AD/A-005 280

MINIMUM FUEL CONTINUOUS LOW THRUSTORBITAL TRANSFER

Eugene A. Smith

Air Force Institute of TechnologyWright-Patterson Air Force Base, Ohio

December 1974 •

I.t

DISTRIBUTED BY:

National Technica infon tion Service

I el - 'JIII

UnclassifiedS.CU,',IY Ct.A A`,,rICATION OF THIS PAGF' (hohn ; ), a , fer,, " d-

RE AD INSTIRUCT. IIONSREPORT DOCUMENTATION PAGE ,:Aoi coNM['l,•I"(N. ' ORMN

I. fI,-'O r NUMIJERi ['. O'OVl' ACCLSSION NO. 3- RE III-'NT ýsC ATALOI NUMBER

GA/MC/711-5 S' 9

'4. TITLE ('id Subtitla J 5. TYPE Or' REPORT & PERIOD COVERZO

qtMNIMlUMI FUEL CONTINUOUS LOW THRUST AFIT ThesisORBITAL TRANSFER 6, PLRFORMING ONG. REPORT NUMBER

7 ?. AU THOR(a) I. CONTRACT ON GRANT NUMUEN(s)

Euqene A. SmithCaptain USAF

4. PERFORMING ORG.iIZATION NAME AND ADUR"SS 10. PRuGRAM ELEMENT. PROJ.CT, TASK

AREA & WI'RK UNIT NUMBERS

Air Force Institute of Technology (AU)ýIriqht-Patterson AFO 011 45133

IC, CONTROLLING OFFICE NAME AND ADOl.ESS 12. REPORT DATE

Air Force In5titute oF Technoloqy (AU) December 1974Wright-Patterson AFB Off 45433 13. NUMOER OF PAGES

4. MONITORINa AGENCY NAME a ADORESS(It diItef vrn tai CutnControllinj Utlic.) 15. SECURITY CLASS. (ot titerport)

Unclassified-Fin. OE-.C L.,SSIFICATION,'/OWNGRADING

SCH.LULE

lb. DISTRIBUTION STAT-MN r (oi thial RKpot)

"Approved for public release; distribution unlimited.

17. 0ISTRIUUTION STATEMENT (ol the Abirtteict entteord In, ilack 20, II dlieririt (ton Roport)

13- SUPPL.MI"NTARY NOT'S

Approved For public releasý IAV AFR/Iq0.-17

,pr-, J LC ,:II Cnrftninn, ISAF'irector oF Inform,1tion

Ii, K[Ei WOIt5 (Continlu.O oon rc'vore ahid* it ocoistawy aid idntilty by block number)

MIinimum Fuel 1201 RT ,.,o1.ducd byLow thrust 2111 RT NATIONAL TECHNICALTransfer orbits 2203 UF INFORMATION SERVICE

U. Dopartimnet of CommailcoSpilnglilld, VA. 2211i

.W. AUSTRAC.T (Gurntrine, on revuerse #Ida It necessary arid Idenlitiy y block number)

;Oifimum fuel continuous low thrust orbit transfer is an optimal control prob-l.:ti which requires the application of iterative numerical methods to solvelhe resulting two point boundary value problem. A combination eradient-neithborinq extremal algorithn gives accurate results when both the Ihitialand terminal orbits are coplanar., These results are then used to extend theproblem to include out-of-plane transfers between orbits oF widely varyinqInclllnati(,n. Control histories, in terms of thrust direction, are plotted

DD `0`3• 1473 EDITION OF I NOV65 1S5 OSOLET- UnclassiiedJAN 7EUTY A IATI N o T r

s SE CUri-TY CEL A5 F4I r iC ON-O-F-TIWI-S-P-A-GI. -IDatn E',itj ,el.,ad

UnclassifiedSCCU RI TY Cl.. Ass'IrICATION OF' THIS PAGE(0hent tlo 'n~efed).

as a function of time for both coplanar and non-coplanar transfers. Steeringprogra,!• for coplanar transfers are characterized ')y a rapid thrust revcrsalwhich is dependent on eccentricity for the time of its occurrence. In the

latter case, out-of-plane thrusting is primarily a function of the inclinationchange between orbits. On the basis of these results, the particular numeri-cal approach used provides an effective metlhod for generating optimal lo%4thrust orbital transfer guidance. I

4

U nclas I f 4

, 1.

CN ' T Si.

, .1

1?'J

o ii

.. .............

''1

, •,I

I

'1I1NIMUM FUEL C0,'.71NUOUS 1.04I TH~RUST

ORBITAL TRANSFER

THES IS

Captain L. SIF

*V

Approved for public releasa; distribution unflimited.

MINIMLUM FUEL CONTINUOUS LOW THRUST ORBITAL TRANSFER

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

ii. Master of Science

"by

Eugene A. Smith

Captain USAF

Graduate Astronautics

December 1974

Approved for public relasc; distribution unlilmited.

GA/MC/74-5

Preface

This thesis on low thrust orbital transfer was suggested by an

article which appeared in a 1971 issue of The Journal of the Astronau-

tical Sciences by J.L, Starr and R._. Sugar. While their a__alysis was

limited to transfer between coplanar orbits, it was my desire to extend

this problem to the more general case of transfer between orbits of dif-

fering Inclination.

The a,)pl icatlon of modern control theory to such a non-linear system

ultimrately necessitated coputer utilization. As is often the case,

this computational phase of engineering was not only time-consuming, but

frustrating as well.I

In my case, the latter difficulty was eventually surmounted through

thu encouragement and understanding of my wife, Glenda. The knowledge

and advice of my advisor, Professor Gerald M. Anderson of the Air Force

Institute of Technology, was also invaluable towards the successful com-

pletion of this work.

rIe *A* S

IEugene A. Smllth :

' t1

Ii

GA/IC/74-5

SiiContents

Page

Preface . . . . . . . . . . . . . . . . . . . .. I.

List of Fligures . . . . . . . . . . . . . . .. . iv

r List of Sy:idbols . . .. .... .... ..... vi

Abstract .. ... . . . . . . . . .. . ix

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

11. Coplanar Transfer . . . .. . . . . . . .. 3

FornLilation of the Co;lanar TPV/P . . . . . . . . . 3Surntmary of the Coplanar TP VP .. ... . .. .. .. 6Gradient Alqori thm ............. 7lNeighboring Lxtrernal ,I'orlthm . 11

IlI. 1.on-Coplanar Transfer .... .................... . 14

.Coordinate System .I.. . . . . . . . .. 14Forinulation of the :on-Conlanir TPBVP ......... .15Sumrmary of the. Non-Co;plariar TPEVP ................ . 19Nleighborini Extrernal Al.r.thr ..... .............. 22

IV. Discussion and Results ............... .................. 24

Computer Implementation ......... . ......... 24Analysis of the Coplanar Problei ........... . 241Results for the Cor)lanar Probler, . . . . . . . . . . . . 27Analysis of the ;ýon-Coolanar Problem .............. ... 36Results for the Non-Corlanar Problem ..... ... . . .lO

V. Conclusions ........... ....................... ..... 47

B ibi looraphy . ................. ......... 49

Appendix A: Derivation of Equations for Coplanar TransFer....... 50

Appendix B: Derivation of Equations for Non-Coplanar Transfer. . 52

Appendix C: Computer Program Listing of CPTRAN ........... ..... 56

Appendix D: Computer Prograrm Listing of NUCPTRANI ...... . 71

•'•,.~V i t , ,. . =• .: . . . .•. , . . . . .. . ..e~ .• .:i .. . . . .•'' .- ' . .' • '-. . . . .. . . . . ." .- = ,. . . •. ,,,. 3.0...• ,• •.,••,-..., •

GA/MC/74-5 ,

List of Finures

Figure Page

I Coplanar Coordinate System ............... 3

2 14on-Coolanar Coordinate System . . . . . . ....... 14

iL 3 Coplanar Transfer Betvicen Circular Orbits . . . . . . . 28

4 Control History for Coolanar Transfer retweenCircular Orbits . . . . . . . .. . . . . . . . . 29

Control Il btory for Co:)]anar Transfer PetweenLilntca O bis,1, ofý- $0. ..) . 30

6 Control History for Conlanar Transfer f3etweenEllipt~cal Orbits (ea as .3, ef - .25) . . 31

7 Coplanar Transfer 8etween Z1 lirtical Orbits(e( m I, er - .25).. . . ........... . . 32

8 Control H1ktory for Coplanar Transfer DotweenElliptical Orbits (c. I .1, ef w .25) . . . . . . 33

9 Coplanar Transfer Letween Elliptical Orbits(c -mi. , of w .25) 311

4 10 Control History for Coplanar Transfer BetweenEl.iptical Orbits (co , .5, of - .25) . . . .. 35

II Variation of x to Inclination Change for TransferBetween Circular Orbits . . ... . .. . ... . .. . . 38

12 VarlaLion of v and tf to Inclination Change forTransfer Uetwoen Circular Orbits . . . .. . . . . . . 39

13 Control History For '.on-Coplanar Transfer BetweenInclined Circular Orbiti (I 5) ........ .... .

14 Control History for 'Tnn-Cconhnar Transfer BetweenInclined Circular Orbits (I - 30') . . . . . . . .... .. 42

15 Control History for ticn-Coplanar Transfer BettwoeenInclined Elliptical Orbit% (I- 3O•0, co = .5, ef -- .25) 1Ii

16 Control History for Non-Coolanar Transfer BotweenInclined Elliptical Orbits(i - 3-)°,0 - I, ef - .25) . . . . . . . . . . . . .. 115

Iv

GA/MC/74-5

Figure Page

17 Control History for Non-Conlanar Transfer BetweenInclined Circular Orbits (I - 60*) . . . . . . . . . . 46

1i Lateral View of Inclined Orbit ....... ............ 54

v

A,

GA/MC/74-5 jList of Symbols

English Symbols

f a Non-dimensional parameter in gradient algorithm

Sbi Linear weighting factors in augmented coste Eccentricity

ei Unit vector in the I direction

H HamIl I ton Ian

FHu Gradient of the Hamlitonlan

h Specific angular momentum

i Inclination angle between non-coplanar orbits

SJ Cost Function

Ja Augmented cost function

ith direction cosine control variable

m Mass

m Constant mass flow rate

, p Semi-latus rectum

r Radial distance from geocenter to spacecraft

Sli Quadratic weighting factors in augmented cost

T Thrust magnitude

"TPBVP Two point boundary value problem

TU (!eocentric time unit

t Time

tf Final time

UII Steering angle measured clockwise from localhorizontal

UZ Steering angle measured clockwise out ofinstantaneuus orbital plane

vi

- *.,''.A~h .~, .. 4@1

GAM/NC,4-5

u Scalar control variable

Vr Radial velocity component

V0 Tangential velocity component

V4 Normal velocity component

X" Vector of state variables

xI ith component of state vector

z Direction normal to Initial orbital plane

Greek Symbols

6 Variation

C Non-dimensional parameter in neighboringextremal algorithm

n Angle in right spherical triangle

8 True anomaly in Initial orbital plane

00 True anomaly in Inclined orbital plane

Vector of costate variables

Al Iith component of costate vector

34 Geocentric gravitational constant

v Vector of constant Laqrange multipliers

v1 ith domponcnt of Lagrange multiplier vector

* Anglo measured clockwlse from z

Ol Angle measured clockwise from initialorbital plane

4" Vector of terminal constraints

1 lith componcnt of terminal constraint vector

Condition on [Hamiltonian at final time

w Angular velocity vector of coordinate system

vii

GA/MC/74-5

Subsc ripts

o Condition at initial time

f Condition at final time

Su perscri tsA

First time derivativw

Second tiie derivative

Optimal control variable

- !Vector quantity

viii

GA/MC/74-5

Abstract

Minimum fuel continuous low thrust orbit transfer is an optimal

1, control problem which requires the application of iterative numerical

methods to solve the resulting two point boundary value prohlen. A

combination gradient-neighboring extremal alnorithin gives accurate re-

suits when both the initial and terrinal orbits are coplanar. These

rusults are then used to extend the proale,-: to include out-of-plane

transfers between orbits of widely varyinn inclination. Control his-

tories, in terms of thrut direction, are olotted as a function of

time for both coplanar and non-coolanar transfers. Steering prcgrams

for coplanar transfers are characterized by a rapid thrust reversal

which is dependent on eccentricity for the time of Its occurrence.

