Stefano Portigliotti a, Massimo Dumontela a, Gianluigi Baldesi b, Donato Sciacovelli b a Guidance Navigation and Control Section,

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1 DCAP (Dynamcs and Control Analyss Pacage) an Effectve Tool for Modellng and Smulatng of Coupled Controlled Rgd Flexble Structure n Space Envronment Stefano Portglott a, Massmo Dumontela a, Ganlug Baldes b, Donato Scacovell b a Gudance Navgaton and Control Secton, System Avoncs and Operatons Dvson, Alena Spazo SpA,Torno, Italy b Structures Secton, Thermal and Structures Dvson, ESA/ESTEC, Noordw, The Netherlands Abstract The paper presents the software DCAP (Dynamcs and Control Analyss Pacage): a sute of fast, effectve computer programs that provdes the user wth capablty to model, smulate and analyze the dynamcs and control performances of coupled rgd and flexble (NASTRAN nterfaced) structural systems subected to possbly tme varyng structural characterstcs and space envronment loads. It uses the formulaton for the dynamcs of mult-rgd/flexble-body systems based on Order(n). Ths avods the explct computaton of a system mass matrx and ts nverson, and t results n a mnmum-dmenson formulaton exhbtng close to Order(n) behavor, n beng the number of system degrees of freedom. A dedcated symbolc manpulaton pre-processor s further used n the codng optmzaton. The modelng capablty s completed wth the possblty of user-defned software, allowng for modelng of specfc control feature not drectly ncluded n the dynamc pacage's lbrary. For the latter control modelng, a new nterface s beng a developed allowng to descrbe the user control drectly by Matlab/Smuln blocs. An applcaton example of 3D traectory flght of a frst stage launch vehcle wth flexble tme varyng structural propertes s presented. Bref overvew of Dynamcs and Control Analyss Pacage DCAP s a sute of fast, effectve computer programs that provdes the user wth a powerful tool for desgnng and verfyng the dynamcs and control performance of coupled rgd and flexble structural systems. The software modules that t contans can be grouped nto four general categores: Pre-Processng, Processng, Post- Processng and Utlty ncludng the MATDCAP I/F to Matlab/Smuln, [1]. Communcaton between the modules s acheved va the dedcated fle structure. The pacage provdes the user wth an outstandng capablty to model, smulate and analyze a complex mult-body system. The latter can be connected n open- and closed-loop topologes, where relatve moton s defned through 'hnges'. Each

2 hnge allows from zero to sx relatve degrees of freedom, as t can be free, loced or constraned to pre-defned moton. Pre-Processng Processng Post-Processng Buld_Model Interactve pre-processor for model defnton Buld_Model -x Pre-processor wth Graphcs User Interface Buld_Model -g 3D graphc modeller Pre_Flex Flexble data pre-processor ASKA-DCAP-NASTRAN Interface Intalse_Orbt Orbtal Data Intalsaton Tme-doman programs Frequency-doman programs Utlty Trans_Matrx Matrx Quadruples Handler MATLAB-DCAP-MATRIXx Interface Plot_Results Smulaton runs graphc post-processor Tme Plots Frequency Plots (Bo de, Nchols, Nyqu st) Root Loc Plots Harmonc (FFT) Plots Anmate 3D anmaton tool Fgure 1: DCAP software modules Transton modes between hnge states can be mplemented as well, allowng locng and releasng commanded through montorng of sutable dynamc states durng system dynamc evoluton. Leaf Bodes Leaf Bodes Leaf Bodes Cut Jont Base Body Branch Body Base Body Branch Body Base Body Base Body (a) Body 1 Hnge 1 Inertal Reference Body 1 Hnge 1 Fgure 2: Open-loop (a) and closed-loop(b) tree topologes (b) Inertal Reference From a general vewpont, the DCAP model s the combnaton of four components: 'structure' or 'plant': the mult- body topology tself 'sensors': the feature that allows the extracton of system moton nformaton for performance montorng 'actuators': the feature through whch forces and torques are appled to the structure 'controller': the part of the system drvng the mult-body structure n order to acheve the allocated obectve. DCAP capabltes can be summarzed to nclude: Rgd and Flexble Body Chans, Open- and Closed- Loop Topology, Sx Degree-of-Freedom Hnges, Large-Angle Rotatons, Lnear and Non-lnear Tme Doman Smulatons, Order (N) Algorthm, Symbolc Code Generaton, Orbtal Envronment, Bult-n Sensors and Actuators,

