A Hamiltonian formulation of the dynamics of spatial mechanisms using Lie Groups and Screw Theory

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1 A Hamltonan formulaton of the dynamcs of spatal mechansms usng Le Groups and Screw Theory Stefano Stramgol Delft Unversty of Technology Dept. of Informaton Tech. and Systems, P.O. Box 5031, NL-2600 GA Delft, NL E-mal: WWW: Bernhard Maschke Unversty of Twente Department of Systems, Sgnals and Control Faculty of Mathematcal Scences 7500 AE Enschede, The Netherlands E-mal: Catherne Bdard Unversty of Twente Department of Bo-engneerng Faculty of Electrcal Engneerng 7500 AE Enschede, The Netherlands E-mal: Abstract In ths paper two man topcs are treated. In the frst part we gve a synthetc presentaton of the geometry of rgd body moton n a projectve geometrcal framework. An mportant ssue s the geometrc approach to the dentfcaton of twsts and wrenches n a Le group approach and ther relaton to screws. In the second part we gve a formulaton of the dynamcs of multbody systems n terms of mplct port controlled Hamltonan system defned wth respect to Drac structures. 1 Introducton The geometry of rgd body moton was one of the man topcs n the developments of geometry n the nneteenth century (see the remarquable hstorcal perpectve by R. Zegler [47]). One of 1

2 the comprehensve theores was proposed by Sr R.S. Ball n the theory of screw exposednhs famous book [3]. In the followng century ths subject remaned atopc of actve research as well for the knematc and statc as well as for the dynamcal models of multbody systems especally n the robotcs communty. It s characterzed by a varety of subtle geometrc approaches. One of them s the theory of Le groups [17] whch turned out to be very useful for the study of moton and robotcs [32]. The theory of screws has the advantage that t s based on the geometry of lnes and s therefore geometrcally attractve, gvng ntutve nsght n the mechansms [22] [37] [19]. On the other hand, Le groups are analytcally very handy. A formal relaton between the two s shown n Sec.2. Concernng the formulaton of the dynamcal models, addtonal structure has to be added. For nstance n the Lagrangan formulaton, the symplectc canoncal form on the space of generalzed coordnates and veloctes or n the standard hamltonan approach the correspondng symplectc Posson bracket. In secton Sec.3we shall brefly ndcate how another geometrcal structure, called Drac structure [9] may be used to formulate the dynamcs of multbody systems n terms of an mplct port controlled Hamltonan system [29]. 2 Rgd body motons Rgd bodes motons have been studed n the past usng dfferent technques lke skrew theory [3] and Le Groups [32]. Screw theory uses the projectve extenson of the Eucledan three dmensonal space. Snce n ths work we relate the two approaches, we frst start wth a formal explanaton relatng the usual concepton of an Eucledan space to ts projectve extenson. In what follows, t s easy to grasp the dea for the lower two dmensonal case consderng a cnema wth an nfntely extended screen (the two dmensonal Eucledan space). A pont on the screen s actually a beam passng throw the projector (the orgn of the vector space of dmenson three whch embeds the screen). The plane parallel to the screen and passng through the projector s called the mproper hyperplane. 2.1 Projectve geometry and Eucledan Spaces An extenton to the concepton we have of the three dmensonal Eucledan spaces can be found n projectve geometry 1. To talk about the 3D Eucledan world n a projectve settng, we need three ngredents: A real vector space V 4 called the supportng vector space of dmenson 4 from whch we exclude the orgn. An equvalence relaton on V 4 {0}: v 1 v 2 α R 0 s.t. v 1 = αv 2. A polarty P, whch s a 2 covarant, symmetrc tensor defned on V 4 whch n the sequel wll be taken sempostve defned and of rank 1. The basc transformatons between ponts of projectve spaces are defned as njectve lnear transformatons between the supportng vector spaces. These transformatons must be njectve to prevent that the subspace correspondng to the kernel of the transformaton s mapped to the 0 element of the codoman whch s NOT a vald element of the projectve space 2. These knds of transformatons are called homographes or collneatons and n our case are mappngs from V 4 to 1 Caley n 1859, by ntroducng the concept of an absolute, showed that projectve geometry s the most general. 2 It wll be shown later that to have a proper defnton of the projectve space, the 0 element must be excluded. 2

3 V 4.MapsfromV 4 to the dual V 4 are nstead called correlatons. A symmetrc correlaton as P s called polarty Improper hyperplane Usng the polarty P, t s possble to consder the vectors p belongng to the quadrc defned by the polarty P whch s called the absolute: P j p p j =0 p V 4. The absolute, s a three dmensonal subspace of V 4 whch s called the mproper hyperplane and t s ndcated wth I 3 V 4. Ths hyperplane represents the ponts at nfnty. The mproper hyperplane splts V 4 n two dsjonted sem-spaces whch we wll call respectvely postve sem-space and ndcate t wth I + and negatve sem-space and ndcate t wth I. The three dmensonal projectve space s defned as the quotent space of V 4 excludng the orgn wth respect to the defned equvalent relaton: P 3 := V4 {0} In a purely projectve settng, wthout consderng the polarty, all ponts are of the same type. Consderng the polarty, we can make a dstncton between fnte ponts and nfnte ponts. Infnte ponts are those whose representatve n V 4 belong to I 3 and fnte ponts are the others. We ndcate wth P F the fnte ponts and wth P the ponts at nfnty. Clearly we have that P = P F P. Furthermore, for each fnte pont p P F (P j p p j 0), there are representatves v V 4 belongng ether to I + or to I. We can defne the sgn functon σ for elements v V 4 : +1 v I + σ(v) := 0 v I 3 (1) 1 v I Adjont polarty Assocated wth the polarty P, one defnes ts adjont 3 Q whch s a 2 contravarant, symmetrc, sempostve tensor of rank 3. Once a proper base {e x,e y,e z,e 0 } for V 4 s chosen, the representatons of P and Q become: P = Q = (2) In the same coordnates, a vector of V 4 has the form (x, y, z, α) T and f α = 0 the vector belongs to I 3. 3 If nstead of 0 on the dagonal elements of P and Q we substtute ɛ, weseethatqp = PQ = ɛi where I s the dentty matrx. 3

