRobotics and Automation, Barcelona, Spain, April 2005.

The authors are with the Laboratory for Integrated Machine Intelligence, De-partment of Robotics, Ritsumeikan University, Japan (e-mail: gr018026@se.rit-sumei.ac.jp; hirai@se.ritsumei.ac.jp).

Digital Object Identifier 10.1109/TRO.2006.886274

1552-3098/$20.00 © 2006 IEEE

1274 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 6, DECEMBER 2006

rolling radius of the soft fingertip [7], [8]. The above studies, however,focused mainly on deriving a control law to realize stable graspingand pose control of the grasped object, not on revealing a physicallyappropriate deformation model, which also contains the nonlinearcharacteristics of a hemispherical soft fingertip.

On the other hand, Xydas et al. proposed an exact deformation modelbased on the mechanics of the materials containing nonlinear charac-teristics, and performed finite element (FE) analysis for a hemispher-ical soft fingertip [9], [10]. Kao et al. experimentally demonstrated thatthe elastic force due to deformation satisfied a power law with respectto the displacement of the fingertip, and insisted that their theory sub-sumes Hertzian contact [11]. These studies, however, did not distin-guish between the material nonlinearity of the soft fingertip and thegeometrical nonlinearity caused by the hemispherical shape of the fin-gertip, and defined a parameter including the effect of both nonlineari-ties. Consequently, the cause of the discrepancy between the results ofthe simulation based on their model and the results of actual experi-ments was not apparent. In addition, because of the complexity of theirproposed models, these studies do not lend themselves to analysis ofequations of motion for the soft-fingered manipulation system overall.While FE analysis may enable us to derive a stress distribution and anelastic force on the soft fingertip, these simulation results depend onthe selected mesh pattern. Although FE analysis based on a certain ar-bitrary mesh pattern may prove the stability for equations of motion ofthe handling system, it does not always provide proof of stability forequations derived from other mesh patterns.

In this paper, we propose a static elastic model of a hemisphericalsoft fingertip in a physically reasonable and straightforward form suit-able for theoretical analysis of robotic handling motions. We distin-guish between geometric nonlinearity due to the hemispherical shapeand material nonlinearity of soft materials, i.e., the nonlinearity of theYoung’s modulus of the soft material, allowing us to focus only on thegeometric nonlinearity of the soft fingertip, and analytically formulateelastic force and elastic potential energy equations for the deformationof the fingertip. We show that each equation is a function of two vari-ables: the maximum displacement of the fingertip and the orientationangle of a contacting object. We also show that when the object is posi-tioned normal to the fingertips, the elastic potential energy is minimum.We finally validate the static elastic model by conducting a compres-sion test of the hemispherical soft fingertip and comparing the results.

II. STATIC ELASTIC MODEL OF HEMISPHERICAL SOFT FINGERTIP

We treat the fingertips as if they were composed of an infinite numberof virtual linear springs standing vertically. Fig. 1 shows one suchspring. We formulate elastic force and elastic potential energy equa-tions for the deformation of the fingertip. In order to simplify the deriva-tion process of both equations, two assumptions associated with mate-rial characteristics are given as follows:

1) the incompressibility of elastomer materials is not dealt with;2) Young’s modulus is constant.

Note that the contact condition being discussed in this paper is re-stricted to the case that an applied force to the fingertip is assumed tobe along z-axis of the fingertip. In addition, we consider that an objectnever comes into contact with the underneath plane of the fingertip.

A. Fingertip Stiffness

Let us apply an infinitesimal virtual spring QR with sectional areadS inside the soft fingertip, as shown in Fig. 1. Let dF be the infin-itesimal elastic force due to the shrinkage PQ of the virtual spring.Let �p be the orientation angle of the contacting object, a be the fin-gertip radius, d be the maximum displacement of the fingertip, ac =a2 � (a� d)2 be the radius of the contacting circle, and P be the

point where the spring is in contact with the object. Furthermore, let �

Fig. 1. Contact mechanism.

