V.G. Petukhov E-mail: [email protected] Khrunichev State Research and Production Space Center.

V.G. PetukhovE-mail: [email protected]

Khrunichev State Research and Production Space Center

CONTENTS

INTRODUCTION

1. CONTINUATION METHOD

2. OPTIMAL PLANETARY TRANSFER VARIABLE SPECIFIC IMPULSE PROBLEM

3. OPTIMAL TRANSFER TO LUNAR ORBIT VARIABLE SPECIFIC IMPULSE PROBLEM

4. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS CONSTANT SPECIFIC IMPULSE PROBLEM

CONCLUSION

V.G. Petukhov. Low Thrust Trajectory Optimization

2

INTRODUCTION


It is presented common methodical approach to computation different problems of low thrust trajectory optimization. This approach basis is formal reduction of maximum principle’s two points boundary value problem to the initial value problem. This reduction is realized by continuation method.

3

INTRODUCTION


Low-thrust trajectory optimization:

T.M. Eneev, V.A. Egorov, V.V. Beletsky, G.B. Efimov,M.S. Konstantinov, G.G. Fedotov, Yu.A. Zakharov,Yu.N. Ivanov, V.V. Tokarev, V.N. Lebedev,V.V. Salmin, S.A. Ishkov, V.V. Vasiliev,T.N. Edelbaum, F.W. Gobetz, J.P. Marec, N.X. Vinh, K.D. Mease, C.G. Sauer,C. Kluever, V. Coverstone-Carroll, S.N. Williams, M. Hechler, etc.

Continuation method:

M. Kubicek, T.Y. Na, etc.

4

INTRODUCTION


Conventional numerical optimization methods shortcomings

• small region of convergence;• computational unstability;• neessity to select initial approximation when it is absent any a-priori information concerning solution.

These problems partially are connected with optimization problem nature (problems of optimal solution stability, existance, and bifurcation). But most of numerical methods introduce own restrictions which are not directly connected with the mathematical problem properties. So the convergence domain of practically all numerical methods is essential smaller in comparison with the extremal point attraction domain in the space of unknown boundary value problem parameters.

Methodical shortcomings are connected with the computational unstability, the convergence domain boundedness, and (in case of direct methods) the big problem dimensionality.

5

INTRODUCTION


Purpose of new continuation method

“Regularization” of numerical trajectory optimization, i.e. elimination (if possible) the methodical deffects of numerical optimization. Particularly, the was stated and solved problem of trajectory optimization using trivial initial approximation (the coasting along the initial orbit for example).

Applied trajectory optimization problems under consideration

1. Planetary low thrust trajectory optimization (the variable specific impulse problem);

2. Lunar low thrust trajectory optimization within the frame of restricted problem of three bodies (the variable specific impulse problem);

3. Optimal low thrust trajectories between non-coplanar elliptical orbits (the constant specific impulse problem).

6

Problem: to solve non-linear system(1)

with respect to vector z

Let z0 - initial approximation of solution. Then

, (2)where b - residuals when z = z0.

Let consider z(), where is a scalar parameter and equation(3)

with respect to z(). Obviously, z(1is solution of eq. (1). Let differentiate eq. (2) on and solve it with respect to dz/d:

(4)

Just after integrating eq. (4) from 0 to 1 we have solution of eq. (1).Equation (4) is the differential equation of continuation method(the formal reduction of non-linear system (1) into initial value problem (4)).

1. CONTINUATION METHOD

f z b0( )

f z b( ) ( ) 1

d

d

zf z b z zz

10

( ) , ( ) 0

0)( zf


7

CONTINUATION METHOD


Application of continuation method to optimal controlboundary value problem

x

p

p

x

Hdt

d

Hdt

d,

kT xxxx )(,)0( 0

kT xxfpz )(),0(

z

p

z

x

z

p

z

p

z

x

z

x

p

x

xpxx

pppx

x

p

HHdt

d

HHdt

d

Hdt

d

Hdt

d

,

,

,

Iz

p

z

xxxxx

,0,)(,)0( 0 kT

z

xf z

)(T

Optimal motion equations(after principle maximum application):

