Elastic Model of Deformable Fingertip for Soft-fingered Manipulation

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1 IEEE TRANSACTION ON ROBOTICS, VOL. 1, NO. 11, NOVEMBER 25 1 Elastic Model of Deformable Fingerti for Soft-fingered Maniulation Takahiro Inoue, Student Member, IEEE, and Shinichi Hirai, Member, IEEE Abstract We roose straightforward static elastic model of a hemisherical soft fingerti undergoing large contact deformation, as occurs when robotic hands with the fingertis handle and maniulate objects, which is suitable for the analysis of soft-fingered maniulation because of the simle form of the model. We focus on formulating elastic force and otential energy equations for the deformation of the fingers which are reresented as an infinite number of virtual srings standing vertically. The equations are functions of two variables: the maximum dislacement of the hemisherical fingerti and the orientation angle of a contacting lanar object. The elastic otential energy has a local minimum (LMEE) in our model. The elastic model was validated by comarison with results of a comression test of the hemisherical soft fingerti. Index Terms Soft fingerti, Maniulation, Grasing, Robotic hand, Elasticity, Deformation model. I. INTRODUCTION To date, various research has been done on maniulations of objects by soft-fingered robotic hands. Most of the studies, articularly earlier studies, focused only on contact mechanisms on various soft fingers. More recently, there has been an increase in studies on sensing mechanisms of human hand and designing control systems in robotic alications to emulate the human caabilities which are alicable to robotic hands. The conventional studies, however, have not been exlicitly roviding any analytical exloration of the simlicity in grasing and maniulating motions in terms of the soft-fingered handling. As a cause of the above mention, it has been substantially difficult to derive a fine elastic model of soft materials used in the fingertis. Yokokohji et al. roosed a control scheme with visual sensors which can cancel the frictional twist/sin moment at the contact oint of soft fingertis, and achieved stable grasing by sherical soft fingertis [1], [2]. Maekawa et al. develoed a finger-shaed tactile sensor covered with a soft and thin material, and roosed a control algorithm based on tactile feedback using the sensor, which needs no information about the geometry of the grased object [3], [4]. They managed to control the osition of an object along a desired trajectory. In these aers, oint-contacts were used to reresent constraints of rolling contact in their theoretical models, although the fingertis were made from soft material such as rubber. Arimoto et al. verified the assivity of equations of motion for a total handling system by using a Lagrangian function incororating the elastic otential energy due to the Manuscrit received October 6, 25; revised October 6, 25. This work was suorted by the IEEE. The authors are with the Laboratory for Integrated Machine Intelligence, Deartment of Robotics, Ritsumeikan University, Jaan ( gr1826@se.ritsumei.ac.j;hirai@se.ritsumei.ac.j). deformation of soft fingertis [5], and comensated for the gravity effect in three-dimensional sace [6]. An elastic force model was also derived for the elastic otential energy of a system in which virtual linear srings were arranged for simlicity in a radial attern inside hemisherical soft fingertis. Doulgeri et al. discussed the roblem of stable grasing with deformable fingertis on which rolling constraints were described as non-holonomic because of change in the effective rolling radius of the soft fingerti [7], [8]. The above studies, however, focused mainly on deriving a control law to realize stable grasing and ose control of the grased object, not on revealing a hysically aroriate deformation model, which also contains the nonlinear characteristics of a hemisherical soft fingerti. On the other hand, Xydas et al. roosed an exact deformation model based on the mechanics of the materials containing nonlinear characteristics, and erformed Finite Element (FE) analysis for a hemisherical soft fingerti [9], [1]. Kao et al. exerimentally demonstrated that the elastic force due to deformation satisfied a ower law with resect to the dislacement of the fingerti, and insisted that