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Trajectory Design Titan Mission Outline The circular restricted - - PDF document
Trajectory Design Titan Mission Outline The circular restricted - - PDF document
Evan Gawlik Trajectory Design Titan Mission Outline The circular restricted three-body problem (CR3BP) Invariant manifolds in the CR3BP Discrete Mechanics and Optimal Control (DMOC) Application: Shoot the Moon Saturnian
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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)
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The Circular Restricted Three-Body Problem (CR3BP)
¨ x − 2 ˙ y = ∂Ω ∂x ¨ y + 2 ˙ x = ∂Ω ∂y Ω(x, y) = x2 + y2 2 + 1 − μ p (x + μ)2 + y2 + μ p (x − 1 + μ)2 + y2
Constant of motion:
E = 1 2( ˙ x2 + ˙ y2) − Ω(x, y)
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The Circular Restricted Three-Body Problem (CR3BP)
Energetically “forbidden” regions
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The Circular Restricted Three-Body Problem (CR3BP)
Lagrange points Li, i = 1, 2, 3, 4, 5 Note the positions of L1 and L2
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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
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Orbits with Prescribed Itineraries
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Interplanetary Transport Network
http://www.jpl.nasa.gov/releases/2002/release_2002_147.html
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Shoot the Moon
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Shoot the Moon
∆V = 163 m/s
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DMOC
Thrust Subject to: D2Ld(qk−1, qk, ∆t) + D1Ld(qk, qk+1, ∆t) + f +
k−1 + f − k = 0
Minimize: ∆V =
N−1
X
k=0
||fk|| ∆t q0 q1 q2 qN
etc.
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Variational Integrators
Arrive at the Euler-Lagrange equations
∂L ∂q − d dt ∂L ∂ ˙ q = 0
Continuous: Extremize the integral Z T L(q, ˙ q) dt q(t) q0 q1 q2 qN
etc.
Discrete: Extremize the sum
N−1
X
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
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DMOC
Thrust Subject to: D2Ld(qk−1, qk, ∆t) + D1Ld(qk, qk+1, ∆t) + f +
k−1 + f − k = 0
Minimize: ∆V =
N−1
X
k=0
||fk|| ∆t q0 q1 q2 qN
etc.
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Shoot the Moon
∆V = 163 m/s
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Shoot the Moon
∆V = 17 m/s
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Saturnian Moon Tour
The invariant manifolds of the various
Saturn-Moon-spacecraft three-body systems do not intersect.
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Saturnian Moon Tour
K = − 1
2a
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Saturnian Moon Tour
µ ωn+1 Kn+1 ¶ = µ ωn − 2π(−2Kn+1)−3/2 mod 2π Kn + μf(ωn) ¶
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Saturnian Moon Tour
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Saturnian Moon Tour
∆V = 13 m/s
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Saturnian Moon Tour
∆V = 13 m/s
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Saturnian Moon Tour
∆V = 13 m/s Rotating Frame
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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
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Acknowledgments
- Dr. Jerrold Marsden