be the angle subtended between line PQ and the z-axis, and � be theazimuthal angle on the xy-plane. Using the contact surface equation,x sin �p + z cos �p = a� d (see Appendix I), the infinitesimal elasticforce dF is given by

dF = k � PQ

= k a2 � (x2 + y2)�a� d� x � sin �p

cos �p(1)

where k is the spring constant of the spring QR. Note that k is propor-tional to the sectional area dS and inversely proportional to the naturallength a2 � (x2 + y2). Letting E be the Young’s modulus of softfinger materials, k is described as (see Appendix II)

k =EdS

a2 � (x2 + y2): (2)

Letting K be the fingertip stiffness on the entire deformed part illus-trated in Fig. 1, K can be expressed from (2) as

K =ell

k = Ea

�a

b (y)

b (y)

dxdy

a2 � (x2 + y2)(3)

where

b1(y) = (a� d) sin �p � cos �p a2c � y2 (4)

b2(y) = (a� d) sin �p + cos �p a2c � y2 (5)

and ell denotes the elliptical region shown in Fig. 2(a). Applying a nu-merical integration to (3), we obtain a constant fingertip stiffness de-picted as continuous lines, as shown in Fig. 3. This indicates that thefingertip stiffnessK is independent of the object orientation �p. Hence,in this paper we additionally provide the third assumption that:

3) the fingertip stiffness is independent of the object orientationas long as the maximum displacement remains constant.

By using the above assumption, we formulate the fingertip stiffnessK in an analytical formula.

Now, performing a substitution that x = r cos� cos �p +(a � d) sin �p and y = r sin�, (3) is then transformed into (seeAppendix III)

K = Ea

0

r2�

0

cos �pd�

a2 � fx2(r; �) + y2(r; �)gdr: (6)

IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 6, DECEMBER 2006 1275

Fig. 2. Integration area.

Fig. 3. Comparison between numerical results of (3) and analytical simulationsof (7) when E = 0:2032 MPa measured in this paper.

Since assumption 3) claims K is independent of �p, we can substitute�p = 0 into (6), and get

K = Ea

0

r2�

0

d�pa2 � r2

dr = 2�Ed: (7)

Plotting the simulation result of (7) as dotted lines onto Fig. 3 togetherwith the results of (3), we find that both lines coincide with each other.This implies that the third assumption due to the numerical observationis appropriate, and additionally, the stiffness is a function of only themaximum displacement d.

B. Elastic Force

Likewise, by using the third assumption associated with the fingertipstiffness, we formulate the elastic force and potential energy equationsin a straightforward way. Using (1), (2), and a geometrical relationshipQT = PQ cos �p , the elastic force F can be written as

F =1

cos �p ell

k �QT

=E

cos �p

a

�a

b (y)

b (y)

QT � dxdya2 � (x2 + y2)

: (8)

Performing the same variable conversion as the derivation process ofK , (8) is then transformed as

F =E

cos �p

a

0

QT (r) � r2�

0

B(r; �)d� dr (9)

Fig. 4. Comparison between the numerical integration and the analytical sim-ulation of F and P , respectively. (a) Elastic force. (b) Elastic energy.

where

QT (r) =pa2 � r2 � (a� d): (10)

In (9), B(r; �) corresponds to the integrand within the braces in (6).Here applying the assumption 3) to B(r; �) as well as (7), F is finallycalculated as

F =E

cos �p

a

0

QT (r) � r2�

0

d�pa2 � r2

dr =�Ed2

cos �p: (11)

C. Elastic Potential Energy

As well as (8), the elastic potential energy P is expressed as

P =1

2 ell

k � PQ2 =1

2 cos2 �p ell

k �QT 2

=E

2 cos2 �p

a

�a

b (y)

b (y)

QT 2 � dxdya2 � (x2 + y2)

: (12)

Performing the same variable conversion as the derivation process ofF , (12) is then transformed as

P =E

2 cos2 �p

a

0

QT2(r) � r

2�

0

B(r; �)d� dr: (13)

Here again, applying the assumption 3) to B(r; �) in (13), P is finallycalculated as

P =E

2 cos2 �p

a

0

QT2(r) � r

2�

0

d�pa2 � r2

dr

=�Ed3

3 cos2 �p: (14)

Note that (11) and (14) clarify that the elastic force and elastic potentialenergy on the entire deformed part of a hemispherical soft fingertip arefunctions of two variables, namely, the maximum displacement d andthe object orientation angle �p. Furthermore, we find that both formulaehave a local minimum when the orientation angle is zero. Especially,we describe the minimum value of elastic energy as local minimum ofelastic potential energy (LMEE).

