Approximate analysis of optimum low thrust transfer between coplanar hyperbolic orbits

< ~ H

von KARMAN INSTITUTE

FOR FLUID DYNAMICS

TECHNICAL NOTE x5

APPROXIMATE ANALYSIS OF OPTIMUM LOW TH RUST TRANSFER BETWEEN COPLANAR HYPERBOLIC ORBITS

by

Lo MOULIN

RHODE-SAINT-GENESE, BELGIUM

MAY

1964

VTH

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losse tijdschriftnummers maximaal 2 weken

.

von KARMAN INSTITUTE FOR FLUID DYNAMICS

TECHNICAL NOTE 15

APPRDXIMAT,E ANALYSIS OF OPTIMUM LOW THRUST

TRANSFER BETWEEN COPLANAR HYPERBOLIC ORBlTS

by

L. MOULIN

The research reported in this document has been sponsored by the Air Force Office of Scientific Research, through the European Office of Aerospace Research, United States

Air Force.

ABSTRACT

Optimization of low thrust transfer between coplanar hyperbolic orbits has been analyzed, on the basis of an

approx-imate analytical solution, in the case of constant thrust set at a constant angle to the velocity vector. The investigation bears upon minimization of fuel consumption and ;selection of an ideal initialorbit. For given initial and final orbits i t is shown that optimum conditions can be obtained with tangen-tial thrust. Further refinement consists of a proper selection of the angular momentum of the init~al orbit, so as to either initiate or terminate l~w thrust propulsion as close as pos-sible to perigee, depending upon the direction of transfer.

TABLE OF CONTENTS 1. 2.

3.

NOMENCLATURE • INTRODUCTION • FORMULATION OF UNIQUENESS OF OPTIMIZATION _• • • • • • • o • THE PROBLEM THE SOLUTION • • • • • • • • • • . 0 0 • • • • • • e 0 • • • •

·

• • • • • • • • • • • • • 0 • • • • • • • • • • • • • • •

4.

SELECT ION OF OPTIMUM INITIAL TRAJ.ECTORIES. _{• • •}

5.

CONCLUDING REMARKS _{• • • • • • • • • • • • •} REFERENCES _{• • • •} • • • 0 • • • • • • • FIGURES _{• • • 0} • e 0 0 . • 0 0 . 0 . 0 • • 1 2

4

8 11 17 20 21 22

1. NOMENCLATURE c e E(k,ep) F(k,cp) h m

u

ó E; p Subscripts o i

ratio of exhaust velocity to circular velocity at sea level

orbit excentricity

incomplete elliptic integral of second kind incomplete elliptic integral of first kind non dimensional angular momentum

mass ratio (actual to initial) non dimensional specific energy

angle between thrust and velocity vector hyperbolic excentric anomaly

ratio of radius vector to the radius of the planet ratio of thrust to initial weight

initialorbit

2.

INTRODUCTION

The prob1em of orbita1 transfer between coplanar and non coplanar orbits has been extensive1y treated in the case of impulse-type propulsion. However, one may consider that future interp1anetary vehic1es wi1l be equipped with advanced propulsion s~stems, which are characterized by low values of the thrust and high specific impulses. The assump-tion of an impulse is then no longer justified. Moreover, i t can reasonab~y be expected that advantage wi11 be taken of the onboard insta11ation of such equipment to use i t for trajeetory corrections.

In that perspective, the kind of problem which is 1ike1y to oocur at first in the field of trajectory correc-tion, is the one of terminal guidance during the fina1 leg of an interplanètary mission, to insure a correet atmospheric entry, for earth in particu1ar. Such a problem must be dealt with in a coordinate system referred to the center of the planet and consequently, attention must be given to trajec-tories of hyperbo1ic type.

The prob1em which has been considered here is the optimization, with re$ard to fuel consumption, of low-thru~t transfer between coplanar hyperbolic orbits.As a first

approach, the prob1emhas been investigated using an approxi-mate ana1ytica1 solution (ref. 1), whereby thrust is assumed

.l

veotoro Although the optimum solution should be derived from

variational calculus, allowing for thrust and thrust setting modulation, the actual solution ean provlde, in a slmple way, basic conolusions in the field.

4.

