Port-Hamiltonian Modeling for Soft-Finger Manipulation

Transcription

1 The 2010 IEEE/RSJ Internatonal Conference on Intellgent Robots and Systems October 18-22, 2010, Tape, Tawan Port-Hamltonan Modelng for Soft-Fnger Manpulaton F. Fcucello, R. Carlon, L.C. Vsser and S. Stramgol Abstract In ths paper, we present a port-hamltonan model of a mult-fngered robotc hand, wth soft-pads, whle graspng and manpulatng an object. The algebrac constrants of the nterconnected systems are represented by a geometrc object, called Drac structure. Ths provdes a powerful way to descrbe the non-contact to contact transton and contact vscoelastcty, by usng the concepts of energy flows and power preservng nterconnectons. Usng the port based model, an Intrnscally Passve Controller (IPC s used to control the nternal forces. Smulaton results valdate the model and demonstrate the effectveness of the port-based approach. I. INTRODUCTION Dextrous manpulaton sklls, for personal and servce robots n unstructured envronments, are of fundamental mportance n performng dfferent tasks. Usually, a robotc hand has to manpulate objects of dfferent shape, sze, weght, materal, and, n some cases, has to nteract wth human bengs. Durng manpulaton, the dynamc propertes of the controlled system change, due to the non-contact to contact transtons and due to the contact vscoelastcty. Therefore, n order to derve the dynamc model of a handobject system durng graspng, the contact model between the fngers and the object s of crucal nterest. In the Lagrangan formulaton, the dynamc model of the hand-object system takes the form of a multbody system. In case of rgd contact, the whole system s a nonholonomc constraned system and the equatons can be obtaned usng the Lagrange-D Alembert formulaton, consderng the graspng constrant equatons [1]. In case of a complant contact model, where the fngers have thck complant layers of vscoelastc materal, the graspng constrant equatons are not vald anymore. Moreover, the dynamcs of the contact are nfluencng the system dynamcs, and have to be consdered. The port-hamltonan framework s based on descrbng a system n terms of energy varables and the nterconnecton of systems by means of power ports. Any physcal system can be descrbed by a set of elements storng knetc or potental energy, a set of energy dsspatng elements, and a set of power preservng ports, through whch energy can only be transferred and not produced [2]. The energy flow Ths work has been partally funded by the European Commsson s Seventh Framework Programme as part of the project DEXMART under the grant agreement ICT Ths work has also been partally funded by The Netherlands Mnstry of Economc Affars as part of FALCON project under the responsblty of the Embedded Systems Insttute (BSIK03021 program wth Vanderlande Industres as the ndustral partner. f.fcucello@unna.t, Dpartmento d Informatca e Sstemstca, Unverstà degl Stud d Napol Federco II, Italy. {r.carlon,l.c.vsser,s.stramgol}@utwente.nl, Department of Electrcal Engneerng, Faculty of Electrcal Engneerng, Mathematcs and Computer Scence, Unversty of Twente, The Netherlands. varables are ntrnscally defned and are ndependent of the partcular confguraton of the physcal systems. The concept of a power port s an effcent and useful way to descrbe the nteracton between physcal systems and between the system and the envronment. The theory of port-hamltonan systems allows to descrbe the system behavon a coordnate-free way and can be naturally extended to nclude constraned systems and complant contact models. Ths approach s useful to model and control the nteracton between a robot and a passve envronment. The robot s a n-dof mechancal passve system wth respect to the controller, that can be modeled as a port-hamltonan system. To preserve a passve behavon the nteracton wth the envronment, both n case of contact and non-contact, an Intrnscally Passve Controller (IPC [3], based on mpedance control [4], can be used. Snce the IPC approach yelds an ntrnscally passve system, the