OPTIMAL ROCKET THRUST PROFILE SHAPING USING.pdf

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Sponso red by-With Part icloation ol -Am erican Inst i tute of Aeronautics and Astronautics (AIAA)/ Am erican Astronautical Society (AAS)

OP TI MA L ROCKET THRUST P RO FI LE SHAPING USINGTHIRD-DEGREE S PL INE FUNCTION INTERPOLAT IONbyIVAN I,. JOHNSONNASA J o h n so n S p a c e C e n t e rHous ton , Texas

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ANAHEIM, CALIFORNIA / AUGUST 59,1974First publication rights reserved by American Institute of Asronaut;cs and Astronautics.1290 Avenue of the Americas, New York, N. Y . 10019, Abstrscts may be published without

permission if credit is given to author and to AIAA. (Price: AlAA Member$1.50. Nonmember $2.001.Note: This paper available at A lAA New York office for si x months;thereafter, photoprint copies ar e available at photocopy prices fromA lAA Library, 750 3rd Avenue. New York. New York 10017


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OPTIMAL ROCKET THRUST PROF ILE SHA P INGUSING THIRD DEGREE SPLINE FUIICTIOH 1IiTERPOI.ATIONIvan L . JohnsonNational Aeronautics and Space AdministrationNa Lyndon B. Johnson Space CenterHouston, Texas

A b s t r ac tO pti ma l s o l i d - ro ck e t t h ru s t p ro f i l e s fo r t h ep a ra l l e l -b u r n , s o l i d - ro ck e t - a s s i s t ed s pace s h u t t l ea r e i n v e s t i g a te d . S o l i d- r o c k e t t h r u s t p r o f i l e sare s imula ted bv us ino th i rd dearee so l ine func-t i o n s , w i t h the-value; o f the t h s t b rd in ate s de-f ined as parameters .P a r a n a t r i c a l l v , usina th e Davidon-F1 etcher-PowellT h e p ro f i l e s a r e o p t i m i zedperialty funct7on metGod, by minimizing propellantweigh t sub jec t t o s t a t e a n d co n t ro l i n eq u a l i t y con -s t r a i n t s an d terminal boundary con di t io ns. Thisstudy shows that opt imizing a co n t ro l v a r i ab l eparamet r i ca l ly by us ing th i rd degree sp l ine func-t ion i n te rp o la t io n a l lows the con t ro l to be shapeds o t h a t i n e q u a l i ty c o n s t r a i n t s a r e s t r i c t l y a dh er edt o and a l l corne rs are e l imina ted . The absence ofco rn e r s , w hich i s r e a l i s t i c i n n a t u re , makes t h i smethod a t t r ac t i v e from t h e v iew p oi nt . o f s o l i d ro ck e tgra in des ign .

In t ro d u c t i o nFor t h i s s t u d y , t h e s h u t t l e co n t r ac t o r 1 9 73so l id rocket th ru s t p ro f i l e was chosen for compar i-

son purpose^.'^) I t i s emp hasi zed t h a t the purposeo f t h i s p ap e r i s t o present a mathematical techniquef o r o p t im i z in g s o l i d r o c ke t t h r u s t p r o f i l e s ; t h epaper i s not meant a s a comparison t o t h e p r e s e n ts h u t t l e c o n f i gu r a t io n .A major c o n t r i b u t o r t o t h e c o s t p er f l i g h t o fa s h u t t l e - ty p e v e h i c le i s t h e c o s t of t h e s o l i drocket boos ters (SRB), w h i c h a s s i s t t h e o r b i t e r i ni n s n r t i n g a speci f i ed pay load . An impor tan t e le-ment of the total SRB co s t i s t h e co s t i n cu r r ed bythe weigh t o f p r ope l l an t , approx imately $1.20 perpound per f l ig h t . Someti iws a s ig n i f i ca n t reduc-t i o n in SRB propel l an t weigh t and therefore cos ti s rea l i zed by op t imal ly shap ing the SRB t h r u s tprof i l e , even wi th regard t o t h e co n s t r a i n t s i m -posed on the f l igh t .Prev ious SRB&sisted space shu t t l e conf igura-ti o n s were shaped opt im ally by assuming th a t thrust