3 Non-lnear Devces (ncludng Coulomb Dampers), Numercal Lnearzaton, Contnuous/Dscrete/User Controllers (Transfer Functon, State Space, Bloc Dagram), Fle Interface to MATLAB, Fle Interface to NASTRAN, Frequency Response, Interactve Pre- and Post-processng and MATDCAP code I/F to Matlab/Smuln. The core of the pacage conssts of the computatonal modules, partcularly orented towards the smulaton of the non-lnear dynamcs of mult-body systems. In addton, programs for numercal lnearzaton, matrx representaton manpulaton (state space) and system combnaton allow lnear problems n both the tme and frequency domans to be handled. Interfaces to popular fnte-element software pacages such as NASTRAN (MSC and COSMIC) and ASKA, are avalable and gve a drect capablty to defne complex flexble structures. Smlar nterfaces are provded to control desgn pacages le MATLAB/Smuln. The modelng capablty s completed wth the possblty of user-defned software ntegraton, allowng for the use of specfc features (sensor and actuator dynamcs, user controller, etc.) not drectly ncluded n the pacage's lbrary. The user frendlness of DCAP s sgnfcantly ncreased usng Matlab. In order to acheve ths challengng target a Matlab S-functon has been created. DCAP allows for code generaton both n FORTRAN and n C. These two optons are both compatble wth codng the S-functons n MATLAB Smuln. Consstently wth avalablty n tme of the DCAP upgrades, the FORTRAN I/F opton s currently mplemented n frst nstance for MATDCAP. Smlar I/F n usng C code s planned to be n a subsequent phase where relatve merts could be envsaged. Ths s-functon can help the user to desgn the system, n fact, used le a Smuln bloc, t can be lned drectly to any other blocs. Ths maes the modelng of the control part much easer and effcent. To let the user the possblty to decde whch ntegrators of DCAP or of MATLAB Smuln would have to tae control of the smulaton, there have been developed two dfferent nterfaces: I. The MATLAB Smuln ntegrator drves the smulaton. In ths case the S- functon s lmted to the calculaton of the tme dervatves, t nterfaces sensors and actuators n the contnuous and possbly n the sampled doman. However under the constrants of beng drven by the tmng mposed by the MATLAB Smuln scheme. Consequently specal features of the DCAP requrng adaptaton of the ntegraton step, such as locng, stcton, could II. not be run n ths scheme. The scheme s run wth constant ntegraton step. The latest one gves the user the possblty to have ndependent ntegrators for the MATLAB Smuln model and for the DCAP model Ths case allows for the two models to exchange nformaton at dscrete tme nstants only. Specal features of the DCAP can be used wthout problem. The analyss of results s supported by a specfc plottng pacage that can handle tme-doman plots, harmonc analyss (drect and nverse FFT), and Bode, Nchols,

4 Nyqust and Root locus charts. Performance can also be nspected va a powerful 3D anmaton tool, whch helps the user to vsualze the smulaton results. The DCAP development has been done on VAX/VMS computers, wth later extenson to UNIX (Sun, Slcon Graphcs) worstatons, recent adaptaton to Lnux machnes and on Wndows (MATDCAP) machnes. The software pacage s complemented by a full set of manuals (User, Theory, Demonstraton and Installaton). Dynamcs formulaton Introducton Early approaches to the dynamcs formulaton for multbody systems led to the equatons of moton, for open-loop tree topologes, of the form M q& = F (1) [ ] T q = q1 q1 L qn where, M s an n n mass matrx, s an n 1 column matrx representng the generalzed coordnates and F s the column matrx contanng the contrbutons from centrfugal, Corols, gravtatonal and external forces. For a numercal smulaton of such a system, the mass matrx must be nverted. Snce the nverson of an n n matrx nvolves operatons of the Order(n3), ths s called an Order(n3) approach. As the number of degrees of freedom ncreases, ths matrx nverson, for every ntegraton step, becomes computatonally expensve. Thus, researchers have sought methodologes to crcumvent the mass matrx nverson, to mprove computatonal effcency. The need for computatonal effcency frst arose n the area of robotcs, where effcent controller desgns ncorporatng the system dynamcs were sought. The research nto mprovements n formulatons that ncrease computatonal speed resulted n - what are today called - Order(n) algorthms. The reason for ths nomenclature s that the computatonal burden n these schemes ncreases only lnearly wth n. The equatons of moton are derved usng Kane's method of generalzed speeds [8,9] for the defnton of dynamcs wth respect to actve degrees of freedom (dofs). The method conssts essentally of two parts: a frontal part, n whch the nerta and actve forces assocated wth each body are shfted to ts nboard body and ths process repeated untl the core body s reached, and a bac substtuton part n whch all the acceleratons are obtaned n terms of the core body acceleratons and all the external forces actng on the system. A multbody system n a topologcal tree s shown n Fgure 2. Body 1 s an arbtrarly selected reference body assumed to be connected to an magnary nertally fxed body, numbered 0. An accountng system s developed that supples a unque procedure for wrtng the nematcs of any body n the system. At the heart of ths system s the defnton of body pars and L(). For Body, L() s the adacent body leadng nward to Body 0 (or to the core body, Body 1). Body L() s then defned as the body drectly nboard of Body. Note that, snce tree topologes are consdered, a body can have more than one outboard body.