their theory subsumes Hertzian contact [11]. These studies, however, did not distinguish between the material nonlinearity of the soft fingerti and the geometrical nonlinearity caused by the hemisherical shae of the fingerti, and defined a arameter including the effect of both nonlinearities. Consequently, the cause of the discreancy between the results of the simulation based on their model and the results of actual exeriments was not aarent. In addition, because of the comlexity of their roosed models, these studies do not lend themselves to analysis of equations of motion for the soft-fingered maniulation system overall. While FE analysis may enable us to derive a stress distribution and an elastic force on the soft fingerti, these simulation results deend on the selected mesh attern. Although FE analysis based on a certain arbitrary mesh attern may rove the stability for equations of motion of the handling system, it does not always rovide roof of stability for equations derived from other mesh atterns. In this reort, we roose a static elastic model of a hemisherical soft fingerti in a hysically reasonable and straightforward form suitable for theoretical analysis of robotic handling motions. We distinguish between geometric nonlinearity due to the hemisherical shae and material nonlinearity of soft materials, i.e., the nonlinearity of the Young s modulus of the soft material, allowing us to focus only on the geometric nonlinearity of the soft fingerti, and analytically formulate elastic force and elastic otential energy equations for the deformation of the fingerti. We show that each equation is a function of two variables: the maximum dislacement of the

2 IEEE TRANSACTION ON ROBOTICS, VOL. 1, NO. 11, NOVEMBER 25 2 fingerti and the orientation angle of a contacting object. We also show that when the object is ositioned normal to the fingertis, the elastic otential energy is minimum. We finally validate the static elastic model by conducting a comression test of the hemisherical soft fingerti and comaring the results. II. STATIC ELASTIC MODEL OF HEMISPHERICAL SOFT FINGERTIP We treat the fingertis as if they were comosed of an infinite number of virtual linear srings standing vertically. Fig.1 shows one such sring. We formulate elastic force and elastic otential energy equations for the deformation of the fingerti. In order to simlify the derivation rocess of both equations, two assumtions associated with material characteristics are given as follows: 1) The incomressibility of elastomer materials is not dealt with. 2) Young s modulus is constant. Note that the contact condition being discussed in the resent study is restricted to the case that an alied force to the fingerti is assumed to be along z-axis of the fingerti. In addition, we consider that an object never comes into contact with the underneath lane of the fingerti. Fig. 1. Contact mechanism A. Fingerti Stiffness Let us aly an infinitesimal virtual sring QR with sectional area ds inside the soft fingerti, as shown in Fig.1. Let df be the infinitesimal elastic force due to the shrinkage PQ of the virtual sring. Let θ be the orientation angle of the contacting object, a be the fingerti radius, d be the maximum dislacement of the fingerti, a c a 2 (a d) 2 be the radius of the contacting circle, and P be the oint where the sring is in contact with the object. Furthermore, let θ be the angle subtended between line PQ and the z-axis, and φ be the azimuthal angle on the xy-lane. Using the contact surface equation, xsinθ + zcosθ a d (see Aendix I), the infinitesimal elastic force df is given by ρ a df k PQ k 2 (x 2 + y 2 ) a d x sinθ ff ; (1) cosθ where k is the sring constant of the sring QR. Note that k is roortional to the sectional area ds and inversely roortional to the natural length a 2 (x 2 + y 2 ). Letting E be the Young s modulus of soft finger materials, k is described as (see Aendix II) E ds a 2 (x 2 + y 2 : (2) ) k Letting K be the fingerti stiffness on the entire deformed art illustrated in Fig.1, K can be exressed from Eq. (2) as where K ell k E ac (y) b2 a c b 1 (y) dx dy a 2 (x 2 + y 2 ) ; (3) b 1 (y) (a d)sinθ cosθ a 2 c y 2 ; (4) b 2 (y) (a d)sinθ + cosθ a 2 c y 2 ; (5) Fig. 2. Integration area. and ell denotes the ellitical region shown in Fig.2-(a). Alying a numerical integration to Eq. (3), we obtain