1. FORMULATION OF THE PROBLEM

The equations of motion for low-thrust propulsion of a vehicle which was initially moving on a hyperbolic orbit have been derived in ref. 1, and can be written as follows:

c ( 1-m)

=

± 1 /

lf (

ç;) -

f

(~

i

)l

(2U ) 3 2

'J

u

=

U o - lCOS 0 + 2U o o

(1.2)

h

=

ho

+

(2U~)572

[·[g(,)-g(l;1)]COSO+[J(')"J('1

J

]Si

n

Ö}

(1.3)

The above equations are valid in the case of constant

I

thrust, set at a constant angle 0 to the velocity vector (0 positive for thrust vector below velocity vector). In the double signs, the upper one correeponds to an initial motion of the vehicle towards perigee (d~ < 0) while the lower one corresponds to a motion from perigee towards infinity (d~ >

0).

The parameter ~ is the hyperbolic excentric anomaly; the

branch of the hyperbolá for which ~ is positive has only been considered. At the point where propulsion is initiated, the value of ~ is given by the expression

1+2U Pi

-1 0

=

cosh ( )

e

(1.4)

o

The auxiliary functions which have been introduced in the above equations are defined as follows:

f (~)

=

E;, - e s i nh E;, _{o .}

H(

d

=

1 tanhE;,

l..j

t;,

--)F(k,<p )-E(k,<p)- (e2 _cosh2_E;,-1)2

2 e 0

e 0

o 0

(1.6)

I( t;,)

=

fE;,

[e~

COSh

21l-1]

-~

dil

= -

el [F(k,<P)] E;,

0 0 0

(1.8)

1 e c osht;, 1

=

22cOSh- (e coshE)-2(1- 0 4 · )(e2 COSh2t;,-1)2

o - 0

where F(k,<jl) and E(k,<p) are the inco~plete eLliptic integrals of first and second kind respectivelyo Argument and modulus are defined by the following relationshipsg

6.

-1 1 (1.10) cp

₌

_{sin (COSh F,;)} k

₌

1 e 0

One shou1d note for further reference that the

derivatives of the auxi1iary functions have the fo11owing signs:

j F,; > 0 f F,; < 0

The writing of the basic equations (1.1) to (1.3) can be somewhat simp1ified if one introduces the coefficients

a = --L 2U

o

and the fo11owing notation for any function y(F,;)s

If in addition one writes

AU

=

U-U, _o Ah

=

h-h

0

the system of equations (1.1) to (1.3) becomes

c(l-m)

=

:i: dAf A U :i: a AHcos ö

=

0 Ah-b(:i:l\gcoSo+Cljsino)

=

0 (1.12) (1.14) (1.15) (1.16) (1.17)

Considering that the characteristics of the propulsion system are given - i.e. Tand c - and if both intttal and final orbits are entirely defined, the above system contains four

unknowns, m, ó, ~ and ~i. Consequently, one of these parameters is free and ean be used for optimization.

It should be noted that although ~. is direetly ~

representative of the value of the radius vector at the point where propulsion is started, eq~ (1.4) does not hold any longer during the subsequent motion. The variable ~ must be regarded as a deseriptive parameter whieh always remains attached to the initialorbit. Consequently, the basic system of equations does not allow for calculatioQ Jof the exact point at whieh the final orbit is reached.

2. UNIQUENESS OF THE SOLUTION

Regarding the quantity ~i as fixed, the prob1em is eomp1ete1y determinedj the solution of the system of equations ean be obtained as fo11ows o

From Eq.(lo16), one obtains immediate1y

cos

₌

_{+ a~H} t.U (2.1)

and, substituting in Eq.(1.17), one wou1d obtain the fo11owing equation

R(~) - 6h-b(*t.geosó+t.jsinó)

=

0 (2.2)

whieh is imp1ieit for ~ as the on1y unknown. The solution

e~rresponds to a zero va1ue of the residue R(~). The solution of the equation is restrieted by a condition of possibi1ity, sinee one must a1ways have leosól ~ 1, or, from Eq.(2 01)

1

6U

I 1 a6H ~

As one varies progressive1y ~ , starting from the initial value ~i' in either direction, Eq.(2.1) shows leospl being equal to infinity at the starting point and then de~

ereasing eontinuously until it beeomes equal to 1 for a par-tieular value of C denoted by ~R. The solution is eonsequently real for ~ < ~R if the vehiele initially moves towards perigee,

or for ~ > ~R if the vehicle moves in the opposite direction.

