CONTROL STRATEGIES FOR FORMATION FLIGHT IN THE VICINITY OF THE LIBRATION POINTS K.C. Howell and B.G. Marchand Purdue University 1
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. 2
2-S/C Formation Model in the Sun-Earth-Moon System ˆ r Deputy S/C ( ) x , y , z d d d ˆ y = ρ r r ˆ β d ξ ˆ ˆ Z z , ˆ x Chief S/C ( ) , , x y z c c c r r r 1 c 2 c c θ ˆ x B ˆ X ˆ y 3
Dynamical Model Nonlinear EOMs: ( ) ( ) ( ) ( ) ( ) = + + + r t f r t 2 Jr t Kr t u t c c c c c ( ) ( ) ( ) ( ) ( ) ( ) ( ) = + − + + + r t f r t r t f r t 2 Jr t Kr t u t d c d c d d d Linear System: ( ) ( ) ( ) ( ) − − 0 I ( ) r t r t r t r t 0 ( ) ( ) = + − d d d d ( ) u t u t ( ) ( ) ( ) ( ) ( ) ( ) Ω − − d d r t r t r t , r t 2 J r t r t I d d c d d d ( ) ( ) ( ) ( ) δ δ δ B x t A t x t u t d d d 4
Reference Motions • Fixed Relative Distance and Orientation – Chief-Deputy Line Fixed Relative to the Rotating Frame ( ) ( ) = = r t c and r t 0 d d – Chief-Deputy Line Fixed Relative to the Inertial Frame ( ) = + x t x cos t y sin t d d 0 d 0 ( ) = − y t y cos t x sin t d d 0 d 0 ( ) = z t z d d 0 • 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) 5
Nominal Formation Keeping Cost (Configurations Fixed in the Rotating Frame) ρ = 5000 km A z = 0.2×10 6 km A z = 0.7×10 6 km A z = 1.2×10 6 km 6
Max./Min. Cost Formations (Configurations Fixed in the Rotating Frame) Minimum Cost Formations Maximum Cost Formation ˆ ˆ z z Deputy S/C ˆ ˆ y y ˆ x ˆ x Deputy S/C Deputy S/C Chief S/C Chief S/C Deputy S/C Deputy S/C Deputy S/C 7
Formation Keeping Cost Variation Along the SEM L 1 and L 2 Halo Families (Configurations Fixed in the Rotating Frame) 8
Nominal Formation Keeping Cost (Configurations Fixed in the Rotating Frame) A z = 0.2×10 6 km A z = 0.7×10 6 km A z = 1.2×10 6 km 9
Quasi-Periodic Configurations (Natural Formations Along the Center Manifold) z ˆ ˆ x ˆ y ∆ V NOMINAL = 0 10
Controllers Considered • LQR t ( ) f 1 ( ) ( ) ( ) ( ) ∫ = δ δ + δ δ T T min J x t Q x t u t R u t d t d d d d 2 0 ( ) ( ) ( ) δ = − − δ 1 T u t R B P t x t d d ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) − = − T − + T − 1 P t A t P t P t A t P t B t R B t P t Q • Input Feedback Linearization ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = − + = + u t f x t g x t x t f x t u t • Output Feedback Linearization ( ) ( ) ( ) ( ) = + x t f x t u t ( ) ( 3 ) ( ) ( 1 ) − − ( ) 2 = − + + = T T T T T r r r 2 r r r r 2 r r r r g r ( ) 1/2 T r r ( ) r ( ) T g r r r = = ( ) ( ) = − − − − y t u t r 2 Jr Kr f r T r r r 2 r r r 11
Dynamic Response to Injection Error ρ = ξ = β = 5000 km, 90 , 0 ( ) [ ] δ = − − T x 0 7 km 5 km 3.5 km 1 mps 1 mps 1 mps IFL Controller LQR Controller 12
Control Acceleration Histories 13
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. 14
Recommend
More recommend
