Available online at ScienceDirect. IFAC PapersOnLine 51-2 (2018)

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1 Availabl onlin at ScincDirct IFAC PaprsOnLin 51-2 (218) Rducd Modls for th Static Simulation of an Elastic Continuum Mchanism Bastian Dutschmann Simon R. Eugstr Christian Ott 218, IFAC (Intrnational Fdration of Automatic Control) Hosting by Elsvir Ltd. All rights rsrvd. Kywords: Robotics, Modl-Rduction 1. INTRODUCTION Institut of Robotics and Mchatronics, DLR (-mail: {bastian.dutschmann, Institut for Nonlinar Mchanics (-mail: Abstract: In this papr, two approachs ar stablishd and compard that simulat th static dformation of a tndon-drivn, lastic continuum mchanism (ECM). Th mchanism at hand is mad out of silicon and dforms in a larg workspac as a rsults of any xtrnally applid wrnch. This yilds high dxtrity and high mchanical robustnss for th systm, but also th commonly usd kinmatic modl is not suitabl any mor. Th discussd modls in this papr ar ssntially diffrnt. At first, th finit lmnt mthod (FEM) is usd to discrtiz th mchanism along its cntrlin. A nonlinar matrial law is stup and idntifid for th axial dirction and it could b shown that th stablishd modl matchs th ral systm vry closly. Th scond modl is mor abstract. Hr, a polynomial rlationship is stup btwn th Cartsian pos of th mchanism and th associatd wrnch ncssary to achiv this dflction. A comparison btwn th two modls show, that th FEM modl is slowr but mor accurat and thrfor usful for offlin computations whras th polynomial modl sms mor suitabl for ral-tim control approachs, with an accptabl accuracy and an fficint computation. Th systm at hand consists of an lastic continuum mchanism (ECM) mad out of silicon and is a planar vrsion of th mchanism dscribd in [8]. Whil on sid of th mchanism is fixd to th ground, on top of th othr sid a robot had is mountd. Th planar mchanism is actuatd by two tndons in an antagonistic fashion. Th tndons ar connctd to th had-plat at ach sid. By pulling at both tndons, a combind loading is xrtd onto th ECM to mov th had into th dsird dirction. Th rquird kinmatical and statical quantitis of th systm modl ar dpictd in Fig. 1. Th common approach to rduc th modl complxity is to assum that th cntr lin of th mchanism dforms lik a circular arc, s.g. [1]. With that, th gomtry of th dformation is havily simplifid and th static quations can b solvd analytically. Basd on that assumption, [5] stablish xprimntally a bnding stiffnss, whras [7] utilizs th manufacturing data for thir stl continuum. If such a kinmatic mapping cannot b usd du to changing loading conditions onto th systm, so calld gomtrically xact modls,.g. [9], which ar usually computationally xpnsiv ar applid. Anothr approach is to hid th gomtry of th dformation and th matrial proprtis within a mapping that nds to b idntifid. [6] tachs a nural ntwork to rlat actuation forcs to static Cartsian positions in th rachabl workspac of th manipulator. Th modl provd to b computationally fficint and it was abl to accuratly prdict th position of th manipulator. In this papr, w introduc a modl-basd on a nonlinar bam formulation which can b solvd using th Finit Elmnt Mthod (FEM). An altrnativ approach is proposd in [2] which aims at a computational fficint modl. Th assumption is that th Cartsian pos at th nd of th ECM is nough to dscrib th dformation bhavior which rducs th modl of th ECM to a nonlinar Cartsian spring. In [3], an xprimntal procdur is xplaind with which such a nonlinar spring can b idntifid xprimntally as a mapping from th Cartsian pos to th Cartsian wrnch using multivariat polynomials. Th prsnt work aims to compar th two aformntiond modls for a tndon-drivn ECM. In th