Boundary conditions (an example):

Boundary value problem parameters and residuals:

Sensitivity matrix:

Associated system of optimal motion o.d.e. andperturbation equations for residuals and sensitivitymatrix calculation:

Extended initial conditions:

8

CONTINUATION METHOD


Using continuation methodfor low-thrust trajectory optimization problem

Optimal control problem reductionto the boundary value problem

by maximum principle

Initialapproximation z0

Initial residuals b calculationby optimal motion o.d.e. integratingfor given initial approximation z0

of boundary value problem parameters

Associated integrating of optimal motion equations and perturbations equations for

current z() to calculate current residuals f(z,) and sensitivity matrix fz(z,)

Continuation method’s o.d.e. integrating

with respect to from 0 to 1

d

d

zf z b z zz

10

( ) , ( ) 0

Integrating of optimal motion equations for current z() to calculate current residuals f(z,)

and for pertubed z() to calculate fz(z,) by finite-difference

Solutionz(1)

CONTINUATION METHOD

1st versionof o.d.e. right

parts calculation

2nd versionof o.d.e. right partscalculation

9

2. OPTIMAL PLANETARY TRANSFERVARIABLE SPECIFIC IMPULSE PROBLEM


10

2.1. TRAJECTORY OPTIMIZATION PROBLEM

Cost function: (constant power, nuclear electric propulsion)

(variable power, solar electric propulsion)

Equations of motion: d2x/dt2=x+aInitial conditions: x(0)=x0(t0), v(0)=v0(t0)+Ve

Boundary conditions1) rendezvous: x(T)=xk(t0+T), v(T)=vk(t0+T)2) flyby: x(T)=xk(t0+T)

where x, v - SC position and velocity vectors, - gravity field force function,a - thrust acceleration vector, x0, v0 - departure planet position and velocity vectors,xk, vk - arrival planet position and velocity vectors, V - initial hyperbolic excess of SC velocity, e -direction of V, N(x,t) - the current power to the initial one ratio.

T

dta0

2

2

1

1

2

2

0

a

N x tdt

T

( , )

OPTIMAL PLANETARY TRANSFER


11



2.2. OPTIMAL MOTION EQUATIONS(CONSTANT POWER)

Hd

dt

1

2a a p

xp p aT

xT

vT

x vT

a p v

~H

d

dt

1

2p p p

xpv

Tv x

TvT

x

d

dt

d

dt

2

2

2

2

xp

pp

x v

vxx v

,

.

fx a a x

v a a v0 0 k

0 0 k

( ; , )

( ; , )

T

T

fx a a x

p a a0 0 k

v 0 0

( ; , )

( ; , )

T

T

za

az z

0

00

: f z b0( )

Hamiltonian:

Optimal control:

Optimal Hamiltonian:

Optimal motion equations:

Residuals:

Boundary value problem parametersand initial residuals vectors:

(rendezvous)

(flyby)

12



f z b( ) ( ) 1

z z z z0 0 1, ~

d

d

zf z b z zz

10

( ) , ( ) 0

d

d,

d

d,

d

d,

d

d,

d

d ,

d

d

o o o

o o o

o o o

o o

2

2

2

2

2

2

2

2

2

2

2

2

xp

pp

x

p

x

p

p

p

p

p xp

x

p

p

p

x

p

x

p

p

p

p

p xp

x

p

p

x v

vxx v

vxx

v

v

v

v

vxx v

vxx

v

v

vxx

v

v

v

v

vxx v

vxx

t

t

t

t

t

t

v

vp.

o

Boundary value problem immersioninto the one-parametric family:

Boundary value problem parametersinitial value and solution:

Differential equations of continuationmethod:

Differential equations for calculation right parts ofcontinuation method’s differential equations:

