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Talk charts from the 2003 Flight Mechanics Symposium, held on October 28-30, 2003, at NASA Goddard Space Flight Center
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1
DESIGN AND CONTROL OF FORMATIONS NEAR THE LIBRATION POINTS
OF THE SUN-EARTH/MOON EPHEMERIS SYSTEM
K.C. Howell and B.G. MarchandPurdue University
2
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
3
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
4
Natural Formations
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Natural Formations:String of Pearls
x y
z z
y
x
6
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
7
Natural Formations:Nearly Periodic Relative Motion
Origin = Chief S/C
10 Revolutions = 1,800 days
8
Evolution of Nearly Vertical OrbitsAlong the yz-Plane
9
Natural Formations:Slowly Expanding Vertical Orbits
Origin = Chief S/C
( )0r ( )fr t
100 Revolutions = 18,000 days
10
Non-Natural Formations
11
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 =
12
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
13
Formation Keeping Cost Variation Along the SEM L1 and L2 Halo Families
(Configurations Fixed in the RLP Frame)
14
Discrete vs. Continuous Control
15
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)
16
Achievable Accuracy via Targeter Scheme
Max
imum
Dev
iatio
n fr
om N
omin
al (c
m)
Formation Distance (meters)
ˆr rY=
17
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
19
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
20
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
21
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= − = −
22
OFL Control of Spherical FormationsRadial Dist. + Rotation Rate
Quadratic Growth in Cost w/ Rotation Rate
Linear Growth in Cost w/ Radial Distance
23
Inertially Fixed Formationsin the Ephemeris Model
e
100 km
Chief S/C
500 m
ˆ inertially fixed formation pointing vector (focal line)e =
24
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