To discuss the actual sign of coso,Eq.(2.l) can be written, to the first order, as

coso

Since H~ > 0, coso has the sign of the quantity ~U. One can consequently define a parameter € as follows

E: = + 1 for ~u > 0

c = - 1 for ~u < 0

One accordingly obtains

( cos 0)

=

E:

~=~ _R

and the particular residual value

which can be positive or negative.

(2.4)

To proceed any further, the derivative of R(~) with respect to ~ should be evaluated. Derivation of eq.(2.2) yields

10.

Using Eq.(2.1) and retaining first order terms only, i.e. regarding the difference between ~ and ~i as a small

quantity, one obtains

de

=

d~

and, substituting in Eq.(2.6) b

sine

j~

(2.8)

Since j~ > 0, the sign of the derivative is dictated by the one of sinó, and since ó is regarded as constant, the function R(~) is monotonous in the real domain.

The secure a root to EQ.(2.2), i t is then obvious that the sign of sinó must be chosen in accordance with that of R(~R)J the following set of conditions can consequently be established:

d~ < 0 (motion towards perigee)

d~ > 0 (moti on towards infini ty): R (~R ) sip ó > 0

The above reasoning establishes the uniqueness of the determination of the solution, and is illustrated by Figs 1 and 20

11.

3. OPTIMIZATION

Optimization bears upon minimization of the fuel consumption, given by the quantity (l~m). For analytical discussion, the system of Eqs.(1.15) to (1.17) can be more conveniently written as

o

(3.1)

The quantity y to be minimized is a function of ~ and

~i only, and it is obvious from Eqs (3.1) and (1.5) that y wil1 assume its smallest ~alue when the difference between ~ and ~.

l-is the smallest, taking into account the constraints (3.2) and (3.3). These twoequations can be used, with 0 as a parameter,' to define the conditions wherepy ~-~i has the smallest possible value.

Introducing the auxiliary variable

x

=

coso

(3.4)

implicit derivation of Eqs.(3.2) and (3.3) with respect to x yields

d( t;- E;i) dx

=

with P(ö,t;,t;i) Q ( ö, t;, t;i ) D ( ö, é;, E;i )

where the auxiliary

F(E;,t;i)

=

G ( E; , E;i )

₌

W ( ~ , t;i ) = p(ö,E;,t;i)-Q(ö,t;,t;i) D( ö, t;, t;i) e bflU 0 F( t;, t;i)

=

_sinöcosö ;

acot2ö(flh~bflgö)G(t;,ti)

= cos = ; abeocotöW(E;,t;i)

functions of t; and t;i are defined by 1

o ,~

r

COShC-T

sinhE;i (eocosht;i- 1 ) eocosht;i+ 1

te COShHf

-sinht;(e cosht;-l) 0 ht 1 o e cos + 0

[e~cosh2t;i

·

-l] ~

-

[e~cosh2

t;-l

J

~

1

t'e coshC-l)(e

I 0 ~ 0

COSM-l~J'

(e cosht.+l)(e coshE;+l) o ~ 0 [sinhti(eOcoShE;i-l)(eocosht;+l) -Sinht;(eocosht;-1)(eocosht;i+1 )]

(3.5)

(3.6'

(3.8)

13.

One checks easi1y that all functions F(~, \ ) , G(~, ~i~

W(~'~i) are positive for ~ < ~i and negative fDr ~ > ~i.

Discussion .of the sign of the derivative

(3.5)

requires to distinguish between 'áll possib1e values of 6, in accordance with conditions

(2.9).

To illustrate the procedure, the ~se of initial motion towardsperigee and decrease in energy wiil be thoroughly discussed. Upper signs should con-sequent1y be used in all equations, and one easily estab1ishes that

ÊlU < 0, cOS6 < 0, F(~,~J.') > 0, G(~,'C) > 0, W(~,C »0

J. J.

but sino can either be positive or negative, depending upon the sign of R(~R)o

Considering first R(~R) >

0,

then sino <

0

and

Eq.(3p 6) yie1ds immediately

On the other hand, R(~R) > 0 implies that

h

~ ~

(l+coso)

ÊI -cos6 > -coso

(3.12)

since ~ < ~i and g~ < 0, Êlg is positivé and the second member is a positive quant~ty. One may thus write

Êlh-

~

> 0 coso

Consequently, from Eq.(3.7)

