dynamical system theory and numerical methods applied to ...€¦ · outline dynamical model crtbp...

Dynamical system theory and numerical methodsapplied to Astrodynamics

Roberto Castelli

Institute for Industrial Mathematics

University of Paderborn

BCAM, Bilbao, 20th December, 2010

AstroNet Dynamical system theory for mission design Roberto Castelli 1 / 70

Introduction

Introduction

In space mission design

I Consider the Force Field acting on the Spacecraft

I Consider Physical and Technical constraints

I Satisfy some mission requirements

I Take care of the fuel consumption and the travelling time

I ....

Genesis Mission


Introduction

Introduction

N-BODY PROBLEM⇓

First guess trajectoriesdesigned in simplified model

I Two-body model

I Restricted Three-bodyproblem

I Bicircular model

I . . .

Numerical Optimisation in Fullsystem

I Direct/Indirect methods

I Multiple shootingtechnique

I Multiobjectiveoptimisation

Different type of Propulsion (Electric - Chemical)

⇓

Low thrust propulsion – Impulsive manoeuvre


Introduction

OUTLINE

Dynamical model CRTBPPeriodic orbitsTube DynamicsPatched CRTBP approximation

Computational methods: Set Oriented NumericsCovering of Invariant SetsGAIO implementation

Examples of mission designEarth to HaloRegions of prevalenceSun-Earth DPO to Earth-Moon DPO

Conclusion


Dynamical model CRTBP

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Conclusion



Circular Restricted Three-Body problem

I Two Primaries move in circular orbits under the mutual gravitationalattraction

I Massless particle moves under the gravitational influence of twoprimaries

In a rotating, adimensional reference frame, µ = m2/(m1 + m2),

(CR3BP)

x − 2y = Ωx

y + 2x = Ωy

z = Ωz

Ω(x , y , z) =12(x2+y2)+ 1−µ

r1+ µ

r2+ 1

2µ(1−µ)



Properties of CRTBP

I Non integrable Autonomous Hamiltonian SystemI Symmetry (x , y , z , x , y , z ; t)→ (x ,−y , z ,−x , y ,−z ;−t)I Jacobi Integral: C = 2Ω(x , y , z)− (x2 + y2 + z2) = −2EI Equilibrium points: Lagrangian Points Lj , j = 1, ..., 5.I Hill’s Region: H(C ) = (x , y , z) : 2Ω(x , y , z)− C ≥ 0


Dynamical model CRTBP Periodic orbits

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Conclusion



Families of Periodic Orbits.

Hamiltonian system⇓

continuous families of periodicorbits.

PO bifurcating from L1 [E. J. Doedel et al.]



Families of Periodic Orbits.

Hamiltonian system⇓

continuous families of periodicorbits.

PO bifurcating from L1 [E. J. Doedel et al.]

Simple Symmetric periodic orbits

Differential correction scheme,based on the variational eq.Find δVy such that the firstx-axis crossing of

φt(X0, 0, 0,Vy + δVy)

is perpendicularAstroNet Dynamical system theory for mission design Roberto Castelli 10 / 70


Periodic Orbits Diagram

Diagram of Simple Symmetric PO in SE system.

(X0,Vy) → (X0, 0, 0,Vy ),Earth < X0 < L2, Vy > 0

Lyapunov and Halo orbits



Periodic orbits: DPO



Resonant Orbits

For which resonances there exist families of periodic orbits?(Collaboration with Prof. P. Zgliczynski)


Dynamical model CRTBP Tube Dynamics

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Conclusion



Dynamics near periodic orbits

I The periodic orbits separates two necks in the Hill’s region

I Linear Dynamics: saddle × center

I 3 types of orbits: asymptotic, transit, non-transit

[W.S. Koon et al.]



Invariant manifolds

The Stable/Unstable Invariant manifoldsSet of orbits asymptotic to the periodic orbit for t → ±∞

[G. Gomez at al.]

