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DESIGN AND CONTROL OF FORMATIONS NEAR THE LIBRATION POINTS

OF THE SUN-EARTH/MOON EPHEMERIS SYSTEM

K.C. Howell and B.G. MarchandPurdue University

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Reference Motions• Natural Formations

– String of Pearls – Others: Identify via Floquet controller (CR3BP)

• Quasi-Periodic Relative Orbits (2D-Torus)• Nearly Periodic Relative Orbits• Slowly Expanding Nearly Vertical Orbits

• Non-Natural Formations– Fixed Relative Distance and Orientation

– Fixed Relative Distance, Free Orientation – Fixed Relative Distance & Rotation Rate– Aspherical Configurations (Position & Rates)

+ Stable Manifolds

RLPInertial

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X

B θ

y

Deputy S/C

x

y

2-S/C Formation Modelin the Sun-Earth-Moon System

ˆ ˆ,Z z

cr

x

Chief S/C

ˆd cr r r rr= − =r

ξβ

( ) ( )( ) ( )Relative EOMs:

r t f r t u t= ∆ +

1L 2L

dr

Ephemeris System = Sun+Earth+Moon Ephemeris + SRP

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Natural Formations

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Natural Formations:String of Pearls

x y

z z

y

x

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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 Relative Motion

Origin = Chief S/C

10 Revolutions = 1,800 days

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Evolution of Nearly Vertical OrbitsAlong the yz-Plane

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Natural Formations:Slowly Expanding Vertical Orbits

Origin = Chief S/C

( )0r ( )fr t

100 Revolutions = 18,000 days

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Non-Natural Formations

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Nominal Formation Keeping Cost(Configurations Fixed in the RLP Frame)

Az = 0.2×106 km

Az = 1.2×106 kmAz = 0.7×106 km

( ) ( )180 days

0

V u t u t d t∆ = ⋅∫

5000 kmr =

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Max./Min. Cost Formations(Configurations Fixed in the RLP Frame)

xDeputy S/C

Deputy S/C

Deputy S/C

Deputy S/C

Chief S/C

y

zMinimum Cost Formations

z

yx

Deputy S/C

Deputy S/CChief S/C

Maximum Cost Formation

Nominal Relative Dynamics in the Synodic Rotating Frame

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Formation Keeping Cost Variation Along the SEM L1 and L2 Halo Families

(Configurations Fixed in the RLP Frame)

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Discrete vs. Continuous Control

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Discrete Control: Linear Targeter

( ) ˆ10 m

0

Yρ

ρ

=

=

Dis

tanc

e Er

ror R

elat

ive

to N

omin

al (c

m)

Time (days)

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Achievable Accuracy via Targeter Scheme

Max

imum

Dev

iatio

n fr

om N

omin

al (c

m)

Formation Distance (meters)

ˆr rY=

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Continuous Control:LQR vs. Input Feedback Linearization

( ) ( )( ) ( )r t F r t u t= + ( ) ( )( ) ( ) ( )( )Desired Dynamic Response

Anihilate Natural Dynamics

,u t F r t g r t r t= − +

• Input Feedback Linearization (IFL)

• LQR for Time-Varying Nominal Motions( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0

1

, , 0

0

T

T Tf

x t r r f t x t u t x x

P A t P t P t A t P t B t R B t P t Q P t−

= = → =

= − − + − → =

( ) ( )

( ) ( ) ( )( )Nominal Control Input

1

Optimal Control, Relative to Nominal, from LQR

Optimal Control Law:

Tu t u t R B P t x t x t− = + − −

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Dynamic Response to Injection Error5000 km, 90 , 0ρ ξ β= = =

LQR Controller IFL Controller

( ) [ ]0 7 km 5 km 3.5 km 1 mps 1 mps 1 mps Txδ = − −

Dynamic Response Modeled in the CR3BPNominal State Fixed in the Rotating Frame

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Output Feedback Linearization(Radial Distance Control)

( ) ( )( )

Generalized Relative EOMs

Measured Output

r f r u t

y l r

= ∆ + →

= →

Formation Dynamics

Measured Output Response (Radial Distance)Actual Response Desired Response

( ) ( ) ( )2

2 , , ,Tp r r q rd ly u r g r rdt

r= = + =

Scalar Nonlinear Functions of and r r

( ) ( )( ) ( ) ( ), 0Th r t r t u t r t− =

Scalar Nonlinear Constraint on Control Inputs

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Output Feedback Linearization (OFL)(Radial Distance Control in the Ephemeris Model)

• Critically damped output response achieved in all cases• Total ∆V can vary significantly for these four controllers

r ( ) ( ) ( )2

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

= − + − ∆

2r

1r

( ) ( ) ( )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( ),y l r r=

( ) ( ),ˆ

h r ru t r

r=

Geometric Approach:Radial inputs onlyr

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OFL Control of Spherical Formationsin the Ephemeris Model

Relative Dynamics as Observed in the Inertial Frame

Nominal Sphere

( ) [ ] ( ) [ ]0 12 5 3 km 0 1 1 1 m/secr r= − = −

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OFL Control of Spherical FormationsRadial Dist. + Rotation Rate

Quadratic Growth in Cost w/ Rotation Rate

Linear Growth in Cost w/ Radial Distance

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Inertially Fixed Formationsin the Ephemeris Model

e

100 km

Chief S/C

500 m

ˆ inertially fixed formation pointing vector (focal line)e =

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Conclusions• Continuous Control in the Ephemeris Model:

– Non-Natural Formations • LQR/IFL → essentially identical responses & control inputs• IFL appears to have some advantages over LQR in this case• OFL → spherical configurations + unnatural rates• Low acceleration levels → Implementation Issues

• Discrete Control of Non-Natural Formations– Targeter Approach

• Small relative separations → Good accuracy• Large relative separations → Require nearly continuous control• Extremely Small ∆V’s (10-5 m/sec)

• Natural Formations– Nearly periodic & quasi-periodic formations in the RLP frame– Floquet controller: numerically ID solutions + stable manifolds