Introduction Flattening the Earth Continuation procedure Flat Earth Numerical simulations Continuation from a flat to a round Earth model in the coplanar orbit transfer problem M. Cerf 1 , T. Haberkorn, Emmanuel Tr´ elat 1 1 EADS Astrium, les Mureaux 2 MAPMO, Universit´ e d’Orl´ eans Congr` es SMAI 2011 23-27 Mai M. Cerf, T. Haberkorn, E. Tr´ elat Continuation from a flat to a round Earth model
Introduction Flattening the Earth Continuation procedure Flat Earth Numerical simulations The coplanar orbit transfer problem Spherical Earth Central gravitational field g ( r ) = µ r 2 System in cylindrical coordinates ˙ r ( t ) = v ( t ) sin γ ( t ) ϕ ( t ) = v ( t ) ˙ r ( t ) cos γ ( t ) v ( t ) = − g ( r ( t )) sin γ ( t ) + T max ˙ m ( t ) u 1 ( t ) � v ( t ) r ( t ) − g ( r ( t )) � T max γ ( t ) = ˙ cos γ ( t ) + m ( t ) v ( t ) u 2 ( t ) v ( t ) ˙ m ( t ) = − β T max � u ( t ) � Thrust: T ( t ) = u ( t ) T max ( T max large: strong thrust) � u 1 ( t ) 2 + u 2 ( t ) 2 � 1 Control: u ( t ) = ( u 1 ( t ) , u 2 ( t )) satisfying � u ( t ) � = M. Cerf, T. Haberkorn, E. Tr´ elat Continuation from a flat to a round Earth model
Introduction Flattening the Earth Continuation procedure Flat Earth Numerical simulations The coplanar orbit transfer problem Initial conditions r ( 0 ) = r 0 , ϕ ( 0 ) = ϕ 0 , v ( 0 ) = v 0 , γ ( 0 ) = γ 0 , m ( 0 ) = m 0 , Final conditions a point of a specified orbit: r ( t f ) = r f , v ( t f ) = v f , γ ( t f ) = γ f , or an elliptic orbit of energy K f < 0 and eccentricity e f : ξ K f = v ( t f ) 2 µ − r ( t f ) − K f = 0 , 2 � 2 1 − r ( t f ) v ( t f ) 2 � ξ e f = sin 2 γ + cos 2 γ − e 2 f = 0 . µ (orientation of the final orbit not prescribed: ϕ ( t f ) free; in other words: argument of the final perigee free) Optimization criterion max m ( t f ) (note that t f has to be fixed) M. Cerf, T. Haberkorn, E. Tr´ elat Continuation from a flat to a round Earth model
Introduction Flattening the Earth Continuation procedure Flat Earth Numerical simulations Application of the Pontryagin Maximum Principle Hamiltonian v � − g ( r ) sin γ + T max � H ( q , p , p 0 , u ) = p r v sin γ + p ϕ r cos γ + p v m u 1 �� v r − g ( r ) � cos γ + T max � + p γ mv u 2 − p m β T max � u � , v Extremal equations q ( t ) = ∂ H p ( t ) = − ∂ H ∂ p ( q ( t ) , p ( t ) , p 0 , u ( t )) , ∂ q ( q ( t ) , p ( t ) , p 0 , u ( t )) , ˙ ˙ Maximization condition H ( q ( t ) , p ( t ) , p 0 , u ( t )) = max � w � � 1 H ( q ( t ) , p ( t ) , p 0 , w ) M. Cerf, T. Haberkorn, E. Tr´ elat Continuation from a flat to a round Earth model
Introduction Flattening the Earth Continuation procedure Flat Earth Numerical simulations Application of the Pontryagin Maximum Principle Hamiltonian v � − g ( r ) sin γ + T max � H ( q , p , p 0 , u ) = p r v sin γ + p ϕ r cos γ + p v m u 1 �� v r − g ( r ) � cos γ + T max � + p γ mv u 2 − p m β T max � u � , v Maximization condition leads to u ( t ) = ( u 1 ( t ) , u 2 ( t )) = ( 0 , 0 ) whenever Φ( t ) < 0 p v ( t ) p γ ( t ) u 1 ( t ) = , u 2 ( t ) = whenever Φ( t ) > 0 � � p v ( t ) 2 + p γ ( t ) 2 p v ( t ) 2 + p γ ( t ) 2 v ( t ) v ( t ) 2 v ( t ) 2 where � p v ( t ) 2 + p γ ( t ) 2 1 Φ( t ) = v ( t ) 2 − β p m ( t ) (switching function) m ( t ) M. Cerf, T. Haberkorn, E. Tr´ elat Continuation from a flat to a round Earth model
Introduction Flattening the Earth Continuation procedure Flat Earth Numerical simulations Application of the Pontryagin Maximum Principle Hamiltonian v � − g ( r ) sin γ + T max � H ( q , p , p 0 , u ) = p r v sin γ + p ϕ r cos γ + p v m u 1 �� v r − g ( r ) � cos γ + T max � + p γ mv u 2 − p m β T max � u � , v Transversality conditions case of a fixed point of a specified orbit: p ϕ ( t f ) = 0 , p m ( t f ) = − p 0 case of an orbit of given energy and eccentricity: ∂ r ξ K f ( p γ ∂ v ξ e f − p v ∂ γ ξ e f ) + ∂ v ξ K f ( p r ∂ γ ξ e f − p γ ∂ r ξ e f ) = 0 Remark p 0 � = 0 (no abnormal) ⇒ p 0 = − 1 no singular arc (Bonnard - Caillau - Faubourg - Gergaud - Haberkorn - Noailles - Tr´ elat) M. Cerf, T. Haberkorn, E. Tr´ elat Continuation from a flat to a round Earth model
Introduction Flattening the Earth Continuation procedure Flat Earth Numerical simulations Shooting method Given ( t f , p 0 ) , one can integrate the Hamiltonian flow from 0 to t f to have ( q ( t f ) , p ( t f )) . Find a zero of r ( t f , p 0 ) − r f ξ K f ( p 0 ) v ( t f , p 0 ) − v f ξ e f ( p 0 ) S ( t f , p 0 ) = γ ( t f , p 0 ) − γ f or ∗ ∗ ∗ , p ϕ ( t f , p 0 ) p ϕ ( t f , p 0 ) p m ( t f , p 0 ) − 1 p m ( t f , p 0 ) − 1 A zero of S ( · , · ) is an admissible trajectory satisfying the necessary conditions. Main problem: how to make the shooting method converge? initialization of the shooting method discontinuities of the optimal control M. Cerf, T. Haberkorn, E. Tr´ elat Continuation from a flat to a round Earth model
