Talk charts from 2004 Flight Mechanics Symposium, held September 14-16 2004, at NASA Goddard Space Flight Center

1. 1
FORMATIONS NEAR THE LIBRATION POINTS:
DESIGN STRATEGIES
USING NATURAL AND NON-NATURAL ARCS
K. C. Howell and B. G. Marchand

2. 2
Formations Near the Libration Points
Moon
1L
2L
ˆx
( )ˆ inertialX
B
θ
ˆy
Chief S/C Path
(Lissajous Orbit)
EPHEM = Sun + Earth + Moon Motion From Ephemeris w/ SRP
CR3BP = Sun + Earth/Moon barycenter Motion Assumed Circular w/o SRP
Deputy S/C
(Orbit Chief Vehicle)

3. 3
Control Methodologies Considered
in both the CR3BP and EPHEM Models
• Continuous Control
– Linear Control
• State Feedback with Control Input Lower Bounds
• Optimal Control → Linear Quadratic Regulator (LQR)
– Nonlinear Control
• Input Feedback Linearization (State Tracking)
• Output Feedback Linearization (Constraint Tracking)
– Spherical + Aspherical Formations (i.e. Parabolic, Hyperbolic, etc.)
• Discrete Control
– Nonlinear Optimal Control
• Impulsive
• Constant Thrust Arcs
– Impulsive Targeter Schemes
• State and Range+Range Rate
– Natural Formations  Impulsive Deployment
– Hybrid Formations  Blending Natural and Non-Natural Motions

4. 4
IMPULSIVE FORMATION KEEPING:
TARGETER METHODS

5. 5
State Targeter:
Impulsive Control Law Formulation
1V∆
2V∆
3V∆
Segment
of Nominal
Deputy Path
0vδ −
0vδ +
0V∆
1rδ
2rδ
3rδ
STM
1 1
1 1
k kk k
k kk k k
A Br r
C Dv v V
+ +
− −
+ +
    
=    
+ ∆    

δ δ
δ δ
( )1
1k k k k k kV B r A r v− −
+∆= − −δ δ δ
Segment
of Chief S/C Path
C(t1)
C(t2)
C(t3)
D(t1)
D(t2)
D(t3)

6. 6
State Targeter:
Radial Distance Error
( ) ˆ10 m Yρ =
DistanceErrorRelativetoNominal(cm)
Time (days)
Max.Deviation
Nominal Separation

7. 7
State Targeter:
Achievable Accuracy
MaximumDeviationfromNominal(cm)
Formation Distance (meters)
ˆr rY=

8. 8
State Targeter:
Maneuver Schedule

9. 9
Range + Range Rate Targeter
Chief S/C
Deputy S/C
r
1V∆
2V∆
3V∆
( )
1/2
Range + Range Rate Constraint:
T
f f
f
T
f
f f
f
f
r r
r
g r rr
r
 
  
= =   
  
  


( ) ( ) ( )
STM
0
0 0 0
0 0
State Relationship Matrix
, ,f
r r
rr r
d g t t x t t
v V
t
r r
r r
−
∂ ∂ 
   ∂ ∂
= Φ = Φ   + ∆∂ ∂   
 
Λ
∂ ∂ 

 


δ
δ
δ

10. 10
Comparison of Range and State Targeters
Chief S/C at Origin
Deputy S/C Path

11. 11
DESIGN OF NON-NATURAL FORMATIONS
USING NATURAL SOLUTION ARCS

12. 12
CR3BP Analysis of Phase Space
Eigenstructure Near Halo Orbit
( ) ( )
( ) ( ) ( )1
,0 0
0 0Jt
x t x
E t e E x−
= Φ
=
δ δ
δ
Reference Halo Orbit
Chief S/C
Deputy S/C
( ) ( ) ( ) ( ) ( ) ( )
6 6
1 1
:Solution to Variational Eqn. in terms of Floquet Modes
j j
j j
jx t x t c t E t ce t t
= =
= = =∑ ∑δ δ
Mode 1  1-D Unstable Subspace
Mode 2  1-D Stable Subspace
Modes (3,4) and (5,6)  4-D Center Subspace
Floquet Modes

