Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Numerical Challenges in Computing Low-energy Low-thrust Trajectories in Multi-Body Environment Richard Epenoy
[email protected] Centre National d’Etudes Spatiales 18 avenue Edouard Belin 31401 Toulouse Cedex 9, France
3rd European Optimisation in Space Engineering Workshop 17th - 18th September 2015, University of Strathclyde, Glasgow
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Outline 1
Sample problem Dynamic model Optimal control formulation and optimality conditions
2
New approach for computing low-energy transfers Numerical issues for shooting methods Three-step solution approach
3
Transfers between Lyapunov orbits of the same energy Statement of the test case Short-duration transfer Long-duration transfers
4
Conclusion and future prospects Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Dynamic model Optimal control formulation and optimality conditions
Sample problem 1
Sample problem Dynamic model Optimal control formulation and optimality conditions
2
New approach for computing low-energy transfers Numerical issues for shooting methods Three-step solution approach
3
Transfers between Lyapunov orbits of the same energy Statement of the test case Short-duration transfer Long-duration transfers
4
Conclusion and future prospects Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Dynamic model Optimal control formulation and optimality conditions
Planar Circular Restricted Three-Body Problem (1/2) Adimensional equations of motion in a synodic reference frame x˙ = vx y˙ = vy (1 − µ) (x + µ) µ (x + µ − 1) v˙x = x + 2vy − − + u1 r13 r23 (1 − µ) y µy v˙y = y − 2vx − − 3 + u2 3 r1 r2 q q r1 = (x + µ)2 + y 2 r2 = (x + µ − 1)2 + y 2 State vector: ξ = (x, y , vx , vy )T T
=⇒ ξ˙ = ϕ(ξξ , u )
Control vector: u = (u1 , u2 ) Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Dynamic model Optimal control formulation and optimality conditions
Planar Circular Restricted Three-Body Problem (2/2) 2 2 Ω(x, y ) = x + y + 1 − µ + µ + µ(1 − µ) 2 r1 r2 2 2 2 J(x, y , vx , vy ) = 2Ω(x, y ) − vx − vy (uu = 0 ) =⇒ (J(x, y , vx , vy ) = C te = Jacobi constant)
Lagrange points V. G. Szebehely. Theory of Orbits - The Restricted Problem of Three Bodies, Academic Press Inc., Harcourt Brace Jovanovich Publishers, Orlando, Florida, 1967, pp. 8-100.
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Dynamic model Optimal control formulation and optimality conditions
Minimum-energy transfer between LPOs (1/2) The fixed time horizon problem to be solved Z 1 tf Find {uu , τ 0 , τ f } = argmin K (uu , τ0 , τf ) = kuu k2 dt 2 u ,τ ,τ 0 f t0 s.t. (P) ξ˙ = ϕ(ξξ , u ) ξ (t0 ) − ξ I (τ0 ) = 0 ξ (tf ) − ξ T (τf ) = 0 ξ I (τ0 ), ξ T (τf ): states on the initial and final Libration Point Orbits computed by means of Lindstedt-Poincar´e techniques† † J. Masdemont. High Order Expansions of Invariant Manifolds of Libration Point Orbits with Applications to Mission Design. Dynamical Systems, 20:1, 2005, pp. 59-113. Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Dynamic model Optimal control formulation and optimality conditions
Minimum-energy transfer between LPOs (2/2) Pontryagin’s Minimum Principle (PMP) Pontryagin’s Hamiltonian H = 12 kuu k2 + λ T ϕ(ξξ , u ) H-minimum control u = − (λ3 , λ4 )T Costate equations ∂ϕ λ˙ = − (ξξ , u )T λ ∂ξξ Transversality conditions λ (t0 )T ξ˙ I (τ0 ) = λ (t0 )T ϕ(ξξ I (τ0 ), 0) = 0 λ (tf )T ξ˙ (τf ) = λ (tf )T ϕ(ξξ T (τf ), 0) = 0 T
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Numerical issues for shooting methods Three-step solution approach
New approach for computing low-energy transfers 1
Sample problem Dynamic model Optimal control formulation and optimality conditions
2
New approach for computing low-energy transfers Numerical issues for shooting methods Three-step solution approach
3
Transfers between Lyapunov orbits of the same energy Statement of the test case Short-duration transfer Long-duration transfers