5 Wth ths defnton of and L(), the remanng system topology varables used for the accountng procedure are defned as follows. Leaf bodes are bodes wth no bodes drectly outboard. Branches are defned as a set of bodes mang up a chan topology. Branch bodes are bodes n a branch, and thus contan outboard as well as nboard bodes. The system confguraton s then completed by defnng the system degrees of freedom (DOF). Body L() Reference L() r p Body L() (nboard of ) L() u p L() R f "p" node r L() L() y "q" node R f Body u q r q R Inertal Reference Fgure 3: Body -L() par Body Reference r 0 u 0 ρ dm "Nodal Body" 0 These are categorzed nto rgd body and elastc degrees of freedom. The former are defned through the defnton of onts that permt relatve moton, and the latter are defned usng the assumed modes method. A ont, shown n Fgure 3, s defned between a par of materal ponts, one on each of the adonng bodes and L(). The rgd-body DOF are then characterzed by the relatve translaton and rotaton of two sets of reference axes located at the two nodes, q on body and p on the body nboard of body -.e., on L(), mang up the th ont. These onts can have up to sx degrees of freedom. The followng symbols are used to descrbe the system confguraton: NS = NB ( NT + NR + NM ) = 1 NB = Number of bodes NT = Number of translatonal DOF at th hnge NR = Number of rotatonal DOF at th hnge NM = Number of deformatonal DOF of Body NS = Total number of degrees of freedom of the tree Formulaton of equatons of moton The equatons of moton are derved va Kane's method. The formulaton and the correspondng soluton algorthm are based on the nematc relatonshps between body pars and L(). Referrng to Fgure 3, we proceed as follows.

6 The vector locatng an elemental mass dm of Body, by a recursve form n the nertal frame, s gven by where L( ) f elemental mass dm n L( ) p L( ) p L( ) ( r + u ) + r u R = R + r + u + y + (2) r s a vector that defnes the undeformed confguraton of the b reference frame, and u q = q NM l= 1 l φ ( r l ) η ( t ) represents the elastc deformaton (vector) experenced by dm as the sum of the product of a set of assumed mode shape vectors φ ( r ) and ther tme-varyng ampltudes l l η ( t ). Usng a set of generalzed speeds W1, K WNS (tme dervatves of coordnates characterzng the confguraton, [9] whch represent the DOF of the system, the R & can always be wrtten as: Where, NS R & = V W + V (3) = 1 V are the coeffcents of the generalzed speeds, and V and functons of the NS generalzed coordnates and tme Newton's law for the tree structure can be wrtten as NB = 1 B V ( d f R& dm) where, df s the force on the dfferental element and of dm. t V t are & + 0, = 1, 2, K, NS (4) R & s the nertal acceleraton The system equatons of moton (Equaton (4)) results n an NS NS mass matrx. Thus, durng numercal smulaton the above algorthm requres the nverson of an NS NS mass matrx, mplyng O(n3) operatons over a reduced set of actve dofs. The steps towards achevement of an Order(n) behavor are descrbed n [4,5,6,7], wth DCAP mplementaton detals n [10] and summarzed n the followng secton. Order(n) soluton algorthm The soluton algorthm, presented n detals n [10], can be thought of as consstng of two basc parts - a frontal part and a bac substtuton part, whch are based on recursve formulaton of the body -L() nematcs. The dsplacement of body- reference can be expressed as

