RD-A±I54 761 OPTIMAL LOW THRUST TRANSFER USING A FIRST-ORDER V/iPERTURBATION MODEL(U) AIR FORCE INST OF TECHWRIGHT-PRTTERSON RFB OH SCHOOL OF ENGINEERING

UNCLASSIFIED T 0 BLACK 06 MAR 85 AFIT/GR/AR/84D-i F/G 12/1 NL

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MICROCOPY RESOLUTION TEST CHARTNATIONAL BUREAU OF STANDARDS-1963-A

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AIR UNIVERSITYUNITED STATES AIR FORCE

OPTIMAL LOW THRUST TRANSFER

USING A FIRST-ORDER PERTURBATION MODEL

THESIS

AFIT/GA/M/84D-I Thomas G. Blak

ILt USAF

DTICSCHOOL OF ENGINEERING LECTEV-i

WRIGHT-PATTERSON AIR FORCE BASE, OHIO

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AFtT/W/M4-

JUN

USING A FIRST-ORM PEURATION M

TESIS

AFIT/BVM/M4- 1 Thame 0. Bluwk

I~t IJ~F Accession ForNTIS- RzLDTIC ?ADUnsnnounood aJustification

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A peia0TI

AFIT/VAM/84D-I

OPTIMAL LOW THRUST TRANSFER

USING A FIRST-ORDER PERTURBATION MODEL

• ,-

THESIS

Presented to the Faculty of the School of Engineering

of the Air Force Institute of Technology

Air University

in Partial Fulfillment of the

Requirements for the Degree of

Master of Science

by

Thomas G. Black, BS.

ILt USA?

Graduate Astronautical Engineering

March 1985

APlPMo for Public rens; dIstrlbutoln unlimited.

~ . ***-*'V.V ~ *~; *** *.* .%: : .. *

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- . , - ... .. .... -

- , -

My gretest thanks gD to Dr. William Wiese, my thess affw, for his help and unfailingsupport In the past year. I found his knowledge and guidance t be absolutely indisansiblethrughout my work an this projeLt I ais want to express my appreciation to my supervisor,Dr. Squitre Brown, wnd to all my co-workers in AFWAL/FIAA. They all helped me to coordinatemy work with my studibs. I also thank all my fivnily vnd friands who have providdemotionalsupport during my studio. Porticular thanks pD to Norman Bom, a true friend who stood -

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bV me In god tiam and badL\

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Tlle of Contents

Acknowedtensts .. ...-

Lst of Fires ............................................ IV

LM of S M bols ........................................................................................................................ vAbstract ................................................................................................................................... vil

I. Introduction ................................................................................................................... I

11. Low Thrust Spiral Soluton ........................................................................................... 2

II1. Equatis of Motion ........................................................................................................ 4

S e l e c tiorn.........................................o.........m.... 4*Simplification to Planar Cas ......................................... 7

IV. Anylytuc Solution ........................................................................................................... 8

Co trol Optim izaion ............................................................................................... a

Small A ngl sumtol e..................um........... ti on.............. 9Orbltal Element Chmps ................................. 10Cotrol Solutio .. I I"

V. Verification......................................................................................... 13

V . sul ... . ................................................................................................................ 24

VII. Co clu*and Rd.om mhn ................................. 31

Bib uliv iphy ........................................................................................................................... 3 2

Vita ........................................................................................................................................... 33

* ... o.. ii

. . . - ~ ** ** *, Jftf~~~ *. .% - *.~ ~. ~• .~ %

. % .... ...... .. . - ~ . ~ ~ . . >

€., -4:4%-.

Figure Pog-

..:,..4. V i , ( ier i ...m...ts ................................................................................ 6

2. vs. u(u r 1.8r .) ......................................................................... .... 15

3. h vs.u (ur " 1.9rev.) .......................................................................................... 16

4. k v u(uor - rw .) .......................................................................................... 17

5. .VS u 1.re ) ........................... ................... 18

7. 0 vL 9(Or w4.9 rw.) ............................................................. 20 1

8. k v. u (or - 4.9 rev.) .................................................................................... 21

9. tvs. v(of 4.9 rev.).............................................................. 22

S10. *f am vs , ............................................................................................................. 23

II. s vs. u ..................................................................................................................... 25

12. G vs. or .................................................................................................................. 26

13. k i vs - r ............................................................................. ..................................... 27 -

14. )2 vs& Ur .................................................................................................................. 28

. . & r ......................................................................................................29

16. * Nm vs of ............................................................................................................. 30

--

• .'- .~-.