Ip

p

p

p

p

pI

p

p

p

xppI

p

x

p

x

p

x

pv

xxx

v0

v0

v0

v0

v0

v0

v0

v0

v0

vv

v0v0v0

v

)0(,0

)0(,0

)0(,

)0(

,0)0(

,)0(

,0)0(

,0)0(

,)0()0(

),0()0(

2

T

00

vv

v

pp

V

pV

dt

d

2.3. EQUATIONS OF CONTINUATION METHOD

13

Earth-to-Mars, rendezvous,launch date June 1, 2000, V= 0 m/s,T=300 days

1 - coast trajectory (1= 0)2-4 - intermediate trajectories (0 < 2 < 3 < 4 < 1)5 - final (optimal) trajectory (5= 1)

2.4. TRAJECTORY SEQUENCEWHICH IS CALCULATED BY CONTINUATION METHOD

USING COASTING AS INITIAL APPROXIMATION

1 23

4

5



14

2.5. NUMERICAL EXAMPLESOPTIMAL TRAJECTORIES TO MERCURY AND NEAR-EARTH ASTEROIDS



15



OPTIMAL ORBITAL PLANE ROTATION EXAMPLES

Optimal 90°-rotationof orbital plane

Optimal 120°-rotationof orbital plane

16

EXAMPLE: INITIAL HYPERBOLIC EXCESS OF VELOCITY IMPACT



17

EXAMPLE: NUCLEAR (RIGHT) AND SOLAR (LEFT) ELECTRIC PROPULSION



18

2.7. METHOD OF CONTINUATION WITH RESPECT TO GRAVITY PARAMETER

Sequence of trajectory calculation using basic continuation method

Sequence of trajectory calculation usingcontinuation with respect to gravity parameter

Reasons of continuation method failure: sensitivity matrix degeneration (bifurcation of optimal solutions)

Mostly bifurcations of optimal planetary trajectories are connected with different number of complete orbits

If angular distance will remain constant during continuation, the continuation way in the parametric space will not cross boundaries of different kinds of optimal trajectories. So, the sensitivity matrix will not degenerate

The purpose of method modification - to fix angular distance of transfer during continuation



19

Let x0(0), x0(T) - departure planet position when t=0 and t=T;xk - target planet position when t=T. Let suppose primary gravity parameter to be linear function of , and let choose initial value of this gravity parameter 0 using following condition:

1) angular distances of transfer are equal when =0 and =1;2) When =1 primary gravity parameter equals to its real value (1 for dimesionless equations)

The initial approximation is SC coast motion along departure planet orbit. Let the initial true anomaly equals to 0 at the start point S, and the final one equals to k=0+ at the final point K ( is angle between x0 and projection of xk into the initial orbit plane).

The solution of Kepler equation gives corresponding values of mean anomalies M0 and Mk (M=E-esinE, where E=2arctg{[(1-e)/(1+e)]0.5tg(/2)} is eccentric anomaly). Mean anomaly is linear function of time at the keplerian orbit: M=M0+n(t-t0), where n=(0/a3)0.5 is mean motion. Therefore, the condition of angular distance invarianct is Mk+2 Nrev=nT+M0, where Nrev is number of complete orbits. So initial value of the primary gravity parameter is

0=[( Mk+2 Nrev - M0)/T]2a3,

and current one is

()=0+(1-0) .

The shape and size of orbits should be invariance witn respect to , therefore

v(t, )=()0.5 v(t, 1).



20

d

dt

d

dt

2

2

2

2

x x

z

p

z

pp

xp

x

z

p

z

x xxv

vxx v xx v xx

v

( ) ,

( ) ,

x

z

x

z

x xv

p

p

p

pE

p

p

p

p

p p

0

v

v0

v

v0

v

v0

v

v0

v v

( ) ( ) ( ),

( )

( ),

( ) ( )

,

( )

( ) ( ) ( )

.

/

0 0 00

0 1

2

0 0 0 0 0 00

1 2

d

dt

d

dt

2

2

2

2

x

z

x

z

p

z

p

z xp

x

z

p

z

xxv

vxx v xx

v

( ) ,

( ) ,

( ) , ( )x p p px v v xx v

x x x v

x x x v

0 0

k k

( ) , ( ) ( ) ,

( ) , ( ) ( ) .

