Information about Formation Flight Near L1 And L2 In The Sun-Earth/Moon Ephemeris System...

Talk Charts from the AAS/AIAA Astrodynamics Specialists Conference, Held Aug 3-7, 2003, in Big Sky, MT.

2 Continuous Control • LQR and Input Feedback Linearization (IFL) – Control Nominal State • Fixed in Rotating Frame • Fixed in Inertial Frame – Critically damped error response – LQR & IFL → similar response & control acceleration histories • Output Feedback Linearization – Radial Distance Control – Free relative orientation • Transition from CR3BP – Ephemeris Model w/ Solar Radiation Pressure (SRP) Formation Keeping via IFL

3 Output Feedback Linearization (Radial Distance Control) ( ) ( )( ) ( ) ( ), 0 T h r t r t u t r t− = r ( ) ( ) ( )2 , Tg r r r r r u t r r f r r rr = − + − ∆ 2,310.0 m/sec 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 = − − + − ∆ 16,442.0 m/sec 49.8 m/sec Control Law Cumulative ∆V (180 days) r ( ) ( ), ˆ h r r u t r r = 16,392.4 m/sec Geometric Approach: Radial inputs only Nonlinear Scalar Constraint on ( )u t ( ),y l r r= Target Nominal: 5 km Mission Time: 180 days r =

4 Partial Feedback Control of Spherical Formations ( ) [ ] ( ) [ ]0 12 5 3 km 0 1 1 1 m/secr r=− =−

5 Formation Keeping in the Ephemeris Model via Discrete Control

6 Linear Targeter Approach ( ) ˆ10 m 0 Yρ ρ = = DistanceErrorRelativetoNominal(cm) Time (days)

7 Achievable Accuracy via Targeter Scheme MaximumDeviationfromNominal(cm) Formation Distance (meters)

8 Natural Formations: String of Pearls ˆx ˆy ˆz ˆz ˆy ˆx

9 Natural Formations: Phased Vehicles Along Halo Orbit C1 D3 D2 D1 Az = 200,000 km Unstable Halo Orbit 1 Unstable Eigenvalue (γ1) 1 Stable Eigenvalue (γ2) 4 Center Eigenvalues (γ3-γ6) L1

10 Eigenstructure Near Halo Orbit ( ) ( ),0 0R R R r x t x r δ δ = = Φ Reference Halo Orbit Chief S/C Deputy S/C ( ) ( ) ( ) ( ),0 0 Floquet Modal Matrix: Jt E t P t S t E e− = = Φ ( ) ( ) ( ) ( ) ( ) 6 6 1 1 :Solution to Variational Eqn. in terms of Floquet Modes j j j j j x t x t c t e t E t cδ δ = = = = =∑ ∑ ( ) ( ) ( ){ } ( ){ } 1 ,0 0 Floquet Decomposition of ,0 : Jt t P t S e P S t − Φ = Φ

11 Natural Solutions: Torus (Associated with Modes 3 and 4) Formation Evolving Along Torus Near L1 (Rotating Libration Point Frame Centered at L1) Chief S/C Centered View (RLP Frame) ˆy ˆx ˆz ˆy ˆx ˆx ˆz ˆy ˆz ˆz ˆx ˆy

12 Natural Solutions: Halo Orbits (Associated with Modes 5 and 6) ˆy ˆx ˆy ˆz ˆx ˆz ˆx ˆz ˆy

13 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 = = = + ∆= + ∆ ∑ ∑

14 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 = = −

15 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 = = −

16 Nearly Periodic Formations Origin = Chief S/C

17 Nearly Vertical Formations Origin = Chief S/C

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

19 Conclusions • Discrete Control of Natural Formations – Floquet controller • Effective in identifying nominal formations + deployment • May lead to feasible control strategies → non-natural formations • Continuous Control of Non-Natural Formations – IFL/OFL effective; LQR computationally inefficient – OFL → spherical configurations + unnatural rates – Low acceleration levels → Implementation Issues • Discrete Control of Non-Natural Formations – Small Formations → Good accuracy – Extremely Small ∆V’s (10-5 m/sec)

20 Backup Slides

21 LQR vs. IFL (CR3BP) 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 =− −

22 LQR vs. IFL (CR3BP) Control Accelerations

23 Output Feedback Linearization (Radial Distance Control) ( ) ( ) ( ) ( )2 1 Generalized Relative EOMs , Measured Output e.g. , , r f r u t y l r r y r y r y r− =∆ + → =→ === Formation Dynamics Measured Output Response 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

24 Targeter Maneuver Schedule

25 Converged Period for OFL Controlled Paths

26 Stability of T-Periodic Orbits ( ) ( ) ( ) ( ) : ,0 0 measured relative to periodic orbit Linear Variational Equation x t t x x t δ δ δ = Φ → Re{ }jγ Im{ }jγ 5,6γ 3γ 4γ 2γ 1γ UnstableStable Center

27 Evolution of Nearly Vertical Orbits Along the yz-Plane

28 Floquet Control (Large Formations – Example 1)

29 Floquet Controller Maneuver Schedule (For Example 1)

30 Nearly Periodic Formations (Inertial Perspective)

31 Nearly Vertical Formations (Inertial Perspective)

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