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Titan Mission Trajectory Designcds.caltech.edu/.../missiontitan/presentation...08.pdf · Titan...
Transcript of Titan Mission Trajectory Designcds.caltech.edu/.../missiontitan/presentation...08.pdf · Titan...
Titan MissionTrajectory Design
Evan Gawlik
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
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)
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)
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 ofL1 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−1Xk=0
||fk||∆t
q0
q1q2 qN
etc.
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
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.
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 Tourµωn+1Kn+1
¶=
µωn − 2π(−2Kn+1)
−3/2 mod 2πKn + 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 SaturnComparison with standard trajectory design techniquesAnalysis of trade-off between Delta-V vs. time-of-flight
Acknowledgments
Dr. Jerrold MarsdenStefano CampagnolaAshley MooreMarin KobilarovSina Ober-BlöbaumSigrid LeyendeckerSURF officeThe Aerospace Corporation