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Two level attitude control for a television and broadcasting satellite
Automatica, Vol. 13, pp. 595 604.
Pergamon Press, 1977.
Printed in Great Britain
Brief Paper Two Level Attitude Control for a Television and Broadcasting Satellite* KLAUS BECKER? + Attitude control; computer application; models; on-line operation; on off control; satellites, artificial; space vehicles: flexible systems. Key Word Index
body. The planes A, B, C represent the antenna cardan bearing. The antenna is fixed on plane A, plane C is fixed on the central body. Plane B is connected with plane A (axis x) and with plane C (axis y ) by joints and electrical actuators. The attitude of planes A and C wilh respect to each other and thus the attitude of the antenna with respect to the central body is determined by the angles A~t,x and A~py which can be easily obtained from the rotation mechanism. The desired attitude of the antenna coordinate system x, y, z with respect to the desired attitude of the body coordinate system X, Y,, Z is determined by the angles a,/3, ~/. Plane C is turned about these angles with respect to body plane D. This is clarified by the planes I, II, III, IV, which are accordingly rotated, where I coincides with C, and IV with body plane D. The solar generator, not shown here, is rotated about Y. For kq, x=kq~y=0 the body has its desired attitude if the antenna has its desired position, too. If the antenna remains in its desired position, then the attitude of the body axis - Z against its desired position - Z s is defined by A~0~and Aq~r. If the antenna is rotated about 3~pxand '~Pr against its desired coordinate system, then the body attitude is defined by
Summary -A two level method for the attitude control of a flexible communication satellite is described. Because of its great inertia the satellite is used as a platform for an antenna fine pointing control system while the satellite body is coarse controlled by jets. A simulation method is used which allows change of a satellite part without changing any other. Complex dynamic models are simplified while keeping the important characteristics. Simulation results are shown. Special forms of oscillations and resonance curves are analytically substantiated. The oscillations prove to be not dangerous. Changes of the solar generator natural frequency are not critical. Introduction THE ATTITUDEcontrol system to be described in the following is intended for a communication satellite§ in a synchronous orbit the antenna of which has to be oriented at ± 0.1 • in pitch and roll and a t _ + 2 in yaw tsee [1]). The required power implies a great solar generator of 24 m of length. This makes the satellite very flexible and the control system has to take account of it. It was intended to use a pure jet control system [2] which showed good properties for the control of rigid bodies. The oscillations to be expected for this project however gave reason to fear that the accuracy o f + 0 . 1 could not be guaranteed with a one-stagecontrol. The control system described here makes use of the fact that the pointing accuracy is much less for the solar generator and satellite body than for the antenna. Therefore body and solar generator are coarse controlled by gas jets as torquers only with an accuracy of_+ 1 resp.-+2L They constitute a kind of platform for the antenna because of their great inertia moment and move very slowly with <0.00l°/sec within the limit cycle range. So no m o m e n t u m storage is needed. The inertia moments about the generator rotation axis e.g. are 569 m2kg for the body, 1687 mZkg for the solar generator and 10 mZkg for the antenna. The antenna is swivelled in pitch and roll against body and solar generator by electrical servo motors and is thus fine controlled with an angular error of_+0.1 '. The solar generator is rotated about 360 ~ within 24 hr against the satellite body. Other problems arise for the simulation of such a complex system. The computer program for only the flexible solar generator[3] is so large that the addition of further satellite parts would lead to impracticable computer times. Universal programs[4] are not useful for the very special control logic applied. Therefore electronic hardware is used for simulation together with an analog computer.
A~pxn= Aq~x + 6~px A~0rn = A~py+ 6¢pr. As the antenna pointing errors 6q~x and 6q~r are small compared with the maximal values of A~0x and Aq~, the antenna attitude about x and y can be used as reference for the central body, because the antenna can be considered to have the desired attitude of the body. The block diagram of the control system is shown in Fig. 3. The logic steering the gas jets gets its signals Aq~x,Aq~r from the antenna position and the z-signal 3~0~from a monopulse sensor. The latter also procures the signals 6~ox and 3q~r for the a n t e n n a fine pointing logic working upon two electrical servo motors. Simulation of the motion of antenna, central body, and solar generator The principle of simulation is based on a u t o n o m o u s blocks for all relevant satellite parts, which get their inputs, moments M i and angles ~p~, as a kind of four terminal network from the neighbour blocks and give back reacting outputs to the same neighbours (Fig. 4). This procedure has the advantage that a block can be studied separately and is exchangeable if due to new facts and perceptions a different mathematical description seems favourable. Although this is not a new approach for simulation, many programs do not allow the exchange of parts because their inputs and outputs, especially the reacting outputs, are not exphcitly accessible. Figure 5 shows the simulation block diagram. All its important blocks have already been separately studied in detail, e.g. by special digital computer programs. These complex programs cannot all be used for the total system simulation, because this would lead to impracticable computer times and possibly to intolerable errors. Therefore simplifications of the complex block models have to be made under keeping the important characteristics. These simplificalions, which are discussed in this chapter, are based on the following considerations: a. Each block has no different properties in the system than it has individually b. Properties of a block which cannot be brought about by other blocks, can be omitted. Therefore all effects possible of a block are individually studied and simplifications are thoroughly checked with regard to the
Angular relations and block diagram Figure 2 shows the angular relations, x , y , z is the antenna coordinate system, X, Y, Z the coordinate system of the central
*Received 22 December 1976; revised 18 May 1977. The original version of this paper was presented at the 7th IFAC Symposium on Automatic Control in Space which was held in Rottach-Egern, Federal Republic of G e r m a n y during May 1976. The published Prodeedings of this IFAC Meeting may be ordered from: Federation IBRA-BIRA, Rue Ravenstein, 3, B1000 Brussels, Belgium. This paper was recommended for publication in revised form by associate editor D. Tabak. tERNO-Raumfahrttechnik, Bremen, Huenefeldstral3e, now with Fachhochschule Hamburg, Berliner Tor 21, FRG. ++Present address: Delbrfickstr. 14, 28 Bremen, Federal Republic of Germany. §(Project HLS of the German Government, Fig. 1, HLS = German abbreviation for High Power Satellite). 595
596
Brief Paper
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Brief Paper
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FIG. 3. Control system block diagram, l: Fine pointing logic; 2: antenna servo motors; 3: antenna; 4: monopulse sensor pitch, roll: 5: monopulse sensor yaw; 6: antenna position sensors; 7: transformation matrix; 8: pulse stagger logic: 9: jets; 10: spacecraft: 11: backup sun sensor; 12: backup earth sensor.
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Brief Paper Central Body
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Generator
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FI(L 6. Principle ol sohlr generator orientalhm.