In the latter case, out-of-plane thrustln- is primarily a function of

the inclination change between orbits.. On the basis of these results,

the particular numerical approach used provides an effective method

for geilerating optimal low thrust orbital transfer guldance.

ix

L_- -. • , ,• ,,'4

- • . : •' • m , , , , , -, ,wm • , i .... ..... ''ii......iil. .. .• .. .i~ i " i •'• .. .... i ......"-''•i i ...

GA/tIC /74-5

MINIMUM FUEL CONTINIUOUS LO'd THRUST ORBITAL TRANSFER

I. Introduction

Wh.lh our present chemical propulsion systems have led to tremen-

dous accomplishments in the field of space exploration, it seems ap-

parent that the day of large, bulky, impulsive-thrust rockets will

soon be ending, The ever Increasing costs of these systens coupled

with the depletion of our energy resources will soon ilake the develop-

ment of a low thrLst propulsion syste.n, such as an Ion or nuclear en-

gine, imperative.

The purpose of this thesis is to examine the minimum fuel orbit

transfer problem In the context of a continuous low thrust propulsion

system. Since the propulsion system is assumed to give constant

magnitude low thrust, a minimum fuel transfer becomes synonymous with

a minimum time transfer. While this particular problem has been

analyzed extens;vely in terms of coplanar orbital transfer (Ref 5:165-

204), the overall purpose of obtaining numerical solutions here is

two-fold: investigate the possibility of using a combined gradient

a ' neighboring extremal algorith:i, for coplanar transfer, and apply

tU :oplanar results as a reference input In extending the prolblem

to non-coplanar transfer.

The problem is to determine the optimal thrust direction to

transfer a vehicle from some.point on an initial orbit, to some de-

sired terminal orbit Inr minimum time, using a constant magnitude low

thrust propulsion system.

In Chapter II both the initial and terminal orbits are assumed

to be coplanar and coapsidal, while the problem is extended to a non-

GA/MC/74-5

coplanar terminal orbit In Chapter III. In each case, the equations

of motion are developed from the restricted two-body problem with addi-

tional terms to account for vehicle thrust. For this analysis a 5000

pound vehicle with a thrust-to-weight ratio of 0.0125 is assumed.

In this paper the problem is aoproached with optimal control

theory. The thrust direction becomes the control variable and, through

the equations of motion, a Two Point Soundary Value Problem (TPUVP) is

formulated. Numerical solutions are then obtained to the resulting

TPBVP using a combination gradient-neighboring extre'ral algorithm for

the case of coplanar orbits. Solutions for the non-coplanar orbits

are obtained by a neighboring extrenal algorithm using the coplanar

results as an initial reference.

The results are presented graphically in Chapter IV in terms ofthe optimal transfer trajectory and optimal control history. The con-

clusions are discussed In Chapter V.

2

GA/MC/74-5

II. Lqpanar Transfer

Formulation of the Coplanar TPBVP

Using the notation as shown in Fig. 1, the state equation as

developed in Appendix A are:

r - Vr (1)

Lo v (2)r

•/ =L 02 L_-- + T sin u 3S2 l (3)r r2 mo-rnt

-VrVO T cosu 4

r mo -rht

where m is a constant mass flow rate and u Is the thrust direction as

measured clockwise from the local horizontal.

Vr T Inrtial Orbit

T 0"O0

Pf, ef .f

Fig. I. Coplanar Coordinate System

. --- a.• ~ tw =• r• ,• V . • J• ,.,•. •,,.. . ..... . . .. .... ., , '3

GA/MC/74-5

Specifying the semi-latus rectum and eccentricity of the initial

and final orbits as Pot 009 Pft and el, respectively, and the true

anomaly of departure from the Initial orbit as 00, the Initial condi-

tions are given by

r. Po (5)I + e cosO

0 0

yVro = 2 sin (6)P0'iO LA

0..-o. (7)ry

where ho is the specific angular momentum of the Initial orbit.

The terminal constraints are then determined as

U rf P 0 (8)I + Of cos Of

r fhf sin O

h2V f0 (10)

ip -VO -h..f 0 (00)rf

where hf is the specific angular momentum of the terminal orbit and

Of, Lhe true anomaly at arrival to the terminal orbit, Is unconsLralned.

Using the Mayer formulation of the cost function In terms of a

minimum time trajectory

J - t f (11)

which yields the following for the Hlamiltonlan

4

GA/MC/74-5

X r VO + V 2 T sin u+ rYO r r 2 0 Mt

+ -VrV 0 + T cos u.12+ (12)

The costote equations thus become

VO 2 VO2 2 7j 3 VrVOX2 -+ A 3 .. 4. (13)

2 .r2 2 3 ) r2

- 0 (14)X3 X- + r- X4 (15)

rr

,N4 = -X2 ... 3 + X` 41•

r r 7rI

with the following transversalilty conditions

V(tf) vl + V3 (17)rf2

k2tf) pfef sin Of nfhf COS Of

- - f V2 (18)(1 + ef cos Of) 2 Pf

X3(tf) v2 (19)

X4(tf) ,V 3 (20)

1 - - (21)

where vj, V2, and v 3 are constant Lagrange multipliers,

Minimizing the Hamiltonian with respect to u and enforcing the

strengthened Legendre-Clebsch condition gives

5

GA/MC/74-5

sinlu* and cosu - - (22)

,32 + A42 /X 32 + X4

as the optimal control.

.Sqummar y of the Coplanar TPBVP

Defining the state vector components r, 0, Vr, and VO as x1 , x2 ,

x3, and x., respectively, the coplanar TPCVP becomes

;I =X3 xi(o) - ro 0(23)

- X2 (0) -0 (24)*x1

A3 = 3 (0) -vro (25)

(in - +it) X 2 x3 4

k4 - x (4) veo0 (26)

TX4010 t 32 + 2

(rho - ilt) h'3 •2 .

.+ X(tf) + V (27)1I x=1 2 1xf

x 2L 2u

X12 - 3

x x

X34ux .. x 4-- -x 4

•'"'" ... •l i ' i • ==, i .i i li'•' I rLii ~ il i ..i ~ ,. i ~ • :,ii i i U lt' i i ''l/ • i i 'i / 'i - I i •*. ''i • ii i l x 1 2•'

GA/MC/74-5 A.

"-pfer sin x2f,x 2 i 0 (tf) I

( + ef COS X2 f)2

.-e hf COS xf. (28)2

pf

x1 X2 x X (tf) (30)

+

X+3VOSZ pf,#+,0 - 0 . . . . . (31I)1+ Of COS X2f

. fhf sin Xp~f 0(2

'P2 - X3 f (32)

h3 m Xf --- f-o (33)

S1 - i1(t ) + I ; 0 (34)

8 constants of integration 4 IniLtial conditions+3 unknown multiplilrs v+ 4-3 territnal cow;tral'tst..f. 4-5 t~ramwr e my,, . l, i. t-...c+,:d ... ton s

• 11 uniw~owns =,12 equaLon1s

Thus in theory it Is possible to solve this TPBVP, However, due

to the non-Iinearity of tho equations and their vc(plllcit dcpenrdence oil

time an analytic solution could not be found.

Grad -ent Al-or"Ith-

"The gradient method proved to be an excellent menns of obtaining

eproducedl fromest availabl copy.

GA/PC/74-5

approximate solutions to this TPGVP. The particular approach used

hiere is the method of multipliers suggested by llestejies (Ref 3:10).

It consists of reformulating the optimal control problem hy forming

an augmented penalty function of the form

tf + (et + (22 12 + 3 3 23 ) + + ?. + b3 :)3

(35)

Swhere both quadratic and linear terms In ,'I have been added to Eq (11)

to enforce the terminal constraints (Re 11:4I09). The S terms In Eq

(35) are welghting terms which may be varied as necessary accordinq to

the terminal error. The multipilers bI are updated from, the rela-

tionship

bi M ip (36)

In such a marner as to Increase in magnitudo as the termifnal condl-

tions become closer to being satisfied.

This chanoe In formiulatkon climinntes hhe tormlnal constraints,

Eqs (3), (9), and (10), from the problem. The only additional chanqe

is then through the transversallty conditions, Eqs (17), (l•), (19),

and (20), which now have the Form

X (tf) S11 fSi1 ( 1 ~ -I + Of cos x 2 7)

Shf h hr

+ S33 x f -.... ,. + U1 + h3 (37)Xlf Xf 2 xlf

I'll,

GA/MC/7 14-5

(f)- S 1 (,I/ -

T; e f COS fh n

22 .. ... -- --_f --f -(1+ aCOS x2 ~ Pf

-Cl cOf COS 2f pfe sin x 2 f

Pf0 (+ f cos X2f) 2

b efhf cos x2 f (.3). \ Pf

. ...... .. t~ + b(3 1,x 3 (t) f 2 2 (x3f Pf + 2

Xlý (tf) *S 33 4f L + b3 (4o)

The qradlent algorithrm is described throut,11 the followlnq steps:

S(I) With somie Initial ostimite for u is a function of

time, e.g., uref(t) m 0, the originall state equations,

(1) thru (0), are Intcgrated forward In time untildJ and d2 Jn > 0 which determines an initial

dtf dtf 2

; final time, tf .

(2) Ecis (37) thru 0IO) arc used as initial c ond itions

with bI -, 0 on the first iteration to intr-norate the

adjuint equations, (13) thru (16), backward In time

using the non-optimal reference trajectory from

step I.

(3) Simultaneously with step 2, the gradient, !u,

Is evaluated at each point alonq this reference

9

GA/lIC /74-5

trajectory.

