1 ASPHERICAL FORMATIONS NEAR THE LIBRATION POINTS IN THE SUN-EARTH/MOON SYSTEM B.G. Marchand and K.C. Howell Purdue University
Jun 22, 2015
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ASPHERICAL FORMATIONS NEAR THE LIBRATION POINTSIN THE SUN-EARTH/MOON SYSTEM
B.G. Marchand and K.C. HowellPurdue University
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Output Feedback Linearization (OFL) in the Ephemeris Model
• Formation Keeping + Deployment– Chief S/C Evolves Along Lissajous Trajectory near Li
– Inertial Formation Geometry• Spherical Configurations
– Deputy Constrained to Orbit Chief S/C– Fixed Radial Distance– Fixed Radial Distance + Rotation Rate
• Aspherical Configurations Inertially Fixed Orientation– Deputy Constrained to Evolve Along Aspherical Surface– Surface may be Offset from Chief S/C
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OFL Controlled Response of Deputy S/CRadial Distance Tracking
2 ( ) ( ) ( )2
, Tg r r r r ru t r r f rr rr
= − + − ∆
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( ) ( ) ( )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
( ) ( ),ˆ
H r ru t r
r=
Geometric Approach:Radial inputs only1
1u
3u
2u
4u
Chief S/C @ Origin (Inside Sphere)
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Impact of Nominal Radial Separationon OFL Controlled Response of Deputy S/C
* 5 kmr = * 5000 kmr =
Radial Inputs Only
3-Axis Control
3-Axis ControlChief S/C @ Origin (Inside Sphere)
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OFL Controlled Response of Deputy S/CRadial Distance + Rotation Rate Tracking
* 5 kmr =
1 rev / 6 hrs1 rev / 1 day
Chief S/C @ Origin (Inside Sphere)
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Impact Commanded Rotation Rate on Cost
1 rev /24 hrs 0.19 mN1 rev /12 hrs 0.76 mN1 rev / 6 hrs 6.40 mN1 rev / 1 hrs 106.50 mN
→→→→
700 kgsm =
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Application to Aspherical Formations
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Parameterization of Parabolic Formation
pa
ph
puν
( )3ˆ focal linee
1e
Nadir
q
1 2 3ˆ ˆ ˆiCDEr xe ye ze= + +
Chief S/C (C)
iDeputy S/C (D )
{ }i i
i i
CD CDTI EE I
CD CDE IE I
r rC
r r
=
: inertially fixed focal frame: inertially fixed ephemeris frame
EI
/ cos
/ sin
p p p
p p p
p
x a u h
y a u h
z u q
ν
ν
=
=
= +
Zenith
Transform State from Focal to Ephemeris Frame
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Controller Development
( ) ( )
( ) ( )
( ) ( )
( )( )( )
( )
2 22 2 2
2 22 2 2
2 2 2 222 2
2 2 20 ,
2 2 21 ,
0 2
u x y
x
y q x y z
z
h h hx y g u u x y x f y fa a au
h h hx y u g q q x y x f y f fa a a
ux y xx yy xy yxgx y x y
x yν ν
− + + ∆ + ∆ − − = + + + ∆ + ∆ −∆ + −− + ++ + +
( )( )2 2
x yy f x f
x y
∆ − ∆ +
( ) ( ) ( )( ) ( ) ( )
( ) ( )
* * 2 *
* * *
*
2
*
, 2
, 2
u p p p n p p n p p
q p p n
n
n
g u
g
u u u u u u
g u u q q q q
k
q
ν ν ν ω ν
ω ω
ω ω
ν
= − − − −
= − − − −
= − −
Desired Response for , , and :u q ν
Solve for Control Law:
, critically dampedu qδ δ →
exponential decayδθ →
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OFL Controlled Parabolic Formation
X
Z
Y0t
0 1.6 hrst +
0 3.8 hrst +
0 8 hrst +
0 5 dayst +
0 6 dayst +
3e
10 km1 rev/day500 m
500 m
Phase I: 200 m
Phase II: 300 m /1 day
Phase III: 500 m
p
p
p
p
p
q
ha
uuu
ν===
=
=
=
=
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OFL Thrust Profile700 kgsm =
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Original Maxim Design
DETECTORSPACECRAFT
CONVERGERSPACECRAFT
200M
COLLECTORSPACECRAFT(32
PLACESEVENLY
SPACED)
200 m
Detector S/C
Converger S/C
Collector S/C
5000 km
10 km
http://maxim.gsfc.nasa.gov/documents/SPIE-2002/spie2002.ppt
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New Maxim Pathfinder
Science Phase #2
High Resolution
(100 nas)
20,000 km
1 km
http://maxim.gsfc.nasa.gov/documents/SPIE-2002/spie2002.ppt
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Maxim Configuration Example
19,500km1 rev/day
500 m
1 km
1 km
p
p
p
q
uha
ν===
=
=
700 kgsm =
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Building Non-Natural FormationsUsing Naturally Existing Solutions
& Impulsive Control
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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 + Slowly Expanding Orbits
Chief S/C @ Origin 1800 days
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Evolution of Nearly Vertical OrbitOver 100 Orbital Periods
Origin = Chief S/C
( )0r ( )fr t
18,000 days
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Enforcing Periodicity in the Ephemeris Model
( )3 5 m/sec
1 maneuver/yearV∆ = −
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Geometry of Natural Solutions in the Ephemeris Model
w/ SRP
Inertial Frame Perspective:
Rotating Frame Perspective:
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Concluding Remarks• OFL Control
– Successful for Formation Keeping & Reconfiguration• Spherical + Parabolic Formations• ↑ Rotation Rate = ↑ Thrust Level
– w/o Rotation Rate Thrust ~ O(nN)– w/ Rotation Rate Thrust ~ O(mN)
• Natural to Non-Natural Formations– Differential Corrector 1 small maneuver/year
• Can work well in the rotating frame– Depends on Impact of SRP
• Difficult in the inertial frame due to Geometry of Initial Guess