1_application of the methods of analytical dynamics to the theory of optimal flights

8/12/2019 1_Application of the Methods of Analytical Dynamics to the Theory of Optimal Flights

1/22

.

b

N A S A T T F-9294,

A P P L I C A T I O N OF T H E M E T H O D S OF A N A L Y T I C A L D Y N A M I C S

T O

THE

THEDRY OF O P T I M A L F ' L I G H T S

V.S.Novoselov

N6 5 - 1 9 5 1 0

ITIIRUI

8

..

-

/

i

TO

(CATIQORYI

NASA CR OR

TMX

OR

AD

NUMBER)

~

-

~~

Transla t ion

of

nPrimeneniye metodov analiticheskoy

dinamiki

v

t e o r i i optimalfnykh poletov".

Vestn. Len. Univ., s e re mat. mekhan.

i astron.,

NO

-3

9 pp -133-146, 1964.

N A T I O N A L A E R O N A U T I C S

AND

S P A C E A D M I N I S T R A T I O N

W A S H IN G T O N MARCH 1965


2/22

NASA TT F-9294

. -

3

PPLICATION

OF

THE MJ3l"ODS

OF

ANALYTICAL DYNAMICS

TO THE

THEDRY

OF OPTIMAL FUGHTS

V

.

Novo s elov

\ q s i o

i

I

I

The author gives an exposition of th e theory of optimal pro-

v

I

I

' ,

ce sses, based on t h e methods

of

analytical dynamics.

teg rat ing th e varia t ional . foxmula of th e act ion fun ct ional

o ve r t h e t r a j e c t o r i e s o r

arcs

of

a

s p e c i a l f i e l d o f

extremals,

the

m a x b u m

p r in c ip l e

i s

proved, and th e suf fi ci en t conditions

of the extremum are established. The po ssi bl e appl icat ion of

t h e Hamilton-Jacobi method t o the construction of optimal

re-

l3y in-

_ _

' i I gimes

i s

poin ted ou t ; s ta r t ing

f r o m

t h a t method, equations

- ', i

-.. b i t s are obtained.

2 5

defining

th e optimal

impulse transfers

between mplanar or-

I

- ,

,Sect ion 1. The Conditions of Stationarity

y .

Given: a control la ble device whose motion i s described by t h e system of

.?

> '

,ordinary

di f fe ren t ia l equa t ions

:

1.1

+ ( I

;where th e 4

t )

are t h e c o n t ro l s o r th e cont rol functions, which may be piece-

/ :

2

wise

continuous; and t i s th e time.

I

, '

The controls must

sa t is fy

t h e

conditions:

' I - 1

' 4

' ,

u 1

\< uk

< I

-

- 7

:>

1

- 2

4

L e t th e functional.

J

have

a

smooth

maximum.

Then,

by equating AU

t o z er o

:we a r r ive

a t

th e necessary conditions of th e

extremum

) .

-

:)

j

i t :

gS

expressing, i n eq.(1.9), th e t o t a l var ia t ion s

of

t h e v a r i a b le s xi

in

-

,terms

of isochronous var ia t ion s by the formulas ibq

= 6 +

q

A t ,

and equating

i t o z e ro t h e

terms i n A t and 6- a t

the

times to

and

T,

the conditions (1.9)

, I

a -

{will e reduced t o t h e form of

/1


5/22

The conditions

1=7),

1.81, and

1.10)

are necessary conditions

for

t h e

v a r i a ti o n a l problem of Bolza (Bibl.4).

o f the

system

1.10) must be omitted and i f

T i s

prescribed, t h e second

equa-

If

to

i s

prescribed, the f i r s t

e q u a t i o

-

- t i on of th at system.

1

) +

' d

W e now no te t h e important problem when t h e f un ct io na l t o be minimized nee

'

0 .

, ; ' n o t

have

a smooth minimurn. T h i s

i s

a rapid -acti on problem. I n this case, Fo

1 2 '

-

T

Hence we obt ain t h e e qu al it y

and t h e condit ions of t ransversal i ty

=

HT

=

-1, as well as eqs.(1.7), (1.8)

The function R , - i n

this

case, i s defined by th e fonnula

I

The necessary cond ition s f o r other optim al problems

w i t h

a

non-smooth ex-

t remum

of

the fundamental functional are obtained similarly.