a constant fingerti stiffness deicted as continuous lines, as shown in Fig.3. This indicates that the fingerti stiffness K is indeendent of the object orientation θ. Hence, in this study we additionally rovide the thir assumtion that: 3) the fingerti stiffness is indeendent of the object orientation as long as the maximum dislacement remains constant. By using the above assumtion, we formulate the fingerti stiffness K in an analytical formula. Now, erforming a substitution that x r cosφ cosθ +(a d)sinθ and y r sinφ, Eq. (3) is then transformed into (see Aendix III) K E ac ( 2π r cosθ dφ a 2 fx 2 (r;φ)+y 2 (r;φ)g ) dr: (6) Since assumtion 3) claims K is indeendent of θ, we can substitute θ into Eq. (6), and get K E ac r ρ 2π dφ ffdr a 2 r 2 2πEd: (7) Plotting the simulation result of Eq. (7) as dotted lines onto Fig.3 together with the results of Eq. (3), we find that both

3 IEEE TRANSACTION ON ROBOTICS, VOL. 1, NO. 11, NOVEMBER 25 3 Fig. 3. Comarison between numerical results of Eq. (3) and analytical simulations of Eq. (7) when E :232 MPa measured in the resent study. Fig. 4. Comarison between the numerical integration and the analytical simulation of F and P, resectively: (a) elastic force, (b) elastic energy. lines coincide with each other. This imlies that the third assumtion due to the numerical observation is aroriate, and additionally the stiffness is a function of only the maximum dislacement d. B. Elastic Force Likewise, by using the third assumtion associated with the fingerti stiffness, we formulate the elastic force and otential energy equations in a straightforward way. Using Eqs. (1), (2) and a geometrical relationshi QT PQcosθ (see Fig.14 in Aendix III), the elastic force F can be written as 1 F k QT cosθ ell E ac b2 (y) cosθ a c b 1 (y) QT dx dy a 2 (x 2 + y 2 ) : (8) Performing the same variable conversion as the derivation rocess of K, Eq. (8) is then transformed as F E ac ρ 2π ff QT (r) r B(r;φ) dφ dr; (9) cosθ where (see Fig.14) QT (r) a 2 r 2 (a d): (1) In Eq. (9), B(r;φ) corresonds to the integrand within the braces in Eq. (6). Here alying the assumtion 3) to B(r;φ) as well as Eq. (7), F is finally calculated as F E ac ρ 2π ff dφ QT (r) r dr cosθ a 2 r 2 πed2 : (11) cosθ C. Elastic Potential Energy As well as Eq. (8), the elastic otential energy P is exressed as P 1 k PQ ell 2cos 2 k QT θ ell 2 E ac b2 (y) 2cos 2 θ a c b 1 (y) QT 2 dx dy a 2 (x 2 + y 2 ) : (12) Performing the same variable conversion as the derivation rocess of F, Eq. (12) is then transformed as E ac ρ 2π ff P 2cos 2 QT 2 (r) r B(r;φ) dφ dr: (13) θ Here again, alying the assumtion 3) to B(r;φ) in Eq. (13), P is finally be calculated as E ac ρ 2π P 2cos 2 QT 2 dφ (r) r ffdr θ a 2 r 2 πed 3 3cos 2 : (14) θ Note that Eqs. (11) and (14) clarify that the elastic force and elastic otential energy on the entire deformed art of a hemisherical soft fingerti are functions of two variables, namely, the maximum dislacement d and the object orientation angle θ. Furthermore, we find that both formulae have a local minimum when the orientation angle is zero. Esecially, we describe the minimum value of elastic energy as Local Minimum of Elastic Potential Energy, which is abbreviated as LMEE. Finally, in order to confirm the transformations of formulae from Eq. (8) to Eq. (11) and Eq. (12) to Eq. (14), we verify the numerical analysis of Eqs. (8) and (12) and simulation results of Eqs. (11) and (14). Fig.4 indicates the result, and concludes that both Eqs. (11) and (14) are mathematically reasonable formulae in the resent study. D. Relationshi between Elastic Force and Elastic Energy While the individual virtual sring used in our study is based on a linear elasticity, the entire fingerti model that is obtained by comleting the double integration on an ellitical region exhibits a geometrical nonlinearity caused by the hemisherical shae of the fingerti. In other words, the comleted fingerti model has a nonlinear fingerti stiffness exressed as Eq. (7). Hence, when we comute the total force Eq. (11) from the energy Eq. (14), we must define an equivalent dislacement and use it for the differentiatial calculation. In the case of normal contact that corresonds to θ, elastic models are given as follows: P πed3 ; 3 (15) P d πed2 F; (16) 2 P 2πEd K ; d2 (17) where d itself corresonds to the equivalent dislacement.