13. 13
Natural Solutions:
Periodic Halo Orbits Near Libration Points
ˆx
ˆz
ˆy
Earth
ˆx
ˆz
Sun

14. 14
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

15. 15
Floquet Controller
(Remove Unstable + 2 Center Modes)
( )
1
2 5 6 3
1 3 4
2 5 6 3
Remove Modes 1, 3, and 4:
0r r r
v v v
x x x
x x x
x x x IV
−
  
+ +   −∆   
δ δ δα
δ δ δ
δ δ δ
( )
1
2 3 4 3
1 5 6
2 3 4 3
Remove Modes 1, 5, and 6:
0r r r
v v v
x x x
x x x
x x x IV
−
  
+ +   −∆   
δ δ δα
δ δ δ
δ δ δ
( )3
2,3,43
or
2,5 6
6
1
,
0
1
:Find that removes undesired response modes
j j
j j
j
j
V x
I
x
V
δ α δ
= =
=
 
+ ∆= + 
 
∆
∑ ∑

16. 16
Sample Deployment into Relative Orbits:
1-∆V at Injection
Origin = Chief S/C
Deputy S/C
Quasi-Periodic
Origin = Chief S/C
3 Deputies
“Periodic”
1800 days

17. 17
Natural Formations:
Nearly Periodic and Drifting Relative Orbits
Chief S/C @ Origin 1800 days

18. 18
Expansion of Drifting Vertical Orbit
Origin = Chief S/C
( )0r ( )fr t
18,000 days = 100 Revolutions

19. 19
Transitioning Natural Motions
into Non-Natural Arcs: Targeter Approach
STEP 1: Identify a suitable initial guess
Target  Orbital Drift Control

20. 20
Application of Two Level Corrector
( )qTΓ
( )0r t
( )mr t
( ) ( )0mr t r t=
( )qV T∆
STEP 2: Apply 2-level corrector (Howell and Wilson:1996) w/ end-state constraint
STEP 3: Shift converged patch states forward by 1 period
STEP 4: Reconverge Solution

21. 21
Drift Controlled Vertical Orbit (6 Revs)
( )
3 5 m/sec
1 maneuver/year
V∆ = −

22. 22
Geometry of Natural Solutions
in the Ephemeris Model
w/ SRP
Inertial Frame Perspective:
Rotating Frame Perspective:

23. 23
Transitioning Natural Motions
into Non-Natural Arcs: IFL Example
(1) Consider 1st Rev Along Orbit #4
as initial guess to simple targeter.
(2) Choose initial state on xz-plane
(3) Target next plane crossing to be ⊥
(4) Use resulting arc as half of the
reference motion.
(5) Numerically mirror solution about
xz-plane and store as nominal.
(6) Use IFL control to enforce a
closed orbit using stored nominal.
Chief S/C @ Origin
Sphere for Visualization Only

24. 24
Hybrid Control:
Natural Motions + Continuous Control
½ Period  Natural Arc
½ Period  IFL Control

25. 25
Concluding Remarks
• Precise Formation Keeping → Continuous Control
– Is it possible?
• Depends on hardware capabilities and nominal motion specified
• Not if thruster On/Off sequences are required & tolerances too high
• Precise Navigation → Natural Formations
– Targeter Methods
• Natural motions can be forced to follow non-natural paths
• Success depends on non-natural motion specified
– Hybrid Methods (Natural Arcs + Continuous Thrust Arcs)
• May prove beneficial for non-natural inertial formation design.

