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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)
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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)
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C)% -. Z CO W, 11 LLL LJ' V) 1%'-~4~
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0 .4 in V) c.)W C)~ I- a: L- a (l -i , !-,- =-i
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II) II I r1 11- i . 0 C) H -
(y ~ ~ n ,-%4-Y ) IN .1 -.J .14 \j = - ri 0A 1-- &--
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4.- Oi IIJ N-~ 44 3~ ~ - ~4U - 4 . N N' 1
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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
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U'.- c c .. -4 & .' 4)
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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
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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
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1- ý4CI N I,- " * tN
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to).* *IL N +I
m l*
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LL. ' a. V4-. I,- LL a1 . -.-. X 0- ?
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GA/MC/ 74-5
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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.-
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66
GA/MC/74-5
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ft 0k fix Z-
4 4 ' 7
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to a0 x i
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l'I - x Id0
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eLL U re J , IL
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N. to M rl, r-. ~ ~ .o f ~ ~~~~~-~~~ '.4 ft Q - )- - ~
N 4 ft~ * Q. L7 ft o i-
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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-
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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
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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' *
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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
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T aroducecj from
72
GA/MC/7L4-51
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P z N%. m4 N Z - ~ .
z- (A NV 2
* -M CJ* (M J I-1 4 .
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-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-
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At. H W>N~. )<I X V-1 N - N m H- C(A Z %. X X. ). CAIi cy C). P-.
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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
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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
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GA/MC/74-5
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4I (A. 0. (A U
Ig W CL t- 1-I- -**- IPIx 0 4J..J
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~sh ftI LL * >-~
NI Ni. (A +I I
94 *- L. H .C~CLO~4i (A) It
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Q) CY m~ 0i 4 9N a z r4.u.~
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x- l'-4 . If .±I LA C L'/ (A (A >. w4
m 'D, >: 0 W jj 0 -j~ -.1 . M-.
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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