general examination for andrew abraham august 31 st , 2012
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General Examination for Andrew Abraham August 31 st , 2012. Journal Article: “Characterization of Trajectories Near the Smaller Primary in Restricted Problem for Applications” by Davis and Howell. Motivation: Mission Planning. Most missions planned using 2-body dynamics - PowerPoint PPT PresentationTRANSCRIPT
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General Examination forAndrew Abraham
August 31st, 2012
Journal Article: “Characterization of Trajectories Near the Smaller Primary in
Restricted Problem for Applications” by Davis and Howell
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Motivation: Mission Planning
• Most missions planned using 2-body dynamics– Space Station (LEO)– GPS (MEO)– Direct TV (GEO)
• Some are planned using 3-body dynamics– SOHO (Sun-Earth L1)
– GRAIL (Sun-Earth L1 to Lunar Orbit)
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Genesis Mission: 2001
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Keplerian Dynamics
Semi-Major Axis, a
Apse Line
r
Periapsis
Primary
Secondary
θ
Apoapsis
rprp
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• At periapsis velocity vector is orthogonal to radius vector.
• Two constants of motion:– Angular Momentum, h– Total Energy, E
Keplerian Dynamics:Need to Know
r
v
periapsis
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Poincare Maps
X
P1(x) S
• Traditional- X maps to P1(x), P2(x), P3(x),…
• Non-Traditional- X maps only to P1(x)- Map many X in set U
P2(x)
• Poincare Surface of Selection- 2-D plane S- Used to reduce dimensions
X in R3 P(x) on S in R2
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Coordinate System: 3-Body
Non-Inertial!
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Definitions / Assumptions
Assume Define
• m1 ≥ m2 >> m3 • m1 and m2 orbit their Barycenter in
perfectly circular orbits1
1 2
mm m
1 23
1,2
G m mG
r
12T tu
2 1 radGT tu
1,2 1r du
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Energy
2 22 23 3
1 12 2
T m v m x y y x z
1 3 2 3
1,3 2,3
Gmm Gm mVr r
2 2 2
3 1,3 2,3
1 12
E x y y x zm r r
2 2 21,3r x y z 2 2 2
2,3 1r x y z
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Equations of MotionL T V
Euler-Lagrange Eq: 0L d Lx dt x
2
2
Vx y xxVy x yy
Vzz
Equations of Motion:
Lagrangian:
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Simplified EOM
2 212
U V x y Pseudopotential:
2
2
Ux yxUy xy
Uzz
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Lagrange Points
0x x y y Look for stationary points from EOM:
5 such points exist: L1-5
4,51 3,2 4
L
Two “triangular” points exist:
Three “co-linear” points exist but are analytically intractable. Numeric approximation, given μ, is the only way to deal with them:
1
2
3
0.8369, 0
1.1557, 0
1.0050, 0
L
L
L
Earth-Moon System
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Lagrange Point Geometry
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Jacobi Energy
2 2 2
1,3 2,3
2 12 2J x y vr r
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Forbidden Regions
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Summary of Article
Items Covered:
• Periapsis Poincare Map• Affect of 3-Body Dynamics on Periapsis• Short Term vs. Long Term Dynamics• yp vs. xp Maps• Titan Mission Example
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Periapsis Poincare MapOriginal State:
Three Conditions:
1. The Poincare Surface of Selection shall be the xy-plane.
2. All trajectories shall have exactly the same Jacobi Energy, J.
3. The initial state shall be the periapsis point of an orbit about P2.
, , , , , ToX x y z x y z
, , , ToX x y x y
2 2 2x y v
Gives direction to the velocity vector
, ToX x y
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Periapsis Poincare Map
State After 1 Revolution State After 6 Revolutions
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Hill Region
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P1 Perturbations
Decrease E, rp Increase e
Increase e, Rotate Apse Line
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yp vs. xp Maps
33 Revolutions, 0.07 0.25oHill p Hillr r r
0or
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Periodicity
CR3BP Coordinates
Inertial Coordinates
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Titan Mission Example
Enter Titan’s Hill Region via L2
Remain Captured After Delta-V Burn
6 Rev. Poincare Map 33 Rev. yp vs. xp Map
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Titan Mission
7.2 m/s Burn at Green Dot: Closes ZVC
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Extension by Abraham
Items Covered:
• List of Possible Extensions• RGB Poincare Map• Programming• Results• Future Work
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Possible Extensions: Abraham
• Numeric Sensitivity– Error Tolerance of Integrator• Affect on: runtime, accuracy
– ODE45 vs. ODE113• Runge-Kutta vs. Adams-Bashforth-Moulton
• Sensitivity to Initial Conditions– w.r.t. to position or Jacobi Energy– Are gradients related? Gravity or psuedopotential
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Possible Extensions: Abraham
• Propagation Time– Sensitivity?– How long should you propagate?– Dependence on J or μ?– Quantify. (Ex: 99% chance Trajectory X won’t impact nor escape moon)
• Spacecraft Dispersions– How vulnerable are the trajectories?– Can the degree of vulnerability be
quantified/mapped?
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RGB Poincare Map
• Working with periapsis is limited– Based on a 2-body definition– Velocity direction may not be orthogonal
• What can replace periapsis?• Average velocity vector
• Can represent any point along trajectory
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How to Assign Colors?
3 Stopping Conditions:1. STOP 1 = Impact Moon = RED2. STOP 2 = Escaped Moon = Blue3. End of Time = Bounded near Moon = Green
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Record Stopping ConditionsLet’s say 36 trajectories are computed per point and the results are:
12 Blue4 Red20 Green
[R, G, B] = [4,20,12]/36
[R, G, B] = [0.111, 0.556, 0.333] on a 0 1 scale
Resulting Color
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First Test
Time = 1tu, Distance Between Pt. = 0.05 du, Angular Separation = 90o, Run-Time = 2min.
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Second Test
Time = 10tu, Distance Between Pt. = 0.05 du, Angular Separation = 90o, Run-Time = 15min.
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Third Test
Time = 10tu, Distance Between Pt. = 0.001 du, Angular Separation = 90o, Run-Time = 10 hours
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Final Test
Time = 10tu, Distance Between Pt. = 0.001 du, Angular Separation = 10o, Run-Time = 30 hours w/ Parallel Computing Using 4-Core Processor
32,000 Pixels,1 Million Trajectories
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Structure
• Flower Petal Pattern– Repetitive– Green where leaves overlap
• Orange = Twice as likely to impact moon as to remain bounded (Orange = 2 red + 1 green)
• Lower left and upper right are same color• Upper left and lower right are same color
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Future Work
• Re-run code w/ better resolution– 1o or even 0.1o angular separation – 10 or 100 million trajectories
• Distributed Computer Cluster– Parallel Computing
• Re-write in C++• Fill in all pixels within Hill Region• Create GUI for manipulation of output
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Future Work
• Restrict the range of velocity vector
• Determine the Maps for various values of J & μ• Make video of map as J evolves from JL1 JL4,5
– Look for bifurcations and chaos – Speaks to spacecraft dispersions
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Questions?
Thank You!