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)

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 2

Multi-impulsive transfer model 3

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. 4

First method (cont.)  First we apply primary impulse so we have, 5

First method (cont.) 6

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

First method (cont.) 8 The main half-axis for (i+1)th orbit and also the flight path angle are calculated as following:

 For the angular momentum and eccentricity we have First method (cont.) 9

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

First method (cont.) 11 With letting out the values of D & K in the last equation we have,

First method (cont.) 12  The total value of (n+1) impulse,

First method (cont.) 13 In this optimization we should calculate the sum of transfer times from initial to final orbit,

First method (cont.) 14

Second method 15

Second method (cont.) 16

The angle between these two vector is, 17 Second method (cont.)

18  Also the eccentricity value,the unit eccentricity vector and the angular momentum are, Second method (cont.)

19

Second method (cont.) 20 The main half-axis of transfer orbits is,  The value of angular momentum for transfer orbits is,

Second method (cont.) 21  The calculation method of velocity vector and the impulses values is

Second method (cont.) 22

Second method (cont.) 23

Second method (cont.) 24  The sum of impulses of this multi-impulse transfer is,

Second method (cont.) 25 Time is calculated like the previous method,

Second method (cont.) 26

Initial and Tangent orbit parameters 27

Dual-impulse transfer (Second method) 28

Pareto-optimal solution of dual-impulse transfer (Second method) 29

Tri-impulse transfer (Second method) 30

Pareto-optimal solution of tri-impulse transfer (Second method) 31

32 Dual-impulse transfer (First method)

Pareto-optimal solution of dual-impulse transfer (First method) 33

34 Tri-impulse transfer (First method)

Pareto-optimal solution of tri-impulse transfer (First method) 35

Quad-impulse transfer (First method) 36

Pareto-optimal solution of quad-impulse transfer (First method) 37