Titan Mission Trajectory Designcds.caltech.edu/.../missiontitan/presentation...08.pdf · Titan...

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Titan Mission Trajectory Design Evan Gawlik

Transcript of Titan Mission Trajectory Designcds.caltech.edu/.../missiontitan/presentation...08.pdf · Titan...

Page 1: Titan Mission Trajectory Designcds.caltech.edu/.../missiontitan/presentation...08.pdf · Titan Mission Trajectory Design Evan Gawlik. Outline zThe circular restricted three-body problem

Titan MissionTrajectory Design

Evan Gawlik

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Outline

The circular restricted three-body problem (CR3BP)Invariant manifolds in the CR3BPDiscrete Mechanics and Optimal Control (DMOC)Application:– Shoot the Moon– Saturnian moon tour

Page 3: Titan Mission Trajectory Designcds.caltech.edu/.../missiontitan/presentation...08.pdf · Titan Mission Trajectory Design Evan Gawlik. Outline zThe circular restricted three-body problem

The Circular Restricted Three-Body Problem (CR3BP)

m1 = 1− μm2 = μ

G = 1

ω = 1

m1 m2x

y

m3

r13 r23

(−μ, 0) (1− μ, 0)

(x, y)

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The Circular Restricted Three-Body Problem (CR3BP)

x− 2y =∂Ω

∂x

y + 2x =∂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(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 ofL1 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

Page 10: Titan Mission Trajectory Designcds.caltech.edu/.../missiontitan/presentation...08.pdf · Titan Mission Trajectory Design Evan Gawlik. Outline zThe circular restricted three-body problem

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−1Xk=0

||fk||∆t

q0

q1q2 qN

etc.

Page 13: Titan Mission Trajectory Designcds.caltech.edu/.../missiontitan/presentation...08.pdf · Titan Mission Trajectory Design Evan Gawlik. Outline zThe circular restricted three-body problem

Variational Integrators

Arrive at theEuler-Lagrange equations

∂L∂q − d

dt∂L∂q = 0

Continuous:Extremize the integralZ T

0

L(q, q) dt

q(t)q0

q1q2 qN

etc.

Discrete:Extremize the sum

N−1Xk=0

Ld(qk, qk+1,∆t)

Arrive at the discreteEuler-Lagrange equations

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

Page 14: Titan Mission Trajectory Designcds.caltech.edu/.../missiontitan/presentation...08.pdf · Titan Mission Trajectory Design Evan Gawlik. Outline zThe circular restricted three-body problem

DMOC

Thrust

Subject to:

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

Minimize:

∆V =N−1Xk=0

||fk||∆t

q0

q1q2 qN

etc.

Page 15: Titan Mission Trajectory Designcds.caltech.edu/.../missiontitan/presentation...08.pdf · Titan Mission Trajectory Design Evan Gawlik. Outline zThe circular restricted three-body problem

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 = − 12a

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Saturnian Moon Tourµωn+1Kn+1

¶=

µωn − 2π(−2Kn+1)

−3/2 mod 2πKn + 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 SaturnComparison with standard trajectory design techniquesAnalysis of trade-off between Delta-V vs. time-of-flight

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Acknowledgments

Dr. Jerrold MarsdenStefano CampagnolaAshley MooreMarin KobilarovSina Ober-BlöbaumSigrid LeyendeckerSURF officeThe Aerospace Corporation