1

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.