controlled system wll be stable, both n case of contact and non-contact phases, for every passve, even unknown, envronment. Ths s n contrast wth a conventonal hybrd controller, whch swtches from poston to force control when a contact occurs. Such controllers can easly become unstable, because of nose affectng the force sensors that detect the contact. Moreover, a hybrd forceposton control requres a perfect plannng of the tasks, whch s only possble f the envronment s known. In ths paper, we present a port-hamltonan model of a mult-fngered robotc hand, wth soft-pads on the fnger tps, graspng an object. The vscoelastc behavor of the contact s descrbed n terms of energy storage and dsspaton. Usng the concept of power ports, the dynamcs of the hand, the contact, and the object are descrbed n a coordnate-free way. Moreover, an IPC s appled to control the moton of the object and to regulate the nternal forces,.e. the forces appled at the contact ponts and not nfluencng the object moton. These forces are mportant to have a stable grasp. In the model of the hand-object system, we assume that the contact forces are always satsfyng the frcton cone condtons,.e. the contact forces are always nsde the frcton cone, and we assume that there s no rollng contact. Ths means that the grasp matrx s constant. The man advantage of the port-hamltonan formulaton for constraned systems s that we do not need to modfy the dynamc equatons when a change occurs n the contact state. Instead, t s possble to represent both cases n a tme-dependent geometrcal structure, that satsfes the power contnuty condtons n every contact state. Ths framework allows to approach the problem n a more ntutve and compact way. The graphc bond graph representaton of /10/$ IEEE 4281

2 the system s based on the energy flow through the ports connectng the sngle components. The papes organzed as follows. In Sec. II, we formulate the problem n a port-based settng and present the mathematcal framework. In Sec. III, we derve the dynamcs of the contact model. In Sec. IV, we propose the port-hamltonan model of the system, based on the analyss of the energetc nterconnectons and the propertes of the subsystems. In Sec. V, smulaton results are presented to valdate the model and to show the behavor of the system when an IPC based controlles appled. Fnally, Sec. VI provdes conclusons and sketches the future work. II. PORT-BASED FORMULATION The port-hamltonan formalsm has been ntroduced by van der Schaft and Maschke n [6]. Port-based modelng s at the bass of network theory, n whch the dfferent parts of the system are nterconnected through power ports and descrbed n terms of power exchange. A power port s defned by a par of dual varables, a flow f and an effort e, whose ntrnsc dual product e f yelds power. If V s the lnear space of flows, then the dual space V s the lnear space of efforts. On the space V V, t s possble to defne a power contnuous structure, called Drac structure, whch defnes the nterconnecton between the power ports,.e. t descrbes how the powes dstrbuted between the ports. The Drac structure s a subspace D V V such that [7]: D = {(f, e e f = 0, (f, e D V V } A generc Drac structure s depcted n Fg. 1. The bonds, connected to the structure, realze the ports through whch energy can be exchanged wth energy storage elements, energy dsspatng elements, the controller, and the nteracton port, wth whch the system nteracts wth the envronment. In order to derve the mathematcal model of the manpulaton system as a port-hamltonan system, we need to defne a state manfold S, of whch the coordnates represent energy varables, and on S a Hamltonan energy functon H : S R descrbng the total energy of the system. Then, by makng explct the Drac structure, the system dynamcs can be derved. Snce an nterconnecton of port- Hamltonan systems s agan a port-hamltonan system, we can proceed by ndvdually modelng the hand, the object that s manpulated, and the contacts, and to defne the nterconnecton of the systems. Regardng the Drac