4
Conclusion and future prospects Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Numerical issues for shooting methods Three-step solution approach
Shooting function S Let θ = λ (t0 )T , τ0 , τf
S:
T
, then
6 R −→ R6
ξ (tf ) − ξ T (τf )
λ (t0 )T ϕ(ξξ I (τ0 ), 0) θ 7−→ S(θ) = T λ (tf ) ϕ(ξξ T (τf ), 0)
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Numerical issues for shooting methods Three-step solution approach
Shooting function S Let θ = λ (t0 )T , τ0 , τf
S:
T
, then
6 R −→ R6
ξ (tf ) − ξ T (τf )
λ (t0 )T ϕ(ξξ I (τ0 ), 0) θ 7−→ S(θ) = T λ (tf ) ϕ(ξξ T (τf ), 0)
Sensitivity of the dynamics and costate equations Inaccuracy when computing the Jacobian of S by Finite Differences
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Numerical issues for shooting methods Three-step solution approach
Shooting function S Let θ = λ (t0 )T , τ0 , τf
S:
T
, then
6 R −→ R6
ξ (tf ) − ξ T (τf )
λ (t0 )T ϕ(ξξ I (τ0 ), 0) θ 7−→ S(θ) = T λ (tf ) ϕ(ξξ T (τf ), 0)
Sensitivity of the dynamics and costate equations Inaccuracy when computing the Jacobian of S by Finite Differences Proposed solution Use of variational equations =⇒ system of 48 ODEs to integrate Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Numerical issues for shooting methods Three-step solution approach
Case of long-duration transfers between orbits of the same energy Ill-conditioning due to the two-time-scale structure of the problem High contraction and expansion rates of the Hamiltonian system Initial thrust arc, equilibrium arc (uu ≈ 0 ), final thrust arc†§
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Numerical issues for shooting methods Three-step solution approach
Case of long-duration transfers between orbits of the same energy Ill-conditioning due to the two-time-scale structure of the problem High contraction and expansion rates of the Hamiltonian system Initial thrust arc, equilibrium arc (uu ≈ 0 ), final thrust arc†§ † G.
G´ omez, W. S. Koon, J. E. Marsden, J. Masdemont, and S. D. Ross. Connecting Orbits and Invariant Manifolds in the Spatial Restricted Three-body Problem. Nonlinearity, 17:5, 2004, pp. 1571-1606.
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Numerical issues for shooting methods Three-step solution approach
Case of long-duration transfers between orbits of the same energy Ill-conditioning due to the two-time-scale structure of the problem High contraction and expansion rates of the Hamiltonian system Initial thrust arc, equilibrium arc (uu ≈ 0 ), final thrust arc†§ † G.
G´ omez, W. S. Koon, J. E. Marsden, J. Masdemont, and S. D. Ross. Connecting Orbits and Invariant Manifolds in the Spatial Restricted Three-body Problem. Nonlinearity, 17:5, 2004, pp. 1571-1606. § B. D. Anderson and P. V. Kokotovic. Optimal Control Problems Over Large Time Intervals. Automatica, 23:3, 1987, pp. 355-363. § A. V. Rao, and K. D. Mease. Eigenvector Approximate Dichotomic Basis Method for Solving Hyper-Sensitive Optimal Control Problems. Optimal Control Applications and Methods, 21:1, 2000, pp. 1-19.
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Numerical issues for shooting methods Three-step solution approach
Case of long-duration transfers between orbits of the same energy Ill-conditioning due to the two-time-scale structure of the problem High contraction and expansion rates of the Hamiltonian system Initial thrust arc, equilibrium arc (uu ≈ 0 ), final thrust arc†§ † G.
G´ omez, W. S. Koon, J. E. Marsden, J. Masdemont, and S. D. Ross. Connecting Orbits and Invariant Manifolds in the Spatial Restricted Three-body Problem. Nonlinearity, 17:5, 2004, pp. 1571-1606. § B. D. Anderson and P. V. Kokotovic. Optimal Control Problems Over Large Time Intervals. Automatica, 23:3, 1987, pp. 355-363. § A. V. Rao, and K. D. Mease. Eigenvector Approximate Dichotomic Basis Method for Solving Hyper-Sensitive Optimal Control Problems. Optimal Control Applications and Methods, 21:1, 2000, pp. 1-19.