I f a

la rg er number of contr ols are being considered, then eqs.(l.8) a r e

replaced by the re la t ion s

(1.81

Now

l e t

the con t ro ls

yk

not be introduced, and

l e t us

inves t iga te the p ro

l e m with only th e or igi nal controls

.

I n t h i s case, eqs.(l.8)must be re-

placed by the conditions


6/22

-ote 1.

The system of equations 1.1)and

(1.7)

i s a canonical system

' ,

ith t h e Hamiltonian H, the pulses h i , and th e coordinates

-

- and

in

that the add i t iona l va r iab les 4 and yk a re present .

. T h i s canonical

system dif fe rs from the

usual

canonical system i n that

H

depends linearly on

A:

L

\

i

Since the system of equations 1.1) and (1.7) i s canonical, w h i l e t h e

1

7

1 :

I :

1

boundary controls

funct ion H

w i l l

be equal t o t h e pa r t i a l time d e r iv a t iv e o f this function with

t he group of v aria bles 1 ,

4

t.

ject ory with continuous contro ls, w e shall have:

1 and 1+2 a r e c on st an t, t h e to t a l time de ri va ti ve of the,&

I -

,

Hence,

on

any phase of th e opt imal tra-

(1 . l l

,

.

2 .where h

=

const and

i s t h e i n i t i a l

time

of that phase.

If th e func t ions Fo and

Fi

do not e xp li ci tl y depend on t , then H w i l l be

consta nt on any such por tion .

i

i

s

-- -

1-

- \

' ,

Note 2. A point

a t

which

t h e

controls undergo

a

discon t inu i ty i s c a l l e d

I

. a

corner.

L e t

us

vary the pos i t ion

and

time

of

some corne r po in t by th e

quan-

Then, from eq.(1.5), t h e va ri at io n of th e fundamental func

. >

t i t i e s

hxi and AT.

4

- < #

dtional

U,

taken on th e opt imal ar c before and a f t e r th e comer,

w i l l

be

I

4 1

This vari a t io n of t he funct ional must vanish f o r th e op timal a rc .

w e

ob ta in th e well-known Weierstrass-Erdmann co nd iti on s of cont inu i ty of i an

H

along th e en t i re opt imal a rc .

Hence

7 4

I

i '

1 ,

'

I

For

t h i s

reason, when

t h e

functions

F,,

and F1

are s t a t io n a ry , t h e Hamil-

\ I

1

I t oni an H

w i l l

be one and th e same constant fo r th e en ti re optimal arc.

For ex

~


7/22

1

ample, f o r th e rapid-action problem, as follows from t h e above discu ssion, it

1

,

w i l l be equa l t o minus unity .

- 4

,

( ,

,S ec tion 2. The

M a x i m u m

Pr inc ip le

-

4

-

1

O 1

,

, over the tubes o f t he t ra jec t o r ie s i s widely used i n an aly tic al mechanics (se e

1 'Etibl.4).

The method of in teg ra tin g th e var ia tio na l formula of th e acti on fun ctio

i

i .?

Let us apply i t t o f ind t he necessary and suff ic i ent conditions of

1 ,

I

t h e

extremum.

Consider any admissible arc AB of the system

1.1)

and

(1.7)

w i t h t h e con-,

This

a r c i s made t o pas s through any two points A and B satisfyin4

1

'

y

t r o l s {h

.

eqs.(l.3)

a t

cert a in t imes To

and

T.

I

( 1

2

In p a r t i c u l a r , t h e points A and

B,

a s

2 ) 1

- - 1

well

as

the times

%

and?, may coincide with

t h e i n i t i a l

and f i na l pos i t ions 1

2 )

_ -

i

-

1

c :E 3 of the opt imal arc-

1

;and times of th e opt imal arc , a n d t h e a r c AB can

di f f e r

from

the optimal only

iover a cer t a in in te rva l fo r which the con t rols

1

_ I ) ~-

differ f rom the controls

~

I c,

I

Let

us

now pass through

t h e

point

A

and the

i n i t i a l

point of t he optimal

.- 1

- I

.,

a rc

i n

t h e

(2n

+ 1)-dimensional space of configurations of the coordinates

a,

impulses h i and time t , a certain curve

(.$:

I

~

+) iWe note th at f o r th e t ime being e q ~ ~ ( 2 . 1 )ay be ar bi t r ar i l y ass igned, subject

- 0

? A X ~ - X ~ ( Q ~ ) ,

- i = ~ ~ ( ~ p ) ,

o = t ( a p , go), a p = a p ( p J . p a -* -1.