Moreover, from

Eq.(3.8)

D ( ö, E:, E:

i ) < 0

Substituting then in Eq.(3.5.) yields

<

°

Considering next the other case where R(E:

R) < 0,

sinö must be positive, and Eqo(~o6) shows immediately that

(3.16)

(3.18)

The discussion of the term Q(ö,E:,E:

i ) is somewhat more complicated. The condition R(E:

R

1

< 0 tields actually

~

~h - cos ö < - _cos b~g Ö (l-+cos ö)

which .shows that the first member, smaller than a positive quantity, can either be positive or negative. If one first

assumes that i t is positive, one necessarily has

~h

-

~

> 0

15.

But Eq.(3~3) must be satisfied with ~j < 0 in the present

situatiog. Eq. {3.20) shows that Eq.(3.3) can on1y be satisfied if sino < 0, which is contrary to the initia1 statement.

Consequent1y, the on1y possibi1ity is

~h

-

~

< 0

cos 0

in what case

Q ( 0, ~, ~i) > 0

Fina11y, from Eq.(3.8) one has

(3.21)

(3.23)

and Eq.(3.5) yie1ds acondition identica1 to Eq.(3.17), name1y

< 0

Since ~ < ~i' the above condition.shows that the smallest possib1e va1ue of the difference wi11 correspond to the smallest possib1e va1ue of x. Consequent1y, since coso is

negative, optimu~ conditions wi11 correspond to

cos Ó -1 (3.24)

or tangentia1 retvothrust. Mathematica11y speaking, optimum

16.

with the boundary of the domain wherein the solution is realo Similar arguments can be repeated for the other cases; they show that whether initial motion takes place towards perigee or towards infinity, optimum conditions correspond to tangen~ tial thrust, with ó

=

0° for increasing energy and ó = 1800 for decreasing energy.

- --

---

---.

4. SELECTION OF OPTIMUM INITIAL TRAJECTORIES

Under the optimum conditions defined in the above seetion, the system of equations (1 015) to (1017) beeomes simp1y e(l-m) :l t:. f t:.U =F-ea t:.H t:.h = : l ebt:.g (4.1) (4.2 ) (4.3)

For given initia1 and fina1 orbits, Eqs.(4.2) and (403) must be used to ea1eu1ate the va1ues of ~ and ~., and Eq.(4.1)

~

yie1ds subsequent1y the va1ue of the fue1 eonsumption.

Optimization oan be earried one step further if one considers that one of the two orbits is not entire1y determined. For instanee, one may regard the energy of the fina1 orbit

as inexorab1y fixed, whi1e the va1ue of the angu1ar momentum ean be ehosen arbitraritYo In sueh a case, the quantity t:.h is not fixed, and one more parameter is eonsequent1y re1eased o

Considering then ~. as a free parameter, the two fir~t equations

~

ean be written as

c(l-m)

d

U(~,c)

=

t:.U :I: e:atlH

=

0

~

inspee-tion of the derivative Y(,i' whieh ean be evaluated from the two above equations. Eq.{4.3) remains then for the ealeulation of L\h.

Using Eqs.(4 0

4)

and (405), one obtains

(4.6)

with

One can easily verify that the function l-Z(~'~i) is positive for ~ < ~iand negative for ~ > ~i. Combination with the double sign shows that one has in aLl cases

or

(4.9)

It follows that;, in order to obtain best conditions, the angular momentum of one of the ~wo orbits should be adjusted so that, for ~nit~al motion towards peri~ee, propulsion term~

nates as close as possible to the perigee; for initial motion towards infinity, propulsion should be initiated as close as possible to the perigeeo The system of Eqs.(4.l) to (403) can

consequently be simplified by putting ~

=

0 in the first case, and ~ 1

=

0 in the seeond one 0

19.

The system of equations must be solved numerically, Attention must be drawn on the fact that the problem becomes more complicated, from the point of view of numerical

calculations, in the special case where the angular momentum h is left arbitrary. S1nce there is one relationship

o

e 2

o 1 + 2U ho 0 2

between excentricity, energy and angular mo.mentum, the numerical calculations become more ~nv~lved because of the presence of the ~actor e in the elliptic integrals.

p

20.

5.

CONCLUDING REMARKS

A simplified approach to the problem of optimum low-thrust trans~er between coplanar hyperbolic orbits has been developed on the basis of an approximate analytical solution, whereby thrust is assumed to be constant and set at a constant angle to the velocity vector.