(4) Since to first order, 6J. HU ISu, a one dimensional

seorch is made on the parameter ai where

6u - "allu, a > 0 (41)

to decrease the augmented cost function, Ja.

(5) The control is Updated through

Unev Lirof + 6u (42)

where Unefw becomes the new estimate in step 1.

(6) The b, are determined from Eq (36) and added to those

used in step 2. This is an Important part of this

procedure because as the terminal constraints become

closer to being satisfied, the linear term~s in Eq

(35) must become dominant to ensure continued con-

vergence4

(7) Steps I thru 6 are repeated as necessary until the

terminal constraints are satisflod to yield the

optimal control, u"(t), and the minimum time trans-

fer trajectoiy.

This method is quite straightforw~ard in approach, but as with

most qradint schemes, it r'quires many iterations arnd Is flow to

converge near the optimum. However, a technique used In the actual

computer program was very effcctive In overcor.nIng this problem and

will be discussed in Chapter IV. ttoreover, the rcsults obtained

served as an excellent reference input to the neig.hboring extremal

algor Ithm.

10

GA/MC/74-5

N e.qhborinq� Ext rerl Ai~jorfthi

A major obstacle In using a neighboring extremal algorithm is the

necessity of a good initial estimate. This difficulty is overcome by

using the results of the gradient method.

The particular noighbr-rIng extree1 a I.qorlthnm used here consi kts

of estimating the unspecified terniinnl conditions of the TPBVP and

generating a transition mtrix, which is used to adjust the terminal

conditions so as to satisfy the specified initial conditions (Ref 1:

219). This itorative procedure is as follows:

(I) Estimate the following elght parameters, x If, X2 f

x3 f, xf, v1 , v$ , v3 , and IP. Initially this is

done using the results fron the gradient method.

(2) The eight paramate,'s in step I determine X•f, ý2f,

X3f, 1PIP ý2 ' and ^. from Eqs (27) thru

(34).

(3) The state and costate equations are integjrated

backward From tf to Jive xIO, x2 0 , Y30, and x40

which are, in general, different fromt the speci-

fied initial conditions.

(4) rhrougih sciall perturbaticns in th,, eight para-

meters of step 1, a transition watrix of the form

(x o0 x20' x3o' x L•. 1.2.

(x if, x 2f, Xý3Fxf , X, V1 V2, V 3 tf)I

is numerically generated such that

produced fromasi available copy 1

GA/MC/74-5,,

SdD" (x 1°' x 2 °' x•°' x o , 'V1 0 '' V 3' 0 f

(x f 2f' 1 3f' 4f •I't2 ' t

dQ L dtf

(413)

(5) The desired changes In 6 " (O), dip, do are deter-

mined from

6 ;" (0) o " x (0)dT =I 0 < 0 )l

donL Li _j

since the TPBVP is completely solved when the error

vector on the right side of Eq (44) is drlvcn to 0,

or equivalently, all boundary conditions are satlis-

fled,

(6) Inverting the transition matrix of step 4 and using

the values of 6 x" (0), dip, and do determined in

step 5, Eq (43) Is employed to solve for (S x" (tf),

dv, and dtf.

(7) The astlniates in step I are now updated according

to the relationship

Lr tow LfJ "=f (45;)

-nows teop I

and the procedure Is repeated until the riqht

side of Eq (411) Is arbitrarily small.

12

GA/i¶C/74-5

Upon convergence of the neighboring extremal. algorithm, u*(t) r-ay

be determined through Eq (22). However, such convergence Is not always

guaranteed. It is actually based on ,Wo unrelated factors which are

covered in Chapter IV: accuracy of the initial estimate and sensitivity

effects.

While both of those factors were encountered to some degree, ex-

treimely accurate results were obtaine,' throu,,ph a combination gradient-

neiahboring extremal algorithm. •iore importantly, these results pro-

vided the necessary basis for the investigation of the non-coplanar or-

bital transfer problem.

13

MA/IC /71, - 5

I1l. .Non-Cop lanar Transfer

Coordinate System

In a natural extension of the coplanar transfer problem djricd in

the preceding chapter, a spherical coordinate system Is introduced as

shown in Fin. 2. The coordinate z is normal to the initial orbital

plane with the angle ¢ described in the usual manner. The orbits are

coapsldal in that a clockwise rotation throumh the inclination angle I

about the semni-latus rectum of the initial crbit results In a comnon

apsidal linn for both orbits.

While this restriction does not al low for the most general orbital

transfer case, it is imposed here as an initial atter'pt to e;•tend this

problem to out-of-plane transfois. Moreover, the nuierical technique

* to be presented could be used to eventually handle thi most rjener'al non-

coplanar transfcr problem.

V r -j

T, r f

Terminal Orbilt . •' -- " O

Pl=, aft h f

Fig. 2. Non-Coplanar Coordinate System

14

GA/MC/74-5

Formulation of the Non-Cor'lanar TPBVP

While this development proceeds In a similar manner to Chapter II,

the spherical triqonometric relationships necessary to formulate this

particular TPBVP are much more complex. For this reason, many of the

details Involved in the actual derivation of this problem are contained

in Appendix B.

"The state equations are as follows

r-Vr (46)

6 = -2.°(47)r

r

Vr + v---2- + .. in2 + u__1÷ (49)r r 2r mO_ T 1 (t

-T 2V0 -VV__0_ Ve cot 1ý + (T0)-VrrVO r-J (ni (50)* YrV r m0 -c (m t) sinT

•, = - ..v_•• + v• . s n • cfs .S + T(13r r mN0 - rt

where ZI, J2, and 9,3 are the directlon cosinas between the thrust vector

and the r-O-ý coordinates, respectively. They are introduced in lieu of

the angle control of Chapter II For ease in establishing optimality cri-

teria later.

In addition to specifying po' eCo Pff eft and 6 as In the coolanar0 0 0

problem, the desired Inclination change I must also be selected. Inl-

tial conditions are determined by

Po (52)

0 0

15

L.

GA/liC /Y7 ;- 5

o 90" (53)

eho s In 9o

PO

ve0 h -_ (55)ro

vo- o (56)

with tcrmninal constraints

rf -0 (57)+ Of Cos Of

S" (c " o+ arctan (tan i cos Of)] I 0 (58)

ý3 a Vrf - e0hf sin (59)Pf

ý4 = VOf .hf (cos I- sin i cos f cot f 0 (I0)

+ hf sin I sin of (61)j 5 IVof + -0(f)'

where: where sin (arctan (tan I cos of)] "

arccos (62),. ,s i n i J

Again the Miayer formulation of the cost function gives

J - tf (63)

and the IHamiltonian becomes

16

GA/IIC/711-5

Ii X Vr + x vP + 3 _23 r

Th 4 -r r r2 1110 -nit

+ + ... . . . . .

+ r" + I

+ cot

r2r2 r 3

I-VýrV' V02 sin Cos .€ 8(65)

\ r2 r2

x 2 0 (66)

k -2 s In cos , V223 -r

T22 COS 1+ .. -. - + 57Sr sin2 (',0 " iL) siný2 -g

V02 (sin2 ( °s )A (67)r

- r r

st available copy.

17

GA/MC/74-5

" - ! _ • z2v0cot •_ Vr (0+'6 X +- r-L +s+ x (70)

6 r 37 r 4 r 5 r 6

and transversality conditions

X (tf) v vI + v4 --!f- (cos i - sin I Cos Of cot @f)rf(

IJ5 ý-f-_ s in i sin 6f (71)

pfef sin Vf , lf

X2(tf) :1 nfi:s~::k2(t) =" v (I+ ef COS O'f)2 aeOf

sin e tan I efhf COS aof;+ v 2 - -...-..- - v 3

2 1 Stf 2 C P t

I + tan2 o2 f•f @f

"+4 h sin I sin 0f cot fr f

vs r- sin I COS ef (72)

hf Sin i COS Ofk3(tf) = V2 - 2i (7o)

rf sin2

X (tf) 3 V5 (76)

N5 (t f) uv4 (75)

A (t ) = v (76)

Ii -( (77)

Minimizing the Hamiltonian with respect to the controls Z1, 12,

and I3 yields

t] =e - (78)1 4 + (h/sin 0)2 + XGZ

S~18Ar

GAIMC/74-5

, -A 5s/sin oL~z = _(79)2 /A2 + (X5/sin A)2 + ) 62

13 - - (80)X42 + (,s/sin Q)2 + X6 2

as the optimal control variables. Note that these conditions automa-

tically satisfy the control constraint, 1 2 2 + 1 + 2 2+3 2 1, as well as

the strengthened Legandre-Clebsch condition.

Surnary of the Nnn-Cop1nar TPBVP

Defining the state vector components r, 0, ,, Vr, VO, and Vý as

x1, x2 , x3 , x4, x4 , and x6, respectively, and letting

A- /X, 2 + (,\ /sin x3) + (80)

the non-coplanar TPBVP becomes

l"x• x•()-r (03)x2 2(0) -r0 (82)

x

3 (0) go *o (84)

3 x 3

+ - sin 2 x3 x4(0) = Vr0 (85)x1 x1

TX 4X 12 (nio - k~) A

* -x4x5 2xsx 6 cot x3x() u va (86)

5 xI xIo

TXS

(NO - ;t) A sin x3

19

- x6() - 0(87)

x S2 sin x 3 Cos X+

(i110 64A) A

-- --- ) + !- x \4)v+ (cs222 3 41 t'

.1XG + -i2 sn x3 21sin i~~ co x ctx~f)

2 3)t1 1

+XX (-.-..-....-X3 X5 v.. ý-- sin I sin x

*-x x + sin x Cos x

2 /

*Pfef? Sinl 00f

2a >.(t~ f) a + ,fCS-,2l

+ , 2 Lýj -I

+ tan'- Icos2 X2f

hlf

+ vs, ý-- in I Cos x 2 ot9

-2 sin x .3 Cos lx2

F-2xsx6 l~ini -In CO S Xit

x, slnA 3Sxf l six3f

Gin0 - i;t ) A Sn 1 .3].rocd, ro

20

GA/iiC/74-5

x 2 (shn2 x3 - cos 2 x3).+ .... 3- (90)xl

XK + - 1 + X" G 4 (tf) - V3 (91)

. 2 2x5 sin 2 x3

x x

X- A (tf) vX2 2 Xl 4 5

x 2x cot x3+ . . . .+

2x5 sin x3 COS X3.... - ----- (92)

16

+f + f 0

0 (94)21

+ rctan (tan cos x) 0 (95)

€ I sin L t (06)J/3 = x f Pf

hxf-f (Cos I"sin I cos x 2f ctot X3f) , 0 (1q7)ý4 5 xf xlf

'• =x6 ~ . sin i sin x ,t0(. )6r f xlf 2f

n .H(tr) + 1 0 (.9-)

WC/ ~kC74-5 :Fi

12 constants of Integration 6 initial conditions+5 unknown multipliers u +5 terminal constraints+t f +7 transvwrsality conditions

= 18 unknowns = 13 equations

As In the coplanar transfer TPLVP, sufficient relationships exist

to obtain numorical solutions for the non-coplanar problem. Further-

more, it Is apparent that a snecified Inclinotlon channe of 0 reduc.s

this TPBVP to tho coplanar case.

