This document discusses using output feedback linearization (OFL) control to maintain formations of spacecraft near libration points. It examines spherical and aspherical configurations, including parabolic formations. OFL is shown to successfully control radial distance and rotation rate tracking for formation keeping and reconfiguration. Increasing the commanded rotation rate increases the required thrust. Natural periodic orbits can be used to build non-natural formations through small periodic maneuvers to enforce periodicity in the ephemeris model.

1. 1
ASPHERICAL FORMATIONS NEAR THE LIBRATION POINTS
IN THE SUN-EARTH/MOON SYSTEM
B.G. Marchand and K.C. Howell
Purdue University

2. 2
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

3. 3
OFL Controlled Response of Deputy S/C
Radial Distance Tracking
2 ( )
( )
( )2
, Tg r r r r r
u t r r f r
r rr
    
= − + − ∆   
   
    
3
4
( )
( )
( )2 2
,1
2
Tg r r r r
u t r f r
r r
  
= − − ∆ 
  
  
( ) ( ) ( )2
, 3
T
r r r
u t rg r r r r f r
rr
   
= − − + − ∆   
  
   
Control Law
( )
( ),
ˆ
H r r
u t r
r
=
 Geometric Approach:
Radial inputs only1
1u
3u
2u
4u
Chief S/C @ Origin (Inside Sphere)

4. 4
Impact of Nominal Radial Separation
on OFL Controlled Response of Deputy S/C
*
5 kmr =
*
5000 kmr =
Radial Inputs Only
3-Axis Control
3-Axis Control
Chief S/C @ Origin (Inside Sphere)

5. 5
OFL Controlled Response of Deputy S/C
Radial Distance + Rotation Rate Tracking
*
5 kmr =
1 rev / 6 hrs
1 rev / 1 day
Chief S/C @ Origin (Inside Sphere)

6. 6
Impact Commanded Rotation Rate on Cost
1 rev /24 hrs 0.19 mN
1 rev /12 hrs 0.76 mN
1 rev / 6 hrs 6.40 mN
1 rev / 1 hrs 106.50 mN
→
→
→
→
700 kgsm =

7. 7
Application to Aspherical Formations

8. 8
Parameterization of Parabolic Formation
pa
ph
pu
ν
( )3
ˆ focal linee
1
ˆe
Nadir
q
1 2 3
ˆ ˆ ˆiCD
Er xe ye ze= + +  
Chief S/C (C)
iDeputy S/C (D )
{ }
i i
i i
CD CD
TI EE I
CD CDE I
E I
r r
C
r r
   
=   
    
: inertially fixed focal frame
: inertially fixed ephemeris frame
E
I
/ 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

9. 9
Controller Development
( ) ( )
( ) ( )
( ) ( )
( )
( )( )
( )
2 2
2 2 2
2 2
2 2 2
2 2 2 2
22 2
2 2 2
0 ,
2 2 2
1 ,
0
2
u x y
x
y q x y z
z
h h h
x y g u u x y x f y f
a a au
h h h
x y u g q q x y x f y f f
a a a
u
x y xx yy xy yx
gx y x y
x y
ν 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 →

10. 10
OFL Controlled Parabolic Formation
ˆX
ˆZ
ˆY
0t
0 1.6 hrst +
0 3.8 hrst +
0 8 hrst +
0 5 dayst +
0 6 dayst +
3
ˆe
10 km
1 rev/day
500 m
500 m
Phase I: 200 m
Phase II: 300 m /1 day
Phase III: 500 m
p
p
p
p
p
q
h
a
u
u
u
ν
=
=
=
=
=
=
=



11. 11
OFL Thrust Profile
700 kgsm =

12. 12
Original Maxim Design
DETECTOR
SPACECRAFT
CONVERGER
SPACECRAFT
200
M
COLLECTOR
SPACECRAFT
(32PLACES
EVENLYSPACED)
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

13. 13
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

14. 14
Maxim Configuration Example
19,500km
1 rev/day
500 m
1 km
1 km
p
p
p
q
u
h
a
ν
=
=
=
=
=

700 kgsm =

15. 15
Building Non-Natural Formations
Using Naturally Existing Solutions
& Impulsive Control

16. 16
Natural Formations:
Quasi-Periodic Relative Orbits → 2-D Torus
ˆy
ˆx
ˆz
Chief S/C Centered View
(RLP Frame)
ˆy
ˆx
ˆx
ˆz
ˆy
ˆz
ˆz
ˆx
ˆy

17. 17
Natural Formations:
Nearly Periodic + Slowly Expanding Orbits
Chief S/C @ Origin 1800 days

18. 18
Evolution of Nearly Vertical Orbit
Over 100 Orbital Periods
Origin = Chief S/C
( )0r ( )fr t
18,000 days

19. 19
Enforcing Periodicity
in the Ephemeris Model
( )
3 5 m/sec
1 maneuver/year
V∆ = −

20. 20
Geometry of Natural Solutions
in the Ephemeris Model
w/ SRP
Inertial Frame Perspective:
Rotating Frame Perspective:

21. 21
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