structure of the contact model, observe that the contact represents the power contnuous nterconnecton between the fnger, the soft-pad and the object, n terms of elastc energy storage and energy dsspaton. Ths nterconnecton can be represented consderng the fnger and the object as two rgd bodes connected trough a vscoelastc soft-pad. Snce n a manpulaton task both contact and non-contact stuatons may occur, the Drac structure s not constant n tme [8]. The contact and non-contact state are both represented n the same swtchng Drac structure, whch s, obvously, tme dependent. Fg. 1. A. Problem statement Drac structure of a generc port-hamltonan system. In the Lagrangan formulaton, the dynamc model of a n-fngered hand, each wth r degrees of freedom, has the form: M(q q + C(q, q q + g(q = τ J T hw c where q = {q 1,...,q n } Q s the vector of the generalzed confguraton varables for the n fngers, wth Q confguraton manfold and q T q Q ther generalzed veloctes, belongng to the tangent space of Q at q. The vector τ Tq Q represents the generalzed actuator forces at the jonts, belongng to the co-tangent space of Q at q. The matrx J h s the hand Jacoban, that maps the jont veloctes to the Cartesan fngertp veloctes. From dualty, t follows that the transpose J T h maps the fngertp forces to generalzed jont forces. Let W c = {W c1,...,w cn } be the vector of the contact wrenches. The dynamcs of the object are gven by: M 0 (x 0 ẍ 0 + C 0 (x 0,ẋ 0 ẋ 0 + g 0 (x 0 = GW c + F env where x 0 X represents the pose of the object, wth X the confguraton manfold of the object, and ẋ 0 ts generalzed velocty. To avod sngulartes due to the local representaton of the pose, we can descrbe the object dynamcs globally by applyng the Newton-Euler equatons to the body confguraton expressed n the Specal Eucldan group SE(3, and then obtan the Lagrange-D Alembert representaton choosng the local coordnates x 0 X for the object confguraton. The matrx G s the grasp matrx, whch s a lnear map between the contact forces, expressed n the contact frame, and the resultant force on the object, expressed n the object frame [9]. The vector GW c descrbes the effect of the fngertp forces on the object, appled at the contact ponts. The external forces actng on the object are descrbed by F env. In the hand and object dynamcs, the matrces M(q, M 0 (x 0 are the symmetrc and postve defnte nerta matrces, the matrces C(q, q, C 0 (x 0,ẋ 0 contan the centrfugal and Corols components, and g(q, g 0 (x 0 are the vectors of generalzed gravty forces actng on the hand and the object, respectvely [1]. In the context of Le group theory, the relatve confguraton of two bodes can be studed usng SE(3. The relatve nstantaneous moton can be studed usng the Le algebra se(3 assocated to SE(3, whch s a 6D algebra, and corresponds to the sx possble motons of a rgd body. The wrenches belong to the dual algebra se (

3 Fg. 2. The geometrcal contact model: the soft-pad s modeled as a sprng and a damper accordng to the nonlnear Hunt-Crossley model. We assume that the fngers have thck complant layers of vscoelastc materal, and the dynamcal behavor of the soft layers s modeled as a sprng and a damper [10]. The model can be extended to a proper generalzaton to the full geometrcal contact descrpton, as proposed n [11]. In ths work, we started from ths geometrcal analyss of the vscoelastc contacts between two objects wthout constrants, and extended ths to a manpulaton context, a complex multbody system ncludng the robotc hand, the soft-pads and the object and subject to constrants. The contact dynamcs between the fngers and the object are represented wth the same geometrcal and energetcally consstent model. Moreover, a nonlnear Hunt-Crossley model of the contact s taken nto account, for a better physcal consstency and descrpton of soft materal behavor [12]. In Fg. 2, a schematc representaton of the object and one fngen contact s shown. Durng the contact, a fnger wth soft-pad s able to transfer to the object four components of the contact wrench W