OPTIMAL TWO IMPULSE EARTH-MOON Trajectories

1. OPTIMAL TWO IMPULSE EARTH-MOON Trajectories John McGreevy Advisor: Manoranjan Majji Committee: John Crassidis TarunSingh

2. Outline • Motivation • Problem Formulation • Results for Planar Trajectories • Comparison with previous work • Effect of varying initial conditions • Effect of varying time of flight • Addition of Terminal Inclination Constraint • Results • Conclusions

3. Motivation • Lunar Missions • Moon observing • Moon landing • Reduce necessary thrust • Save fuel/money • Increase payload • Short duration • Not low-energy trajectories

4. Problem Statement • Find trajectory that minimizes cost function • Mayer cost • Quadratic cost allows closed form solutions • Fixed initial conditions • Circular, low Earth orbit, in Earth-Moon plane • Terminal manifold • Circular, low lunar orbit • Fixed time • Non-dimensional units • Distance ~ Moon orbit radius • Time ~ Inverse of angular frequency of lunar orbit

5. System Dynamics

6. Solution Method • Earth centered non-rotating frame • Using necessary conditions of optimal control theory • Hamiltonian:

7. Solution Method

8. Terminal Constraints • Form augmented cost function: • are terminal Lagrange multipliers

9. Terminal Constraints • Solve internal optimization problem for • Take first variation of with respect to • Because states are free at final time, the co-states are fixed • Transversality conditions:

10. Necessary Conditions • Can also verify solution using direct method • Take variation of augmented cost function with respect to and set = 0

11. Necessary Conditions

12. Computations • Utilized Shooting Method • Guess values of unknowns • 6 initial values of co-states • 3 Lagrange multipliers • Solve 12 differential equations • Evaluate terminal boundary conditions • 6 transverality conditions • 3 terminal constraint equations • Update guess using Newton’s Method and repeat until convergance

13. Computations • Provided initial conditions based on Miele1 • Fixed initial conditions • f0 = -116.88° • Orbit altitude of 463 km • Circular orbit • In Earth-Moon plane • Fixed terminal manifold • Orbit altitude of 100km • Circular orbit • Did not require orbit to be in Earth-Moon plane [1] Miele, A. and Mancuso, S., “Optimal Trajectories For Earth-Moon-Earth Flight,” ActaAstronautica, Vol. 49, No. 2, 2001, pp. 59 – 71.

14. Comparison Results • Verify accuracy of current method • Reasons for some expected differences: • Different cost function • Weighted quadratic with respect to state variables instead of 2-norm • Additional term in dynamics

15. Comparison Results Inertial Frame, 4.5 days, -116.88°, 463km

16. Comparison Results Rotating Frame, 4.5 days, -116.88°, 463km

17. Comparison Results Rotating Frame, 4.5 days, -116.88°, 463km

18. Comparison Results Inertial Frame, 4.5 days, -116.88°, 463km • Comparable results for both trajectories • Small increase due to additional term in dynamics

19. Varying Initial Conditions • Vary initial true anomaly • Range of 50° from -116.88° to -166.88° • Initial altitude of 160 km • Match Apollo • Same terminal manifold as before • Orbit altitude of 100km • Circular orbit • Did not require orbit to be in Earth-Moon plane • Use previous solution to provide initial guess of unknowns for similar initial conditions

20. Varying Initial Conditions

21. Varying Initial Conditions

22. Varying Time of Flight • Want to find time of flight which minimizes cost function • Use initial true anomaly of -135° based on previous section • Initial altitude of 160 km • Same terminal manifold as before • Orbit altitude of 100km • Circular orbit • Did not require orbit to be in Earth-Moon plane

23. Varying Time of Flight Retrograde: Prograde: • Inertial: • Rotating:

24. Varying Time of Flight

25. Varying Time of Flight

26. Summary So Far • Take advantage of quadratic cost formulation • Multiple solutions for each set of initial conditions • Retrograde and Prograde • Retrograde orbit is more sensitive to initial guess • Lowest cost occurs at tf=3.15 days • Next: • Non-planar orbits • Specify terminal orbit inclination

27. Additional Constraint • Using equation for orbital inclination: • Put into quadratic form: • Form new augmented cost:

28. Terminal Constraints • New equation for • where • Because states are free at final time, the co-states are fixed • Transversality conditions:

29. Varying Terminal Inclination

30. Varying Terminal Inclination

31. Varying Terminal Inclination

32. Varying Terminal Inclination

33. Varying Terminal Inclination

34. Solution failure at 90° inclination • Inverted matrix in becomes ill-conditioned at 90° • M is singular (rank 1) • Large value of • M dominates • Possible cause is constraint formulation

35. Summary • Take advantage of quadratic cost formulation • Multiple solutions for each set of initial conditions • Retrograde and Prograde • Obtained solutions for desired terminal inclinations • Discovered convergence problem at 90° inclination • Future Work • Vary multiple initial conditions and time simultaneously • Non-planar orbits • Specify initial orbit inclination • Multiple revolution trajectories

36. Questions?