(vacuum) could be simulated by a sequence o f p i ece -wise cont inuous l in ea r segments . with the values ofthrust (vacuum) a t each junct ion po in t t rea ted asparameters . ( l* ') This type of model ing o r in te r-p o l a t i o n p re s en t s s ev e ra l i n e f f i c i en c i e s or prob-lems. The cor ner s formed a t the jun ctur e poin tsw i t h i n t h e sequence of piecewise l inear segments i na f e a s i b l e t h r u s t p r o f i l e r e p r e s e n t d i s c o n t i n u i t i e si n d e r i v a t i v e s o f t h r u s t . T he se d i n c o n t i n u i t i e sa r e u n d es i r ab l e t o an en g i n ee r t r y i n g t o determinean actual SRB t h r u s t p r o f i l e i n o r d e r t o s i m u l a t eth e sequence of l i ne ar segments. Floreover, the

L.

v

l inear-segments approach cons t ra ins the th rus t be-tween data ooints to a s t r a i a h t l i n e : t h i s o r e v e nti n r e s p ec t i o opt imal ly reduhng SRB.propel i an tw e i qh t , e f f i c i e n t h and li nq o f s t a t e a n d co n t ro l i nequ al i ty c on s t r a in t s on dynamic pressure and ax ia la c c e l e r a t i o n .In refererices 1 a n d 2 and i n t h e s h u t t l e co n -tractor 's solut ion, the maximum dynamic pressurei n e q u a l i t y c o n s t r a i n t

$ 650 ps fwas sa t i s f i ed in t e rms o f dynamic pressure passagethrouah 65 0 o sf a t o n lv o n e i n s t an t . T h is i s b ecathe con s t r a in t was appbx inate d a s a po in t' i nequali t y c o n s t r a i n t a t t h e i n s t a n t dynam ic p r e s s u r ereached i t s maximum and beca use of t h r u s t r a t el i m i t a t i o n s .

References 1 and 2 a n d t h e c o n t r a c t o r t r e a t -m e n t did no t con tend wi th the ax ia l accelera t ionc o n s t r a i n ta, 2 39,

However, use of a s i n g l e s t r a i g h t l i n e f o r t h r u s td u r i n g t h i s t i m e r e s u l t s i n o n l y a s m a l l l o s s i nperformance.T h rus t p ro f i l e s d esi g ned i n th i s manner loseco n s i d e rab l e p e r fo rm n ce i n terms of SRB p r o p e l l a n

weight . On t h e o t h e r h a n d , an opt imal solut ionwould y i e l d a th rus t - t im e curve which , a f t e r en-counter ing a cons t ra in t boundary , m i g h t r id e i tu n t i l some o t h e r co n s t r a i n t i s m et. T h is i s d i f f -i c u l t t o ap p ro x im a te i f p i ecew is e co n ti nu o u s l i n e asegments are used f o r th rus t model ing un less a verlarge nuinber of s eg ments i s ch os en. S t i l l , co rn e rwould probably remain a problem, and the number ofparameters req uire d may be qu it e la rg e . Kelley anOenhaml3) exp la in tha t the use o f cub ic sp l ine funt ions fo r model ing may o f f e r a n at t rac t ive comprombetween number of parameters ( re su l t in g from use omany segments) a n d overa l l smoothness o f the resu li n g co n t ro l . I n add i t ion , because of th e na tureof th i s sequence of t h i r d degree polynomials interp o l a t i n g t h ru s t , en t ry t o and ex i t f ro m in eq u a l i t yco ns t ra in t boundar ies wi l l t end t o be smooth.

This paper p resen t s the resu l t s o f a s tudyt h a t employed third-degree o r cu b i c - s p l i n e fu n c t i oi r : e r p o l : r i m ( 4 ) 1 3 rclLcc S23 prope l l J - . l Meiqht bparamet r i c , ! ly o r r i r r i z i n o rn e S RB t n r . . s t - t i i - e c . .r vTh e parameter opt imizat ion method used i s theD av ido n-F le tch er-P ow ell per.al t y f u n ~ t i o n ( ~ ' ~ Jmethod. Also used was a n e r r o r f u n c t i o n methodt h a t e f f i c i e n t l y s a t i s f i e s i n e q ua l it y c o n s t r a i n t son dynamic pressure a nd ax ia l accelera t ion and de-sign c on st ra in ts on maximum th ru st and th ru st ra te

(8 )


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Reference Tra jeccox

gl:g2:g3: f l i gh t-p ath angle = 0.54313"g4: angle o f in c l ina t ion = 90'hl: a mu nt of OMS propel lant

burned 5 11 904 pounds

a l t i t u d e = 50 n. mi .ve loc i t y = 25 920 fps

1

I n order to .s tudy reductions i n SRB propel lantThe data pointsweight, a reference tr aj ec tor y was generated bysolving a maximum payload problem.for the thrust p ro fi l e were taken from ref. 9 and.interpolated by using third-degree spline funct ions.The resu l t ing curve i s shown i n f ig ure l ( a) . Enginecha racter i st ics and weight prope rt ies are presentedi n the tab le . A f t e r finding the maximum payload,

the payload w i l l then be held constant and thethrust data points w i l l be allowed to vary as para-meters to reduce SR8 prop ell ant.