Finally, in order to confirm the transformations of formulae from (8)to (11), and (12) to (14), we verify the numerical analysis of (8) and(12) and simulation results of (11) and (14). Fig. 4 indicates the result,and concludes that both (11) and (14) are mathematically reasonableformulae in this paper.

D. Relationship Between Elastic Force and Elastic Energy

While the individual virtual spring used in our study is based ona linear elasticity, the entire fingertip model that is obtained by com-pleting the double integration on an elliptical region exhibits a geomet-rical nonlinearity caused by the hemispherical shape of the fingertip.In other words, the completed fingertip model has a nonlinear fingertip

1276 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 6, DECEMBER 2006

stiffness expressed as (7). Hence, when we compute the total force (11)from the energy (14), we must define an equivalent displacement anduse it for the differential calculation.

In the case of normal contact that corresponds to �p = 0, elasticmodels are given as follows:

P =�Ed3

3(15)

@P

@d= �Ed

2= F (16)

@2P

@d2= 2�Ed = K (17)

where d itself corresponds to the equivalent displacement. Continu-ously, let us consider the case of diagonal contact when �p 6= 0. Wedefine �zeq as an equivalent displacement, and it must satisfy

@P

@�zeq=

�Ed2

cos �p= F (18)

@2P

@�z2eq= 2�Ed = K: (19)

The displacement �zeq to fulfill (18) and (19) can be found such thata geometrical relationship d = �zeq cos �p is maintained as shown inFig. 14. It is obvious that �zeq means a true maximum displacementamong all the virtual springs in any case that includes �p = 0 and�p 6= 0.

III. COMPARISON WITH HERTZIAN CONTACT

In 1881, Hertz proposed a contact theory for two elastic objectshaving arbitrary curved surfaces [12]. He showed that a normal contactforce generated between an elastic sphere and a plane whose Youngmodulus is infinity can be expressed as

F =4pR

3

E

1� �2d (20)

whereR is the radius,E the Young modulus of the object,� the Poissonratio, and d the maximum displacement of the sphere. Since the aboveequation is useful from a practical viewpoint, it has been widely usedfor computing the contact stress between, for example, a wheel and arail, a roll and material, and a retainer and a ball in a bearing. However,in Hertzian contact, it is assumed that both elastic objects are openelliptic paraboloids with an arbitrary radius of curvature. Consequently,no boundary conditions are used in the Hertzian contact model.

Kao et al. defined the parameter cd corresponding to a material andgeometric nonlinearity [11], and transformed (20) into

F = cdd�: (21)

They conducted a vertical compression test using a hemisphericalsoft fingertip, and estimated the parameter cd empirically by usinga weighted least-squares method (LSM). It has been shown that � isapproximately 2.3 or 1.75 when the rate of deformation of the fingeris above or below 20%, respectively. In other words, the parameter� is not identical to 3/2 in the contact model of soft fingertips. Thusthe Hertzian contact theory cannot be adopted for deriving the elasticmodel of the hemispherical soft fingertip.

Fig. 5 shows a comparison result in which the elastic force valuewith respect to the displacement d is plotted when a hemispherical softfingertip is compressed vertically, whose radius is 20 mm. It is obviousthat our vertically oriented spring model is more suitable for derivingan elastic force up to the midrange displacement of the fingertip. It is

Fig. 5. Comparison between the Hertzian contact model and the elastic forcemodel when � = 0 and E = 0:2032 MPa.

Fig. 6. Stress-strain diagram of polyurethane rubber. (a) Stress-strain diagram.(b) Enlarged diagram.

because that our model contains the geometric nonlinearity due to thehemispherical shape of the fingertip, that is, this model could indicatethat � becomes 2 by only adopting appropriate natural length to theindividual springs.