'.'

N - number of revolutions of the central bWd In an orbital transer -

a - semimajor axis

- time

- graitational peaamete

F - vehicle thrust vector

V - vehicle veocity vector

VcirC - circular orbit velocity

a - vehiclemass

V - trueEanamlyR~~~~~~~q~ - ailaceeain apnn

T - radgial acceleration component

T -tnntmal acceleration component

a - cetricity

a manmotion

r - distance from center of mas of the central boii

i - inclination

(. -erument of perfIas

-longtudeofuown dingnode

V - eguemen of latitude

P - aem-latusectum

h - alternate orbital element

k - alternate orbital element

H - megnitude of angular mommntum

A - mnitude of vehicle sacleration

e - pitch angle

4? - yaw angle

e - dimensionlI ratio of vehicle acceleration to local grnvltationel accIlration

H - Hamiltonian

- lagrange multiplier aciated with a

X2 lagrange multiplier aciated with h

X- lagang multiplier eaciate with k

U - uantity cmbining d X I

U2 - qunttitycombining )3 and X,

ao - initial value of a

or finalvalusof a

* Uo - Initial value of u

-r firmalvalue of u

uvi

K~' .v.*v*. ---. ° 1

An anlytlIc solution to the Optlmal low thrust transfe" problem Is developed using a

flrst-ordhr perturbation spprooh for planr orbit transfers. This solutlon is verified by - -

omparlaon with the solution obtained by interating the equations of motion. For low vehiclO

ecclerstluw the analytic aolution yields results quite cloe to thon obtained by the IntegratedSolution. At higher aeleratlon levels, however, the anMlytic solution diverges from the

integrated solution. For very low accelration levels the analytic solution pproechs the

limiting cu of the infinitsimaly low thrust spiral solutlon.

7.-

vii%

* ?Ai.:?.:_

OPTIMAL LOW THRUST ORBIT TRANSFERUSING A FIRST-ORDER PERTURBATION MODEL

I.. Intr:..,.io

In performing any orbital transfer it Is essential to determine the optimum thrust vector

profile This problem is easily solved for two exreme c he t i otlnuous thrust spiraltramfer for a vehicle having infiniesimlly small acceleration, and the two burn Impulsivetransfer for vary high vehicle acceleartion. Between these two extremes, the optimal transferproblem has been eddeswed by a variety of individuals. Several authors .2.3 have consideredcontinuous thrust transfers requiring one revolution or less to complete, while others4 have --

atiptd trawsfers reqiuiring many revolutions using -ome version of the method ofevereging This letter approach usually leads to a series of small, nearly impulsive, burns andIs subaptimel.

It is alwus possible, In principle, to solve the transfer problem through integration ,.the equations of motion. Unfortunately, the amount of computation required for transfersInvolving many revolutions is prohibitive, and numerical errors rapidly become unacceptably

large. Wiesel and Alfano-6 have developed an analytical aproch which las to a moretractable solution of thm optimal transfer proe for many revolutions. Their approachinvolves splitting the control problem into a "fast" timesca problem over one or a fbw orbits,and a *slow" timescale problem over an entire transfer. The slow timescale solutionIncorporates a series of solutions to the fast timemle problem. This approach leads to anoptimal solution for an infinite number of revolutions. Wimel andAlfano found that the errorsIn obtaining desired final conditions were proportional to I /N, where N was the number ofrevolutions In the transfer. This result suggests that the solution of the optimal control rproblem for finite revolution transfers should be emenable to a perturbation theory approach.

This paper will develop an anaytic solution to the fast timascale problem using afirst-order perturbation approach. This analytic solution Is compared with the solutionobtained by iMtedating the quations of motion to determine Its range of validity. BeW on theeresults, sm conclusions are dawn regarding the application of perturbetion theory to the lowthrust trans*f problem.

.:7

' '. ' ' '. ' . ' ' .". ,. ' ,', '. " ,, ' p . ' ., ,. . . - .. . . . ' . ' , " . . - .. - . ' . ' ' , . ' . . - .. - . . % . . .' .