v

40K payload south polar mission(Western Test Range)( a ) Weights, l b

Booster propellant (nominal) . . . . . 2 829 449Booster dry weight . . . . . . . . . . . . 423 266Boo ster GLOW . . . . . . . . . . . . . 3 252 715Orbiter ascent propellant . . . . . . 1 696 854O rbite r dry weight ( with EOHT) . . . . 269 734OMS and RCS propellant weight . . . . 14 654ASRM . . . . . . . . . . . . . . . . . 99 050Payload (actual) . . . . . . . . . . . 39 418Orbiter GLOW (with ASRM) . . . . . . . 2 119 008T ota l GLOW . . . . . . . . . . . . . . 5 371 723

(b) P ropulsio n (each engine)v Booster

Sea level thrust . . . . . . . . . . 4 642 420Vacuum th ru s t (sax) . . . . . . . . 4 089 110Ispacuum, sec . . . . . . . . . . 273.238

Orb i te rSea level thrust . . . . . . . . . . 375 000Vacuum th ru s t . . . . . . . . . . . 470 000Ispacuum, sec . . . . . . . . . . 450.647

The reference traj ec tory used i n the studysa t is f ie d once-around abort condit ions f o r thepolar mission and included the following events:a . 5-second ver t ica l r iseb. 10-second pi tch ove rc . A t t = 30 seconds the abort solid rocketd.

motors ( ASRM) are dropped.

(max Q) = 630 psf i s reached.

t i on of 2 . 2 3 ~ ~s reached and t a i l o f f begins.

and iterative guidance mode ( I G N ) s t e er i ng i sin i t ia ted .

A t t = 38 seconds maximum dynamic pressure

e . A t t = 116 seconds maximum a x i a l acce le ra-

f. A t t = 128.6 seconds SRB cases are staged--

(1 )

g. A t t = 281 seconds an or b it e r engine i sturned off; the other two engines are brought to ale vel o f 109 percent; and the orb ita l maneuverinsystem (OMS) engines and reation control system TRCS )engines are turned on.h. A t t = 439.1 seconds RC S prope l lan t i sdepleted.i.(T/W) reaches 3 and the orbiter enqines w i l l begin toth ro t t le .

A t t = 625.4 seconds thrus t-to- wei ght r a ti o

j. A t t = 641.5 seconds the o rb it e r in s er ts.The steering a f t e r the p i tchover unt i l s tag ing i sCn = 0 (aerodynamic normal force coefficient).

To simul ate the refere nce tra je c tor y, a maximumpayload op timi za tio n problem was so lved.meter vec tor x had the fo ll ow in g parameters ascomponents:The para-

a. P ayload weightb. Azimuth after ver t ic al r i s ec.. Amount o f 0% iropellant burned after oneorb iter engine outd.( t = 15 sec)Value o f p i tc h a t end of pitchover maneuvere. Value o f p i t ch ang le a f te r f i r s t s tageseparat ion ( t = 128.6 sec)f.(t = 281 sec)g.(t = 439.1 sec)h.( t = 625.4 sec)

Value o f p i tc h a t beginning of abort

Value o f p i tch a t RCS propel lant depletion

Value o f p itc h when T/W reaches 3

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figure 6(a).Q = 1/2pv2 reaches 650 ps f a t t = tl, then an i n -Ztantaneous decrease to a l ev el corresponding toQ = 0.i n th e Q curve. From tl t o t2, while Q = 650 psf,the thr us t i s computed from 6 = 0. The reason f o rt hi s i s that Q li 650 i s a f i r s t -o rder s ta te va r i -able inequa lity constrain t. (10) The con trol,thrust, does not app?ar exp li c it ly i n 9 = 1!2?v2,but appears i n i t s f i r s t de r iva tive i n the v te rm

It cons is ts o f a imaximum th ru s t u n t i l

F igu re 6( b] shows the correspo nding changesv

;1 =$4pv2/2 = 0 (2)From figureThe equation for th is th ro tt l e was derived fo ra s i mi la r s i tu at ion i n reference 11.6(c) , a t t ime t2 the axial accelerat ion reaches

3g0 befo re the thr us t can r i s e back t o the maximumlevel . A t that po int the thrust w i l l begin to de-crease as it r ides the ax ia l acce lerat ion const ra int. . boundary

( T CO S a - D ) l m = 3g0 (3)u n t i l t = t3where ta il of f begins.tl -c t3would have t o be determined by sati s fyi nga i l o f the necessary conditions for an extremum.

