Optimization of Preliminary Low-Thrust Trajectories From GEO-Energy Orbits To Earth-Moon, L1, Lagrange Point Orbits Using Particle Swarm

Optimization

Andrew AbrahamLehigh University

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Introduction:The Importance of Lagrange Points

L1L2

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

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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

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Assume:

- m1 & m2 orbit their barycenter in perfectly circular orbits

- m1≥m2>>m3

Define:µ≡

𝑚2𝑚1+𝑚2

Synodic Reference Frame

CR3B Problem Setup

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Ω=12

(𝑥2+𝑦2 )+1−μ𝑟1

+μ𝑟2

CR3BP Low-Thrust Equations of Motion

𝑟1=√ (𝑥+μ )2+𝑦2+𝑧 2

𝑟2=√ (𝑥+μ−1 )2+𝑦2+𝑧 2

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Lagrange Points

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L-Point Orbits and Their Manifolds

1. Pick a point on the orbit, X0

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

𝑋𝑃𝑒𝑟𝑡𝑢𝑟𝑏𝑒𝑑=𝑋 0±ϵυ

X0

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Overview

L1L2

Goal: Get From GEO (A) to a L1 Halo (C) via some trajectory (B)

A

B

C

?- Low Thrust

Patch Point

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Mingotti et al.

L1L2

A

B

C

1. Begin with a reasonable guess trajectory2. Trajectory will join a low-thrust arc with the

invariant stable manifold3. 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 converge2. 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

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New Approach1. Select a “Patch Point” on the Stable Manifold

2. Propagate a low-thrust trajectory backwards in time from that point

3. 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

4. Stop propagation once Jacobi Energy of the spacecraft is equal to a GEO orbit

5. Repeat steps 1-4 for a new Patch Point

6. Find the Patch Point that minimizes some cost/fitness/performance function

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Fitness Function

𝐽 (𝑋 𝑠 .𝑚 .)=𝑐1‖𝑒𝐺𝐸𝑂 (𝑋 𝑠 .𝑚 .)−𝑒𝑑𝑒𝑠𝑖𝑟𝑒𝑑‖+𝑐2∆𝑚 (𝑋 𝑠 .𝑚 . )+𝑐3∆𝑇 (𝑋 𝑠 .𝑚 .)

= 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

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How to Find the Optimal Patch Point?

k = 610+

τs.m. = -18.31τ

s.m. = -17.71

τs.m. = -21.56

Patch Point

k = 583+

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Particle Swarm Optimization (PSO)

= Number of Particles = Position of Particle i during the jth iteration = Velocity of Particle i during the jth iteration = “Global Best” value for any Particle= “Personal Best” value for Particle i during the jth iteration = Random number with range zero to one and uniform distribution

χ❑( 𝑗+1 ) (𝑖 )= χ❑

( 𝑗 ) (𝑖 )+𝜔   ( 𝑗+1 ) (𝑖 )

= Inertial Weight = Cognitive Weight = Social Weight

= 0.15 * = 1.5 * = 1.5 *

* Pontani and Conway, “Particle Swarm Optimization Applied to Space Trajectories,” Journal of Guidance, Navigation, Control, and Dynamics, Vol. 33, Sep.-Oct. 2010

(1)

(2)

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Application of PSO: Nominal Orbit

χ❑( 𝑗 ) (𝑖 )≡X 0 (k¿ , τ s . m . )

k = 1k = 2

k = 3

k = N/2

k = N

k = …

𝑋 0=[0 .866224052875085 [𝑑𝑢 ]0 .011670195668094 [𝑑𝑢 ]0 .186912185139037 [𝑑𝑢 ]0 .013870554690931 [𝑣 𝑢 ]0 .245270168936540 [𝑣𝑢 ]0 .021792775971957 [𝑣 𝑢 ]

]𝑃=2.31339 [𝑡𝑢 ]

Earth-Moon L1 Northern Halo Orbit: Defined by… χ

= …

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

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Optimal Trajectory

Optimal Patch Point:

k=610+, τs.m.=-18.101[tu], eGEO = 0.000930

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

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

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Future Work

χ❑( 𝑗 ) (𝑖 )≡X 0 (k¿ , τ s . m . )

χ❑( 𝑗 ) (𝑖 )≡X 0 (k , τ s .m .)

Repeat Study… hope is to further reduce

run-time by using less particles

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Thank You!

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

𝐽 (𝑋 𝑠 .𝑚 .)=𝑐1𝑒𝐺𝐸𝑂 (𝑋 𝑠 .𝑚 . )

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