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

Introduction: The Importance of Lagrange Points 2 L1L1 L2L2 Applications of Earth-Moon L 1 Orbits: Communications relay Navigation Aid Observation & Surveillance of Earth and/or Moon Magnetotail Measurements (ARTIMIS mission) Parking Orbits for Space Stations or Spacecraft

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

4 Assume: -m 1 & m 2 orbit their barycenter in perfectly circular orbits -m 1 ≥m 2 >>m 3 Define: CR3B Problem Setup

5 CR3BP Low-Thrust Equations of Motion

6 Lagrange Points

7 L-Point Orbits and Their Manifolds 1.Pick a point on the orbit, X 0 2.Integrate the EOM and STM for 1 period. The STM = Monodromy Matrix 3.Calculate the Eigenvalues (λ) and Eigenvectors (υ) of the Monodromy Matrix 4.Find the stable Eigenvector/value 5.Perturb the original state by a small amount along the stable Eigenvector and propagate that perturbation backwards in time to generate a trajectory 6.Repeat steps 1-5 for multiple points along the nominal orbit X0X0

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Overview 9 L1L1 L2L2 Goal: Get From GEO (A) to a L 1 Halo (C) via some trajectory (B) A B C ?- Low Thrust Patch Point

Mingotti et al. 10 L1L1 L2L2 A B C 1.Begin with a reasonable guess trajectory 2.Trajectory will join a low-thrust arc with the invariant stable manifold 3.Use Non-Linear Programming (NLP) with Direct Transcription and Collocation 4.Fast algorithm with reasonable convergence Mingotti’s Technique * ?- Low Thrust Shortcomings: 1.Requires a reasonable guess solution to converge 2.Prone to locating local minima instead of global minima *G. Mingotti et al, “Combined Optimal Low-Thrust and Stable-Manifold Trajectories to the Earth-Moon Halo Orbits,” AIP Conference Proceedings, 2007

11 New Approach

12 Fitness Function

13 How to Find the Optimal Patch Point? k = τ s.m. = τ s.m. = τ s.m. = k = 583 +

14 Particle Swarm Optimization (PSO) * Pontani and Conway, “Particle Swarm Optimization Applied to Space Trajectories,” Journal of Guidance, Navigation, Control, and Dynamics, Vol. 33, Sep.-Oct (1) (2)

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

16 Study A: c 1 =1, c 2 =c 3 =0

17 Optimal Trajectory Optimal Patch Point: k=610+, τ s.m. = [tu], e GEO =

18 Study B: c 1 =1, c 2 =10 -3, c 3 =0

19 Study C: c 1 =1, c 2 =10 -3, c 3 =10 -4

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

21 Thank You!

22 Study A: c 1 =1, c 2 =c 3 =0

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