The quantitative coincidence \\ith Flexpan is good. lhe simplification is achieved b~ a 4-masses-model which is shm~n ira Fig. ,~. File geomehic parameters h and L the masses m. the spring c o n s t a R l K, aiR| the dal'npmg c o l ] s l a n t D arc dete~ mined b~ the l"lexpan restllls. W i t h the aid ol | lg. 9 we ilbltlill the et_ltlzl/ious ol motion of the model m~ -l)l), fkq~ mq, • 1942 + / x q 2 1",(3. 5. Block diagram of the total s p a c e c r a f t silnulation I. 2.5, 7. I 1 coordinatc transformations : 3 : al]tenlla : 4: ztnlelma fine pointing system: 6: central body: 8: solar generator orientation mechanism; 9: solar generator: I0: sensor system: 12 : pulse stagger logic: 13 :.jets.
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and from there the ll]OI]lClllS about axes ~"a n d ./ possible inputs coming from tile neighboLu blocks, lb~ the 'smallness" assumptions cannot be completely checked xia ~l]c simulation based upon them. No simplifications must be made for the control logics. "file blocks of the simulation diagram are presented in the I'ollo~ ing. I. Central body. It can be considered as rigid and is expressed by' the Euler equations realized on the analog computer. In the normal phase the minimal angular acceleration will be 9 x l0 '~ sec 2. leading to a minimal acceleration term of the body of 3 x 10 ONm. With a maximal velocity of l.5 ×10 Ssec 'lhe maximal product term o f t h e b o d y i s 5 × 1 0 u Nm. the ratio ofthe terms is 1.7 x 10 2. Generally this ratio will be smaller. So with sufficient acctnacy the product terms can be neglected. 2. Solar ,Acncralor oriclltalion nlechutti.',ni. The solar generatoi is rotated about body axis Y by angular steps A~p = ~,)t~rv, here ~ is the time betv,,een succeeding step:, and ,)t: is the earth rotalh,l/ veloci D. Stator and rotor of tile step-motor are COlmectcd ~sith central body and solar generator axis respecti\ ely [Fig 0f The simulation model of the mechanism is realized ,aith thc aid of all electronic pulse genefator. 3. Solar gem:tutor. The dynamic behaxiour of thc SOil1 generator has been studied with the aid of a special computer p r o g r a m "Flexpan" based on the analytical method ,a.hich is explained [n t31. Will| this nonlinear program 16 mode shapes and natural frequcncic,~ ha~c been studied. In normal naode Ollb. 3 of these mode shapes can bc brought about by the on board actuators, its there are the jets and the solar generator orieutation drive. The other shapes can o n h be excited by' special aCttlator arrangements 'ahtch are not a,,aihlble on board, some of tbem being ~ery sophisticated. Figure 7 left sho,as 3 possible mode shapes, Fig. 7 right 3 examples of mode shapes not excitable with c,n board means. If the complex p r o g r a m glexpan v, ouM bc used its a part o f t h c comf, lete satellite simtllation program, onl? the 3 mode shapes of Fig. 7 left could be brought about b) the torero] actuator models. Therefl~re a simplified model is establisiaed carrying lhc follov, ing l'catures: The model is linear, for the ['lcxpan result:, pioxcd linea, behaviour of the solar generator The model h a s t h c ~ possible mode shapes of the }:lexpan pr~gram.
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As damping is ver', small, a time constalll abo0t 100 sec. tile a bin. c equations can be simplified by setting D = 0 ~hich makes the judgement of the resuh~, more pessimistic. Solar generator oscillations about axes ~ and J are excited by' the coarse jct control system, about ~" b', the orientation m e c h a n i s m As the solar panels are much less llexible about q than about ,.'. the p r o g r a m Flexpan as ~ell as the 4-masses-model assume tile panels to be rigid about q. The corase control system cannot produce resonance oscilhttions because the average time between t~,o impulses is ,.~ith 1000 5000see much greater than the damping time constant of 100 sec and far away from ~.lny natural frequenc',,. Moreoxer the pulse sequence is random. "File sokn- generator steps hov, ever [na? produce resonallces. lheir frequenc 3 must be greater than 0.15 Hz in order to prexent mlolerable attitude errors b,, 1oo g r e a t steps. So care has w be - laken that the step frequent 5 does i]ot coincide with one of the solar generators natural frequencies lying bet,leon 0.1 and t).2H/. An analytical considcrathm ot fllesc ,~scillations is presented hi a sepatat¢ chapter 4. /IITlOlHU laricllhHiotl t~1,9{haHi.'ml. Jt col]sJsts oJ a ~v~.o-hamc cardan bearing the frames of,ahich are rotated about - ! v, ith a constant angularxelocily of 0.01 s e c b y electrical motols. The l o v : e s t nalLlral freqnency being 486 Hz, it can bc considered as rigid. 5. Anlenllu. The lo,hest frequency of the antenna ts more than 20Hz. Therefore it can be considered as rigid and expressed bx the Euler equations. Their cross products can be neglected. 6. ('ontro[ xl.s[cltl /of t'odl'~,< , ol'idtlldlioll. ( ' c n u a l
bodv
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soh.u" generator are controlled at * I resp.~ 2 by using.jets as torquers only. 1 he control logic V~lth staggered pulsesL21 used here prm, ides quasi optimal behaviour concerning fuel consumption and jet switching frequency, and adapts to changing perturbation moments. The logic which needs no angu[ar late information, selects the impulses to be sv. itehed Otlt Of :t II/lmbcr
Brief Paper
599
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FIG. 7. Solar generator mode shapes, lelt: excitable with on board means; right: not excitable with on board means.
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600
Brief Paper
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tq(,. 10. Pulse stagger electronic for coarse control with jets. 1 : scnsol signal: 2: rehLv characteristics: 3: realization (fl'lomca] c o n d i t i o n s 4 : j e t steering signals: 5: d c m u h i p l e x e r : 6: stagger coLiia|el-: 7: multiplexer: g: oscillator: 9: frequency d i x i d e )
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The body oscillations are inputs of the sensors and act as m)b,e h/r the COlatrol system. This system d e m a n d s lhat the nol>,c a m p l i t u d e s arc not greater than a p p r o x i m a t c l > 1 5 ~fi' the admissible a n g u l a r error. As will bc shown in a later chapter the body oscillation a m p l i t u d e s can rise up to 0.015 . "[ his v, ould bc v, ilhin the limi! if the d e a d z o n c of the pulsc stagger c o m r o l systcm ,a ould be chosen 4 0. I . So it seems possible to control the flexible satellite with a onestage jet control only. However there is practicall 3 no margin. ] h e r e f o r e a + ft.1 line COlllrol system is established as a secoml stage control v, hile a first stage coarse control i~, acificxcd with a ± 1 resp. + 2 deadzone. For Sillltllation tile c,.',lqI)~l elcclrvmic is ~m Imc ~ i l h the a n a l o g c o m p u t e r 7 Antupmaji'n,.,pointing,X!'stunr A rcla} conHol s ) s l c m with d e a d / o n e and hxslcrcsis (Fig. 12) sv, itches lilt lllOIOf, it" tile
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lhrce axes el conirol air-bearing simulation resuh,,.