c,.e. the three components of the lnear contact force and the component of the contact torque around the drecton orthogonal to the surface of the object n the pont of contact. Our goal s to descrbe the dynamcs of ths system n the port-hamltonan framework, ncludng the contact dynamcs. Ths allows to present the problem n a more ntutve and compact way. Moreover, gven the port-hamltonan system representaton, an IPC can be easly derved to control the system. III. CONTACT MODEL In ths Secton, we derve the dynamc model of the contact, based on the Hunt-Crossley contact model [12]. The Hunt-Crossley model ncorporates a sprng n parallel wth a nonlnear damper to model the vscoelastc dynamcs. In order to obtan a local representaton of the Drac structure of the contact n a matrx form, we have to defne a contact frame n the contact pont c on the object, as ndcated n Fg. 2. The contact frame Σ c related to the fnge, has the orgn n the contact pont c, and the axs z c s normal to the object surface, pontng nsde the object. There s a unque plane O orthogonal to z c and passng trough c, spanned by the axs x c and y c of the contact frame. In Fg. 2, the object reference frame Σ o, and the world frame Σ b are also depcted. If we chose a bass of the two screws (r x,r y, representng pure rotatons around the two axes x c and y c of the contact frame, and a bass of the screws representng rotaton around z c and the three translatons (r z,t x,t y,t z, we can decompose se(3 n the drect sum of the two subspaces S := span {r z,t x,t y,t z } and N := span {r x,r y }, representng the subspace of the transferable wrenches n the contact pont and the non transferable wrenches respectvely. In partcular, the motons n S nvolve a change n storage of potental energy n the vscoelastc contact [11]. A tangent map P exsts, that projects a twsts n se(3 n the subspace S of motons, and the dual cotangent map P : P : se(3 S, and P : S se (3 Consequently, the elastc storage element, representng the elastc energy stored n the compressed surface of the softpad n contact wth the object, s a 4D port wth power c,c c varables T and W c,store, that are, respectvely, the relatve twst and the contact elastc wrench expressed n the contact frame and projected n the subspace of motons nvolvng elastc storage of energy. In partcular: T c,c = PT c,c, W c c,store = P Wc c,store Snce the dynamcs of the object are dependent of the set of all wrenches actng on t, t s necessary to measure the contact wrenches n order to compute the poston of the object center of mass and of the ponts of contact. Once the measurements of the contact wrenches are avalable, and the stffness of the elastc storage element s known, the deformaton (x, y, z,θ of the soft-pad can be computed n the bass of the screws spannng the subspace of relatve motons nvolvng elastc storage of energy S. Wrtng ths deformaton, relatve to the contact coordnate frame, as an element H SE(3: cos(θ sn(θ 0 x H = sn(θ cos(θ 0 y z the storage of potental energy n the element can be represented by a functon V ( H : SE(3 R. If H(t s known, the relatve twst Tr c,c of the object at the contact pont wth respect to the fnge, can be expressed, n theory, n the contact frame as Tr c,c = H H 1. In practce we can obtan ths twst by measurng the deformaton of the soft-pad usng and estmate the dervatve usng an observer or by numercal dfferentaton. The wrench generated due to a deformaton δtr c,c, related to the relatve poston of the rgd fnger tp x c and the object contact pont x c c, has the followng expresson n the contact frame, 4283