For a s ol id rocket booster, the optimal thrus tpr o fi le o f f igure 6(a) suggests the use of two di f-ferent sets rather than a s ing le set o f so l id roc-ke t boosters. Both sets of rocke ts would burn i np a r a l l e l u n t i l time ti, a t which time one s et,having a constant thrust of (Tmax - T*), upon beingspent, would be dropped, while the other set, havingan in i t ia l cons tant th rus t T*, would continue t oburn.

The times

-P jr am er ri ck il y Optimiz>n3J !>r,st Prof-_ -s i n 2 Tnird-?cqree S? l ine rm: io n Interpo ldr ion

I n the design of a s ol id roc ket motor, a.thrust-time curve and an envelope limitation arefurnished. (12)storage requirements and vehicle overall require-ments, are also involved i n the se lec t ion of apropel lant. The to ta l impulse and propel lant weightof the des ire d motor i s determined as a funct ion ofthe thru st- time curve. Based uoon to ta l impulse.

Other factors, such as temperature

pr op el la nt weight, and parameters such as chamberpressure, burnin g rate, surface area, and nozz lethro at area, the desired so li d roc ket motor can bedescribed.on the amunt of prope llan t required, i f selectedef fic ie nt ly (opt imally) , can save a s i gnif ica ntamount of pr op el la nt weigh t and thereby reduce cos t.A comparison of figures l ( a ) and 6(a) revea ls tha tthe reference thrust prof i le i s far from optimal.

The thrust-time curve, an important influence

Problem Definit ioni-he payload was s pe ci fie d as 39 418 pounds,the amount obtained for the reference trajectory.The value o f thrust i n each data po int (22 p o f i t sfor th is case) and the values of the f i r s t der iva-t ive s a t the heginning and end o f the spl ine in ter -pol at ion interva l o f thrust versus t ime were trea tas parameters.+t = 30 seconds to thrust ta i lo f f in i t ia t i on and thetime th at the o ptimal s tee rin g was begun were al s otrea ted as parameters. This brought the to ta l num-ber of parameters to 33, including the ones definedi n the reference case. I n addit ion to the const ra indefined by equation ( l) , the fol lowing constraints

The l eng th o f the burn time from

h2: 5 25 psf

were imposed to keep the optimal s tee rin g from be-ginning too ear ly.A measure o f performance was d ef in ed as th eweight of SRB propel lant , W .

to find the parameters described which l o c al ly mini-mize W while sat isfy ing the constraints (1) and ( 4 )and the inequality constraints on dynamic pressure,a xi a l ac cel eratio n, and maximum thrus t. This wouldeffec t ively shape an ef f i ci en t thrust-t ime curve.P arametr ic T hrust P ro fi le I4odeline

The problem then wasPP

Pmust be known so tha t it can be added in to the to ta lconfiguration weight.t r e a t i t a s a parameter, so tha t i t can be added i nin i t ia l l y , and cons t ra in i t t o be the same as thenumerically integrated value obtained a t staging.This would take care o f the s i tuat io n i n which Ivaries . But fo r the s i tuat io n in which I i s con-sidered constant this i s no t necessary. Fo r theSRB's

Before an ascent simulation can take place, WOne scheme f o r th i s i s t o

S PS P

fi = (106/ ISp)T( t ) (5)and

whereTmx, 0 5 t 5 15

(7)T ( t ) = TSpln, 15 I t 5 ttail' ta i l . ' ta i l 5 t 5 tstage

Thrust during the time of the sp l ine interpo lat ion,Tspln*[ti+,, T(titl)l, ti 2 t 5 titi, i s defined bybetween the two data points [ti, T( t i ) l and

\*- t The second der iva tives c ould have been spec ified instead of the f i rs t; howpver, the thrus t- ti ne curveshaping i s much more s en s iti ve to changes i n the f i r s t derivat ives than i t i s to changes i n the secondderivat ives.3


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T s p l n ( t ) = T ( t i ) t a l i z f a t i z 2 + a3i23;2 -- ( t - t i ) / ( t j f l t i ) ,i = 1,. ..,21 (8 1

The c o e f f i c i e n t s a l i , a Z i , and a3i a r e de terminedby a s e t of eq uat ions based on the use o f a th i rd -Also required i n t h e s p l i n e i n t e r p o l a t i o n a r e t h es l o p es f ( t l ) and f ( t z 2 ) .T t a i l , i s def ined by

d q r c e n a tu r a l s p l i n e f un c ti o n for i n t e r p o l a t i o n . ( 4 )D u r i n g t a i l o f f t h e thrust,

