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