4 IEEE TRANSACTION ON ROBOTICS, VOL. 1, NO. 11, NOVEMBER 25 4 Continuously, let us consider the case of diagonal contact when θ 6. We define z eq as an equivalent dislacement, and it must satisfy P z eq πed2 cosθ F; (18) 2 P z 2 eq 2πEd K : (19) The dislacement z eq to fulfill Eqs. (18) and (19) can be found such that a geometrical relationshi d z eq cosθ is maintained as shown in Fig. 14. It is obvious that z eq means a true maximum dislacement among all the virtual srings in any case that includes θ and θ 6. Fig. 5. Comarison between the Hertzian contact model and the resent elastic force model when θ and E :232 [MPa]. III. COMPARISON WITH HERTIAN CONTACT In 1881, Hertz roosed a contact theory for two elastic objects having arbitrary curved surfaces [12]. He showed that a normal contact force generated between an elastic shere and a lane whose Young s modulus is infinity can be exressed as F 4 R E 3 1 σ 2 d 3 2 ; (2) where R is the radius, E the Young s modulus of the object, σ the Poisson s ratio, and d the maximum dislacement of the shere. Since the above equation is useful from a ractical viewoint, it has been widely used for comuting the contact stress between, for examle, a wheel and a rail, 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 oen ellitic araboloids with an arbitrary radius of curvature. Consequently, no boundary conditions are used in the Hertzian contact model. Kao et al. defined the arameter c d corresonding to a material and geometric nonlinearity [11], and transformed Eq.(2) into F c d d ζ : (21) They conducted a vertical comression test using a hemisherical soft fingerti, and estimated the arameter c d emirically by using a weighted least-squares method. It has been shown that ζ is aroximately 2.3 or 1.75 when the rate of deformation of the finger is above or below 2%, resectively. In other words, the arameter ζ is not identical to 3/2 in the contact model of soft fingertis. Thus the Hertzian contact theory cannot be adoted for deriving the elastic model of the hemisherical soft fingerti. Fig.5 shows a comarison result in which the elastic force value with resect to the dislacement d is lotted when a hemisherical soft fingerti is comressed vertically, whose radius is 2 mm. It is obvious that our vertically-oriented sring model is more suitable for deriving an elastic force u to the midrange dislacement of the fingerti. It is because that our model contains the geometric nonlinearity due to the hemisherical shae of the fingerti, that is, the resent model could indicate that ζ becomes 2 by only adoting aroriate natural length to the individual srings. Fig. 6. Stress-strain diagram of olyurethane rubber: (a) stress-strain diagram, (b) enlarged diagram. Soft materials exhibit nonlinear characteristics, even for infinitesimal deformations. In fact, Tatara newly derived a nonlinear Young s modulus with resect to comressive strain [13]. Furthermore, the concet for the contact angle of the object is not incororated in the Hertzian contact theory. While the Hertzian contact theory can be utilized for a simle contact attern corresonding to the normal contact, no contact at any other arbitrary angle or rolling contact can be defined. On the other hand the elastic models roosed in this aer cover any contact angle of the object, and therefore, these models can be used to analyze grasing and maniulating motions containing varied ossible contact forms by soft-fingered robotic hand. IV. MEASUREMENT OF YOUNG S MODULUS In the resent study, the Young s modulus of the soft fingerti was measured by conducting a comression test on 6 cylinders of olyurethane gel. Three cylinders were 2 mm in diameter and 15, 2, and 25 mm in height, and three were 3 mm in diameter and also 15, 2, and 25 mm in 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 art of the diagram is shown in Fig.6-(b). Numerical values shown in both grahs denote the secimen height on the left side and the secimen diameter on the right side. The data were averaged and smoothed using the least-squares method (LSM), as shown in Fig.7. We assumed that the maximum deformation of the soft fingerti is 5% in the radius. Furthermore, in order to focus redominantly on the geometric nonlinearity due to the hemisherical shae, we avoided the issue of the material s nonlinearity which is directly related to the Young s modulus of soft materials.