__. .* % / .% .',_% _ .. .% . . ... . . . . , , ,' - . -= .. .. *,. - , *. , - . • . • . . *± ,. . -- . . . .

11. Low Thrust 5plral Solution

As a limiting cas, the optimal control solution for a vehicle having Infinitesimally smallImationy is developed for a planar tradser beginning in a circular orbit. The solution isoptimal In the sea of minimizing tOn time required to perform the transfer and, hence,mmcimizing the time rae of chaneor the semimajor axis, a. Therate orchange of theuinmor auis is given ty

dt.dt .2E2

where the en er unit mass is

(2)

&Ww the time rate of chmnge of the energ is

(3)

dt a (dRrT

In the above equations, the gravitational parameter is represented by )I, a is the vehiclemms, Y Is the true anomaly, &nd F and V we the vehicle thrust and velocity vectors,respectively. The orbital radius, r, is the sue as the emimajor axis for a circular orbit Rad T are, respectively, the radial aT tangential components of the vehicle acceleration. The

2 2.

derivaltive rwtsenspe to iiizn t Iserqie opromteted ecL,"

wine iteor b is given byL -...

- -. -. ,o. -o

For a circular orbit eccentricity Is zero, so the right hand sid of Equation (4) Is zero.

Hence, the radial term in Equation (3) is Initially zero and the vehicle directs Its thrusttangentially. Since the vehicle has infinitesimally small acceleration, orbital eccentricity must L

remain infinitesimally small and the right hand side of Equation (4) will always be nearly zero.As a consequence, in wordmnoe with Equation (3), a radial acceleration component will notresult In any change In energy. Since energy is affected only by the tangential accelerationcomponent, the optimal control solution Is to direct the thrust vector tangent to the orbit. This

corresponH to a yaw angle of zero.For zero yaw angle, the dot product of the vehicle thrust vector and the vehicle velocity

vector Is(5):-

F-V- FV

so the rate of change of the semimajor axis is given by

(6)

where Vdrc is the circular orbit velocity. Separating variebles and Integrating:

(7)o 3 / 2 i l i a / , .2 d t :

yields the semimaor ais as a function of time:

(8)

0(t) - -1/-1/2 2

002 M__ /2 (t- to) :

I06

3 ::

I.r

111. Eauattons of Motion

,Si¢4.on of Form ..

It is desirable to select a form of the equations of motion that will make analysis of the low

thrust transfer problem as easy as possible. A useful form is the Lagrange planetary equations

expressed in terms of force components.7 .8

(9)

dl n i Y ar - T

dt n na2 e --e a .

dl n o u V

rcsu N

dt no2,71 . 2 ,..

R)Isi ~i+T N:.

r sinu Ndt na271 e2 sin i

Unfortunately, this form Involves the argument of periee, c, and the eccentricity, , so it Is

singular for small eccentricity. This singularity problem can be eliminated by making the

change of variables:

(10)hi e cos

k= esin-

Another problem is that the true anomaly, v, is not well defined for nearly circular orbits. The

true anomaly can be replaced by the argument of latitude, u, where:

(11)g=N,

This change of variables transforms the independent variable from time (implicit in v) to u.

4

, - .. . .. .. ....--

Aplying thse variable changes to the Lagrange planetay equations, expanding for smalleccentricity. and retaining only the lowest order terms yields:9

(12)

du a- aP 11 rut""duT

du 2a sin i

In these equations Rt, T, and N wre the acceleration comlort in the radial, tangential, and ".

qi ~normal directions, respectively., i s the inclination. and I s the right asesion of the' ""

aedng node. The quanity a is the mean motion.

nN

_, The mass and acceleration of a low thrust vehicle making gml orbit chianges can be•,-..1[ treated as aroast This assumption can rnot be made when considering the long0 timesale

problem. For a vehicle with constant acceleration A, pitch angle 0, and yew angle t,(see "]""" Figure I 1 the accelertio componets ire:

(14)T- ILR

R - -A cos e sin*4 -.-ntNa Nre t A sin po t hr a-

- N-.

" omldrcins epciey 5s th"nliainad;)i h r'tasesono,

+'T'; -. +,- "' ascending'..'- d.-.,+.." -The '. /,antit.y ' ."- is" the'-mea:' m to .:-,' ' ' ' ." -- .- . . .. ,. . . .,.. ,

1%. h'.