FIt, I ~ A n l e l m a l m e p ( m a t i n g l o g l , ~ of availablc pulses 'qaggered a c c o r d i n g to a gcomctric series, and switches them at 2 discrete positive and 2 negalive a n g u l a r lexels. Ten logical statements are lhe basis of the pulse selection slralegy. Eleclronic s i m u l a l i o n h a r d w a r e is achiexed ,Mth digital 1(% (Fig. l()). but digita.I processors also could bc used Figure 11 shov,,s results of rigid body air-bearing sin> uhltions[5] tlSitlg original sensors, 0.5 N jets. and the electronius of |rib!. 10. |:Of tile SillatllatiolaS the ratio oJ" pel-Iul-bati(dl illomelll thl'ough inertia 111o111e11[was gOllaC orders greater tila.i1 has to be expected for the orbit case. This leads to c o m p a r a t i v e l y very sholl limit cycle pcriods for dae simul-ltion. In orbit, periods of 1000 5000 sec ha',c to bc expected. For two axes a control acctlr4c~ Of +0.001 was uchicved. This is also the limit of the ,,Cl>,~l acCtlYaCy. These simuh.Hi,.',ns dem,.msmtte that for a rigid satellite a one stage attitude c,,'mlrol '.; ith the pulse stagger logic would proc/Hc suflicierd accuracy and no fine a.diusting of the a n t e n n a would be necessary. Things change however with the presence ol'l]cxibilil).
angular error r e a c h e s t h c d e a d z o n c l i m i t of 0.05 to 0.ON . I'he m,.',t()r sv~ivels Ihe a n t e n n a with an uncontrolled x eh>cit3 o | about O.01 scc until Ihe error is zero. The lilac control h,gic i., realized for simuhttion by electronic hardy, arc. Fhe simuJati,:m results oblailted with the ahoxc dcsc:ibed blocks are presented m a later chapter. tlldlyli(a[ , ,,*l.sideralitnl ffi HIv s
The step m o t o r ol lhe solar g e n e r a t o r o r i e n l a l i o n mechanism I'ofccs a difl'crence between central body angle (,0 and solar getleral,.',r shaft ala,~lc q)' according 1o a step functi,.;n !|'iS. 6) to(t)
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lnlnledJatcJ~ a]'tcr o r e Ao-slep the s o | a t gCllCla[oI bar IN distorted accc, r d i n - to curve d of I'i,g. N. In this fhst tn,.',ment the
Brief Paper
601
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1 FIG. 13. Generation of amplitude modulated body oscillations.
central body and the masses have the same position as before because of their inertia. Then they begin to move, and as the inertia moment of the solar generator 4 mb 2 is greater than that of the central body 0> the body oscillations are correspondingly greater. Due to damping they come to rest according to curve c, which shows the relative position of body and generator only. In a space fixed coordinate system the solar generator makes a step much smaller than A~, while the body realizes a step little smaller than Aq) in opposite direction, the difference being Aq). The motion of the body is the most interesting one because it carries the antenna to be orientated. As the motion is symmetric about ~, we have q~ =q3 = -r/2 = - ~1,*= r/• Introducing this with equation (7)into equations 11) to (5) under consideration of the reacting moment M; - 0:4i, and with m, = m, D - O, we obtain the answer upon one Aq)-step 4 m h "-Aq?
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~#(t)=
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FIG. 14. Polygon evaluation of oscillation amplitudes.
n vectors close a regular polygon Iollowing the relation
O.m
T
n,%=m'2n=tl ,)o = natural frequency
2n
where m is the smallest integer for which n is integer. From this the number of polygon corners
O~A~p
q)'(t)=
(1 - c o s (~ot)
(9)
4rob z + O;
r(I
n =m From equations (8) and (9) results ( D ' ( t ) - ( p ( t ) = A ~ p in accordance with equation (7). Now a step series is considered. In Fig. 13 ~,0~ to q% are the responses upon the first to the fifth orientation step which follow with a delay r upon each other and havingamplitudes oh. For this example r r 0 = 3/4 where z0 is the natural oscillation period. The resulting curve (p is amplitude modulated because with the beginning of~0a the sum of(p~ to ~., is constant 4+. q~3 cancels q)t and ~pa cancels q):. so that with q% the motion repeats. The vector representation of Fig. 13 clarifies this fact. The result of the first Aq)-step is vector al rotating with ")0- After r seconds the next step brings a2. Meanwhile a, has passed 3/4 of a full rotation. The resultant is r2 while the third step results in a3 and the resultant r3. The resultant after the fourth step is zero. Generally the angular difference between succeeding vectors (Fig. 14) is ,%=
(11)
T0
r
ra
2re
1101
(12) r
The resultants ~0~ are obtained from the polygon by connecting one corner with all the others. The nth resultant is zero. This means that n is the number of steps of one amplitude modulated period. The maximal resultant of the polygon is
+ (13) =
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ao
si-n
4
Taking from equation (8) the amplitudc 4m]~?2
0=4rob 2 + O: &o
(14)
and introducing equations (1(11 and (14) into (13) under consideration of A(p = ~oer, tile maximal resultant 0,. is obtained
602
Brief Paper
as f u n c t i o n OfT r u
The response of the system to a step velocity of magnitude ~'h~has an oscillation component of the same magnitude as expressed by equation {17). This amplitude can be considered as a limit ,,; hich cannot be crossed to lower values independent of ~,,hcthcr the solar generator orientation control is linear o r n,mlmcar
ror
4robe
~ m ~ 4 m h 2 + O LI
115) sin 7r O)E i o
S i m u l a t i o n resull.s
This function, which is shown in Fig. 15, has singularities at r/r0 = v (v = 1,2 .... ). For these resonance points the polygon angles are 0o = v "2z, and the polygon vectors are all in one line, so that the polygon diameter becomes infinity. For r changing from zero to infinity and for constant ro we have an infinity of resonance points for the multistep excitation. The resonances can also be interpreted with the aid of the Fourier series representation of the muhistep input with period z. which has a component in the resonance frequency ~oo giving rise to a resonance in the system, if r/r 0 is a positive integer. The classical resonance curve for the same plant with sinusoidal input goes to zero with step period r, while for multistep input the resonance curve goes to a finite wdue.
17 V
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1"he simulation was based on the block diagranl of t"% 5 and exccuted by analog computer using clcctronic hardware I'or the COaISE c o n t r o l systcln, IhE fine COlltl-l~[ syMeln, a i m Ihe s o l a r generator orientation mechanism. Figures 16 to 2{) s]lov, lhe normal phase resuhs for the critical #-axis about which tile solar generator is rotated and which brings the grealest oscillation amplitudes. Open Ioo1~ .qmututions. The analytical considEralion', brought lhe result that the solar generator adjusthlg procedure is a nn~re severe source of oscillations than thc jet impulses of the coal sc control system. In order to substantiate the analysis the sohngenerator step period r was changed from r . . I scc n~ r = 2% - 10.28 sec. kigures 16 and 17 show the central body respol>e ~p rcveahng that the oscillations are amplitude modulated, very well to realize in curves 10, 1 I, and 17, with resonances at % and 2q,. Also the quantitative coincidence with the analysis is good. In Fig. 17. bottom right, there is a plot of the maximal amplitudes over the step frequency obtained from the curves of Figs 16 and 17. Here we find a good coincidence with Fig. 15. file curves of Figs 16 and 17 have a mean rate ol angular change of 0.00312 /sec for the central body. The solar generator shaft motion ~p', shown in Fig. 18, has a mean velocity, of 0.00105 sec. So the mean relative velocity between solar generator and body is equal to the difference 0.00417 sec of these values, which corresponds to the earth rotation velocity. F'igure 18 shows a group of dynamic magnitudes for the chosen step period of 2.266sec. The solar generator shaft motion O' is very rugged due to the steps, but the long panels smooth this n l o t i o n ;-IS call be seen from the mass deviations % The step magnitude was 0.00946 .