2.1

.

,

a t

3V

,:

: t o t h e requirements that-> 0, and that the contour of ($ passes through the

( i

I

I -

r ;below.

above two points.

The addit ional res tr ic t ions on t h e choice of

C$ w i l l

be

giv

I

I

We s ha ll apply the term extremal t o any ar c of t h e

system 1.1)

and

. 1.7).

;curve t included between the

in i t ia l

point

of

th e optimal curve and th e point

Let us now pas s through po in ts

of

the curve

AB

and

that

p a r t o f t h e

I

-

i

7


8/22

, A , t h e

extremals corresponding t o cont ro ls equa l t o th e con t ro ls { & I on the

.opt imal arc .

itremals

crossing th e curve

AB

shall

i n t e r s e c t

i t .

Let us continue the contour C$ beyond the point A

so

that t h e ex

I

L e t t h e r e l a t io n s

2.1)

be such

that

for

v = v

th e curve

6 shall

be

,

. transformed

by

t h e se extremals in to the curve

Gr

passing through the end point

,o f t he opt imal a rc and the po in t

B.

,

Lo

.

1 1

The contour of G

w i l l

be defined by the

2.2

1

Through th e poi nt s of this segment of t h e curve

I

l e t us draw extremals

w i t h c o n t ro l s

ctst)

such t h a t

t h e s e

extremals, for v

=

v, shall completely f ix

th at segnent of t he curve G

between t h e end point

of

th e op t imal a r c and the

point

B, i f

th e following condit ions are s a t i s f i e d : ~

-

a;

t

Q p

(PI,

uo))=u&

u;

( p(l.4*4

k t ap(PI.

d .

(PI, *),

A t

any t h e

t

t h e

boundary

values

f o r t h e c on tr ol s

{d]

i l l be and

- Gj.

* I

1 % +

\

I

1- final

condition:

Let

us

now sub je ct th e choice of the curve

and the controls

{d]

o thc

I ,

- i

Let

us

write out the condit ion (2.3) i n t h e expanded form:

( 2 . 3

2.4,

I

7


9/22

The following notation

has

been used in

eq. 2.4):

I

A l l th e above conditions can be sa ti sf ie d, because of t he f ac t that t h e A

functional dependence

of x

and

1 on aP is

arb i t ra rg , the func tions

up

of the

parameter

P

and of th e time t a r e ar bi tr ar y, and th e funct ion t(Q',, v)

i s

- l ikewise a rb i t ra ry .

- t

-

..

i l

1

4

For any

poss ible in te rp lane ta ry f l ig h ts , th e acce le ra t ion

developed by

modern rocket engines considerably exceeds the other terms on the r igh t s ides

of eqs.(L.l).

impulsive c r i t er i on of veloci ty:

1

Therefore, f or t he boost phase, l e t us take t h e equations of t h

I


14/22

~-

~ ~ Z , I - = ~ ~ C O S + . vF==i j s int , i = ~ ,if=o. (4.1

Star t ing ou t from the capab i l i t i e s

of

rocket engineering and th e allowabl

accelera t ions,

a

r e s t r i c t i o n on th e c o nt ro l r e l a t i v e t o t h e

value

of

B

m u s t

be

adopted :

- I

'

-

'

0

(4.2

I i

' I

The equations of motion

on

the passive phases a re obtained f rom the sys te

I _

,

'

(4.1) by s e tt i n g

= 0.

Understanding the term optimal i n the sense of minimizing

t h e

consumption

of mass, w e

s h a l l minimize

t h e

functional

T

(4.3

_ I

.

where

i s

t he i n i t i a l

mass

and

mrc tne

final mass of t h e ship, while and T

- 1

~~

- ,

' a r e the respective times of s t a r t and flnish.

I

The fu nc ti on al (4.3) must be minimiized by means of th e co ns tr ai nt s

4.1)

2 ) .

:,>

-

.

