Continuation from a flat to a round Earth model in the coplanar orbit transfer problem

Transcription

1 Continuation from a flat to a round Earth model in the coplanar orbit transfer problem M. Cerf 1, T. Haberkorn, Emmanuel Trélat 1 1 EADS Astrium, les Mureaux 2 MAPMO, Université d Orléans First Industrial Workshop, SADCO 2011, March 2nd

2 The coplanar orbit transfer problem Spherical Earth Central gravitational field g(r) = µ r 2 System in cylindrical coordinates ṙ(t) = v(t) sin γ(t) ϕ(t) = v(t) cos γ(t) r(t) v(t) = g(r(t)) sin γ(t) + Tmax m(t) u 1(t) v(t) γ(t) = r(t) g(r(t)) «cos γ(t) + Tmax v(t) m(t)v(t) u 2(t) ṁ(t) = βt max u(t) Thrust: T (t) = u(t)t max (T max large: strong thrust) Control: u(t) = (u 1 (t), u 2 (t)) satisfying u(t) = p u 1 (t) 2 + u 2 (t) 2 1

3 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 Kf = v(t f ) 2 µ 2 r(t f ) K f = 0, ξ ef = sin 2 γ + 1 r(t f )v(t f ) 2 «2 cos 2 γ ef 2 = 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)

4 Application of the Pontryagin Maximum Principle Hamiltonian H(q, p, p 0 v, u) = p r v sin γ + p ϕ cos γ + pv r + p γ v r g(r) v g(r) sin γ + Tmax m u 1 ««cos γ + Tmax mv u 2 p mβt max u, «Extremal equations q(t) = H p (q(t), p(t), p0, u(t)), ṗ(t) = H q (q(t), p(t), p0, u(t)), Maximization condition H(q(t), p(t), p 0, u(t)) = max w 1 H(q(t), p(t), p0, w)

5 Application of the Pontryagin Maximum Principle Hamiltonian H(q, p, p 0 v, u) = p r v sin γ + p ϕ cos γ + pv r + p γ v r g(r) v g(r) sin γ + Tmax m u 1 ««cos γ + Tmax mv u 2 p mβt max u, «Maximization condition leads to u(t) = (u 1 (t), u 2 (t)) = (0, 0) whenever Φ(t) < 0 u 1 (t) = r p v (t) p v (t) 2 + pγ (t)2 v(t) 2, u 2 (t) = v(t) r p γ(t) p v (t) 2 + pγ (t)2 v(t) 2 whenever Φ(t) > 0 where s Φ(t) = 1 p v (t) 2 + pγ(t)2 βpm(t) (switching function) m(t) v(t) 2

6 Application of the Pontryagin Maximum Principle Hamiltonian H(q, p, p 0 v, u) = p r v sin γ + p ϕ cos γ + pv r + p γ v r g(r) v g(r) sin γ + Tmax m u 1 ««cos γ + Tmax mv u 2 p mβt max u, «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 Kf (p γ v ξ ef p v γξ ef ) + v ξ Kf (p r γξ ef p γ r ξ ef ) = 0 Remark p 0 0 (no abnormal) p 0 = 1 no singular arc (Bonnard - Caillau - Faubourg - Gergaud - Haberkorn - Noailles - Trélat)

7 Shooting method Find a zero of r(t f, p 0 ) r f ξ Kf (p 0 ) 1 v(t f, p 0 ) v f ξ ef (p S(t f, p 0 ) = B γ(t f, p 0 ) γ f p ϕ(t f, p 0 ) A or 0 ) B p ϕ(t f, p 0 ) A, p m(t f, p 0 ) 1 p m(t f, p 0 ) 1 Main problem: how to make the shooting method converge? initialization of the shooting method discontinuities of the optimal control

8 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. Wächter, L.T. Biegler On the implementation of an interior-point lter line- search algorithm for large-scale nonlinear programming, Mathematical Programming 106 (2006), but usual flaws of direct methods (computationally demanding, lack of numerical precision).

9 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 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), L.W. Neustadt, A general theory of minimum-fuel space trajectories, SIAM Journal on Control 3, no. 2 (1965),

10 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),

11 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), P. Martinon, J. Gergaud, Using switching detection and variational equations for the shooting method, Optimal Cont. Appl. Methods 28, no. 2 (2007), but not adapted for high-thrust transfer.