4 2.1.3 Ponts and Free vectors It s possble to assocate to each par of fnte vectors n P F a unque element of I: f : P F P F I;(p, q) p p σ(p) q q σ(q) (3) where p, q are any representatves and represents the P -norm. It s possble to see that the prevous operaton s ndeed ndependent of the representatves of the ponts and therefore well defned. Note that the dfference of two ponts can be calculated usng the vector structure of V 4. Furthermore, the mproper hyperplane wthout the equvalence relaton and wth the orgn of V 4, gets the meanng of the vector space of free-vectors. In the usual coordnates, ths means that f α p x p α q x q p = α p y p α p z p and q = α q y q α q z q α p α q we have p σ(p) =α p and q σ(q) =α q and therefore x p x q f(p, q) = y p y q z p z q 0 Note that ndeed the last component s equal to zero whch confrms the fact that the element belongs to I. It s usual to use the notaton: p q := f(p, q) for obvous reasons, but the normalsaton of Eq.3before the subtracton s essental to make the operaton ntrnscally defned Lnes A lne l P 3 n a projectve context s nothng else than a one dmensonal subspace whch corresponds to a two dmensonal subspace L V 4 (wthout the orgn) of the supportng vector space V 4. In a projectve context t s also possble to talk about lnes at nfnty when L I 3.As a consequence, a lne can be descrbed both as the subspace spanned by two ponts of V 4 or as the ntersecton of two hyperplanes of V 4.Asnevery2ndmensonal vector space an n dmensonal subspace s self-dual, so s a lne n P 3 a self dual entty. It s possble to show that usng the prevous coordnates, gven two dstnct ponts x, y belongng to a lne where ) (ȳ ) ( x x = and where x, ȳ R the correspondng lne can be dentfed homogeneously to a vector of the form (ȳ ) x α where α R {0}. (4) x ȳ 4

5 Eq.4 can be clearly wrtten as: ( ) ( ) ȳ x ū =. (5) x (ȳ x) r ū It s easy to see that choosng nstead of x and ȳ other two dstnct ponts on the same lne n the usual geometrcal sence, the vector l defned as n Eq.4 s the same f a proportonal multplcaton constant s used. Ths mples that the space of lnes has somehow a projectve nature by tself, but t should be notced that the lnear combnaton of two lnes as expressed n Eq.4 s n general NOT a lne Eucldean product on the mproper hyperplane It s now possble to defne a proper non-sngular, nternal product for I 3 whch gves rse to the scalar product whch characterses a proper Eucldean Space. In the prevous canoncal coordnates, an element v I 3 s charactersed by the last component equal to zero. We can assocate to v I 3 the subspace of hyperplanes H v for whch v s what s called the polar wth respect to Q: h H v v = Qh where Q s the adjont of P. We can now defne the scalar product of two vectors v 1,v 2 n I 3 as: v 1,v 2 := h T 1 Qh 2 h 1 H v1,h 2 H v2 (6) It s easy to see that the prevous defnton s well posed snce t s ndependent on the elements h 1 and h 2 due to the structure of Q The Eucldean space It s now possble to defne the followng Eucldean three dmensonal space as the par (E 3, ) such that E 3 := P 3,and, s the scalar product just defned on E 3 := I 3 whch s treated as a vector space. As wth P, we ndcate wth E F fnte ponts and wth E nfnte ponts and agan: E = E F E. Proper collneatons of the Eucldean space are defned as the ones whch keep the polarty P,and therefore the mproper hyperplane I 3 nvarant. It can be seen that n the prevous coordnates a proper Eucldean collneaton has the form: ( ) c : R 4 R 4 σr p ; q q (7) 0 1 where R SO(3), the set of postvely orented orthonormal matrces, σ can be ether equal to +1 for an orentaton preservng transformaton or 1 for a transformaton changng orentaton and p R 3. Lookng at the expresson of Eq.4 for a lne, t s possble to fnd how these coordnates transform. Usng Eq.7 and Eq.4, t s possble to see that the mappng of a lne corresponds to the mappng of ponts as reported n Eq.7 becomes: ) ( ) (ȳ2 x 2 σr 0 )(ȳ1 x = 1 (8) x 2 ȳ 2 σ pr R x 1 ȳ 1 5

6 where the operator tlde s such that p s a skew-symmetrc matrx and t s such that pa = p a for each a R 3. Usng the matrx rotaton group SO(3) and the ntrnscally defned operaton of Le brackets [21], t s possble to orent E by defnng a 3 form Ω on I 3. We suppose that the coordnates (e x,e y,e z,e 0 ) chosen are such that Ω(e x,e y,e z ) = +1 and e z =[e x,e y ] where [, ] represents the Le algebra commutator of so(3). In these coordnates, the operaton of vector product nr 3 results the usual one. 2.2 Dsplacements of rgd bodes Eucldean system As shown n [24] and [39], to descrbe the relatve moton of 3D rgd parts n a geometrcally ntrnsc way, t s necessary to consder a set of Eucldean spaces of dmenson three. Ths set s called a Eucldean system n [38] and ndcated wth S m := {E 1, E 2,...,E m } (9) The concept of an observer s very mportant and s coupled to the concept of space: an observer s dentfed wth a Eucldan space n whch he s rgdly connected together wth all those objects whch are always statonary n relaton to hm. A set of objects whch are never changng poston n relaton to one another can be consdered as a sngle entty from a knematc pont of vew. For these reasons and for more formal ones that should become clear n the sequel, we consder as many Eucldan spaces as there are bodes movng relatve to one another. All these Eucldean spaces are such that they have the mproper hyperplane of ther projectve extenson n common. An object wll be a constant subset of a Eucldan space together wth a mass densty functon whch assocates a value to each of the body s ponts. It s also possble to defne, usng the Kllng form [38], an Equvalence Inertal Relaton on the set of all possble Eucldean spaces. Elements of the same class can only translate wth constant veloctes wth respect to one another. One Eucldean class whch s partcularly mportant and s called Inertal class s the class where physcal laws are nvarant. We wll usually ndcate a partcular representatve of ths class wth E 0. In the projectve settng, these Eucldean spaces can be seen as a set of four dmensonal vector spaces wth a common polarty P and a common mproper hyperplane A rgd body In the presented settng, a rgd body s s a subset of a Eucldean space E S m together wth a mass densty functon whch assocates a value to each of the body s ponts: ρ : E R + The mass densty functon s relevant for dynamc consderatons. For knematc consderatons, t s suffcent to defne a proper body B as a constant subset of a Eucldan space E nstead. 6