~

'or 4 , l v ) , r e s t r i c t i o n of th e con t rol o f i n t h e form

of

eq.(4.2), and th e

- I

. boundary conditions (Bibl.8):

Here the subscript H i n d ic at e s t he c h a ra c t e r i s t i c s of t h e in i t i a l o rb i t

and

the subscr ip t K those of th e terminal o rb i t .

used i n eqs.(h.l+) and (4.5): e

=

eccentr ic i ty ; p = parameter; w = angular d i s -

tance of the per icen ter f r o m t h e polar

ax i s ;

f = true anomaly.

I n t he case under consideration,

"I,,

=

0

and the

boundary

condit ions are

The following notation

i s

expressed by the exp lic i t re la t i ons (4.4) and (4.5).

not introduce t he indeterminate constants Vp, and shall use t he conditions

of

t r a n sv e r sa l i t y i n t h e g e n eral form:

For this reason w e s h a l l

~~ ~

0.

I

13

I


15/22

~~ ~

Here the to ta l va r ia t ions of the var iables ar e calcula ted along the limit

-

'

except

A t

are

expressed respectively

i n

terms of

A f

and

bfK

.

In formulating

in g o rb i ts .

I n

v i e w

of eqs.(4.4) and

(4.5),

th e var ia t ions of all var iab les

-

th e problem w e do not consider t h e specif i c motions a long the l im it i ng o rbi t s ,

so that th e var ia t ions of AfH ,

AK ,At and A t w i l l be independent.

obtain & , H K =

0

and, s in ce eqs.(l+.l) a r e st ati on ar y, we obt ain H = 0 along

the en t i re op timal t ra jec to ry .

the l imi t ing or bi ts are proport ional to the r ight s ide s of eqs .(4.1) , calcu-

Hence w e

The variations of vel oci t ie s and coordinates o

.

l a t e d on th e l imi t in g o rb i t s f o r B = 0.

W e

a r r i v e

a t

the following conditions

of

t ransversa l i ty :

-i

j

2 :bits, respec t ivdy .

- I

- t

S t a rt i n g out from eqs.(L.l') l e t

us

set up

1

1

t i v e phases:

H=b(--- l

+A,cos*j,+).,s

the funct ion

H

for the ac- A

We now w r it e th e Hamilton-Jacobi equat ion:

* .

+I(-]W

+2;F;W

at

The complete i n t e g r a l of

th i s

equation

is

The i n t eg r a l s of e q ~ ~ ( 3 . 3 )nd

(3.4)

y i e l d

f

(4.7

W e reca l l the condi t ions of nxdmm H on the opt imal t ra jectory.

Writing

out t h e

f i r s t

and second p a r t i a l derivatives of th e function

(4.7)

with respec


16/22

~

t o 5 and

,

we come t o th e conclusion that the ac t iv e phases can include th e

tlE

tr aj ec to ri es : ( a ) of th e programmed

t h ru s t

a t -

da

o r

(b) phases of mardmMl react ive accelera t ion G a t

.

(4.5

Since H =

C

and H is a linear homogeneous function of B th e s ign of ineq ual i ty for t h e

der iva t ive

-% i s

dropped and

t h e

inde te rmhate mul t ip l ie rs he re a re also de-

%

telmined

from

eqs.(4.9).

1,

Equations (4.8) and (4.9) show

that

the inc l ina t ion ang le f of th e th rus t

,

remains

consta nt f o r any powered sec tio n

of

t h e t r a j e c to ry .

For

each such

- , ,

phase

or

section, eqs.(4.l')

and

(4 .8) w i l l y i e l d

(4.1

- 1

-.

I t follows

more

p a r t i c d a r l y

from e q s . ( i + . l O f t f i a t

tim

consumption--ofmass

during

t h e

ac tiv e phase i s determined by t h e requi red increment of ve lo ci ty an

7

1

-

does riot depend on t h e p a r t i c u l a r law of burning.

On th e passiv e phases, t h e controls over

B

and are shut

down.

These

phases admit

of the

maximum

principle.

L e t us f ind an expression f o r th e inde

terminate factors

on

the passive phases

by

t h e Hamilton-Jacobi method.

The Hamilton-Jacobi equ ation i s of t h e form

dW

d t

-

4.u.

