Trajectory Design in a PlanetMoon Environment Using the Controlled Keplerian Map

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Trajectory Design in a Planet-Moon Environment
Using the Controlled Keplerian Map

• Piyush Grover
• Shane Ross
• Engineering Science and Mechanics, Virginia Tech
• www.filebox.vt.edu/groverp
• www.esm.vt.edu/sdross
• 01/30/2008

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Outline of the work

• Aim To obtain trajectories for multi-moon
  orbiter visiting various moons of Jupiter, and
  estimates of fuel required.
• Patched three body approach (P3BA).
• Use of Analytical Keplerian Map approximation for
  fast propagation of initial conditions in PCR3BP.
• Discrete low thrust control in the form of
  instantaneous velocity inputs.

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PCR3BP Dynamics

• Recall the phase space in the planar circular
  restricted three body problem motion of a
  spacecraft in the field of two large bodies in
  circular motion.

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PCR3BP Dynamics II
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Planet Moon Environment An n(3) body problem

• Jupiter-Europa-Ganymede considered here.
• Four body problem.
• Desirable to spend considerable time around each
  moon and not just a hyperbolic flyby.
• Need fuel efficient methods which also minimize
  time spent for inter moon travel Patched conic
  methods would not suffice.

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Patched Three Body Approach

• Introduced by Koon et al.(2000) and developed in
  Ross et al. (2003).
• Jovicentric orbit of the spacecraft.
• The perturbations of the dominant moon is
• considered (occurs in the form of gravity
  assists).
• Solutions are calculated and appropriately
  patched.

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Patched Three Body Approach II

• Spacecraft gets a gravity assist from outer moon
  M1
• when it passes through apoapse if near a
  resonance.
• When periapse close to inner moonM2s orbit is
  reached,
• it takes control this occurs for ellipse E

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Patched Three Body Approach III
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Actual Fuel Optimal Trajectory
The transfer between three-body systems occurs
when energy surfaces intersectcan be seen on
semimajor axis vs. eccentricity diagram
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Actual Fuel Optimal Trajectory II

• Obviously not time optimal.
• Transfer times can be too long to be feasible
  (several years).
• Spacecraft gets stuck in resonances for long
  periods.

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Closer Look at Gravity Assists

• Sensitive dependence on the periapse angle.
• The jump in semi-major axis is almost
  instantaneous.

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Keplerian Map

• Captures the kick in a received by the
  spacecraft during every periapse passage
  denoted by kick function f
• Derived by integrating the perturbation due to
  the moon over an unperturbed Keplerian orbit
  around Jupiter.

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Keplerian Map II

• Can be thought of as an Poincare Section at
  periapse reduction in phase space by 1
  dimension.
• Preserves the Hamiltonian structure of the
  PCR3BP.
• Kick function is odd w.r.t periapse angle, and is
  significant over a very small range of periapse
  values

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Controlled Keplerian Map

• Can add low thrust discrete control, in the form
  of instantaneous velocity inputs.
• Assume Hamiltonian to be preserved since the
  control is very small.
• Map takes the form

u denotes the control input.
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Trajectory design

• Involves choosing appropriate control input
  sequence.
• Use a two fold strategy .
• Coarse control for rapid change of the semi-major
  axis.
• Fine-control for targetting specific areas in the
  phase space, i.e. entry/exit regions.
• Use the well known forward backward method for
  fine control.

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Coarse control

• Philosophy
• Go with the flow Control input to drive
  the spacecraft into regions with rapid decrease
  in semi-major axis(A-), and away from regions of
  high increase(A)

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Algorithm for Coarse Control
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Sample trajectory for a single PCR3BP
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Multi-Moon Orbiter(MMO) Trajectory

• Start with initial conditions near exit from
  Ganymede.
• Use coarse control to move towards the switching
  region with apoapse map.
• Switch to periapse map and continue with coarse
  control.
• Use fine control to target the image of stable
  manifold of periodic orbit around L2 for
  Jupiter-Europa system

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Sample MMO Trajectory

• Uses about 150 m/s of velocity input and takes
  about 1.5 years.
• Time taken is less than 10 of the fuel-optimal
  trajectory.

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Fuel Vs Time-of-Flight

• Fuel expected to be proportional to the size of
  regions A- and A, upto a limit.

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Summary and Conclusions

• Described a quick method to get good initial
  guesses for the multimoon orbiter trajectory.
• Uses the analytical Keplerian Map with patched
  three body approximation.
• Algorithmic approach.
• Can be fed into more sophisticated programs to
  get end-to-end trajectories.
• Can get low order estimates of fuel required for
  mission completion in a certain time frame.