7 2.2.3 Coordnate functons A coordnate Ψ for the space E s a 4-tuple of the form Ψ =(o, ˆx, ŷ, ẑ ) o E, ˆx, ŷ, ẑ E where ˆx, ŷ, ẑ should be lnear ndependent. If furthermore ˆx, ŷ, ẑ are orthonormal, we call the coordnates orthonormal. Eventually we call the coordnates postvely orented ff Ω(ˆx, ŷ, ẑ )= +1 where Ω s the form orentatng the Eucldean space. If not otherwse specfed, we wll always consder postvely orented, orthonormal coordantes systems. We can now defne a base for V 4 dependent on Ψ usng the four lnear ndependent vectors: ˆx, ŷ, ẑ, o o σ(o ) V4 of V 4. The map whch assocate a representatve of a pont of E to ts numercal representaton usng ths base, s ndcated wth ψ : Clearly we have that: ψ (p) = ( ) p x,p y,p z,α p p = pxˆx + p y ŷ + p z ẑ + α o σ(o ) ψ 1 (,,, 0) = I 3 =1,...,m (10) and therefore ths coordnates are charactersed by the fact that the last component equal to zero represents a pont at nfnty Dsplacements of rgd bodes We can assocate the relatve postons of bodes belongng to two dfferent Eucldean spaces E and E j to a postve sometry between E and E j [24] [38]: h j : E E j ; p h j (p ) o such that h j s an sometry: and t s orentaton preservng (h j (p ) h j (q )) = (p q ) p,q E (h j (p ) h j (o )) (h j (q ) h j (o )) = h j ((p o ) (q o )+o ) h j (o ) We ndcate the set of postve sometres from E to E j wth SE j (3). In the case = j, SE (3) wll be ndcated wth SE (3) and s a Le group [21]. Once we choose a reference relatve poston r j SE j (3), usng the commutaton dagram shown n Fg.1, t s possble to assocate wth each element h j SEj (3) an sometry n the Le group SE (3): h,j = r j 1 h j (11) 7

8 E r j E j h h j hj E r j E j Fgure 1: The commutaton dagram of the relaton among elements of the Le groups and the reference. In an analoguous way, one may assocate wth each element h j SEj (3) an sometry of SE j(3): h j,j = h j rj 1 (12) Note that by NO means s ths mappng ntrnsc [39]: t depends on the reference choce r j! An element h j SEj s an ntrnsc, coordnate free representatve of the relatve poston of E wth respect to E j. If we assgn to each space E a coordnate ψ wth the propertes reported n Eq.10 and we assume that n a certan nstant the relatve poston s h j SEj, a pont p E corresponds unquely to the pont h j (p) E j. The usual change of coordnate from Ψ to Ψ j for a certan relatve confguraton s therefore: c : R 4 R 4 : p (ψ j oh j oψ 1 )(p ) If the coordnate systems Ψ and Ψ j are postvely orented and orthonormal and snce h j SEj (3) s a postvely orented sometry, then the mappng c s lnear and s equal to a matrx of the form: ( H j R j = p j ) (13) 0 1 where R j SO(3), pj R3 and therefore H j SE(3). Note that pj characterses how the orgn of Ψ s mapped to n frame Ψ j. It s mportant to notce that we can easly relate the restrctons that n the coordnates Ψ the polarty P gets the form of Eq.2 together wth the Eucldean product defned n Eq.6, to the usual concept of orthonormal bases for the Eucldean spaces. Ths can be seen consderng the three vectors ˆx := ψ 1 ((1, 0, 0, 0)), ŷ := ψ 1 ((0, 1, 0, 0)) and ẑ := ψ 1 ((0, 0, 1, 0)) belongng to E 3. It s easy to see that for the defned coordnates we have that ˆx,ŷ and ẑ are an orthonormal set snce they have norm one and are orthogonal wth respect to each other. Furthermore, the vector ψ ((0, 0, 0, 1)) belongs to the dual space of I 3 and has P -norm equal to 1. It s possble to see that we can defne a bjectve relaton between elements of E 3 and elements of V 4 whch have, usng a coordnate ψ, the last component 4 equal to α>0. If we consder the 4 Note that both element wth the last component equal to α and α have P -norm equal to α 8

9 last component as a scalng factor respect to the P -norm, t s reasonable to gve elements wth the last component equal to one a specal role. For what sad, usng the coordnates ψ and consderng elements wth α = 1 as specal representatve of the equvalent classes whch are elements of E, we have all the propertes of an orthonormal base for the elements ˆx, ŷ and ẑ. 2.3 Twsts Veloctes and twsts So far we have consdered relatve postons wthout ntroducng moton. The latter s clearly a change of relatve poston. To be able to talk about change, we need to consder a tme set I whch wll be an open nterval of R. We can then consder curves n SE j (n) parameterzed by t I. Defnton 1 (Relatve Moton). We call a dfferentable functon of the followng form 5 : a relatve moton of space wth respect to j h j : I SEj (n) Snce relatve motons are dfferentable, we can consder the velocty vector of the curve at h j whch we ndcate as ḣ j T h j SE j (n). The vector ḣj (t) s a bounded vector dependent on the current confguraton hj (t). Usng the dentfcaton of relatve postons wth sometres n SE j (3) reported n Eq.12, one can assocate to a local velocty ḣj (0) an element of se j(n), the Le algebra of the Le group SE j (3) of E j called twst n j frame. Analogously, usng the dentfcaton reported n Eq.11 one can assocate an element of se (n), the Le algebra correspondng to the Le group SE (n) ofe called twst n frame. These elements may be defned n a geometrcal way [21, 24, 38], but n ths text we wll defne them n ther numercal representaton below. We denote wth t j,k se j (3) the twst whch descrbes the moton of E wth respect to E k as an element of se j (3). It s shown n [39, 38] that specal care should be consdered n the defntons of ntrnsc twsts Numercal representaton of twsts We have shown n the prevous secton that after we have chosen coordnates, we can relate a relatve confguraton h j SEj (n) to a matrx belongng to SE(3). Namely, we can assocate the relatve poston h j SEj (n) to the matrx Hj whch corresponds to the change of coordnates from Ψ to Ψ j. Consder now a pont p E. If we take the numercal representaton P of ths pont p usng the coordnates Ψ, we clearly have that P = 0 snce p s a fxed pont of E. We can then wrte the numercal representaton of ths pont n the coordnate Ψ j fxed wth E j. Usng the coordnates change matrx, we obtan: P j = H j P 5 Note the abuse of notaton here: h j has been used both as an element of SEj (n) and as a functon from I to SE j (n). 9