26. 26
BACKUP SLIDES

27. 27
Hybrid Control: Accelerations

28. 28
( )
1/2
Range + Range Rate Constraint:
T
f f
f
T
f
f f
f
f
r r
r
g r rr
r
 
  
= =   
  
  


( ) ( ) ( ) ( )
STM
*
0 0 0 0 0
0
State Relationship Matrix
, ,f f f
r r
g x r r
d g g g x t t x t t x
r rx
r
t
x
r
δ δ δ
∂ ∂ 
 ∂ ∂ ∂ ∂
= − = = Φ = Φ 
∂ ∂∂ ∂  
 ∂
Λ
∂ 

 


*
0
0
First Order Approximation:
f f
g
g g x
x
δ
∂
= +
∂
Radial Targeter:
Control Law Formulation
*
*
Desired Range + Range Rate:
0
f
f
r
g
 
=  
 

29. 29
Dynamical Model
( )
( ) ( )
( )
( )
2
22
2
2
3 3
1, 2,
s j j
s
js
s j j
s
P P P P
PP
N
P
PP P P P P P
sr
PI
j j s
p
r r
u tr
r r r
ft
µ
µ
= ≠
 
 − + −

=

 
 
+ +∑
Gravity Terms
Solar Radiation Pressure
Control InputGeneralized Dynamical Model for Each S/C:
( )
( )
Assumptions:
Chief S/C Evolves Along Natural Solution 0 (Nominal)
Deputy S/C Evolves Along Non-Natural Solution 0
c
d
u t
u t
→ ∴ =
→ ∴ ≠

30. 30
Chief S/C Motion:
Natural Solutions Near L1 and L2
“Halo”OrbitsNearLi
LissajousTrajectoriesNearLi
Sun Sun

31. 31
Controlled Deputy S/C Motion (Example 1):
Formation Fixed in the Rotating Frame
Chief S/C
Deputy S/C
ˆyρ ρ=
C(t1)
C(t2)
C(t3)
C(t4)
D(t1)
D(t2)
D(t3)
D(t4)

32. 32
X [au]
Y[au]
Controlled Deputy S/C Motion (Example 2):
Formation Fixed in the Inertial Frame
ˆYρ ρ=Chief S/C
Deputy S/C
C(t1)
C(t2)C(t3)
C(t4)
D(t1)
D(t2)D(t3)
D(t4)

33. 33
MAXIM:
APPLICATIONS OF IFL AND OFL

34. 34
MAXIM Mission Sequence

35. 35
MAXIM:
THRUSTER ON-OFF SEQUENCE

36. 36
Free Flyer Configuration
FF2
FF1
FF3
FF4
FF5 FF6
Hub

37. 37
Radial Error wrt. Hub S/C
Thrusters off = 100,000 sec
W Wr r− 
Nominal Radial Vector
in UVW Coordinates
Actual Radial Vector
in UVW Coordinates

38. 38
Free Flyers
UV-Plane Angular Drift (DEG)
ν ν− 
ε ε− 
Nominal
Actual
Nominal
Actual

39. 39
Thrust Profile
Thrusters Off Between t1 = 1 day & t2 = t1 + 100,000 sec.
A
B
C
( )O 0.05 Nµ ( )O 0.05 Nµ
3 mN≈ 3 mN≈

40. 40
MAXIM:
FORMATION RECONFIGURATION

41. 41
Target Reconfiguration
ˆX
ˆZ
ˆY
Detector Target #11
ˆw
1
ˆu
2
ˆw
2
ˆu
Target #2
1
ˆv
2
ˆv
Hub (t1): α = 0°, δ = 0°
Hub (t2): α = 0°, δ = 0°

42. 42
Graphical Representation
of Reconfiguration for Free Flyers
1
ˆˆ ||w X
2
ˆ ˆ||w Y
INITIAL ORIENTATION OF UV-PLANE
FINAL ORIENTATION OF UV-PLANE

43. 43
Thrust to Reconfigure
From α = 0o
to α = 90o
with δ = 0o
1 | ˆˆ |w X
2
ˆ ˆTarget: ||w Y
Reconfiguration Time Increased to
7 days to reduce Detector S/C Control Thrust

44. 44
Mission Specifications
• Hold periscope positions to within 15-µm
• Detector pointing accuracy – arcminutes
• ∠ Periscope-Detector-Target alignment – µas
• Phase 1  1 Target /week
• Phase 2  1 Target/ 3 weeks
• Hub  inter. comm. port between detector & freeflyers
• Reconfiguration (Formation Slewing) Times:
– 1 Day for Phase 1
– 1 Week for Phase 2
• Propulsion
– Formation Slewing  0.02 N (Hydrazine)
– Formation keeping  0.03 mN (PPTs)
Frequent Reconfigurations

