Wave-Based Attitude Control of Spacecraft with Fuel Sloshing Dynamics

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1 ECCOMAS Themtic Conference on Multiody Dynmics June 29 - July 2, 25, Brcelon, Ctloni, Spin Wve-Bsed Attitude Control of Spcecrft with Fuel Sloshing Dynmics J. W. Thompson, W. J. O Connor #, School of Mechnicl nd Mterils Engineering University College Dulin Belfield, Dulin 4, Irelnd joseph.thompson@ucdconnect.ie # willim.oconnor@ucd.ie ABSTRACT Mny mechnicl systems re inherently flexile, mking it difficult to chieve rpid, controlled motion. The control chllenge is even greter when the system is not well modelled, hs dynmics tht chnge with time, or is under-ctuted. A rocket with sloshing fluid propellnt is n extreme cse. Mny control strtegies struggle with such systems. However wve-sed control method hs een shown to cope well with these chllenges. The key ide is tht the motion of the ctutor cn e seprted into two notionl components, one trvelling from the ctutor into the system, the other leving the system through the ctutor. Intuitively the ctutor simultneously lunches mechnicl wves into system while it sors returning wves. When the lunching nd soring is finished virtions hve een dmped nd the desired reference motion is left ehind. A mthemticl model is developed for n upper stge ccelerting rocket moving in single plne. An equivlent mechnicl model in the form of pendulum is included to represent the fuel sloshing dynmics. In numericl simultions the controller successfully suppresses the sloshing motion. A mjor dvntge of the strtegy is tht no mesurement of the pendulum sttes (sloshing motion) is required. However it is found tht when the effective sloshing mss ecomes smll reltive to the rocket ody, it tkes longer to fully suppress sloshing motions. This is cceptle, however, ecuse in this cse, y definition, the sloshing does not cuse mjor prolem for the rocket controller. Keywords: Spcecrft Dynmics, Attitude Control, Flexile Systems, Sloshing, Mechnicl Wves INTRODUCTION The filure of n erly Jupiter llistic missile in 957 ws cused y the interction of the control system nd the sloshing liquid fuel on ord []. The prolem ws originlly solved y luminium drink cns, which could e fitted into the fuel tnk nd which floted on the surfce of the fuel. Their friction oth ginst the wlls of the tnk nd ginst ech other dmped out the fuel oscilltions. This experience motivted the eventul solution to the prolem, which ws the ddition of ffles to the tnk wlls. However ffles cn only provide certin level of dmping of the liquid motion nd cn only e optimized for one tnk fill level, so their effectiveness is reduced s the propellnt is depleted. They lso dd complexity nd mss to the vehicle nd so increse costs. [2]. It therefore ecomes necessry to design control system for rocket which ctively tkes into ccount the effects of propellnt sloshing nd tries to compenste for them. Mny mechnicl systems re inherently flexile, mking it difficult to chieve rpid, controlled motion. The control chllenge is even greter when the system is not well modelled, hs dynmics tht chnge with time, or is under-ctuted. A rocket with sloshing fluid propellnt on ord is n extreme cse of such system. A wve-sed control method hs een shown to cope well with the chllenges outlined ove [3, 4]. The key ide is tht the motion of the ctutor cn e seprted into two notionl components, one trvelling from the ctutor into the system, the other leving the system through the ctutor. Intuitively the ctutor simultneously lunches mechnicl wves into system while it sors returning wves. When the lunching nd soring is finished, virtions

2 hve een dmped nd the desired reference motion is left ehind. The method hs een demonstrted to work well for -D nd 2-D lumped flexile systems nd in rootic nd crne pplictions [5, 6]. The im of this pper is to extend the ppliction to the control of spcecrft with flexile structures nd ppendges (e.g. solr pnels), nd with on-ord liquid propellnt. This new re of ppliction presents mny new chllenges. The spcecrft systems re often nonliner, their ssocited flexiility is non-uniform, the sloshing dynmics re difficult or impossile to predict, nd sensors nd ctutors cn ehve fr from the idel. In this pper the exmple of n ccelerting upper stge rocket is exmined. A wve-sed controller is designed for the upper stge AVUM of the Europen luncher Veg nd is tested y numericl simultion. 2 MATHEMATICAL MODEL In this section mthemticl model is developed for n upper stge ccelerting rocket moving in single plne. The rocket is ssumed to e in microgrvity environment nd free from erodynmic effects. The sloshing fuel mss is represented y mechnicl nlog in the form of simple pendulum ttched to the min rocket ody. The fuel mss is prtitioned, ccording to the tnk fill level, into fixed point mss nd moving pendulum mss [7]. 2. Plnr upper stge model The model of the upper stge rocket with single pendulum is shown in figure. x f φ M z F Figure. Upper stge rocket model The rocket ody nd pendulum re isolted nd free ody digrms for ech re shown in figure 3. x R z R x M f x R x R z φ z F z Figure 2. Free ody digrms The equtions of motion for the two odies my e written s: F R x = m x ()