10 T Ḣ j Notaton T := H jḣj Ḣ j = Hj T P j = H j ( TP ) T,j T := Ḣj H j Ḣ j = TH j P j = T (H j P ) T j,j Table 1: The possble forms of twsts Dfferentatng over tme, we have: P j = Ḣj P where we used the fact that P = 0 and where we consder the followng representatve for ponts: ( ) P k p k := k =, j 1 Before proceedng, we need a result whch s well known and useful for what follows. Theorem 1. Gven a matrx H SE(3) contnuously dependng on a scalar t,we have that: ḢH 1,H 1 Ḣ se(3). where ) } {( ω v se(3) := R 4 4 s.t. ω, v R and ω s the skew-symmetrc matrx such that for any vector x, ωx = ω x. The set of matrces se(3) whch result from the product of a matrx belongng to SE(3) tmes the dervatve of ts nverse (or vce versa) s called the Le algebra of SE(3). The elements belongng to se(3) are the numercal representatve of the twsts n frame or n j frame. We wll use the followng notaton: ( ) ) ω ( ω T := v T v = se(3) 0 0 The vector T s the vector of Plücker coordnates of the twst T and we wll abusvely also call a twst. We have therefore a vector representaton of a twst and a matrx representaton: respectvely T R 6 and T R 4 4. For the twst of E wth respect to E j we consder P j = Ḣj P 10

11 and we have the two possbltes reported n Tab.1. Furthermore (ṗj ) ( P j = H j ( TP R j ) = p j )( ) )( ω v p ṗ j = R j (ω p )+R j v and P j = T (H j P ) (ṗj ) ( ω = 0 v 0 0 )( R j p j )( ) p ṗ j = ω (R j p + p j )+v The rotatonal part s the usual angular rotaton. The lnear velocty part of the twst T, j, v = p j ω, j +ṗ j s the velocty of a pont q rgdly connected to Ψ j and passng through the orgn of Ψ Changes of coordnates for twsts: the adjont representaton The twsts n frame and n j frame are related by a lnear map called adjont mappng or adjont representaton of the Le group SE(3) on se(3) [21, 38] and have a numercal representaton relatng the vector representatons of the twsts as follows: where T j,j Ad H j = Ad H j T,j ( R j := 0 p j Rj R j ). The matrx Ad H j s dependent on H j and ths mples that f the relatve poston of E wth respect to E j changes, H j wll change and as a consequence, the matrx Ad H j wll be tme varng. We wll need the tme dervatve of Ad H j for expressng the dynamcs of a rgd body. ( ) Ad H j = Ad H j ad T,j where ad s called the adjont representaton of the Le algebra [21] k,j ) ( ω ad T k,j := 0 Note that the matrx ad T s a functon of a twst T and not of a relatve poston H as Ad H. ṽ k,j ω k,j 2.4 Wrenches and ther dentfcaton wth twsts In order to defne properly the nteracton of a rgd body wth the rest of the mechansm, force varables, n the sense of the acton of a set of forces on a body, have to be consdered. The acton of a set of forces on a body s called wrench. The wrench s a vector of se (3), the dual space of the space of twsts, whch allows to express the power resultng from the acton of the wrench W on the body undergong a trajectory H(t) wth twst T, s the scalar product: Power = W, T 11

12 where, denotes the dualty product. Usng Plücker coordnates for the wrenches (n the dual bass to the Plücker bass of twsts) one obtan a vector representaton of the wrenches: W = ( m f ) where m represents the torque and f the lnear force, the power may be expressed: Power = WT Changes of coordnates for Wrenches As the wrenches belong to the dual space of twsts, changes of coordnates nduces the adjont change of wrenches. Hence n the numercal representaton, the transformaton of wrenches s gven by the transposton of the adjont mappng: (W ) T = Ad T (W j ) T H j It s mportant to note that f the mappng Ad H j was mappng twsts from Ψ to Ψ j, the transposed maps wrenches n the opposte drecton: from Ψ j to Ψ! Ths s because Wrenches are duals of Twsts and they are pulled-back B-nvarant forms on se(3) and recprocty of twsts It has been shown n [23] that the space of b-nvarant, and therefore ntrnsc, 2-forms defned on se (3) s a lnear combnaton of the hyperbolc metrc H j and the Kllng form K j. In the coordnates we used so far, ther numercal representatons are: H = 1 2 ( 0 ) I3 I 3 0 And a general b-nvarant metrc s gven by: and K = 4 ( ) I B = hh + kk h, k R Among all the possble metrcs, there are two whch have specal propertes whch allow to drectly relate Le groups to the theory of screws. If we consder the kernel of B, ths s dfferent than the trval zero element of se (3) ff h =0. A proportonal element of the Kllng form has therefore a frst specal property. Furthermore, among all the Bs there s just one metrc whch s generated by an nvolutve map, whch means that an expresson of ts nverse s equal to the form tself and ths s namely obtaned when k =0 and h = 2. Ths nvolutvty property results fundamental to relate Le groups to screws and corresponds to the selft-dualty of lnes. We ndcate H =2Hand K = 1 4 K. These two specal 2-forms are fundamental for what follows Identfcaton of twsts and wrenches Usng the nvolutve map H and the dualty product on se(3), one may dentfy the twsts wth the wrenches. In the sequel we shall use ths dentfcaton n order to dentfy both twsts and wrenches wth screws. 12

13 se(3) L a Fgure 2: The relaton of the sets n se(3). Therefore one may defne the followng map, denoted by H # from twsts T se(3) to wrenches W = H # (T ), as follows: W, T = H(T,T ) T se(3) (14) It s remarkable that, usng ths dentfcaton, one may dentfy the adjont mappng wth ts contragradent map, n other words the coordnate change on twsts and wrenches. 2.5 From Le Groups to Screws The relaton between twsts (and wrenches) and lnes n P 3 s stated by the classcal Chasles (and Ponsot) theorem. For expressng these theorems, some subspaces and subsets of se (3) need to be defned usng the b-nvarant forms [22] Decomposton of se (3) Defnton 2 (Self recprocal sub-set). We call self recprocal sub-set the followng sub-set of se (3): R := {t se (3) s.t H j t t j =0}. Defnton 3 (Axal sub-space). We call axal sub-space the followng sub-space of se (3): a := {t se (3) s.t. K j t t j =0}. It s the bggest subspace whose numercal representaton s nvarant for changes of the orgn of the coordnates system. Defnton 4 (Lnes sub-set). We call lnes sub-space the followng sub-set of se (3): L := R a Note that R and L are NOT subspaces, but a s. Furthermore, a R. Theorem 2 (Geometrcal nterpretaton of L). Any element ( ω r ω ) T L s charactersed by the fact that t leaves a lne of E nvarant n the drecton αω and passng through r. From the prevous theorem, we can conclude that a twst l L s therefore correspondng to a rotaton around ths lne. 13

