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Autonomous Navigation in Libration Point Orbits
Keric A. Hill
Thesis Committee:
George H. Born, chairR. Steven NeremPenina AxelradPeter L. Bender
Rodney Anderson
27 April 2007
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Why Do We Need Autonomy?
Image credit: http://solarsystem.nasa.gov/multimedia/gallery/
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Measurement Types
Measurement TypeAccuracy
• Horizon Scanner angles to Earth
• Stellar Refraction angles to Earth
• Landmark Tracker angles to Landmark km
• Space Sextant scalar to the Moon km
• Sun sensors angles to the Sun
• Star trackers angles to stars
• Magnetic field sensors angles to Earth km
• Optical Navigation angles to s/c or bodies
• X-ray Navigation scalar to barycenter km
• Forward Link Doppler scalar to groundstation km
• DIODE (near Earth) scalar to DORIS stations m
• GPS (near Earth) 3D position, time cm
• Crosslinks (LiAISON) scalar to other s/c m
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Crosslinks
SST picture
Image credit: http://www.centennialofflight.gov/essay/Dictionary/TDRSS/
• Scalar measurements (range or range-rate)
• Estimate size, shape of orbits
• Estimate relative orientation of the orbits.
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Crosslinks
• Scalar measurements (range or range-rate)
• Estimate size, shape of orbits
• Estimate relative orientation of the orbits.
Image credit: http://www.centennialofflight.gov/essay/Dictionary/TDRSS/
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Two-body Problem SST
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Two-body Problem SST
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Two-body Problem SST
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Two-Body Symmetry
The vector field of accelerations in the x-y plane for the two-body problem.
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Two-Body Solutions
Initial Conditions
000 ,, zyx
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Two-Body Solutions
• All observable:
– a1, a2, e1, e2, v1, v2
• NOT all observable:
– Ω1, Ω2, i1, i2, ω1, ω2
Initial Conditions
Radius20
20
20 zyx
000 ,, zyx
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J2 Symmetry
The vector field of accelerations in the x-z plane for two-body and J2.
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J2 Solutions
Initial Conditions
000 ,, zyx
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J2 Solutions
• Observable:– a1, a2, e1, e2, v1, v2,
– ΔΩ, i1, i2, ω1, ω2
• NOT observable:– Ω1, Ω2
Initial Conditions
000 ,, zyx
Radius:
Height:
20
20 yx
0z
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Circular Restricted Three-body Problem
P1 P2x
y
Barycenter
z
μ 1-μ
r1 r2
spacecraft
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Three-body Symmetry
The vector field of accelerations in the x-z plane for the three-body problem.
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Lagrange Points
x
y
L1 L2
L4
L5
L3
P1 P2
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Three-body Solutions
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Proving Observability
• Orbit determination with two spacecraft.
• One spacecraft is in a lunar halo orbit.
• Observation type: Crosslink range.
– Gaussian noise 1 σ = 1.0 m.
• Batch processor :
– Householder transformation.
• Fit span = 1.5 halo orbit periods (~18 days).
• Infinite a priori covariance.
• Observations every ~ 6 minutes.
• LOS checks.
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OD Accuracy Metric
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Position Along the Halo
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Initial Positions
Sat 1
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Spacecraft Separation
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Out of Plane Component
LL1 Halo2 constellations
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LL3 Results: Weak
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Halo-Moon
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Monte Carlo Analysis
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Constellation Design Principles
• At least one spacecraft should be in a libration orbit.
• Spacecraft should be widely separated.
• Orbits should not be coplanar.
• Shorter period orbits lead to better results.
• More spacecraft lead to better results.
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Some Interesting Questions
• How does orbit determination work for unstable orbits?
• Why do the phase angles of the spacecraft affect the orbit determination so much?
