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FORMATION FLIGHT NEAR L1 AND L2IN THE SUN-EARTH/MOON EPHEMERIS SYSTEM

INCLUDING SOLAR RADIATION PRESSURE

B.G. Marchand and K.C. Howell

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Continuous Control• LQR and Input Feedback Linearization (IFL)

– Control Nominal State• Fixed in Rotating Frame• Fixed in Inertial Frame

– Critically damped error response– LQR & IFL → similar response

& control acceleration histories

• Output Feedback Linearization– Radial Distance Control– Free relative orientation

• Transition from CR3BP– Ephemeris Model w/ Solar Radiation Pressure (SRP)

Formation Keeping via IFL

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Output Feedback Linearization(Radial Distance Control)

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

r ( ) ( ) ( )2

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

= − + − ∆

2,310.0 m/sec

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 = − − + − ∆

16,442.0 m/sec

49.8 m/sec

Control LawCumulative ∆V

(180 days)

r ( ) ( ),ˆ

h r ru t r

r=

16,392.4 m/secGeometric Approach:

Radial inputs only

Nonlinear Scalar Constraint on ( )u t

( ),y l r r=

Target Nominal: 5 kmMission Time: 180 days

r =

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Partial Feedback Control of Spherical Formations

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

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Formation Keeping in the Ephemeris Model

via Discrete Control

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Linear Targeter Approach

( ) 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)

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Natural Formations:String of Pearls

x y

z z

y

x

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Natural Formations:Phased Vehicles Along Halo Orbit

C1

D3

D2

D1

Az = 200,000 km Unstable Halo Orbit

1 Unstable Eigenvalue (γ1)1 Stable Eigenvalue (γ2)4 Center Eigenvalues (γ3-γ6)

L1

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Eigenstructure Near Halo Orbit

( ) ( ),0 0RR

R

rx t x

rδ δ

= = Φ

Reference Halo Orbit

Chief S/C

Deputy S/C

( ) ( ) ( ) ( ),0 0Floquet Modal Matrix:

JtE t P t S t E e−= = Φ

( ) ( ) ( ) ( ) ( )6 6

1 1

:Solution to Variational Eqn. in terms of Floquet Modes

j j jj j

x t x t c t e t E t cδ δ= =

= = =∑ ∑

( )( ) ( ){ } ( ){ } 1

,0 0

Floquet Decomposition of ,0 :Jtt P t S e P S

t−

Φ =

Φ

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Natural Solutions:Torus (Associated with Modes 3 and 4)

Formation Evolving Along Torus Near L1(Rotating Libration Point Frame Centered at L1)

Chief S/C Centered View(RLP Frame)

y

x

z

y

x

x

z

y

z

z

xy

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Natural Solutions:Halo Orbits (Associated with Modes 5 and 6)

y

x

y

z

x

z

x

z

y

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Floquet Controller(Remove Unstable + 2 Center Modes)

( )1

2 5 6 31 3 4

2 5 6 3

Remove Modes 1, 3, and 4:

0r r r

v v v

x x xx x x

x x x Ivδ δ δα

δ δ δδ δ δ

−

= + + −∆

( )1

2 3 4 31 5 6

2 3 4 3

Remove Modes 1, 5, and 6:

0r r r

v v v

x x xx x x

x x x Ivδ δ δα

δ δ δδ δ δ

−

= + + −∆

( )3

2,3,43 or

2,5 6

6

1

,

01

:Find that removes undesired response modes

j jj j

j

j

v xI

x

v

δ α δ= =

=

+ ∆ = +

∆

∑ ∑

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Deployment into Torus(Remove Modes 1, 5, and 6)

Origin = Chief S/C

Deputy S/C( ) [ ]( ) [ ]0 5 00 0 m

0 1 1 1 m/sec

r

r

=

= −

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Deployment into Natural Orbits(Remove Modes 1, 3, and 4)

Origin = Chief S/C

3 Deputies

( ) [ ]( ) [ ]

00 0 0 m

0 1 1 1 m/sec

r r

r

=

= −

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Nearly Periodic Formations

Origin = Chief S/C

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Nearly Vertical Formations

Origin = Chief S/C

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Evolution of Nearly Vertical OrbitOver 100 Orbital Periods

Origin = Chief S/C

( )0r ( )fr t

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Conclusions

• Discrete Control of Natural Formations– Floquet controller

• Effective in identifying nominal formations + deployment • May lead to feasible control strategies → non-natural

formations

• Continuous Control of Non-Natural Formations– IFL/OFL effective; LQR computationally inefficient– OFL → spherical configurations + unnatural rates– Low acceleration levels → Implementation Issues

• Discrete Control of Non-Natural Formations– Small Formations → Good accuracy – Extremely Small ∆V’s (10-5 m/sec)

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Backup Slides

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LQR vs. IFL (CR3BP) Dynamic Response to Injection Error

5000 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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LQR vs. IFL (CR3BP) Control Accelerations

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Output Feedback Linearization(Radial Distance Control)

( ) ( )( ) ( )2 1

Generalized Relative EOMs

, Measured Output e.g. , ,

r f r u t

y l r r y r y r y r−

= ∆ + →

= → = = =

Formation Dynamics

Measured Output ResponseActual 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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Targeter Maneuver Schedule

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Converged Periodfor OFL Controlled Paths

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Stability of T-Periodic Orbits

( ) ( ) ( )( )

:,0 0

measured relative to periodic orbit

Linear Variational Equationx t t x

x t

δ δ

δ

= Φ

→

Re{ }jγ

Im{ }jγ

5,6γ3γ

4γ

2γ 1γ

UnstableStable

Center

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Evolution of Nearly Vertical OrbitsAlong the yz-Plane

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Floquet Control(Large Formations – Example 1)

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Floquet Controller Maneuver Schedule(For Example 1)

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Nearly Periodic Formations(Inertial Perspective)

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Nearly Vertical Formations(Inertial Perspective)