multiobjective optimization of multiple-impulse transfer between two coplanar orbits using genetic...

Multiobjective Optimization of Multiple-Impulse Transfer between Two Coplanar Orbits using Genetic Algorithm

Nima Assadian*, Hossein Mahboubi Fouladi†, Abbas Kafaee Razavi‡,Vahid Hamed Azimi¶

*†‡ Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran¶ Mathematical group, Department of Science, K. N. Toosi University of Technology, Tehran, Iran(graduated)

2

Introduction

•Optimal transfer between two coplanar elliptical orbits

•Multiobjective optimization▫Transfer time▫Total impulse

•Multiple-impulse maneuvers•Two type of parameterization:

▫Based on true anomaly, normal and tangential velocities of impulse points

▫Position vector & true anomaly of impulse points

3

Multi-impulsive transfer model

4

First method

•First we are at the initial orbit then with choosing

optional true anomaly ,parallel velocity and normal

velocity of impulse point we enter the first transfer

orbit and with repeating this process for next

transfers finally we arrive at the target orbit.

5

First method (cont.)

21

0 11 1

(1 )

1 ( )

er a

e cos

First we apply primary impulse so we have,

21 1 1(1 )h a e

6

First method (cont.)

2(1 )

1 ( )i

i ii i i

er a

e cos

• In the moment of (i+1)th impulse applied at the random position of we are at the reference axis,

7

First method (cont.)

•And for the velocity before (i+1)th impulse we have,

2 2

iiiV V V

The values of and are chosen from genetic algorithm in random way so we obtain the value of total impulse for each (i),

0

2( ) ( )

i

i i

Va r

8

First method (cont.)

•The main half-axis for (i+1)th orbit and also the flight path angle are calculated as following:

2 21 0( )i i iiV V V V

Also is the value of velocity after applying (i+1)th impulse:

1 212( )

2

i

i

i

aV

r

1=cos ( )i

i

i

i

h

rV

9

1 0 0= + V cos( )- V sin( )i i iii i i ih h r r

For the angular momentum and eccentricity we have

1

1

2

1

1 ii

ia

he

First method (cont.)

10

First method (cont.)

•The condition of answer being, arriving to the final orbit, is equivalent to S ≥ 1

22 1

22 1

D cos(- )+ cos(k))1 1

D sin(- )+ sin(k)1

(

)1

(

n n

n n

D D

D

e eS

e e

D

11

First method (cont.)

•With letting out the values of D & K in the last equation we have,

21

1 22 2

1

(1 )n

nn n

eD a

a e

1 1k= - ;n n

12

First method (cont.)

The total value of (n+1) impulse,

1 1 1 1 1 1

2 20 1 0 1 1 0

1

2 ( )n n n n n n

n

t ii

V V V V V cos V

13

First method (cont.)

•In this optimization we should calculate the sum of transfer times from initial to final orbit,

1 11

1 1

-rE =cos ( )

ei i

ii i

a

a- +

++ +

1 1 11 1

1 1

-rE =cos ( )

ei i

ii i

a

a- + +

++ +

14

First method (cont.)

11

n

ii

T t

1 1

1.51

1 1 1 1 1 1 0.5( ( ( ) ( )))

i i

ii i i i

at E E e sin E sin E

1 0t

15

Second method

• In the following subject the other method for

optimal transfer between two arbitrary orbits is

presented, With this difference in this method

(position vector) and (true anomaly of impulse

points) are entered into the calculations

16

Second method (cont.)

•The values of and are chosen random

• from genetic algorithm method which have two

components &.

17

•The angle between these two vector is,

1 1

1

cos ( )i ii

i i

R R

RRj - +

+

×D =

v v

Second method (cont.)

18

Also the eccentricity value ,the unit eccentricity vector and the angular momentum are,

11

1 11

1

cos( ) cos( )

i

ii

ii i i

i

R

Re

R

Rq j q

++

+ ++

-

=

+D -

Second method (cont.)

19

Second method (cont.)

1

1 1

i ji

i i i

R Rh

h RR+

+ +

´=

v

1 11 1

1 1

sin( ) cos( )i i i ii i

i i i i

e R h R

e Rh Rq q+ +

+ +

+ +

´= +

v v vv

20

Second method (cont.)