7 and dfferentated to L L L ( p p ) ( q q ) L( ) ( ) ( ) ( ) R = R + r + u + y r + u f f f o ( r 0 + u 0 ) + u 0 + ω0 ρ R& = R& + ω (6) where the open dot denotes dfferentaton wth respect to tme n Body- frame and ω = ω ω 0 = ω L( ) + u o + u o L( ) p 0 + L( ) where the prme dentfes the rotaton due to flexblty, as detaled n later sectons. Wth the generalzed veloctes for Body- defned as { q } ω T & = y& 1 K y& & θ 1 K 1 NT & θ & η K & η NR NM (8) for the th hnge NT translatonal and NR rotatonal (Euler angle) dofs and th body modal coordnates η, the expresson of (3) becomes R& q& R& = q& f ω + q& o q u u 0 ω0 ( r + u + ρ ) + + ρ 0 0 or R& f η& = V + ω ( r 0 + u ρ 0 + ) + V 0 q& (10) Wth the coeffcents of generalzed speeds defned as o q& q& f R& ω η& u u f 0 0 V = ω = V = + ρ y& θ& η& η& The generalzed nertal forces can then be expressed as f * T = b R&& dm f * R = b ( r + u ) R&& dm o o * { fη } = b V &η R& dm And equvalently the generalzed actve forces determned by dottng the actve forces df and torques dτ to coeffcents of generalzed speeds f { fη } = f & η = V df f = ω ( dτ + + b R b ( r T u ) d f ) d f V b The above expressons for the nertal forces wll be functon of { R& },{ & ω },{ && y },{ && θ },{ & η }. L( ) L( ) (5) (7) (9) (11) (12) (13)

8 The frontal part conssts of the steps needed to perform the elmnaton of a body's modal { η& & } and relatve ont degrees of freedom { & y },{ & θ } n terms of ts nboard body motons { R& ( ) },{ ω& L( ) } L and nown external forces. Once performed the step for a leaf body-, the obtaned equatons are shfted to nner body-l() reference obtanng equatons augmented due to the contrbuton of outboard body-. The process s repeated untl the core Body-1 s reached, for all branches, and augmented dynamc equatons are then solved wth respect to 1 1 { y },{ & θ & } dofs. Note that, for a core body, R & L( ) and ω& L( ) are null vectors. The bac substtuton part conssts of ndeed bac-substtutng along the tree topology and determnng the body/ont- DOF acceleratons n terms of the moton varables of ts nboard body-l(). The soluton algorthm, whch s demonstrated to behave as Order(n), can then be summarzed as follows: Step 1: Defne the system topology and ntalze the ont varables and ther rates. Step 2: Compute all nematc relatonshp and generalzed nertal and actve force quanttes. Step 3: Frontal Part: a) For all leaf bodes n the branch, perform the modal and ont DOF elmnaton s. Setup the nerta forces of body and the actve forces. b) For a generc branch body, frst obtan the nerta forces of all of ts outboard bodes n terms of the current branch body and ts nboard body acceleratons. Follow steps 2 and 3a, except that the contrbuton from ts outboard bodes must be consdered for the nerta and actve forces, respectvely. A generc branch body may have more than one outboard body and hence the order n whch these elmnaton s are carred out s mportant. Complete steps 2-3 for all the bodes n the system untl the core body s reached. Step 4: Solve for the acceleratons assocated wth the augmented core body DOF n terms of all the actve forces on the system. Step 5: Assemble the system state vector dervatves for numercal ntegraton, ncludng addtonal states comng from user code Step 6: Bac Substtuton: Worng from Body 1 outward, solve for all the body. Note that, n the present algorthm the system mass matrx s never explctly nverted, thus avodng the need for O(n 3 ) operatons. The computatonal burden ncreases only lnearly as the number of bodes n ncreases, because the present algorthm requres computatons of Order(n). The Order(n) dynamcs formulaton s extended to closed-loop topologes (Fgure 2.b). The method conssts of renderng the system open-loop by ntroducng a cutont n the system In addton to the actve forces present n the system, the (unnown) constrant forces and torques ntroduced by the cut-ont must be accounted for. These unnowns wll be elmnated by usng the compatblty condtons at the cut-ont. Thus, ths procedure wll nvolve two frontal and two bac substtuton steps for every closed-loop n the system. Detals on DCAP Order(n) formulaton for closed-loop topologes s gven n [10].