Proposed solution: the three-step solution approach Continuation method to solve the ill-conditioned problem Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Numerical issues for shooting methods Three-step solution approach
First step: determination of a feasible solution A quadratic-zero-quadratic feasible control ai (t − t1 )2 if t ∈ [t0 , t1 ] 0 if t ∈ [t1 , t2 ] ui0 (t) = bi (t − t2 )2 if t ∈ [t2 , tf ]
(i = 1, ..., 2)
Find a zero-cost solution of the feasibility problem T Let w = a1 , a2 , b1 , b2 , t1 , t2 , τ00 , τf0 , then w ) = 12 kξξ (tf ) − ξ T (τf0 )k2 Find w = argmin G (w w s.t. (FP) 0 ˙ ξ = ϕ(ξξ , u ) ξ (t0 ) − ξ I (τ00 ) = 0 Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Numerical issues for shooting methods Three-step solution approach
Second step: determination of a suboptimal solution Solve a family of problems by continuation on ∈ [0, 1] Starting from { = 0, u = u 0 , λ = λ 0 = 0 }, Z 1 tf Find u = argmin K (uu ) = (1 − )kuu − u 0 k2 + kuu k2 dt 2 t0 u s.t. ξ˙ = ϕ(ξξ , u ) ξ (t0 ) − ξ I (τ00 ) = 0 ξ (tf ) − ξ T (τf0 ) = 0 Finally, for = 1, {uu 1 , τ00 , τf0 } is a suboptimal solution of (P)
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Numerical issues for shooting methods Three-step solution approach
Third step: determination of the optimal solution (1/2) Minimum-energy problem with fixed values of τ0 and τf Let τ0 and τf be given and let L(τ0 , τf ) = K (uu τ0 ,τf , τ0 , τf ), where Z 1 tf kuu k2 dt u τ0 ,τf = argmin K (uu , τ0 , τf ) = 2 t0 u s.t. ξ˙ = ϕ(ξξ , u ) ξ (t0 ) − ξ I (τ0 ) = 0 ξ (tf ) − ξ T (τf ) = 0 Bilevel programming formulation of problem (P) (P) ≡ {τ 0 , τ f } = argmin L(τ0 , τf ), τ0 ,τf
Richard Epenoy
u τ 0 ,τ f = u ,
u τ 0 ,τ 0 = u 1 0
f
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Numerical issues for shooting methods Three-step solution approach
Third step: determination of the optimal solution (2/2) Solution of (P) by means of a gradient algorithm Gradient of function L derived from the PMP λ τ0 ,τf (t0 )T ϕ(ξξ I (τ0 ), 0)
∇L(τ0 , τf ) =
!
λτ0 ,τf (tf )T ϕ(ξξ T (τf ), 0) −λ
Gradient algorithm
τ0k+1 τfk+1
=
τ0k τfk
− β∇L τ0k , τfk
Convergence criterion k∇L τ0N , τfN k ≤ ν Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Statement of the test case Short-duration transfer Long-duration transfers
Transfers between Lyapunov orbits of the same energy 1
Sample problem Dynamic model Optimal control formulation and optimality conditions
2
New approach for computing low-energy transfers Numerical issues for shooting methods Three-step solution approach
3
Transfers between Lyapunov orbits of the same energy Statement of the test case Short-duration transfer Long-duration transfers
4
Conclusion and future prospects Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Statement of the test case Short-duration transfer Long-duration transfers
Transfers in the Earth-Moon PCR3BP Initial and final Lyapunov orbits Lagrange point L1 L2
Jacobi constant 3.178 3.178
Adimensional x-amplitude 0.135159595 0.100411240
Adimensional period 2.776024945 3.385292341
Short transfer duration: mere use of the indirect shooting method Case 1: Tf = 12 days =⇒ tf = 2.759659 Long transfer duration: the three-step approach is essential Case 2: Tf = 29 days =⇒ tf = 6.669175 Case 3: Tf = 44 days =⇒ tf = 10.118748 Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Statement of the test case Short-duration transfer Long-duration transfers
Case 1: control history
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Statement of the test case Short-duration transfer Long-duration transfers
Case 1: optimal trajectory
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Statement of the test case Short-duration transfer Long-duration transfers
Case 2 - First step: feasible control
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Statement of the test case Short-duration transfer Long-duration transfers
Case 2 - Second step: suboptimal control
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Statement of the test case Short-duration transfer Long-duration transfers
Case 2 - Third step (1/2): optimal low-energy control
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Statement of the test case Short-duration transfer Long-duration transfers
Case 2 - Third step (2/2): optimal low-energy trajectory
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Statement of the test case Short-duration transfer Long-duration transfers
Case 2: synthesis of the results Evolution of τ0 , τf , and performance index Step 1 2 3
τ0 0.355639452 0.355639452 0.668190968
τf 0.725904853 0.725904853 1.015627159
K (uu , τ0 , τf ) 1.462274560x10−5 4.760839250x10−6 2.014050772x10−6
Low-energy solution consistent with the one-revolution heteroclinic connection G. G´ omez, W. S. Koon, J. E. Marsden, J. Masdemont, and S. D. Ross. Connecting Orbits and Invariant Manifolds in the Spatial Restricted Three-body Problem. Nonlinearity, 17:5, 2004, pp. 1571-1606. E. Canalias, and J. Masdemont. Homoclinic and Heteroclinic Transfer Trajectories Between Lyapunov Orbits in the Sun-Earth and Earth-Moon Systems. Discrete and Continuous Dynamical Systems - Series A, 14:2, 2006, pp. 261-279.