Introduction Flattening the Earth Continuation procedure Flat Earth Numerical simulations Shooting method Main problem: how to make the shooting method converge? initialization of the shooting method discontinuities of the optimal control Several methods: use first a direct method to provide a good initial guess, e.g. AMPL combined with IPOPT: R. Fourer, D.M. Gay, B.W. Kernighan, AMPL: A modeling language for mathematical programming , Duxbury Press, Brooks-Cole Publishing Company (1993). A. W¨ achter, L.T. Biegler On the implementation of an interior-point lter line- search algorithm for large-scale nonlinear programming , Mathematical Programming 106 (2006), 25–57. but usual flaws of direct methods (computationally demanding, lack of numerical precision). M. Cerf, T. Haberkorn, E. Tr´ elat Continuation from a flat to a round Earth model
Introduction Flattening the Earth Continuation procedure Flat Earth Numerical simulations Shooting method Main problem: how to make the shooting method converge? initialization of the shooting method discontinuities of the optimal control Several methods: use the impulse transfer solution to provide a good initial guess: P . Augros, R. Delage, L. Perrot, Computation of optimal coplanar orbit transfers , AIAA 1999. but valid only for nearly circular initial and final orbits. See also: J. Gergaud, T. Haberkorn, Orbital transfer: some links between the low-thrust and the impulse cases , Acta Astronautica 60 , no. 6-9 (2007), 649–657. L.W. Neustadt, A general theory of minimum-fuel space trajectories , SIAM Journal on Control 3 , no. 2 (1965), 317–356. M. Cerf, T. Haberkorn, E. Tr´ elat Continuation from a flat to a round Earth model
Introduction Flattening the Earth Continuation procedure Flat Earth Numerical simulations Shooting method Main problem: how to make the shooting method converge? initialization of the shooting method discontinuities of the optimal control Several methods: multiple shooting method parameterized by the number of thrust arcs: H. J. Oberle, K. Taubert, Existence and multiple solutions of the minimum-fuel orbit transfer problem , J. Optim. Theory Appl. 95 (1997), 243–262. M. Cerf, T. Haberkorn, E. Tr´ elat Continuation from a flat to a round Earth model
Introduction Flattening the Earth Continuation procedure Flat Earth Numerical simulations Shooting method Main problem: how to make the shooting method converge? initialization of the shooting method discontinuities of the optimal control Several methods: differential or simplicial continuation method linking the minimization of the L 2 -norm of the control to the minimization of the fuel consumption: J. Gergaud, T. Haberkorn, P . Martinon, Low thrust minimum fuel orbital transfer: an homotopic approach , J. Guidance Cont. Dyn. 27 , 6 (2004), 1046–1060. P . Martinon, J. Gergaud, Using switching detection and variational equations for the shooting method , Optimal Cont. Appl. Methods 28 , no. 2 (2007), 95–116. but not adapted for high-thrust transfer. M. Cerf, T. Haberkorn, E. Tr´ elat Continuation from a flat to a round Earth model
Introduction Flattening the Earth Continuation procedure Flat Earth Numerical simulations Flattening the Earth Observation: Solving the optimal control problem for a flat Earth model with constant gravity is simple and algorithmically very efficient. In view of that: Continuation from this simple model to the initial round Earth model. M. Cerf, T. Haberkorn, E. Tr´ elat Continuation from a flat to a round Earth model
Introduction Flattening the Earth Continuation procedure Flat Earth Numerical simulations Simplified flat Earth model System x ( t ) = v x ( t ) ˙ ˙ h ( t ) = v h ( t ) v x ( t ) = T max max m ( t f ) ˙ m ( t ) u x ( t ) t f free v h ( t ) = T max ˙ m ( t ) u h ( t ) − g 0 � u x ( t ) 2 + u h ( t ) 2 ˙ m ( t ) = − β T max Control Control ( u x ( · ) , u h ( · )) such that u x ( · ) 2 + u h ( · ) 2 � 1 initial conditions: x ( 0 ) = x 0 , h ( 0 ) = h 0 , v x ( 0 ) = v x 0 , v h ( 0 ) = v h 0 , m ( 0 ) = m 0 final conditions: h ( t f ) = h f , v x ( t f ) = v xf , v h ( t f ) = 0 M. Cerf, T. Haberkorn, E. Tr´ elat Continuation from a flat to a round Earth model
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