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ASPHERICAL FORMATIONS NEAR THE LIBRATION POINTSIN THE SUN-EARTH/MOON SYSTEM

B.G. Marchand and K.C. HowellPurdue University

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Output Feedback Linearization (OFL) in the Ephemeris Model

• Formation Keeping + Deployment– Chief S/C Evolves Along Lissajous Trajectory near Li

– Inertial Formation Geometry• Spherical Configurations

– Deputy Constrained to Orbit Chief S/C– Fixed Radial Distance– Fixed Radial Distance + Rotation Rate

• Aspherical Configurations Inertially Fixed Orientation– Deputy Constrained to Evolve Along Aspherical Surface– Surface may be Offset from Chief S/C

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OFL Controlled Response of Deputy S/CRadial Distance Tracking

2 ( ) ( ) ( )2

, Tg r r r r ru t r r f rr rr

= − + − ∆

3

4

( ) ( ) ( )2 2

,12

Tg r r r ru t r f rr r

= − − ∆

( ) ( ) ( )2, 3Tr r ru t rg r r r r f r

rr = − − + − ∆

Control Law

( ) ( ),ˆ

H r ru t r

r=

Geometric Approach:Radial inputs only1

1u

3u

2u

4u

Chief S/C @ Origin (Inside Sphere)

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Impact of Nominal Radial Separationon OFL Controlled Response of Deputy S/C

* 5 kmr = * 5000 kmr =

Radial Inputs Only

3-Axis Control

3-Axis ControlChief S/C @ Origin (Inside Sphere)

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OFL Controlled Response of Deputy S/CRadial Distance + Rotation Rate Tracking

* 5 kmr =

1 rev / 6 hrs1 rev / 1 day

Chief S/C @ Origin (Inside Sphere)

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Impact Commanded Rotation Rate on Cost

1 rev /24 hrs 0.19 mN1 rev /12 hrs 0.76 mN1 rev / 6 hrs 6.40 mN1 rev / 1 hrs 106.50 mN

→→→→

700 kgsm =

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Application to Aspherical Formations

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Parameterization of Parabolic Formation

pa

ph

puν

( )3ˆ focal linee

1e

Nadir

q

1 2 3ˆ ˆ ˆiCDEr xe ye ze= + +

Chief S/C (C)

iDeputy S/C (D )

{ }i i

i i

CD CDTI EE I

CD CDE IE I

r rC

r r

=

: inertially fixed focal frame: inertially fixed ephemeris frame

EI

/ cos

/ sin

p p p

p p p

p

x a u h

y a u h

z u q

ν

ν

=

=

= +

Zenith

Transform State from Focal to Ephemeris Frame

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Controller Development

( ) ( )

( ) ( )

( ) ( )

( )( )( )

( )

2 22 2 2

2 22 2 2

2 2 2 222 2

2 2 20 ,

2 2 21 ,

0 2

u x y

x

y q x y z

z

h h hx y g u u x y x f y fa a au

h h hx y u g q q x y x f y f fa a a

ux y xx yy xy yxgx y x y

x yν ν

− + + ∆ + ∆ − − = + + + ∆ + ∆ −∆ + −− + ++ + +

( )( )2 2

x yy f x f

x y

∆ − ∆ +

( ) ( ) ( )( ) ( ) ( )

( ) ( )

* * 2 *

* * *

*

2

*

, 2

, 2

u p p p n p p n p p

q p p n

n

n

g u

g

u u u u u u

g u u q q q q

k

q

ν ν ν ω ν

ω ω

ω ω

ν

= − − − −

= − − − −

= − −

Desired Response for , , and :u q ν

Solve for Control Law:

, critically dampedu qδ δ →

exponential decayδθ →

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OFL Controlled Parabolic Formation

X

Z

Y0t

0 1.6 hrst +

0 3.8 hrst +

0 8 hrst +

0 5 dayst +

0 6 dayst +

3e

10 km1 rev/day500 m

500 m

Phase I: 200 m

Phase II: 300 m /1 day

Phase III: 500 m

p

p

p

p

p

q

ha

uuu

ν===

=

=

=

=

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OFL Thrust Profile700 kgsm =

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Original Maxim Design

DETECTORSPACECRAFT

CONVERGERSPACECRAFT

200M

COLLECTORSPACECRAFT(32

PLACESEVENLY

SPACED)

200 m

Detector S/C

Converger S/C

Collector S/C

5000 km

10 km

http://maxim.gsfc.nasa.gov/documents/SPIE-2002/spie2002.ppt

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New Maxim Pathfinder

Science Phase #2

High Resolution

(100 nas)

20,000 km

1 km

http://maxim.gsfc.nasa.gov/documents/SPIE-2002/spie2002.ppt

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Maxim Configuration Example

19,500km1 rev/day

500 m

1 km

1 km

p

p

p

q

uha

ν===

=

=

700 kgsm =

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Building Non-Natural FormationsUsing Naturally Existing Solutions

& Impulsive Control

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Natural Formations:Quasi-Periodic Relative Orbits → 2-D Torus

y

x

z

Chief S/C Centered View(RLP Frame)

y

x

x

z

y

z

z

xy

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Natural Formations:Nearly Periodic + Slowly Expanding Orbits

Chief S/C @ Origin 1800 days

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Evolution of Nearly Vertical OrbitOver 100 Orbital Periods

Origin = Chief S/C

( )0r ( )fr t

18,000 days

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Enforcing Periodicity in the Ephemeris Model

( )3 5 m/sec

1 maneuver/yearV∆ = −

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Geometry of Natural Solutions in the Ephemeris Model

w/ SRP

Inertial Frame Perspective:

Rotating Frame Perspective:

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Concluding Remarks• OFL Control

– Successful for Formation Keeping & Reconfiguration• Spherical + Parabolic Formations• ↑ Rotation Rate = ↑ Thrust Level

– w/o Rotation Rate Thrust ~ O(nN)– w/ Rotation Rate Thrust ~ O(mN)

• Natural to Non-Natural Formations– Differential Corrector 1 small maneuver/year

• Can work well in the rotating frame– Depends on Impact of SRP

• Difficult in the inertial frame due to Geometry of Initial Guess