first part, th FEM modl togthr with th xprimntal idntification of th stiffnss paramtrs, as.g. th bnding stiffnss, ar discussd. In th scond part, th idntification procdur for th polynomials is summarizd. For brvity, w will focus hr only on th planar cas, s Fig. 1, although, th polynomial mapping is asily xtndabl to th spatial cas [3]. In contrast, a spatial FEM modl can b drivd by th discussd formalism of this work. Howvr, th discrtization and intrpolation of th rotation paramtrization for th spatial modl is not trivial and nds spcial tratmnt. Th main contribution of this work is twofold: To th bst of th authors knowldg, this is th first work that stablishs a nonlinar FEM modl of a tndondrivn ECM with nonlinar input coupling and nonlinar matrial law in th comprssion dirction. Scond, th polynomial modl is compard with th FEM modl using xprimntal data rgarding accuracy and computation tim. 2. FINITE ELEMENT MODEL Th dformation of th systm is mbddd in th Euclidan thr-spac E 3 with origin O and coordinat fram I i E 3,i = {x, y, z} and taks plac xclusivly in th I x- I z-plan. Th ECM is modld as a planar nonlinar Timoshnko bam. According to th Timoshnko bam assumptions, th dformation of th thr-dimnsional continuum can b dscribd only by th dformation of a cntrlin and th rotations of plan rigid cross-sctions attachd to vry point of th cntrlin. Th cntrlin r = r(ν) E 3 is a curv paramtrizd by ν = [,L] IR, , IFAC (Intrnational Fdration of Automatic Control) Hosting by Elsvir Ltd. All rights rsrvd. Pr rviw undr rsponsibility of Intrnational Fdration of Automatic Control /j.ifacol

2 44 Bastian Dutschmann t al. / IFAC PaprsOnLin 51-2 (218) which appars in th xprssion ( I C x ) =(A ICC C x ) = A ICA T ICI C x = I ˆkICI C x (5) and holds analogously for th I C y - and I C z -dirction. Not that ( ) dnots th drivativ with rspct to th argumnt. Insrting (1) into (4), straightforward computation lads to th matrial curvatur vctor Ik IC (ν) =( I ˆkIC (ν))ˇ= ( θ (ν) ) T. (6) Fig. 1. Schmatic drawing of th systm. whr ν is th arclngth of th undformd bam with lngth L. Th cross-sctions of th bam ar rprsntd by th cross-sction-fixd frams C i = C i (ν) E3,i = {x, y, z} continuously varying along th cntrlin. Th bam is fixd to th ground such that r() = and C i () = I i,i = {x, y, z}. On top of th bam at ν = L a rigid and masslss plat with a width of 2d is attachd. In P and R two masslss tndons ar connctd to th plat. Both tndons ar rdirctd by a pully and subjctd at thir nds to th tnsil forcs λ l and λ r, rspctivly. Th had is modld as a rigid body with cntr of mass (CoM) S, mass m H and is rigidly connctd to th bam in Q such that had-fixd fram H i := C i (L),i= {x,y,z}. Th Cartsian coordinat rprsntation of a vctor a E 3 in an arbitrary orthonormal B-systm rotatd against th I-systm is dnotd as B a = (a B x a B y a B z ) T IR 3 with a = a B x B x + a B y B y + a B z B z E 3. Th orthogonal transformation matrix A IB R 3 3 rlats th rspctiv coordinats in accordanc with I a = A IBB a and corrsponds with th coordinats of th B-fram basis vctors in th I-systm, i.. A IB =( I B x I B y I B z ) IR 3 3. Th invrs of th transformation matrix A IB is dnotd as A BI = A 1 IB = AT IB. 2.1 Kinmatics Th cntrlin r(ν) and th basis vctors of th crosssction-fixd frams C i (ν) ar rprsntd in th I-systm as ( ) x(ν) Ir(ν) =, A IC (ν) = z(ν) ( ) cos θ(ν) sin θ(ν) 1, sin θ(ν) cos θ(ν) (1) and dtrmind by th ral-valud gnralizd position functions x = x(ν), z = z(ν) and θ = θ(ν). Hnc, th position vctor r OQ and th basis vctors of th H-systm ar givn in th I-systm by Ir OQ = I r(l), A IH = A IC (L). (2) With (2) and th dimnsions from Fig. 1, th position of th CoM of th had is Ir OS = I r OQ + h I H x + b I H z. (3) Th chang of th cross-sction orintation along ν is dscribd by th matrial curvatur, th skw-symmtric matrix I ˆk IC (ν) =A IC(ν)A T IC(ν) IR 3 3 (4) Virtual Displacmnts and Rotations Lt x = x(ν, ε), z = z(ν, ε) and θ = θ(ν, ε) b variational familis of th gnralizd position functions, i.. diffrntiabl paramtrizations with rspct to a paramtr ε IR such that th actual positions (w ar looking for) ar mbddd in th family and ar obtaind for ε = ε. Insrting ths functions in (1), th variational familis r = r(ν, ε) and Ā IC = ĀIC(ν, ε) ar inducd. Th virtual displacmnt of th cntrlin δr and th virtual rotations of th crosssctions δ ˆφ IC ar thn dfind by Iδr = I ˆr, I δ ε ˆφ IC = ĀIC ε A T IC. (7) ε=ε ε=ε Explicit computation using (1) and (2) givs Iδr(ν) =(δx(ν) δz(ν) ) T, I δr Q =(δx L δz L ) T, (8) whr δx L = δx(l), δz L = δz(l). Analogously to th matrial curvatur vctor (6), computation of th scond quality in (7) lads to th virtual rotation vctor Iδφ IC (ν) =( I δ ˆφ IC (ν))ˇ= ( δθ(ν) ) T. (9) To stablish th virtual work of th xtrnal forcs acting onto th systm, w will nd th virtual displacmnt of th CoM and th virtual displacmnts of th tndon connction points which ar for δθ L = δθ(l) Iδr S = I δr Q + δθ LI H y (h I H x + b I H z ), (1) Iδr P = I δr Q + δθ LI H y d I H z, (11) Iδr R = I δr Q δθ LI H y d I H z. (12) Strain Masurs Th strain masurs ar dfind as th cntrlin s tangnt in th cross-sction-fixd C-systm Cγ(ν) = ( γ C x (ν) γ C z (ν) ) T = A T IC (ν) I r (ν) (13) togthr with th matrial curvatur vctor in th C- systm Ck IC (ν) =A T IC(ν) I k IC (ν) =( θ (ν) ) T. (14) 2.2 Static Equilibrium Equations of th Systm To dtrmin th quilibrium quations of th systm, w us th principl of virtual work which postulats that th total virtual work δw tot of th systm is zro for all admissibl virtual displacmnts. For static quilibrium, th total virtual work is composd of intrnal and xtrnal virtual work contributions δw int and δw xt, rspctivly. For th systm at hand this rsults in th rquirmnt δw tot = δw int + δw xt = δr adm,δφ IC,adm (15) whras th admissibl virtual displacmnts and rotations ar givn by th ons of (7), (8) and (9) which additionally rspct th clamping condition at ν =, i.. I δr adm ()= and I δφ IC,adm () =. In th following, w introduc all virtual work contributions of th systm.

3 Bastian Dutschmann t al. / IFAC PaprsOnLin 51-2 (218) Virtual work of intrnal forcs Th only intrnal forcs of th systm com from th nonlinar Timoshnko bam whos intrnal virtual work contribution is according to [4] givn by δw int { = In T ( I δr I δφ IC I r ) (16) + I m T Iδφ } IC dν. Thrin th rsultant contact forcs n = n(ν) E 3 and rsultant contact coupls m = m(ν) E 3 can b idntifid. Sinc for th planar systm th contact forc in I y-dirction dos not contribut to th virtual work, only th contact forcs in I x- and I z-dirction ar of intrst. Ths can b xprssd as ( n I ) ( ) x (ν) N(ν) In(ν) = = A ICC n(ν) =A IC, (17) n I z(ν) Q(ν) whr N = N(ν) IR and Q = Q(ν) IR ar th normal and shar forcs acting at ach cross-sction. In th contact coupl I m(ν) =( M(ν) ) T only th I y-dirction with th bnding momnt M = M(ν) IR is rlvant. Using th planar kinmatics (8) and (9) togthr with th just introducd contact forcs and coupls, th intrnal virtual work xprssion (16) taks th form δw int { = δx n I x + δz n I z δθ(z n I x x n I z) + δθ M } (18) dν. Th constitutiv laws dscribing th matrial bhavior of th bam ar formulatd btwn forc and strain componnts of th cross-sction-fixd C-systm, i.. N = EA ( γx C 1 ) 3 (γx C ) 2,Q= GAγz C,M= EIθ, (19) with th axial stiffnss EA IR, th shar stiffnss GA IR and th bnding stiffnss EI IR. Not th nonlinar No-Hookan matrial law for th normal