Variational integrators for constrained dynamical systems

Transcription

1 ZAMM Z. Angew. Mah. Mech. 88, No. 9, (28) / DOI.2/zamm.2773 Variaional inegraors for consrained dynamical sysems Sigrid Leyendecker,2,, Jerrold E. Marsden,, and Michael Oriz 2, Conrol and Dynamical Sysems, California Insiue of Technology, USA 2 Graduae Aeronauical Laboraories, California Insiue of Technology, USA Received 28 December 27, acceped 27 May 28 Published online Sepember 28 Key words Variaional ime inegraion, consrained dynamical sysems, differenial algebraic equaions, flexible mulibody dynamics. A variaional formulaion of consrained dynamics is presened in he coninuous and in he discree seing. The exising heory on variaional inegraion of consrained problems is exended by aspecs on he iniializaion of simulaions, he discree Legendre ransform and cerain posprocessing seps. Furhermore, he discree null space mehod which has been inroduced in he framework of energy-momenum conserving inegraion of consrained sysems is adaped o he framework of variaional inegraors. I eliminaes he consrain forces (including he Lagrange mulipliers) from he imesepping scheme and subsequenly reduces is dimension o he minimal possible number. While reaining he srucure preserving properies of he specific inegraor, he soluion of he smaller dimensional sysem saves compuaional coss and does no suffer from condiioning problems. The performance of he variaional discree null space mehod is illusraed by numerical examples dealing wih mass poin sysems, a closed kinemaic chain of rigid bodies and flexible mulibody dynamics and he soluions are compared o hose obained by an energy-momenum scheme. 28 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Inroducion The disinguishing feaure of variaional inegraors is ha ime-sepping schemes are derived from a discree variaional principle based on a discree acion funcional ha approximaes he coninuous one. This is opposed o he sandard derivaion of inegraion mehods ha sar wih a coninuous equaion of moion (which iself migh have been derived from a coninuous variaional principle) and replace he coninuous quaniies, in paricular he derivaives wih respec o ime, by discree approximaions. The variaional heory of discree mechanics provides a heoreical framework ha parallels coninuous variaional dynamics. Discree analogues o he Euler-Lagrange equaions, Noeher s heorem, and he Legendre ransform are derived from a discree Lagrangian by performing similar seps as in he coninuous heory. The resuling ime-sepping schemes are srucure preserving, i.e. hey are symplecic-momenum conserving and exhibi good energy behavior, meaning ha no arificial dissipaion is presen and he energy error says bounded over long-ime simulaions. There exis many works on symplecic inegraors like [ 7] o menion jus a view. A deailed inroducion plus a survey on he hisory and lieraure on he variaional view of discree mechanics is given in [8]. Tha work inroduces also he heoreical background o deal wih holonomically consrained sysems in he framework of discree variaional mechanics by enforcing consrains using Lagrange mulipliers or formulaing he problem in generalized coordinaes direcly in he consrain manifold. A more applied approach o he variaional inegraion of consrained problems, ha already involves he idea of formulaing he discree Lagrangian in erms of a reparamerized redundan configuraion variable, is given in [9]. However, many issues ha are imporan from a pracical poin of view when using variaional inegraors for he simulaion of consrained dynamics, like a consisen iniializaion or posprocessing seps necessary o evaluae he obained discree rajecory, have no been addressed ye. The firs purpose of his work is o fill his gap by providing deails on he discree Lagrangian, he discree Legendre ransform, he calculaion of energy along a discree rajecory for consrained problems and he fulfillmen of consrains on configuraion and on momenum level. Secondly he recenly in he conex of energy-momenum conserving ime-sepping schemes developed discree null space mehod [] is adaped o he framework of variaional inegraors, in paricular Corresponding auhor, sleye@calech.edu, Phone: , Fax: Research parially suppored by AFOSR gran FA ; jmarsden@calech.edu oriz@aero.calech.edu 28 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

2 678 S. Leyendecker e al.: Consrained variaional inegraors i is shown ha he resuling scheme is no only equivalen o he corresponding scheme using Lagrange mulipliers, bu ha i can be derived iself via a discree variaional principle. Using he discree null space mehod, he consrain forces are eliminaed from he sysem and is dimension is reduced o he minimal possible number by he inroducion of a discree reparamerizaion in one ime inerval only. Besides leading o lower compuaional coss for many applicaions, his procedure removes he well-known condiioning problems associaed wih he use of Lagrange mulipliers [,2] while fulfilling consrains exacly. Thus i combines a number of advanageous properies compared o many oher possibiliies o handle consrains [3]. The discree null space mehod has been applied successfully o he energy-momenum conserving inegraion of he dynamics of mulibody sysems consising of rigid bodies conneced by joins [4,5] and o flexible mulibody sysems being composed of geomerically exac beams and shells [5,6]. Symplecic-momenum conserving inegraion of examples from boh fields are presened in his work and he performance is compared o ha using an energy-momenum scheme. For consrained problems, he energy-momenum conserving ime-sepping equaions involve he evaluaion of more erms a he unknown configuraion, wherefore he linearizaion is more complicaed and he ieraive soluion is compuaionally more expensive. No aemp o place he employed consrained formulaion of rigid body and beam dynamics in he exising lieraure is made in his inroducion since such surveys can be found in he inroducions of he works cied in he corresponding secions. This paper sars wih a presenaion of coninuous consrained Lagrangian dynamics in Sec. 2. Then, he discree counerpars are shown in Sec. 3. In paricular, he discree null space mehod wih nodal reparamerizaion is inroduced in he variaional framework and he quesions of consisen iniializaion, consrained discree Legendre ransforms and hidden consrains are addressed. The developed heoreical aspecs are illusraed by he simple example of a mahemaical pendulum in Sec. 3.. Sec. 4 recalls he formulaion of rigid body dynamics in erms of a consrained configuraion variable and presens he null space marix and nodal reparamerizaion required for he inegraion using he discree null space mehod. The procedure is exended o rigid mulibody sysems in Sec. 5 and an example of a closed kinemaic chain is invesigaed in deail. Finally, aspecs of geomerically exac beam dynamics in he framework of variaional inegraion are discussed in Sec. 6, an example of a mulibody sysem, consising of a beam and rigid bodies is invesigaed and he resuls are compared o he hose obained using an energy-momenum scheme. 2 Consrained Lagrangian dynamics Consider an n-dimensional mechanical sysem in a configuraion manifold Q R n wih configuraion vecor q() Q and velociy vecor q() T q() Q,where denoes he ime variable in he bounded inerval [, N ] R. In general, he Lagrangian of a mechanical sysem consiss of he difference of he kineic energy T ( q) and a poenial V (q) accouning for elasic deformaion and for exernal loading (if presen). Le he moion be consrained by he vecor valued funcion of holonomic, scleronomic consrains requiring g(q) = R m. I is assumed ha R m is a regular value of he consrains, such ha C = g () ={q q Q, g(q) =} Q () is an (n m)-dimensional submanifold, called consrain manifold. Jus as C can be embedded in Q via i : C Q, is 2(n m)-dimensional angen bundle TC = { (q, q) (q, q) T q Q, g(q) =, G(q) q = } TQ (2) can be embedded in TQin a naural way by angen lif Ti: TC TQ. Here and in he sequel G(q) =Dg(q) denoes he m n Jacobian of he consrains. Noe ha admissible velociies are consrained o he null space of he consrain Jacobian. A Lagrangian L : TQ R can be resriced o L C = L TC : TC R. To invesigae he relaion of he dynamics of L C on TC and he dynamics of L on TQ, he following noaion is used. C(Q) =C([, N ],Q,q, q N ) denoes he space of smooh funcions saisfying q( )=q and q( N )=q N,whereq, q N C Q are fixed endpoins. Le C(C) denoe he corresponding space of curves in C and se C(R m )=C([, N ], R m ) o be he space of curves λ :[, N ] R m wih no boundary condiions. This noaion has been inroduced in [8], where a large par of he heory presened here can be found. Theorem 2. Suppose ha is a regular value of he scleronomic holonomic consrains g : Q R m and se C = g () Q.LeL : TQ R be a Lagrangian and L C = L TC is resricion o TC. Then he following saemens are equivalen: (i) q C(C) exremizes he acion inegral S C (q) = N for L C. L C (q, q) d and hence solves he Euler-Lagrange equaions 28 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