4 accordng to the Hunt-Crossley model 1 : W c c = K s ( δt c,c + Ds ( δt c,c T c,c (1 where K s s the two-covarant stffness tensor [14], such that K s = P K s P wth K s the stffness matrx that( relates the 4D port varables ( T r c,c, W c c,store, and D s δt c,c s the dampng matrx that depends on the deformaton, and s defned locally as ( T δtr c,c D, wth D a constant dagonal matrx dependng on the structure and on the materal of the softpad. By consderng the map P, we can express the vector of contact wrenches wth respect to the contact deformaton: Wc c = P ( K ( s δ Tc,c ( + D s δ Tc,c Tc,c where D s s defned analogous to D s, but wth respect to the 4D vector of the deformatons: c,c δ T = [x,y,z,θ] T whch represents the deformaton of the soft-pad n the allowed drectons. IV. PORT-HAMILTONIAN MODEL In ths Secton, we ntend to derve the port-hamltonan equatons of the complete system. Therefore, startng from the general representaton of Fg. 1, we can characterze the Drac structure by explctly descrbng the three dfferent sub-systems,.e. the hand, the object and the contact. The complete system s composed by a number of storage and dsspatve elements and nteracton ports, connected together, as shown n Fg. 3. The power port between the Drac structure of the fngers and the Drac structure of the contact s a mult-bond port. In partcular, the nterconnecton between the hand and the contact s dentfed by the effort Wc b f,.e. the vector of the wrenches exerted by the soft-pads (due to the object on the fngers, and by the flow T b r = J h q R 6n,.e. the vector of the twsts of the (rgd fnger tp n the base frame. The power port between the Drac structure of the object and the Drac structure of the contact s a mult-bond port as well. The port s characterzed by the effort Wc b o,.e. the vector of wrenches exerted by the fngers on the object, and by the flow T b c = G T (x 0 ẋ 0 R 6n,.e. the vector of the twsts of the object n the contact ponts n the base frame. The power port between the storage element of the soft-pads and the Drac structure of the contact s characterzed by the effort Wc c,store,.e. the vector of the wrench transferable component due to the (soft-pad sprngs, represented by the vector of the partal dervatves of the total Hamltonan energy functon wth respect to the vector of the states of the (soft-pad sprngs,, and by the flow s T c T c,c r = ṡ c,.e. the vector of the relatve twsts between the fngers and the object, projected n the subspace S of 1 In the Hunt-Crossley model, the stffness and dampng forces are taken proportonal to δ m, wth δ the scalar deformaton and m usually close to unty. For smplcty, we have assumed m = 1 n ths work. Fg. 3. Drac structure of the whole constraned port-hamltonan system. The fngers and the object are connected wth two effort sources, that are, respectvely, the control acton and the external forces actng on the object. the moton nvolvng elastc storage of energy. The power port between the dsspatve element of the soft-pads and the Drac structure of the contact s characterzed by the effort W c c,ds,.e. the vector of the wrench transferable component c,c due to the (soft-pad dampers, and by the flow T r. To complete the descrpton of the system represented n Fg. 3, we need to consder the nerta of the fngers, connected to the fnger Drac structure by means ( of the power flow defned by the conjugate varables ṗ,, and the control port descrbed by ( q,τ. For the object, we need to consder the nerta of the object, connected to the object Drac structure by means ( of the power flow defned by the conjugate varables ṗ o,, and the envronment nteracton port descrbed by (ẋ o,f env. For the sake of smplcty, we neglect Corols effects. Ths smplfcaton s justfed because of the small work space we consder, and the relatvely low veloctes of the system. For clarty, n the fgure we have omtted the power port of the gravty external force, that we take n account n the port-hamltonan ( equatons, ( for the fngers and the object,.e. q w, and ẋ q T o,w, respectvely. x T o A. Drac Structure of the fngers The Hamltonan energy functon of the fngers s H(s c,p,q = 1 2 pt M 1 (qp st c K s s c + V (q where the frst term s the knetc energy of the fnger, the second s the potental energy of the sprngs that model the soft-pads and V (q s the potental energy of the fngers due to gravty. The Hamltonan equatons of the fngers are: ṗ = q T + JT hw b c f + τ, q = The Drac structure can be represented by a skew- 4284