/

/

0 0 1 2

1 2

T T

fx x

x vk

k

( )

( ) ( )/

T

T 1 2

fx

xv k

( )/

1

2 1 2

01

z zzf

bzfz

)0( ,)(d

d

z = (pv(0), dpv(0)/dt)T = p pv vo o

T,

fx p x p

x p x pz

v v

v v

( ) ( ) d

d( )

d

d( )

o o

o o

T T

tT

tT

b = f(z0)

Equations of motion:

Boundary conditions:

Residuals:

Boundary value problem parameters:

Equation of continuation method:

where



21

Numerical example: Mercury rendezvousConstant power, launch date January 1, 2001, transfer duration 1200 days

All solutions are obtained using coasting along the Earth orbit as initial approximation

Basic versionof continuation method

Continuation with respect to gravity parameter

5 complete orbits 7 complete orbits



22

EXAMPLES: OPTIMAL TRAJECTORIES TO MAJOR PLANETS OF SOLAR SYSTEM



23

3. OPTIMAL TRANSFER TO LUNAR ORBITVARIABLE SPECIFIC IMPULSE PROBLEM

It is considered the transfer of SC using variable specific impulse thruster from a geocentric orbit into an orbit around the Moon.The SC trajectory is divided into the 4 arcs:

1) Geocentric spiral untwisting from an initial orbit up to a geocentric intermediate orbit;

2) L2-rendezvous trajectory;

3) Trajectory from the point L2 of Earth-Moon system to a selenocentric intermediate orbit;

4) Selenocentric twisting down to a final orbit.

The 1st and 4th arcs can be eliminated if initial and final orbits have high altitude.Trajectories of 2nd and 3rd arcs are defined by continuation method.


24

VALIDATION OF TRAJECTORY DIVIDING INTO ARCS

Region of SC motion for critical Jacoby’s constant

Region of SC motion for SC relative velocity 10 m/s on the Hill’s sphere

opening width ~60000 km

Hill’s sphere

Region of satellite motion

Moonto Earth

Curves of zero velocity(contours of Jacoby’s integral)

1. Typical duration of hyperbolic motion within Hill’s sphere of Moon is ~1 days.

2. Typical velocity increment due to thrust acceleration is ~10 m/s for 1 day if thrust acceleration is ~0.1 mm/s2.

3. Opening width in the L2 vicinity is ~60000 km for SC relative velocity 10 m/s on the Hill’s sphere.

To capture SC into the Moon orbit using electric propulsion (thrust acceleration ~0.1 mm/s2) SC relative velocity should be not greater ~10 m/s when distance from L2 is less ~30000 km.

OPTIMAL TRANSFER TO LUNAR ORBIT


25

EARTH_MOON L2 RENDEZVOUSModel problem of transfer from circular Earth orbit

(altitude 250000 км, inclination 63°, right ascension of ascending node 12°,lattitude argument 0°; launch date January 5, 2001)

0.5

0.0

a,mm/s2

0.0 95.0t, days 0.0 95.0t, days 0.0 95.0t, days 0.0 95.0t, days

4 complete orbits 5 complete orbits 6 complete orbits 7 complete orbits



26

EARTH-MOON L2 RENDEZVOUS USING MOON GRAVITY ASSISTED MANEUVER

Th

rust

acc

ele

ratio

n,

mm

/s2

0.5

0.0 0 Т, days 95

Moon orbit

Final Moon position

Initial Moon position

Initial L2 position

Final L2 position

Initial orbit

Gravity assisted maneuver

Th

rust

acc

ele

ratio

n,

mm

/s2

1.0

0.0 0 Т, days 95

2.5 orbits 7.5 orbits



27

TRANSFER FROM EARTH-MOON L2 INTO CIRCULAR MOON ORBIT

Final orbit: r = 30000 km, i = 0.Transfer: 1.5 orbits, T = 10 days



Thr

ust

acce

lera

tion

0.5 mm/s2

0 mm/s2

0 Time, d 10

0 Time, d 15

0 Time, d 20

Moon

Final (intermediate) orbit

Initial L2 position Final L2 position



28

TRANSFER FROM EARTH-MOON L2 INTO ELLIPTICAL MOON ORBIT (i=90°, hp=300 km, ha=10000 km, 10.5 orbits)