Soft materials exhibit nonlinear characteristics, even for infinites-imal deformations. In fact, Tatara newly derived a nonlinear Youngmodulus with respect to compressive strain [13]. Furthermore, theconcept for the contact angle of the object is not incorporated in theHertzian contact theory. While the Hertzian contact theory can be usedfor a simple contact pattern corresponding to the normal contact, nocontact at any other arbitrary angle or rolling contact can be defined.On the other hand, the elastic models proposed in this paper cover anycontact angle of the object, and therefore, these models can be used toanalyze grasping and manipulating motions containing varied possiblecontact forms by soft-fingered robotic hand.

IV. MEASUREMENT OF YOUNG’S MODULUS

In this paper, the Young modulus of the soft fingertip was measuredby conducting a compression test on six cylinders of polyurethane gel.Three cylinders were 20 mm in diameter and 15, 20, and 25 mm inheight, and three were 30 mm in diameter and also 15, 20, and 25 mmin height, as shown in Fig. 8(a).

Fig. 6(a) shows the overall view of a measured stress-strain diagram,and an enlarged view of part of the diagram is shown in Fig. 6(b). Nu-merical values shown in both graphs denote the specimen height on theleft side and the specimen diameter on the right side. The data were av-eraged and smoothed using the LSM, as shown in Fig. 7. We assumedthat the maximum deformation of the soft fingertip is 50% in the radius.Furthermore, in order to focus predominantly on the geometric nonlin-earity due to the hemispherical shape, we avoided the issue of the ma-terial’s nonlinearity which is directly related to the Young’s modulus

IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 6, DECEMBER 2006 1277

Fig. 7. Average value of stress-strain diagram.

Fig. 8. Compression test of a hemispherical soft fingertip. (a) Several speci-mens. (b) Compression test.

of soft materials. Consequently, we performed a linear approximationfor a 50% strain, as in Fig. 7, and estimated the Young’s modulus as0.2032 MPa.

V. COMPRESSION TEST

By compressing a hemispherical soft fingertip made of polyurethanegel along the normal direction, as shown in Fig. 1 and Fig. 8(b), we ver-ified the validity of our elastic force model represented in (11). Further-more, by conducting multiple experiments with various contacting an-gles, we demonstrated the existence of the local minimum of the elasticforce. In the compression test, we used a fingertip with a diameter of40 mm, and contacting rods with 13 different shapes. The rods wereinclined from 0� to 30� in increments of 2.5�, as shown in Fig. 8(b).Fig. 9 compares experiments with simulation results. The horizontalaxis represents the maximum displacement of the compressed fingertip,while the vertical axis represents the elastic force measured by a load-cell placed in the compression machine.

In all the graphs in Fig. 9, the simulation and experimental resultsare almost identical to each other up to d = 6.0 mm, after which thediscrepancies increase with the magnitude of the displacement. Thediscrepancy comes from the linear approximation of the experimentalstress-strain diagram shown in Fig. 7. The effect leads directly to non-linearity of Young’s modulus, which is outside the scope of this paper.

Figs. 10(a) and 11(a) show simulation and experimental results,respectively. Enlarged views of both results are also shown inFigs. 10(b) and 11(b). The numerical values in each graph denote theinclined angle of the contacting object, and both results are plotted atintervals of 5.0�. The elastic force increases as the orientation angleincreases under constant maximum displacement. As confirmation, weplotted the elastic force against �p of (11) in Fig. 12. The numericalvalues shown in the graph denote the maximum displacement d. Atabout 0�, there is a clear local minimum of the elastic force, and thechange in elastic force with �p is greatest when the displacement is

Fig. 9. Elastic forces in experiments. (a) 2.5�. (b) 5.0�. (c) 7.5�. (d) 10.0�.(e) 12.5�. (f)15.0�. (g) 17.5�. (h) 20.0�. (i) 22.5�. (j) 25.0�. (k) 27.5�. (l) 30.0�.

maximum, that is, 8.0 mm. The same tendency can also be seen inthe simulation results. The results therefore indicate that our proposed

1278 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 6, DECEMBER 2006

Fig. 10. Simulation results of elastic force. (a) Simulations. (b) Enlarged view.