5 symmetrc matrx: ṗ 0 1 J T h 1 q T b = r J h q w τ W b c f q T where 0 and 1 denote zero and dentty matrces of approprate dmensons. B. Drac Structure of the object The Hamltonan energy functon of the object s gven by H(p,x o = 1 2 pt o M 1 p o + V o (x o where V o (x o s the potental energy of the object due to gravty. The Hamltonan equatons of the object are: ṗ o = x T + GWc b o + F env, ẋ o = o The Drac structure can be represented by a skewsymmetrc matrx: ṗ o q o T b c ẋ o,w = 0 1 G G T C. Drac Structure of the contact F env W b c o x T o In order to obtan a local representaton of the Drac structure of the contact n a matrx form, we have to choose a reference frame of the contact, and consder the transformaton between the contact frame Σ c and the world frame Σ b. In the case of soft fnger contact model, the Drac structure of the contact s the followng: Wc b f Wc b o T c,c r ṡ c 0 0 A A = 0 0 A A A A 0 0 A A 0 0 T b r T b c W c c,ds o s T c where A s the matrx representaton of the operator: A := dag {A 1,...,A n}, A = (s, 1Ad T H c P b The matrx A s the dual operator to A, and s expressed as the transpose of matrx expresson for A. The bnary sgnal s, s defned as s, = 1 f there s no contact, and s, = 0 f there s contact [11]. If the poston of the object s unknown, as n an unstructured envronment, the bnary sgnal can be related to the sgnal of a force sensor located n the soft-pad. The adjont operator Ad c H b transforms the relatve twst between the rgd part of the fnge and the contact pont on the object from coordnates n Σ b, the world frame, to Σ c, the contact coordnate frame. The frst two equatons gve the expresson of the contact forces actng on the fnger and on the object n case of contact and non-contact, thanks to the bnary sgnal. In case of non-contact they are zero and n the other case they are obvously opposte, and they have the expresson of Hunt-Crossley model. D. Drac Structure of whole system In ths Secton, we derve the Drac structure of the whole system by combnng the three Drac structures derved for the dfferent sub-systems. It follows that the skew-symmetrc Drac matrx, from Eq. (2 at the top of next page, descrbes the energy flows through the system. From Eq. (2 t s possble to obtan the nput-state-output port-hamltonan system of the whole system. The flow and effort varables of control and nteracton port are splt nto conjugated nput-output pars. The general form s the followng: ẋ = (J(x R(x x T + g(xu y = g T (x x T where (u, y are the nput/output pars correspondng to the control port, the matrx J(x s a skew-symmetrc matrx representng the port topology defned by the Drac structure, whle the matrx R(x = R T (x 0 specfes the energy dsspaton. In our case we have the state x = [ p q po x o s c ] T and energy functon H(x = 1 2 pt M 1 (qp st c K s s c + V (q pt o M 1 p o + V o (x o Furthermore, we obtan: J T h A J(x = GA A T J h 0 A T G T 0 0 J T h A DA T J h 0 J T h A DA T G T R(x = GA DA T J h 0 GA DA T G T [ ] T g(x = u = [ τ F env ] T, y = [ q ẋo ] T The matrx J(x s obtaned from Eq. (2 by nspecton, as well as matrx g(x. The matrx R(x s obtaned consderng the expresson of the dampng force n Eq. (1. V. SIMULATIONS For the valdaton of the model, a smple example modeled n 20-sm smulaton software [15] s consdered. We mplement the bond graph representaton of two fngers wth soft-pads, that nteracts wth an object n a plane. An IPC s mplemented to control the jont torques: τ = g(q + J T h[k c (x rd x r ] D c q where the terms K c and D c are the proportonal and dervatve control gans. Here the desred rgd fnger tps postons 4285