The analysis shows that for given initial and

final orbits, optimum transfer with regard to fuel eonsu~ption is always obtained with tangential thrust.

In the case where the angular momentum of one of the two orbits can be ehosen arbitrarity, fuel consumption ean be further reduced by a proper selection of' that param-eter, whieh eorresponds to initiating propulsion as close as possible to the perigee of the initialorbit, in the case where the vehicle moves fromperigee towards infinity, or terminating propulsion .as close as possible to the perigee of the final orbit, if the vehicle moves towards the perigee.

In optimum conditions, the problem reduees to a system of non ,linear equations, which can be solved numeri-cally without difficulty.

21.

REFERENCES

MOULIN, Le: Approximate analytical solution for low thrust propulsion in spaceo

(AF EOAR Grant 63-33, Scientific Report n° 1, December 1963) e

R ( ~)

R

~~

(

~R)

'imaginary I

~:

~R

FIG. 1. INITIAL MOTION TOWARDS PERIGEE

R (~ )

~R«)

imaginary R

~

v.K.I. TN 15 AF EOAR ORANT 63-39

SR-2

von Karman Inst1tute for Flu1d Dynam1os,

May

1964.

APPROXIMATE ANALYSIS OF OPTIMUM LOW THRUST

TRANSFER BETWEEN COPLANAR HYPERBOLIC ORBITS.

By L. Mou11n.

Opt1m1zat1on ot low thrust transfer between

oop1anar hyperbo1io orbits has been ana1yzed, on the bas1s of an approx1mate analytical

solut10n, 1n tha oase of oonstant thrust set at a constant angle to the veloc1ty vector.

t.o.P.

The 1nvest1gat10n bears upon m1n1m1zat10n of

fuel consumpt1on and selectton of an 1deal 1n1t1al orb1t. For g1ven 1n1t1al and f1nal orb1ts i t is shown that opt1mum cond1t10ns

can be obta1ned w1th tangential thrust.

Fur-ther ref1nement consists of a proper se1eo-tion of the angular momentum of the 1n1t1al orb1t, so as to e1ther 1n1t1ate or term1nate

low thrust propuls10n as close as poss1b1e

to perigee, depending up on the direction of transfer.

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'SOlWVUÄQ PInlä ~oJ 9.n.I.9uI uvm~v~ UOA

~-HS

v.K.I. TN 15 AF EOAR GRANT

63-39

SR-2

von Karman Inst1tute for Fluid Dynamics,

May

1964.

APPROXIMATE ANALYSIS OF OPTIMUM LOW THRUST TRANSFER BETWEEN COPLANAR HYPERBOLIC ORBITS. By L. Moulin.

Opt1m1zat10n of low thrust transfer between ooplanar hyperbo110 orblts has been analyzed, on the basis of an approximate analytioal solution, in the case of constant thrust set at a constant angle to the velocity veotor.

t.o.P.

·JeJSUVJ~

JO uot~OeJIP e~~ uodn ~utpuedep 'ee~tJed o~

elqtssod sv asol0 sv uOtslndoJd ~snJ~~ MOL

e~vutwJe~ JO a~vt~tut ~e~~ta o~ sv os '~tq~o

lvt~tut e~~ JO wn~uawow ~vln~uv a~~ JO UOI~

-oales ~edoJd v JO s~sISUOO ~ueweutJeJ Je~~

-~n~ ·~snJ~~ lvt~ue~uv~ ~~lM peutv~qo eq Uvo

suOt~tPUOO wnwl~do ~v~~ UMO~S Bt ~t s~tqJO

lvulJ pu. 1.t~tut ueAt~ JO~ ·~lqJO lvt~tut

lvept uv JO aOl~oales pu. uot~dwnsuoo lenJ

JO UOt~vztWtutw uodn SJveq UOt~.~l~seAUt e~~

The investigat10n bears upon m1nim1zat10n of fuel consumpt10n and selection of an 1deal

initial orb1t. For g1ven initial and final

orblts l t is shown that optimum condltlons can be obtained wlth tangential thrust. Fur-ther reflnement consists of a proper seleo-tion of the angular momentum of the initial orbit, so as to either init1ate or terminate low thrust propulsion as close as poss1ble to perigee, depending upon the direotlon of transfer.