Neil bori, ExtreialAloorithm

This similarity suggests the likelihood of obtaininq solutions to

the non-coplartar TPBVP by Initially r'aking a small inclination chanCe

to an optimal coplanar transfer trajectory. The use of a giradlent al-

gorithm to obtain a reference Input for the neigIborinj extremal a,-

gorithm can thus be obviated.

The particular neighboring extreral al iorithn used is Identical

In nature to that outlined in Chapter II and for that reason will not

be detailed again here. In fact, the only changes arise through an

Inclusion of the four added initial and terminal conditions to this

enlarged 12 state system.

The orig inal technilqu of specifyinq a small Inclinatlon change

with an optimal coplanar trajectory as a rnference input, proved quite

successful In obtaining accurate numerical solutlons through this

neighboring extrernal alc.orithm. Moreover, once a solution is deter-

mined for a small Inclination change, It Is then possible to us'e this

solutrion as a reference Input and Increase the snuc:ifled Inclination

change. This procedure Is repeated to obtain numerical results for

relatively larqe values of inclination change.

s~prodw rectfromm22 tbst ava ia le copy.

22)

GA/tC/74-5

The major drawrback of this approach, however, Is the excessive

'number of iterations required for converqence, especially in view of

sensitivity effects which may severely limit the size of the variation

in inclination. This problem will be discussed In the next chapter,

where a curvo-fit'tinj technique is presented and shown to be very of-

fective in overcoming this obstacle.

I

I

I

,I23 '

GA/AlC/74-5

IV. Discussion and Results

Comp2ucer _l.T. I n tat Ion

In applying the numerical algorithm of Chnpters II and III, two

computer programs were written: CPTRAN for the coplanar transfer and

NCPTRAN for the non-coplanar problem. A separate listing for these

two programs is contained in Appendix C and D, respectively.

For practical reosons alone, the semi-latus rectum of both the

initial and final orbits remains fixed in this analysis. With one

except Ion, trransfer (roii the Initial orbit was always specified to

start at perigee. Thus the only parameters that varied were the eccen-

tricities of the Initial and terminal orbits as well as the Inclina-

tion angle In the non-coplanar problem.

Geocentric canonical units were used for all computational work

with the results displayed accordingly. Thus, the unit of length was

the radius of the earth and the gravitational constant 11 was normalized

to one. This resulted in a tim, unit (TU) or 13.447 milnutes, Vehicle

mass was also normalized to one to produce a constant mass Nlow rate of

.00. In all cases, accuracy wns required to the fifth docimal place

before convergence of a solution was presumed.

Analysis of the Coplan.'r Projle

As previously mentioned, the effectlvcness of the neighborinrg ex-

tremal algorithoa is based, to a large extent, on the reference input

provided through the oradlent method. It was thus desirable to achieve

approximate solutions to the coplanar TPUVP from the gradient algorithm

in as few Iterations as possible. This was accomplished In the'follow-

Ing manner.

214

GA/fIC/7't-5

The first few iterations show the auqmented cost function of Eq

(35) to decrease rapidly, after which a pliteau is reached where

further reductions In the cost become quite small, It Is at this point

that the b multipliers or Eq (36) are exploited to overcome this dif-

ficulty and to speed up the convergence liocess. In CPTRAN a ¢deck

was thus made on the net decrease in Ja, If this decrease was less than

10-1 the time consuming process of updating the control, uref(t), and

starting a new iteration was bypassed. Instead, the b, were uoJated

accordingi to step 6 of the alqorithi until a reasonable decrease in

Ja wag achieved. This stiall adjustrient was Instrumental In obt:iininq

results accurate to the second decinal place in 50 iterations,

In transitioninq to the neighboring extremal algorithm, týe final

values of u and tF obtained above are used to compute ii reference

trajectory for the coplanar TPBVP. The eight parameters needed in

step I of the procedure are thus determined, In evary case th'is ini-

tial estimate was sufficiently close to the optimal for convergence

of the algorithm.

The linear assumption Inherent in Eq (43) places great erphasis

on the selection of E in Eq (44). A's the boundary conditions become

closer to being satisfied this linc.irity a~sumption is more valid and

the value of c can be increased accordingly, up to a maximum value of 1.

On the other hand, a poor Initial estimate co;,;pletely destroys such a

supposition, and no matter how scmall c is picked, us. of this rethod

would be questionable. The value of r. used In CPTRAN , thus con-

tinuously updated by dividing .005 by the root-i;iean-squaro of the

error vector in Eq (44) until c reached Its maximum value of 1.

25

GA/MC/71i-5

While a complete sensitivity analysis is beyond the scope of this

'thesis, a short note on sensitivity effects Is in order. Due to the

nature of the costate or Euler-Lagrange equations, as x increases

along an extreral path N decreases in magnitude. This property in turn

causes large Inaccuracies in the transition matrix of the neighboring

oxtreanal algorithm through round-off and Integration errors (Ref I:

215). It Is then possible for an ovwrcorrection In the terminal con-

ditions of step 6 In the neighboring extremal algorithr, and diverqence

From the solution.

It was found that a perturbation on the order of 10-2 in stop 4

of the neighboring extremal alqorlthi produced a transItlon matrix

equlvalent to ono produced using any soaller perturbation. For this

reason a perturbatlon of .01 was usa. In CPTRAN to generate the transi-

tion matrix. The slight delay encountered In final convergence of the

solution Is thu4 explained by the uFfect of the above noted sensitivity

In forming a transition matrix of sufficient accuracy to improve the

previous est imate.

The effectiveness of the combination gradlent-neighboring extre-

mral algorithm is that no knowledgi of the solution is required before-

hand. For any initial estimate of the control, no matter how poor,

accurate solutions to the TPBVP are ultimately achieved. However,

this flexibility resulted in a penalty of increased computer time.

Thus, once an optimal solution was obtained for a particular set of

orbital parameters, It was more efficient to use this solution as an

estimate In the neighboring extrernal algorithm when smiai1 variations

in these orbital parameters were desired. In sorne instances, such as

a small change In the eccentricity of the terminal orbit, It was then

26

GA/MC/74-5

possible to bypass the gradient portion of this method. This caused

a 50% reduction In the computational time required for convergence.

Results for the Coplanar Problem

Transfer between two circular orbits was chosen as an obvious

starting place. The well known spiral transfer trajectory was obtained

as shown in Fig. 3, with the control history plotted as a function of

transfer time In Fig. 4. This steering program compared favorably with

that obtained by Faulders in a similar problem (Ref 2:45). On all

figures PO, EO, PF, and EF refer to the semi-latus rectum and eccen-

tricity of the Initial and terminal orbit, respectively.

Results for various combinations of eccentricity were also cb-

tained. As one might suspect, there i. a remarkable similarity in con-

trol histories when ucccntricities are the so•me or even approxiwately

so. This effect Is denionstrated by comparing Fig. )I with the control

history plots of Figs. 5 and 6. Note that all three cases are charac-

terized by a rapid reversal In thrust direction through the halfway

point In transfer time.

This is in sharp contrast to orbits of differing eccentricity.

Correspondinij trajectories and stecrinq progrrams lor t o such cases

are displayed in Figs. 7 thru 10. Fig. 8 shows a delay In the thrust

reversal noted above whereas tho opposite effect is observewd in Fig.

10. That Is to say, In transforing to an orbit of gjreater eccentri-

city the net effect Is a shift of the control plot to the right,

Similarly, a shift to the left results when the Initial orbit Is of

larger eccentricity.

Fl¶,. 9 shows the only case considered where departure occurred

from other than perigee, 00 45 to be exact. In this particular

27

GA/IMC/7 11-5

ti

COPLRNRR TRANSFER

PD=2, 56

i PF=4, 00, EF=O. 00

Fig. 3. Coplanar Transfer Between Circular Orbits

28

GA/MZ /711-5

_______c

U 3 e C 3 c o

0 U t i L "

CL JLLJ l Ju

U'3 L

j(DU

LOL

U:)

LL

I-

LL1-

MU 0

(i,

z

CL,C)C

WOOS 00 0Lu" 0,00 00,6 0(siýýoioL 29I

GA/MC/74-5

Es.. •

I, II I 1 II

Mt) Lt - -aCL, L4J 0-•. u

I-

'-1

(-o

. .2

uu%

(Li

'41

00 09 ODE 00 O1 OU D 10 9fe 00"06 O

30

.- .%' . .

GA/MC/7b-5

0O m 0 N

N 4)

LU

LO

0,

a:

Lii

00.. 10 C 0 19Z 0 18 0 060 Ln

31-

GA/MG /711j-5I

COPLRNRR TRRNSFER

PLJ=2. 56

EO=C. 20

Fig. . Copla~nar frconsfer Be tvrin ElliptIca1 Orbl'ts(c * .1, ef m.25)

32

ciAftC/74-5

to00 0UU') II CI NI'

Cý Cý N

cz 0 U.4U

Q. uItCb.

uJ

CL

1% LLJ

cr. 1= I

0-

CL

00 09 0 ILZ 0,81 OD06 Do00'(9J9DiOo)n

:33

GA/1-IC /711-5

COPLRN'RR T~RANSFER

Fig. 9. Coplanar Tronsfcr Bctwecn Elliptical1 Orbits(e, * .50 of 6 25)

34

GA/MC/ 711-5

LIN

Mi U 0 t" J

II 0- wI I

U'-

Cc IA

C CL

Q--

0.

ClC

35.

CA/MC/7'o-5

Instance a minimal change' In the control law resulted. However, total

time of transfer was some .71 TU's less than the same problem where

departure was specified at perioee. This fact is mentioned here only

to show that no general statement can be made concernino transfer time.

It Is strictly a function of the eccentricities of the orbits and the

starting point from the initial orbit.

Ln.!ý_11..of the Non- op j.n a r Problem

In Chapter III the recursive method of obtaininq solutions for

Increasinq values of Inclination between specified orbits was exoleined.

This technique was initially eriployad here inasmuch as reference solu-

tions were already available Fron the coplanar analysis, Very slow

convergence of a gradient algorithm in this twelve state system with

three control variables was also anticipated.

On the othcr hand, solutions for an inclination change approach-

Ing 90* wore an ultimate objective. An alternative approach vias thus

necessary to avoid the arbitrariness of the recursive technique. For

Instance,after successfully achieving an Inclination change (INC) of

360, this procedure would only work for an Increase in Inclination of

less than one-fourth of a degree using the 36* transfer as the initial

estimate. This was clearly unacceptable.