I are topologically equivalent toN − 2 dimensional cylinders inthe N − 1 dim. energy manifold

I act as separatrices in the phasespace between transit andnon-transit orbit



Invariant manifolds

The Stable/Unstable Invariant manifoldsSet of orbits asymptotic to the periodic orbit for t → ±∞

I are topologically equivalent toN − 2 dimensional cylinders inthe N − 1 dim. energy manifold

I act as separatrices in the phasespace between transit andnon-transit orbit

I approach the smaller primary

I tangent to the eigenspace of thelinearized system (monodromymatrix)


Dynamical model CRTBP Patched CRTBP approximation

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Conclusion



Mission Design: dynamical system theory

Dynamical system theory in low energy trajectory designPatched 3-body problem

• The 4-Body system isapproximated with thesuperpositions of two RestrictedThree-Body problems

• The invariant manifold structuresare exploited to design legs oftrajectory

• The design restricts to theselection of a connection point ona suitable Poincare section



Some examples

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.])


Computational methods: Set Oriented Numerics

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Conclusion



Set Oriented Numerics

Main Goal: Study the long termbehavior of complex and chaoticDynamical Systems

2010

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Set Oriented Numerics

I Computation of several short term trajectories instead of single longterm trajectory

I Approximation of Global structureI Invariant Sets : global attractors, Invariant manifoldsI Invariant measures, almost invariant setsI Transport operatorsI Multiobjective optimization (Pareto set)



Methodology

Consider a discrete dynamical system

xk+1 = f (xk), k = 0, 1, 2, . . . , f : Rn → Rn

Aim: Approximation of a structure within a bounded set Q.Method: Generate a sequence of collections B1,B2 . . . of subsets of Q s.tB0 = Q, and iteratively Bk from Bk−1

I Subdivision: define a new collection Bk such that⋃B∈Bk

B =⋃

B∈Bk−1

B

diam(Bk) ≤ θkdiam(Bk−1), θk ∈ (0, 1)

I Selection: define Bk as

Bk = B ∈ Bk : such that g(B) ∩ B 6= ∅ for some B ∈ Bk


Computational methods: Set Oriented Numerics Covering of Invariant Sets

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Conclusion



Relative Global Attractor

Definition Relative Global attractor

Let Q ⊂ Rn be a compact set. Define global attractor relative to Q by

AQ =⋂j≥0

f j(Q)

PropertiesAQ ⊂ Qf −1(AQ) ⊂ AQ but not necessarily f (AQ) ⊂ AQ .

Selection step: define Bk as

Bk = B ∈ Bk : such that f −1(B) ∩ B 6= ∅ for some B ∈ Bk



Henon map xk+1 = 1− ax2

k + byk a = 1.4, b = 0.3yk+1 = xk

Covering of the attractor relative to [−2, 2]2, k = 8, 12, 16, 20



Computing the unstable manifold

Initialization - Continuation Algorithm

Aim: Compute the unstable manifold of a point p into a (large ) compactset Q, ( p ∈ Q)

1 Given Q, compute P0,P1, . . .Pl nested sequence of fine partitions ofQ. Select the element C ∈ Pl such that p ∈ C and AC = W u

loc(p)∩C

2 Initialization Starting from B0 = C, refine the approximation of

W uloc(p) ∩ C by subdivision, yielding B(0)

k ⊂ Pl+k

3 Continuation From B(j−1)k compute

B(j)k = B ∈ Pl+k : f (B ′) ∩ B 6= ∅, for some B ′ ∈ B(j−1)

k



Lorenz systemx = σ(y − x)y = ρx − y − xzz = −βz + xy

(0, 0, 0) is fixed pointσ = 10, ρ = 28, β = 8/3

Covering of the two-dimensional stable manifold of the originLeft: 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]



Box covering of the part of unstable manifold of an Halo orbit in theSun-Earth CRTBP.


Computational methods: Set Oriented Numerics GAIO implementation

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Conclusion



Implementation – GAIO package

GAIO: Global Analysis Invariant Object ([M. Dellnitz et al.])