T t a i l ( t ) = T ( t Z 2 ) + bly f b2Y2 f b3y3 (9)where

Y = ( t - t t a j l ) / A t T

The Error Function Modelss r a n s,i nce t h e s t a t e and co n t ro l i n eq u a l i t y con-

s1 = p v 2 / 2 - 650 psf 2 0s2 = T ( t ) - T I Os = ( T cos a - O)/m - 3g0 5 0 (1 3 )3

must he maintained during this t ime, p o i n t inequal -i t y c o n s t r a i n t s w ere d e f in e d a t e ac h i n t e g r a t i o nt ime, t:3 's l j z p ( t j ) v ( t j ) 2 / 2 - 650 5 0s E T ( t j ) - Tmx I O (14)s3 j 1 [ T ( t j ) c 0 S [ a ( t . ) ] - D ( t j ) ] / m ( t j ) - 39, 5 023 3

- t . (10) Error fu n c t i o n s e l , e2 , and e3 are def ined by' s t a g e t a i ln.T h e co e f f i c i en t s b l , b 2 , and b3 were obtained by

m od elin g t h e t a i l o f f e x a c t l y a s t h e r e f e r e n c e t a i l -1

e, = g1 syj u ( s l j )o f f e x c e p t t h a t T ( t Z 2 ) was u sed a s t h e i n i t i a l v a lu eo f thrust r a t h e r t h a n the reference value . j = ln .

e2 = h2C s u(sZj)1b l = -0.367620493 A t Tb2 = 3(0.203743 - b l ) f 0.735240986 A t T (11) j = I

n.e3 =h3C i j ~ ( 5 ~ ~ )b3 = -0.367620493 A t T - 2[0.203743 - T ( t Z 2 ) ] 1- In i t i a l ly , wi thout hav ing been shaped? a t h rus t -t ime curve modeled with equations ( 7 ) t h r o u g h (11)could look l ike the fo l lowing: j=1n i 5 number o f in tegra t ion s t eps

and U O denotes a u n i t s t e p f u n c t i o n .funct ion of the form An e r r o r(16)= e + e1 z t e 3

i s accumulated and added t o t h e p en a l t y fu n c t i o n . (6)Th e t echn ique descr ibed i n r e f e r en ce 8 i s a co n cep tu sed t o h and le s t a t e i n eq u a l i t y co n s t r a i n t s , b u t i ti s v e ry s i m i l ar t o an i n t eg ra l p en a l t y fu n c t i o n :

Then,The e r ro r fu n c t i o n (1 6 ) s a t i s f i e s t h e con -s t r a i n t s ( 1 4 ) i n the same manner t h a t t e rminal an d'1i a 2 i i n t e r m d i a t e b o u n d a r y p o i n t c o n s t r a i n t s a r e s a t i s -

f ie d i n the p en al ty f ~ n c t i o n . ( ~ " ~ )t y f u n c ti o n f o r t h i s c a s e i swp = ( 10 G/I sp ) I f [ T ( t i ) + ~f -j- So, t h e p en a l -i =1

4


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The solut ion for the problm descr ibed pre-v ious ly y ie lded the SRB th ru s t, dynamic pressure,axi al acceleration, thrus t rate, and pitc h controlpr of i l es depicted i n f igures l(h!, 2(b), 3(b), . l(b),- and 5( b) . The amount o f SKB propellant requiredwas reduced by 161 776 pounds.f igures 6(a) and l ( b ) shows that th is s olu t ionlooks very much li k e the optiilial so lution exceptthat the corners o f the thrus t p ro fi le v iere roundedo f f when co ns trai nt boundaries were intersec ted.A comparison o f

The corners were rounded as l i t t l e as the curve f i twould permit.Observation o f these tw o f igures and o f f igure2(b) indicates that the discontinuity i n vacuumthru st apparently tended to occur very near t = 30seconds, which corresponds to t, i n f igu re 6(a ), t o

throttle down fo r th e 9 = 650 ps f dynamic press ureconstraint boundary. A t t = 30 seconds the ASRMare dropped, causing an instantaneous inc reas c i nacceleration, an increase whick i s enough to h i t9 = 650 ps f. Without a method t o c ut back ac ce l-e ra t ion ju s t be fo re ( t = 30- sec) thi s drop i n mass,i t appears tha t ei the r t hi s would be the optimaltime to encounter the con strai nt o r another i nstan -taneous drop i n thru s t should bc allowed to occur.The l a tt e r reason i s probably the ca% and can bedone mathematically, although the res ul ting th rus t-time curve would probably not be physically feasi-ble. The so lution obtained sa tis fie d the conditionsgiven by the foriner reason. C urrent confi gurationsare not carrying ASRM's, so this problem no longerex is ts .