9 Symbolc processng Symbolc processng of the equatons of moton can result n a substantally more effcent smulaton. Ths ncrease n effcency s acheved through smplfcatons that are possble because of specal confguraton characterstcs, as well as arthmetc and algebrac smplfcatons. A schematc of the symbolc processng and smulaton modules, along wth ther relatonshp wth the DCAP pacage, s shown n Fgure 4. They receve ther nput from three sources: ` Fgure 4: DCAP Smulaton Context dagram a confguraton data fle whch descrbes the mult-body system beng smulated, ts topology and propertes, generated through DCAP command lne or Graphc User Interface modules a flexble-body data set that contans data relatng to the flexblty propertes of each flexble body n the system, obtaned from dedcated nterface to general fnte element codes or generated nternally for smplfed flexble body types (see later secton) an equaton fle contanng the templates of the equatons of moton of a generc mult-body system, mplementng the formal codng of all the nematc, frontal and bac substtuton equatons consstent wth the formalsm outlned n prevous secton The symbolc processor, coded n C-language, produces n output a set of FORTRAN source fles contanng the mplementaton of the specfc set of equatons of moton that are applcable to the mult-body confguraton defned. Ths source code s then compled and lned wth the smulaton lbrary to generate the executable module. An addtonal set of user-defned routnes s conceved n DCAP n terms of user-contnuous-controller, user dscrete-controller or userfuncton generator. The user-controllers mplementaton s revsted latter n ths paper hen descrbng the Matlab nterface n MATDCAP secton.

10 Flexble bodes The modal formulaton used n DCAP for flexble bodes depends on the defnton of assumed mode shape vectors φ ( r ) beng multpled by tme-varyng ampltudes η ( t ) for th mode. The local deformaton s then expressed, conformng the nodal-body2 notaton shown n Fgure 3, wth flexble dsplacement and rotaton u 0 NM NM Φ 0 0 = 1 = 1 ( r ) η u = Φ ( r ) η = (14) The modal shapes φ and modal slopes φ can be ether obtaned from popular fnte element codes, or expressed analytcally for smple flexble body confguraton such as beams or cables. Modal representaton allows the defnton of a mass matrx for body- that s confgurable up to 2 nd order nonlnear dependency on modal coordnates. NM [ M ] [ L ] [ M + N ] [ Y ] [ ] R M = + T η + T [ L ] [ m ] = 1 [ Y ] [0 ] (15) NM NM [ P ] [0 ] + η η T = 1 = 1 [0 ] [0 ] Wth the sub-matrx parttons applyng for the 6 6 rgd body components dentfed by the rgd body mass matrx [ M R ], the NM NM modal mass matrx [ m ]. At zero order, the couplng wth flexble dofs s defned through the modal partcpatons [ L ] szed 6 NM, defned wth respect to body reference as [ r φ Kr φ ] dm = b 1 NM [ L ] [ φ Kφ ] dm b 1 NM wth the ntegraton defned at fnte element code level. It should be notced that n DCAP formulaton, the rotatonal terms are located n row/colum 1 3, the translatonal n 4 6. At frst order, the [ M + N ] 6 6 matrces allow the mplementaton of frst order dependency of the center of mass and nerta tensor on deformaton η, whle the [ Y ] 6 NM matrces gves the frst order change n modal partcpatons. Wth the 3 3 parttonng 0 (16)

11 and ~ ( φ ~ r + ~ ~ r φ ) ~ dm φ dm b b [ M + = N ] ~ (17) φ dm 0 b = [ φ φ Kφ φ ] dm [ Y b 1 NM ] (18) 0 At second order, the [ P ] matrces allow the mplementaton of second order dependency of the nerta tensor on deformatons η η. ( ~ φ ~ φ ) = dm 0 [P b ] (19) 0 0 It s worth pontng out that selecton of modal bass s left to the user defnton of the constrants n the fnte element model. In the general approach, selecton of normal modes of vbraton allows mplementaton of dagonal modal mass [ m ] and modal stffness [ ] matrces, wth the former selectable to dentty. The general purpose Alter routnes provded wth DCAP pacage allow drect computaton wthn NASTRAN of the requred modal ntegrals to be handled by pre_flex flexble data pre-processor. As a very useful extenson, Crag-Bampton approaches are also mplemented n the nterface routnes. The fxed-nterface normal modes bass s augmented wth statc modes (unt dsplacement at nterface dofs), whch allow usage of reduced modal bases for mproved convergence when redundant constrants are dentfed for the flexble body n the tree topology. Clearly, for Crag-Bampton approaches the modal mass matrx wll be no longer dagonal. Propertes of the egenfunctons of a fxed-free beam The assembly of modal ntegrals startng from fnte element models allow the ntegraton n DCAP models of complex flexble bodes. Nevertheless, for smple shapes such as beams and cables, the modal ntegrals can be obtaned nternally wth mplementaton of analytcal defnton of mode shapes. The boundary value problem assocated wth a unform beam wth fxed-free end condtons s defned by EI u ( x ) λµ u( x ) = 0 (20) wth the assocated boundary condtons, u (0)=0 and EIu'''(L)=0 where, EI and µ are the stffness and mass per unt length, respectvely, and u(x) s the elastc dsplacement of a pont along the beam wth spatal coordnate x. Prmes denote dfferentaton wth respect to x.

