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OPTIMAL TWO IMPULSE EARTH-MOON Trajectories. John McGreevy Advisor : Manoranjan Majji Committee: John Crassidis Tarun Singh. Outline. Motivation Problem Formulation Results for Planar Trajectories Comparison with previous work Effect of varying initial conditions

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optimal two impulse earth moon trajectories

OPTIMAL TWO IMPULSE EARTH-MOON Trajectories

John McGreevy

Advisor: Manoranjan Majji

Committee:

John Crassidis

TarunSingh

outline
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
motivation
Motivation
  • Lunar Missions
    • Moon observing
    • Moon landing
  • Reduce necessary thrust
    • Save fuel/money
    • Increase payload
  • Short duration
    • Not low-energy trajectories
problem statement
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
solution method
Solution Method
  • Earth centered non-rotating frame
  • Using necessary conditions of optimal control theory
  • Hamiltonian:
terminal constraints
Terminal Constraints
  • Form augmented cost function:
  • are terminal Lagrange multipliers
terminal constraints1
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:
necessary conditions
Necessary Conditions
  • Can also verify solution using direct method
    • Take variation of augmented cost function with respect to and set = 0
computations
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
computations1
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.

comparison results
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
comparison results1
Comparison Results

Inertial Frame, 4.5 days, -116.88°, 463km

comparison results2
Comparison Results

Rotating Frame, 4.5 days, -116.88°, 463km

comparison results3
Comparison Results

Rotating Frame, 4.5 days, -116.88°, 463km

comparison results4
Comparison Results

Inertial Frame, 4.5 days, -116.88°, 463km

  • Comparable results for both trajectories
  • Small increase due to additional term in dynamics
varying initial conditions
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
v arying time of flight
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
v arying time of flight1
Varying Time of Flight

Retrograde:

Prograde:

  • Inertial:
  • Rotating:
summary so far
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
additional constraint
Additional Constraint
  • Using equation for orbital inclination:
  • Put into quadratic form:
  • Form new augmented cost:
terminal constraints2
Terminal Constraints
  • New equation for
    • where
  • Because states are free at final time, the co-states are fixed
    • Transversality conditions:
solution failure at 90 inclination
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
summary
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
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