5 IEEE TRANSACTION ON ROBOTICS, VOL. 1, NO. 11, NOVEMBER 25 5 Fig. 7. Average value of stress-strain diagram. Fig. 8. Comression test of a hemisherical soft fingerti: (a) several secimens, (b) comression test. Consequently, we erformed a linear aroximation for a 5% strain, as in Fig.7, and estimated the Young s modulus as.232 MPa. V. COMPRESSION TEST By comressing a hemisherical soft fingerti made of olyurethane gel along the normal direction, as shown in Fig.1 and Fig.8-(b), we verified the validity of our elastic force model reresented in Eq. (11). Furthermore, by conducting multile exeriments with various contacting angles, we demonstrated the existence of the local minimum of the elastic force. In the comression test, we used a fingerti with a diameter of 4 mm, and contacting rods with thirteen different shaes. The rods were inclined from to 3 deg in increments of 2.5 deg, as shown in Fig.8-(b). Fig.9 comares exeriments with simulation results. The horizontal axis reresents the maximum dislacement of the comressed fingerti, while the vertical axis reresents the elastic force measured by a loadcell laced in the comression machine. In all the grahs in Fig.9, the simulation and exerimental results are almost identical to each other u to d 6: mm, after which the discreancies increase with the magnitude of the dislacement. The discreancy comes from the linear aroximation of the exerimental stress-strain diagram shown in Fig.7. The effect leads directly to nonlinearity of Young s modulus, which is outside the scoe of the resent study. Fig.1-(a) and Fig.11-(a) show simulation and exerimental results, resectively. Enlarged views of both results are also shown in Fig.1-(b) and Fig.11-(b). The numerical values in each grah denote the inclined angle of the contacting object, and both results are lotted at intervals of 5. deg. The elastic force increases as the orientation angle increases Fig. 9. Elastic forces in exeriments: (a) 2.5 [deg], (b) 5. [deg], (c) 7.5 [deg], (d) 1. [deg], (e) 12.5 [deg], (f)15. [deg], (g) 17.5 [deg], (h) 2. [deg], (i) 22.5 [deg], (j) 25. [deg], (k) 27.5 [deg], (l) 3. [deg].

6 IEEE TRANSACTION ON ROBOTICS, VOL. 1, NO. 11, NOVEMBER 25 6 Fig. 1. view. Simulation results of elastic force: (a) simulations, (b) enlarged Fig. 12. Local minimum of elastic force: (a) simulations, (b) exeriments. Fig. 13. Sring constant inside the soft fingerti. Fig. 11. view. Exerimental results of elastic force: (a) exeriments, (b) enlarged under constant maximum dislacement. As confirmation, we lotted the elastic force against θ of Eq. (11) in Fig.12. The numerical values shown in the grah denote the maximum dislacement d. At about deg, there is a clear local minimum of the elastic force, and the change in elastic force with θ is greatest when the dislacement is maximum, that is, 8. mm. The same tendency can also be seen in the simulation results. The results therefore indicate that our roosed elastic model can resent a distinctive henomenon, i.e., a local minimum elastic force, even when the deriving rocess is reresented simly by bringing linear virtual srings standing in the normal direction. On the other hand, the discreancy in the large dislacement shown in Fig.12 would be reduced if the Young s modulus could be defined as a nonlinear function of comression strain, and be used to adot the model to accommodate the nonlinearity of the material. However, the resent study focuses on the geometric nonlinearity, and the deriving rocess including both nonlinearities will be addressed in future studies. VI. CONCLUDING REMARKS We have formulated a static elastic force model and an elastic otential energy function based on virtual srings inside a hemisherical soft fingerti. We have also roven the existence of an LMEE and exerimentally