and l ( b ) ] reveals tha t the burn time was shortenedalso stages the 423 266-pound SR B cases ear l ier .The p i t c h p ro f i le s [ f igs . 5 ( a ) and 5(b)] indicatethat the t imc of optimal s teer ing i ni t i a t i o n wasreduced by 14.5 seconds, and the value of the pi tc hwas higher than the reference pitch , ind ic ati ng thevehicle was f l y i ng close r to the loca l hor iz ontalto reduce g rav ity losses . (14)Thrust-Rate-Limi ted Optimal Solutions

Apparently, i n the design of a so li d rocketmt o r , one condi tion which must be sa tis fie d by athrust-time curve i s a rest r ict ion on thrust ratea t any ins tan t i n time. The optimal thrust- timec urve i n fi g ur e l ( b ) i s n o t a feasible curve bywhich to design a solid rocket motor because of thelarae thrust rate maanitudes associated with it.

A comparison o f thrust. pro fi l es [ f i gs l ( a )-- by 20.1 seconds, which, besides reduc ing pro pe ll an t,

TheFefore, another i i eq ua l i ty cons t ra int i s int r o -duced to constrain the slope of the thrust-timecurve during the main burn (the spline interpola-t ion) :

S ince th is const ra int w i l l be maintained via penaltyfunct ion, anothcr er ro r function, ea, i s defined:

e =4

and added t o e i n equation (16).i s obtained by evalu ating the de riv ati ve of equa-t i o n (8):The value of i

TSpln(t) = (ali + 2a 2i z + 3a3iZ2)/(t i+1 - ti)(21 1

The bounds fo r Tnn and Tmax are:T . ( t ) = -1.5 percent o f T /secmi n maxTmx(t) = 0.5 percent o f TmaX/sec (22)

For th is case the reduct ion i n W w as 56 088 poundsrepresenting a loss of 65 percent i n the reduct ionobtained by the optimal thrus t-time curve. F iguresl( c ) , 2(c) , 3(c) , 4(c) , and 5(c) feature thrus t,dynamic pressure, a xial a cceleration, thru s t rate,and p i tc h for th i s case.

the spline curve f i t and l e t t i ng the interpo l at ionbegine a t t = 0 rathe r than a t t = 15 seconds,another case was run which als o s a ti s fi e d the con-s t ra in t s (22). The reduc t ion i n propella nt weightwas 118 572 pounds.and 5(d) feature thrust, dynamic pressure, axialacce lerat ion, thrus t rate, and pitch fo r thi s case.

Examination o f fig ure l ( d ) shows tha t thethrust increased (progressed) i n i t i a l l y a long themaximum ra te boundary. Comparison w i th 1 ( c ) showsthat the ef fect of al loh ing thru st t o progress i n i -t i a l l y shortened the thrus t dura t ion by 3 seconds,and the th r o tt l i n g down f o r maximum dynamic press urbegins about 2 seconds earlier.

P

Adding f i ve more data po ints (parameters) t o

Figures l( d) , 2(d) , 3(d), 4(d)

Discuss ion o f R e a

Several ooints can be observed bv studvina a l lo f t h e th r u s t ' p r o f i l e s i n f i gu r e 1.t o ineet the max Q constraint, the reference case"n t h r i t ti i n g1 (a) obviously vio lates the thrus t rate c ons traint,equation (22), and appears m r e l i k e I (b ) ratherthan l ( c ) .makes no attempt t o s a ti s fy th e maximum a xi a l acc elera t ion constraint, equation (3) .reveals several interes t ing things. F irs t , curve2(a), the referen ce curve di d not reach the maximumQ le vel 650 psf, but instead reached 630, re s ul tin gi n a loss of performance.the optimal, very much except f o r the rounding ofthe corners where the boundary i s encountered.Curves 2(d) and 2(c ) resemble each other i n tha tthey peak a t maximum Q ra the r than ri d e the maximum0 boundarv. This i s because of the thr us t ra te

Also , observ ing 3(a) , p rof i le l ( a )

Figure 2, featuring the dynamic pressure curve

Curve 2(b) resembles 6 ( b

constrain&.t o be higher a t almost a l l in sta nts and apPrOXlmateThe dynamic pre ss ure i n 2 ( d ) i s !hewnthe optimal 2(b) by becoming a f l a tt e r curve nearQ = 650 psf.For curveswas below 25 ps f and tSteer

2(b) and 2(d), however, Q = 25 psf was reached 6seconds and 3.5 seconds after tstage,espect ively.