45. 45
NATURAL FORMATIONS

46. 46
Natural Formations:
String of Pearls
ˆx ˆy
ˆz ˆz
ˆy
ˆx

47. 47
Deployment into Torus
(Remove Modes 1, 5, and 6)
Origin = Chief S/C
Deputy S/C
( ) [ ]
( ) [ ]
0 5 00 0 m
0 1 1 1 m/sec
r
r
=
= −

48. 48
Deployment into Natural Orbits
(Remove Modes 1, 3, and 4)
Origin = Chief S/C
3 Deputies
( ) [ ]
( ) [ ]
00 0 0 m
0 1 1 1 m/sec
r r
r
=
= −

49. 49
SPHERICAL FORMATIONS

50. 50
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)

51. 51
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)

52. 52
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 =

53. 53
ASPHERICAL FORMATIONS

54. 54
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

55. 55
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 →

56. 56
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
=
=
=
=
=
=
=



57. 57
OFL Thrust Profile
700 kgsm =

58. 58
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

59. 59
Maxim Configuration Example
19,500km
1 rev/day
500 m
1 km
1 km
p
p
p
q
u
h
a
ν
=
=
=
=
=

700 kgsm =

60. 60
NONLINEAR OPTIMAL CONTROL

61. 61
Formulation
( ) ( ) ( ) ( )
11 1
0 0
min , , , ,
i
i
tN N
N i i i N
i i t
J x L t x u x L t x u d tφ φ
+− −
= =
=+ =+∑ ∑ ∫ 
Subject to:
( ) ( )1 0, , ; Subject to 0i i i ix F t x u x x+ =
( ) ( ) ( )
( )
( )
( ) ( )
( )
1
1
1
1 1 1
0
0
min
, ,
, , ;
, ,
Subject to
0
N n N N
i i i i
i i i i
n i n i i i i
J x x t x
x F t x u
x F t x u
x t x t L t x u
x
x
φ φ+
+
+
+ + +
= + =
  
= = =  
+   
 
=  
 
 
 

Equivalent Representation as Augmented Nonlinear System:

62. 62
Euler-Lagrange Optimality Conditions
(Based on Calculus of Variations)
( )
1
1
Condition #1: 1
Condition #2: 0 ; 0, , 1
NT T Ti
i i N
i N
Ti
i
i
i
i
i
i
xH
x x
F
x
F
N
u
H
i
u
φ
λ λ λ
λ
+
+
∂ ∂ ∂
∂
∂
= = → =  
∂ ∂ 
∂
∂
= = = −
∂



  

 
( )1 , ,T
i i i i iH F t x uλ +=  
Identify and from augmented linear system.i i
i i
F F
x u
∂ ∂
∂ ∂
 


63. 63
Identification of Gradients
From the Linearized Model
( ) ( ) ( ) ( ) ( )x t A t x t B t u tδ δ δ= +   
( )
( ) 0
0
A t
A t L
x
 
 = ∂ 
 ∂ 
  ( )
3
3
0
0T
B t I
 
 =  
  

( )
( )
( )
( )
0
11
0, ,
;
0 0, , nn
x xf t x ux
xL t x ux ++
      
= =      
     


Augmented Nonlinear System:
Augmented Linear System:

64. 64
Solution to Linearized Equations
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
0
0 0
0 0 0 0 7
, ,
, , ; ,
t
t
x t t t x t B u d
t t A t t t t t I
δ δ τ τ δ τ τ=Φ + Φ
Φ = Φ Φ =
∫ 
  
i
F
x
∂
∂


Relation to Gradients in E-L Optimality Conditions:
( ) ( ) ( ) ( )
1
1 1 1, ,
i
i
t
i i i i i
t
x t t x t B u dδ δ τ τ δ τ τ
+
+ + +=Φ + Φ∫ 

