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Transcript of Titan Mission Trajectory Titan Mission Trajectory Design Evan Gawlik. Outline zThe circular...

  • Titan Mission Trajectory Design

    Evan Gawlik

  • Outline

    The circular restricted three-body problem (CR3BP) Invariant manifolds in the CR3BP Discrete Mechanics and Optimal Control (DMOC) Application: – Shoot the Moon – Saturnian moon tour

  • The Circular Restricted Three-Body Problem (CR3BP)

    m1 = 1− μ m2 = μ

    G = 1

    ω = 1

    m1 m2 x

    y

    m3

    r13 r23

    (−μ, 0) (1− μ, 0)

    (x, y)

  • The Circular Restricted Three-Body Problem (CR3BP)

    ẍ− 2ẏ = ∂Ω ∂x

    ÿ + 2ẋ = ∂Ω

    ∂y

    Ω(x, y) = x2 + y2

    2 +

    1− μp (x+ μ)2 + y2

    + μp

    (x− 1 + μ)2 + y2

    Constant of motion:

    E = 1

    2 (ẋ2 + ẏ2)− Ω(x, y)

  • The Circular Restricted Three-Body Problem (CR3BP)

    Energetically “forbidden” regions

  • The Circular Restricted Three-Body Problem (CR3BP)

    Lagrange points Li, i = 1, 2, 3, 4, 5

    Note the positions of L1 and L2

  • Invariant Manifolds

    Ross, S.D., “Cylindrical Manifolds and Tube Dynamics in the Restricted Three-Body Problem" (PhD thesis, California Institute of Technology, 2004), 109.

    Stable and unstable manifolds of the L1 and L2 Lyapunov orbits belonging to a particular energy surface

    Projection of cylindrical tubes onto position space

  • Orbits with Prescribed Itineraries

  • Interplanetary Transport Network

    http://www.jpl.nasa.gov/releases/2002/release_2002_147.html

  • Shoot the Moon

  • Shoot the Moon

    ∆V = 163 m/s

  • DMOC

    Thrust

    Subject to:

    D2Ld(qk−1, qk,∆t) +D1Ld(qk, qk+1,∆t) + f+k−1 + f − k = 0

    Minimize:

    ∆V = N−1X k=0

    ||fk||∆t

    q0

    q1 q2 qN

    etc.

  • Variational Integrators

    Arrive at the Euler-Lagrange equations

    ∂L ∂q − ddt ∂L∂q̇ = 0

    Continuous: Extremize the integralZ T

    0

    L(q, q̇) dt

    q(t) q0

    q1 q2 qN

    etc.

    Discrete: Extremize the sum

    N−1X k=0

    Ld(qk, qk+1,∆t)

    Arrive at the discrete Euler-Lagrange equations

    D2Ld(qk−1, qk,∆t) +D1Ld(qk, qk+1,∆t) = 0

  • DMOC

    Thrust

    Subject to:

    D2Ld(qk−1, qk,∆t) +D1Ld(qk, qk+1,∆t) + f+k−1 + f − k = 0

    Minimize:

    ∆V = N−1X k=0

    ||fk||∆t

    q0

    q1 q2 qN

    etc.

  • Shoot the Moon

    ∆V = 163 m/s

  • Shoot the Moon

    ∆V = 17 m/s

  • Saturnian Moon Tour

    The invariant manifolds of the various Saturn-Moon-spacecraft three-body systems do not intersect.

  • Saturnian Moon Tour

    K = − 12a

  • Saturnian Moon Tour µ ωn+1 Kn+1

    ¶ =

    µ ωn − 2π(−2Kn+1)−3/2 mod 2π

    Kn + μf(ωn)

  • Saturnian Moon Tour

  • Saturnian Moon Tour

    ∆V = 13 m/s

  • Saturnian Moon Tour

    ∆V = 13 m/s

  • Saturnian Moon Tour

    ∆V = 13 m/s

    Rotating Frame

  • Further Study

    Continuation to inner moons of Saturn Comparison with standard trajectory design techniques Analysis of trade-off between Delta-V vs. time-of-flight

  • Acknowledgments

    Dr. Jerrold Marsden Stefano Campagnola Ashley Moore Marin Kobilarov Sina Ober-Blöbaum Sigrid Leyendecker SURF office The Aerospace Corporation

    Titan Mission�Trajectory Design Outline The Circular Restricted Three-Body Problem (CR3BP) The Circular Restricted Three-Body Problem (CR3BP) The Circular Restricted Three-Body Problem (CR3BP) The Circular Restricted Three-Body Problem (CR3BP) Invariant Manifolds Orbits with Prescribed Itineraries Interplanetary Transport Network Shoot the Moon Shoot the Moon DMOC Variational Integrators DMOC Shoot the Moon Shoot the Moon Saturnian Moon Tour Saturnian Moon Tour Saturnian Moon Tour Saturnian Moon Tour Saturnian Moon Tour Saturnian Moon Tour Saturnian Moon Tour Further Study Acknowledgments