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 moon tour

The Circular Restricted Three-Body Problem (CR3BP) y m 3 ( x, y ) m 1 = 1 − μ r 13 r 23 m 2 = μ m 1 m 2 G = 1 x ω = 1 ( − μ , 0) (1 − μ , 0)

The Circular Restricted Three-Body Problem (CR3BP) ∂ Ω x − 2 ˙ ¨ y = ∂ x ∂ Ω y + 2 ˙ ¨ x = ∂ y x 2 + y 2 1 − μ μ p p Ω ( x, y ) = + ( x + μ ) 2 + y 2 + ( x − 1 + μ ) 2 + y 2 2 Constant of motion: 1 x 2 + ˙ y 2 ) − Ω ( x, y ) E = 2( ˙

The Circular Restricted Three-Body Problem (CR3BP) Energetically “forbidden” regions

The Circular Restricted Three-Body Problem (CR3BP) Lagrange points L i , i = 1 , 2 , 3 , 4 , 5 Note the positions of L 1 and L 2

Invariant Manifolds Stable and unstable manifolds of the L 1 and L 2 Lyapunov orbits belonging to a particular energy surface Projection of cylindrical tubes onto position space Ross, S.D., “Cylindrical Manifolds and Tube Dynamics in the Restricted Three-Body Problem" (PhD thesis, California Institute of Technology, 2004), 109.

Orbits with Prescribed Itineraries

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

Shoot the Moon

∆ V = 163 m/s Shoot the Moon

DMOC Minimize: N − 1 X ∆ V = || f k || ∆ t q 2 q N q 1 k =0 q 0 etc. Subject to: D 2 L d ( q k − 1 , q k , ∆ t ) + D 1 L d ( q k , q k +1 , ∆ t ) + f + k − 1 + f − k = 0 Thrust

Variational Integrators q 2 q N q 1 q ( t ) q 0 etc. Continuous: Discrete: Extremize the integral Extremize the sum Z T N − 1 X L ( q, ˙ q ) d t L d ( q k , q k +1 , ∆ t ) 0 k =0 Arrive at the Arrive at the discrete Euler-Lagrange equations Euler-Lagrange equations ∂ L ∂ q − d ∂ L D 2 L d ( q k − 1 , q k , ∆ t ) + D 1 L d ( q k , q k +1 , ∆ t ) = 0 q = 0 d t ∂ ˙

DMOC Minimize: N − 1 X ∆ V = || f k || ∆ t q 2 q N q 1 k =0 q 0 etc. Subject to: D 2 L d ( q k − 1 , q k , ∆ t ) + D 1 L d ( q k , q k +1 , ∆ t ) + f + k − 1 + f − k = 0 Thrust

∆ V = 163 m/s Shoot the Moon

∆ V = 17 m/s Shoot the Moon

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

2 a K = − 1 Saturnian Moon Tour

Saturnian Moon Tour µ ¶ µ ¶ ω n − 2 π ( − 2 K n +1 ) − 3 / 2 ω n +1 mod 2 π = K n +1 K n + μ f ( ω n )

Saturnian Moon Tour

∆ V = 13 m/s Saturnian Moon Tour

∆ V = 13 m/s Saturnian Moon Tour

∆ V = 13 m/s Saturnian Moon Tour 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