CONTROL STRATEGIES FOR FORMATION FLIGHT IN THE VICINITY OF THE LIBRATION POINTS. K.C. Howell and B.G. Marchand Purdue University

Transcription

1 CONTROL STRATEGIES FOR FORMATION FLIGHT IN THE VICINITY OF THE LIBRATION POINTS K.C. Howell an B.G. Marchan Purue University 1

2 Previous Work on Formation Flight Multi-S/C Formations in the 2BP Small Relative Separation (10 m 1 km) Moel Relative Dynamics via the C-W Equations Formation Control LQR for Time Invariant Systems Feeback Linearization Lyapunov Base an Aaptive Control Multi-S/C Formations in the 3BP Consier Wier Separation Range Nonlinear moel with complex reference motions Perioic, Quasi-Perioic, Stable/Unstable Manifols Formation Control via simplifie LQR techniques an Gain Scheuling -type methos. 2

3 2-S/C Formation Moel in the Sun-Earth-Moon System Deputy S/C ( x, y, z ) ˆr ŷ Zˆ, zˆ r = ρrˆ β ξ Chief S/C ( x, y, z ) c c c ˆx r 1c r c r 2c B θ ˆx ŷ ˆX 3

4 Dynamical Moel Nonlinear EOMs: = = r t f r t Jr t Kr t u t c c c c c r t f rc t r t f rc t Jr t Kr t u t Linear System: Ω r 0 t r t I r t r t 0 = + r t r t ( rc t, r t ) 2J r t r t I δ x ( t) ( u ( t) u ( t) ) At δ x ( t) B δu ( t) 4

5 Reference Motions Fixe Relative Distance an Orientation Chief-Deputy Line Fixe Relative to the Rotating Frame r t = c an r t = 0 Chief-Deputy Line Fixe Relative to the Inertial Frame x t = x cost+ y sin t 0 0 y t = y cost x sin t 0 0 z t = z Fixe Relative Distance, No Orientation Constraints Natural Formations (Center Manifol) 0 Deputy evolves along a quasi-perioic 2-D Torus that envelops the chief spacecraft s halo orbit (boune motion) 5

6 Nominal Formation Keeping Cost (Configurations Fixe in the Rotating Frame) ρ = 5000 km A z = km A z = km A z = km 6

7 Max./Min. Cost Formations (Configurations Fixe in the Rotating Frame) Minimum Cost Formations ẑ Maximum Cost Formation ẑ Deputy S/C ŷ ˆx ŷ ˆx Deputy S/C Chief S/C Chief S/C Deputy S/C Deputy S/C Deputy S/C Deputy S/C 7

8 8 Formation Keeping Cost Variation Along the SEM L 1 an L 2 Halo Families (Configurations Fixe in the Rotating Frame)

9 Nominal Formation Keeping Cost (Configurations Fixe in the Rotating Frame) A z = km A z = km A z = km 9

10 Quasi-Perioic Configurations (Natural Formations Along the Center Manifol) ˆx ẑ ŷ V NOMINAL = 0 10

11 LQR t f 0 Controllers Consiere T T ( δ δ δ δ ) 1 min J = x t Q x t + u t R u t 2 Input Feeback Linearization 1 T = δ δu t R BPt x t x ( t) = f x( t) + u t Output Feeback Linearization T 1 T P t = A t P t P t A t + P t B t R B t P t Q ( ) u t = f x t + g x t 11 ( r T r) 1/2 x t = f x t + u t y t r = T r = r r r T 2 2 T T 2 T T 3 1 r = r r r r + r r r r + r r = g r T g r r r u ( t) = r 2Jr Kr f r 2 r r

12 Dynamic Response to Injection Error ρ = 5000 km, ξ = 90, β = 0 [ ] δ x 0 = 7 km 5 km 3.5 km 1 mps 1 mps 1 mps T LQR Controller IFL Controller 12

13 13 Control Acceleration Histories

14 Conclusions Natural vs. Force Formations The nominal formation keeping costs in the CR3BP are very low, even for relatively large non-naturally occurring formations. Above the nominal cost, stanar LQR an FL approaches work well in this problem. Both LQR & FL yiel essentially the same control histories but FL metho is computationally simpler to implement. The require control accelerations are extremely low. However, this may change once other sources of error an uncertainty are introuce. Low Thrust Delivery Continuous vs. Discrete Control Complexity increases once these results are transferre into the ephemeris moel. 14