3 ZAMM Z. Angew. Mah. Mech. 88, No. 9 (28) / (ii) q C(Q) and λ C(R m ) saisfy he consrained Euler-Lagrange equaions L(q, q) q d d ( L(q, q) q ) G T (q) λ =, g(q) =. (3) (iii) (q, λ) C(Q R m ) exremize S(q, λ) =S(q) λ, g(q) and hence, solve he Euler-Lagrange equaions for he augmened Lagrangian L : T (Q R m ) R defined by L(q, λ, q, λ) =L(q, q) g T (q) λ. (4) The proof given in [8] makes use of he Lagrange muliplier heorem (see e.g. [7]). The erm G T (q) λ (TC) in (3) represens he consrain forces ha preven he sysem from deviaion of he consrain manifold. See Corollary 3.5 for furher explanaion on he space (TC). The coninuous null space mehod. For every q C, he basis vecors of T q C form an n (n m) marix P (q) wih corresponding linear map P (q) :R n m T q C. This marix is called null space marix, since range (P (q)) = null (G(q)) = T q C. (5) Hence admissible velociies can be expressed as q() =P (q) ν() (6) wih he independen generalized (quasi-) velociies ν R n m. Thus a premuliplicaion of he differenial Eq. (3) by P T (q) eliminaes he consrain forces including he Lagrange mulipliers from he sysem. The resuling d Alember-ype equaions of moion read [ L(q, q) P T (q) d ( )] L(q, q) =, q d q (7) g(q) =. They are called d Alember-ype equaions of moion, since he eliminaion of he consrain forces from he sysem by premuliplicaion wih he null space marix is closely relaed o d Alember s principle saying ha he virual work done by consrain forces is zero. Admissible virual variaions in T q C can be expressed as δq = P (q) δw wih δw R n m. Wih hese preliminaries, D Alember s principle reads δq T G T (q) λ =(P (q) δw) T G T (q) λ = δw R n m. (8) Remark 2.2 (Coninuous null space marix) Noe ha he null space marix is no unique, a necessary and sufficien condiion on P (q) is (5). The null space marix can be found in differen ways, eiher by velociy analysis (i.e. corresponding o (6), he map mapping he independen generalized velociies o he redundan velociies represens a viable null space marix) or by performing a QR-decomposiion of he ransposed coninuous consrain Jacobian G T = Q R =[Q, Q 2 ] [ R (n m) m ] wih he nonsingular upper riangular marix R R m m and he orhogonal marix Q O(n), which can be pariioned ino he orhogonal marices Q R n m and Q 2 R n (n m).thenp (q) =Q 2 (q) serves as null space marix, which is someimes called naural orhogonal complemen (see [8]). The hird way o obain a coninuous null space marix as he Jacobian of he reparamerizaion of he consrain manifold is ofen possible, bu he resuling coninuous null space marix can in general no be used o infer a discree null space marix. This is due o he fac ha he respecive discree values of he generalized coordinaes are no available in he presen approach. (9) 28 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

4 68 S. Leyendecker e al.: Consrained variaional inegraors Reparamerizaion in generalized coordinaes. For many applicaions i is possible o find a reparamerizaion of he consrain manifold F : U R n m C in erms of independen generalized coordinaes u U. Then he Jacobian DF (u) of he coordinae ransformaion plays he role of a null space marix. In he sequel, he (n m)-dimensional manifold U will be ermed generalized manifold. Since he consrains are fulfilled auomaically by he reparamerized configuraion variable q = F (u), he sysem is reduced o n m second order differenial equaions. The equaions of moion of minimal possible dimension for he presen mechanical sysem (which consiss of precisely n m degrees of freedom) hen read [ L(q, q) DF T (u) d ( )] L(q, q) =. () q d q Using (), one can wrie d d ( DF T (u) ) L(q, q) ( ( ) ) = D DF T L(q, q) (u) u + DF T L(q, q) (u). () q q q On he oher hand, defining a Lagrangian in generalized coordinaes L U : TU R by L U (u, u) =L(F (u),df T (u) u), is parial derivaives read L U (u, u) = DF T L(q, q) ( ( ) ) (u) + D DF T L(q, q) (u) u, u q q L U (u, u) = DF T L(q, q) (u). u q (2) Thus () is equivalen o he equaions of moion in erms of generalized coordinaes ( ) L U (u, u) d L U (u, u) =. (3) u d u Corollary 2.3 The equaions of moion (3), (7), (), and (3) are equivalen, hey yield he same moion q() in [, N ]. Remark 2.4 (Resriced Lagrangian) I is imporan o noe ha even hough he resriced Lagrangian L C : TC R can also be wrien as L C (q, q) =L(q, q) wih q = F (u) and q = DF (u) u, i is differen from L U because he wo funcions are defined on differen domains. Remark 2.5 (Projecions) The premuliplicaion of he Euler-Lagrange equaions by he ransposed null space marix in (7) can be inerpreed as a projecion ono he coangen space of he generalized manifold since P T (q) :Tq Q T u U. The same holds for he special case in () where he Jacobian of he reparamerizaion serves as a null space marix. Alernaively, one could hink of premuliplying (3) by he projecion Q(q) :Tq Q η(tq C) where η(t q C) = { } (q, p) (q, p) Tq Q, g(q) =, G(q) (FL(p)) = T Q. (4) Even hough he consrain forces are eliminaed by his projecion, he resuling equaions of moion are redundan, since hey have been projeced ono a lower dimensional submanifold. η : T C T Q is he embedding defined by requiring ha he following diagram commues (see [8] for furher deails). TQ C Ti TC FL T Q η FL C T C Such a projecion can be calculaed as [ ] Q = I n n G T G M G T G M, (6) where I n n is he n n ideniy marix and all quaniies are evaluaed a q. (5) 28 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