14 Theorem 3 (Geometrcal nterpretaton of a). Any element ( 0 v ) T a leaves free-vectors nvarant. From the prevous theorem, we can conclude that a twst a a corresponds to a translaton parallel to v. Based on the specal metrcs H and K, we can defne an ntrnsc surjectve but not njectve operaton whch s called polar map: p:se (3) a ; t H jl Kj t Note that we have nverted H n order to get a proper tensor map. It s now possble to express Chasles theorem as follows: Theorem 4 (Chasles). For any t se (3),there exst a l L and two constants α l,α a R such that: t = α l l + α a p(l) (15) In partcular,when t se (3) a, l s unque. Usng the coordnates we have ntroduced prevously, the form of an element of L wll be: ( ) ω l = L where ω 0 v Snce ω 0 due to the fact that L = R a, we can consder n these coordnates the orthogonal complement ω of ω. The fact that l R results n the algebrac condton ω T v =0 n these coordnates. Ths mples that v ω and therefore there exst a unque r R 3 such that v = r ω. It s therefore possble to express ANY element of L n the followng form: ( ) ω (16) r ω where ω 0. Furthermore, as a consequence of Th.4, we can always wrte a twst as: ( ) ( ) û 0 t = ω + v r û û where û s a unt vector. The followng theorem s also very mportant because t allows to express a pure translaton as the sum of two rotatons. Theorem 5 (Axal decomposton). For any a a,there exst l 1,l 2 L such that a = l 1 l 2. As a drect consequence of Th.5, we can express any element a a as: ( ) ( ) ( ) 0 ω ω =. (r 1 r 2 ) ω r 1 ω r 2 ω One of the conclusons we can draw from what sad so far, s that: se (3) = span{l} and therefore we can get any twst as a lnear combnatons of elements of L. 14

15 2.5.2 Decomposton of se (3) The decomposton whch has been shown for se (3), can be drectly mapped to se (3) defnng R = H # (R), a = H # (a) andl = H # (L) where H # corresponds to the hyperbolc metrc and t s defned n Eq.14. The Ponsot theorem s then expressed: Theorem 6 (Ponsot). For any w se (3),there exst a l L and two constants α l,α a R such that: In partcular,when w se (3) a, l s unque. A pure moment may be obtaned as the sum of two forces. w = α l l + α a p (l ) (17) Theorem 7 (Dual axal decomposton). For any a a,there exst l 1,l 2 R such that a = l 1 l 2. One of the conclusons we can draw from what sad so far, s that: The vector space of screw-vectors se (3) = span{l } We have seen that any element of se(3) belongng to the lnes sub-set L has a lne assocated to t, namely the lne whch stays nvarant durng ths rgd rotaton. To do ths bjectvely, we need to assocate to a lne a drecton and a magntude whch would characterse the angular velocty of the correspondng twst belongng to L. In ths way, we leave the homegeneous character of lnes, and obtan the vectors L (or L ). These vectors are called bound lne vectors (or rotors). Conversely, lnes (not at nfnty) are members of the projectvzaton 6 of L. Consderng the whole set of projectve lnes, ncludng lnes at nfnty, and leavng ther homogeneous character, the set R of self-recprocal twsts s obtaned. The vectors of R are called lne vectors [22]. Those correspondng to lnes at nfnty, member of a, are then called free lne vectors [22] (or also free vectors). Conversely, projectve lnes are members of the projectvzaton of R. Nether L nor R are vector spaces, but L spans a 6-dmensonal vector space, that corresponds to se(3). The vectors of ths space are usually called screws [32], screw vectors [22] or also motors. Geometrcal screws, defned by a lne and a ptch, are members of the projectvzaton of se(3) [36] Operatons on screw-vectors Operatons on twsts, or on wrenches, or between wrenches and twsts can now be mapped nto operatons on screws. Gven two screws s 1 and s 2 assocated respectvely to two elements t 1,t 2 se(3), ther recprocal product s defned as: s 1 s 2 := t T 1 Ht 2 6 Wth projectvzaton s ntended the process of consderng elements a, b L equvalent ff there exst a scalar α 0suchthata = αb. 15

16 The recprocal product corresponds to the dualty product of twsts and wrenches. Let s 1 and s 2 assocated respectvely to t 1 se(3) and w 2 se(3), s 1 s 2 := t 1,w 2 When s 1 s 2 = 0, the two screws are sad to be recprocal. Recprocty of screws s used to express non-workng condton. Lne-vectors are self-recprocal, ths corresponds to the self-dualty of lnes. Let consder the scalar t T 1 Kt 2. The correspondng screw operaton s called Perpendcular product and t s ndcated wth: s 1 s 2 := t T 1 Kt 2 The vector space se(3) s a Le algebra and therefore t has an nternal product called a commutator and correspondng to the Le brackets of the assocated Le group SE(3). Wth the prevous assocaton between screws and twsts, we can defne the screw-vector product as: Usng the usual coordnates we have: s 1 s 2 := [( ω1 v 1 s 1 s 2 := [t 1,t 2 ] ), ( ω2 v 2 )] ( ) ω = 1 ω 2 ω 1 v 2 + ω 2 v 1 As already mensoned, we can assocate to a screw s a screw correspondng to a lne at nfnty whch s called ts polar and wll be ndcated wth s : s = H 1 Ks As screws may be decomposed as sum of bound and free lne-vectors, operatons on screws may be reduced to operatons on lne-vectors. Geometrc dstances and relatons between lnes may then be used very effcently to perform calculaton [22] [6]. In partcular the recprocty of lne-vectors s equvalent to the ncdence of ther lnes (axes). 3 Knestatc and dynamc model In ths secton we shall brefly present the dynamcal model of spatal mechansms as the nterconnecton of elementary subsystems consstng of rgd bodes, spatal sprngs and knematc pars, through a power contnuous nterconnecton. Indeed formulatons of dynamcal models of spatal mechancal systems usng network representatons (lnear graphs or bond graphs [33] [15]) have been proposed as an extenson of the network representaton of electrcal crcut models or ther analogue n mechancs, thermcs or hydraulcs [16] [35], [20], [2], [42]. However these representatons use as power varables, (.e. the varables of the nterconnecton network analoguous to the voltages and currents n Krchhoff s laws) ether scalars (n whch case one has to choose a pror some coordnate systems) or real vectors (whch means that one decomposes the translatons and the rotatons). Usng these varables may obscure and complcate consderably the modelng and the analyss of the mechancal systems as real vectors or scalar do not reflect the geometrc propertes of rgd body veloctes (and dsplacements). Therefore other network representatons of spatal mechancal systems were proposed usng as power varables the ntrnsc representaton of veloctes and forces of rgd bodes as twsts and wrenches. T.H.Daves [11] used the orented graph descrbng the topologcal 16

