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CONTROL STRATEGIES FOR FORMATION FLIGHTIN THE VICINITY OF THE LIBRATION POINTS

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

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Previous Work on Formation Flight

• Multi-S/C Formations in the 2BP– Small Relative Separation (10 m – 1 km)

• Model Relative Dynamics via the C-W Equations• Formation Control

– LQR for Time Invariant Systems– Feedback Linearization– Lyapunov Based and Adaptive Control

• Multi-S/C Formations in the 3BP– Consider Wider Separation Range

• Nonlinear model with complex reference motions– Periodic, Quasi-Periodic, Stable/Unstable Manifolds

• Formation Control via simplified LQR techniques and “Gain Scheduling”-type methods.

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Deputy S/C( ), ,d d dx y z

x

y

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

X

ˆ ˆ,Z z

B

1cr2cr

cr

xθ

y

Chief S/C( ), ,c c cx y z

ˆdr rρ=

r

ξβ

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Dynamical Model

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

2

2c c c c c

d c d c d d d

r t f r t Jr t Kr t u t

r t f r t r t f r t Jr t Kr t u t

= + + + = + − + + +

Nonlinear EOMs:

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

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

0 0, 2

d d d dd d

c dd d d d

Ir t r t r t r tu t u t

r t r t J Ir t r t r t r t

− − = + − Ω− −

Linear System:

( )A t B ( )du tδ( )dx tδ( )dx tδ

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Reference Motions• Fixed Relative Distance and Orientation

– Chief-Deputy Line Fixed Relative to the Rotating Frame

– Chief-Deputy Line Fixed Relative to the Inertial Frame

• Fixed Relative Distance, No Orientation Constraints• Natural Formations (Center Manifold)

– Deputy evolves along a quasi-periodic 2-D Torus that envelops the chief spacecraft’s halo orbit (bounded motion)

( ) ( ) and 0d dr t c r t= =

( )( )( )

0 0

0 0

0

cos sin

cos sind d d

d d d

d d

x t x t y t

y t y t x t

z t z

= +

= −

=

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

Az = 0.2×106 km

Az = 1.2×106 kmAz = 0.7×106 km

5000 kmρ =

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

xDeputy S/C

Deputy S/C

Deputy S/C

Deputy S/C

Chief S/C

y

z

Minimum Cost Formations

z

yx

Deputy S/C

Deputy S/CChief S/C

Maximum Cost Formation

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Formation Keeping Cost Variation Along the SEM L1 and L2 Halo Families(Configurations Fixed in the Rotating Frame)

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

Az = 0.2×106 km

Az = 1.2×106 kmAz = 0.7×106 km

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Quasi-Periodic Configurations(Natural Formations Along the Center Manifold)

x

y

z

∆VNOMINAL = 0

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Controllers Considered

( ) ( ) ( ) ( )( )0

1min2

ftT T

d d d dJ x t Q x t u t R u t d tδ δ δ δ= +∫

• LQR

( ) ( )( ) ( )x t f x t u t= + ( ) ( )( ) ( )( )u t f x t g x t= − +

• Input Feedback Linearization

( ) ( )( ) ( )

( )( )1/ 2T

T

x t f x t u t

r rry t r rr

r

= +

= =

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

3 122 2

2 2

T T T T T

T

r r r r r r r r r r r g r

g r r ru t r Jr Kr f rr r

− −= − + + =

= − − − −

• Output Feedback Linearization

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

1

1

Td d

T T

u t R B P t x t

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

δ δ−

−

= −

= − − + −

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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 = − −

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Control Acceleration Histories

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Conclusions• Natural vs. Forced Formations

– The nominal formation keeping costs in the CR3BP are verylow, even for relatively large non-naturally occurring formations.

• Above the nominal cost, standard LQR and FL approaches work well in this problem. – Both LQR & FL yield essentially the same control histories but

FL method is computationally simpler to implement.• The required control accelerations are extremely low.

However, this may change once other sources of error and uncertainty are introduced. – Low Thrust Delivery– Continuous vs. Discrete Control

• Complexity increases once these results are transferred into the ephemeris model.