5 ZAMM Z. Angew. Mah. Mech. 88, No. 9 (28) / Consrained discree variaional dynamics Corresponding o he configuraion manifold Q, he discree sae space is defined by Q Q which is locally isomorphic o TQ. For a consan ime-sep h R,apahq :[, N ] Q is replaced by a discree pah q d : {, + h,,..., + Nh = N } Q, N N, whereq n = q d ( + nh) is viewed as an approximaion o q( + nh). Similarly, λ n = λ d ( n ) approximaes he Lagrange muliplier a n = + nh. According o he key idea of variaional inegraors, he variaional principle is discreized raher han he resuling equaions of moion. The acion inegral is approximaed in a ime inerval [ n, n+ ] using he discree Lagrangian L d : Q Q R via n+ L d (q n, q n+ ) L(q, q) d. (7) n In his work, he midpoin approximaion L d (q n, q n+ )= ( ) 2h (q n+ q n ) T qn+ + q n M (q n+ q n ) hv 2 is used. Variaion of he discree acion sum reads S d = N n= L d (q n, q n+ ) N δs d = δq T D L d (q, q )+ δqn T (D 2 L d (q n, q n )+D L d (q n, q n+ ))+δqn T D 2 L d (q N, q N ). (9) n= Requiring is saionariy for all {δq n } N n= and δq = δq N = yields he discree (unconsrained) Euler-Lagrange equaions D L d (q n, q n+ )+D 2 L d (q n, q n )=. (2) The inegral in [ n, n+ ] of he scalar produc of he consrains and he corresponding Lagrange muliplier is approximaed by n+ 2 gt d (q n) λ n + 2 gt d (q n+) λ n+ g T (q) λ d, (2) n whereby gd T (q n)=hg T (q n ) is used and le G T d (q n)=dgd T (q n). Analogue o Theorem 2., he relaion beween he consrained discree Lagrangian sysem on Q Q and ha corresponding o a discree Lagrangian resriced o C C is saed in he following heorem which has again been aken from [8]. Le C d (Q) =C({, + h,..., + Nh = N },Q,q, q N ) denoe he space of discree rajecories saisfying q d ( )=q and q d ( N )=q N for given q, q N C. LeC d (C) denoe he corresponding se of discree rajecories in C and se C d (R m )=C({, + h,..., +(N)h = N }, R m ) o be he se of maps λ d : {, + h,..., +(N)h = N } R m wih no boundary condiions. Theorem 3. Suppose ha is a regular value of he scleronomic holonomic consrains g : Q R m and se C = g () Q. LeL d : Q Q R be a discree Lagrangian and L C d = L d C C is resricion o C C. Then he following saemens are equivalen: (i) q d = {q n } N n= C d(c) exremizes he discree acion Sd C = S d C C and hence solves he discree Euler-Lagrange equaions for L C d. (ii) {q n } N n= C d(c) and {λ n } N n= C d(r m ) saisfy he consrained discree Euler-Lagrange equaions D L d (q n, q n+ )+D 2 L d (q n, q n ) G T d (q n) λ n =, g(q n+ )=. (iii) (q d, λ d ) C d (Q R m ) exremize S d (q d, λ d )=S d (q d ) λ d, g d (q d ) and hence, solve he Euler-Lagrange equaions for he augmened Lagrangian L d :(Q R m ) (Q R m ) R defined by L d (q n, λ n, q n+, λ n+ )=L d (q n, q n+ ) 2 gt d (q n ) λ n 2 gt d (q n+ ) λ n+. (23) (8) (22) 28 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

