1

FORMATIONS NEAR THE LIBRATION POINTS: DESIGN STRATEGIES

USING NATURAL AND NON-NATURAL ARCSK. C. Howell and B. G. Marchand

2

Formations Near the Libration Points

Moon

1L

2L

x

( )ˆ inertialXB

θ

y

Chief S/C Path(Lissajous Orbit)

EPHEM = Sun + Earth + Moon Motion From Ephemeris w/ SRPCR3BP = Sun + Earth/Moon barycenter Motion Assumed Circular w/o SRP

Deputy S/C(Orbit Chief Vehicle)

3

Control Methodologies Consideredin 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

IMPULSIVE FORMATION KEEPING:TARGETER METHODS

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 rC Dv v V

+ +− −+ +

= + ∆

δ δδ δ

( )11k 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

State Targeter: Radial Distance Error

( ) ˆ10 m Yρ =

Dis

tanc

e Er

ror R

elat

ive

to N

omin

al (c

m)

Time (days)

Max.Deviation

Nominal Separation

7

State Targeter:Achievable Accuracy

Max

imum

Dev

iatio

n fr

om N

omin

al (c

m)

Formation Distance (meters)

ˆr rY=

8

State Targeter: Maneuver Schedule

9

Range + Range Rate Targeter

Chief S/C

Deputy S/C

r

1V∆

2V∆

3V∆

( )1/ 2

Range + Range Rate Constraint:T

f ff

Tf f f

f

f

r rr

g r rrr

= =

( ) ( ) ( )STM

00 0 0

0 0

State Relationship Matrix

, ,f

r rrr rd g t t x t t

v Vt

r rr r

−

∂ ∂ ∂ ∂= Φ = Φ + ∆∂ ∂

Λ

∂ ∂

δδ

δ

10

Comparison of Range and State Targeters

Chief S/C at Origin

Deputy S/C Path

11

DESIGN OF NON-NATURAL FORMATIONSUSING NATURAL SOLUTION ARCS

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 jj j

jx t x t c t E t ce t t= =

= = =∑ ∑δ δ

Mode 1 1-D Unstable SubspaceMode 2 1-D Stable SubspaceModes (3,4) and (5,6) 4-D Center Subspace

Floquet Modes

13

Natural Solutions:Periodic Halo Orbits Near Libration Points

x

z

y

Earth

x

z

Sun

14

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

15

Floquet Controller(Remove Unstable + 2 Center Modes)

( )1

2 5 6 31 3 4

2 5 6 3

Remove Modes 1, 3, and 4:

0r r r

v v v

x x xx x x

x x x IV

−

= + + −∆

δ δ δαδ δ δ

δ δ δ

( )1

2 3 4 31 5 6

2 3 4 3

Remove Modes 1, 5, and 6:

0r r r

v v v

x x xx x x

x x x IV

−

= + + −∆

δ δ δαδ δ δ

δ δ δ

( )3

2,3,43 or

2,5 6

6

1

,

01

:Find that removes undesired response modes

j jj j

j

j

V xI

x

V

δ α δ= =

=

+ ∆ = +

∆

∑ ∑

16

Sample Deployment into Relative Orbits:1-∆V at Injection

Origin = Chief S/C

Deputy S/C

Qua

si-P

erio

dic

Origin = Chief S/C

3 Deputies

“Per

iodi

c”

1800 days

17

Natural Formations:Nearly Periodic and Drifting Relative Orbits

Chief S/C @ Origin 1800 days

18

Expansion of Drifting Vertical Orbit

Origin = Chief S/C

( )0r ( )fr t

18,000 days = 100 Revolutions

19

Transitioning Natural Motions into Non-Natural Arcs: Targeter Approach

STEP 1: Identify a suitable initial guess

Target Orbital Drift Control

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 periodSTEP 4: Reconverge Solution

21

Drift Controlled Vertical Orbit (6 Revs)

( )3 5 m/sec

1 maneuver/yearV∆ = −

22

Geometry of Natural Solutions in the Ephemeris Model

w/ SRP

Inertial Frame Perspective:

Rotating Frame Perspective:

23

Transitioning Natural Motions into Non-Natural Arcs: IFL Example

(1) Consider 1st Rev Along Orbit #4as 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 @ OriginSphere for Visualization Only