The complete integral

i s

W=a,t+ccpp+@(v,, &H r . %

where

@

i s

a

certain function of these arguments.

On

the bas is of eq. 3.4), t h i s yields


17/22

'For his

reason,

ha =

const f o r both passi ve and ac ti ve phases.

v i r t u e

of

t h e Weierstrass-Erdmann con ditio ns,

1

is the same constant over the entire o p

timal

t ra jec to ry .

-

.

For the passive phases, w e have

/complete in te gr al

i s

writ ten

i n

t h e

form

of

= 0, hence L

=

0, and by 9 . (3 . 6 ) t h e

.

I

i o I

2

S t a r t i n g

from

eq.(3.4),

w e

obtain expressions

for

the indeterminate multi-

( p l i e r s :

'

.-

- -

,')

1 The contin ui ty of th e indeterminate mu lt ip l ier s , t h e i r exp l ic i t expres-

,

The general representation

[

eq. (4.22)] ob ta in ed above

i s

p a r t i c u l a r l y

convenient fo r consider ing quasic ircular t ra ns fer orb i ts .

*.

r 4

isions given by

eq~~(4.9)

nd (4.22), the boundary conditions (L.4)

-

( 4 . 5 ) ,

I

, t h e condi t ions (4 .10) fo r ve loc i ty

jumps

f o r each ac ti ve phase, and t he condi-

t i o n s

(4.6)

of t ransversa l i ty , y i e ld

a

number of eq uations su ff ic ie nt fo r

de-

termining a l l unknown qu ant i t ie s f o r

a

pr es cr ib ed number of impulses.

An in v e s tig a t io n

of

these equations would be a se pa rat e problem,

beyond

th e scope of t h i s paper .

2


22/22

NASA 'M' F-9294

I

i

BIBLIOGRAPHY

,

1. Ykiele, A.:

General Variational Theory

of

the F l igh t

Pa ths of

Rocket-

I

Powered Aircra ft . Missi les and S a t e l l i t e Carriers. Astronautica acta,

-

I

\ *

VoLIV, No.4, 1958.

$

i

1 2 1 malfnykh trayektoriyakh raket ). Priklad. Matem. Mekhan.,

Vol.XXV,

j

2. Leytman, G. [George Leitmann]:

On Optimal- Rocket T ra je ct or ie s (Ob op ti-

1 :

' .

No.6, 1961.

3. Ui tt ek er , Ye.T. [E.T.Whittakerl: Ana ly ti ca l Dynamics (Anal itiche ska ya

I *

dinamika).

O N T I ,

Moscow

-

Leningrad,

1937.

I

4.

B l i s s ,

G.A.:

Lectures on th e Calculus

of

Varia t ions (Lekts i i PO var ia t s -

ionnomu ischisleniyu).

International Publishing House

( I L ) ,

Noscow, 19504

~

8.

,

Ye.F.: Mathematical Theory

of

0pti.ma.l Processes (matemakicheskaya t e w ,

ptimalfnykh pr ot se ss ov ). Fizmatgiz, Moscow,

1961.

Pon tryagin, L.S., Boltyansk iy,

V.G.,

Gamkrelidze, R.V.,

and

Mishchenko,

i

I

Brekvell [J .V .Breakwell] : Optimization of Trajectories (Optimizatsiya

i

I

tray ekto riy ). Vopr. raket. t e k h . , No.1, 1961. [North American Aviation, '

Inc. , Report NO.AL-2706, August,

19571

.

I

Gyunter, N.M.: In tegra t ion of First-Order Pa r t i a l Dif fere nt ia l Equations 1

I

integrirovaniye uravneniy pervogo poryadka v chastnykh proizvodnykh)

ONTI, Gostekhizdat, Moscow, 1934. I

I

I

Subbotin , M.F.: Course i n Cele st ia l Mechanics (Kurs nebesnoy mekhaniki). 1

I

Vol.1,

OGTZ,

Gostekhizdat, Leningrad

-

Moscow,

1941.

- 1

9. Lawden, D.F.: In te rp lan eta ry Rocket Tr aj ec to rie s. Advances i n Space I

I

Science,

Vol.1, 1959.

I

,

Received

10

May

1963.

1_application of the methods of analytical dynamics to the theory of optimal flights

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