3 f R z = m z (2) I = M + (R z ) (3) R x sinφ + R z cosφ = (4) m f [ x 2 + cosφ( + φ) 2 + sinφ( + φ)] = R x (5) m f [ z sinφ( + φ) 2 + cosφ( + φ)] = R z (6) All symols re descried in tle. Symol m I m f φ F f M R x R z x z Tle. List of symols Description ody mss ody moment of inerti pendulum mss length of pendulum distnce from COM to pivot point ody pitch ngle ngle of the slosh pendulum w.r.t. ody xil force lterl force moment pplied to ody internl constrint force t pivot point internl constrint force t pivot point xil ody ccelertion lterl ody ccelertion Sustituting for R x nd R z from equtions 5 nd 6 into equtions -4 elimintes these internl constrint forces, nd gives mimiml set of four equtions for the four degree of freedom system. (m + m f ) x m f 2 + m f cosφ( + φ) 2 + m f sinφ( + φ) = F (7) (m + m f ) z m f m f sinφ( + φ) 2 + m f cosφ( + φ) = f (8) (I + m f 2 ) m f z + m f sinφ( + φ) 2 m f cosφ( + φ) = M (9) (m f 2 )( + φ) m f sinφ 2 m f cosφ + m f ( z cosφ + x sinφ) = () 2.2 Choice of ctutors The model descried is generl in tht the ody is ctuted y two forces, F nd f, nd moment, M. In relity the rocket my hve one or mny ctutors, ut in ny configurtion these ctutors my e resolved to these two forces, xil nd lterl, nd moment pplied to the rocket ody. In some cses these inputs my not e independent of ech other, ut insted function of some lesser numer of inputs. For exmple, in the cse of rocket s shown in figure 3() with single gimled engine the forces nd moment re no longer independent nd re given y: M = T ( + c)sinδ, F = T cosδ, f = T sinδ () where T is the constnt thrust developed y the rocket engine, c is the distnce of the giml from the mss centre, nd the single input is the engine giml ngle δ. Similirly the rocket my e ctuted y lterl thrusters s shown in figure 3(). In this cse: M = T l d, F = T, f = T l (2) where gin T is the constnt thrust developed y the non-gimlling rocket engine, d is the xil distnce from the thrusters to the mss centre, nd the single input is the mgnitude of the lterl thrust T l.

4 x φ δ x φ T l z c T z d T () () Figure 3. Possile ctutor configurtions 2.3 Linerized model Solving equtions 7 nd 8 for x nd z respectively gives: x = F + m f 2 m f cosφ( + φ) 2 m f sinφ( + φ) m + m f (3) z = f + m f + m f sinφ( + φ) 2 m f cosφ( + φ) m + m f (4) Sustituting these expressions into equtions 9 nd gives simplified system of two equtions descriing the pitch nd slosh dynmics: where: [I + mm ( 2 cosφ)] mm φ cosφ + mm ( + φ) 2 sinφ = M + m f (5) m ( 2 cosφ) + mm 2 φ + m (F m 2 )sinφ = m f cosφ (6) m = After lineriztion out [, φ,, φ] = equtions 5 nd 6 ecome: m f m + m f (7) [I + mm ( 2 )] mm φ = M + m f (8) m ( 2 ) + mm 2 φ + m Fφ = m f (9) The stte vector of the linerized system consists of the pitch nd slosh ngles nd their derivtives: x = [,φ,, φ] T (2) It is ssumed tht the xil thrust F is constnt, so tht the input vector consists of the lterl force nd moment M: u = [ f,m] T (2) Now equtions 8 nd 9 my e rewritten in stte spce form: where: A = Fm f I(m+m f ) Fm f ( 2 ) I(m+m f ) ẋ = Ax + Bu (22) F m, B = m I ( ) I (23)