17 structure of a mechansms and showed that a generalsaton of Krchhoff s laws apples to the twsts and wrenches of a mechansm. Ths work was extended to knestatc models of mult-body systems by C.Bdard [7], who defned a bond graph model usng the geometrcal defnton of twsts and wrenches as screw-vectors. Fnally an extenson of these bond graph models to the complete dynamcal model of a mechancal network (ncludng spatal sprngs and rgd bodes), but usng the Le-algebrac defnton of twsts and wrenches was proposed n [5, 26, 24, 38]. Wth such network models one may assocate an analytcal expresson of the dynamcs, essentally n terms of port controlled Hamltonan systems [25]. Indeed t may be shown that, n general, one may map the nterconnecton structure of network models wth the geometrcal structure of Hamltonan systems (Posson brackets for explct Hamltonan systems [30] or Drac structures for mplct Hamltonan systems [45]) and the total energy wth the Hamltonan functon. For bond graph models of spatal mechancal systems usng the Le-algebrac defnton of twsts and wrenches, two formulatons of the dynamcs assocated wth the network structure were proposed: as constraned Hamltonan system n [24] and as an mplct Hamltonan systems n [29]. In the sequel, we shall brefly present the dynamcal model of a spatal mechancal system as the nterconnecton of the elementary subsystems: rgd body, spatal sprng and knematc par, usng as power varables the twsts and wrenches expressed as screw-vectors. 3.1 Energy storng elements and port controlled Hamltonan systems Port controlled Hamltonan systems Arzng from the bond graph formalsm, more precsely the generalzed or thermodynamcal bond graph formalsm [8], so-called port controlled Hamltonan systems were defned [30]. For the sake of smplcty, hereafter we shall present the defnton on R n whch corresponds actually to local defnton of such systems, assocated wth some local choce of coordnates. On R n,aport controlled Hamltonan system s defned by a skew-symmetrc structure matrx, denoted by J(x), a real functon, called Hamltonan functon, denoted by H(x), an nput matrx, denoted by G(x) and the followng equatons: ẋ = J(x) H(x) + G(x)u x y = G(x) T H(x) (18) x where the state x R n, the nput u R p, the output y R n where R n s dentfed wth ts dual vector space R n. Due to the skew-symmetry of the structure matrx : J T (x) = J(x), the system s ndeed conservatve and the change of ts nternal energy s only due to the power suppled by the nterconnectng port charactersed by V V : Ḣ = H T x l=1 ẋ = H x T J(x) H x }{{} 0 J T (x)= J(x) + H x T G(x) u = y T u } {{ } y T Hence, f furthermore the Hamltonan functon s bounded from below, then the system s passve and lossless. If furthermore the structure matrx J(x) satsfes the relatons n ] l Jjk kl Jj jl Jk [J x l + J x l + J x l 0 (19) 17

18 for, j, k =1,.., n, whch s called the Jacob denttes, then t corresponds to the local defnton of a Posson tensor [21]. It was shown that the structure of port controlled Hamltonan systems may be related to the structure of network or bond graph models of physcal systems wthout energy storng elements n excess: the structure matrx J(x) represents the power contnuous nterconnecton structure of the network and the Hamltonan functon corresponds to the total energy of the system [30] [31] [31]. In the case n whch J(x) satsfes the Jacob dentty, the drft dynamcs of the port controlled Hamltonan system may be related to standard,.e. symplectc, Hamltonan systems. Indeed, f one some neghbourhood the rank of the structure matrx s constant and m =rankj(x), then there exst so-called canoncal coordnates (q, p, r) such that the structure matrx gets the followng form: J = 0 I m 0 I m 0 0 (20) p On ths canoncal expresson of the structure matrx, t appears clearly that t may be splt nto a frst bloc-dagonal term beng the standard symplectc matrx and a second one whch determnes the kernel. The coordnates r, called redundant coordnates, corresponds to the kernel of the structure matrx and seen as coordnate functons, they generate the so-called Casmr functons [21]. It s clear that whatever the Hamltonan functon s, the redundant coordnates are dynamcal nvarants of the drft dynamcs of the port controlled Hamltonan system n Eq. 18. These dynamcal nvarants, the Casmr functons, play an essental role n the reducton of Hamltonan systems wth symmetres [21], but they play also an essental role for the passve control [10] [27] as shall be presented heareafter. Dsspaton can be added by defnng the structure not only wth the skew-symmetrc structure matrx J(x) but also wth a postve sem-defnte structure matrx denoted by R(x) [10]. Wth ths extenton, one obtans the port controlled Hamltonan systems wth dsspaton: And the energy balance gves ẋ =(J(x) R(x)) H(x) x y = G(x) T H(x) x + G(x)u Ḣ = H T R(x) H x x + yt u whch shows that part of the nternal energy s dsspated and modeled by R(x) The rgd body element The frst elementary subsystem of a spatal mechancal system s naturally the rgd body. In network (bond graph) models, a rgd body may be represented by a 1-port element [2] wth consttutve relatons beng ts dynamcal model. Ths model may be expressed n terms of a port controlled Hamltonan system whch we shall present hereafter. Its bond graph representaton was proposed n [24] but note that t dffers from ths model by usng now the representaton of twsts and wrenches as screws. The dynamcs of a rgd body s defned on the state space composed of the poston of the body wth respect to the nertal space H 0 SE(3) and of ts momentum n body frame P se (3) 18