12 By solvng the egenvalue problem, the characterstc equaton s obtaned as: cos βl coshβl = 1 (21) and, the egenfunctons assocated wth each of the solutons of the characterstc equaton are gven by 1 φr( x ) = [( cos βr x coshβr x) + Ar ( sn βr x snh βr x) ] (22) µ L Where the r th egenvalue s defned as 4 λµ β = (23) EI and sn βrl snhβrl Ar = (24) cos βrl + coshβrl and the egenfunctons are unt modal mass normalzed such that ( ( x ), µφ L 2 ( x )) µφ ( x )dx = 1 φ (25) r r The vbraton frequences are obtaned from the relaton, ω 2 =λ. An mportant observaton s n order wth regards to obtanng the frequences of fxed-free beams wth dfferent lengths, whch s the case when a beam wth a prsmatc end condton s travellng whle the ont tself remans statonary. The characterstc equaton, Eq.(21), s a functon of the product βl, and not a functon of β or L alone. It mples that the characterstc equaton can be solved once and for all, the solutons of the characterstc equaton, namely βl's can be tabulated, and the frequences of a unform fxed-free beam of any length can be obtaned from such a table and Eq.(23). Mathematcally, d dl 0 r ( β L) = 0 (26) Expandng Eq.(26), the varaton of β wth respect to L can be obtaned as dβ β = dl L (27) Or, n terms of the frequences, dω 2ω = dl L (28) A closed-form soluton for ω s gven by ω ( L ) = D 2 L (29) where the length dependency of the frequences s stated explctly and D = ω( L 0 ) L 2 0 s a constant. Snce the frequences and the assocated egenfunctons for any length of the beam can be obtaned, a queston s rased as to

13 how the egenfunctons can be related for two beams of dfferent lengths. To ths end, t can be shown that dφr( x,l ) 1 x dφr( x,l ) = φr( x,l ) (30) dl 2L L dx where, the length L on the rght sde of Eq.(22) s also treated as a varable n obtanng the above result. The above analyss s vald even f the boundary condtons are dfferent but stll represent the classcal end condtons such as pnned or guded; then, Eqs.(20) and (22) wll be approprately modfed. Although the normalzaton constant wll not retan ts smple form as n Eq.(23), t wll contan the product βland hence d( βl ) = 0. dl Dynamcs of flexble structures durng deployment An extenson allowable n DCAP wth respect to the analytcal modelng of beams and cables s represented by the mplementaton of models for deployng structures. The complex formulaton of a flexble beam or cable n deployment s nternally handled by DCAP through the symbolc processor, allowng the automatc mplementaton of the tme-varyng flexblty of the body beng deployed. The equatons of moton for a structure durng deployment from and retracton nto a base that s part of an open-loop mult-body chan. The contnuously changng elastc behavor of such a structure s modeled through nstantaneous frequences and mode shapes, usng a contnuum approach. The propertes of the egenfunctons of a fxed-free beam are exploted to express varous doman ntegral terms as explct functons of the nstantaneous deployed length. The equatons reflect the effects of rgd-body and elastc motons on each other. Thus, ths complete model can be used n the controller desgn, f desred. The deployment rate and the base/hnge rotatonal moton can also be prescrbed, f one s only nterested n studyng the effect of rgd-body moton on the elastc behavor of the structure. A maor obstacle to the modelng of structures durng deployment s that the elastc behavor of the structure undergoes contnuous change. Consder the deployment of a space truss from a canster. As the deployed length ncreases, the flexblty of the structure ncreases. Snce ths s a contnuous process, the frequences of the structure also undergo contnuous change. Consder the elastc behavor of a beam under deployment from a fxed base. It s usually assumed that translatng flexble lns can be modeled as beams (Euler- Bernoull model) n flexure wth fxed-free end condtons [11,12]. Tabarro et al. [11] derved certan propertes of the mode shapes of fxed-free beams n flexure. The results of [11] have been used n [12] to study the effect of the couplng between the translatonal (rgd-body) and flexble motons.

