Andrew Abraham Lehigh University

1. Optimization of Preliminary Low-Thrust Trajectories From GEO-Energy Orbits To Earth-Moon, L1, Lagrange Point Orbits Using Particle Swarm Optimization Andrew Abraham Lehigh University

2. Introduction:The Importance of Lagrange Points L2 L1 • Applications of Earth-Moon L1 Orbits: • Communications relay • Navigation Aid • Observation & Surveillance of Earth and/or Moon • Magnetotail Measurements (ARTIMIS mission) • Parking Orbits for Space Stations or Spacecraft

3. Introduction:The Importance of Low-Thrust • Advantages of Low-Thrust Dynamics: • Low fuel consumption • Better Isp (order of magnitude) • High payload fraction delivered to target • Power source arrives at target

4. CR3B Problem Setup • Assume: • m1 & m2 orbit their barycenter in perfectly circular orbits • m1≥m2>>m3 Define: Synodic Reference Frame

5. CR3BP Low-Thrust Equations of Motion

6. Lagrange Points

7. L-Point Orbits and Their Manifolds Pick a point on the orbit, X0 Integrate the EOM and STM for 1 period. The STM = Monodromy Matrix Calculate the Eigenvalues (λ) and Eigenvectors (υ) of the Monodromy Matrix Find the stable Eigenvector/value Perturb the original state by a small amount along the stable Eigenvector and propagate that perturbation backwards in time to generate a trajectory Repeat steps 1-5 for multiple points along the nominal orbit X0

9. Overview A L2 L1 C B ?- Low Thrust Patch Point Goal: Get From GEO (A) to a L1 Halo (C) via some trajectory (B)

10. Mingotti et al. A L2 L1 C B ?- Low Thrust Mingotti’s Technique* Begin with a reasonable guess trajectory Trajectory will join a low-thrust arc with the invariant stable manifold Use Non-Linear Programming (NLP) with Direct Transcription and Collocation Fast algorithm with reasonable convergence Shortcomings: Requires a reasonable guess solution to converge Prone to locating local minima instead of global minima *G. Mingottiet al, “Combined Optimal Low-Thrust and Stable-Manifold Trajectories to the Earth-Moon Halo Orbits,” AIP Conference Proceedings, 2007

11. New Approach • Select a “Patch Point” on the Stable Manifold • Propagate a low-thrust trajectory backwards in time from that point • Use a “tangential thrust” control law to steer the spacecraft • Instantaneous 3-body velocity of the spacecraft • This control law is the most fuel and time efficient law because it maximizes the Jacobi Energy • Stop propagation once Jacobi Energy of the spacecraft is equal to a GEO orbit • Repeat steps 1-4 for a new Patch Point • Find the Patch Point that minimizes some cost/fitness/performance function

12. Fitness Function = eccentricity of GEO-energy Earth orbit = fuel consumed during low-thrust transfer = time of flight from GEO to the nominal L-point orbit = a patch point on the stable manifold

13. How to Find the Optimal Patch Point? τs.m. = -21.56 k = 583+ τs.m. = -18.31 τs.m. = -17.71 Patch Point k = 610+

14. Particle Swarm Optimization (PSO) = Inertial Weight = Cognitive Weight = Social Weight = 0.15 * = 1.5 * = 1.5 * (1) (2) = Number of Particles = Position of Particle iduring the jth iteration = Velocity of Particle iduring the jthiteration = “Global Best” value for any Particle = “Personal Best” value for Particle iduring the jthiteration = Random number with range zero to one and uniform distribution *Pontani and Conway, “Particle Swarm Optimization Applied to Space Trajectories,” Journal of Guidance, Navigation, Control, and Dynamics, Vol. 33, Sep.-Oct. 2010

15. Application of PSO: Nominal Orbit k = 1 Earth-Moon L1 Northern Halo Orbit: Defined by… k = N k = 2 k = 3 = … k = … k = N/2

16. Study A: c1=1, c2=c3=0

17. Optimal Trajectory Optimal Patch Point: k=610+, τs.m.=-18.101[tu], eGEO = 0.000930

18. Study B: c1=1, c2=10-3, c3=0

19. Study C: c1=1, c2=10-3, c3=10-4

20. Future Work Repeat Study… hope is to further reduce run-time by using less particles

21. Thank You!

22. Study A: c1=1, c2=c3=0