Talk charts from the 2003 Flight Mechanics Symposium, held on October 28-30, 2003, at NASA Goddard Space Flight Center

1. 1
DESIGN AND CONTROL OF FORMATIONS
NEAR THE LIBRATION POINTS
OF THE SUN-EARTH/MOON EPHEMERIS SYSTEM
K.C. Howell and B.G. Marchand
Purdue University

2. 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
RLP
Inertial




3. 3
ˆX
B θ
ˆy
Deputy S/C
ˆx
ˆy
2-S/C Formation Model
in the Sun-Earth-Moon System
ˆ ˆ,Z z
cr
ˆx
Chief S/C
ˆd cr r r rr= − =
ˆr
ξ
β
( ) ( )( ) ( )
Relative EOMs:
r t f r t u t=∆ +
1L 2L
dr
Ephemeris System = Sun+Earth+Moon Ephemeris + SRP

4. 4
Natural Formations

5. 5
Natural Formations:
String of Pearls
ˆx ˆy
ˆz ˆz
ˆy
ˆx

6. 6
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

7. 7
Natural Formations:
Nearly Periodic Relative Motion
Origin = Chief S/C
10 Revolutions = 1,800 days

8. 8
Evolution of Nearly Vertical Orbits
Along the yz-Plane

9. 9
Natural Formations:
Slowly Expanding Vertical Orbits
Origin = Chief S/C
( )0r ( )fr t
100 Revolutions = 18,000 days

10. 10
Non-Natural Formations

11. 11
Nominal Formation Keeping Cost
(Configurations Fixed in the RLP Frame)
Az = 0.2×106 km
Az = 1.2×106 km
Az = 0.7×106 km
( ) ( )
180 days
0
V u t u t d t∆= ⋅∫
 
5000 kmr =

12. 12
Max./Min. Cost Formations
(Configurations Fixed in the RLP Frame)
ˆx
Deputy S/C
Deputy S/C
Deputy S/C
Deputy S/C
Chief S/C
ˆy
ˆz
Minimum Cost Formations
ˆz
ˆy
ˆx
Deputy S/C
Deputy S/CChief S/C
Maximum Cost Formation
Nominal Relative Dynamics in the Synodic Rotating Frame

13. 13
Formation Keeping Cost Variation
Along the SEM L1 and L2 Halo Families
(Configurations Fixed in the RLP Frame)

14. 14
Discrete vs. Continuous Control

15. 15
Discrete Control: Linear Targeter
( ) ˆ10 m
0
Yρ
ρ
=
=
DistanceErrorRelativetoNominal(cm)
Time (days)

16. 16
Achievable Accuracy
via Targeter Scheme
MaximumDeviationfromNominal(cm)
Formation Distance (meters)
ˆr rY=

17. 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 T
f
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:
T
u t u t R B P t x t x t−
 
 
= + − − 
  
 


18. 18
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
T
xδ =− −
Dynamic Response Modeled in the CR3BP
Nominal State Fixed in the Rotating Frame

19. 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
, , ,T
p r r q r
d l
y u r g r r
dt
r== + =  
Scalar Nonlinear Functions of andr r
( ) ( )( ) ( ) ( ), 0
T
h r t r t u t r t− =
Scalar Nonlinear Constraint on Control Inputs

20. 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 r
u t r r f r
r rr
    
= − + − ∆   
   
    
2
r
1
r
( )
( )
( )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( ),y l r r= 
( )
( ),
ˆ
h r r
u t r
r
=
 Geometric Approach:
Radial inputs only
r

21. 21
OFL Control of Spherical Formations
in 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. 22
OFL Control of Spherical Formations
Radial Dist. + Rotation Rate
Quadratic Growth in Cost w/ Rotation Rate
Linear Growth in Cost w/ Radial Distance

23. 23
Inertially Fixed Formations
in the Ephemeris Model
ˆe
100 km
Chief S/C
500 m
ˆ inertially fixed formation pointing vector (focal line)e =

24. 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