Cale"le

Ante sflreltionsilp Is L.

where f is the dimensionless ratio of vehicle amelration to local gw ittiona e1 miration.Substtuing the expresions for the acceertion components and e Into the equiations of motionyflla

(16)

do

Elcoso case sin* 2 sino cosO xstidu

- c -Eco sine

-~ - sin U sine

Simplification to plinaer CMit was decided to develop only the more limited cms of orbital changes within a single

plane. This is eulivalent to satting te pitch angle 0 equal to zero In the equtions above. Thisassumption results in the following tat of three setios.

A EG- stol sint* + 2 em costi

doelcoso sin* + 2 sinuocosti

This is the final form of the equations of motion used to develop the analytical solution to the .

optimal low thrust transfer pr"lem.

LiZ" IV. Analytic Solution "-.

Control Optlmlanlo

To solve the low thrust transfer problem, it is nbceuery to determine the optimal yaw

angle program to perform an orbital transfer from a given at of initial onditions (oO, ho , k ) L

to a given at of final conditions (or, hr. k). For a vehicle under comstent thrust, an optimal

trajectory is eluivalent to a minimum time trajectory.9 If the vehicle's orbital ei ntricity Is

very small at all points in the trajectory then time Is simply proportional to u. Boud on this

asumption, the Hamiltonian for the optimal transerd 3

(18)H~l~ dX,A dis& + X3& --;

du 2du du '

Since only small orbital chan os are being considered, the dependent veriables on the right sides

of the equations of motion can be treated acons Using this usumption, the coanoil

meqtioms for the lorange multipliers are

(19)

,...-a -

• -im -~ -

so all three of the l8range multipliers era constants. The solution to the long timescele problem

would Incorporate a erie of such small chenig Inserting tie eutiom of motion Into the

Hamiltonlin and computing the yw optlmality condition

(20)

.1.0

- U*,,°% °'

o 2o • ., .at

8F' °

°, o o ..• . - ••• • -. . " "" ° "" " "" • -" " • " " * * ~** •. " % " *" "" " ° " ° . . . " % . . ' ' .

yils the control low:

(21),._,I,, -X29inu + kX, su..,tit 2(a),l X2cosu + k3 sinu)

Note that, even though the )s we constant, * will very because of its dependence on u. Thedirection or the trafer, inward or outward, is used to resolve the ambigity in the solution for1'. For outwartrefers * will takw on valum in the ranp -11/ i /2, while forinward transfers * will take on values in the other two quadrants.

Since the lrangs multipliers are constant throughout a given orbital transfer so long asthe change in a Is small, and * is dependent on these multipliers, the multipliers re

effectively the control variables for this problem. The yaw control low can be simplified byletting

(22)

The yaw control law then bco mes:

(23) I-

-U, snu + U2 cosu2(a + Ulcosu + U2 sinu)

= 8mollYW A sAsumption .:.ItI,T o further simplify the analytical solution to the low thrust transfer problem, It Is

assumed that the yw angle ' is small. Since the optimal yaw angle for a vehicle havingInfinituslmal thrust Is *-O, this isa reaonble assumption for a low thrust vehicle. Note thatif * Is amiell, UI and U2 must be small. Using this small angle aeumption, the equation ofmotion becm-

• .;..._

* ,'* "-li.

o*. % ",

(24)

"u - 2aE

u EI-tslnu *2cosuj llii.,,

du Eftcosu * 2sinu.

and the ye control law becomes

(25)- -Utsinu + U2cosu

Note that the equation for the mimqjor axis drns not involve *, At this level of approximationthe analytical solution is identical to the spiral solution for a. The differential equation for acan be solved directly in terms of u to yield

(26)

IIL0 ao -4 a(- 0

Orbital Element ChanesThe chae in orbital elements over an entire orbit transfer are computed by Integring

thequtlons of motion over u. For the smimaor axis this IS

(27)

AO- or- 0 - J.. du

and similarly for the other two orbital elemonts Owrying Mu them ntei atlons for thesptluns of motion raulting from the small mmpton yields

1...,.

10 S *"

*..

bo %'.o" ,- .- - o. . . -. . *. * °* . S - . o 5 . -. . .

(28)Am- 2ao6u

Ah 10 1 q-snu sin V ONV u) - -sjn2v

ONO +osn *11~ + BSn~ ~

I"'('8 82(sn

Notethat Equilon (26) Is valid for large ctanges In a while the first of Equetions (28) is

valid only for small ch mis.