14 The canster s assumed to be attached to the nboard body through a hnge that L ( ) y = 0 permts only relatve rotatonal moton. Thus. Let the deployed porton of the space truss be modeled as a beam havng unform stffness and mass propertes. Assumng that the staced porton of the structure behaves le a rgdbody, the deployed porton of the structure can be modeled as a flexble beam that s nstantaneously fxed to the staced part. Let the translatonal (deployment) velocty of the beam be denoted by L &. As the beam s extended, the length of the vbratng secton of the beam ncreases. We denote the length of the beam outsde the support by L and assgn the spatal varable x measured from the fxed end to denote the materal pont on the deployed porton of the beam. Let x=0 and x=l correspond to the fxed- and free-ends, respectvely. Now, the part of the beam outsde the support can be modeled as a fxed-free beam at any tme nstant. Varable mass and varable stffness bodes In later mplementaton, the capablty for havng bodes changng ther mass and stffness propertes durng the dynamc smulaton has been extended to both rgd bodes (for the mass, center of mass locaton and nerta tensor varaton) and generc flexble bodes obtaned from fnte element models (for both mass propertes and stffness propertes). The mplementaton has been allowng the confguraton of specfc rgd-varable and flexble-varable body types. For these body types, mass, center of mass, nerta tensor, modal mass and stffness matrces are made tme-dependent through the defnton of a specfc number of user-defned confguraton data. For flexble bodes, ths means avalablty of a set of fnte element code runs for a fxed structural topology at some reference ey confguraton. For rgd bodes, a dedcated fle hostng several confguraton data ponts d mplemented. The varablty can be selected to be ether functon of tme or functon of mass. It s relevant to notce that no nternal consstency chec s made for the forces arsng from mass varaton. In other words, when defnng the dynamcs symbolcally as d F ( t ) = ( M R & ) = M( t )R( && t ) + M( & t )R( & t ) (31) dt The rght sde term M & ( t )R( & t ) should be mplemented by the user as external force consstently wth the specfed M(t). A typcal case s represented by a launch vehcle. In the user-defned FORTRAN routnes, the engne can be modeled by the user though the specfc mpulse and the mass flow rate rates. The DCAP user contnuous controller allows nternal codng of states to be ntegrated n lne wth system dynamcs, and therefore the modeled mass flow rate can be ntegrated to obtan the nstantaneous mass of fuel burnt and the nstantaneous mass of the vehcle. The latter can be made avalable as output of the user-controller and lned as varable drvng the tabular mass/stffness nterpolaton for tme-varyng data.

15 The number of allowable confguraton can be selected by the user, and the runtme update performed. Ths mplementaton s clearly optmal n terms of run tme for the Order(n) formulaton, where no mass matrx nverson s drectly requred. In classcal Order(n 3 ) formulaton the approach s nstead rather tme consumng, requrng besdes mass matrces assembly ther explct nverson at each ntegraton step. An applcaton: LV GNC smulator The LV gudance, navgaton and control functon s performed by an On-Board program flght Computer (OBC), whch executes the navgaton calculatons and mplements the gudance law (Fgure 5). Ths functon s performed wth the ad of an Inertal Measurement Unt (IMU), whch delvers navgaton and atttude data to the computer. Atttude control s performed: o Durng frst three stages flght by vectorng of the motor thrust acheved by deflecton of the motor nozzle. Ths functon s mplemented by the Thrust Vector Control system o Durng exo-atmospherc flght by atttude control thrusters actuaton Motor nozzle deflecton commands are transmtted to the TVC system electronc control unt IPDU (Integrated Power Dstrbuton Unt), whch mplements a local feed-bac control system to deflect the nozzle and eep t n the desred poston worng aganst loads appled to the nozzle. Computatonal tools are sought necessary to perform a) non-lnear traectory dynamcs tme smulaton under varable mass, b) control desgn evaluaton at dscrete traectory ponts n lnear doman, c) stage separaton dynamcs. The DCAP software has been selected for the tass nvolvng non-lnear dynamcs smulaton, ncludng modelng capabltes of contact dynamcs, and stcton-frcton. For the control desgn analyss, as for the case of TVC related actvtes, advantages exst also here n usng the MATLAB envronment. Partcularly wth respect to control tools, the post processng analyss and use of the numerous computatonal tools (tool boxes) of MATLAB. I M U NAVIGATION CURRENT LV Poston & atttude GUIDANCE DESIRED TVC ptch & yaw actual pston 1 dplacement TVC DESIRED pstons dplacement TVC angles actual pston 2 dplacement TVC pstons dsplacement Fgure 5: LV GNC descrpton