demonstrated that the elastic force due to the deformation has a local minimum. Our model requires us to only measure the Young s modulus of a corresonding material to be used in robotic fingertis. In future studies, we will consider constant volume deformation of incomressible elastomer materials, and derive elastic models incororating a nonlinear Young s modulus. These new findings suggest a quasi-static maniulation theory based on the LMEE for a minimum d.o.f. robotic hand [14]. By exanding the new idea of LMEE in the develoment of grasing and maniulation theory using softfingered robotic hand, it is exected that the stable grasing and the ose control of a grased object by a minimum d.o.f. two-fingered hand may be achieved and a succinct control system will be designed. APPENDIX I CONTACT PLANE FORMULA As illustrated in Fig.1, the oint C is described in a vector form as 2 3 (a d)sinθ! OC 4 5 : (22) (a d)cosθ In addition, a normal unit vector with resect to the contact surface is reresented as 2 n 4 sinθ cosθ 3 5 : (23) Since nthe contact lane can be written by an inner roduct form, [x;y;z] T! o OC n, the lane equation is therefore described as follows: xsinθ + zcosθ a d: (24) APPENDIX II SPRING CONSTANT FORMULATION As shown in Fig.13, letting k,ds, and L resectively be the sring constant, the sectional area, and the natural length of a secimen for measuring the Young s modulus, and E be the Young s modulus obtained from an aroriate comression

7 IEEE TRANSACTION ON ROBOTICS, VOL. 1, NO. 11, NOVEMBER 25 7 Fig. 14. Equivalent fingerti stiffness with resect to -coordinate system. Fig. 15. Fingerti stiffness on a certain erimeter: (a) ellitical erimeter, (b) circular erimenter. test, we can derive the following equations according to linear material mechanics: () σ Eε (25) F E δx S L (26) () E L δx F L kδx L k S δx S ; (27) S where F denotes an alied force to the secimen and δx is a dislacement in the identification test. Since this aer assumes that the Young s modulus is an invariant hysical value for individual material, the following equation is satisfied: k L ds k L S E (28) () k E ds L EdS a 2 (x 2 + y 2 ) : (29) APPENDIX III COORDINATE CONVERSION TO DERIVE THE FINGERTIP STIFFNESS As illustrated in Fig.14, let be the coordinate system translated to O from the frame, and be the cylindrical coordinate system inclined by θ from z -axis. Let r be the arbitrary radius on the contact circle that has an origin C, and φ be the common rotational angle around the z-, z -, and z - axes. The relationshi between (x ;y ) on the frame and (r;φ) on the frame is then exressed as x r cosφ cosθ ; (3) y r sinφ: (31) Since the relationshi between (x;y) and (x ;y ) is described as x x +(a d)sinθ and y y, the variable transformation through the coordinate systems and can be exressed as x r cosφ cosθ +(a d)sinθ ; (32) y r sinφ: (33) Simultaneously, the ellitical region at the bottom surface of the fingerti shown in Fig.14 can be converted to a circular region according to the above transformation rule, that is, the integration area of (r;φ) varies at [;a c ] and [,2π], resectively. Next, let us consider the hysical meaning of the double integration of B(r; φ ) used in Eqs. (9) and (13), which is detailed in Eq. (6) as: 2π 2π B(r;φ) dφ cosθ dφ a 2 fx 2 (r;φ)+y 2 (r;φ)g (34) : Eq. (34) corresonds to a stiffness on an ellitical erimeter whose longitudinal radius is r, as shown in Fig.15-(a). Additionally, substituting θ into Eq. (34) enables to obtain an equivalent stiffness on a circular erimeter of radius r shown in Fig.15-(b). REFERENCES [1] Y.Yokokohji, M.Sakamoto, and T.Yoshikawa, Vision-Aided Object Maniulation by a Multifingered Hand with Soft Fingertis, in Proc. IEEE Int. Conf. Robot. Autom., , [2] Y.Yokokohji, M.Sakamoto, and T.Yoshikawa, Object