forc which taks into account th xprimntally obsrvd stiffning bhavior of th matrial in comprssion. Linarization around th undformd configuration, i.. γx C = 1, lads dirctly to Hook s law N = EA(γx C 1). Virtual work of xtrnal forcs Th virtual work contributions of th xtrnal forcs of th systm is additivly composd of th virtual work du to gravity and th virtual work of th tndon actuation, δw xt = δw xt,g + δw xt,t. (2) For a cross-sction dnsity ρa, th virtual work du to gravity with gravity constant g δw xt,g = (ρag ) I δrdν (m H g ) I δr S, (21) can b furthr simplifid using (1) to δw xt,g = (ρag ) I δrdν m H gδx L m H gδθ L ( I I x) T ( I H y (h I H x + b I H z )). (22) Th virtual work of th tndon actuation δw xt,t = λ li T l Iδr P + λ RI T r Iδr R (23) is rwrittn using (11) and (12) to ) δw xt,t =(δx δz) (I l I r ( λ (24) + δθ( I H y ) T d I H z I l d I H z I r )λ with th tndon forc vctor λ =(λ l λ r ) T and th unit tndon dirction vctors for lft I H l IR 3 and right I H r IR 3 tndon as dpictd in Fig. 1, i.. I H l = ( ) I l,x I T l,z = I r p,l I r P / I r p,l I r P, I H r = ( ) I r,x I T r,z = I r p,r I r R / I r p,r I r R. (25) Not that th position vctors r p,l and r p,r, which dnot th points whr th tndons run onto th corrsponding pully, dpnd on r Q and θ L. Within th idntification procss in Sct. 2.4, instad of th tndon forcs and th gravity forcs, an xtrnal forc IF H id IR 3 is xrtd at point Q whos virtual work is δw xt,id = I F H idδr Q. (26) Th principl of virtual work (15) with th contributions (18), (22) and (24) corrsponds to a wak variational xprssion of a nonlinar ordinary diffrntial quation which can b obtaind by intgration by parts of (18). Sinc this ODE cannot b solvd analytically, w introduc in th subsqunt sction th finit lmnt mthod discrtizing (15) in th sns of Bubnov-Galrkin. In doing so, w rduc an infinit dimnsional systm to a finit dimnsional on. 2.3 Finit Elmnt Mthod Aiming for linar Lagrangian shap functions, th paramtr spac of ν, i.. [,L], is dividd by th nods n 1 =<...<n < <n kl+1 = L into k l lmnt sts Ω =[n,n +1 ]. Using th rlation ν 2 (ν) = n +1 n (ν n ) 1, (27) it is convnint to introduc in vry lmnt st Ω th lmnt coordinat ν [ 1, 1]. Thn th gnralizd position function x is approximatd by th linar intrpolation of th nodal coordinats x = x(n ). Accordingly th position function taks th form k l x(ν) = χ Ω (ν)n(ν (ν)) T ( x x +1 ), (28) whr χ Ω is th charactristic function bing on for ν Ω and zro lswhr and N(ν )=.5 (1 ν 1+ν ) T (29) corrsponds to th linar shap function which linarly intrpolats two subsqunt nodal coordinats. Th gnralizd position functions z and θ ar approximatd analogously such that th continuous formulation can b rprsntd by th finit st of coordinats q = ( x 1 z 1 θ 1 x kl+1 z kl+1 θ k l+1 ) T. (3) Th lmnt connctivity matrix C x IR 3(kl+1) 2 bing dfind by q x = ( x x ) +1 T =(C x ) T q (31) xtracts from th gnralizd coordinats q th rlvant coordinats for th linar intrpolation within on lmnt. Bsids th analogous dfinition for z and θ, w introduc also th connctivity matrix C L IR 3(kl+1) 3 givn by q kl+1 = ( x kl+1 z kl+1 θ ) k T l+1 =(C L ) T q. (32) Applying th chain rul togthr with (27), th drivativ of (28) with rspct to ν is obtaind as k l x (ν) = χ Ω (ν)n (ν (ν)) T 2 L q x (33)