6 682 S. Leyendecker e al.: Consrained variaional inegraors Remark 3.2 (Augmened discree Lagrangian) The paricular choice of he augmened discree Lagrangian (23) has several consequences ha are illusraed by numerical examples in Sec. 3.. Firs of all, he negaive sign in fron of he scalar produc of consrains and Lagrange mulipliers causes he consrain forces G T (q n ) λ n o have he righ orienaion. Secondly, including he scalar produc of he consrains wih he Lagrange muliplier a boh ime nodes yields conribuions of he consrain forces in boh discree Legendre ransforms (34), (35). One consequence of ha is ha he firs Lagrange muliplier λ which is compued ogeher wih q using (34) has he correc absolue value. Anoher consequence is ha in he absence of a poenial, he discree Legendre ransforms (34), (35) yield conjugae momena which are consisen wih he consrains on momenum level. Discree null space mehod. The reducion of he ime-sepping scheme (22) can be accomplished in analogy o he coninuous case according o he discree null space mehod. This idea has firs been inroduced in he conex of energymomenum conserving inegraion and applied o he consrained dynamics of mass poin sysems in [], hen i has been furher developed in [4 6] for rigid and flexible mulibody sysems. In order o eliminae he discree consrain forces from he equaions, a discree null space marix fulfilling range (P (q n )) = null (G d (q n )) (24) is employed. Analogue o (7), he premuliplicaion of (22) by he ransposed discree null space marix cancels he consrain forces from he sysem, i.e. he Lagrange mulipliers are eliminaed from he se of unknowns and he sysem s dimension is reduced o n. P T (q n ) [D 2 L d (q n, q n )+D L d (q n, q n+ )] =, g(q n+ )=. (25) Proposiion 3.3 The d Alember-ype ime-sepping scheme (25) is equivalen o he consrained scheme (22). P r o o f. Recapiulaing he consrucion procedure of he d Alember-ype scheme from he consrained scheme, i is obvious ha for given values (q n, q n ), a soluion (q n+, λ n ) of he consrained scheme (22) is also a soluion of he d Alember-ype scheme (25). Assume ha q n+ solves he d Alember-ype scheme (25) for given (q n, q n ). Noe ha condiion (24) on he discree null space marix implies null ( P T (q n ) ) = range ( G T d (q n) ) (see e.g. [9]). Togeher wih (25) i follows ha ( ) ( ) [D 2 L d (q n, q n )+D L d (q n, q n+ )] null P T (q n ) = range G T d (q n ). (26) Accordingly, here exiss a muliplier λ n R m such ha (q n+, λ n ) solve he consrained scheme (22). An explici formula o compue λ n is given in (38). Therefore, he d Alember-ype scheme has he same conservaion properies as he consrained scheme. Sympleciciy and momenum maps are conserved along a discree rajecory q d of (25) and he consrains are fulfilled exacly a he ime nodes. Remark 3.4 (Difference o energy-momenum scheme) I is imporan o noe, ha he choice o evaluae he consrains and he Lagrange mulipliers a he ime nodes in (23) causes he evaluaion of he consrain Jacobian in (22) a he ime nodes. Therefore a discree null space marix wih he propery (24) can simply be found by evaluaion of he coninuous null space marix a he ime nodes. Acquainance of he coninuous null space marix for a specific mechanical sysem always yields an explici represenaion of he discree null space marix for he variaional ime-sepping scheme emanaing from he discree variaional principle in conjuncion wih he chosen approximaions. This is in conras o energy-momenum conserving ime-sepping schemes based on he concep of discree derivaives [,2] or on finie elemens in ime [2], where he discree consrain Jacobian G(q n, q n+ ) depends on boh he presen and he unknown configuraion. As a consequence of he more rare appearance of q n+, he linearizaion of he variaional schemes (22) and (25), necessary o solve he nonlinear algebraic sysem ieraively, is simpler and less compuaionally cos-inensive. Nodal reparamerizaion. Similar o he coninuous case, a reducion of he sysem o he minimal possible dimension can be accomplished by a local reparamerizaion of he consrain manifold in he neighborhood of he discree configuraion variable q n C. A he ime nodes, q n is expressed in erms of he discree generalized coordinaes u n U R n m, such ha he consrains are fulfilled q n = F d (u n, q n ) wih g(q n )=g(f d (u n, q n )) =. (27) 28 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

7 ZAMM Z. Angew. Mah. Mech. 88, No. 9 (28) / Noe ha he discree generalized coordinaes u n are incremenal variables ha describe he evoluion of he configuraion variable in one ime inerval only. This avoids he danger of encounering singulariies which are ofen presen in absolue reparamerizaions. Inserion of he nodal reparamerizaions for he configuraion (27) ino he scheme redundanises (25) 2. The resuling scheme n m-dimensional scheme P T (q n ) [D 2 L d (q n, q n )+D L d (q n, F d (u n+, q n ))] = (28) has o be solved for u n+,henq n+ is obained by he reparamerizaion (27). (28) is equivalen o he consrained scheme (22), hus i also has he key properies of exac consrain fulfillmen, sympleciciy and momenum conservaion. While he consrained scheme becomes increasingly ill-condiioned for decreasing ime-seps, he condiion number of (25), (28) is independen of he ime-sep. Saring wih he discree reparamerizaion q n = F d (u n, q n ), i is possible o derive (28) direcly in one sep. The variaion of a redundan configuraion variable can be expressed in erms of variaions of he discree generalized coordinaes as δq n = F d u n δu n + n k= k i=n F d q i F d δu k + u k i=n F d q i δq. (29) Here, he discree variaional principle (9) requires saionariy for all {δu n } N n= wih δq = δq N =. Aferinsering (29) ino (9), he variaion δu N appears only in he las erm of he sum in (9) implying Fd ( ) T u N [D 2 L d (q N 2, q N )+D L d (q N, q N )] =. Repeaing his argumen, one arrives a he variaional d Alember-ype scheme wih nodal reparamerizaion ( ) T Fd [D 2 L d (q n, q n )+D L d (q n, F d (u n+, q n ))] =, (3) u n where F d u n is he discree null space marix. A similar procedure is followed in [9] o derive a reduced variaional ime-sepping scheme for consrained sysems. However, an absolue reparamerizaion q n = F (u n ) is used here, hus he variaional principle is differen and he danger of singulariies is no excluded. Corollary 3.5 The discree ime-sepping schemes (22), (25), (28), and (3) are equivalen, hey yield he same discree rajecory q d in [, N ]. More specifically, (22), (25), (28), and (3) represen differen possibiliies o realize he condiions q n C and δq T n (D L d (q n, q n+ )+D 2 L d (q n, q n )) = δq n T qn C. (3) Equivalenly, one can reques q n C and D L d (q n, q n+ )+D 2 L d (q n, q n ) T qn C, (32) whereby he orhogonaliy condiion only makes sense when D L d (q n, q n+ )+D 2 L d (q n, q n ) T q n Q is idenified wih is represening elemen in T qn Q (using Riez s heorem, see e.g. [22]). Then (32) means D L d (q n, q n+ )+D 2 L d (q n, q n ) ann(t qn C)={S T q n Q S Tqn C = }. (33) Remark 3.6 (Projecions) As menioned for he coninuous case in Remark 2.5, insead of using he discree null ( space ) marix P T (q n ):Tq n Q T U, one could realize condiion (3) or (32) using he projecion Q(q n ):Tq n Q η Tq n C where Q(q n ) is given by formula (6) and fulfils Q(q n ) G T d (q n)= n m. Thereby he consrain forces (including he Lagrange mulipliers) are eliminaed from he sysem while he number of equaions is no alered. Thus i can no be employed o deermine he rajecory since he projecion ono he lower dimensional submanifold yields redundan equaions. However, his projecion will be useful laer for cerain posprocessing seps of he discree rajecory where i is imporan o know conjugae momena ha are consisen wih he hidden consrains. Remark 3.7 (Γ-convergence) The Γ-convergence of discree acion for consrained sysems o he corresponding coninuum acion funcional is proven in [23] and he convergence properies of soluions of he discree Euler-Lagrange equaions o saionary poins of he coninuum problem is sudied. This exends he resuls in [24] o consrained sysems. In [23], he convergence resul is illusraed wih examples of mass poin sysems and flexible mulibody dynamics ha make use of he discree null space mehod described in deail here in Secs. 3., 4, 5, and WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