19 [21]. The rgd body s endowed wth a knetc energy E K (P ) whch s a quadratc functon of the momentum n body frame P : E K (P )= 1 2 P (I ) 1 (P ) T where I s a constant tensor called the nerta tensor. It may also be endowed wth a potental energy whch s a functon E P (H 0 ) of the dsplacement H 0. The potental energy may be, for nstance, due to the gravty or may be zero7. Ths element has two (power conjugated) port varables beng the twst n nertal frame T 0,0 of the body and the conjugated wrench n nertal frame W 0. The consttutve relatons of ths rgd body elements may then be wrtten n the followng form of a port controlled Hamltonan system: (see [29, equaton (2.1)]): ( ) ( ) (dep d H0 0 TLH 0 ( dt P = H TL T P 0) ) ( ) 0 H 0 T,0 + Ad T W 0 H 0 T 0,0 = ( ) ( de 0 Ad P ( H 0) ) H 0 T,0 (21) ( ) Pω where P Pv := s a 6 6 matrx composed of 3 3skew-symmetrc matrces representng P v 0 the vector product of the angular and lnear momenta, H 0 s a 6 dmensonal vector representaton of H 0,andTL H 0 s such that TL H 0T,0 s equal to the parameterzed form of Ḣ 0 = H0,0 T. Note that the twst T,0 s actually seen n these equatons as the gradent of the knetc energy E K (P ) wth respect to the momentum P. It may also be noted that the frst equaton n ths system represents smply the defnton of the twst of the body n nertal space and the second one s an extenson of the Newton Euler equatons to a rgd body whch s also endowed wth some potental energy [21] The sprng element The second type of energy storage n a spatal mechansms conssts n spatal sprngs that means sprngs wth a dsplacement beng a relatve poston H j SE(3) between the Eucldean frames and j. In general t may be defned by some potental functon E P (H j ). In bond graph models t may be represented by a 1-port element wth port varables beng the twst T,j and elastc force W,j of the sprng n [24]. Hence the consttutve relatons of the sprng element may be expressed as the followng port controlled Hamltonan system: d H j dt = TL H j T,j W,j = TLT de H j P ( H j ) (22) where the ndex notaton s the one used n [41]. Note that the actual defnton and parameterzaton of the potental functons for spatal sprngs s far from beng obvous [13] [14]. 7 Even when the potental energy s zero, the state H 0 s necessary to be able to calculate changes of coordnates. 19

20 3.2 The knestatc model, Drac structures and Hamltonan dynamcs Drac structures on vector spaces Consder a m-dmensonal real vector space V and denote by V ts dual. A Drac structure [9] on the vector space V s an m-dmensonal subspace L of V V such that: (f,e) L, e, f = 0 (23) where, denotes the dualty product. In order to gve some constructve defnton of dfferent Drac structures n the sequel, we shall use a defnton of Drac structures n terms of two lnear maps and called kernel representaton of a Drac structure [31]. Every Drac structure L V V s unquely defned n a bass B = {v 1,.., v m } by the par of real matrces (F, E) called structure matrces and satsfyng the condton: by: EF T + FE T = 0 (24) L = {(f,e) V V s.t. F f + Eē =0} (25) where f s the coordnate vector of f n the bass B of V and ē s the coordnate vector of e n the dual bass to B of V. A Drac structure defned on a vector space V encompasses the two structures of a Posson vector-space and a pre-symplectc vector-space [9]. Such Drac structures arse n electrcal crcuts (or ther mechancal analogue) where Krchhoff s laws actually endows the set of currents and voltages of the crcut or the rate varables and the co-energy varables of an LC-crcut (possbly wth elements n excess) wth a Drac structure [28] [1]. In the sequel we shall see how Drac structures may be assocated wth the knestatc model of a mechansm The port connecton graph Intally a mechansm s descrbed by the topology of the set of rgd bodes, sprngs and knematc pars that compose t. Ths topology s descrbed by an orented graph, called prmary graph. Its vertces are assocated wth the bodes (more precsely wth the Eucldean space assocated wth t) and ts edges are assocated wth the generalzed sprngs, knematc pars and dampers wth arbtrary orentaton. However n order to represent explctly the port varables of the rgd bodes on the topology, one has to augment the prmary graph n order to assocate an edge also to every body. Ths augmented graph s called port connecton graph [18], whch s obtaned by augmentng the prmary graph by a reference vertex and a Lagrangan tree [34] connectng the reference vertex wth the set of vertces of the prmary graph. Examples can be found n [38]. If the nerta of some body s neglected (.e., s set to zero), ths s taken nto account by erasng the correspondng edge n the Lagrangan tree. In ths way all the elements of the mechancal network are assocated wth an edge and the port varables of the elements may be consdered as through and across varables of the port connecton graph lke voltages and currents n an electrcal network [34]. Then t may be shown that these varables obey a generalzaton of Krchhoff s laws appled to the port nterconnecton graph [11]: the sum of the twsts along any cycle (or loop) n the port connecton graph vanshes, the sum of the wrenches along any co-cycle (or cutset) n the port connecton graph vanshes. 20

21 The Drac structure assocated wth the port connecton graph G may be defned as follows. Denote by T the maxmal tree n G correspondng to the Lagrangan tree and by T the complementary co-tree n G. Denote by B { 1, 0, 1} q n the ncdence matrx of the prmary graph, where q s the number of rgd bodes correspondng to the order of the Eucledan system and n s the number of edges of the prmary graph. Denote by C the fundamental loop matrx defned as: C = ( ) B T I n n and by Q the fundamental cutset matrx assocated wth T [34]: Q = ( I q q B ) It s easy to see that CQ T = 0 (26) Then Krchhoff s loop rules s equvalent to the followng constrant relatons on the across and through varables of the graph, that s, the twsts T T := (T T 1,.., T T n ) se(3) n and the wrenches W := (W 1,.., W n ) se (3) n : CT = 0 (27) QW T = 0 (28) It s now possble to defne the (n + q) (n + q) structure matrces of Eq.24 for the correspondng Drac structure as: ( ) ( ) C 0n (q+n) F = and E = Q 0 q (q+n) Krchhoff s laws are then equvalent to defne the set of admssble twsts and wrenches as: D G,T = {(f,e) se(3) n se(3) n s.t. Ff + Ee =0} (29) From Eq.26 follows that ker C and ker Q are dual spaces and t can be seen that dm D G,T =6n. (Ths s exactly Tellegen s theorem). Hence the admssble set of twsts and wrenches assocated wth the constrant relaton of the port connecton graph form a constant Drac structure on se(3) n Drac structures on dfferentable manfolds The defnton of Drac structures on vector spaces may be extended to dfferentable manfolds as follows. Let M be a dfferentable manfold, a Drac structure [9] on M s gven by a smooth vector sub-bundle L T M T M such that the lnear space L(x) T x M T x M s a Drac structure on the tangent vector space T x M for every x M. More ntrnsc defntons n terms of dualty and some characterstc dstrbutons and codstrbutons may be found n [9] [10]. In the sequel we shall use only a local a kernel representaton of Drac structures. And the Drac structure may also be locally characterzed by a par of structure matrces (dependng now smoothly on the pont x n M). Indeed choosng some coordnates (x 1,.., x m )n some neghborhood of a pont x M, a Drac structure s defned by a par of matrces F (x),e(x) dependng smoothly on x such that: L(x) ={(f,e) T x M T x M s.t. F (x)f + E(x)e =0} (30) 21