4 46 Bastian Dutschmann t al. / IFAC PaprsOnLin 51-2 (218) with N =(.5.5 ) T and L = n +1 n. Morovr, th approximation (28) inducs togthr with (31) also k l δx(ν) = χ Ω (ν)n(ν (ν)) T (C x) T δq, k l δx (ν) = χ Ω (ν)n (ν (ν)) T 2 L (C x) T δq. (34) Providing th sam computations for z and θ, th virtual work of th intrnal forcs (18) is approximatd by insrting (33) and (34). Togthr with a chang of coordinats in th rspctiv intgral xprssion, w thn obtain k l δw int = δq T whr f int, x f int, θ = δq T f int, 1 = = k l f int = [ C xf int, x + C zf int, z N n I xdν, f int, z N M + N ( x n I z z n I x [ C xf int, x + C zf int, z + C θf int, θ ] 1 = N n I zdν, 1 ) L 2 dν, + C θf int, θ ]. (35) (36) Not that th approximation of th position functions ar also usd within th valuation of n I x, n I z and M which is why f int = f int (q) is a nonlinar vctor valud function. All intgral xprssions ar valuatd numrically using Gauss quadratur. Th discrtization of th xtrnal virtual work du to gravity (22) lads to δw xt,g = δq T f xt,g, (37) whr k l 1 f xt,g = C xρag N L 1 2 dν ( ) (38) 1 m H g C L ( I I x) T ( I H y (h H x + b I H z ) ). Using (32), th discrtization of th virtual work of th tndon forcs (24) is δw xt,t =(δq kl+1 ) T Pλ= δq T C L Pλ (39) with th tndon coupling matrix I l,x I r,x P = I l,z I r,z. (4) d( I H y ) T I H z I l d( I H y ) T I H z I r Using (35), (37) and (39) in th principl of virtual work (15), th infinit dimnsional variational xprssion is rducd to ( th finit dimnsional xprssion ) δq T f int + f xt,g + C L Pλ = δq adm (41) which inducs a nonlinar vctor valud quation which can b solvd numrically. 2.4 Idntification of th stiffnss paramtrs In this sction, th xprimntal idntification procss is discussd in which th stiffnss paramtrs EA, GA and Fig. 2. Planar tstbd usd for th idntification procss. EI for th constitutiv laws (19) ar idntifid. Two diffrnt xprimnts ar carrid out to xcit indpndntly th axial stiffnss EA as wll as th shar and bnding stiffnss GA and EI, rspctivly. Th lastic paramtrs ar incorporatd in th FEM modl bing dscribd by a nonlinar function. Thus, a nonlinar last squar optimization is applid for th idntification procss using th lsqnonlin routin from MATLAB. Within this nonlinar optimization, th rror function (ξ) IR m, is minimizd to find th dsird paramtrs ξ IR p, min (ξ) 2 2, (42) ξ whr m, p IR ar th numbr of masurmnts and th numbr of idntifid paramtrs, rspctivly. Th idntification procss is prformd for k l = 2 numbr of lmnts. Comprssion tst for idntification of EA In th comprssion tst, a cylindrical spcimn mad of silicon is comprssd up to 2% of th undformd lngth L. Th tst is conductd for two diffrnt lngths L = [28mm, 4mm]. Th masurd quantitis ar th axial forc f x [N] and th displacmnt L [m] which ar illustratd in Fig. 3 as a strss-strain diagram with th axial strss σ x = fx A and th axial strain γx C = L L. For ξ = EA, th rror function usd for th idntification is (EA)=( x 1 x 1 (EA)... x m x m (EA) ) T, (43) whr x i is th masurd and x i (EA) is th computd axial position for an xtrnal forc IF H id =( f x ) T (44) with th virtual work contribution (26). In th lft diagram of Fig. 3, a clar nonlinarity in th masurd strssstrain curv (rd curv) can b obsrvd, which can b rproducd much bttr by th No-Hookan matrial law (19) than by th linar Hookan law. Tab. 1 shows that th linar Hookan law ovrstimats th stiffnss of th spcimns. Tabl 1. Lft: Idntifid EA for Hookan and No-Hookan matrial laws. Right: Idntifid GA and EI. comprssion shar L [mm] EA nh [N] EA H [N] f t,max GA [N] EI [Nm 2 ] N N Shar tst for idntification of GA and EI For th shar tst, a forc in I z-dirction is applid at th tip of th ECM as dpictd in Fig. 2 which lads to a bnding dformation about th I y-axis and a shar dformation along th I z-axis. For all displacmnts incrmnts, th