The approach that was adopted was strailqhtrorward and very of-

foctive in surmounting iuch uncertainty. It con.s;ted of using the

twelve parameters xf, v, and tf obtained from the previous three solu-

tions for respective values of inclination, A least square curve-fit

was then made to estimate these parameters for the newi value of in-

clination. To illustrate, consider the above rientioned problem where

36

= , •--i,,.11 d=J........................................... ,...... •.......... 1 a;, , ...... I * *.•' *•l-,-* '<W.4

GA/MC/74-5

the recursive technique was stalled at 36*. The solutions obtained

for inclination change of 30*, 330, and 36' provides reference points

for each of the twelve parameters needed in step I of the neighboring

extremal algorithm. Curve-fitting thus provideid a second order poly-

nomial which ýn turn produced an estimate of Vf, V, and tf for an in-

clination chanqe of 39'o

Figs. I and 12 show a plot of these parameters as a function of

inclination change for a transfer between the same circular orbits

used In the coplanar analysis, 6esides displayinj the obvious non-

linear characteristics of the problem, thfse graphs may be Interpreted

as sensitivity plots, since they show the effect of a small variation

in inclination, Note that while small changes in Inclination resulted

in relatively small changes in xf and tf, such Is not the case for v,

Alain using the 36* value of inclination, it can be observed from Fig.

1' that v2 , v3, v4, and v5 are rapidly changing for even a small In-

crease In Inclination. This explains the difficulty encountered in the

recursive miethod since results for an inclination chanqe of 36' are

thus shown to provide a poor estimate for any larner value of Inclina-

tion.

As expected, upon approachino a specified inclination change of

35' convergence became extremely slow. This was so for two reasons:

the extremely long transfer times caused round-off and Integration

errors to have a much greater effec. than In the coplanar problem

where this factor was (1Iscussed, and, at an Inclination change of 90,

the state equations go undefined as ., approaches 0. This In itself

causes extreme sensitivity to any variation in a near optimal trajec-

tory. The overall result Was a co'ilote breakdown in the nnighboring

37

N0-) t Ud) 4),

EI 0 4 X +4-

ILJ

ci .l

CL: ciu:LC

GA/IIC/7 11-5

I: 3 Uz z zl z LaI-

00.4 +X'Q

U) 40

-j

0

L))

4-:3

.o L

-C3

C;C)IO~~ 4nc- Dlv 009 w 4

39~

GALM:C/74-5

extremal algorithm and for this reason no solutions were obtained be-

yond an Inclination change of 85. Actually, this apparent limitation

could have been overcomie by a simple rotation of the coordinate system

to avoid this singularity.

ResuIts for the Non-Coplanar Problem

For ease In understanding, the control histories are displayed

graphically as tv.o thrust directions, Ul, In keeping with the co-

planar problem for cotroarative purposes, represents the thrust direc-

tion as measured clockw.'Iso frorn the local horizontal In the r-O plane.

U2 Is then defined as the thrust direction measured clockwise out of

the r-O plane. Previous notation is used on all figures with the

addition of IIIC to denote Inclination change in degrees.

Fig. 13 shows the control histories for a small out-of-plane

transfer beteon the same circular orbits used In the coplanar analy-

sis, Hote that Ul is almost identical to the control shown In Fig. 4,

as should be the case. U2 Is also reasonable since very little out-of-

plane thrusting would be required.

As demonstrated In Figs. 14 and 15, a significant change in con-

trol histories occurred as Inclination was increased. In cach case,

Ul bears little resermblance to any steering program obtained in the

coplanar problem, since most of the thrust is now directed out-of-

plane, On the other hand, U2 displayed a remarkable similarity for

an inclination change of 300, even though the transfer was between

circular orbits In Fig, 14 and widely varying elliptical orbits in

Fig. 15.

These results su(Igest the following relationships: out-of-plane

control history (U2) is primarily a function of specified inclination

*.0

GA/IC/7 1i-5

In~w II

C) m.4

U U

04

U.

-2o ;OD-96 0 'ýZ O 0i Do a o 0ý1 0 ., u Dootl CO 00OG

GA/tiC/7tI- 5

U,4 0 O il, : U- LA 2.(I

c~Jo':oiiIn

liii Ivi

O 0

a: m

-j-

0

6s w

CD -4 L.

o

424

GA/flC/74-5

(NO W IC4

M 0 U.w

di

up 0

dd

01

r 0L

* 4 ?

N N 4 j

I- -

4-0U, ~j 4

corf o- i o-Y

epodca Wio113 est vailble opy

GA/1IC/74-5

change while the In-plane steering program (Ul) varies with a change

in eccentricity in much the same manner as a coplanar transfer. Fig.

16 is further confirmation of these relationships.

Further inspection of Figs. 14-16 reveals that U2 is not unlike

a bang-bang controller in that the out-of-plone thrust Is, for the

most part, nearly normal (± 0O0) to the instantaoeous orbital plane of

the transfer trajectory. This particular result was obtainad in all

transfers of 30W-4g5 Inclination change and is not surnrislng In view

of the coordinate system used. It Is rientioned here because transfers

in this Inclination range are often required to achieve a polar or

equatorial orbit for vehicles launched from the United States.

Graphical presentation of transfer trajectories ,as prohibitive

In view of the added dimensionality of thLo non-coplanar problem. How-

ever, for values of inclination greater than 45*, transfer trajectories

In excess of one earth revolution were obtained. Fig. 17 portrays a

typical control history for a transfer of this type. Due to the long

transfer time required, the previous bang-bang property of U2 Is no

longer present.

Unlike the coplanar problem, transfer time between non-coplanar

orbits is very much dependent on required ncl ination chancle. In fact,

the TF plot of Fig. 12 shows transfer time as a. monotonically Increas-

ing function of specified inclination change for the niven circular

orbits.

44

GA/lC/7'$-5

ON N

Z`4-u

*k Q

0--m .

Pu Pu

OD-Oig~~~~ 0004 00 1 0.0 c oo~ 0 c 00 O10 0 )US N O14,ýj0

C;U

N N

.4.. GA/MC/7'4-5

0. La 0. w

c.J0~. IIwtit 0 CJ~LZ N '~9

O.WQ.LiJ'

m ~ 3-

4 0 L

a : Val

CDC

C')

P11 -Nc

U.00 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ N .0C 0 ozD 4 0 0 OZ7ýID-CP Pa E

pGJ00 in (SD.O z

I . ! pouc fo

46 Ol -vlit oy

GA/MC/74-5

V. Conclusions

The problem of minimum time transfer to some desired terminal

orbit using a constant inarinitude, low thrust propulsion system has

been Investigated,

lFor the case of a coplan-ir transfer, the application of a combina-

Stion gradient-neighboring extremal algorithmi was shown to be a most

effective means of obtaining numerical solutions to the resulting non-

linear TPBVP. The particular appeal of this method was that no know-

ledge of the solution was required ai priori to obtain extremely ac-

c'Jrate results. Thus, a combination gradient-neighboring extremal

al]orithm would seem to be applicable to a wide class of orbital

transfer problems.

On the other hand, for greater efficiency in computer time, It Is

possible to bypass the gradient portion of this method once an optimal

solution is obtained for transfer between specified orbits. This Is

accomplished by using the above sulution as a direct input to the

neighboring extremal aliorithni when making some small variation, such

as increasing eccentricity, in thu qiven orbital parameters.

With this approach, the cop'lanar analysis was instrumental In

extendino the problem to non-coplanar terminal orbits. Upon achiev-

Ing solutions to the non-coplanrar problem for smal values of Inclina-

tion, a method of usin~g these Initial r'ciults woa developed. I t con-

sisted of generating new estimates to the neighboring extremal algo-

rithm by a polynomial approximation of paraimterrs as o function of

inclination. In this manner, solutions to the non-coplanar problem

were achieved between orbits of widely varying inclination.

GA/MC/74-5

In view of the accuracy of these results, this approach forms the

basis of an effective numerical technique for completely general opti-

mal low thrust orbital transfer.

43

GA/MC/74-5

Bib liorap hy

I1 Bryson, A.E. and Y.-C. Hao. pl le.d .Op timal Control. Waltham MA:Ginn and Company, 1969.

2. Faulders, C.R. "'Minimuni-Time Steerinq Proqrarns for Orbital Trans-

fer with Low Thrust Rockets.'' Astronaut. Acta, 7:35-49 (1961).

3. tMilela, A., et al. "On the 'Iethod of Multipliers for MathematicalProgramming Probemrns." Aero.Astronautics Report No.99. RiceUniversity: 1972.

4. Sage, A,P. Optimai Systems Control, En~Iewood Cliffs 'iJ: Pren-tice-Hal I , Inc.- , 19T_ . -...... ..... . ..

5. Starr, J.L, and R.D. Sugar. 'Some Computational Aspects of MIni-mum Fuel Continuous Low Thrust Orbit Transfer ProbIres.ms The Jour-nal of the Astronautical Sciencýýs, lq:169-204 (Ptovember-D&ocember197 . ....................

: •produ.ed, from$Ie~~ avalllable, COpy.

49

•"'hhib 'iui "w . ... .. ' iii .u .-. ,s.•..& 4 . : ,. _-' ~I 4&,'••- M.g-s='.......k.h,,,<, .. .... . . . .. . ... , .,•. 4SS •,,....M , k ._ .. .. _. _ d.

GA/ 74IC -5

Derivation of ELqcit ions for Col•anar Transfer

St.te. E u.tions

Using the standard vector notation of Z'r and W,1 to denote unit

vectors in the r-O plane., the position vector of the vehicle at any

time may be written as

r rer (\-l)

w;L:, the angular velocity of the cocrdinate system given by

;7- i•n (A-2)

wheren is a unlt vector normal to the r-O Plano. Differentiating

Cq (A-1) twica yields

r - rOr + rH 3

'- (r- r-) 0r + (2a. + re) 0 -4)

as vehicle velocity and acceleration, rcspectIvely.

The restricted two-body cqucLJon w,.-ith terims to account for

vehicle thrust has the form

2 o - rtt 2 mo - rtm t

(A-s)

whore u Is measured clockwise froni the e direction.

Now I1t

- vr (A-6)

50

GA/MC/7 14-5

6 - Vy (A-7)r

and equatinq vector cormponents on the right hand side of Eqs (A-4) and

(A-5)

Vr . - - + s sIn u (A-8)r r2 mo e,-r v

-VrVO r-+ os u (A-9)111 - nit

which are the four state equations.

B•oundary Condit ions

The general conic solution to the restricted two-body equation

has the form

r (A-10)I + 0 cos 0

Differentiating Eq (A-1O) y'alds

Vr ,h sin e (A-Il)P

whore h is conserved specific angular momentum such that

h - r 2 0 (A-12)

and applying Eq (A-7) in Eq (A-12) results In

VO -h (A-13)r

Thus, Eqs (A-1i), (A-1i), and (A-13) are the three conditions

which must be satisfied In both the Initial and terminal orbits,

51

GA/PIC/74-5

...,%ndL)x B

Derivation of Equations for Non-Coplanar Transfer

State Ecuat ions

The standard vector notation of i0 r) cO, and iý is aqaIn used to

denote unit vectors in the r-O-¢ spherical coordinate system, which has

angular velocity

w". O cos * e'r + P 'O -c iln ' (s-n)

Differentlatlnq the position vector, re'r, givas

r m rer + ru sin o e0 + re eo (D-2)

ius vehicle velocity and

"r- (1: - rW2 - r6 2 sin 2 +) ar

+ (re sin € + 2r6 sin 0 + 2r,'6 cos

+ (r6 + 2• - r6 2 sin ý cos (n-3)

as vehicle a=aleration.