24

Hybrid Control:Natural Motions + Continuous Control

½ Period Natural Arc½ Period IFL Control

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

BACKUP SLIDES

27

Hybrid Control: Accelerations

28

( )1/ 2

Range + Range Rate Constraint:T

f ff

Tf f f

f

f

r rr

g r rrr

= =

( ) ( ) ( ) ( )STM

*0 0 0 0 0

0

State Relationship Matrix

, ,f f f

r rg x r rd g g g x t t x t t x

r rxr

tx

r

δ δ δ

∂ ∂ ∂ ∂ ∂ ∂= − = = Φ = Φ ∂ ∂∂ ∂ ∂

Λ

∂

*0

0

First Order Approximation:

f fgg g xxδ∂

= +∂

Radial Targeter: Control Law Formulation

**

Desired Range + Range Rate:

0f

fr

g

=

29

Dynamical Model

( )( ) ( )

( ) ( )2

22

22

3 31, 2,

s j js

js s j j

s

P P P PPP

NP

PP P P P P P srPI

j j sp

r r u trr r r

ftµ

µ= ≠

− + −=

+ +∑

Gravity TermsSolar 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 0c

d

u t

u t

→ ∴ =

→ ∴ ≠

30

Chief S/C Motion: Natural Solutions Near L1 and L2

“Hal

o” O

rbits

Nea

rLi

Liss

ajou

s Tra

ject

orie

s Nea

rLi

Sun Sun

31

Controlled Deputy S/C Motion (Example 1):Formation Fixed in the Rotating Frame

Chief S/CDeputy S/C

yρ ρ=

C(t1)

C(t2)

C(t3)

C(t4)

D(t1)

D(t2)

D(t3)

D(t4)

32

X [au]

Y [a

u]

Controlled Deputy S/C Motion (Example 2):Formation Fixed in the Inertial Frame

Yρ ρ=Chief S/CDeputy S/C

C(t1)

C(t2)C(t3)

C(t4)

D(t1)

D(t2)D(t3)

D(t4)

33

MAXIM:APPLICATIONS OF IFL AND OFL

34

MAXIM Mission Sequence

35

MAXIM:THRUSTER ON-OFF SEQUENCE

36

Free Flyer Configuration

FF2

FF1

FF3

FF4

FF5 FF6

Hub

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

Free FlyersUV-Plane Angular Drift (DEG)ν ν−

ε ε−

NominalActual

NominalActual

39

Thrust ProfileThrusters Off Between t1 = 1 day & t2 = t1 + 100,000 sec.

A BC

( )O 0.05 Nµ ( )O 0.05 Nµ

3 mN≈ 3 mN≈

40

MAXIM:FORMATION RECONFIGURATION

41

Target Reconfiguration

X

Z

Y

Detector Target #11w1u

2w

2u

Target #2

1v

2v

Hub (t1): α = 0°, δ = 0°

Hub (t2): α = 0°, δ = 0°

42

Graphical Representation of Reconfiguration for Free Flyers

1ˆˆ ||w X

2ˆ ˆ||w Y

INITIAL ORIENTATION OF UV-PLANE

FINAL ORIENTATION OF UV-PLANE

43

Thrust to Reconfigure From α = 0o to α = 90o with δ = 0o

1 | ˆˆ | w X2ˆ ˆTarget: || w Y

Reconfiguration Time Increased to7 days to reduce Detector S/C Control Thrust

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

NATURAL FORMATIONS

46

Natural Formations:String of Pearls

x y

z z

y

x

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

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

SPHERICAL FORMATIONS

50

OFL Controlled Response of Deputy S/CRadial Distance Tracking

2 ( ) ( ) ( )2

, Tg r r r r ru t r r f rr rr

= − + − ∆

3

4

( ) ( ) ( )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)

51

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)

52

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 =

53

ASPHERICAL FORMATIONS

54

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

55

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 →

56

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===

=

=

=

=

57

OFL Thrust Profile700 kgsm =

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

Maxim Configuration Example

19,500km1 rev/day

500 m

1 km

1 km

p

p

p

q

uha

ν===

=

=

700 kgsm =

60

NONLINEAR OPTIMAL CONTROL

61

Formulation( ) ( ) ( ) ( )