65. 65
Control Gradient for Impulsive Control
( )
( )( )
( ) ( )
1 1
1
1 1
,
,
, ,
i i i i
i i i i
i i i i i i
x t t x
t t x B V
t t x t t B V
δ δ
δ
δ
− +
+ +
−
+
−
+ +
= Φ
=Φ + ∆
= Φ + Φ ∆
 
 
 
i
F
u
∂
∂

( )1,i i
i
F
t t B
u
+
∂
= Φ
∂



66. 66
Control Gradient for Constant Thrust Arcs
( ) ( ) ( )
1
1 1 1, ,
i
i
t
i i i i i i
t
x t t x t B d uδ δ τ τ τ δ
+
+ + +
 
=Φ + Φ 
  
∫ 
i
F
u
∂
∂

( ) ( ) ( )
1
1
1, ,
i
i
t
i i i
i t
F
t t t B d
u
τ τ τ
+
−
+
 ∂
=Φ Φ 
∂   
∫

( )
( )
( )
( )
( ) ( )
( )
1
1*
, ,
, ,
,,
,,
n
i
i
ii
x f t x u
x L t x u
A t t tt t
t t Bt t
+
−
   
   
   =   ΦΦ   
   ΦΦ   


 

Equations to Integrate Numerically

67. 67
Numerical Solution Process: Global Approach
( )
0(1) ; ( 0,1,..., 1)
(2)1-Scalar Equation to Optimize in 3 1 Control Variables
Use optimizer to identify optimal
Input , , and initia
given .
During each iteratio
l guess for
n
i
i
N i
i
i N
N
H
u
x u
u
t = −
−
∂
∂

( )
( )
( ) Sequence forward and store
of the optimizer, the following steps are followed:
( ) Evaluate cost functional,
( ) Evaluate 1
; 1
(
, , 1
N
NT N
N
N
i
N
b J x
x
c
x x
a x i N
d
φ
φ φ
λ
= … −
=
∂  ∂
= =  
∂ ∂ 
 
 

( )
) Sequence backward and compute ; 1, ,1
and used in next update of control input.
i
i
i
i
i
H
i N
u
H
e J
u
λ
∂
= −
∂
∂
∂
 

68. 68
Formulation
( ) ( ) ( ) ( )
11 1
0 0
min , , , ,
i
i
tN N
N i i i N
i i t
J x L t x u x L t x u d tφ φ
+− −
= =
=+ =+∑ ∑ ∫ 
Subject to:
( ) ( )1 0, , ; Subject to 0i i i ix F t x u x x+ =
( ) ( ) ( )
( )
( )
( ) ( )
( )
1
1
1
1 1 1
0
0
min
, ,
, , ;
, ,
Subject to
0
N n N N
i i i i
i i i i
n i n i i i i
J x x t x
x F t x u
x F t x u
x t x t L t x u
x
x
φ φ+
+
+
+ + +
= + =
  
= = =  
+   
 
=  
 
 
 

Equivalent Representation as Augmented Nonlinear System:

69. 69
Impulsive Optimal Control
Minimize State Error with End-State Weighting
( ) ( ) ( ) ( )
11
0
1 1
min
2 2
i
i
tN
T T
N N N N
i t
J x x W x x x x Q x x d t
+−
=
= − − + − −∑ ∫
   

70. 70
State Corrector vs. Nonlinear Optimal Control:
Cost Function

71. 71
State Corrector vs. Nonlinear Optimal Control:
Impulsive Maneuver Differences

72. 72
Impulsive Radial Optimal Control
( )
11
2
0
1
min
2
i
i
tN
i t
J q r r dt
+−
=
= −∑ ∫


73. 73
Radial Optimal Control:
Maneuver History

74. 74
State Corrector vs. Nonlinear Optimal Control:
Cost Function

75. 75
RANGE + RANGE RATE TARGETER

76. 76
Comparison of Range and State Targeters

77. 77
Range Targeter:
Spatial Behavior of Corrected Solution
Chief S/C at Origin
Deputy S/C Path