5 Procdings of th 9th MATHMOD Vinna, Austria, Fbruary 21-23, 218 Bastian Dutschmann t al. / IFAC PaprsOnLin 51-2 (218) [1 5 N/m 2 ] -1-2 L = 4mm masurmnt modl-no Hook modl-hook rror [mm,dg] F [N] Fig. 3. Lft: Idntification of EA. Right: Error of th position x, z and orintation θ in th shar tst btwn masurd and modld data. x [m] x [m] z [m] z [m] Fig. 4. Workspac P with clos-up viw (right) of th planar ECM for 1 randomly applid tndon forcs (λ l,λ r ) in th rang of [N-16N]. position and orintation ar masurd by a camra as wll as th applid tndon forc. Th idntifid valu EA = 75[N] from th comprssion tst is usd within th shar tst as an initial guss for th bnding stiffnss EI = 75/A I = 8718 I [Nm 2 ] with scond momnt of ara I. Hr, ξ =(GA EI ) T IR 2 and th rror function for th idntification is (ξ) = ( z1 z 1 (ξ) θ 1 θ 1 (ξ)... z m z m (ξ) θ m θ m (ξ) ) T, whr z i, θ i ar th masurd and z i,θ i ar th computd z-position and tip angl for th xtrnal forc IF H id =(f t ) T. (45) Th rsults of th idntification procdur ar dpictd in Fig. 3 and in Tab POLYNOMIAL MODEL As mntiond in th Introduction, an abstract modling approach basd on polynomials is suggstd by [2] for a modl-basd control approach for an ECM. Th modl considrs that th mapping from th Cartsian pos (x L,z L,θ L ) IR 3 in th workspac P to an associatd Cartsian wrnch applid at th had I h H IR 3 which is ncssary to dflct th ECM into this configuration, can b rprsntd by a polynomial function. In [3], a mthod for th idntification of such a mapping is proposd and th rsults show that polynomials of ordr 3 provid th bst match for simulation and xprimntal data. For ach wrnch componnt (h 1 h 2 h 3 ) T =(f x f z τ y ) T, a multivariat polynomial mapping of th form h j = x T β j, j =1, 2, 3, (46) is idntifid, whr β j IR 2 rprsnts th polynomial cofficints for th j-th wrnch componnt and x IR 2 is x z θ Occurrnc 3 f 2 x f z 1 τ y (ĥj h j )/h j Fig. 5. Histogram of rl computd with 3 tsting points. rr [mm / dg] z x θ θ L [dg] rror [N, Nm] fx fz τy θ L [dg] Fig. 6. Lft: Error of th computd had position by th FEM modl and th masurd had position for a positiv bnding motion. Right: Error of th prdictd wrnch from th polynomial map and masurd had wrnch for all wrnch componnts. x =(1 x L z L θ L (x L ) 2 x L z L (θ L ) 2 (x L ) 3 (x L ) 2 z L x L z L θ L (θ L ) 3 ) T. (47) To find th cofficints via a polynomial rgrssion according to [1], th training st C with 1 poss is calculatd using th calibratd FEM modl from th formr sction. Th input of th FEM simulation is a random wrnch from th st {h IR 3 h min < h < h max }, (48) with h max = [2, 8, 11] and h min =[ 33, 8, 11]. Th maximum and minimum valus ar chosn to nsur that th training st C covrs wll th rachabl workspac P of th ECM, s Fig. 4. To valuat th accuracy of th polynomial modl, 3 data pairs from C ar randomly chosn to srv as th tst st. Th rlativ rror distribution of ach h j of a 3- dgr polynomial rgrssion modl is dpictd in Fig. 5, in which ĥj and h j dnot th stimatd and th obsrvd valu of th polynomial modl. Th vrtical axis rprsnts th amount of prdiction points rfrrd to a prdiction rror lvl. Most of th prdictd wrnch componnts hav a rlativ rror lss than.2. Th points with rlativ rror byond th intrval [.2,.2] ar not displayd as locatd in a small rang around zro (from 2[N] to 2[N] for th forc and from.2[nm] to.2[nm] for th torqu). 4. COMPARISON This sction will compar th two stablishd modls rgarding thr prdiction accuracy and computation tim. For th valuation, w us xprimntal data from th planar tstbd whr a combind loading is applid, i.. th mchanism is movd with two antagonistically arrangd tndons, cf. Fig. 2. As th stablishd modls ar static ons, th data is masurd at poss in static quilibrium. 4.1 Accuracy Th accuracy of th FEM modl is mainly dpndnt on th idntifid matrial paramtrs providd that th assumptions about th gomtry of th dformation and th 5