Boxes : Generalized rectangle B(c , r) ⊂ Rn

B(c , r) = y ∈ Rn : |yi − ci | ≤ ri , i : 1 . . . n

identified by a centre and vector of radii, c , r ∈ Rn.

Subdivision : by bisection along one of the coordinate direction.

Storage of boxes: Binary tree



Image of a Box: Choice of test-points

The image of a box B in the collection B is defined

FB(B) = B ′ ∈ B : f (B) ∩ B ′ 6= ∅

In low dimensional phase space (d ≤ 3)I N points on the edges of the boxes + the centerI on uniform grid within the box

In higher dimensionI 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.AstroNet Dynamical system theory for mission design Roberto Castelli 32 / 70

Examples of mission design

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Conclusion


Examples of mission design Earth to Halo

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Conclusion



Leo to Halo mission design

I Scientific purposes: Solar observer [ISEE, SOHO, Genesis], Lunarfar-side data relay

I Low energy ballistic transfers made up of impulsive manoeuvres.

I two coupled Restricted Three-Body Problem Planar + Spatial

I 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:

Planar Sun-Earth modelLaunch point on LEO (167 km)Tangential manoeuvre (∆V )

Halo orbit arrival

Spatial Earth-Moon modelStable manifoldBallistic capture to the Halo

Poincare section along a line in configuration space



Transfer Points

I Properties of transfer points:

I Necessary condition for a feasible transfer:

the pair of points on the section must have

the same location in configuration space.

I The discontinuity in terms of ∆v has to be small.



Poincare maps →Transfer Points



Technique: Box approach

Box Covering of the EMPoincare section

Intersection with the SEPoincare section



Sample first guess trajectory

I First guess trajectories with JEM = 3.159738 (Az = 8000 km)

and JEM = 3.161327 (Az = 10000 km) are later optimized

in the bicircular Sun-perturbed EM model.



Designed trajectories



SOLUTION PERFORMANCES

Name Type ∆vi [m/s] ∆vf [m/s] ∆vt [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


Examples of mission design Regions of prevalence

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Conclusion



Choice of the Poincare section

How to chose the Poincare section in the Patched CRTBP approximation?

I Usually: on a straight line

I Sometimes: on the boundary of the sphere of influence

I Here: The PS is set according with the prevalence of each CRTBP[R. Castelli]



The regions of prevalence

How to chose the Poincare section in the Patched CRTBP approximation?

I Usually: on a straight line

I Sometimes: on the boundary of the sphere of influence

I Here: The PS is set according with the prevalence of each CRTBP

Comparison: Bicircular model ⇔ two CRTBP∆SE (z) =‖ BCP − CR3BPSE ‖, ∆EM(z) =‖ BCP − CR3BPEM ‖



The regions of prevalence

The Regions of Prevalence of each CR3BP is defined according with bythe sign of

∆E (z) = (∆SE −∆EM)(z)

For a choice of the relative phase of the primaries θ

RPEM(θ) = z ∈ C : ∆E (z) > 0 EM Region of Prevalence

RPSE (θ) = z ∈ C : ∆E (z) < 0 SE Region of Prevalence

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.



Plot of the curve ∆E = 0

Figure: Γ(θ) for θ = 0, 2/3π, 4/3π in SE (left) and EM (right) coordinatesframe.



* [J.S. Parker]



Coupled CR3BP Approximation

Choice of the Poincare section as the boundary of the Regions of Prevalence

For every value of θ

• The points LEM1,2 are in

the EM Region ofInfluence

• The points LSE1,2 are in

the SE Region ofInfluence



Trajectory design

Plot of W sEM,2(γ1) and W u

SE ,2(γ2) until the curve Γ(π/3) is reached



Detection of the connection points

Box Covering Approach, GAIOI Compute the Poincare map W s

EM,2(γ1) ∩ Γ(θ) and cover it with BoxStructures

I Cover the transfer region



Detection of the connection points

Box Covering Approach, GAIOI Compute the Poincare map W s

EM,2(γ1) ∩ Γ(θ) and cover it with BoxStructures

I Cover the transfer regionI Intersect the Box Covering with W u

SE ,1,2(γ2)