Thr

ust

acce

lera

tion

1 mm/s2

0 mm/s2

0 Time, d 30

Moon

Final orbitInitial L2 position

Final L2 position



29

TRANSFER FROM ELLIPTICAL EARTH ORBIT INTO CIRCULAR MOON ORBIT.TRAJECTORY ARCS

Geocentric spiral untwisting Earth-Moon L2 rendezvous Transfer from Earth-Moon L2 into equatorial 30000-km

circular Moon orbit

Moon

Earth

Earth

Thr

ust

acce

lera

tion

0.5 mm/s2

0 mm/s2

0 Time, d 95

Thr

ust

acce

lera

tion

0.5 mm/s2

0 mm/s2

0 Time, d 95



30

4. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS

CONSTANT SPECIFIC IMPULSE PROBLEM


31

Equations of SC motion are written in the equinoctial elements which have not singularty when eccentricty or inclination is nullified. The optimal control problem is reduced into the two-point boundary value problem by maximum principle.

This boundary value problem is reduced into the initial value problem by continuation method. It is necessary to integrare system of optimal motion o.d.e. (P-system) and to calculate partial derivatives of final state vector of P-system on the initial value of co-state variables to calculate right parts of continuation method’s o.d.e.

The right parts of the P-system are numerically averaged over true lattitude during the P-system integration. Partial derivative of final state vector of P-system on the initial value of co-state vector is calculating using finite differences.

The boundary value problem residual vector are calculated as result of first integration of P-system. 6 additional integrations of P-system is required to calculate sensitivity matrix using finite differences. As result, the right parts of the continuation method’s o.d.e. are calculated after solving correspoding linear system.

System of continuation method’s o.d.e. is numerically integrated on continuation parameter from 0 to 1. As a result, the optimal solution is calculated.

OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS


32



- thrust switching function, P - thrust, m - SC mass, - pitch, - yaw

System of equinoctial elements:

- primary gravity parameter; p, e, , , i, - keplerian elements.

coscosm

Pa cossin

m

Pa r sin

m

Pan

p

h coseex sineey cos2

tani

ix sin2

tani

iy F

,

,sin

,sinsin~2

1

,sincos~2

1

,sincoscossin1cossincos

,sincoscoscos1cossinsin

,coscos

3

2

w

P

dt

dm

h

m

P

hdt

dF

Fh

m

P

dt

di

Fh

m

P

dt

di

eeFFh

m

P

dt

de

eeFFh

m

P

dt

de

hh

m

P

dt

dh

y

x

xxy

yxx

Thrust acceleration components in the orbital reference frame:

Equation of motionin the equinoctialelements:

FeFe yx sincos1

FiFi yx cossin 221~yx ii

w - exhaust velocity

4.1. EQUATION OF MOTION

Boundary conditions: t = 0: t = T: ykyxkxykyxkxk iiiieeeehh ,,,,000000 ,,,,, mmiiiieeeehh yyxxyyxx

33



Averaged Hamiltonian does not depends on F, so after averaging . So as orbit-to-orbit transfers are

considered, the final value F=F(T) is not fixed pF(T)=0 (transversality condition)

it can be missed terms including pF , where

Cost function: T

dtw

PJ

0

min

0

F

H

dt

dpF

0Fp

sincossincoscos13

2

nrFm AAAh

m

Pp

hp

w

PH

eyyexxh peFpeFhpA sin1cos1

eyexr pFpFA cossin

Fiyixeyxexyn ppFpFpepeA sincos~2

1

22cos

r

r

AA

A

22sin

r

r

AA

A

222

22

cosnr

r

AAA

AA

222sin

nr

n

AAA

A

0 ,0

0 ,1

s

s

212221nr

ms AAA

m

h

w

p

21222 ~~~1 nrm AAAm

Pp

w

PH

212223

2

1 nrFm AAAh

m

Pp

hp

w

PH

Hamiltonian:

Optimal control:

nnrr Ah

AAh

AAh

A ~,~,~

Optimal Hamiltonian:

или 1

4.2. OPTIMAL CONTROL

34



4.3. EQUATIONS OF OPTIMAL MOTION (P-SYSTEM)

,~~~

,~

~~

~~

~~~~

,

,~

~~

~~

~~~~

212222

21222

21222

nrm

nn

nnnr

m

nn

nnnr

AAAm

P

m

H

dt

dp

AA

AA

AAAAA

m

PH

dt

d

m

P

p

H

dt

dm

AA

AA

AAAAA

m

PH

dt

d

xxxx

p

pppp

x

TT ,,,,,,,,, iyixeyexhyxyx pppppiieeh px

.,,,~

nriAhA ii

pp

.,,,cossin~

;~

;~

;sin~

;cos~

;1~

nriF

AA

FeFeh

F

A

i

Ah

i

A

i

Ah

i

A

e

AA

Fh

e

A

e

AA

Fh

e

A

h

AhA

h

A

ii

yxi

x

i

x

i

x

i

x

i

y

ii

y

i

x

ii

x

iii

i

where - state and co-state vectors,

35



.;sin~2

1;cos~

2

1;;;0

;0;cos;sin;0

;0;sin1;cos1;

F

n

iy

n

ix

nx

ey

ny

ex

n

h

n

F

r

iy

r

ix

r

ey

r

ex

r

h

r

Fiyixy

eyx

exh

p

AF

p

AF

p

Ae

p

Ae

p

A

p

A

p

A

p

A

p

AF

p

AF

p

A

p

A

p

A

p

A

p

AeF

p

AeF

p

Ah

p

A

;cos1sinsincossin1cossincos

;0

;sincos1sin;sincos1cos; 22

eyxyexxy

yx

exeyy

eyexx

h

pFFFeFepFFFeFeF

A

i

A

i

A

pFFpFe

ApFFpF

e

Ap

h

A

;sincoscossincossin;0

;cossinsin;cossincos;0

FpFpFpFpFeFeF

A

i

A

i

A

FpFpFe

AFpFpF

e

A

h

A

eyexeyexyxr

y

r

x

r

eyexy

reyex

x

rr

.sincossincos

sincos~2

1sincos

;0;sin;cos;0

FyxFxy

ixiyexyeyxyxn

y

n

x

nFex

y

nFey

x

nn

pFiFipFeFe

FpFppepeFiFiF

A

i

A

i

AFpp

e

AFpp

e

A

h

A

36



Continuation method’s equation: , where (minimum time) or z=p (fixed time);

b=f(z0) - residual vector for initial z (when =0). The boundary value problem is solved by integration of continuation method’s equation on from 0 to 1. Partial derivatives of residual vector f on vector z and linear system solving for computation right parts of o.d.e. are processed numerically.

4.4. BOUNDARY VALUE PROBLEM

Within the minimum time problem 1 and equations for m and pm are eliminated by substitusion expression m = m0 - (P/w) t into other equations. Equation of residuals is following:

This equation should be solved with respect to unknown initial value of co-state vector p(0) and transfer duration T.

0

)(

)(

)(

)(

)(

)(

TTH

iTi

iTi

eTe

eTe

hTh

yky

xkx

yky

xkx

k

f

0

)(

)(

)(

)(

)(

)(

Tp

iTi

iTi

eTe

eTe

hTh

m

yky

xkx

yky

xkx

k

fWithin the fixed-time problem equation of residuals is following:

This equation should be solved with respect to unknown initial value of co-state vector p(0), pm(0).

bz

fz1

d

d

T

p

p

p

p

p

iy

ix

ey

ex

h

z

37

4.5. DETAILS OF BOUNDARY VALUE PROBLEM SOLVING



Boundary value problem is solved by continuation method.

The averaged with respect to true lattitude equations of optimal motion are used to calculate residuals f. These equations have singularity when co-state vector p=0, so it is impossible to use zero initial co-state vector (coast motion) as initial approximation.

Within the minimum time problem the following initial approximation was used: ph(0)=1 if the final semi-major axis greater than the semi-major axis of initial orbit and ph(0)=-1 otherwise. The rest vector p components were picked out equal to 0 and the initial approximation of transfer duration was T|=0=1 (dimensionless). Using this initial approximation there were found the minimum-time transfers to GEO from the elliptical transfer orbits having inclination 0°-75° and apogee altitude 10000-120000 km. If initial apogee altitude was not match with this range, the solution for a transfer from close initial orbit was used as the initial approximation.