40
fir o
FIG. 15. Maximal oscillation amplitude as function of step period.
D •
~
With fine control deadzone limits of_+0.08 , maximal oscillation amplitudes ~m<0.03 can be tolerated. This leads to a ratio ( a ~ / r o R < O . O 0 7 3 /sec for ro=5.188sec and R = 4 mbZ/(4 mb 2 + 0c) = 0.748. Figure 15 shows that the condition ~b,, < 0.03 is met within the ranges 0 < r/r0 < 0.85 and 1.22 < r / % < 1.65. A value of r/r0 =0.44 has been chosen, which secures a good safety margin. The result is a maximal amplitude of 0.007 . A further decrease of r/'c 0 does not bring considerably smaller amplitudes. Changes of the solar generator adjusting frequency o f - 50%,+ ~c % or changes of its natural frequency of_+ 100% keep the oscillation amplitudes within the tolerated range of _<0.03. Figure 15 reveals another point of interest. A decrease of step periods z near zero or a step frequency near infinity which is accompanied by step angles near zero, doesn't bring the oscillation amplitudes below 0.00515. Generally this amplitude is 419l[) 2
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1161
In steady state the amplitude modulation of the oscillations disappears for finite damping• The amplitudes become constant as has been proven in [6] for a similar case, but they remain smaller than the maximal amplitudes for the undamped case. In the case of very small but nonzero linear damping the amplitudes become half of the maximal amplitudes of the undamped case. Correspondingly the limit amplitude for very small damping and for z.'ro-,O is
!
t
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- .
5,124 s e c
i
4rob 2 lira
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-
l'n~. 16. Central body' oscillation for'r ~ r,,.
Brief Paper
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,~15 I=7 s ~
16
4 5 sec.
I\1I'\i
0,01° I'- 4
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.... r AA
4 I000
0
/f +
I 2000
l 3000
FIG. 19. Large scale motion of solar generator, central body, and antenna in normal mode.
10,282 sec I ] { l l ! J~
esonance c~urve ob]tained from ,
s,'Tul°tI°n1! 5 _~o--
FIG. 17. Central body oscillations for :0 < : < 2:0-
1 : : 2 , 2 6 6 sec
±
q~
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,
20
30
t LO
50
60
70 sec
FIG. 20. Small scale motion of solar generator, central body, and antenna in normal mode.
604
Brief Paper
:~lolJotl ql s~J[ar gett~'ralor, hody aml ~ltllt'llll~l about .~ i*7 closed loop. Figure 19 shows in small scale tile motion of tile three satellite parts mo~ing against each other about tile most critical axis ,£. which lies parallel to body axis I,'. "File angles ~,~. ~?, and +?l of solar generator, body, and alllCnlla are measured against axis Z~ p o e / r a g to tile center of earth. The perturbation moment arisiug mainly fronl solar radiation is practically constant I\~r tile presented period. Therefore the molten of the bed 3 is composed of parabolas. <,0as well :isl eonlalU oscillations with amplitudes<0.01 Jor tile chosen solar generator step Irequeuc}. lhese oscillations alC so small that tile) cannot be shov, n in Fig. 17. In larger scale the) are c o m m g m sight in Fig. 20. If tile body angle of l-ig. 19 exceeds the error limits o f + I , let impulses J~.d,_ are given throwing back the body into thc deadzone. Time betweeu two impulses lies between 1000 and 5000 sec. Between two impulses body aud solar generator can be considered as to be in open loop. The antenna angle tp4 is kept inside the deadzone linlits el' q 0.08 with a margin of 0.02 againsl tile permitted error of f 0. I . This is made possible by two degrees of antenna fi'eedom o f f 3 against tile body. The antenna follows tile body motion ~ itll a constant angular difference nntil it meets a deadzone limit. Here tin antenna reorientation process is initiated runnhlg with eotlstant speed to t i l e / e r e - l e e , thus shil'tin# the alltcnlla a g a i u s t
tile b e d ) by a small augle. For reasons of scale tllcsc pt-ocessc> appear as vertical lines in Fig. 19, Tile antenna fine control, which ix directly mitiated by the fine sensor, works without using information about tile body attitude as i1"the body were a stable platform s u p p o r t m g the anlenna. This consideration is justified because body and solar generator are movhlg for some 0.001 only duriug a fine control process while the antenna is i-enlovcd t',) approximately O. 1 . So body and solar g e n e r a t o r a r c relaiivel\ at rest compared with tile alltenua. {rod no infornlation about the body attitude is needed for fine control. The solar generator set angle ~2~ is turned about the body set angle
Pergamon Press, 1977.