Given the Initial ad final conditions for an orbital transfer It Is nenceey to compute the

controls (i.t the ) s) required to achieve thor transfer. It Is moure thot ech transfer begins

an ands In a circular orbit. The total Pngs In h and k ovr an entire transfr o this type

must therefore be aro, since them two elemets are directly proportional to the oetrlclit.

Using this fact, the verlables U1 and U2 can be determined by simultaneously solving the

fol lowing two saations.

(29)

A&h O sflnu -- (-1 ursin u f r)- U2 sin2u•

Ak - 0 Weo - csouf) - 15n2u1r * (sfnUrCmUr * Ur)

3olution of them equattm yeld-"

(30)

(U2 slnuf - 8.)sinofrOf - stnu osu,

e%(uIn3lfur- (I - osr)(ur - stnur Ofu r.))U..., - sln2urcos2ur .

U11 ":

The Iew multiplier mc*ea with the semtmajor axis is computed using the

truw~rwIity udaonm-

S 0 o I U I U2 do

Solving his agintio for X,1 Yields"-...

(32)

Aa(40(o + cOsu+ sinu) * (ilsinu - U2 0oSu)(sinu - aou)I

where a asafunction of u is given by qWoimn26. The vulumof k2 and X3 are computed

from X , UpwidU2. Wtththe )'sin hand, theyew ngle, ,can be cmputed for anyvalue ofousing Equatio (25).

12

* * ** .

-- * .*.~* %.*. .-- *."

- 'V. N, % . "I k-

V. _ fLL Zio

The analytiml solution to the low thrust transfer problem developed In Section IV wasverified by cmparison with the integrated equations of motion (Eqrs. 13) for a set of orbitaltraonsfer All computations were carried out in canonical units, for which the distance unit is

* one Earth radius and the gravtatimnl parameter, p, Is numerically equal to one. All of thepverification treectorles boan at u - 0. The Initial and firal conditions for all of theme

tralectories were

(33)

(a%, h , ko) 1(, o, 0)

1 (or, hr, kO) 1 ( 0 1, 0, 0) .::

Note the small cha in the semimajor ais. Solutions for these conditions can be extended toother valuesof ao and a by scaling E.

Using the above conditions and specifying u, the 's and the required acceleration werecomputedi In order to make a valid compaison, these computed values were used in both theanalytical solution and the solution vie direct integration of the equations of motion. For theanalytical solution, Equation (26) was used instead of the first of Equations (28) to providehlgher accuracy for changes in the semimajor ais. In the Integrated solution, however, theanalytcal result of Equation (26) was not used; all three of Equations (17) were integrated.Four hundred integration steps per revolution were used The small angle asumption was ot"made for the interated solution.

Orbital traiisfers computed using both the analytical solution and the Integrated solutionare compared In Figures 2-9. Figures 2-5 show the orbital elements and yaw angle as afunction of u for a transfer to uo - 1.8 revolutions. The correspondence between the analyticand Integraed solutions Is poor. Note that for small changes In a the yaw angle computed usingthe two solution mathods Is essentially Identical, so Figure 5 shows only one curve FIgures 6-9atiw a transfr to ur - 4.9 revolutions, and In this case the cor ndnce between the two

*solutions Is very god. Figures 2-9 Indicate tha the analytical solution Is valid for lar U,but breeks down at small values of u7 (i.e. higher accelratos).

~* % * * * * .. . . -. . * .* . ~ ... * ~ . *-.-

Using the analytical solution, the maximum yaw angles for transers involving largevalues of r (now 100 revolutions) were cmputed. Since % and are fixed (Es. 33),

large ur corresponds to a very low acceleration level. As shown in Figure 10, the maximumyew angles for these transflers were very small. Note the continuing doce in the maximum

F.

* 14

p2'-

Z,*-_. .::. +'.

-a.

1.010

1.009

1.006 Fm

1.007

1.006

a1.005

1.004

1.003

1.002

1.001

0.0 0.2 0.4 Oh6 0.6 1.0 1.2 I.4 16 La

Figure 2. a vs. U (Ur - 1.8 rev.)

15,

1.010 .. -

2.0

1.0

h0.5

Nx 10'-3)

0.0

-05

-1.00.0 0.2 OA 0.6 0.6 1.0 1.2 IA lb6 B.