5 We find tht the condition for stility of this linerized system is: mm f ( 2 ) m + m f + I (24) Possile pendulum configurtions re shown in figure 4. For < (pendulum pivot point ehind the mss centre) (figure 4()) or > (whole pendulum in front of mss centre) (figure 4()) this condition is lwys stisfied, ut in the region < < stility is chieved only if: I(m + m f ) mm f (25) Physiclly this mens tht the pendulum must strddle the point P shown in figure 4(c) which is distnce I(m+m f ) mm f from the centre of mss long the x-xis. I(m+m f ) mm f P () () Figure 4. Possile pendulum configurtions (c) 3 WAVE-BASED MODEL The first step in developing wve model for the rocket system is to express the equtions of motion in form resemling cscded lumped flexile system. The rocket system descried y eqution 22 hs two degrees of freedom nd so it is required to trnsform this to pper like 2-DOF lumped flexile system, i.e two msses/inertis with n interconnecting spring. It is lso required tht there e single control input which ctutes just the first degree-of-freedom, i.e. single luncher nd sorer of wves. The system should hve the following form: ż = Âz + ˆB f (26) where: Â = m k m k k m 2 k m 2, ˆB = m, z = x x 2 x x 2 (27) These equtions descrie the dynmics of 2-DOF mss-spring system, where x nd x 2 re the displcements of the msses nd f ct is the ctuting force on the first mss. Assume tht the lterl force f on the rocket ody is zero nd just the pure moment M is ville for control purposes. In relity this could e imgined s gimlled rocket engine with lterl thrusters t mss centre to cncel the lterl forces from the engine. The input mtrix B from eqution 23 then ecomes

6 column vector: B = I ( ) I (28) To trnsfrom the system to the required form the chnge of sis z = Mx is used, where: The sttes re now: M = nd the prmeters of the system re given y:, Â = MAM, ˆB = MB (29) ( ) x =, x 2 = + φ (3) k = Fm f ( ) (m + m f ), m = I, m 2 = mm f ( 2 ) m + m f (3) Figure 5 shows the equivlent mss-spring system with notionl mss m nd notionl spring of stiffness k ppended to the system. The force in the first spring is considered to e the ctution force f. The system my now e considered s ctuted y the displcement x of notionl mss m such tht: f = k (x x ) (32) The wve model ssumes tht the displcement of ech mss x i is cn e seperted into leftwrd nd rightwrd trvelling components i nd i respectively or A i nd B i in the Lplce (complex frequency) domin [8]. The propgtion of the rightwrd nd leftwrd trvelling wves is descried y wve trnsfer functions G i, H i nd F respectively such tht: A i = G i A i, B i = H i B i+, B 2 = FA 2 (33) For controller design it is esier to work with trnsfer functions tht del with the ctuting force F rther thn the notionl displcement X. The spring force F cn lso e seperted into rightwrds nd leftwrds trvelling components F A nd F B respectively. Then the cross-over wve trnsfer functions P nd Q relte displcements to forces y: A = P F A, F B = Q B (34) nd these cn e clculted from the ordinry wve trnsfer functions s: P = G k ( G ), Q = k (H ) (35) 4 CONTROL DESIGN 4. Wve-Bsed Controller A WBC3 (force ctuted) controller ws designed to control the rocket ttitude. The controller uses only the trnsfer functions G nd H. The second-order uniform system pproximtions [8] re used where: ωg 2 2k G = s 2 + ω G s + ωg 2, ω G = (36) m

7 G G A A A 2 k k m m m 2 F B B B 2 H H Figure 5. Wve model of the rocket re f + X + M re f 2 + k B ( ) P M B = H P Q Figure 6. Wve-sed control system M ω 2 H H = s 2 + ω H s + ωh 2, ω H = 2k m (37) The wve-sed control scheme is shown in figure 6. re f is the desired reference pitch ngle. The control input is the input torque M re f. Two vriles re mesured for feedck. These re the pitch ngle nd the ctul chieved torque M. In this pper the ctutor is ssumed idel except for the sturtion limits. The wve sed control strtegy lunches wve equl to hlf the reference signl re f. The mesured vlues of nd M re then used to clculte the returning wve component t the ctutor which cn e clculted s: ( ) P M B = H (38) P Q The ctutor is then moved to mtch this returning wve component nd therey sor it. When the soring is finished the system will hve een displced y twice the specified lunch wve, i.e. will e t the reference displcement. 4.2 Time-Optiml Controller For comprison, the torque-limited ng-ng solution for rest-to-rest mneuver of rigid rocket ws clculted. With the slosh pendulum frozen in position the moment of inerti of the rocket ody out the overll mss centre is given y: I rigid = I + mm f ( ) 2 (m + m f ) (39)