22 On ths local representaton t may be seen that Drac structures defned on dfferentable manfolds generalze as well as (generalzed) Posson manfolds as pre-symplectc manfolds n the same way as Drac structures do generalze pre-symplectc and Posson vector spaces [9]. Remark 1. On Drac structure there exst also some addtonal closeness condton whch generalze the Jacob denttes of Posson brackets takng nto account also the nput port. They play also an essental role n the geometrcal structure and representaton of Drac structures; the nterested reader s referred to [9] [12] [10] The knematc par elements The elementary constrants between two rgd bodes are defned by so-called knematc pars. A knematc par s the knematc dealzaton of a set of contacts that occur between two rgd bodes at some confguraton of the bodes [22]. The wrench W transmtted by a knematc par s constraned to a lnear subspace of the space of wrenches se (3) called the space of constrant wrenches [22] and denoted by W C. A relatve twst between the two bodes s allowed by the knematc par when t produces no work wth any transmssble wrench. The relatve twst s thus constraned to belong to a lnear subspace of the space of twsts se(3), called the space of freedom twsts [22] and denoted by T A. It s the space orthogonal to the space of transmtted wrenches W C, n the sense of the dualty product, because an deal knematc par produces no work: w t =0 t T A, w W C (31) It s mportant to realse that the Dract structure here used s actually defned on a submanfold of the state manfold (SE 0 (3) se (3)) m correspondng to the m rgd bodes from whch the mechansm s composed of (see Eq.21). Ths sub-manfold K, s actually: K := SE j(1) (1) (3)... SEj(k) (k) (3) where k s the number of knematc pars and (l),j(l) 1,...,n are the two rgd bodes whch the l-th knematc par connects [38]. It s shown n [38], that after references are chosen, t s possble to fnd a dffeomorphsm π : SE k (3) K where SE k (3) s a Le group wth Le algebra se k (3). Usng ths dffeomorphsm, Le rght and left translatons, t s therefore possble to map se(3) k se(3) k to T x K T x K wth x K: Π x : se(3) k se(3) k T x K T x K ;(T,W) ((π ol x ) T,(L x oπ ) 1 W ) We can now defne the confguraton dependent par of admssble twsts and transmtted wrenches of a knematc par at a confguraton x Kas elements of the followng Drac structure characterzng the knematc par: D P (x) =Π x (T A (x), W C (x)) (32) It s also possble, once coordnates are used, to gve an expresson of the structure matrces expressed for the elements n se k (3) se k (3): ( ) ( ) T (x) 0q n F (x) = and E(x) = W (x) 0 p n 22

23 where p =dmt A (x), q =dmw C (x), E(x),F(x) aren n, T (x),w(x) are full-rank matrces and such that ker W (x) = range T T (x), ker T (x) = range W T (x) andx s a knematc state whch represents the confguratons of the knematc pars. Remark 2. Note that knematc pars whch nteract wth ther envronment (for nstance actuators or other mechansms along the freedom twsts also defne a Drac structure on the space of freedom twsts,constrant wrenches and nteracton twsts and wrenches. The reader s referred to [24,29,38] for detaled defntons The knestatc connecton network By combnng the two types of elementary constrants presented here above (.e., the port connecton graph and the knematc pars) a complex power conservng nterconnecton (n the sense of the defnton gven n the part I of [29]), relatng the rgd bodes and the spatal sprngs of a mechancal system, may be obtaned. Ths nterconnecton structure defnes on ts port varables some constrant relatons whch are called the knestatc model of the mechancal system [22] as they consst of constrant relatons on the wrenches and ther dual constrant relatons on the twsts. Therefore we shall call the nterconnecton structure composed of the port connecton graph and the knematc pars, the knestatc connecton network. The nterested reader s referred to [7, 4, 38] for the detaled analyss of ths network and for ts use n the analyss and desgn of mechansms. In ths paragraph we shall characterze t n terms of Drac structures deduced from the Drac structure assocated wth the knematc pars and wth the port connecton graph. In the sequel, we shall gve an expresson of a Drac structure assocated wth a knestatc model n the case where the nertas of all the bodes are taken nto account. Hence a maxmal tree of the port connecton graph G s gven by the Lagrangan tree T B on whch the bodes are connected. Denote by T P the co-tree composed of the edges of the port connecton graph wth whch the knematc pars are assocated. Note that the co-tree T P s not necessarly maxmal f some edges T S of the port connecton graph correspond to spatal sprngs or ports: G = T B T P T S. Denote by D G,TB the Drac structure on se(3) n defned by the port connecton graph (see Sec.3.2.2) and by D P the Drac structure for the knematc pars (see Sec.3.2.4). The power pars belongng to D G,TB must also satsfy the state dependent constrants of the knematc pars. The resultng Drac structure s therefore: D K (x) = { (T,W) se(3) n se(3) n s.t. (T,W) D G,TB and (T,W) (Π x ) 1 D P (x) } (33) The space D K (x) s therefore defned by assemblng the constrants correspondng to the port connecton graph descrbed n Eq.28 and the constrants assocated wth each knematc par. The admssble twsts and wrenches satsfy: W T = W T + W T + W T = 0 (34) G T B T P T S where the frst sum represents the power exchanged to modfy the knetc energy, the second the power exchanged to modfy the potental energy and the thrd one the power exchanged through the knematc pars wth the envronment. By Proposton 1.1 n [46] D Kx s a Drac structure. Remark 3. Note that for the sake of smplcty we dd not consder that the knematc pars are actuated. In the case that some of the knematc pars are nteractng wth the envronment,the defnton of the Drac has to be changed by augmentng the wrenches and twsts by the wrenches and twsts at the ports of the knematc pars [29]. 23