Figure 3. h VS. U (Ut - 1 .8 rev.)

16

I."

2.0

1.5 ArosyUc

1.0

k0.5

(x 10'-3)

:ill 0.0

-0.5

-1.00.0 02 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

u (rms)

Figure 4. k vs. u (Uf 1.8 rev.)

17........................................ .

50.

40-

50

20

10 dog

-10

-20

-40

0.0 02 OA O6 0.8 1.0 12 IA 1. 1.8u (rws)-'"

Figure 5. 1, vs. U (Ur 1.8 rev.)

%- ,'.- *

• - .-i'"8

o4

1.010

1.009

1,008- nesd1.007 - Analytic

1.007

81.0056

a1.005

1.004

1.002

1.002

1.000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Figure 6. a vs. u ( - 4.9 rev.)

19

b ....... *

0.5

0.3'

0.2

0.1t

(X W0-3) 0.0.I

-0.1

-0.4

OA0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

FigUre 7. h VS. U (uf 4.9 rev.

20

L~AL

7..

0.6

05

0.7 -- hn taic

h OA.-..

(x 10*-3) 0.3

0.2

0.1

0.0

-0.10.0 0.5 1.0 15 2.0 2. 3.0 35 4.0 45 5.0

~U 0")

Figure 8. k VS. (o - 4.9 re.)

21

5

3

21

PSI

(dog)

-2

-3

-5%

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Figure 9 f vs. . (of - 4.9 rev.)

22

0.7

0.5

(dsg)0.3

0.2

0.1

0.0 -100 101 102 103 104 105 106 107 106

FSr lo'*fdXVS) U

23

V1. Results

* .-- VI.B~l .

A solution to the low thrust bundary value problem, for a given transfer, occus whenthe value of the Hamiltonian first reachas arO. A typical plot of the Hamlltonian as a function ofu isshown in Figure 11. The Hamiltonln aclletes up anddown, but its mien valuedicre nns

a the milnajor axis lnceeaseL If the Hamlltonian Is ntnded Weond the point at which it

first equals zero, It will dip below zero with each subment revolution. This Indicates that aminimum fuel solution to the low thrust tser problem, as opnmd to minimum time

mlution, can be developedL For a minimum fuel solution, the vehicles engine would be ,m when

the Hamiltonian was positive end off when it was negative. The turn on and turn off point 1o an

Inward transfer from u to uo would simply be the mirror Image of the turn on and turn off

pointsfo an outward transfr from u. to o.Thequantities , I1, A2 X, and the absolute value of 4nld areshown asa functionof

Uf in Figures 12-16. Note that iutkm do not exist for some values of of below 2.5

revolutions. All three X's are periodic with a period of 211 (one revolution) and, In eachrevolution, they areall singular at the 0.5 end 0.75 revolutiai points. Bcaie of dependence

on the inverse rate of change of the semimajor axis, the mean magnitude of )\ incr e as o Uf

Sincreases. Themenvaluesof #\2 and \3 do not chnge. tmax also hes periodof 2W, but

' Its maximum value per revolution asymptotically aprc zero as uf Incr and vehicle

acceleration decreaes. The dcIeae in tn as ur incr is shown in both Figure 16 andFigure 10.

One major advantage of the anelyticel solution over the Intepted solution is that it

requires much less computation. For any transfer, using the analytical solution, the values ofthe orbital elements for any given value of u can be computed directly from Equations (28). In

comparion, obtaining the orbital elements for the same velue of u using the integratd solutionrequires ntegrating over the entire range from u* 0 to u. Computing and outputting the

orbital elements forty times per revolution, the analytical solution was able to complete onerevolutio about ninety times a quickly as the Integrated solution. This gives a rough Indication

of the computational advantage of the analytical solution.

24

2,1 .

0.5

0.305

0.20-

H 0.15-

0.05.

0.00

-0.0610 1 2 3 4 5 6

U-rmul (urw)

Figure I1. H vs.u

25

600

Nx 10^-6) 400

300

200

100

0 I I0 1 2 3 4 5 6 7 8

Moare 12 E vs. UfOI,

26

25-

20-

15,

10

5L I

Nx 10'3) ~- 4 ---- 4-

-5,

-15

-20

-250 1 2 3 4 5 6 7S

Figure 13. X Vs o

27

• .- s-- -o * " '.

*- -. 7.'