Printed in Great Britain
Brief Paper Two Level Attitude Control for a Television and Broadcasting Satellite* KLAUS BECKER? + Attitude control; computer application; models; on-line operation; on off control; satellites, artificial; space vehicles: flexible systems. Key Word Index
body. The planes A, B, C represent the antenna cardan bearing. The antenna is fixed on plane A, plane C is fixed on the central body. Plane B is connected with plane A (axis x) and with plane C (axis y ) by joints and electrical actuators. The attitude of planes A and C wilh respect to each other and thus the attitude of the antenna with respect to the central body is determined by the angles A~t,x and A~py which can be easily obtained from the rotation mechanism. The desired attitude of the antenna coordinate system x, y, z with respect to the desired attitude of the body coordinate system X, Y,, Z is determined by the angles a,/3, ~/. Plane C is turned about these angles with respect to body plane D. This is clarified by the planes I, II, III, IV, which are accordingly rotated, where I coincides with C, and IV with body plane D. The solar generator, not shown here, is rotated about Y. For kq, x=kq~y=0 the body has its desired attitude if the antenna has its desired position, too. If the antenna remains in its desired position, then the attitude of the body axis - Z against its desired position - Z s is defined by A~0~and Aq~r. If the antenna is rotated about 3~pxand '~Pr against its desired coordinate system, then the body attitude is defined by
Summary -A two level method for the attitude control of a flexible communication satellite is described. Because of its great inertia the satellite is used as a platform for an antenna fine pointing control system while the satellite body is coarse controlled by jets. A simulation method is used which allows change of a satellite part without changing any other. Complex dynamic models are simplified while keeping the important characteristics. Simulation results are shown. Special forms of oscillations and resonance curves are analytically substantiated. The oscillations prove to be not dangerous. Changes of the solar generator natural frequency are not critical. Introduction THE ATTITUDEcontrol system to be described in the following is intended for a communication satellite§ in a synchronous orbit the antenna of which has to be oriented at ± 0.1 • in pitch and roll and a t _ + 2 in yaw tsee [1]). The required power implies a great solar generator of 24 m of length. This makes the satellite very flexible and the control system has to take account of it. It was intended to use a pure jet control system [2] which showed good properties for the control of rigid bodies. The oscillations to be expected for this project however gave reason to fear that the accuracy o f + 0 . 1 could not be guaranteed with a one-stagecontrol. The control system described here makes use of the fact that the pointing accuracy is much less for the solar generator and satellite body than for the antenna. Therefore body and solar generator are coarse controlled by gas jets as torquers only with an accuracy of_+ 1 resp.-+2L They constitute a kind of platform for the antenna because of their great inertia moment and move very slowly with <0.00l°/sec within the limit cycle range. So no m o m e n t u m storage is needed. The inertia moments about the generator rotation axis e.g. are 569 m2kg for the body, 1687 mZkg for the solar generator and 10 mZkg for the antenna. The antenna is swivelled in pitch and roll against body and solar generator by electrical servo motors and is thus fine controlled with an angular error of_+0.1 '. The solar generator is rotated about 360 ~ within 24 hr against the satellite body. Other problems arise for the simulation of such a complex system. The computer program for only the flexible solar generator[3] is so large that the addition of further satellite parts would lead to impracticable computer times. Universal programs[4] are not useful for the very special control logic applied. Therefore electronic hardware is used for simulation together with an analog computer.
A~pxn= Aq~x + 6~px A~0rn = A~py+ 6¢pr. As the antenna pointing errors 6q~x and 6q~r are small compared with the maximal values of A~0x and Aq~, the antenna attitude about x and y can be used as reference for the central body, because the antenna can be considered to have the desired attitude of the body. The block diagram of the control system is shown in Fig. 3. The logic steering the gas jets gets its signals Aq~x,Aq~r from the antenna position and the z-signal 3~0~from a monopulse sensor. The latter also procures the signals 6~ox and 3q~r for the a n t e n n a fine pointing logic working upon two electrical servo motors. Simulation of the motion of antenna, central body, and solar generator The principle of simulation is based on a u t o n o m o u s blocks for all relevant satellite parts, which get their inputs, moments M i and angles ~p~, as a kind of four terminal network from the neighbour blocks and give back reacting outputs to the same neighbours (Fig. 4). This procedure has the advantage that a block can be studied separately and is exchangeable if due to new facts and perceptions a different mathematical description seems favourable. Although this is not a new approach for simulation, many programs do not allow the exchange of parts because their inputs and outputs, especially the reacting outputs, are not exphcitly accessible. Figure 5 shows the simulation block diagram. All its important blocks have already been separately studied in detail, e.g. by special digital computer programs. These complex programs cannot all be used for the total system simulation, because this would lead to impracticable computer times and possibly to intolerable errors. Therefore simplifications of the complex block models have to be made under keeping the important characteristics. These simplificalions, which are discussed in this chapter, are based on the following considerations: a. Each block has no different properties in the system than it has individually b. Properties of a block which cannot be brought about by other blocks, can be omitted. Therefore all effects possible of a block are individually studied and simplifications are thoroughly checked with regard to the
Angular relations and block diagram Figure 2 shows the angular relations, x , y , z is the antenna coordinate system, X, Y, Z the coordinate system of the central
*Received 22 December 1976; revised 18 May 1977. The original version of this paper was presented at the 7th IFAC Symposium on Automatic Control in Space which was held in Rottach-Egern, Federal Republic of G e r m a n y during May 1976. The published Prodeedings of this IFAC Meeting may be ordered from: Federation IBRA-BIRA, Rue Ravenstein, 3, B1000 Brussels, Belgium. This paper was recommended for publication in revised form by associate editor D. Tabak. tERNO-Raumfahrttechnik, Bremen, Huenefeldstral3e, now with Fachhochschule Hamburg, Berliner Tor 21, FRG. ++Present address: Delbrfickstr. 14, 28 Bremen, Federal Republic of Germany. §(Project HLS of the German Government, Fig. 1, HLS = German abbreviation for High Power Satellite). 595
596
Brief Paper
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s
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l, Explosive ~icw o f H LS ~ i t h solar generator rolled tip,
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Brief Paper
597
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FIG. 3. Control system block diagram, l: Fine pointing logic; 2: antenna servo motors; 3: antenna; 4: monopulse sensor pitch, roll: 5: monopulse sensor yaw; 6: antenna position sensors; 7: transformation matrix; 8: pulse stagger logic: 9: jets; 10: spacecraft: 11: backup sun sensor; 12: backup earth sensor.
M
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-->. FIG. 4. Principle of simulation by autonomous parts.
598
Brief Paper Central Body
Solar
Generator
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FI(L 6. Principle ol sohlr generator orientalhm.
The quantitative coincidence \\ith Flexpan is good. lhe simplification is achieved b~ a 4-masses-model which is shm~n ira Fig. ,~. File geomehic parameters h and L the masses m. the spring c o n s t a R l K, aiR| the dal'npmg c o l ] s l a n t D arc dete~ mined b~ the l"lexpan restllls. W i t h the aid ol | lg. 9 we ilbltlill the et_ltlzl/ious ol motion of the model m~ -l)l), fkq~ mq, • 1942 + / x q 2 1",(3. 5. Block diagram of the total s p a c e c r a f t silnulation I. 2.5, 7. I 1 coordinatc transformations : 3 : al]tenlla : 4: ztnlelma fine pointing system: 6: central body: 8: solar generator orientation mechanism; 9: solar generator: I0: sensor system: 12 : pulse stagger logic: 13 :.jets.