Fig. 11. Experimental results of elastic force. (a) Experiments. (b) Enlargedview.

Fig. 12. Local minimum of elastic force. (a) Simulations. (b) Experiments.

elastic model can present a distinctive phenomenon, i.e., a localminimum elastic force, even when the deriving process is representedsimply by bringing linear virtual springs standing in the normaldirection. On the other hand, the discrepancy in the large displacementshown in Fig. 12 would be reduced if Young’s modulus could bedefined as a nonlinear function of compression strain, and be usedto adopt the model to accommodate the nonlinearity of the material.However, this paper focuses on the geometric nonlinearity, and thederiving process including both nonlinearities will be addressed infuture studies.

VI. CONCLUDING REMARKS

We have formulated a static elastic force model and an elastic po-tential energy function based on virtual springs inside a hemisphericalsoft fingertip. We have also proven the existence of an LMEE and ex-perimentally demonstrated that the elastic force due to the deformationhas a local minimum. Our model requires us to only measure Young’smodulus of a corresponding material to be used in robotic fingertips. Infuture studies, we will consider constant volume deformation of incom-pressible elastomer materials, and derive elastic models incorporatinga nonlinear Young modulus.

These new findings suggest a quasi-static manipulation theorybased on the LMEE for a minimum degree-of-freedom (DOF) robotichand [14]. By expanding the new idea of LMEE in the development of

Fig. 13. Spring constant inside the soft fingertip.

grasping and manipulation theory using a soft-fingered robotic hand,it is expected that the stable grasping and the pose control of a graspedobject by a minimum DOF two-fingered hand may be achieved and asuccinct control system will be designed.

APPENDIX ICONTACT PLANE FORMULA

As illustrated in Fig. 1, the point C is described in a vector form as

�!OC =

(a� d) sin �p0

(a� d) cos �p

: (22)

In addition, a normal unit vector with respect to the contact surface isrepresented as

n =

sin �p0

cos �p

: (23)

Since the contact plane can be written by an inner product form,

f[x; y; z]T ��!OCg � n = 0, the plane equation is therefore described

as follows:

x sin �p + z cos �p = a� d: (24)

APPENDIX IISPRING CONSTANT FORMULATION

As shown in Fig. 13, letting k0; dS0, and L0, respectively, be thespring constant, the sectional area, and the natural length of a spec-imen for measuring Young’s modulus, and E be Young’s modulus ob-tained from an appropriate compression test, we can derive the fol-lowing equations according to linear material mechanics:

� = E� (25)

()F

S0= E

�x

L0

(26)

() E =L0

�x�F

S0=

L0

�x�k0�x

S0= k0

L0

S0(27)

where F denotes an applied force to the specimen and �x is a displace-ment in the identification test. Since this paper assumes that Young’smodulus is an invariant physical value for individual material, the fol-lowing equation is satisfied:

kL

dS= k0

L0

S0= E (28)

() k = EdS

L=

EdS

a2 � (x2 + y2): (29)

IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 6, DECEMBER 2006 1279

Fig. 14. Equivalent fingertip stiffness with respect to -coordinate system.

Fig. 15. Fingertip stiffness on a certain perimeter. (a) Elliptical perimeter. (b)Circular perimeter.

APPENDIX IIICOORDINATE CONVERSION TO DERIVE THE FINGERTIP STIFFNESS

As illustrated in Fig. 14, let 0 be the coordinate system translatedto O0 from the frame, and 00 be the cylindrical coordinate systeminclined by �p from z0-axis. Let r be the arbitrary radius on the con-tact circle that has an origin C , and � be the common rotational anglearound the z-, z0-, and z00-axes. The relationship between (x0; y0) onthe 0 frame and (r; �) on the 00 frame is then expressed as

x0 = r cos� cos �p (30)

y0 = r sin�: (31)

Since the relationship between (x; y) and (x0; y0) is described as x =x0 + (a � d) sin �p, and y = y0, the variable transformation throughthe coordinate systems and 00 can be expressed as

x = r cos� cos �p + (a� d) sin �p (32)

y = r sin�: (33)

Simultaneously, the elliptical region at the bottom surface of the fin-gertip shown in Fig. 14 can be converted to a circular region accordingto the above transformation rule, that is, the integration area of (r; �)varies at [0; ac] and [0; 2�], respectively.