8 684 S. Leyendecker e al.: Consrained variaional inegraors Consrained discree legendre ransform. So far, he derivaion of variaional ime-sepping schemes for consrained sysems is based solely on he discree pah of configuraions, i.e. he discree rajecory q d C. Thereby, he discree Lagrangian (7) involves a finie difference approximaion for he velociy in he kineic energy and an evaluaion of he poenial energy a some midpoin. Such approximaed velociies do no fulfill he emporal differeniaed form of he consrains, he so-called hidden consrains. However, informaion on he sysems evoluion on velociy or momenum level migh be of ineres. This can be obained using a discree consrained Legendre ransform. Based on he augmened discree Lagrangian (23), he consrained discree Legendre ransforms F c L d : Q Q Tq n Q and F c+ L d : Q Q Tq n Q read F c L d :(q n, q n+ ) (q n, p n ) p n = D L d (q n, q n+ )+ 2 GT d (q n) λ n, F c+ L d :(q n, q n ) (q n, p + n ), p + n = D 2 L d (q n, q n ) 2 GT d (q n ) λ n. (34) (35) Wih (34), (35), he consrained ime-sepping scheme (22) can be inerpreed as enforcing he maching of momena p + n p n = such ha along he discree rajecory, here is a unique momenum a each ime node n which can be denoed by p n. When he discree null space mehod is used, he Lagrange mulipliers are no a hand as an oupu of he simulaion. They can be recovered easily using he n m marix ( R d (q n )=G T d (q n) G d (q n ) G T d n)) (q (36) which obviously fulfills G d (q n ) R d (q n )=I m m, (37) where I m m denoes he m-dimensional ideniy marix. Then he Lagrange mulipliers can be recovered by premuliplying (22) by R T (q n ) and accouning for (37). In paricular, his yields λ n = R T d (q n) [D L d (q n, q n+ )+D 2 L d (q n, q n )], (38) whereupon he consrained Legendre ransforms can be used. On he oher hand, if no informaion on he consrain forces is needed, one ( can avoid ) o recover he Lagrange mulipliers ( ) and use he projeced discree Legendre ransforms Q F c L d : Q Q η Tq n C and Q F c+ L d : Q Q η Tq n C reading Q p n = Q(q n ) D L d (q n, q n+ ), (39) Q p + n = Q(q n ) D 2 L d (q n, q n ), (4) where Q(q n ) is given by formula (6) and fulfills Q(q n ) G T d (q n)= n m. Boh projeced discree Legendre ransforms yield he same momenum vecor denoed by Q p n. Noe ha for he consrained discree Legendre ransforms and for he projeced discree Legendre ransforms, he oupu is an n-dimensional momenum vecor. In he projeced case, i lies in he (n m)-dimensional submanifold η ( T q n C ). Ye anoher possibiliy is o compue an (n m)-dimensional momenum vecor by projecing wih he discree null space marix. The reduced discree Legendre ransforms P F c L d : Q Q T U and P F c+ L d : Q Q T U are given by P p n = P T (q n ) D L d (q n, q n+ ), (4) P p + n = P T (q n ) D 2 L d (q n, q n ). (42) Hidden consrains. The emporary differeniaed form of he configuraion consrains yields he so-called secondary or hidden consrains, which have been inroduced already wihou menioning in (2) and (4) f(q, q) = d g(q) = G(q) q =, d h(q, p) = d g(q) (43) = G(q) (FL(p)) =. d 28 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

9 ZAMM Z. Angew. Mah. Mech. 88, No. 9 (28) / Similar o (4), in a coninuous augmened Lagrangian hey could be included as follows L(q, λ, q, λ, μ, μ) =L(q, q) g T (q) λ f T (q, q) μ. (44) Their discree form could be enforced during he simulaion by inclusion of discree versions in he augmened discree Lagrangian. However, he invesigaion of several numerical examples dealing wih he energy-momenum conserving inegraion of poin mass sysems, rigid bodies and geomerically exac beams (e.g. in [25], [26]) has brough forward ha he incorporaion of he emporally differeniaed form of he consrains has no lead o crucial advanages (besides he fulfillmen of he secondary consrains hemselves). This fac is also repored in [27] and references herein. The discree rajecory has no been influenced considerably by heir fulfillmen. Thus, heir inclusion in he rajecory based approach of variaional inegraors seems an exaggeraion besides, hey are fulfilled o some exen by he discree rajecory. On velociy level, one possibiliy is o wrie he discree hidden consrains in he form ( ) qn+ + q n f d (q n, q n+ )=G qn+ q n =. (45) 2 h In he example of he mahemaical pendulum where he consrain manifold is an equicurved circle, hey are fulfilled exacly in presence and in absence of he poenial energy. However, for general consrain manifolds, hey need no be fulfilled. For Lagrangians of he form L(q, q) = 2 qt M q V (q), where he conjugae momena compued by he coninuous Legendre ransform are given by p = M q, i can be inferred from (43) 2 ha he discree hidden consrains on momenum level read h d (q n, p n )=G(q n ) M p n =. (46) ( ) Of course, h d (q n, Q p n )=holds exacly, since null(g(q n )) = T qn C and elemens in η Tq n C can be idenified wih heir represening elemen in T qn C (using Riez s heorem, see e.g. [22]). To dae, experience shows ha h d (q n, p n )=is fulfilled in he absence of a poenial bu no when a poenial is presen (see Sec. 3.). Iniializaion of he simulaion. Anoher reason for which a discree consrained Legendre ransform is required is he iniializaion of he simulaion. To simulae he moion of a consrained dynamical sysem wih one of he equivalen imesepping schemes, i is necessary o specify an iniial configuraion q C and a second configuraion q C. Togeher wih he ime-sep h, heir difference quoien represens he velociy of he sysem in he firs ime inerval [,h]. For non-rivial mechanical sysems, i migh be difficul or even impossible o come up wih a reasonable q ha fulfills he consrains. Since he purpose of numerical ime-inegraion schemes is o approximae a coninuous rajecory, i seems appropriae o iniialize he simulaion by prescribing q() C and q() T q() C. The laer can e.g. be found by choosing generalized velociies ν() and employing formula (6). Obviously, an Euler sep q = h q() does no yield a second configuraion ha fulfills he consrains. On he oher hand, consisen iniial momena can be calculaed via he coninuous Legendre ransform p() = L q T q() C. Finally, seing p = p(), one can solve (34) ogeher wih g(q )=for q C and λ. Alernaively, using he discree null space mehod, he reduced coninuous momena are given by P p = P T (q()) p(). Afer insering P p and q = F d (u, q()) ino (4), he laer can be solved for u, hen q follows by nodal reparamerizaion. In his case, he corresponding Lagrange muliplier can be recovered as λ =2R T d (q()) [p() + D L d (q(), q )]. (47) Remark 3.8 (Unconsrained sysems) For unconsrained sysems, one can simply choose q Q. Neverheless, one has o keep in mind ha he difference quoien q q h deermines he iniial kineic energy of he sysem. Also here, one can use he coninuous iniial sae o compue q from p n = D L d (q n, q n+ ). Only when no poenial is presen and he moion is recilinear, an Euler seps coincides wih he calculaion of q from he discree Legendre ransform. 3. Numerical example: mahemaical pendulum As an easy bu illusraive example, a wo-dimensional (n =2) mahemaical pendulum wih mass M =, yielding he 2 2 mass marix M = MI 2 2, and rod lengh l =issudied. The configuraion space is Q = R 2 and he consrain manifold is C = Sl. Firs of all, he graviaion is se o zero such ha he mass poin moves on he uni circle wih consan angular velociy. The iniial posiion is q() = [, ] T and he iniial generalized velociy is ν() =, hus he oal energy of he sysem is.5. The m =consrain funcion, he consrain Jacobian and he null space marix read [ ] g(q) = ( q T q l 2) =, G(q) =q T q, P (q) = y. (48) 2 q x 28 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