The box covering approach

I Allows to close the Poincare section

I Allows to control the accuracy of the intersection

I Find systematically all the possible intersections within a certaindistance


Examples of mission design Sun-Earth DPO to Earth-Moon DPO

OUTLINE




Conclusion



First stage: Leo to SE-DPO

Look for impulsive manoeuvre transfer from Leo to DPO in SE-CRTBP

I Integrate backwards thestable manifold

I Intersect the manifoldwith Leo

I Select those intersectionsthat are tangent to Leow.r.t geocentriccoordinates



∆V at Leo

Remark A manoeuvre |v | appliedin the same direction of the motionproduces the maximal change ofJacobi 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, Vt orbital

velocity

The Jac.const on a Leo ( 167 Km alt.) is about 3.070352The Jac.const. of family g is in the range [3.00014;3.00092]



Jacobi- DPO h-LEO (Km) ∆V (m/s)

3.000464798057 220 3190

3.000464798057 160 3212



Second stage: SE-DPO to EM-DPO

Procedure for design the transfer:

1) Select two DPOs

2) Compute the Poincare map of (un)-stable manifold on a section( line through the Earth with slope θSE and θEM).

Left: Stable manifold in the interior region for a DPO in the EM-CRTBP.

Right: Unstable manifold of a DPO in SE-CRTBP



Design


I Select two DPOsI Compute the Poincare map of (un)-stable manifold on a section

3) Write the two maps in the same system of coordinates, beingθ = θSE − θEM the relative phase of the primaries at the transfer time



Design


I Select two DPOsI Compute the Poincare map of (un)-stable manifold on a sectionI Write the two maps in the same system of coordinates, beingθ = θSE − θEM the relative phase of the primaries

4) Look for possible connections on the Poicare section

Projection of the Poincare maps onto the (x , vx) plane and (x , vy ) plane, in EM-rf



Results: Interior connection



Results: Exterior connection



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


Conclusion

OUTLINE




Conclusion


Conclusion

Conclusion

I Dynamical model

I The CRTBP is introduced to model the dynamics

I The families of periodic orbits have been investigated

I The invariant manifolds provide low energy transfers in the phase space

I Design technique:

I The patched CRTBP approximation has been formalized.

I Immediate definition of the transfer points in the phase space throughthe box covering approach.

I A technique to design impulsive transfers has been developed.

I Designed trajectory:

I Efficient trajectories in terms of ∆v have been designed in the planarand spatial case

I A non-classical Poincare section has been presented in terms of Regionsof 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 withapplication 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. Heteroclinicconnections between periodic orbits and resonance transitions in celestial mechanicsChaos, 10(2):427–469, 2000.

G. Gomez, 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 tothe Moon – Celestial Mech. Dynam. Astronom, Vol 81, pp 63-73, 2001

R. Castelli, G. Mingotti, A. Zanzottera and M. Dellnitz, Intersecting InvariantManifolds in Spatial Restricted Three-Body Problems: Design and Optimization ofEarth-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 ProblemsApproximation Submitted to Commun. Nonlinear Sci. Numer. Simulat.

M. Dellnitz; G. Froyland; O. Junge The algorithms behind GAIO – Set orientednumerical 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 DynamicalSystems – Handbook of dynamical systems, 2002

J.S. Parker– Families of low-energy lunar halo trasfer – Proceedings of theAAS/AIAA Space Flight Mechanics Meeting, pp 483–502, 2006.

G.Mingotti, F. Topputo, and F. Bernelli-Zazzera, Exploiting Distant Periodic Orbitsand 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, ActaAstronautica, Vol.65, pp. 853–850, 2009


Conclusion

Thank you


Conclusion

⇒ Dynamical model

⇒ Set oriented Numerics

⇒ Mission Design

⇒ References


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