11 1

0 0min , , , ,

i

i

tN N

N i i i Ni 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

11

1 1 1

00

min

, ,, , ;

, ,

Subject to 0

N n N N

i i i ii i i i

n i n i i i i

J x x t x

x F t x ux F t x u

x t x t L t x u

xx

φ φ+

++

+ + +

= + =

= = = +

=

Equivalent Representation as Augmented Nonlinear System:

62

Euler-Lagrange Optimality Conditions(Based on Calculus of Variations)

( )1

1

Condition #1: 1

Condition #2: 0 ; 0, , 1

NT T Tii i N

i N

Ti

i

i

i

ii

i

xHx x

Fx

F Nu

H iu

φλ λ λ

λ

+

+

∂ ∂ ∂∂

∂

= = → = ∂ ∂ ∂

∂= = = −∂

( )1 , ,Ti i i i iH F t x uλ +=

Identify and from augmented linear system.i i

i i

F Fx u∂ ∂∂ ∂

63

Identification of Gradients From the Linearized Model

( ) ( ) ( ) ( ) ( )x t A t x t B t u tδ δ δ= +

( )( ) 0

0

A tA t L

x

= ∂ ∂

( )3

3

0

0T

B t I =

( )( )

( )( )

0

11

0, ,;

0 0, , nn

x xf t x uxxL t x ux ++

= =

Augmented Nonlinear System:

Augmented Linear System:

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

Fx∂∂

Relation to Gradients in E-L Optimality Conditions:

( ) ( ) ( ) ( )1

1 1 1, ,i

i

t

i i i i it

x t t x t B u dδ δ τ τ δ τ τ+

+ + += Φ + Φ∫

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

Fu∂∂

( )1,i ii

F t t Bu +

∂= Φ

∂

66

Control Gradient for Constant Thrust Arcs

( ) ( ) ( )1

1 1 1, ,i

i

t

i i i i i it

x t t x t B d uδ δ τ τ τ δ+

+ + +

= Φ + Φ

∫

i

Fu∂∂

( ) ( ) ( )1

11, ,

i

i

t

i i ii t

F t t t B du

τ τ τ+

−+

∂= Φ Φ

∂ ∫

( )( )

( )( )( ) ( )( )

1

1*

, ,, ,

,,,,

n

ii

ii

x f t x ux L t x u

A t t tt tt t Bt t

+

−

= ΦΦ ΦΦ

Equations to Integrate Numerically

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

ii

N i

i

i NN

Hu

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 NN

N

i

N

b J x

xc

x x

a x i N

d

φ

φ φλ

= … −

=

∂ ∂= = ∂ ∂

( )

) Sequence backward and compute ; 1, ,1

and used in next update of control input.

ii

i

i

i

H i Nu

He Ju

λ ∂= −

∂∂∂

68

Formulation( ) ( ) ( ) ( )

11 1

0 0min , , , ,

i

i

tN N

N i i i Ni 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

11

1 1 1

00

min

, ,, , ;

, ,

Subject to 0

N n N N

i i i ii i i i

n i n i i i i

J x x t x

x F t x ux F t x u

x t x t L t x u

xx

φ φ+

++

+ + +

= + =

= = = +

=

Equivalent Representation as Augmented Nonlinear System:

69

Impulsive Optimal ControlMinimize State Error with End-State Weighting

( ) ( ) ( ) ( )11

0

1 1min2 2

i

i

tNT T

N N N Ni t

J x x W x x x x Q x x d t+−

=

= − − + − −∑ ∫

70

State Corrector vs. Nonlinear Optimal Control:Cost Function

71

State Corrector vs. Nonlinear Optimal Control:Impulsive Maneuver Differences

72

Impulsive Radial Optimal Control

( )11 2

0

1min2

i

i

tN

i t

J q r r dt+−

=

= −∑ ∫

73

Radial Optimal Control:Maneuver History

74

State Corrector vs. Nonlinear Optimal Control:Cost Function

75

RANGE + RANGE RATE TARGETER

76

Comparison of Range and State Targeters

77

Range Targeter:Spatial Behavior of Corrected Solution

Chief S/C at Origin

Deputy S/C Path