l)(hd, I)I
mij~ + l)t).~-f Kq~
t,~:'~
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and from there the ll]OI]lClllS about axes ~"a n d ./ possible inputs coming from tile neighboLu blocks, lb~ the 'smallness" assumptions cannot be completely checked xia ~l]c simulation based upon them. No simplifications must be made for the control logics. "file blocks of the simulation diagram are presented in the I'ollo~ ing. I. Central body. It can be considered as rigid and is expressed by' the Euler equations realized on the analog computer. In the normal phase the minimal angular acceleration will be 9 x l0 '~ sec 2. leading to a minimal acceleration term of the body of 3 x 10 ONm. With a maximal velocity of l.5 ×10 Ssec 'lhe maximal product term o f t h e b o d y i s 5 × 1 0 u Nm. the ratio ofthe terms is 1.7 x 10 2. Generally this ratio will be smaller. So with sufficient acctnacy the product terms can be neglected. 2. Solar ,Acncralor oriclltalion nlechutti.',ni. The solar generatoi is rotated about body axis Y by angular steps A~p = ~,)t~rv, here ~ is the time betv,,een succeeding step:, and ,)t: is the earth rotalh,l/ veloci D. Stator and rotor of tile step-motor are COlmectcd ~sith central body and solar generator axis respecti\ ely [Fig 0f The simulation model of the mechanism is realized ,aith thc aid of all electronic pulse genefator. 3. Solar gem:tutor. The dynamic behaxiour of thc SOil1 generator has been studied with the aid of a special computer p r o g r a m "Flexpan" based on the analytical method ,a.hich is explained [n t31. Will| this nonlinear program 16 mode shapes and natural frequcncic,~ ha~c been studied. In normal naode Ollb. 3 of these mode shapes can bc brought about by the on board actuators, its there are the jets and the solar generator orieutation drive. The other shapes can o n h be excited by' special aCttlator arrangements 'ahtch are not a,,aihlble on board, some of tbem being ~ery sophisticated. Figure 7 left sho,as 3 possible mode shapes, Fig. 7 right 3 examples of mode shapes not excitable with c,n board means. If the complex p r o g r a m glexpan v, ouM bc used its a part o f t h c comf, lete satellite simtllation program, onl? the 3 mode shapes of Fig. 7 left could be brought about b) the torero] actuator models. Therefl~re a simplified model is establisiaed carrying lhc follov, ing l'catures: The model is linear, for the ['lcxpan result:, pioxcd linea, behaviour of the solar generator The model h a s t h c ~ possible mode shapes of the }:lexpan pr~gram.
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q'~ +i/2
q~ ! q~)
151
4
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~)~ q4'
l("t
J
As damping is ver', small, a time constalll abo0t 100 sec. tile a bin. c equations can be simplified by setting D = 0 ~hich makes the judgement of the resuh~, more pessimistic. Solar generator oscillations about axes ~ and J are excited by' the coarse jct control system, about ~" b', the orientation m e c h a n i s m As the solar panels are much less llexible about q than about ,.'. the p r o g r a m Flexpan as ~ell as the 4-masses-model assume tile panels to be rigid about q. The corase control system cannot produce resonance oscilhttions because the average time between t~,o impulses is ,.~ith 1000 5000see much greater than the damping time constant of 100 sec and far away from ~.lny natural frequenc',,. Moreoxer the pulse sequence is random. "File sokn- generator steps hov, ever [na? produce resonallces. lheir frequenc 3 must be greater than 0.15 Hz in order to prexent mlolerable attitude errors b,, 1oo g r e a t steps. So care has w be - laken that the step frequent 5 does i]ot coincide with one of the solar generators natural frequencies lying bet,leon 0.1 and t).2H/. An analytical considcrathm ot fllesc ,~scillations is presented hi a sepatat¢ chapter 4. /IITlOlHU laricllhHiotl t~1,9{haHi.'ml. Jt col]sJsts oJ a ~v~.o-hamc cardan bearing the frames of,ahich are rotated about - ! v, ith a constant angularxelocily of 0.01 s e c b y electrical motols. The l o v : e s t nalLlral freqnency being 486 Hz, it can bc considered as rigid. 5. Anlenllu. The lo,hest frequency of the antenna ts more than 20Hz. Therefore it can be considered as rigid and expressed bx the Euler equations. Their cross products can be neglected. 6. ('ontro[ xl.s[cltl /of t'odl'~,< , ol'idtlldlioll. ( ' c n u a l
bodv
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soh.u" generator are controlled at * I resp.~ 2 by using.jets as torquers only. 1 he control logic V~lth staggered pulsesL21 used here prm, ides quasi optimal behaviour concerning fuel consumption and jet switching frequency, and adapts to changing perturbation moments. The logic which needs no angu[ar late information, selects the impulses to be sv. itehed Otlt Of :t II/lmbcr
Brief Paper
599
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FIG. 7. Solar generator mode shapes, lelt: excitable with on board means; right: not excitable with on board means.
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FIG. 8. 4-Masses-model of the solar generator.
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600
Brief Paper
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tq(,. 10. Pulse stagger electronic for coarse control with jets. 1 : scnsol signal: 2: rehLv characteristics: 3: realization (fl'lomca] c o n d i t i o n s 4 : j e t steering signals: 5: d c m u h i p l e x e r : 6: stagger coLiia|el-: 7: multiplexer: g: oscillator: 9: frequency d i x i d e )
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The body oscillations are inputs of the sensors and act as m)b,e h/r the COlatrol system. This system d e m a n d s lhat the nol>,c a m p l i t u d e s arc not greater than a p p r o x i m a t c l > 1 5 ~fi' the admissible a n g u l a r error. As will bc shown in a later chapter the body oscillation a m p l i t u d e s can rise up to 0.015 . "[ his v, ould bc v, ilhin the limi! if the d e a d z o n c of the pulsc stagger c o m r o l systcm ,a ould be chosen 4 0. I . So it seems possible to control the flexible satellite with a onestage jet control only. However there is practicall 3 no margin. ] h e r e f o r e a + ft.1 line COlllrol system is established as a secoml stage control v, hile a first stage coarse control i~, acificxcd with a ± 1 resp. + 2 deadzone. For Sillltllation tile c,.',lqI)~l elcclrvmic is ~m Imc ~ i l h the a n a l o g c o m p u t e r 7 Antupmaji'n,.,pointing,X!'stunr A rcla} conHol s ) s l c m with d e a d / o n e and hxslcrcsis (Fig. 12) sv, itches lilt lllOIOf, it" tile
~
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v
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lhrce axes el conirol air-bearing simulation resuh,,.
FIt, I ~ A n l e l m a l m e p ( m a t i n g l o g l , ~ of availablc pulses 'qaggered a c c o r d i n g to a gcomctric series, and switches them at 2 discrete positive and 2 negalive a n g u l a r lexels. Ten logical statements are lhe basis of the pulse selection slralegy. Eleclronic s i m u l a l i o n h a r d w a r e is achiexed ,Mth digital 1(% (Fig. l()). but digita.I processors also could bc used Figure 11 shov,,s results of rigid body air-bearing sin> uhltions[5] tlSitlg original sensors, 0.5 N jets. and the electronius of |rib!. 10. |:Of tile SillatllatiolaS the ratio oJ" pel-Iul-bati(dl illomelll thl'ough inertia 111o111e11[was gOllaC orders greater tila.i1 has to be expected for the orbit case. This leads to c o m p a r a t i v e l y very sholl limit cycle pcriods for dae simul-ltion. In orbit, periods of 1000 5000 sec ha',c to bc expected. For two axes a control acctlr4c~ Of +0.001 was uchicved. This is also the limit of the ,,Cl>,~l acCtlYaCy. These simuh.Hi,.',ns dem,.msmtte that for a rigid satellite a one stage attitude c,,'mlrol '.; ith the pulse stagger logic would proc/Hc suflicierd accuracy and no fine a.diusting of the a n t e n n a would be necessary. Things change however with the presence ol'l]cxibilil).