6 48 Bastian Dutschmann t al. / IFAC PaprsOnLin 51-2 (218) Fig. 7. Histograms of th computational tim usd by th FEM modl (lft) and th polynomial modl (right) computd with 1 randomly slctd poss from C. applid matrial laws match wll. In Fig. 6, a comparison is shown btwn static poss of simulatd and masurd data. Th input to th FEM modl ar th masurd tndon forcs λ for which (x L,z L,θ L ) is computd and compard with th masurd had pos by th camra systm. Th lft plot of Fig. 6 stats, that th rrors stay blow 5% with rspct to th masurd workspac (z: 2.6%, x: 4.7%, θ: 1.8%) yilding an accurat modl. To valuat th accuracy of th polynomial modl, th sam masurd data is usd, howvr, th input is th masurd had-pos to comput th associatd had wrnch. Th masurd had wrnch is computd basd on th masurd tndon forcs and th tndon coupling matrix P introducd in (4). Th rsults ar dpictd in th right of Fig. 6. Compard with th accuracy of th FEM modl, th polynomial modl is lss accurat (x: 9.4% z: 11.6% θ: 9.7%). 4.2 Computational Efficincy As statd in th motivation, a primary goal of th idntifid mapping is to b computationally fficint for th us in ral tim control. To invstigat th computational fficincy, 1 pos-wrnch pairs ar randomly slctd from P and ar usd as input of th polynomial modl and th FEM modl and th computational tim of ach modl is masurd. Fig. 7 dpicts th histogram of th computation tim, which stats that th polynomial modl is at last 2 tims fastr than th FEM modl. 4.3 Discussion Both modls prov to rprsnt th dformation charactristic of th tndon-drivn ECM. Th FEM modl is mor accurat but nds mor tim for on simulation run. By rducing th numbr of lmnts, this computation tim can b dcrasd and it is an opn rsarch qustion in this fild to which xtnd th numbr of lmnts can b rducd without losing accuracy. Also, mor sophisticatd shap functions can b applid which incorporat th charactristic of a dformd cntr lin. By that, it could b possibl to rduc th numbr of lmnts still achiving high accuracy. Th polynomial modl is computationally fast but lacks accuracy. To incras th accuracy, th polynomial modl can b xprimntally traind to match th ral systm as proposd in [3]. Howvr, maximum modl rrors of 11% ar still in a vry good rang. From a modling prspctiv, it should b invstigatd th validity of th assumption that th dformation of th ECM can b dscribd by (x L,z L,θ L ) only. 5. CONCLUSION In this work, two modls ar invstigatd to simulat th static dformation of a tndon-drivn lastic continuum mchanism. At first, a nonlinar FEM modl is stup and th paramtrs ar xprimntally idntifid. Th incorporatd gomtry of th dformation matchs wll th ral systm. Furthrmor, it could b obsrvd during th xprimnts that a No-Hookan matrial law for th axial comprssion nds to b implmntd to gnrat an accurat modl. To th authors knowldg, this is th first tim that a FEM modl with nonlinar matrial law and nonlinar tndon-coupling is applid for simulation and idntification of a tndon-drivn ECM. In th futur, th numbr of lmnts and altrnativ shap functions shall b idntifid to incras th computational fficincy. Th scond modl is basd on multivariat polynomials which maps from th Cartsian had position of th ECM to th associatd wrnch ncssary for this dformation. Th modl provd to match th xprimntal platform wll, howvr with lss accuracy compard to th FEM modl. In th futur, th idntification procss in which th polynomial cofficints ar found nds to b updatd to incorporat that physical proprtis ar displayd by th polynomial modl,.g. th symmtry of th associatd stiffnss matrix. This proprty could b incorporatd in th idntification procss which would bnfit from a rduction of th to b idntifid polynomial cofficints. REFERENCES [1] D. Camarillo, C. Carlson, and J. Salisbury. 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