Information about Control Strategies For Formation Flight In The Vicinity Of The Libration...

Talk charts from 13th AAS/AIAA Space Flight

Mechanics Meeting, held Feb 9-13, 2003, in Ponce, PR.

Mechanics Meeting, held Feb 9-13, 2003, in Ponce, PR.

2 Previous Work on Formation Flight • Multi-S/C Formations in the 2BP – Small Relative Separation (10 m – 1 km) • Model Relative Dynamics via the C-W Equations • Formation Control – LQR for Time Invariant Systems – Feedback Linearization – Lyapunov Based and Adaptive Control • Multi-S/C Formations in the 3BP – Consider Wider Separation Range • Nonlinear model with complex reference motions – Periodic, Quasi-Periodic, Stable/Unstable Manifolds • Formation Control via simplified LQR techniques and “Gain Scheduling”-type methods.

3 Deputy S/C ( ), ,d d dx y z ˆx ˆy 2-S/C Formation Model in the Sun-Earth-Moon System ˆX ˆ ˆ,Z z B 1cr 2crcr ˆxθ ˆy Chief S/C ( ), ,c c cx y z ˆdr rρ= ˆr ξ β

4 Dynamical Model ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 c c c c c d c d c d d d r t f r t Jr t Kr t u t r t f r t r t f r t Jr t Kr t u t = + + + = + − + + + Nonlinear EOMs: ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 0 0 , 2 d d d d d d c dd d d d Ir t r t r t r t u t u t r t r t J Ir t r t r t r t − − + − Ω− − Linear System: ( )A t B ( )du tδ( )dx tδ( )dx tδ

5 Reference Motions • Fixed Relative Distance and Orientation – Chief-Deputy Line Fixed Relative to the Rotating Frame – Chief-Deputy Line Fixed Relative to the Inertial Frame • Fixed Relative Distance, No Orientation Constraints • Natural Formations (Center Manifold) – Deputy evolves along a quasi-periodic 2-D Torus that envelops the chief spacecraft’s halo orbit (bounded motion) ( ) ( )and 0d dr t c r t= = ( ) ( ) ( ) 0 0 0 0 0 cos sin cos sin d d d d d d d d x t x t y t y t y t x t z t z = + = − =

6 Nominal Formation Keeping Cost (Configurations Fixed in the Rotating Frame) Az = 0.2×106 km Az = 1.2×106 km Az = 0.7×106 km 5000 kmρ =

7 Max./Min. Cost Formations (Configurations Fixed in the Rotating 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

8 Formation Keeping Cost Variation Along the SEM L1 and L2 Halo Families (Configurations Fixed in the Rotating Frame)

9 Nominal Formation Keeping Cost (Configurations Fixed in the Rotating Frame) Az = 0.2×106 km Az = 1.2×106 km Az = 0.7×106 km

10 Quasi-Periodic Configurations (Natural Formations Along the Center Manifold) ˆx ˆy ˆz ∆VNOMINAL = 0

11 Controllers Considered ( ) ( ) ( ) ( )( )0 1 min 2 ft T T d d d dJ x t Q x t u t R u t d tδ δ δ δ+∫ • LQR ( ) ( )( ) ( )x t f x t u t= + ( ) ( )( ) ( )( )u t f x t g x t=− + • Input Feedback Linearization ( ) ( )( ) ( ) ( ) ( ) 1/2T T x t f x t u t r r r y t r rr r = + = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 1 2 2 2 2 2 T T T T T T r r r r r r r r r r r g r g r r r u t r Jr Kr f r r r − − =− + + = = − − − − • Output Feedback Linearization ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 T d d T T u t R B P t x t P t A t P t P t A t P t B t R B t P t Q δ δ− − = − =− − + −

12 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 =− −

13 Control Acceleration Histories

14 Conclusions • Natural vs. Forced Formations – The nominal formation keeping costs in the CR3BP are very low, even for relatively large non-naturally occurring formations. • Above the nominal cost, standard LQR and FL approaches work well in this problem. – Both LQR & FL yield essentially the same control histories but FL method is computationally simpler to implement. • The required control accelerations are extremely low. However, this may change once other sources of error and uncertainty are introduced. – Low Thrust Delivery – Continuous vs. Discrete Control • Complexity increases once these results are transferred into the ephemeris model.

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