angular error r e a c h e s t h c d e a d z o n c l i m i t of 0.05 to 0.ON . I'he m,.',t()r sv~ivels Ihe a n t e n n a with an uncontrolled x eh>cit3 o | about O.01 scc until Ihe error is zero. The lilac control h,gic i., realized for simuhttion by electronic hardy, arc. Fhe simuJati,:m results oblailted with the ahoxc dcsc:ibed blocks are presented m a later chapter. tlldlyli(a[ , ,,*l.sideralitnl ffi HIv s
The step m o t o r ol lhe solar g e n e r a t o r o r i e n l a l i o n mechanism I'ofccs a difl'crence between central body angle (,0 and solar getleral,.',r shaft ala,~lc q)' according 1o a step functi,.;n !|'iS. 6) to(t)
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u
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lnlnledJatcJ~ a]'tcr o r e Ao-slep the s o | a t gCllCla[oI bar IN distorted accc, r d i n - to curve d of I'i,g. N. In this fhst tn,.',ment the
Brief Paper
601
0
3
= 4rob2
:1
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1 FIG. 13. Generation of amplitude modulated body oscillations.
central body and the masses have the same position as before because of their inertia. Then they begin to move, and as the inertia moment of the solar generator 4 mb 2 is greater than that of the central body 0> the body oscillations are correspondingly greater. Due to damping they come to rest according to curve c, which shows the relative position of body and generator only. In a space fixed coordinate system the solar generator makes a step much smaller than A~, while the body realizes a step little smaller than Aq) in opposite direction, the difference being Aq). The motion of the body is the most interesting one because it carries the antenna to be orientated. As the motion is symmetric about ~, we have q~ =q3 = -r/2 = - ~1,*= r/• Introducing this with equation (7)into equations 11) to (5) under consideration of the reacting moment M; - 0:4i, and with m, = m, D - O, we obtain the answer upon one Aq)-step 4 m h "-Aq?
4rob 2 + O-
~#(t)=
11 - cos ~,)ot),c')o = K 4rob 2 + t)~
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(8)
o=19 vn= (
f
A
4
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FIG. 14. Polygon evaluation of oscillation amplitudes.
n vectors close a regular polygon Iollowing the relation
O.m
T
n,%=m'2n=tl ,)o = natural frequency
2n
where m is the smallest integer for which n is integer. From this the number of polygon corners
O~A~p
q)'(t)=
(1 - c o s (~ot)
(9)
4rob z + O;
r(I
n =m From equations (8) and (9) results ( D ' ( t ) - ( p ( t ) = A ~ p in accordance with equation (7). Now a step series is considered. In Fig. 13 ~,0~ to q% are the responses upon the first to the fifth orientation step which follow with a delay r upon each other and havingamplitudes oh. For this example r r 0 = 3/4 where z0 is the natural oscillation period. The resulting curve (p is amplitude modulated because with the beginning of~0a the sum of(p~ to ~., is constant 4+. q~3 cancels q)t and ~pa cancels q):. so that with q% the motion repeats. The vector representation of Fig. 13 clarifies this fact. The result of the first Aq)-step is vector al rotating with ")0- After r seconds the next step brings a2. Meanwhile a, has passed 3/4 of a full rotation. The resultant is r2 while the third step results in a3 and the resultant r3. The resultant after the fourth step is zero. Generally the angular difference between succeeding vectors (Fig. 14) is ,%=
(11)
T0
r
ra
2re
1101
(12) r
The resultants ~0~ are obtained from the polygon by connecting one corner with all the others. The nth resultant is zero. This means that n is the number of steps of one amplitude modulated period. The maximal resultant of the polygon is
+ (13) =
~" <
ao
si-n
4
Taking from equation (8) the amplitudc 4m]~?2
0=4rob 2 + O: &o
(14)
and introducing equations (1(11 and (14) into (13) under consideration of A(p = ~oer, tile maximal resultant 0,. is obtained
602
Brief Paper
as f u n c t i o n OfT r u
The response of the system to a step velocity of magnitude ~'h~has an oscillation component of the same magnitude as expressed by equation {17). This amplitude can be considered as a limit ,,; hich cannot be crossed to lower values independent of ~,,hcthcr the solar generator orientation control is linear o r n,mlmcar
ror
4robe
~ m ~ 4 m h 2 + O LI
115) sin 7r O)E i o
S i m u l a t i o n resull.s
This function, which is shown in Fig. 15, has singularities at r/r0 = v (v = 1,2 .... ). For these resonance points the polygon angles are 0o = v "2z, and the polygon vectors are all in one line, so that the polygon diameter becomes infinity. For r changing from zero to infinity and for constant ro we have an infinity of resonance points for the multistep excitation. The resonances can also be interpreted with the aid of the Fourier series representation of the muhistep input with period z. which has a component in the resonance frequency ~oo giving rise to a resonance in the system, if r/r 0 is a positive integer. The classical resonance curve for the same plant with sinusoidal input goes to zero with step period r, while for multistep input the resonance curve goes to a finite wdue.
17 V
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7o x
15 Io
R
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0.8510122
30
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=
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1"he simulation was based on the block diagranl of t"% 5 and exccuted by analog computer using clcctronic hardware I'or the COaISE c o n t r o l systcln, IhE fine COlltl-l~[ syMeln, a i m Ihe s o l a r generator orientation mechanism. Figures 16 to 2{) s]lov, lhe normal phase resuhs for the critical #-axis about which tile solar generator is rotated and which brings the grealest oscillation amplitudes. Open Ioo1~ .qmututions. The analytical considEralion', brought lhe result that the solar generator adjusthlg procedure is a nn~re severe source of oscillations than thc jet impulses of the coal sc control system. In order to substantiate the analysis the sohngenerator step period r was changed from r . . I scc n~ r = 2% - 10.28 sec. kigures 16 and 17 show the central body respol>e ~p rcveahng that the oscillations are amplitude modulated, very well to realize in curves 10, 1 I, and 17, with resonances at % and 2q,. Also the quantitative coincidence with the analysis is good. In Fig. 17. bottom right, there is a plot of the maximal amplitudes over the step frequency obtained from the curves of Figs 16 and 17. Here we find a good coincidence with Fig. 15. file curves of Figs 16 and 17 have a mean rate ol angular change of 0.00312 /sec for the central body. The solar generator shaft motion ~p', shown in Fig. 18, has a mean velocity, of 0.00105 sec. So the mean relative velocity between solar generator and body is equal to the difference 0.00417 sec of these values, which corresponds to the earth rotation velocity. F'igure 18 shows a group of dynamic magnitudes for the chosen step period of 2.266sec. The solar generator shaft motion O' is very rugged due to the steps, but the long panels smooth this n l o t i o n ;-IS call be seen from the mass deviations % The step magnitude was 0.00946 .
40
fir o
FIG. 15. Maximal oscillation amplitude as function of step period.