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Statement of the test case Short-duration transfer Long-duration transfers
Case 3 - First step: feasible control
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Statement of the test case Short-duration transfer Long-duration transfers
Case 3 - Second step: suboptimal control
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Statement of the test case Short-duration transfer Long-duration transfers
Case 3 - Third step (1/2): optimal low-energy control
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Statement of the test case Short-duration transfer Long-duration transfers
Case 3 - Third step (2/2): optimal low-energy trajectory
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Statement of the test case Short-duration transfer Long-duration transfers
Case 3: synthesis of the results Evolution of τ0 , τf , and performance index Step 1 2 3
τ0 1.632183471 1.632183471 2.768373812
τf 0.677910923 0.677910923 1.812860237
K (uu , τ0 , τf ) 3.545166535x10−6 2.289363671x10−6 2.543914352x10−8
Low-energy solution consistent with the two-revolution heteroclinic connection G. G´ omez, W. S. Koon, J. E. Marsden, J. Masdemont, and S. D. Ross. Connecting Orbits and Invariant Manifolds in the Spatial Restricted Three-body Problem. Nonlinearity, 17:5, 2004, pp. 1571-1606. E. Canalias, and J. Masdemont. Homoclinic and Heteroclinic Transfer Trajectories Between Lyapunov Orbits in the Sun-Earth and Earth-Moon Systems. Discrete and Continuous Dynamical Systems - Series A, 14:2, 2006, pp. 261-279.
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Conclusion and future prospects 1
Sample problem Dynamic model Optimal control formulation and optimality conditions
2
New approach for computing low-energy transfers Numerical issues for shooting methods Three-step solution approach
3
Transfers between Lyapunov orbits of the same energy Statement of the test case Short-duration transfer Long-duration transfers
4
Conclusion and future prospects Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Conclusion Low-energy transfers between Libration Point Orbits Approach based on single shooting and variational equations Long-duration transfers between orbits of the same energy Hypersensitive optimal control problem Necessity of a specific solution methodology The three-step approach Numerically effective alternative to direct methods Unlike other methods§ no prior knowledge is enforced Direct method with an enforced coast arc § J. R. Stuart, M. T. Ozimek, and K. C. Howell. Optimal, Low-Thrust, Path-Constrained Transfers between Libration Point Orbits using Invariant Manifolds. AIAA/AAS Astrodynamics Specialist Conference, Toronto, Canada, August 2010. Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Future prospects Next evolutions of the method Real propulsion model for the spacecraft with mass variation Three-dimension transfers between LPOs in the CR3BP Low-thrust Earth-Moon transfers with Sun perturbation Earth-Moon based Bicircular Four-Body Problem (B4BP) Apply the three-step approach to find low-energy trajectories Comparisons with direct methods§ Direct method with an initial guess based on manifolds § G. Mingotti, F. Topputo, and F. Bernelli-Zazzera. Efficient Invariant-manifold, Lowthrust Planar Trajectories to the Moon. Communications in Nonlinear Science and Numerical Simulation, 17:2, 2012, pp. 817-831.
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
Sample problem New approach for computing low-energy transfers Transfers between Lyapunov orbits of the same energy Conclusion and future prospects
Thank you for your attention
Richard Epenoy
Low-Energy Transfers Between Libration Point Orbits