Next, let us consider the physical meaning of the double integrationof B(r; �) used in (9) and (13), which is detailed in (6) as

2�

0

B(r; �)d� =2�

0

cos �pd�

a2 � fx2(r; �) + y2(r; �)g: (34)

Equation (34) corresponds to a stiffness on an elliptical perimeterwhose longitudinal radius is r, as shown in Fig. 15(a). Additionally,substituting �p = 0 into (34) enables obtaining an equivalent stiffnesson a circular perimeter of radius r, shown in Fig. 15(b).

REFERENCES

[1] Y. Yokokohji, M. Sakamoto, and T. Yoshikawa, “Vision-aided objectmanipulation by a multifingered hand with soft fingertips,” in Proc.IEEE Int. Conf. Robot. Autom., 1999, pp. 3201–3208.

[2] Y. Yokokohji, M. Sakamoto, and T. Yoshikawa, “Object manipulationby soft fingers and vision,” in Proc. 9th Int. Symp. Robot. Res., 2000,pp. 365–374.

[3] H. Maekawa, K. Komoriya, and K. Tanie, “Manipulation of an un-known object by multifingered hands with rolling contact using tac-tile feedback,” in Proc. IEEE Int. Conf. Intell. Robots Syst., 1992, pp.1877–1882.

[4] H. Maekawa, K. Tanie, K. Komoriya, M. Kaneko, C. Horiguchi, andT. Sugawara, “Development of a finger-shaped tactile sensor and itsevaluation by active touch,” in Proc. IEEE Int. Conf. Robot. Autom.,1992, pp. 1327–1334.

[5] S. Arimoto, P. Nguyen, H. Y. Han, and Z. Doulgeri, “Dynamics andcontrol of a set of dual fingers with soft tips,” Robotica, vol. 18, pp.71–80, 2000.

[6] S. Arimoto, Z. Doulgeri, P. Nguyen, and J. Fasoulas, “Stable pinchingby a pair of robot fingers with soft tips under the effect of gravity,”Robotica, vol. 20, pp. 241–249, 2002.

[7] Z. Doulgeri, J. Fasoulas, and S. Arimoto, “Feedback control for objectmanipulation by a pair of soft tip fingers,” Robotica, vol. 20, pp. 1–11,2002.

[8] Z. Doulgeri and J. Fasoulas, “Grasping control of rolling manipulationwith deformable fingertips,” IEEE/ASME Trans. Mechatron., vol. 8, no.2, pp. 283–286, Mar. 2003.

[9] N. Xydas and I. Kao, “Modeling of contact mechanics and frictionlimit surfaces for soft fingers in robotics, with experimental results,”J. Robot. Res., vol. 18, no. 8, pp. 941–950, 1999.

[10] N. Xydas, M. Bhagavat, and I. Kao, “Study of soft-finger contact me-chanics using finite elements analysis and experiments,” in Proc. IEEEInt. Conf. Robot. Autom., 2000, pp. 2179–2184.

[11] I. Kao and F. Yang, “Stiffness and contact mechanics for soft fingers ingrasping and manipulation,” IEEE Trans. Robot. Autom., vol. 20, no.1, pp. 132–135, Feb. 2004.

[12] J. L. Johnson, Contact Mechanics. Cambridge, U.K.: CambridgeUniv. Press, 1985, ch. 4.

[13] Y. Tatara, “On compression of rubber elastic sphere over a large rangeof displacements—Part I: Theoretical study,” ASME J. Eng. Mater.Technol., vol. 113, pp. 285–291, 1991.

[14] T. Inoue and S. Hirai, “Study on hemispherical soft-fingered handlingfor fine manipulation by minimum D.O.F. robotic hand,” in Proc. IEEEInt. Conf. Robot. Autom., 2006, pp. 2454–2459.