It is used numerical averaging the equations of optimal motion on the true lattitude F during these equations integration.

The partial derivatives of residuals f with respect to p(0), T, which are necessary for continuation method, are processed numerically using finite differences.

So, there are used numerical integration of numerically averaged equations of optimal motion and numerical differentiating of residuals to calculate right parts of continuation method’s o.d.e.

38

4.6. OPTIMAL SOLUTION IN NON-AVERAGED MOTION



The real and averaged evolutions of orbital motion are close each to other due to the relatively low thrust acceleration level.

To check accuracy of optimal averaged solution, the obtained optimal p(0) and T were used for numerical integration of non-averaged equations of motion. The initial value of true lattitude F was chosen arbitrary (the perigee or apogee values mostly). The initial value of pF was equals to 0 (see note above).

The optimal thrust steering and insertion errors were calculated as result of numerical integration of the non-averaged equations. The relative errors due to averaging did not exceed 0.1% for transfer from an elliptical orbit to GEO when thrust acceleration was 0.1-0.5 mm/s2.

An optimal thrust steering examples are presented below.

39



4.7. OPTIMAL ORBITAL EVOLUTIONAND OPTIMAL THRUST STEERING

(MINIMUM-TIME PROBLEM)

0

10000

20000

30000

40000

50000

60000

70000

80000

0 50 100 150

Время, сут

Ра

ссто

ян

ие

, км

Радиус перигея

Радиус апогея

Большая полуось

0

10

20

30

40

50

60

70

80

0 50 100 150

Время, сут

На

кло

не

ни

е,

гра

дус

ы

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0 50 100 150

Время, сут

Экс

це

нтр

иси

тет

1. Average apogee, semi-major axis, and eccentricity have maximum during transfer.

2. Perigee distance increases monotonously.

Orbital evolution for suboptimal apogee altitude of initial orbit(ha = 30000 km, i = 75°)

Time, days

Time, days

Time, days

Dis

tanc

e, k

mIn

clin

atio

n, d

egE

ccen

tric

ity

Perigee distanceApogee distanceSemi-major axis

40



-180-150-120-90-60-30

0306090

120150180

0 20 40 60 80 100 120 140 160

Время, сутки

Тан

гаж

, гр

адус

ы

-180-150-120-90-60-30

0306090

120150180

0 20 40 60 80 100 120 140 160

Время, сут

Уго

л а

таки

, гр

адус

ы

-90

-60

-30

0

30

60

90

0 20 40 60 80 100 120 140 160

Время, сутки

Уго

л р

ыск

ань

я,

град

усы

Optimal thrust steering for suboptimal apogee altitude of initial orbit(ha = 30000 km, i = 75°)