•The main half-axis of transfer orbits is,

1 11 2

1

(1 cos( ))

(1 )i i i

i

i

R ea

e

q+ +

+

+

+=

-

The value of angular momentum for transfer orbits is,

2(1 )i i ih a em= -

21

Second method (cont.)

The calculation method of velocity vector and the impulses values is

1

, 1 1 11

-sin( + )

v e +cos( + )

0

i i

bi i i iih

q jm

q j+

+ + ++

é ùDê úê ú= Dê úê úê úûë

1

, 1 11

-sin( )

v e +cos( )

0

i

a i i iih

qm

q+

+ ++

é ùê úê ú= ê úê úê úûë

22

Second method (cont.)

• is the perigee angle for the orbit transfer point to

the reference axis,

1

1

11

cos (ˆ)i

i

i

e i

ew

+

++- ×

=v

23

Second method (cont.)

1, ,=Q v

bi biV - ´v

cos( ) sin( ) 0

-sin( ) cos( ) 0

0 0 1

i i

i i iQ

w w

w w

é ùê úê ú= ê úê úê úë û

24

Second method (cont.)

1, ,

V =Q va i a i

- ´v

1 1

, , , ,1 1

(V ) (V )n n

t i a i bi a i bii i

V V V V+ +

= =

D = D = - × -å åv v v v

The sum of impulses of this multi-impulse transfer is,

25

Second method (cont.)

•Time is calculated like the previous method,

1 1

1 1

a -RE =cos ( )

(a e )i i

ii i

- +

+ +

1 1 1

1 1

a -RE1=cos ( )

a ei i

ii i

- + +

+ +

26

Second method (cont.)

1.51

1( ( ( ) ( )E1 E E )1 )Ei

i ii i i i

at e sin sin

m+

+= - - -

1

n

ii

T t=

= å

27

Initial and Tangent orbit parameters

1a = 7000

1e = 0.4

2a = 130000

2e = 0.1

= 22.92 (deg)

28

Dual-impulse transfer (Second method)

-1.5 -1 -0.5 0 0.5 1 1.5

x 105

-1.5

-1

-0.5

0

0.5

1

1.5x 10

5

X (km)

Y (k

m)

Initial orbit

Target orbit

29

Pareto-optimal solution of dual-impulse transfer (Second method)

2 4 6 8 10 12 14 168

10

12

14

16

18

20

22

24

26

28

V (km/s)

Tra

nsfe

r tim

e (

s) T =27.81627 (h)

minV =3.37820 (km/s)

30

Tri-impulse transfer (Second method)

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5

x 105

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5x 10

5

X (km)

Y (k

m)

31

Pareto-optimal solution of tri-impulse transfer (Second method)

3 3.5 4 4.5 5 5.5 62

3

4

5

6

7

8

9

10x 10

5

V (km/s)

Tra

nsfe

r tim

e (s

)

T = 251.396 (h)

minV =3.16955 (km/s)

32

-1.5 -1 -0.5 0 0.5 1 1.5

x 105

-1.5

-1

-0.5

0

0.5

1

1.5x 10

5

X (km)

Y (

km)

Dual-impulse transfer (First method)

33

Pareto-optimal solution of dual-impulse transfer (First method)

3 3.5 4 4.5 5 5.53

4

5

6

7

8

9

10x 10

4

V (km/s)

Tra

nsfe

r tim

e (s

)

minV = 3.265168 (km/s) T =26.984 (h)

34

-1.5 -1 -0.5 0 0.5 1 1.5

x 105

-1.5

-1

-0.5

0

0.5

1

1.5x 10

5

X (km/s)

Y (k

m/s

)

Tri-impulse transfer (First method)

35

Pareto-optimal solution of tri-impulse transfer (First method)

3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.24

5

6

7

8

9

10x 10

4

V (km/s)

Tran

sfer

tim

e (s

)

minV = 3.27381 (km/s) T =26.557 (h)

36

Quad-impulse transfer (First method)

-1.5 -1 -0.5 0 0.5 1 1.5

x 105

-1.5

-1

-0.5

0

0.5

1

1.5x 10

5

X (km/s)

Y (k

m/s

)

37

Pareto-optimal solution of quad-impulse transfer (First method)

3 3.5 4 4.5 5 5.53

4

5

6

7

8

9

10x 10

4

V (km/s)

Tran

sfer

tim

e (s

)

minV = 3.26815 (km/s) T = 26.162 (h)