Maniulation by Soft Fingers and Vision, in Proc. 9th Int. Symosium of Robotics Research, , 2. [3] H.Maekawa, K.Komoriya, and K.Tanie, Maniulation of an Unknown Object by Multifingered Hands with Rolling Contact Using Tactile Feedback, in Proc. IEEE Int. Conf. Intell. Robots Syst., , [4] H.Maekawa, K.Tanie, K.Komoriya, M.Kaneko, C.Horiguchi, and T.Sugawara, Develoment of a Finger-Shaed Tactile Sensor and its Evaluation by Active Touch, in Proc. IEEE Int. Conf. Robot. Autom., , [5] S.Arimoto, P.Nguyen, H.Y.Han, and.doulgeri, Dynamics and Control of a set of Dual Fingers with Soft Tis, Robotica, vol.18,.71-8, 2. [6] S.Arimoto,.Doulgeri, P.Nguyen, and J.Fasoulas, Stable Pinching by a air of Robot Fingers with Soft Tis under the Effect of Gravity, Robotica, vol.2, , 22. [7].Doulgeri, J.Fasoulas, and S.Arimoto, Feedback Control for Object Maniulation by a air of Soft Ti Fingers, Robotica, vol.2,.1-11, 22. [8].Doulgeri and J.Fasoulas, Grasing Control of Rolling Maniulation with Deformable Fingertis, IEEE/ASME Trans. Mechatronics, vol.8, no.2, , 23. [9] N.Xydas and I.Kao, Modeling of Contact Mechanics and Friction Limit Surfaces for Soft Fingers in Robotics, with Exerimental Results, J. Robotics Research, vol.18, no.8, , [1] N.Xydas, M.Bhagavat, and I.Kao, Study of Soft-Finger Contact Mechanics Using Finite Elements Analysis and Exeriments, in Proc. IEEE Int. Conf. Robot. Automat., , 2. [11] I.Kao and F.Yang, Stiffness and Contact Mechanics for Soft Fingers in Grasing and Maniulation, IEEE Trans. Robot. Automat., vol.2, no.1, , 24.

8 IEEE TRANSACTION ON ROBOTICS, VOL. 1, NO. 11, NOVEMBER 25 8 [12] J.L.Johnson, Contact Mechanics, Cambridge University Press, Chater 4, [13] Y.Tatara, On Comression of Rubber Elastic Shere Over a Large Range of Dislacements-Part I:Theoretical Study, ASME J. Engineering Materials and Technology, vol.113, , [14] T.Inoue and S.Hirai, Study on Hemisherical Soft-Fingered Handling for Fine Maniulation by Minimum D.O.F. Robotic Hand, in Proc. IEEE Int. Conf. Robot. Autom., , 26. PLACE PHOTO HERE Takahiro Inoue was born in Osaka, Jaan, on February 14, He received the B.S. degree in Mechanical Engineering at Osaka Institute of Technology, Jaan, in 22, at which time he was resented with the Hatakeyama Award from JSME. He then received the M.S. degree in 24 and is currently working toward the Ph.D. degree in both Information Science and Engineering at Ritsumeikan University, Jaan, where he is now a COE Research Fellow funded by Jaan Society of the Promotion of Science (JSPS). He received the finalist of Best Maniulation Paer Award of IEEE Int. Conf. on Robotics and Automation at 25 and 26. His research interests include soft-fingered maniulation, soft object modeling, and MEMS technology. He is a student member of IEEE, RSJ, SICE, and JSME. PLACE PHOTO HERE Shinichi Hirai was born in Fuji, Jaan, in He received the B.S., M.S., and Doctoral degrees in alied mathematics and hysics from Kyoto University in 1985, 1987, and 1991, resectively. He is currently a Professor in the Deartment of Robotics at Ritsumeikan University. He was a Visiting Researcher at Massachusetts Institute of Technology in 1989 and was an Assistant Professor at Osaka University from 199 to His current research interests are modeling and control of deformable structures, realtime comuter vision, and soft-fingered maniulation. He received SICE (Society of Instrument and Control Engineers) Best Paer Award at 199, JSME (Jaan Society of Mechanical Engineers) Robotics and Mechatronics Div. Achievement Awards at 1996, the finalist of Automation Best Paer Award at 21 IEEE Int. Conf. on Robotics and Automation, and the finalist of Maniulation Best Paer Award at 25 and 26 IEEE Int. Conf. on Robotics and Automation. He is a member of IEEE, RSJ, JSME, and SICE.