12 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.

13 Simplified flat Earth model System ẋ(t) = v x (t) ḣ(t) = v h (t) v x (t) = Tmax ux (t) m(t) v h (t) = Tmax m(t) u h(t) g 0 max m(t f ) t f free ṁ(t) = βt max qu x (t) 2 + u h (t) 2 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 x0, v h (0) = v h0, m(0) = m 0 final conditions: h(t f ) = h f, v x (t f ) = v xf, v h (t f ) = 0

14 Modified flat Earth model Idea: mapping circular orbits to horizontal trajectories 8 >< >: x = r ϕ h = r r T v x = v cos γ v h = v sin γ 8 >< >: r = r T + h x ϕ = qr T +h v = vx 2 + vh 2 γ = arctan v h v x «««ux cos γ sin γ u1 = u h sin γ cos γ u 2 x f (v xf, v hf) (r T + h f)ϕ f x f v f γ f v h0 h f v 0 γ 0 h f v x0 h 0 h 0 r T ϕ f

15 Modified flat Earth model Plugging this change of coordinates into the initial round Earth model: leads to... ṙ(t) = v(t) sin γ(t) ϕ(t) = v(t) cos γ(t) r(t) v(t) = g(r(t)) sin γ(t) + Tmax m(t) u 1(t) v(t) γ(t) = r(t) g(r(t)) «cos γ(t) + Tmax v(t) m(t)v(t) u 2(t) ṁ(t) = βt max u(t)

16 Modified flat Earth model Modified flat Earth model x(t) ẋ(t) = v x (t) + v h (t) r T + h(t) ḣ(t) = v h (t) v x (t) = Tmax m(t) ux (t) vx (t)v h(t) r T + h(t) v h (t) = Tmax m(t) u vx (t)2 h(t) g(r T + h(t)) + r T + h(t) ṁ(t) = βt max u(t) Differences with the simplified flat Earth model (with constant gravity): the term in green: variable (usual) gravity. the terms in red: correcting terms allowing the existence of horizontal (periodic up to translation in x) trajectories with no thrust.

17 Continuation procedure Simplified flat Earth model (with constant gravity) continuation modified flat Earth model: procedure x(t) ẋ(t) = v x (t) + λ 2 v h (t) r T + h(t) ḣ(t) = v h (t) v x (t) = Tmax m(t) ux (t) λ v x (t)v h (t) 2 r T + h(t) v h (t) = Tmax m(t) u h(t) µ (r T +λ 1 h(t)) 2 + λ v x (t) 2 2 r T + h(t) 2 parameters: 0 λ λ 2 1 ṁ(t) = βt max qu x (t) 2 + u h (t) 2 λ 1 = λ 2 = 0: simplified flat Earth model with constant gravity λ 1 = 1, λ 2 = 0: simplified flat Earth model with usual gravity λ 1 = λ 2 = 1: modified flat Earth model (equivalent to usual round Earth)

18 Continuation procedure Simplified flat Earth model (with constant gravity) continuation modified flat Earth model: procedure x(t) ẋ(t) = v x (t) + λ 2 v h (t) r T + h(t) ḣ(t) = v h (t) v x (t) = Tmax m(t) ux (t) λ v x (t)v h (t) 2 r T + h(t) v h (t) = Tmax m(t) u h(t) µ (r T +λ 1 h(t)) 2 + λ v x (t) 2 2 r T + h(t) 2 parameters: 0 λ λ 2 1 ṁ(t) = βt max qu x (t) 2 + u h (t) 2 Two-parameters family of optimal control problems: (OCP) λ1,λ 2

19 Continuation procedure (OCP) 0,0 flat Earth model, constant gravity (linear) continuation on λ 1 [0, 1] final time t f free (OCP) 1,0 flat Earth model, usual gravity (linear) continuation on λ 2 [0, 1] final time t f fixed (OCP) 1,1 modified flat Earth model (equivalent to round Earth model) Application of the PMP to (OCP) λ1,λ 2 series of shooting problems.

