Optimal two impulse earth moon trajectories
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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:


V arying time of flight2
Varying Time of Flight


V arying time of flight3
Varying Time of Flight


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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