The restricted two-body equation with additional terms for

vehicle thrust becomes

r nitWr m

TI. TI.3 -+ ' o+ %(-m - mt 11 - t.

where z.1, I.2, and I.3 are the direction cosines between the thrust

vector and er, , and respctively.

52

G.A/M~C/74- 55!

In addition to

- Vr (0-5)

r

lot

and again equating acceleration terms in Eqs (8-3) and (0-.11)

VVO sin 2• - _ + . _8)r r r.2 Me -ti

vo--iL - Mk SRAr r (111C( -. it) sin

TL,3-_VrVO +V02 S in '-Lc__OSJ__.+"--;-3 BIO

r r nMo . I t

which are the six state equations.

Initial Conditions

Three of the five unspecified initial conditions are the same as

those derived In Appendix A, Eqs (A-10), (A-I1), and (A-13). 0 of the

Initial orbit Is fixed at 900 In view of the deflnition of the coor-

dinate system in Chapter Iii. This In turn sots the value of Vý at 0

everywhrre In the initial orbit.

Terrnir, il ConditIons

The establishment of terminal conditions for an inclined ellipti-

cal orbit requires the uSO of Sphricail trigonometric relat lonships

to solve for the angles (p' and 0' as shown in Fig. 18.

From ;oapior's rule for right splihrical triangles

.5,3

GA/;C/74-,

S• " " Term. ina~l Orbit8 e 90 a- 90°

: ~ ~ n 900 Initia l l Orbit,

Fig. 18. Lateral View of Inclined Orbit

sin I cos 0 cossin n sin - cos -

or

arctan (tan i cos o) (1-12)

Similarly, since

C o s 5_1s.n .•sins1- (D-13)

. arctan (tan

Determrina tion of 0 thus allows use oF the general conic solution

In tho ternmlnal orbit where

1 + a cos 0'

Vr oh sin '(D-16)p

Furthermore, @ may now be determined as

0 0 + 0' 6 90* + arctan (tan I cos o) (1-17)

5',

'ii

The final two conditions are established throuqh the definitton

of conserved speclfic angular momentum

h - r x rVO r O - rVO sin e (B-18)

In the terminal orbit, Ihmay be written as

I h tI sin Iex + h cos I ez (6-1I)

where

cos 0sin ~ 3.+ cos co eS -sin 0 (o10-20)

;Z cos sin O e~ (8-21)

After substitution of Eqs (B-20) and (B-21), Eq (0-19) is equated

to Eq (D-18) to yield

VO is ( cos I - sin I cos 0 cot (0-22)

VO sin I sin o (B-23)rI

55

GA/MC/74-5

A pjend ix C

The following contains a listing of the computer program, CPTRAN,

which was used to obtain results to the coplanar TPBVP of Chapter II.

Besides'those subroutines listed, SET, STEP, and MINV were also em-

ployed. SET and STEP were called for integration purposes, while MINV

was needed to invert the transition watrix of the nelghborinqi extremal

algorithm. All three of these subprograms are found on the general

package of AFITSUBROUTINES which is on permanent file zt the Air Force

Institute of Technology.

i+ 56

GA/MC/74-5

a. a. T

4 '-4 (1*N

10 v 4 9.4a

n. a cl U.a. W- v4* .4lW

9 4 Ifi aICI x CD II*

ELI Q 64 * La..q-1 z 0 0 * *

40 V 4 al x Z r7%I*IN~ 1-4 V- X 04

-r 1-4 11 ia

H a Cl

Id -.0 I ' 04A

0L I-- N H 0 ' 041 l* >0 (' N4 vf u4H

0 ýrf W4 H 11 0 If0v4 w be 0 (\Ji 11-4 H

a.. ~-ii . ~.j * x LiiX4 V4 a C 0 4T w xtr4) x

M 4:3 CL -H II * 9>H N LA Ll. Ld v4 .~ 0 Q % ."

o * .- r- CA 0 0- C 'Jav4 M~ ler ) z .M C 4 x G C

f, ob u/ )(~ -r4 C/ % X-'-- il.

CL c a a \ (' LL 0 0 aý -L

Z- r0. - U. '. (N (/.41 'r M.'J C) * L 0 -3 UA

6-4 D~u a.. .-4 C ~ VI H .i ft #Ai N J- 0 L (In)

I- > a~ L ~ ~ 1-(ILL C C W' X U.' I- H -4a.~ý( V, LL.t' -I LL-.- ~ .4--

C)% -. Z CO W, 11 LLL LJ' V) 1%'-~4~

1 ~ ~ i 04 WI Q IU'I C I - U"I- Lf1- W- + U0 Vo H *4

0 .4 in V) c.)W C)~ I- a: L- a (l -i , !-,- =-i

7 -) 0.- .I, - It i C - U- ci LL I H . >4LAC3H ý C- _jrlV4 - o* 1

0 7' 4 i ittr 1 1 -- 1 UN0 '? 1,-i f- .- 4 ,: i

CA.0 1 ( -c -< = - < 1-1 f 1 57U

GA/MC/7 1 '-5

*J

N,.

(A')+ 0

*U.

-TU- U. zt

%-(4 W3

0 .-* 1`

x* CLaILIn " 1 ) :

M U." X CLI N !

U. .,-4 (A N

•Z ,31., 0 .

X-- I U.H=I

1 - H+ LL .0)MX 11c

fl MI- * 4(4 1 _M >e. I-I .N a(.

- Iv . - (A, -) N N, N z - N4

0 c- U .Z " IL - ,.* H ILA X 1,_ Lj "vh '• -- 4 *--- L I

- I C ,-l-X- I . .+'- ,- (A 1 X V ")

4- . C. ' ' \I* * " -0. * N- -M0..Z~~n f, x.. -4 . U.. G I.I-JA( C. W J M Jo C.L Z, 11'

+' ,-) 4 t- , U,,• ý-i (' " " X_ I• N • 7 N

(/I -- C ,. - , ,.. 0 W 4

,i. W "/' - .w4 II -.4 1 1 c I. n .O i In. a) I- v •

I-t G'wI ý4 • IA ,, Lk' r Il *t O\~l I * - -w .4 o# t

I - -- q-. 4L4 * V. 1,= t \J v4 m ' ( N

o,•* Lor"l , l- P. 4 + .. I,-• ý4 ,-4 1. 4f C\ jL -i EL -- ("I' C- ,.L-

0-"i * P )) 1 "l~.. L - 0.. -.4 * I 4, J 4 ' " I *• -

!1' .) - '' m . h 1,- (/ VI X X V,• W1 -, ^ / (. * .L 4" 1,-4 +- ti

II) II I r1 11- i . 0 C) H -

(y ~ ~ n ,-%4-Y ) IN .1 -.J .14 \j = - ri 0A 1-- &--

l- M i i / C)I ,- I :t 11i i i I / Irli-4 r\/ V) I-. ll4 .... 11 1 -0ii.0. r III I IL NI' l/ * H

4.- Oi IIJ N-~ 44 3~ ~ - ~4U - 4 . N N' 1

1. *~jH C + U . -~- - ' (J 4 '-Cl *N (N 58-C

rCA/IIC/711-5

N x

+0 I0. N

- ~ *4 U.+ +r4

x W44

v -4 x II U= - o% $

II 4. €- I X Q, X V

Z 6-4 r , - . ILL Z x .14

H. x.- •- =o' '• . 1 'I.4 1- 1 WI &'SI.-- - -- + W

10- .*• " ,

0 1 0 -it I C)

Itd Z10 . -?LLL Cl 0

M ,0 .I M, 10 ,,.4 v U, + M v4. *-4" . x O . -

r.. 'd ) ro , •0 - . - 43.

l ~'- -V)(~ Hj> : 0. W

V. II N3 -, g N I t, (A I-I + I.

ct . •. W.o0

IL it. .. "

p . 10- nJ NQ. Z N ri (A

C.) I.I, 0 V-4 C LLL ý-- Lc=.0 U) N CL *

i~ -I .-- ~ 4 .Ir-LL . N +9- L- 4.n~ L(~44 0' '' *-t- L.1'- .r:' N = *

N0 z ýL 1 0 69 [1 U-.lf~a .l-f L a.i XI U.-4 L "< " -- LL ZL\4 1, lf C3

0 VQ a b-i ftl -I J o-'4 .A LL.i t4- LL VI 01-- pAii f ~ 4 - i I i I -C. I + 0. 4-1 (/ Li I LL '

4- J 4- Ml IJ S C) * C~J (L /~ 3

a d t ILI~~ N C -f--N. v l 'I.-4qr ifLAIf-4 C' U. ý VA %4 v. 4

N o N o -L tL 6 N x V vJ P-1IC! 1,-4

Ik--1 WQ

GA/MC/74-5

LAI+N

+0 *

T. $ kl .

C..

C3 0-4 X: 0I-L4 r mjo

3 * b- t .1 - 4

III H LL r ILI I;.4 C- LL sh

u + 1, • ,o I I. %PoT v-4Q V * T44.C L(.L. •I.X • X - c• 1 .J--2 I I

M .M • W. P,..)' P-4 ̂ •- x tb T-4 b-4-r + e l.- . L i..- - 1)N. -, .. U- (. 14

O' v .La . d -• •. %0 C'I .- :

U- V)7+r) H,L., C + 0 • - ., ) ,,0 h t

C- - -I•• •4-+,,,.I CI-.Li-,C4 v I. e. H •" " ,4 € t* * 1-o~'r 4!:14 " uHLLV)j -j " 0 -J .U 11.- o% 2

I-4 • C rl . II "I, - X .- i- c -"c. CSI",, I-I C, N .M. _ (7 L 0I

1..4 O #.-4* _ (. 4* .• I -4t'. . -- "•' , J !.4 . 0 . .+., )- X ,a r t 4 $'

| f^ i.. , -. b i'. j. f LS + * + 7 _ Io- ) () .X M W

cr, v, C 'n vt T c-. T- I) Z * i: -I .'A v I0 a. ~- I -- .i a v- to) (i y~ I.Id 4 LL (v 4 Li. L. -1 -4 0 % 11. IL *.* r7 ý4~ 1 " .