A t tstageor the constant rate casetstage*

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The axia l acc elera tion and thru s t ra te curves(fi gs . 3 and 4) are se l f explanatory, showing thelarge thru s t ra te magnitudes as expected i n 4(b).

in te re s tin g property. The reference curve [5(b)land curve 5(d), show the pi tch p rof il es decreasingafter staging, whereas for the constant thrust ratecase 5(c ) the pi tch pr o fi l e increased. Comparisono f 5( b) and 5(c ) shows th at fo r 5(b) the p itc h wasmu'ch higher, es pec ial ly a f te r pitchover. Th is in-dicates, with reference to f igure 2, that the vehi-c le rose, pitched over more, and fleir clos er t o theloca l hor izonta l and then throt t le d ea r l i er to meetthe 4 constraint.

The p i t ch p r o f i l e s i n f i g u r e 5 point out an

ConclusionsThe re su lts i n th is study show tha t the methodo f th i rd-degree sp l ine funct ion interpo lat ion o fthe control ( thrus t i n th is case). connected withthe e rro r function technique, i s ' i reasonable wayo f handling s tate and contro l inequ al i ty const ra intsopt ima l ly ,~ad i f f i c u l t p roblem i n op timiza tion .Accordina t o the contrac tor. one undesirable fe aturei n tlirus t-time curve shaping i s corners. The tech-nique presented eliminated th is feature by roundingthe corners of the thrust-time curve smoothly goingfrom one con stra int boundary to another. Also, theco ns traint boundaries are rounded a t entry' and e xi t.The way that the interval for the spline func-t ion in te rpo la t ion o f the SRB th ru s t was modeled i sonl y one o f several ways. It appears from figure

6(1) th dt a f uture method o f modeling the optimalpr o fi le v i gh t be to parameterize the length of themaximum th rus t tl, the length (t3 - tl), and thethru st T(t;) along wi th the slope i ( t7 ) . The sp lineinterpo la t ion would s t i l l be from tl t o t3, but th istime the discontinuity would not be fa i r ed in acrosstl as i n f igure l ( b) , an addit iona l obstacle.reason f o r th i s change i s th at the SR8 profiles,,ob-served have allowed discontinuities or "cornersa t th is time. The modeling i n th i s reg ion i s veryc r i ti c a l i n gai nin g maximum performance. To handlethe f i rs t k ind o f thrust rate constra ined caseunder th i s new formulation, T ( ti ) i s set equal toTmax, taking the jump out o f T versus t a t t = tl,and the rate c onstraint i s applied as previouslyvi a the e rr o r func tion method. This method w i l lals o req uire the e rro r fun ction method to maintainthe other state and control ine qua lity constraints(14) i n the same manner des crib ed. F or the para-me tri c a ll y optimal s ol ufi on the maximum thr us t con-s t r a i n t w i l l be a ll e vi a te d somewhat, and performancew i l l be improved because the profile w i l l be morenearly optimal.

the minimum number of data po in ts re qu ire d t o modelthe thrust-t ime curve and s t i l l maintain int e gr it yo f the c on s tra in ts and o bta in maximum performance.A red uc tion i n the number of paraitieters i n th i scase could mean a larg e reduc tion i n computationt i m e and a l e ss d i f f i c u l t p roblem to so lve f r o m theviewpoint of an ite ra ti on problem.This method o f pa rametrical ly d ef in in g a con-trol function such as thrust by using spline func-t io n inte rpol at io n can be applied t o the modeling

L-

The

F uture re se arch i s recommended t o determine

L-

C

of vehicle att i tude control funct ions such as pi tc ho r angle o f attack and bank angle, o r any othe rcontrol. The er ro r funct ion technique for s at i s -fying inequality constraints can be used wheneverthe c on tro ls are approximated by a sequence o fpiecewise continuous li n e a r segments o r any oth erdes irable parametric repre se ntation of the co ntrolas well.

References1. J ohnson. I . L.: and Kamm J. L .: PerformanceAna lysis o f 049AlSRM S huttle for MinimumGLOW. JSC I N 72-FM-202, Sept. 5, 1972.2 .

3.

4.

5.

6 .

7.

8.

9.10.

11.

12.