10 686 S. Leyendecker e al.: Consrained variaional inegraors Fig. (online colour a: Mass poin on uni circle: snapshos of he moion and consrain forces (h = ). During he ime sepping scheme, hey are evaluaed a q n. In he ime inerval [ n, n+ ], he mass poins roaes abou he ou of plane axis by he angle u n+ R, herefore he discree reparamerizaion is given by [ ] cos(u q n+ = F d (u n+, q n )= n+ ) sin(u n+ ) q n. (49) sin(u n+ ) cos(u n+ ) Using he consrained ime-sepping scheme (22), n + m =3equaions have o be solved for q n+ and λ n while he number of equaions is reduced o one if he discree null space mehod wih nodal reparamerizaion (28) is used. Fig. shows configuraions of he moion and he corresponding consrain forces G T (q n ) λ n ha poin owards he cener of he uni circle. The diagram on he lef in Fig. 2 shows he evoluion of he discree energy E d in he upper graph, calculaed in erms of subsequen configuraions as E d (q n, q n+ )= ( ) T ( ) qn+ q n qn+ q n M 2 h h and he lower graph depics he discree Hamilonian H d in erms of he momena as H d (q n, p n )= 2 pt n M p n. (5) Noe ha he expression for he energy in erms of he momena exacly preserves he iniial energy while ha in erms of velociy preserves a value which is slighly differen, see Fig. 2. This indicaes ha in he presence of consrains, posprocessing seps should involve he momena obained by he discree Legendre ransform raher han he velociies. Furhermore, one can see in Fig. ha besides poining ino he correc direcion, he consrain forces do all have he same value λ n = , in paricular, λ makes no excepion. This is a consequence of he discree Legendre ransform (34) being based in he augmened discree Lagrangian (23) (see also Remark 3.2). (5) x x x x Fig. 2 (online colour a: Mass poin on uni circle: oal energy and fulfillmen of he consrains on configuraion level, on velociy level, and on momenum level (h = ). 28 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

11 ZAMM Z. Angew. Mah. Mech. 88, No. 9 (28) / Fig. 3 (online colour a: Mass poin on uni circle: illusraion of he discree Legendre ransform and he maching of momena (h =8 ) x x x Fig. 4 (online colour a: Pendulum: snapshos of he moion and consrain forces and fulfillmen of consrains on configuraion level, on velociy level, and on momenum level (h = ). The righ hand diagram in Fig. 2 reveals ha he primary as well as he secondary consrains (45), (46) are fulfilled exacly in he absence of a poenial. This is no surprise since one can observe from Fig. 3 ha velociies are angenial a he midpoins and momena are angenial o he consrain manifold a he configuraions hemselves. The resuls obained using he consrained scheme are indisinguishable from hose using he d Alember ype scheme wih nodal reparamerizaion. Secondly, he moion is influenced by he poenial V (q) =(M g) T q whereby g =[, 9.8] T. The pendulum is released wih no iniial velociy from he posiion q() = [, ] T. Fig. 4 shows ha he consrain forces again poin owards he pendulum s suspension poin and heir absolue value varies according o he acual configuraion. The energies E d (q n, q n+ )= ( ) T ( ) ( ) qn+ q n qn+ q n qn+ + q n M + V (52) 2 h h 2 depiced in he upper graph and H d (q n, p n )= 2 pt n M p n + V (q n ) (53) depiced in he lower graph of Fig. 5 on he lef look very much alike, however, he plo of he oal energy on a finer scale on he righ reveals ha H d oscillaes wih smaller ampliude han E d and ha deviaions occur wih a differen sign. In he presence of a poenial, he disances beween wo subsequen configuraions are no equal as can be observed from Fig. 4 and 6. I is observable from hese picures ha q n+ q n is always angenial o he consrain manifold a he midpoin wherefore he consrains on velociy level are fulfilled exacly (see Fig. 4). However, he momenum vecor p n is no in he direcion of he angen (see Fig. 6), hus i is no in he null space of he consrain Jacobian and he hidden consrains are no fulfilled. Of course, he projeced momenafulfill hem. Despie being no exacly equal, he plos of he evoluion of oal energy being compued from p n and Q p n are indisinguishable, wherefore only one of hem is shown in Fig WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

12 688 S. Leyendecker e al.: Consrained variaional inegraors energy kineic energy poenial energy oal energy kineic energy poenial energy oal energy energy Fig. 5 (online colour a: Pendulum: energy in erms of (q n, q n+) and (q n, p n) (h = ) Fig. 6 (online colour a: Pendulum: illusraion of he discree Legendre ransform and he maching of momena. 4 Rigid body dynamics This work makes use of a consrained formulaion of rigid body dynamics [28], ha direcly fis in he framework of DAEs. I circumvens he need o deal wih roaional parameers, angular velociies and acceleraions in he Lagrangian. This formulaion is explained in deail in [4,5] where i is also shown ha he reduced equaions of moion (7) represen he well-known Newon-Euler equaions for he rigid body dynamics. The reamen of rigid bodies as srucural elemens relies on he kinemaic assumpions is illusraed in Fig. 7 (see [29]). The fac ha he placemen of a maerial poin in he body s configuraion X = X I d I B R 3 relaive o an orhonormal basis {e I } fixed in space can be described as x(x,)=ϕ()+x I d I () (54) Fig. 7 (online colour a: Configuraion of a rigid body wih respec o an orhonormal frame {e I} fixed in space. 28 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