D •
~
With fine control deadzone limits of_+0.08 , maximal oscillation amplitudes ~m<0.03 can be tolerated. This leads to a ratio ( a ~ / r o R < O . O 0 7 3 /sec for ro=5.188sec and R = 4 mbZ/(4 mb 2 + 0c) = 0.748. Figure 15 shows that the condition ~b,, < 0.03 is met within the ranges 0 < r/r0 < 0.85 and 1.22 < r / % < 1.65. A value of r/r0 =0.44 has been chosen, which secures a good safety margin. The result is a maximal amplitude of 0.007 . A further decrease of r/'c 0 does not bring considerably smaller amplitudes. Changes of the solar generator adjusting frequency o f - 50%,+ ~c % or changes of its natural frequency of_+ 100% keep the oscillation amplitudes within the tolerated range of _<0.03. Figure 15 reveals another point of interest. A decrease of step periods z near zero or a step frequency near infinity which is accompanied by step angles near zero, doesn't bring the oscillation amplitudes below 0.00515. Generally this amplitude is 419l[) 2
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B
:+ Sue, . , , ~
9 ,"
i
TO(UE
4rob 2 + (}.: rr
1161
In steady state the amplitude modulation of the oscillations disappears for finite damping• The amplitudes become constant as has been proven in [6] for a similar case, but they remain smaller than the maximal amplitudes for the undamped case. In the case of very small but nonzero linear damping the amplitudes become half of the maximal amplitudes of the undamped case. Correspondingly the limit amplitude for very small damping and for z.'ro-,O is
!
t
•
- .
5,124 s e c
i
4rob 2 lira
: -,, ~0
d),l=
%~o~:
i
(17} 4mb= +0; 2re
-
-
l'n~. 16. Central body' oscillation for'r ~ r,,.
Brief Paper
603
~13 1&
,~15 I=7 s ~
16
4 5 sec.
I\1I'\i
0,01° I'- 4
J,
.... r AA
4 I000
0
/f +
I 2000
l 3000
FIG. 19. Large scale motion of solar generator, central body, and antenna in normal mode.
10,282 sec I ] { l l ! J~
esonance c~urve ob]tained from ,
s,'Tul°tI°n1! 5 _~o--
FIG. 17. Central body oscillations for :0 < : < 2:0-
1 : : 2 , 2 6 6 sec
±
q~
T ± T
~' % 113~4
M~
J__ T
_t_
T
lmm
lmms-I
1rams-2
-095 ° ~nt
÷01 °
_1 o
0,1Nm
M~
+005'
~0Lqo
[se
-005' _L __
T
0,01°
-01 o
÷-/-
-
~ I0
,
,
20
30
t LO
50
60
70 sec
FIG. 20. Small scale motion of solar generator, central body, and antenna in normal mode.
604
Brief Paper
:~lolJotl ql s~J[ar gett~'ralor, hody aml ~ltllt'llll~l about .~ i*7 closed loop. Figure 19 shows in small scale tile motion of tile three satellite parts mo~ing against each other about tile most critical axis ,£. which lies parallel to body axis I,'. "File angles ~,~. ~?, and +?l of solar generator, body, and alllCnlla are measured against axis Z~ p o e / r a g to tile center of earth. The perturbation moment arisiug mainly fronl solar radiation is practically constant I\~r tile presented period. Therefore the molten of the bed 3 is composed of parabolas. <,0as well :is
tile b e d ) by a small augle. For reasons of scale tllcsc pt-ocessc> appear as vertical lines in Fig. 19, Tile antenna fine control, which ix directly mitiated by the fine sensor, works without using information about tile body attitude as i1"the body were a stable platform s u p p o r t m g the anlenna. This consideration is justified because body and solar generator are movhlg for some 0.001 only duriug a fine control process while the antenna is i-enlovcd t',) approximately O. 1 . So body and solar g e n e r a t o r a r c relaiivel\ at rest compared with tile alltenua. {rod no infornlation about the body attitude is needed for fine control. The solar generator set angle ~2~ is turned about the body set angle
-() by the earth rotation velocit> 0.00417 see. The real geuerator angle ~p' is the addition of a step function appr,.~ximating tile set angle CUI'lC fD~ Oil Olle side and the body angle tO on file other side. So tile solar generator motion has long tcrmparabolas~ithinilsdeadloned~.d o l ' + l similar lo those of lhe bod). Figure 20 sht',\t,,s in [arger scale sinlulatJon resu[is I'or a short period of 70 sec. The angles
0.001 s c c tintil a
let Impulse is gl\en. The declined cur,+e l-epresentmg lhc solar gencralor inotion O' shows this as ~ell :ix the central body moliou t#. Also Ihe antellna angle ~/0t reino\es fFoln the /eltl-elTor line with this vclocity until it meets in 4 the dcadzone limit \~ hich here \
oscillation adds to the steady oscillations caused by the solar generator orieutation steps. Depending on the phase the amplitude can al most be doubled for less then 100 scc. Tl~is \verst c;isc is sho\~n in Fig. 20.
Fhc oscillation anlplitude>, cauised b) line control title < 0.0006 . because tile moment o[" rilomentu111 of the small serve inolor, ~hich has to shift the ailiellna abont I n l m o11l), is <:70.01 N msec. Tile anlenua lnotnent tim is <:::().(ill] N UlSCC.Therefore tile effects of fine control (ill the bed 3 Call be neglected. Altogether Fig. 20 shnws that Ihc oscillations lmrdl 3 :.lfl'ect lhc cotltrol properties, if the solar general o, step freq ueilc+', ix c h o s e n appropriately
( )~llc]llSiolls The investigations of a two level control eonccpt had the FCSII]I dmt the oscillations of the flexible spacecraft arc not dangerous for fine p o e t mg o f + 0. ! . The greatest source of oscillations is the solar geueralor orientation s>stcm+ (Tomparcd to th{s tile jet impulses cite relatively unimportalli. Tile simulatiou restllts, which arc m good coiucidencc ~lth anal),tical considerations, showed thai the desired~ 0.1 pointing accurac) i.sjust the limit I+ora oue stage.jet conlro]. Ito,ac~et there is no safety margin, but + 0.2 should be i-eali/~ible \~ith sufficient margin. Tile two stage control has great safety margins. An aCctlrac 5 i)l 4 0.03 seems possible with the presented concept. ( ' h a n g e of the solar generator orientation principle brings no ad\alllage. because the stepwise t~tientation does not procure more oscillations t h a n a l i n e a r c o n t r o l i f t h e s t c p f r e q u c n c y i x +2
RUIU#'t>tlt'UX [ 1 ] AEG, Dornier System. E R N O Raumfahrtlechnik: Project High Pov+er Satellite. Final Report fin G e r m a n ) i P r o j e k t HLS. Abschlul3bericht ) DFVLR, K61n-Porz {1975). 12] K. BtcKf 14: Attitude control system for long time missions usmg pure .jet control torquing with staggered pulses. 5th If"A (" Svmpo.siurn oft Automatic Control in Space, (ienox a. 4 8 June (1973i. [ 31 R. F~i+KAI and H. Sent? rzti: D y n a m i c behaviour of satellites with large area solar panels, Final Report (in GermanL i Dynamisches Verhaheu yon Satelliten mit grol$1];