I -4 ~-4 lId * N wL 11 to 'r * ( I -L t .. .Adl 11 ,4 ~-'In I -- I - n k C. ýi-

M- .4 \ - C M 11 X- 1.4 (1 - .T: .4 iU. "4 j_,-~ ..l-4 $"4

L 60

rAM/7-

U. U. a

0

*v - Wit 4

z% 4MW0l-0; r

*. 0 000

oftI lo- 1,- 4

1,-0 %'.0 q- 0 00 0V4 C; -

$1 0 wt 4 4

10 ( '4-4V 0004 00 a14 A U.- 0L2 t~ W '1-4 m 0cW . 0)LL .x 0 4 - II)

_ji at W I %- 30 1-~ w.0 N In)WE LL + Z W U) DII OW LL -..5

U. * t.1,-4 '1 .1,- IL 0 C.

W4 1V -- i .C) z' 4- 1- 0 '73 I~jrU. =1~ m = ~- ar It 04

r- t 4L. a: U~ 0L~ 0 a 1 ia C) a C Q) LA. ~#4 - 44ýf.C C * ozNe(P -. -0% a

I PA4 tL I-~ PI (. * t". (2 C~0 4 2 a - U 1

I-I I .t.-7 Z. *a .0 cl C.~ Lo

..J . .4 'P4 ) 1-1 0 C o, 4-4 It II

. i ~'. 1 .- - ')(' ' 44 LL C" 0\ ' 4% at v t( ~4~'0 t -4 CL - D m~ =) In 1C1- 40 XW=I0 I

U'.- c c .. -4 & .' 4)

3 t't- 1 f4 1 . v.. 0-o 4.

GA/MC/ 74-5

CYI

* 0 ).Caz (Iu

* w w* -LL + C3 IA. T4ii0 ~I 1-4 1- 1

.V. W N

_j M

'4U.1' X - 0

> LiJ .1* 11 14 =i I4 M_ Oil- l i- N N

r li-- re 1- QT~ *x 0y Q LL Ii N z 4

LU'' -.40 -j l'- 1-4~ 7) 1) 46 ~ -- 4 x~ fi II 1- 14 Xt7t~

l' N C9 . 9-I. U. (fl * xInrz -1* 1- L 0-U

+ u d 4 0. Um LL 1-4-U. Mj.- 41i'-aL V

UJ * 14. w Cý'*m1- (A/ -00 M +mU t Li M~*I X ~I %,i 0% 1

'-a 01. -V1 t re)'f- W it '4

11 L I ft(/', x f>01 M94J' N .J4 FI - r - 111 0 -'-- 0t-Ic -. =i4 C * Oft oJ(J L. *

w 4 ?"l,('J' & " =) .- Ii X w

Li) is 1-I W- X 11. 4%4( %0fV)( ,

b-4~- -N =* U)N w14 0% , I

1- 1- x I I l zI v -4 U IACC 1-,w U

w44 '4,

0t W4 i.

LO 11If C L. L.it Il x t j j u1 624

GA/MC/74-5

m LI. 4. 1DCV

(A L i .jUN LINI 0 0-

0 0. 4. 1. 0 f

1.4 cl X )C* r- I L I I% .

w4 114 1l 0

0

0o Oi It z ri7(f- ,4 4) *U *. 4 4z

-z II 1-4 V) Z VD

H- m x-i Jb * .1~

10Z 8 t - .0 X I a * 1-1,cm ~ ~ ~ ) (0 T 1-lX llo 00011%.011 .0 . Owl 0.j If.-I..

H C. c) x \ ~N ILIOJO 2A. C - .4, I f wL4c .4 ~

ml NO . W XC mc IA.I.Q H

X-'~ I'd V)L j V) w w * V4. ~ X ~ ~ ) i *U' I.P -11 ,1)

H W U. oftr a:~(\ N) u 7?-' CCYdf- ~ ~ L J U. t t6 ? 0 1-4C *. I. *ll

1- ý4CI N I,- " * tN

bes avial copyco,

63* I M I V r

CA/-/71-

00 1- W

V-41

LL~

4. 4 U.b r ILL C>4wLLi *ti M LM I

Oft0 f 1-4 N NMai -10- 00 41- .z 64 -t %.

*IN co. z X1X U1.

W4 *. I

- L I H M ~ 4Ik H i I.L . . Ct +' 'I (A 4 x. L ý %.0 4 - U. W ' %.0

if,- - C - C L. U,. Lj. V-41Ci. -4-4 ~ I.14 P . ~ t.

Lb~0 tAj - .- I 1-VU ~ > ý4 '. . - 4V4

W (A Ot t-4f- X, C 1 1 0

64 c CLM 111 -4 x < L X

k~~~i~ -1 o XtiS, rO m

if r t LL01 N -6I

GA/MC/ 74-5

N x

4 +

(P3 X +

to).* *IL N +I

m l*

Z N *>C- '.)0% V- + * +..La..L . N- . atZ +LU

LL. ' a. V4-. I,- LL a1 . -.-. X 0- ?

?.+ C3 Q~ (mrx m9-

C X U. -.-. L ZNLI . kL .L~Im VA V). Lt.~ LL(hI U. - ** - 4:

m !" X Z>e . L InU. 2 a.- x+- *~. LA. Li.- 1 L 1 C ' % ý IiH110

( 4s I. 7-ý ** + . , 1f> ,M V ýIor

V)~ I LLC '.% '%0 o~.ti.X 2-. N 1) ^- 0IIj t- .) L ) : (

0.,CJ*C C3.- C' So XLUC ~ e(A L 0 0

4. +I A -4 rI I L` ( % 11 - L65

GA/MC/ 74-5

Wo.

oz

H ce

zz

zz10 Z~

Z. CI- U ý 1-

1-4 1 "4 M Hltt w .o = H 1

C, r.'- 0 Z -% w-

Cl 41 X

v-S~0 44 C) ~ 40k *CX.-

uI, Lis 4O i.A. M 0 0X L) j X w 4v

66

GA/MC/74-5

J- LL

ft 0k fix Z-

4 4 ' 7

ffi

ce LD .S CL

x . 04 0 .

ce D +wf UN

to a0 x i

H t Lis 4: (

X 7

Ne - I--. 0. 7. a

w 0% 0 0I

l'I - x Id0

zj. 4 2 to H4 10 l' 0 0. se j11% " l - 0 . * t -. 0

Xt 6. L.

4 t1 L , 4P)0 N CL 7. s. e

eLL U re J , IL

t/) Vg ft3 0 .

'! 11 04 11 C J LI7: * D .t M - 1.-2 j iW 1-

(A -4 . I- -r) i- " , 1_ 0t 0.. C.). .I00 ft 0.0C CD .LL- 0 CCCLw e0*1VzD :T10 IL

LLU I -4LU U.t (.:: o-X Mf HýC ,x WL

LU ~ ~ Ob....%-4 .. ~ a3

N. to M rl, r-. ~ ~ .o f ~ ~~~~~-~~~ '.4 ft Q - )- - ~

N 4 ft~ * Q. L7 ft o i-

L~ c:* ~ t I g ~ '~' r~ 6..7

GA/MC/ 74-5

WV A

I- X

lo -

lUJ

0

LiH

+ V44

** x

X l'

H H. N(> *JS

LL~~ x ) tr tLItavIa

6 @ coy

64t

GA/MC/7 11-5

N Xo - m

r- -

,-, ,,,. ,I-LL. ft)*

1-4 IN

W * WAI

H u

I.- -H -,Ift * %0 : I

I-e- X X* oft

1 ," "., W. >

'- I •J _ • 4" * I I €

W I-I. '-' "d "'N~ -. < •' -. L ,bI -0 I-

I-- .ZL.) -4 U.%•," -''-""{l )• X .>': ) I 3-- =) ,-,'NS{,' a ("%, T "" X X I •. l • I'.. ia **} !

c t- Va ' a- vi • .j - • -. -X . • -, O. -Il - :I~ C,"• (-- .- •

W,4-z

m .) u oa H x ui~ .•ro - l r 0 a-j

4.4

(A vu 0i- 4

Q U C., 1,4 - * >

ill CL V- N L VA9

*N

C*

NH

* 0v4 j LL

C).

* 04

+ I I -ý . - - ) l

LL 3'0LZD - 0" 0W7

* -- 1C'4~~V NXO.

*L **u

* prdue fo

sI avalabe cpy

>( ~ Q70

GA/MC/74-5 D

Append ix D

The following contains a Ilisting of the computer program, HICPTRAN,

which was used to obtain results tI; the non-cop:anar TPBVP of Chapter

Ill. The AFITSUM3ROUTINES, SET, STEP, and Hil:V were again employed In

the saine manner as explained in Appendix C.

71

A. .

GA/tMC /74-5

%7,

It

It

M. N -9M

LL (0 -L X U.

:3 4%NLN454 N- d. eCu4 CLI

-±11.-LL It 5-4 4

0~- Xa S1j1 "'a Vo U. N wLt '.7: -D M re

0. 0 ft MO~ II.) ..q

. w-I LA 1-O. CL.? a W 0 4. 0 -

CL D. U 1 .4~ 44-co N 11 -0 M- t

CL. 7 =v* N M M 1 I-A I. .Z

~~-4v- 7I~ I'A' oxI(~ ~ V

Clt r4 r- z 01 0 y -)* 1-. Z V 0 Xo * zZ z 1 -ILA. f aL-.> Z it -5 U- <1--4 U. LLI.- 0 C, C il

i-. Q~ a 1:: a a-I a~. CX -4 o-4O + wtf-ý W It a-U. V X- 4 C )c 11 ii T 0 c 4.+

l" w4r UNJ~' W) ^. *(..- ta 1-4.. a. 6 v

C. t, ft--o ~ - f s. X -j , -- N .- ta. 3 (L y.. J,-lI~Tr ** . ~v -U .Q +U 4"-n. ILL C

C\ IN (r a* #0 (~~~~ -4-4.I -4 - ta-J fNJ%~ 4 -C' *

) 0 .4 Li %ý c. ýd 0 .n . c- : - f4 lý *- N as Qi is i *- ", M

D? 1) 1-4 X~ 0 lit 0 IA 1. 1I Iit 0 C11 0l 0 M "% Ii t - 4 D LA. LL LL. 0 Wi (

Q.c. W Q . C LL. C. .U- Wk Q. Cl.' Cr LA- : . CL X O> >< X~ Z - '~.4~C

~~rr. W.-viv

T aroducecj from

72

GA/MC/7L4-51

* .L t4- (19-

11% LZ - I X IL

P z N%. m4 N Z - ~ .

z- (A NV 2

* -M CJ* (M J I-1 4 .

o 0 -Ci) %-,C) -m

_* . z * N-,0yC J L + = ) = -1 If 11 .1 6% 4

-i * YLU. M LLQ

zN 0 -0 4. L N X LXH u ( '* - * +f 09-%

(A~ 1-J */L. H- H4C~'% r.9-(A LL xLJ- (/ 3: ~L..X reO 0. 1-

N~ (A U- 1- Xi ~I-..X- %,z 0V) X~ 0 L.d vt c~IN-u- LL 00 ~ ( 1iJý I < *X*%- l (YQ .