13 ZAMM Z. Angew. Mah. Mech. 88, No. 9 (28) / is used. Here X I R, I=, 2, 3 represen coordinaes in he body-fixed direcor riad {d I }. The ime-dependen configuraion variable of a rigid body ϕ() d q() = () d 2 () R2 d 3 () consiss of he placemen of he cener of mass ϕ R 3 and he direcors d I R 3,I =, 2, 3 represening he orienaion of he rigid body. Due o he body s rigidiy, he direcors are consrained o say orhonormal during he moion. Thus one works wih he embedding of he consrain manifold C = R 3 SO(3) ino he configuraion manifold Q = R 2.These orhonormaliy condiions peraining o he kinemaic assumpions of he underlying heory are ermed inernal consrains. There are m in =6independen inernal consrains for he rigid body wih associaed consrain funcions 2 [dt d ] 2 [dt 2 d 2 ] g in (q) = 2 [dt 3 d 3 ]. d T d 2 d T d 3 For simpliciy, i is assumed ha he axes of he body frame, i.e. he direcors, coincide wih he principal axes of ineria of he rigid body. Then he body s Euler ensor wih respec o he cener of mass can be relaed o he ineria ensor J via (55) (56) E = 2 (r J)I J, (57) where I denoes he 3 3 ideniy marix. The principal values of he Euler ensor E I ogeher wih he body s oal mass M ϕ build he rigid body s consan symmeric posiive definie mass marix M ϕ I E M = I E 2 I, (58) E 3 I where denoes he 3 3 zero marix. This descripion of rigid body dynamics has been expaiaed in [4] where also he null space marix I d P in (q) = d 2 d 3 corresponding o he consrains (56) has been derived. Here â denoes he skew-symmeric 3 3 marix wih corresponding axial vecor a R 3. The derivaion of he null space marix in (59) makes use of (6) and he fac ha he independen generalized velociies of a rigid body are he ranslaional velociy ϕ R 3 and he angular velociy ω R 3. They can be comprised ino he wis of he rigid body [ ] ϕ =, ω whereupon (6) yields he direcor velociies d I = ω d I for I =, 2, 3. When he nodal reparamerizaion of unknowns is applied, he configuraion of he free rigid body is specified by six unknowns u =(u ϕn+, θ n+ ) U R 3 R 3, characerizing he incremenal displacemen and incremenal roaion, respecively. Accordingly, in he presen case he (59) (6) 28 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

14 69 S. Leyendecker e al.: Consrained variaional inegraors nodal reparamerizaion F d : U C inroduced in (27) assumes he form ϕ n + u ϕn+ exp( θ q n+ = F d (u n+, q n )= n+ ) (d ) n, (6) exp( θ n+ ) (d 2 ) n exp( θ n+ ) (d 3 ) n where Rodrigues formula is used o obain a closed form expression of he exponenial map, see e.g. [3]. 5 Rigid mulibody sysem dynamics The consrained descripion of rigid bodies in erms of direcors is excepionally well suied for he coupling of several bodies in a mulibody sysem. The body fixed direcors offer he possibiliy o specify he coupling of neighboring bodies by joins consraining heir relaive moion in a sraighforward way. These couplings are ermed exernal consrains. As menioned in Remark 3.4, using a variaional inegraor based on he augmened Lagrangian (23) simplifies he eliminaion of he consrain forces from he sysem (compared o he use of cerain energy-momenum mehods), since a discree null space marix can direcly be inferred from a coninuous one by evaluaion a q n. Therefore, his secion summarizes in a concise way he necessary ingrediens for he variaional inegraion of he dynamics of kinemaic pairs. The idea of his procedure has been presened already in he framework of energy-momenum conserving ime inegraion in [4,5]. Simple kinemaic chains as well as ree-srucured mulibody sysems ha can be composed by lower kinemaic pairs are considered in he sequel. Le a mulibody sysem consis of N +rigid bodies numbered by α =,...,N and N axes n,...n N,wheren α is specified in he α-h body frame by n α = n α I d α I. (62) The N joins connecing he bodies are numbered by α =,...,N and he locaion of he α-h join in he (α )-s and α-h body is characerized by ϱ α,α = ϱ α,α I d α I, ϱ α,α = ϱ α,α I d α I (63) as depiced schemaically for wo neighboring links in Fig. 8. Noe ha for ree-srucured mulibody sysems, he wo bodies forming a kinemaic pair are no necessarily numbered consecuively. Assuming ha none of he links is fixed in space, he mulibody sysem can be described in erms of n = 2(N +) redundan coordinaes q () q() =.. (64) q N () generalizing (55). The corresponding consan mass marix is given by M M M = , (65) M N Fig. 8 α-h pair in a kinemaic chain. 28 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