20 Change of coordinates Remark: Once the continuation process has converged, we obtain the initial adjoint vector for (OCP) 1,1 in the modified coordinates. To recover the adjoint vector in the usual cylindrical coordinates, we use the general fact: Lemma Change of coordinates x 1 = φ(x) and u 1 = ψ(u) dynamics f 1 (x 1, u 1 ) = dφ(x).f (φ 1 (x 1 ), ψ 1 (u 1 )) and for the adjoint vectors: p 1 ( ) = t dφ(x( )) 1 p( ). Here, this yields: p r = x r T + h px + p h p ϕ = (r T + h)p x p v = cos γ p vx + sin γ p vh p γ = v( sin γ p vx + cos γ p vh ).

21 Analysis of the flat Earth model System ẋ = v x Initial conditions Final conditions ḣ = v h x(0) = x 0 x(t f ) free t f free v x = Tmax m ux v h = Tmax m u h g 0 ṁ = βt max qu 2 x + u 2 h h(0) = h 0 v x (0) = v x0 v h (0) = v h0 m(0) = m 0 h(t f ) = h f v x (t f ) = v xf v h (t f ) = 0 m(t f ) free max m(t f ) Theorem If h f > h 0 + v2 h0 2g 0, then the optimal trajectory is a succession of at most two arcs, and the thrust u( ) T max is either constant on [0, t f ] and equal to T max, or of the type T max 0, or of the type 0 T max.

22 Analysis of the flat Earth model Main ideas of the proof: Application of the PMP The switching function Φ = 1 m q p 2 v x + p 2 v h βp m satisfies: Φ = p h p vh q m pv 2 x + pv 2 h Φ = β u m Φ has at most one minimum Φ m qp 2 v x + p 2 vh Φ 2 + strategies T max, T max 0, 0 T max, or T max 0 T max The strategy T max 0 T max cannot occur ph 2 q m pv 2 x + pv 2 h

23 Shooting method in the flat Earth model A priori, we have: 5 unknowns p h, p vx, p vh (0), p m(0), and t f 5 equations h(t f ) = h f, v x (t f ) = v xf, v h (t f ) = 0, p m(t f ) = 1, H(t f ) = 0 but using several tricks and some system analysis, the shooting method can be simplified to: 3 unknowns p vx, p vh (0), and the first switching time t 1 3 equations h(t f ) = h f, v x (t f ) = v xf, v h (t 1 ) + g 0 t 1 = g 0 p vh (0) very easy and efficient (instantaneous) algorithm and the initialization of the shooting method is automatic (CV for any initial adjoint vector) automatic tool for initializing the continuation procedure

24 Numerical simulations T max = 180 kn Isp = 450 s λ λ Initial conditions ϕ 0 = 0 (SSO) h 0 = 200 km v 0 = 5.5 km/s γ 0 = 2 deg m 0 = kg λ p h p vx λ p vh p m Final conditions h f = 800 km v f = 7.5 km/s γ f = 0 deg (nearly circular final orbit) Evolution of the shooting function unknowns (p h, p vx, p vh, p m) (abscissa) with respect to homotopic parameter λ 2 (ordinate) continuous but not C 1 path: λ , λ 2 0.8, and λ : 0 λ : T max λ 2 0.8: T max 0 T max 0.8 λ : T max λ 2 1: T max 0 T max

25 Numerical simulations 2 x x 104 h (km) x (km) v x (km/s) v h (km/s) t (s) u x u h u m (kg) t (s) Trajectory and control strategy of (OCP) 1,0 (dashed) and (OCP) 1,1 (plain). t f 1483s Remark In the case of a final orbit (no injecting point): additional continuation on transversality conditions.

26 Numerical simulations Comparison with a direct method: Heun (RK2) discretization with N points combination of AMPL with IPOPT needs however a careful initial guess Continuation method 3 seconds: (OCP) 0,0 : instantaneous from (OCP) 0,0 to (OCP) 1,0 : 0.5 second from (OCP) 1,0 to (OCP) 1,1 : 2.5 seconds Accuracy: Direct method N = 100: 5 seconds N = 1000: 165 seconds Accuracy: 10 6

27 Conclusion Algorithmic procedure to solve the problem of minimization of fuel consumption for the coplanar orbit transfer problem by shooting method approach Does not require any careful initial guess Open questions Is this procedure systematically efficient, for any possible coplanar orbit transfer? Extension to 3D