8 The switching time for rest to rest mneuver is: re f I rigid t s = (4) M mx where M mx is the mximum torque. Then the control input for the ng-ng mneuver eginning t t = t is: t < t M mx t < t < t +t s M = (4) M mx t s < t < t + 2t s t > t + 2t s 5 RESULTS 6 Simultion of rel rocket The wve-sed controller ws tested y numericl simultion. Suitle prmeters for the presented rocket model were chosen to represent AVUM, upper stge of the Europen Veg luncher [9] (tle 2). The included slosh pendulum represents the primry sloshing mode for the rocket s fuel tnk when hlf full. The sturtion torque M mx ws clculted from the mximum giml ngle of the AVUM engine. The vlues chosen for notionl mss nd spring stiffness were m = m nd k = k (m /m 2 ), however the choice for these prmeters is ritrry to some degree nd rnge of vlues will give good control response. Results re shown in figure 7 for five degree step chnge in commnded pitch ngle re f. The wve-sed controller is compred to the torque-limited time-optiml solution for the rigidized rocket. Tle 2. Summry of model prmeters representtive of AVUM upper stge Prmeter Vlue Unit m 25 kg m f 88 kg I 883 kg m 2.53 m.43 m F 245 N M mx 93 N m 7 DISCUSSION It cn e seen tht the fuel slosh dynmics cuse the open-loop time optiml controller to lnd off trget nd drift wy from the trget over time. The fuel sloshing persists for long times in the sence of dmping in the model. The wve-sed controller lnds on trget nd supresses the sloshing motion. However the sloshing persists for severl oscilltions. The reson for this is the non-uniformity of the system, i.e. unequl inertis m nd m 2. In this cse m is much less thn m 2. From wve perspective there is chnge in wve impednce etween the two different msses nd some wves ecome trpped on the right hnd side of this oundry. For this reson the ctutor only sors frction of the motion on ech oscilltion cycle, ut over severl cycles cn sor it ll. When the rtio of inertis m m 2 is much less thn one, the effect of the pendulum on the ody is much reduced nd so it tkes longer to fully suppress sloshing motions. This is cceptle, however, ecuse in this cse, y definition, the sloshing does not cuse mjor prolem for the rocket controller. On the other hnd, the control chllenge is gretest when the fluid inerti rtio is lrge, nd this is precisely when the new strtegy delivers much

9 improved performnce. An interesteing venue of future reserch is developing wve-models nd controllers which tke into ccount the non-uniformity of the system to e controlled. A cler dvntge of wve-sed control is tht ll mesuring is done t the ctutor, in this cse the rocket ody, so no mesurement of the pendulum sttes is necessry, which is significnt onus given the chllenge of mesuring or modelling them in rel rocket. Future reserch includes mking the controller roust to externl disturnces such s erodynmic, stge sepertion or grvity forces; considering non-idel ctutor nd sensor ehviour; including multiple slosh pendulums representing either multiple fuel tnks or multiple modes of sloshing in single tnk; nd extending the nlysis to 6-DOF model where roll, pitch nd yw must e simultneously controlled. REFERENCES [] W. Heussermnn. Developments in the field of utomtic guidnce nd control of rockets. Journl of Guidnce, Control, nd Dynmics, 4(3): , My 98. [2] M. Reyhnoglu. Modeling nd Control of Spce Vehicles with Fuel Slosh Dynmics. 24. [3] Willim J O Connor. Excellent control of flexile systems. Control, pges , 25. [4] Willim O Connor nd Donogh Lng. Position Control of Flexile Root Arms Using Mechnicl Wves. Journl of Dynmic Systems, Mesurement, nd Control, 2(3):334, 998. [5] Willim J. O Connor nd Alessndro Fumglli. Refined Wve-Bsed Control Applied to Nonliner, Bending, nd Slewing Flexile Systems. Journl of Applied Mechnics, 76(4):45, 29. [6] Willim J. O Connor. A Gntry Crne Prolem Solved. Journl of Dynmic Systems, Mesurement, nd Control, 25(4):569, 23. [7] Frnklin T Dodge. the New "Dynmic Behvior of Liquids in Moving Continers". Southwest Reserch Institute, 2. [8] Willim J. O Connor. Wve-like modelling of cscded, lumped, flexile systems with n ritrrily moving oundry. Journl of Sound nd Virtion, 33(3):37 383, 2. [9] Edourd Perez. Veg User s Mnul, Issue 3, Revision. (3):88, 26.

10 8 (deg) 6 4 Reference 2 Bng-ng WBC time (s) 4 2 φ (deg) time (s), 5 M (Nm) 5, time (s) Figure 7. Pitch nd pendulum ngle φ for 5 degree step chnge in re f.