At. H W>N~. )<I X V-1 N - N m H- C(A Z %. X X. ). CAIi cy C). P-.

z-* *- 0- 4. - 0-. Nci (A

1-1Z- 04 4 (\ X) U'S ~+ u* )+ . H9 -0I -4 Z U -UL) -C~4 4- U/- t.-- n LL. V) -j

7I L - - *- I--I * ý * %. C%. *) X X U) 1..1-4 <i U.X N x 11 U\KI Lt&H C*'~1 ' 1 U=-',* il-x'1

* LL*~~~~ c~lCjt I'LL~ (A (\ 7-ý -- s- ~ ~NI ;-1* t-'-JC X. +fl-

9 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ C 4. -CC3' 4.4- ~ .r".<~,~4 U

CL C44 XI ('. V. i i ) X II- i /D4~. A C 1-4 4)C

Ct~ 'H A I -. 2. Lt -2 C) E/1 .......

C11. ~ ~ ~ ~ Rerdue fr10om \j ý4f- l1ý-Xe~ avilbl -copy.

GA/MC/74-5

N a u ~ x94 II 9

4I (A. 0. (A U

Ig W CL t- 1-I- -**- IPIx 0 4J..J

0 - N 0 4, Nt+

~sh ftI LL * >-~

NI Ni. (A +I I

94 *- L. H .C~CLO~4i (A) It

~~cr on ,. X~u t l-

Q) CY m~ 0i 4 9N a z r4.u.~

.2o-. i j CL cz ax K% .4 4. a1^*~f 4-0U- w4U.1 h4--.

x- l'-4 . If .±I LA C L'/ (A (A >. w4

m 'D, >: 0 W jj 0 -j~ -.1 . M-.

~f~\j II-i 1-4 0i 7: f~t N 10,-~~( wUI4X III N C()* 10 X~E HT>4 *lI=4m.x

i-~~ v~~ 6.-4 w l 'U . 1, 1, to- X + = 1

i. "L: .. q IC' *4 0. 0~ .0 41 CN +m~i to(\ C.6 .9 (/6. ILL 11C~ -,%oI--

*cpue l V-4 1,1 *4-4 M ~ jC~C1f .-. ,.c' N 44 D

> ~ ~ 611 clý U .) 1. 1. 4. - a IV)~

II Cie (m V) v) (nI (A H 0-

(-. )4* kc '. X .X UN 6.* UNtJi0 1ic 4, +I +I 21 LL -. .

PV) 0 04xA xI)x ( -U 4>e S<tr _I '- ,IN r7 + l

m I(% CV L krwX c -1O- I)J (NJ

ý4 64 w4 ( from

(\j ,L) M L V ti- J - ýJ11 7 1

GA/MC/74- 5

4. 4 --. -. -(•, -MX..-

O N xl Z-. t/, x 4

S,• !l

-4 04.. . ". M . W4

0~~ %. V_) = 4'( z•" W

0- 4V) N x W4of

=. ~ -

M. N V * -Y

- * %-I m %% **A On Xr *C (A

...) L) (A -0. Z C3 ,.S- L• I +

~* + LL

•.~ LL ,z- X) v4 o% 4 •O E•• '"{V " & ,- ,',• '

N L -P4 %. I% v- %-0 o%

%W U N 'Lt. Hx

S. . . -X 0-4_ C) CY UP) *Mt m .

C' 0• * x ".0 ",,4 N N•• •H

t~~b V ; <,X >,• I(.I I)- Q' m I .,• • ' I % x , x7 * ,-", .(A • t•

o N C. *+ I z x

o 4) ..j4 L.) - * - ) - H4*a 'X IN -. 4* %ý (1- ;! -- >4c.- G1

II •" *I CAII I II I %< I.•X . - -. I -, . . -- ,", € .' II - -.~ ' .--',' I -

-o - Co X .. x

, -. 0 4 h.~ / *i ~ C)') H ~ ' .

C.) NJ (lepr 7!c C''J X Uh q4 ° - lý-l 0 a - 0 fl)* axa* vi II% X W H

U - - V) 0 V -

c z ~ Z L 4' 0 C. (A W~- 4 4 ,) 0~ C)

(A) !,P- U C)OZ 0~\ I)* "-Z Z-) X* N\J t )

.-.40 X LJZ.... L)* V, X- 0~L 1- Lit L). C I Z . + * L~ &

0 LLC4 Z +t AZ ~ u4 u* 44* x * 1 +-4- >j L

(L M '4LL >< I0 f* HT - C X-l-; 0 t

~ .~ * L. +. 4 7!: L 7. -1 :-.4. Lt -4 + x. + 'C ,-NCA " X Vi-u,- (.:) 1-4* .6 * -< X-~ X :c ' '

(I ft ZI U. tl. 4. T LL o) It *- C. n? I ~4I 4- ( X - 1- + r

~~~~~~~ X..4 V' " '4.. (4 I 4 4) (,A 5< nI #-) 44C? I 4~I s

X. Se~ I II 44 11 11 11 li X +i x * - %- t-- :1. 11Z - wt ('4 1 4.

ti. U.4' 1- "-4 i-4 4I :!LL- ý ~-4* 11'.:c.' 'J- - L 1: - * 4 ~ -

l-I prodtNI

75

GA/MC/74-5

X 4S1-0

II

I H

LL J%

II II O'--

0. ,-,.'t"

e,+ • +,' zl +

ul w, X>- X 1

W) .j (A)(/,J C Uo M H Z IULEU. ,Li. LAI HC3 C3 -. 0 ILI ft a

C: C) L Is (Y N- V) (II

0. (. e. 0 C T CI

L. U.. LLA_ ,,. L. LL U. LL L. N I l. .I li

' % %. % . -.. J X I.. ' , --1c+ • ++•je eI L .LiILLLU.•i-.I--+ I- . -/ tr II , ,..I I t iI"+ ... - -. -. " rM ,c_, . c c-I r , I 'f .- 4 N ( "I",r ,-I )-1 "' X 3 J -IIJ I,41 Ww 5. -1 N, P01 lc W 1C\ 1 ' 4 H fI-ir/ (A 0 11 e

J -. I x. I . 10. 0-, .- I/% h, ,-. ,.! I- I X X II- .

ýt LAJ c.7 I EN V

1-. s'!I IU I II II II II II II L-, .,, 2, , I '- o x II •. ,-4 :,( •' ii4 It ,, i o.. ,m. J*u, tc. 4 N, re ý•, . ,4 1-1 1: x '. - . L ~ (11 l I.,I C-1 t+•; N• Ul j

.o ,-l".'"•- '. . . - I 1h' (-I • ( .'1 0 N ('. . t C%I..J.- 1'J~ 4 U*d)~ ,III. - 4* 0~ T . O~ 1,- 4) W 0

U. U k ,-- 1 - 1,-+ 1 - 1,- 1,- J,- I-- X Z' Z. :C LL 0 :" 4 U.• L~, _L: W;•,,. C.) ;A

epro u., ,,-, ¢ tr L• ., ,'o

• best availablu copy. UW 117 1., . c )

76

GA/MC/74-5

xXX

U' 1;

LL~%

act W4( ,4 4-4 N U

LA H

LL A* N-" V4 =11

C3 LL 4 O

Z ~t14 Ujz w4 .1 t

U)N1 NC + +Z (A

*,-, 0 W.,

o-- 7. <- -

H V

bJ >

In-+

C., ý- > ? -41, 1 -4"

rr n C.(1 . . - (..I

W Y' e 1 ',- ,, 1-4 0 m C, . a z. 7- -, " c, ,,X V 7 , U N,

U'" ., tu I.A. It, 1,1 tJ (I. C :1. CL Y, 0 r X %1.0 X 1. ,,-4 ,% r4- t 4 1

*H 0 L/)i.4 j a#LLC

I-. Q. C1.J M C' CL (I- CI C .IA III . k

IL Mk III U l'' lIl tO 111 01 u,{ I I I o-, I+t ¢ C " , r' • L,C(• .I.' . 1z+ ¢• r,,.€ +.

-q I- I 1. ýN i : -l + + L

-5 If I f A I I I I I If It I. A 0" i I I

,..J ~ 1 II 1I I 1I I f 11 II Ij ,-,, :t . 1 w/•+ •. ,,w •U •+"' • +++4 - O •

W .s~0 .. u-4W C~Z ..LL

-A -44N4 4Ch1~ W-4J 1 lL1-4 CV _ 1-.4f Hi -t a

GJ U w re 0 n, 0" 0 - 7) =) ' 0>- W4 ;Zi L U - .l~YI .a m 4. :t U. C~- .A Zi Z :z W .6C

L) 0

=1ro' ' = ro

ecpt aalucilabl, opy+o

77

GA/MC/ 74-5

2 ,,4 *4 wX%1

* x

do N pi N-~ 4r

w4~IP Lii * i U ~LL LL - *

I- N N

0 x - ~ l Ij9~+ V- I%

1-4

0 H) IV)

rel IL -UtO)

U. IinX Ii. H 0

in H " : 0 - % >C

UN. x .

U. U. 1)W Vi 4 N" C. Z 'U

C\J M 0 1LN X

C LN .l (L w 4 C ~ L.L: x I r U)*

*L LL4L.L Ll :4 9

ii k4 Ot V ~~ Ž .4q-4 xM oe -J X f A

-V4 M'V 10< X * *G- I

ci u00V0%0 0

tL -IU. ý t, A I i- ' IA%()~j~stv tobl C. 0 0kc ý x W

78 *14V41- '0D 4LK)-!x

GA/MC/ Th -5

a. x

* -1

V4 (A W - v-4 "IX '4

V-4 V) 0

N N X L)* *~ * U'%

* - *%) 14 - ,-

M~ *

+. bf XA< NI~ .

1- **N0)X =- 0

L~~~~L X.>*-'- ko~

*~ 0 v

--. n .. r 1-

N/ - 4--. >< 0-4 1-4 A jI

0Z - AYCl f

(\ #4N 1' - 0- I.V g-. h-I .4 1C C

* d C" ý-4 f': ? r: n- 0 --7 0 j

N T4N U-)i

F~dLI~rorn

,ails copy

GA/MC/7 11-5

Eugene A. Smith was born 27 July 19415, I'n Mlddlaville NY. Ile

attended West Canada Valley Central School where he was valedictorian

of his 1963 graduatirln class. An Air Force Academy appointment fol-

lowed, and upon graduation in 1967, he was concommissioned a second

lieutenant and sent to Reese AFD for Undergraduate Pillot Training.

After UPT, he served In South ECast Asia flylng A-i's and 0-2's. From

this ovursea5 Lour he received the Silver Star, the Distinguishod

Flying Cross, and eliht Air Mledals,

Upon returning to tho United States, Lt Smith was asslgned to

',lllhams AFB• as a T-38 instructor pilot. Hfl Instruct.d for two years

and was in Check Section for one year.

In Au,,just of 1972 Captain Smtith was notLiflod of acceptance to

the Air .orco InstiLuto of Toclinology In the Astronautical Engineering

proogram.

Captain So.I th currentIy I iv..s In Forest Ridg]c, Dayton 0H with hIs

%I fe, Glenda and dau-hto r , Andrea,

This thesis was typed hy Katherine Pannal I

80