Kanim, J . L.; and J ohnson, I . L.: An OptimalS oli d Rocket Motor Thrust P ro fi le to M inimize040A S hu ttle GLOW. JSC I N 72-FM-159,J une 16, 1972.Kelley, Henry J . ; and Denham, Wal ter F .: Exam-in a t i o n o f Accelerated F i rs t Order Methodsfo r A irc raf t F l igh t Path Optimization.No. 68-19, Contract NAS 1-7987, Analytical

Mechanics As soc iates, Inc., Oct. 1968.G r e v i l l e , T. N. E.: S pline function, interpo-la ti o n, and nu nrri ca l quadrature, M athematicalMethods for U i i ta 1 Com uters, edi te ym t F a h . + l . 11, :h!p 8,Wiley, 1967.

Report

F letcher , R.; and Powell, PI. 3. D . : A RapidlyConvergent Oescent Method f o r Min imiz atio n.Computer J ourna l, Vol. 6, No. 2, pp. 163-168,J uly 1963.Kelley, Henry J . ; Denham, Walter F . ; J ohnson,Ivan L.; and Wheatley, P atr ic k 0.: An Accel-era ted G radie nt Elethod fo r P arameter Optimi-za tion wi th Non-Linear C onstraints. TheJ ournal o f the Astronautical Sciences, Vol.Vol. X I I I , no. 4, pp. 166-169, J uly-Aug., 1966.

Automatica, Vo1J ohnson, I van: The E rro r F unction Method f o rHandling Inequality Constraints. JS C I N73-FM-114, J ul y 25, 1973.North A wr ica n Rockwell Carp., Data Sheet, 1973.Bryson, A. E.; Oenham, W . F.; and Dreyfus, S. E .:O p t i m a l P rograming P roblems with Inequal i ty

Constraints I : Necessarv C onditions f o rExtrema1 S olu tions . A I A i J ournal, Vol. 1,No . 11, p p . 2544-2550, Nov. 1963.J ohnson, Ivan L.: Space S huttle Ve hicl e S iz ingwi th Maximum Dynamic P ressure C ons train t I n-cluded. JSC I N 72-FM-196, Aug. 15, 1972.Lambert. C. H.: S oli d Rockets. E nqineerinaDesign and O pera tion o f Manned Spa ce craft-Seminar Series, Lecture 25, S pring, 1964.


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13. Johnson, Ivan L . : Impulsive Orbit T ran s fe rOot imizat ion b v an Accelerated GradientMethod. Journal o f Spacecraft and Rockets ,Vo1. 6 , no. 5 , May 1969, pp . 630-632.-. 14. Johnson, Ivan L .: and Karnm, J a m s L.: NearOptimal Sh ut t le Tra je ct or ie s Using Acceler-ate d Gradien t Methods. Presented a tA A S I A I A A Ast rodynamics Specia l i s t s Confer -e n c e , Aug. 17-19, 1971, F t . Lauderdale,Flor ida .

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W

, . ' . . . . . .I , 12: 10

i i r . 3-( a ) Reference. I,-.Lb) Parametric opt imal

v Figure 1. - SRB vacuum thrust versus t ime.

,,-, 3 ,lrr. %I( a ) Reference. ( b ) Parametric opt imal

T l l . in ,#*,/L( c ) Co n st an t t h r u s t r a t e . ( d ) With in i t i a l p roqress ion .

Figure 2 . - Dynamic pressure versus time8


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,,*e, -

( b ) Parai!ietl'ic optimal.. . . . . .

. . . . . .,. .. .. . ._ _ , , _ , .... ,...

. . . . . . . . . . . . . . . . !:?-.....

20 4" h o 80 100 I20 I40,,e,*

(c) Co n s t an t t h ru s t r a t e .F i g i i r e ' 3 . - Ax ia l B

20 10 60 BO LO O 12 0 14 0Tl-. *

( d ) W i t h i n i t i a l progress ion .i cce lera t ion versus t ime.

20 10 bo 80 LO O 1 2 0 1 4 0Tin.. I*

( b ) Parametric opt imal .

- . f

- ..40 20 4 0 60 BO 100 12 0 140I,-,-

( c ) Co n s t an t t h ru s t r a t e .Figure 4 . - Vacuum

- 2

( d ) W i t h i n i t i a l p r o g r e s s i o n .t h r u s t r a t e v e r s u s t i m e .

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,/"e, ,cc(a) Reference .. . . . . . . . . , . . . . . .. . . . . .. . . . . . . . ... r u . . .

. . . . . . . . . . .

,oo io0 ,oo 40 0 500 600 7009-

( d ) With in i t i a l p rogress ion .Fiqurc. 5.- Pi tch versus t ime.

T h . I* i t- . %

( a ) SRB vacuum thrust versus time. ( b ) Dynamic pressure vcrsus time.

( c ) Axial accelera t ion NASA-JSC

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