Design And Control Of Formations Near The Libration Points Of The Sun-Earth/Moon Ephemeris System
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Transcript of Design And Control Of Formations Near The Libration Points Of The Sun-Earth/Moon Ephemeris System
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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