15 ZAMM Z. Angew. Mah. Mech. 88, No. 9 (28) / 69 where each submarix M α R 2 2 coincides wih (58). The rigidiy of each link gives rise o six inernal consrains gin α (qα ) R 6 of he form (56) for α =,...,N. They can be combined o he m in =6(N +)-dimensional vecor of inernal consrains g in (q) and he m in n inernal consrain Jacobian marix G in (q). Similar o he case of a single rigid body reaed in Sec. 4, he wis of a N +free rigid bodies reads =., N where, analogous o (6), he wis of he α-h body α R 6,isgivenby [ ] α ϕ = α. ω α Now he redundan velociies q R 2(N+) of he mulibody sysem may be expressed as q = P in (q), wherehe 2(N +) 6(N +)marix P in (q) is given by Pin (q )... P P in (q) = in (q ) (68). Pin N (qn ) and Pin α (qα ) is he null space marix associaed wih he α-h free body, which wih regard o (59) reads I Pin(q α α d )= α d α. 2 d α 3 Noe ha by design G in (q) P in (q) =,he6(n +) 6(N +)zero marix. 5. Kinemaic pairs The coupling of wo neighboring links in Fig. 8 by a specific join J yields m (J) ex exernal consrains gex α ([qα, q α ] T ) R m(j) ex. In [4,5], lower kinemaic pairs J {R, P, C, S, E}, i.e. revolue, prismaic, cylindrical, spherical and planar pairs have been invesigaed. Depending on he number of exernal consrains m (J) ex hey give rise o, he degrees of freedom of he relaive moion of one body wih respec o he oher is decreased from 6 o r (J) =6 m (J) ex. Afer recalling he derivaion of he null space mehod and nodal reparamerizaion for kinemaic pairs briefly, deails are given for he spherical and he revolue pair only, since hese are used in he numerical examples presened in Secs. 5.5 and 6.2. Alogeher, m = m in + m ex consrains peraining o he mulibody sysem and he corresponding consrain Jacobians can be combined o [ ] [ ] g g(q) = in (q) R m G, G(q) = in (q) R m n. (7) g ex (q) G ex (q) The remainder of his secion presens deails of he exernal consrains caused by lower kinemaic pairs (composed of body and body 2) and heir reamen in he framework of he discree null space mehod. Wih he null space marices for kinemaic pairs a hand, a generalizaion o mulibody sysems being composed by pairs can be performed easily by respecing formula (6). Null space marix. In a kinemaic pair, he moion of he second body wih respec o an axis fixed in he firs body (or wih respec o a plane for he planar pair) can be accouned for by inroducing r (J) join velociies τ (J). Thus he moion of he kinemaic pair can be characerized by he independen generalized velociies ν (J) R 6+r(J) wih [ ] ν (J) =. (7) τ (J) (66) (67) (69) 28 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

16 692 S. Leyendecker e al.: Consrained variaional inegraors In paricular, inroducing he 6 (6 + r (J) ) marix P 2,(J) ex (q), he wis of he second body 2 R 6 can be expressed as 2,(J) = P 2,(J) ex (q) ν (J). (72) Accordingly, he wis of he kinemaic pair can be wrien in he form (J) = P (J) ex (q) ν(j) (73) wih he 2 (6 + r (J) ) marix P (J) ex (q), which may be pariioned according o P (J) ex (q) = I r (J). (74) P 2,(J) ex (q) Once P (J) ex (q) has been esablished, he oal null space marix peraining he kinemaic pair under consideraion can be calculaed from P (J) (q) =P in (q) P (J) ex (q) = P in (q ) P 2 2 r (J). (75) in (q2 ) P 2,(J) ex (q) Remark 5. (Naural orhogonal complemen) Similar o he procedure for he design of appropriae null space marices oulined above, he relaionship beween rigid body wiss and join velociies is used in [8] o deduce he naural orhogonal complemen in he conex of simple kinemaic chains comprised of elemenary kinemaic pairs. Nodal reparamerizaion. Corresponding o he independen generalized velociies ν (J) R 6+r(J) inroduced in (7), he redundan coordinaes q R 24 of each kinemaic pair J {R, P, C, S, E} may be expressed in erms of 6+r (J) independen generalized coordinaes. Concerning he reparamerizaion of unknowns in he discree null space mehod, relaionships of he form q n+ = F (J) d (μ (J) n+, q n) (76) are required, where μ (J) n+ =(u ϕ n+, θ n+, ϑ (J) n+ ) R6+r(J) (77) consiss of a minimal number of incremenal unknowns in [ n, n+ ] for a specific kinemaic pair. In (77), (u ϕ n+, θn+ ) R 3 R 3 are incremenal displacemens and roaions, respecively, associaed wih he firs body (see Sec. 4). Furhermore, ϑ (J) n+ denoe incremenal unknowns which characerize he configuraion of he second body relaive o he axis (or Rr(J) plane in case of he E pair) of relaive moion fixed in he firs body. In view of (64), he mapping in (76) may be pariioned according o q n+ = F d (u ϕ n+, θ n+, q n ), qn+ 2 = F 2,(J) d (μ (J) n+, q n). (78) Here, Fd (u ϕ n+, θn+, q 2,(J) n ) is given by (6). I hus remains o specify he mapping Fd pair under consideraion. (μ (J) n+, q n) for each kinemaic 5.2 Spherical pair The S pair, shown in Fig. 9, prevens all relaive ranslaion beween he wo bodies, and hus i gives rise o m (S) ex =3 exernal consrains of he form g (S) ex (q) =ϕ 2 ϕ + ϱ 2 ϱ =. (79) While he ranslaional moion of he pair can be accouned for by he velociy of one body s cener of mass, say by ϕ, boh bodies can roae independenly. Thus he roaional moion of body 2 is characerized by r (S) =3degrees of freedom. 28 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

17 ZAMM Z. Angew. Mah. Mech. 88, No. 9 (28) / Fig. 9 Spherical pair. Specifically, wih regard o (7) τ (S) = ω 2, he angular velociy of he second body. Accordingly, in he presen case, he vecor of independen generalized velociies reads [ ] ν (S) =. (8) ω 2 Recall ha he wis of he firs rigid body given in (67) consiss of is ranslaional velociy ϕ and is angular velociy ω. Taking he ime derivaive of he exernal consrains (79) and expressing he redundan velociies in erms of he independen generalized velociies (8) yields ϕ 2 = ϕ + ω ϱ ω 2 ϱ 2. (8) Now i can be easily deduced from he relaionship 2,(S) = P 2,(S) ex [ ] P 2,(S) ex (q) = I ϱ ϱ 2 I (q) ν (S),ha (82) and finally P 2 in (q2 ) P 2,(S) ex (q) = I ϱ ϱ 2 d 2 d 2 2. (83) d 2 3 To specify he reduced se of incremenal unknowns (77) for he S pair, (8) induces ϑ (S) n+ = θ2 n+ R 3, he incremenal roaion vecor peraining o he second body. Then he roaional updae of he body frame associaed wih he second body can be performed according o (d 2 I) n+ =exp( θ 2 n+ ) (d2 I) n. (84) Enforcing he exernal consrains (79) a he end of he ime-sep implies ϕ 2 n+ = ϕ n+ + ϱ n+ ϱ2 n+. (85) Evenually, he las wo equaions can be used o deermine he mapping ϕ n + u ϕ n+ +exp( θ n+ ) ϱ n exp( θ n+ 2 ) ϱ2 n qn+ 2 = F 2,(S) d (μ (S) n+, q n)= exp( θ n+ 2 ) (d2 ) n exp( θ n+ 2 ) (d2 2) n. (86) exp( θ 2 n+ ) (d2 3) n 28 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim