Dynamical system theory and numerical methods applied to Astrodynamics

Dynamical system theory and numerical methods applied to Astrodynamics Roberto Castelli Institute for Industrial Mathematics University of Paderborn BCAM, Bilbao, 20th December, 2010

Introduction Introduction In space mission design Consider the Force Field acting on the Spacecraft Consider Physical and Technical constraints Satisfy some mission requirements Take care of the fuel consumption and the travelling time... Genesis Mission

Introduction Introduction First guess trajectories designed in simplified model Two-body model Restricted Three-body problem Bicircular model... N-BODY PROBLEM Numerical Optimisation in Full system Direct/Indirect methods Multiple shooting technique Multiobjective optimisation Different type of Propulsion (Electric - Chemical) Low thrust propulsion Impulsive manoeuvre

OUTLINE Introduction Dynamical model CRTBP Periodic orbits Tube Dynamics Patched CRTBP approximation Computational methods: Set Oriented Numerics Covering of Invariant Sets GAIO implementation Earth to Halo Regions of prevalence Sun-Earth DPO to Earth-Moon DPO Conclusion

OUTLINE Dynamical model CRTBP Dynamical model CRTBP Periodic orbits Tube Dynamics Patched CRTBP approximation Computational methods: Set Oriented Numerics Covering of Invariant Sets GAIO implementation Earth to Halo Regions of prevalence Sun-Earth DPO to Earth-Moon DPO Conclusion

Dynamical model CRTBP Circular Restricted Three-Body problem Two Primaries move in circular orbits under the mutual gravitational attraction Massless particle moves under the gravitational influence of two primaries In a rotating, adimensional reference frame, µ = m 2 /(m 1 + m 2 ), (CR3BP) ẍ 2ẏ = Ω x ÿ + 2ẋ = Ω y z = Ω z Ω(x, y, z) = 1 2 (x 2 +y 2 )+ 1 µ + µ r 2 + 1 2 µ(1 µ) r 1

Properties of CRTBP Dynamical model CRTBP Non integrable Autonomous Hamiltonian System Symmetry (x, y, z, ẋ, ẏ, ż; t) (x, y, z, ẋ, ẏ, ż; t) Jacobi Integral: C = 2Ω(x, y, z) (ẋ 2 + ẏ 2 + ż 2 ) = 2E Equilibrium points: Lagrangian Points L j, j = 1,..., 5. Hill s Region: H(C) = {(x, y, z) : 2Ω(x, y, z) C 0}

OUTLINE Dynamical model CRTBP Periodic orbits Dynamical model CRTBP Periodic orbits Tube Dynamics Patched CRTBP approximation Computational methods: Set Oriented Numerics Covering of Invariant Sets GAIO implementation Earth to Halo Regions of prevalence Sun-Earth DPO to Earth-Moon DPO Conclusion

Dynamical model CRTBP Families of Periodic Orbits. Periodic orbits Hamiltonian system continuous families of periodic orbits. PO bifurcating from L 1 [E. J. Doedel et al.]

Dynamical model CRTBP Families of Periodic Orbits. Periodic orbits Hamiltonian system continuous families of periodic orbits. PO bifurcating from L 1 [E. J. Doedel et al.] Differential correction scheme, based on the variational eq. Find δvy such that the first x-axis crossing of φ t (X 0, 0, 0, Vy + δvy) is perpendicular Simple Symmetric periodic orbits

Dynamical model CRTBP Periodic Orbits Diagram Periodic orbits Diagram of Simple Symmetric PO in SE system. (X 0, Vy) (X 0, 0, 0, V y ), Earth < X 0 < L 2, Vy > 0 Lyapunov and Halo orbits

Periodic orbits: DPO Dynamical model CRTBP Periodic orbits

Resonant Orbits Dynamical model CRTBP Periodic orbits For which resonances there exist families of periodic orbits? (Collaboration with Prof. P. Zgliczynski)

OUTLINE Dynamical model CRTBP Tube Dynamics Dynamical model CRTBP Periodic orbits Tube Dynamics Patched CRTBP approximation Computational methods: Set Oriented Numerics Covering of Invariant Sets GAIO implementation Earth to Halo Regions of prevalence Sun-Earth DPO to Earth-Moon DPO Conclusion

Dynamical model CRTBP Tube Dynamics Dynamics near periodic orbits The periodic orbits separates two necks in the Hill s region Linear Dynamics: saddle center 3 types of orbits: asymptotic, transit, non-transit [W.S. Koon et al.]

Dynamical model CRTBP Tube Dynamics Invariant manifolds The Stable/Unstable Invariant manifolds Set of orbits asymptotic to the periodic orbit for t ± are topologically equivalent to N 2 dimensional cylinders in the N 1 dim. energy manifold act as separatrices in the phase space between transit and non-transit orbit [G. Gomez at al.]

OUTLINE Dynamical model CRTBP Patched CRTBP approximation Dynamical model CRTBP Periodic orbits Tube Dynamics Patched CRTBP approximation Computational methods: Set Oriented Numerics Covering of Invariant Sets GAIO implementation Earth to Halo Regions of prevalence Sun-Earth DPO to Earth-Moon DPO Conclusion

Dynamical model CRTBP Patched CRTBP approximation Mission Design: dynamical system theory Dynamical system theory in low energy trajectory design Patched 3-body problem The 4-Body system is approximated with the superpositions of two Restricted Three-Body problems The invariant manifold structures are exploited to design legs of trajectory The design restricts to the selection of a connection point on a suitable Poincaré section

Some examples Dynamical model CRTBP Patched CRTBP approximation Low energy transfer to the Moon (Fig. from [W.S. Koon et al.]) Petit Grand Tour of the moons of Jupiter, (Fig. from [G. Gomez at al.])

OUTLINE Computational methods: Set Oriented Numerics Dynamical model CRTBP Periodic orbits Tube Dynamics Patched CRTBP approximation Computational methods: Set Oriented Numerics Covering of Invariant Sets GAIO implementation Earth to Halo Regions of prevalence Sun-Earth DPO to Earth-Moon DPO Conclusion

Computational methods: Set Oriented Numerics Set Oriented Numerics Main Goal: Study the long term behavior of complex and chaotic Dynamical Systems 50 40 30 20 10 0 20 10 0 10 20 40 20 0 20 40 Set Oriented Numerics Computation of several short term trajectories instead of single long term trajectory Approximation of Global structure Invariant Sets : global attractors, Invariant manifolds Invariant measures, almost invariant sets Transport operators Multiobjective optimization (Pareto set)

Computational methods: Set Oriented Numerics Methodology Consider a discrete dynamical system x k+1 = f (x k ), k = 0, 1, 2,..., f : R n R n Aim: Approximation of a structure within a bounded set Q. Method: Generate a sequence of collections B 1, B 2... of subsets of Q s.t B 0 = {Q}, and iteratively B k from B k 1 Subdivision: define a new collection B k such that B = B B B k B B k 1 Selection: define B k as diam( B k ) θ k diam(b k 1 ), θ k (0, 1) B k = {B B k : such that g(b) B for some B B k }

OUTLINE Computational methods: Set Oriented Numerics Covering of Invariant Sets Dynamical model CRTBP Periodic orbits Tube Dynamics Patched CRTBP approximation Computational methods: Set Oriented Numerics Covering of Invariant Sets GAIO implementation Earth to Halo Regions of prevalence Sun-Earth DPO to Earth-Moon DPO Conclusion

Computational methods: Set Oriented Numerics Relative Global Attractor Covering of Invariant Sets Definition Relative Global attractor Let Q R n be a compact set. Define global attractor relative to Q by A Q = j 0 f j (Q) Properties A Q Q f 1 (A Q ) A Q but not necessarily f (A Q ) A Q. Selection step: define B k as B k = {B B k : such that f 1 (B) B for some B B k }

Hénon map Computational methods: Set Oriented Numerics Covering of Invariant Sets { xk+1 = 1 ax 2 k + by k a = 1.4, b = 0.3 y k+1 = x k Covering of the attractor relative to [ 2, 2] 2, k = 8, 12, 16, 20

Computational methods: Set Oriented Numerics Computing the unstable manifold Covering of Invariant Sets Initialization - Continuation Algorithm Aim: Compute the unstable manifold of a point p into a (large ) compact set Q, ( p Q) 1 Given Q, compute P 0, P 1,... P l nested sequence of fine partitions of Q. Select the element C P l such that p C and A C = Wloc u (p) C 2 Initialization Starting from B 0 = {C}, refine the approximation of Wloc u (p) C by subdivision, yielding B(0) k P l+k 3 Continuation From B (j 1) k compute B (j) k = {B P l+k : f (B ) B, for some B B (j 1) k }

Computational methods: Set Oriented Numerics Lorenz system ẋ = σ(y x) ẏ = ρx y xz ż = βz + xy Covering of Invariant Sets (0, 0, 0) is fixed point σ = 10, ρ = 28, β = 8/3 Covering of the two-dimensional stable manifold of the origin Left: l = 9, k = 6, j = 4 initial box Q = [ 70, 70] [ 70, 70] [ 80, 80], Right:l = 21, k = 0, j = 10, initial box Q = [ 120, 120] [ 120, 120] [ 160, 160]

Computational methods: Set Oriented Numerics Covering of Invariant Sets Box covering of the part of unstable manifold of an Halo orbit in the Sun-Earth CRTBP.

OUTLINE Computational methods: Set Oriented Numerics GAIO implementation Dynamical model CRTBP Periodic orbits Tube Dynamics Patched CRTBP approximation Computational methods: Set Oriented Numerics Covering of Invariant Sets GAIO implementation Earth to Halo Regions of prevalence Sun-Earth DPO to Earth-Moon DPO Conclusion

Computational methods: Set Oriented Numerics Implementation GAIO package GAIO implementation GAIO: Global Analysis Invariant Object ([M. Dellnitz et al.]) Boxes : Generalized rectangle B(c, r) R n B(c, r) = {y R n : y i c i r i, i : 1... n} identified by a centre and vector of radii, c, r R n. Subdivision : by bisection along one of the coordinate direction. Storage of boxes: Binary tree

Computational methods: Set Oriented Numerics GAIO implementation Image of a Box: Choice of test-points The image of a box B in the collection B is defined F B (B) = {B B : f (B) B } In low dimensional phase space (d 3) N points on the edges of the boxes + the center on uniform grid within the box In higher dimension randomly distributed Remark: Rigorous choice of test point in such a way that no boxes are lost due to the discretization could be done if the Lipschitz constant of the map f is known.

OUTLINE Dynamical model CRTBP Periodic orbits Tube Dynamics Patched CRTBP approximation Computational methods: Set Oriented Numerics Covering of Invariant Sets GAIO implementation Earth to Halo Regions of prevalence Sun-Earth DPO to Earth-Moon DPO Conclusion

OUTLINE Earth to Halo Dynamical model CRTBP Periodic orbits Tube Dynamics Patched CRTBP approximation Computational methods: Set Oriented Numerics Covering of Invariant Sets GAIO implementation Earth to Halo Regions of prevalence Sun-Earth DPO to Earth-Moon DPO Conclusion

Earth to Halo Leo to Halo mission design Scientific purposes: Solar observer [ISEE, SOHO, Genesis], Lunar far-side data relay Low energy ballistic transfers made up of impulsive manoeuvres. two coupled Restricted Three-Body Problem Planar + Spatial Statement of the problem: Optimisation theory, with dynamics described by the Restricted Four-Body model - bicircular, spatial - with the Sun gravitational influence (Sun perturbed CRTBP). [R. Castelli et al.]

Mission Design Earth escape stage: Earth to Halo Halo orbit arrival Planar Sun-Earth model Launch point on LEO (167 km) Tangential manoeuvre ( V ) Spatial Earth-Moon model Stable manifold Ballistic capture to the Halo Poincaré section along a line in configuration space

Earth to Halo Transfer Points Properties of transfer points: Necessary condition for a feasible transfer: the pair of points on the section must have the same location in configuration space. The discontinuity in terms of v has to be small.

Poincaré maps Transfer Points Earth to Halo

Technique: Box approach Earth to Halo Box Covering of the EM Poincaré section Intersection with the SE Poincaré section

Earth to Halo Sample first guess trajectory First guess trajectories with J EM = 3.159738 (Az = 8000 km) and J EM = 3.161327 (Az = 10000 km) are later optimized in the bicircular Sun-perturbed EM model.

Designed trajectories Earth to Halo

SOLUTION PERFORMANCES Earth to Halo Name Type v i [m/s] v f [m/s] v t [m/s] t [days] sol.1.1 Two-Imp. 3110 214 3324 106 sol.1.2 Sing-Imp. 3161 0 3161 105 sol.2.1 Two-Imp. 3150 228 3378 128 sol.2.2 Sing-Imp. 3201 0 3201 134 Mingotti Two-Imp. 3676 65 Parker Two-Imp. 3132 618 3750 Parker Sing-Imp. 3235 3235 Mingtao Three-Imp. 3120 360 3480 17

OUTLINE Regions of prevalence Dynamical model CRTBP Periodic orbits Tube Dynamics Patched CRTBP approximation Computational methods: Set Oriented Numerics Covering of Invariant Sets GAIO implementation Earth to Halo Regions of prevalence Sun-Earth DPO to Earth-Moon DPO Conclusion

Regions of prevalence Choice of the Poincaré section How to chose the Poincaré section in the Patched CRTBP approximation? Usually: on a straight line Sometimes: on the boundary of the sphere of influence Here: The PS is set according with the prevalence of each CRTBP [R. Castelli]

Regions of prevalence The regions of prevalence How to chose the Poincaré section in the Patched CRTBP approximation? Usually: on a straight line Sometimes: on the boundary of the sphere of influence Here: The PS is set according with the prevalence of each CRTBP Comparison: Bicircular model two CRTBP SE (z) = BCP CR3BP SE, EM (z) = BCP CR3BP EM

Regions of prevalence The regions of prevalence The Regions of Prevalence of each CR3BP is defined according with by the sign of E(z) = ( SE EM )(z) For a choice of the relative phase of the primaries θ RP EM (θ) = {z C : E(z) > 0} RP SE (θ) = {z C : E(z) < 0} The curve Γ(θ) = {z C : E(z) = 0} is a closed, simple curve is defined implicitly as a function of (x, y) depends on θ changes in time. EM Region of Prevalence SE Region of Prevalence

Plot of the curve E = 0 Regions of prevalence Figure: Γ(θ) for θ = 0, 2/3π, 4/3π in SE (left) and EM (right) coordinates frame.

Regions of prevalence * [J.S. Parker]

Regions of prevalence Coupled CR3BP Approximation Choice of the Poincaré section as the boundary of the Regions of Prevalence For every value of θ The points L EM 1,2 are in the EM Region of Influence The points L SE 1,2 are in the SE Region of Influence

Trajectory design Regions of prevalence Plot of W s EM,2 (γ 1) and W u SE,2 (γ 2) until the curve Γ(π/3) is reached

Detection of the connection points Regions of prevalence Box Covering Approach, GAIO Compute the Poincaré map W s EM,2 (γ 1) Γ(θ) and cover it with Box Structures Cover the transfer region

Detection of the connection points Regions of prevalence Box Covering Approach, GAIO Compute the Poincaré map W s EM,2 (γ 1) Γ(θ) and cover it with Box Structures Cover the transfer region Intersect the Box Covering with W u SE,1,2 (γ 2)

Regions of prevalence The box covering approach Allows to close the Poincaré section Allows to control the accuracy of the intersection Find systematically all the possible intersections within a certain distance

Regions of prevalence

OUTLINE Sun-Earth DPO to Earth-Moon DPO Dynamical model CRTBP Periodic orbits Tube Dynamics Patched CRTBP approximation Computational methods: Set Oriented Numerics Covering of Invariant Sets GAIO implementation Earth to Halo Regions of prevalence Sun-Earth DPO to Earth-Moon DPO Conclusion

First stage: Leo to SE-DPO Sun-Earth DPO to Earth-Moon DPO Look for impulsive manoeuvre transfer from Leo to DPO in SE-CRTBP Integrate backwards the stable manifold Intersect the manifold with Leo Select those intersections that are tangent to Leo w.r.t geocentric coordinates

V at Leo Sun-Earth DPO to Earth-Moon DPO Remark A manoeuvre v applied in the same direction of the motion produces the maximal change of Jacobi constant. It holds J = v 2 + 2 C v z where C = Vt z 1 depends on the Leo altitude and z is the geocentric distance, V t orbital velocity The Jac.const on a Leo ( 167 Km alt.) is about 3.070352 The Jac.const. of family g is in the range [3.00014;3.00092]

Sun-Earth DPO to Earth-Moon DPO Jacobi- DPO h-leo (Km) V (m/s) 3.000464798057 220 3190 3.000464798057 160 3212

Sun-Earth DPO to Earth-Moon DPO Second stage: SE-DPO to EM-DPO Procedure for design the transfer: 1) Select two DPOs 2) Compute the Poincaré map of (un)-stable manifold on a section ( line through the Earth with slope θ SE and θ EM ). Left: Stable manifold in the interior region for a DPO in the EM-CRTBP. Right: Unstable manifold of a DPO in SE-CRTBP

Design Sun-Earth DPO to Earth-Moon DPO Procedure for design the transfer: Select two DPOs Compute the Poincaré map of (un)-stable manifold on a section 3) Write the two maps in the same system of coordinates, being θ = θ SE θ EM the relative phase of the primaries at the transfer time

Sun-Earth DPO to Earth-Moon DPO Design Procedure for design the transfer: Select two DPOs Compute the Poincaré map of (un)-stable manifold on a section Write the two maps in the same system of coordinates, being θ = θ SE θ EM the relative phase of the primaries 4) Look for possible connections on the Poicaré section Projection of the Poincaré maps onto the (x, v x) plane and (x, v y ) plane, in EM-rf

Results: Interior connection Sun-Earth DPO to Earth-Moon DPO

Results: Exterior connection Sun-Earth DPO to Earth-Moon DPO

Sun-Earth DPO to Earth-Moon DPO SE-Jacobi EM-Jacobi Time, (days) V (m/s) Interior 3.0004647980 3.02599 115 339 Exterior 3.00043012418 3.026764 116 8 Connection between two DPOs Type Start Target Time V (m/s) Mingotti Ext- Optim. LEO DPO Jac=? 90 3160 Ming Exterior LEO Retr. DPO 101 3207 Ming Interior LEO Retr. DPO 33 3802 [G. Mingotti et al.] : Earth to EM-DPO with low thrust propulsion [X. Ming at el.]: Earth to retrograde stable orbit around the Moon

OUTLINE Conclusion Dynamical model CRTBP Periodic orbits Tube Dynamics Patched CRTBP approximation Computational methods: Set Oriented Numerics Covering of Invariant Sets GAIO implementation Earth to Halo Regions of prevalence Sun-Earth DPO to Earth-Moon DPO Conclusion

Conclusion Conclusion Dynamical model The CRTBP is introduced to model the dynamics The families of periodic orbits have been investigated The invariant manifolds provide low energy transfers in the phase space Design technique: The patched CRTBP approximation has been formalized. Immediate definition of the transfer points in the phase space through the box covering approach. A technique to design impulsive transfers has been developed. Designed trajectory: Efficient trajectories in terms of v have been designed in the planar and spatial case A non-classical Poincaré section has been presented in terms of Regions of prevalence.

Conclusion REFERENCES E. J. Doedel, R. C. Paffenroth, H. B. Keller, D. J. Dichmann, J. Galan, A. Vanderbauwhede, Computation of periodic solutions in conservative systems with application to the 3-Body problem, Int. J. Bifurcation and Chaos, Vol. 13(6), 2003, 1-29 W. S. Koon, M. W. Lo, J. E. Marsden, E. Jerrold, and S. D. Ross. Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics Chaos, 10(2):427 469, 2000. G. Gómez, W.S. Koon, M.W. Lo, J.E. Marsden, J. Masdemont and S.D. Ross Invariant Manifolds, the Spatial Three-Body Problem and Space Mission Design Advances in the Astronautical Sciences, 2002 W. S. Koon, M. W. Lo, J. E. Marsden and S. D. Roos, Low energy transfer to the Moon Celestial Mech. Dynam. Astronom, Vol 81, pp 63-73, 2001 R. Castelli, G. Mingotti, A. Zanzottera and M. Dellnitz, Intersecting Invariant Manifolds in Spatial Restricted Three-Body Problems: Design and Optimization of Earth-to-Halo Transfers in the Sun Earth Moon Scenario, submitted to Commun. Nonlinear Sci. Numer. Simulat.

Conclusion REFERENCES R. Castelli Regions of Prevalence in the Coupled Restricted Three-Body Problems Approximation Submitted to Commun. Nonlinear Sci. Numer. Simulat. M. Dellnitz; G. Froyland; O. Junge The algorithms behind GAIO Set oriented numerical methods for dynamical systems B. Fiedler (ed.): Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pp. 145-174, Springer, 2001 M. Dellnitz and O. Junge Set Oriented Numerical Methods for Dynamical Systems Handbook of dynamical systems, 2002 J.S. Parker Families of low-energy lunar halo trasfer Proceedings of the AAS/AIAA Space Flight Mechanics Meeting, pp 483 502, 2006. G.Mingotti, F. Topputo, and F. Bernelli-Zazzera, Exploiting Distant Periodic Orbits and their Invariant Manifolds to Design Novel Space Trajectories to the Moon, Proceedings of the 20th AAS/AIAA Space Flight Mechanics Meeting, San Diego, California, 14-17 February, 2010 Ming X. and Shijie X. Exploration of distant retrograde orbits around Moon, Acta Astronautica, Vol.65, pp. 853 850, 2009

Conclusion Thank you

Conclusion Dynamical model Mission Design Set oriented Numerics References