16 Results Lnear analyss Ths analyss opton s mplemented n DCAP by means of numercal lnearsaton of the non lnear dynamc equatons about an assumed equlbrum state. The model s mplemented for a 6DOF confguraton wth two pstons now actuatng on the nozzle. Fgure 6 shows examples of frequency response from demanded pston 1 dsplacement to actual pston 1 dsplacement, whle Fgure 7 shows the correspondng cross-couplng from demanded pston 1 dsplacement to pston 2 dsplacement. These results are relevant to a case wth LV flexblty begnnng of lfe, bg loop open, and no force feedbac n the pston control. Fgure 6: Frequency response from demanded pston 1 dsplacement to actual pston 1 dsplacement Fgure 7: Cross-couplng from demanded pston 1 dsplacement to pston 2 dsplacement Non-lnear analyss DCAP also allows the non-lnear tme smulaton of the LV traectory. Optons are avalable for ncludng rgd and flexble tme varyng dynamcs. Currently the software runs on DCAP Lnux Platform and on MATDCAP n Smuln/Matlab Wndows envronment. The followng fgures show the man results of a 1 st stage 3D non-lnear traectory smulaton.

17 Fgure 8: X-nertal poston of LV CoG Fgure 9: Desred vs Actual Pston 1 Poston Fgure 10: Fly Alongsde vew Fgure 11: Fxed Poston vew Concluson Generc modelng capabltes and computatonal speed are strong assets of the DCAP. Recent advances address senstvty modules towards MonteCarlo type of analyss and MATLAB-Smuln I/F. The latter maes the control desgn tas much smpler thus when mplemented us a user dedcated functon wthn DCAP. Ths way of operaton s expected to enhance very much the user frendlness and thus the number of users of the pacage.current developments are expected to focus on extendng further functonaltes of the modelng, GUI nterface, and 3D modelnganmaton. References [1] DCAP Release 7, ESA Contract 7971/88/NL/JG, Alena, [2] R. Franco,M, L. Dumontel, S. Portglott & R. Venugopal, The Dynamcs and Control Analyss Pacage (DCAP) - A versatle tool for satellte control, ESA Bulletn, Nr. 87, August, ( )

18 [3] R. Franco et al, Enhanced Dynamcs and Control Analyss Pacage (DCAP), Proceedngs of ESA WPP-041, ESA Worshop on Spacecraft Gudance Navgaton and Control, September [4] Sngh, R.P., VanderVoort, R.J., and Lns, P.W., Dynamcs of Flexble Bodes n Tree Topology - A Computer Orented Approach, Journal of Gudance, Control and Dynamcs, Vol. 10, No. 5, September-October [5] Sngh, R.P. and Schubele, B., Computatonally Effcent Algorthm for Dynamcs of Mult-ln Mechansm, AAIA GN&C Conference, Boston, MA, August [6] Irons, B.M., A Frontal Soluton Program, Internatonal Journal of Numercal Methods n Engneerng, N. 2, pp. 5-32, [7] Nelan, P.E., Effcent Computer Smulaton of Moton of Mult-body Systems, PhD Dssertaton, Stanford Unversty, September [8] Kane, T.R., and Levnson, D.A., Dynamcs, Theory and Applcatons, McGraw-Hll, New Yor, [9] Kane, T.R., Lns, P.W., and Levnson, D.A., Spacecraft Dynamcs, McGraw- Hll, New Yor, [10] M. L. Dumontel, S. Portglott, R.Venugopal, DCAP: A Tool for Analyss and Smulaton of Mult-Body Systems, 45 th IAF Conference, Oslo, October [11] B. Tabarro, Leech C.M. and Y.I. Km, On the dynamcs of an axally movng beam, Journal of the Franln Insttute, 297(3): , March [12] S.K. Tadonda and H. Baruh, Dynamcs and Control of a Translatng Flexble Beam, Journal of Dynamc Systems, Measurement and Control, to appear. [13] Hppmann, G., An Algorthm for complant contact between complexly shaped surfaces n mult-body dynamcs, Proceedngs of the Internatonal Conference on Advances n Computatonal Multbody Dynamcs, Lsbon, Portugal, July 1-4, [14] D. Scacovell, S. Kryeno, G. Baldes, A. Threttle, R. Redondo & P. D. Resta, Vega Prototype 3D Smulaton Software wth Tme Varyng structural characterstcs, 5 th Inter. Conference on Space Launchers: Mssons, Control and Avoncs, Madrd, Span, November 25-27, [15] G. Baldes, Modellng Launch Vehcle Nozzle and TVC loop wth DCAP & Importng the Non-Lnear Dynamcs n Smuln/Matlab Envronment, report n Fulfllment of Master n Satelltes and Orbtng Platforms, Unversty La Sapenza, Rome, Italy