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Observation Effectiveness
Accumulating the Information Matrix:
The effectiveness of the observation at time ti:
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Observation Effectiveness for Two-body Orbits
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Observation Effectiveness for Three-body Orbits
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Two-body Orbits, by Components
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510
-8
10-7
10-6
10-5
10-4
10-3
10-2
Time (days)
com
pone
nts
of
1
xyz
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Three-body Orbits, by Components
0 2 4 6 8 10 12 1410
-5
10-4
10-3
10-2
10-1
100
101
102
Time (days)
com
pone
nts
of
1
xyz
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Observation Effectiveness Dissected
Uncertainty Growth
Observation Geometry
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Instability and Aspect Ratio
Larger Aspect Ratio
Smaller Aspect Ratio
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Uncertainty Growth
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Observation Geometry
Most Effective Observation Vector
Axis of Most Uncertainty
Least Effective Observation
Vector
Axis of Least Uncertainty
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Local Unstable Manifolds
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Realistic Simulations
• Truth Model:
– DE403 lunar and planetary ephemeris
– DE403 lunar librations
– Solar Radiation Pressure (SRP)
– LP100K Lunar Gravity Model
– 7th-8th order Runge-Kutta Integrator
– Stationkeeping maneuvers with execution errors
• Orbit Determination Model:
– Extended Kalman Filter with process noise
– SRP error ~10-9 m/s2
– LP100K statistical clone
– Stationkeeping maneuvers without execution errors
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Halo Orbiter:4 Δv’s per period5% Δv errorscR error -> 1 x 10-9 m/s2
position error RSS ≈ 80 m
Snoopy-Woodstock Simulation
Lunar Orbiter:50x 95 km, polar orbit cR error -> 1 x 10-9 m/s2
5% Δv errors1σ gravity field cloneposition error RSS ≈ 7 m
Propagation: RK78 with JPL DE405 ephemeris, SRP, LP100K Lunar Gravity (20x20)
Orbit Determination: Extended Kalman Filte
Observations: Crosslink range with 1 m noise every 60 seconds
Moon
EarthThe lunar orbiter could hold science
instruments and be tracked to estimate the far side gravity field.
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Snoopy
0 5 10 15 20 25 30-200
0
200
x sa
t 1
(m)
Day (RMS = 37.94 m) 64.7% < 1
EKF sat 1 position error with state noise 2.68e-009 m/s2, RSS = 77.78 m
x error
2 stdev
0 5 10 15 20 25 30-200
0
200
y sa
t 1
(m)
Day (RMS = 42.57 m) 52.3% < 1
y error
2 stdev
0 5 10 15 20 25 30-200
0
200
z sa
t 1
(m)
Day (RMS = 52.90 m) 66.1% < 1
z error
2 stdev
L2 halo orbiter EKF position
error
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Woodstock
0 5 10 15 20 25 30-20
0
20
x sa
t 2
(m)
Day (RMS = 5.70 m) 90.8% < 1
EKF sat 2 position error with state noise 1.34e-008 m/s2, RSS = 6.87 m
x error
2 stdev
0 5 10 15 20 25 30-20
0
20
y sa
t 2
(m)
Day (RMS = 2.40 m) 71.3% < 1
y error
2 stdev
0 5 10 15 20 25 30-20
0
20
z sa
t 2
(m)
Day (RMS = 2.99 m) 91.0% < 1
z error
2 stdev
Lunar orbiter EKF position
error
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L2-Frozen Orbit Simulation
0 5 10 15 20 25 30
-200
-100
0
100
200
x sa
t 1
(m)
Day (RMS = 16.66 m) 89.9% < 1
EKF sat 1 position error with state noise 1.34e-011 m/s2, RSS = 19.82 m
x error
2 stdev
0 5 10 15 20 25 30
-200
-100
0
100
200
y sa
t 1
(m)
Day (RMS = 7.49 m) 96.4% < 1
y error
2 stdev
0 5 10 15 20 25 30
-200
-100
0
100
200
z sa
t 1
(m)
Day (RMS = 7.71 m) 99.6% < 1
z error
2 stdev
L2 halo orbiter EKF position
error
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L2-Frozen Orbit Simulation
Frozen orbiter EKF position
error
0 5 10 15 20 25 30-20
-10
0
10
20
x sa
t 2
(m)
Day (RMS = 4.23 m) 33.0% < 1
EKF sat 2 position error with state noise 1.34e-010 m/s2, RSS = 5.38 m
x error
2 stdev
0 5 10 15 20 25 30-20
-10
0
10
20
y sa
t 2
(m)
Day (RMS = 2.53 m) 56.6% < 1
y error
2 stdev
0 5 10 15 20 25 30-20
-10
0
10
20
z sa
t 2
(m)
Day (RMS = 2.16 m) 80.7% < 1
z error
2 stdev
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Frozen Orbit Constellation
Frozen orbiter EKF position
error
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L1-LEO
L1 halo orbiter EKF position
error
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Application: Comm/Nav for the Moon
Image credit: http://photojournal.jpl.nasa.gov
L1 L2
South Pole/
Aitken Basin
Far
Side
EarthMoon
6 out of 10 of the lunar landing sites mentioned in ESAS require a communication relay.
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Future Work
• Perform navigation simulations using independently validated software (GEONS was not quite ready).
• Compare ground-based navigation with space-based navigation at the Moon.
• Obtain and process crosslink measurements for any of the following situations:
– Halo Orbiter – Halo Orbiter
– Halo Orbiter – Lunar Orbiter
– Lunar Orbiter – Earth Orbiter
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Acknowledgements
• This material is based upon work supported under a National Science Foundation Graduate Research Fellowship. Any opinions, findings, conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation.
• The idea for this research came from Him for whom all orbits are known.