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30

VII. Conclusions and Recommendations

In this paper an analytic solution to the optimal low thrust transfer problem was

developed using a first-order perturbation approach. For low vehicle accelerations this

solution yielded results quite close to those obtained by integrating the vehicle equations ofmotion. At hlier acceleration levels, however, the analytic solution diverged from the

integrated solution. For very low acceleration levels the analytic solution was found to appr h

the limiting case of the infinitesimally low thrust spiral solution.

The results described in this paper could be further developed by incorporating a higher

order perturbation model. Such a higher order model would extend the validity of the analytic

solution to higher levels of vehicle acceleration. Another useful activity would be to develop an

analytical solution for the full, three dimensional orbital transfer problem, eliminating thelimitation to planar transfers Using the minimum time transfer solution as a basis, the

minimum fuel solution should also be developed By Incorporating results from all of these

recommended areas of study, there is potential for the development of powerful and efficient

tools for the analysis of low thrust transfers.

31... . . . . . ....-..

* - . --.. * - .' ..-. . . . . . . .

I. KOpp, R.E. and MOill, R. "Several Trajectory Optimization Technlques, Part I,Discussion," Comoutino Methods in Optimization Problems. New York: AcademicPress, 1964.

2. Mover, H.6. and Pinkhem, 0. "Several Trajectory Optimization Techniques, Part II,Applications," Comouting Methods in Optimization Problems. New York: AcaemicPress, 1964.

3. Edelbaum, T.N. "Propulsion Requirements for Controllable Satellites," ARS Journal, Vol.31,pp. 1079-1089, 1961.

4. Redding, D.C. "Highly Efficient, Very Low-Thrust Transfer to Oeosynchronous Orbit: Exactand Approximate Solutions," Journal of Ouidence, Control, and Dynamics, Vol. 7,pp. 141-147, 1984.

5. Wiesel, W.E. end Alfano, S. "Optimal Mny-Revolution Trensfer," Journal of Guidance,

Control, and Dynamics, Vol. 8, pp. 155- 157, 1985.

6. Alfano, S. "Low Thrust Orbit Transfer," Mesters Thesis, Air Force Institute of Technology,

1982.

7. Bate R.R., Mueller 0.0. and White J.E. Fundamentals of Astrodnmics. Now York: DoverPublications, Inc., 197 1.

8. Denby J.MA Fundamentals of Celestial Mechanics New York: MacMillan Publishing Co.,Inc., 1962.

9. Wlesel, W.E. Unpublished paper in preparation.

32

e7. .... . . . . ........ . . .... . .

!!! , -- ..

* A.-,

Thomas Black was born on 31 January 1958 in Euclid, Ohio. He graduated from high

school in Canandaigua, New York in 1976 and attended the University of Virginia from which he

received the deee of Bachelor of Science in Awepece Engineering in May 1980. While at the

University of Virginia he entered Air Force ROTC, and was commissioned upon graduation. His

first Air Force assignment was at the Rocket Propulsion Laboratory, Edwards AFB, CA, where he

served as a project manager. He entered the School of Engineering, Air Force Institute of

Technology, in May 1983. Since May 1984 he has worked in the Technology Assessment Office . -

of the Flight Dynamics Laboratory.

Permanent Address: 100 Oroe Drive

Candleigue, NY 14424

? 33

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17. COSATI CODES l8. SUBJECT TERMS (Continue on reverse if necetary &WgkW "&b Wi#j 3

FIELD GROUP SUB. GR. -)Low Thrust Perturbation)03 03 Orbit Transfer Optimization.

19. ABSTRACT (Continue on reverse if neceaaary and identify by block number)

TITLE: OPTIMAL LOW THRUST TRANSFER USING A FIRST-ORDER PERTURBATION MODEL

Thesis Advisor: Dr. William Wiesel

>An analytic solution to the optimal low thrust transfer problem is developed usinga first-order perturbation approach for planar orbit transfers. This solution isverified by comparison with the solution obtained by integrating the equations ofmotion. For low vehicle accelerations the analytic solution yields results quite closeto those obtained by the integrated solution. At higher acceleration levels, however,the analytic solution diverges from the integrated solution. For very low accelerationlevels the analytic solution approachs the limiting case of the infinitesimally lowthrust spiral isolution. - c 4 , \.

(."I-

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Dr. William Wiesel (513) 255-2362 AFIT/EN

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