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

1. Multiobjective Optimization of Multiple-Impulse Transfer between Two Coplanar Orbits using Genetic Algorithm NimaAssadian*, HosseinMahboubiFouladi†, Abbas KafaeeRazavi‡,VahidHamedAzimi¶ *†‡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.) • First we apply primary impulse so we have,

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

7. And for the velocity before (i+1)th impulse we have, First method (cont.) • The values of and are chosen from genetic algorithm in random way so we obtain the value of total impulse for each (i),

8. The main half-axis for (i+1)th orbit and also the flight path angle are calculated as following: First method (cont.) • Also is the value of velocity after applying (i+1)th impulse:

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

10. First method (cont.) • The condition of answer being, arriving to the final orbit, is equivalent to S ≥ 1

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

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

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

14. First method (cont.)

15. 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 Second method

16. The values of and are chosen random from genetic algorithm method which have two components &. Second method (cont.)

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

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

19. Second method (cont.)

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

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

22. is the perigee angle for the orbit transfer point to the reference axis, Second method (cont.)

23. Second method (cont.)

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

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

26. Second method (cont.)

27. Initial and Tangent orbit parameters

28. Dual-impulse transfer (Second method)

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

30. Tri-impulse transfer (Second method)

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

32. Dual-impulse transfer (First method)

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

34. Tri-impulse transfer (First method)

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

36. Quad-impulse transfer (First method)

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