6 ṗ J T h q A JT h A q w ṗ o ẋ o = GA GA ẋ o,w T c,c r A T J h 0 0 A T G T ṡ c A T J h 0 0 A T G T τ q T F env x T o W c,ds c s T c (2 Fg. 4. Smulaton results - The smulatons valdate the model wth control of the nternal force and control of the object moton. When the bnary sgnal swtches from 1 to 0, the fngers are n contact wth the object. x rd are derved consderng the desred forces at the contact and the model of the soft-pad. By means of the vrtual poston, we are able to regulate the moton of the object and, through the null space of the grasp matrx, to regulate the nternal forces as well by controllng the relatve poston of the fnger tps. The soft-pad s modeled as a sprng and a damper, wth a stffness constant k = 50 N/m, and dampng constant d = 3 Ns/m. When the soft-pad touches the object, the bnary sgnal s swtches from 1 to 0. The smulaton s dvded n three phases. Frst, the fngers move, touch the object and push aganst t untl the nternal forces are regulated at 16.5 N. Then, the fngers move the object, whle mantanng the regulaton of the nternal forces at the same value. The dfference from the contact force of the two fngers are due to the nerta of the fngers and of the object. In the thrd step the fngers move away from the object. The object contnues to move n the left drecton, due to the deformaton n the soft-pads, and t wll stop due to the frcton. Fg. 4 shows the dagrams of the postons of the fnger tps and of the object, and the contact forces at the fnger tps. For a better understandng of the smulaton results, a vdeo of a 3D anmaton n 20-sm has been made and attached to the paper. VI. CONCLUSIONS AND FUTURE WORK In ths paper, we presented a port-hamltonan analyss and model of a mult-fngered robotc hand wth softpads whle graspng and manpulatng an object. The man advantage of ths formalsm s that t allows to descrbe the behavor of the system n terms of energy storage and energy flow, usng the concept of power ports. Ths formalsm represents an effcent and useful way to descrbe the nteracton between the object and the fngers, as well as the nteracton of the whole system wth the envronment. Usng screw theory and Le group theory, the vscoelastc descrpton of the contact can be expressed as a power contnuous nterconnecton, represented by a Drac structure. Therefore, we are able to descrbe non-contact to contact transton, and contact vscoelastcty wthout changng the dynamcal equatons of the model. Future work wll focus on the passve control of the nternal and external forces, also n case of sharng tasks wth a human beng. Moreover, dfferent models of the softpad wll be used, lke soft-pads wth nonlnear stffness [13]. REFERENCES [1] R.M. Murray, Z. L and S.S. Sastry, A Mathematcal Introducton to Robot Manpulaton, CRC Press, [2] P.C. Breedveld, Physcal Systems Theory n Terms of Bond Graphs, PhD thess, Unversty of Twente, Enschede, [3] S. Stramgol, Modelng and IPC Control of Interactve Mechancal Systems: A Coordnate-free Approach, Sprnger-Verlag, [4] N. Hogan, Impedance control: An approach to manpulaton: Parts I- III, ASME Trans. Jour. Dynamc Systems and Measurement Control, vol. 107, pp.1-24, [5] S. Stramgol and V. Dundam, Varable Spatal Sprngs for Robot Control, IEEE/RSJ Int. Conf. on Intellgent Robots and Systems, [6] A.J. van der Schaft and B.M. Maschke, On the Hamltonan formulaton of nonholonomc mechancal systems, Reports on Mathematcal Physcs, vol. 34, no. 2, pp , [7] A.J. van der Schaft, L 2 -Gan and Passvty Technques n Nonlnear Control, Sprnger, [8] S. Stramgol and V. Dundam, Port Based Modelng of Spatal Vsco- Elastc Contacts, European Journal of Control, pp , [9] B. Sclano and O. Khatb (Eds., Sprnger Handbook of Robotcs, Chapter 28 Graspng, Sprnger-Verlag, [10] L. Bagott, P. Tezz, C. Melchorr and G. Vassura, Modellng and controllng the complance of a robotc hand wth soft fnger-pads, IEEE Int. Conf. on Robotcs and Automaton, [11] S. Stramgol and V. Dundam, Modelng the knematcs and dynamcs of complant contact, IEEE Int. Conf. on Robotcs and Automaton, [12] N. Dolat, C. Melchorr and S. Stramgol, Contact Impedance Estmaton for Robotc Systems, IEEE Trans. on Robotcs, vol. 21, pp , [13] G. Bersell and G. Vassura, Dfferentated layer desgn to modfy the complance of soft-pads for robotcs lmbs, IEEE Int. Conf. on Robotcs and Automaton, [14] M. Zefran and V. Kumar, Affne connectons for the Cartesan stffness matrx, IEEE Int. Conf. on Robotcs and Automaton, [15] Controllab Products B.V., 20-sm,