Acceleration-braking

Braking-acceleration

Acceleration

Time, days

Time, days

Time, days

Yaw

, deg

Pitc

h, d

egA

ngle

of

atta

ck, d

eg

41



OPTIMAL THRUST STEERING

-90

-60

-30

0

30

60

90

0 0.2 0.4 0.6 0.8 1

Время, сут

Уго

л р

ыск

ань

я,

град

усы

-90

-60

-30

0

30

60

90

80 80.2 80.4 80.6 80.8 81 81.2 81.4 81.6 81.8 82

Время, сут

Уго

л р

ыск

ань

я,

град

усы

-90

-60

-30

0

30

60

90

141 141.2 141.4 141.6 141.8 142 142.2 142.4 142.6 142.8 143

Время, сут

Уго

л р

ыск

ань

я,

град

усы

-180-150-120-90-60-30

0306090

120150180

0 0.2 0.4 0.6 0.8 1

Время, сут

Уго

л, г

рад

усы

тангаж

угол атаки

траекторный угол

-90

-60

-30

0

30

60

90

80 80.2 80.4 80.6 80.8 81 81.2 81.4 81.6 81.8 82

Время, сут

Уго

л, г

рад

усы

тангаж

угол атаки


-180-150-120-90-60-30

0306090

120150180

141 141.2 141.4 141.6 141.8 142 142.2 142.4 142.6 142.8 143

Время, сут

Уго

л, г

рад

усы

тангаж

угол атаки


Yaw

, deg

Time, days

Time, days

Time, days

Time, days

Time, days

Time, days

Yaw

, deg

Yaw

, deg

Ang

le, d

egA

ngle

, deg

Ang

le, d

eg

pitchangle of attackpath angle



42



OPTIMAL THRUST STEERING

-180-150-120

-90-60-30

0306090

120150180

0 30 60 90 120 150 180 210 240 270 300 330 360

Истинная аномалия, градусы

Та

нга

ж,

гра

дус

ы

141-е сутки

80-е сутки

2-е сутки

-30-20-10

01020304050607080

0 30 60 90 120 150 180 210 240 270 300 330 360


Ры

ска

нь

е,

гра

дус

ы

141-е сутки

80-е сутки

2-е сутки

-180-150-120

-90-60-30

0306090

120150180

0 30 60 90 120 150 180 210 240 270 300 330 360


Уго

л а

таки

, гр

ад

усы

141-е сутки

80-е сутки

2-е сутки

t=141 d

t=80 d

t=2 d

t=141 d

t=80 d

t=2 d

t=141 d

t=80 d

t=2 d

True anomaly, deg

True anomaly, deg

True anomaly, deg

Ang

le o

f at

tack

, deg

Yaw

, deg

Pit

ch, d

eg

43



Orbital evolutionand optimal thrust steeringfor optimal apogee altitudeof initial orbit(ha = 140000 км, i = 65°)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70 80 90 100 110

t, days

Ecc

entr

icity

0100002000030000400005000060000700008000090000

100000110000120000130000140000150000

0 10 20 30 40 50 60 70 80 90 100 110

t, days

Dis

tanc

e, k

m

rp

ra

a

-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

180

0 10 20 30 40 50 60 70 80 90 100 110

t, days

Ang

le °

pitch

yaw

Perigee & apogee distance and semi-major axis

Eccentricity Eccentricity

44

Eccentricity Eccentricity



0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 10 20 30 40 50 60 70 80 90 100 110 120

t, days

Ecc

entr

icity

0

50000

100000

150000

200000

250000

0 10 20 30 40 50 60 70 80 90 100 110 120

t, days

Dis

tanc

e, k

m

rp

ra

a

-180-150-120-90-60-30

0306090

120150180

0 10 20 30 40 50 60 70 80 90 100 110 120

t, days

An

gle

°

pitch

yaw

Perigee & apogee distance and semi-major axis

Orbital evolutionand optimal thrust steering forsuperoptimal apogee altitudeof initial orbit(ha = 240000 км, i = 65°)

Braking-accelerationBraking

45



4.8. OPTIMIZATION OF TRANSFER FROM ELLIPTIC ORBIT TO GEO

Initial perigee altitude 250 km,SC mass in the GEO 450 kg, thrust 0.166 N, specific impulse 1500 s

Initial apogee altitude, thousands km

Init

ial

incl

inat

ion

°

Initial apogee altitude, thousands km

Tra

nsf

er d

ura

tio

n, d

ays

i0=75°

i0=65°i0=51.3°

i0=0°

47


48

CONCLUSION

The developed continuation method demonstrated extremely effectiveness for variable specific impulse problem. The combination of two continuation versions (basic continuation method and continuation with respect to gravity parameter) allows to process planetary mission analysis fast and exhaustevely.

The L2-ended low thrust trajectories were optimized using the continuation method. These solutions were used to construct quasioptimal trajectories between Earth and Moon orbits.

The version of continuation method allows to carry out full-scale analysis of the low-thrust mission to GEO from the inclined elliptical transfer orbit.

So, the continuation method performances make this method an effective and useful tool for analysis the wide range of electric propulsion mission

V.G. Petukhov E-mail: [email protected] Khrunichev State Research and Production Space Center.

Documents

Transcript of V.G. Petukhov E-mail: [email protected] Khrunichev State Research and Production Space Center.