Design of Low Energy Space Missions

using Dynamical Systems Theory

Koon, Lo, Marsden, and Ross

W.S. Koon (Caltech) and S.D. Ross (USC)

CIMMS Workshop, October 7, 2004

1

� Acknowledgements

I H. Poincare, J. Moser

I C. Conley, R. McGehee, D. Appleyard

I C. Simo, J. Llibre, R. Martinez

I B. Farquhar, D. Dunham

I E. Belbruno, B. Marsden, J. Miller

I K. Howell, B. Barden, R. Wilson

I S. Wiggins, V. Rom-Kedar, F. Lekien

� Outline

I Main Theme

• how to use dynamical systems theory of 3-body problemin low energy trajectory design.

I Background and Motivation:

• NASA’s Genesis Discovery Mission.

• A Low Energy Tour of Jupiter’s Moons.

I Restricted 3-Body Problem.

I Main Results.

I Ongoing Work.

• Low Thrust Trajectories in a Multi-Body Environment.

• Parking a Satellite near an Asteroid Pair.

� Motivation: Genesis Discovery Mission

I Genesis spacecraft

• collected solar wind sample from a L1 halo orbit,

• returned them to Earth.

I Halo orbit, transfer/ return trajectories in rotating frame.

L1 L2

x (km)

y(k

m)

-1E+06 0 1E+06

-1E+06

-500000

0

500000

1E+06

� Motivation: Genesis Discovery Mission

I Designed using dynamical systems theory(Barden, Howell, and Lo).

I Followed natural dynamics, little propulsion after launch.

I Return-to-Earth portion utilized heteroclinic dynamics.

L1 L2

x (km)

y(k

m)

-1E+06 0 1E+06

-1E+06

-500000

0

500000

1E+06

� Motivation: Petit Grand Tour of Jupiter’s Moons

I Construct a low energy trajectoryto visit several moons in one mission.

I Instead of flybys, can orbit each moon for any duration.

I NASA is considering a Jupiter Icy Moon Orbiter (JIMO).

� Design Strategy: Patched 3-Body Solutions

I Jupiter-Ganymede-Europa-SC 4-body system approximatedas 2 coupled 3-body systems

I 3-body solutions of each 3-body systemsare linked in right order to construct orbit with desired itinerary.

I Try to minimize ∆V at each transfer patch point.

I Initial solution refined in 4-body model.

I 3-body solutions offer a large class of low energy trajectories.

Jupiter Europa

Ganymede

Spacecrafttransfertrajectory

∆V at transferpatch point

Jupiter Europa

Ganymede

� Planar Circular Restricted 3-Body Problem

I 2 main bodies

• Total mass normalized to 1: mJ = µ, mS = 1− µ.

• Rotate about center of mass, angular velocity normalized to 1.

I Choose rotating coordinate system with origin at center of mass,2 main bodies fixed at (−µ, 0) and (1− µ, 0).

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

x (nondimensional units, rotating frame)

y (n

ondi

men

sion

al u

nits

, rot

atin

g fr

ame)

mS = 1 - µ mJ = µ

S J

Jupiter's orbit

L2

L4

L5

L3 L1

comet

� Equilibrium Points

I Equations of motion for SC are

x− 2y = −∂U

∂x, y + 2x = −∂U

∂y,

where U(x, y) = −x2+y2

2 − 1−µrs

− µrj

.

I Five equilibrium points:

• 3 unstable collinear equilbrium points, L1, L2, L3.

• 2 equilateral equilibrium points, L4, L5.

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

x (nondimensional units, rotating frame)

y (n

ondi

men

sion

al u

nits

, rot

atin

g fr

ame)

mS = 1 - µ mJ = µ

S J

Jupiter's orbit

L2

L4

L5

L3 L1

comet

� Hill’s Realm

I Energy integral: E(x, y, x, y) = (x2 + y2)/2 + U(x, y).

I E can be used to determine (Hill’s ) realm in position spacewhere SC is energetically permitted to move.

I Effective potential: U(x, y) = −x2+y2

2 − 1−µrs

− µrj

.

� Hill’s Realm

I To fix energy value E is to fix height of plot of U(x, y).Contour plots give 5 cases of Hill’s realm.

-1 0 1

-1

0

1

-1 0 1

-1

0

1

-1 0 1

-1

0

1

-1 0 1

-1

0

1

S J S J

S JS J

Case 1 : C>C1 Case 2 : C1>C>C2

Case 4 : C3>C>C4=C5Case 3 : C2>C>C3

� The Flow near L1 and L2

I For energy value just above that of L2,Hill’s realm contains a “neck” about L1 & L2.

I SC can make transition through these equilibrium realms.

I 4 types of orbits:periodic, asymptotic, transit & nontransit.

x (rotating frame)

y (

rota

tin

g f

ram

e)

J ML1

x (rotating frame)

y (

rota

tin

g f

ram

e)

L2

ExteriorRegion

InteriorRegion

MoonRegion

ForbiddenRegion

L2

� Invariant Manifold as Separatrix

I Asymptotic orbits form 2D invariant manifold tubesin 3D energy surface.

I They separate transit and non-transit orbits:

• Transit orbits are those inside the tubes.

•Non-transit orbits are those outside the tubes.

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-0.06

-0.04

-0.02

0

0.02

0.04

0.06

x (rotating frame)

y (r

otat

ing

fram

e)

L1 PeriodicOrbit

StableManifold

UnstableManifold

Jupiter

StableManifold Forbidden Region

Forbidden RegionUnstableManifold

Moon

� Invariant Manifold as Separatrix

I Invariant Manifold Tubes associated with periodic orbits aboutL1, L2 control ballistic capture and escape.

Moon

L2Ballistic

Capture Into Elliptical Orbit

EarthL2

orbit

Moon

P

� Heteroclinic Connection

I Found heteroclinic connection between pair of periodic orbits.

I Found a large class of orbits near this (homo/heteroclinic) chain.

I SC can follow these channels in rapid transition.

-6 -4 -2 0 2 4 6

-6

-4

-2

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6

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y (A

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r rot

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x (AU, Sun-Jupiter rotating frame)

y (A

U, S

un-J

upite

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atin

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� Existence of Transitional Orbits

I Main Theorem: For any admissible itinerary,e.g., (. . . ,X,J;S,J,X, . . .), there exists an orbit whosewhereabouts matches this itinerary.

I Can even specify number of revolutions the comet makesaround Sun & Jupiter (plus L1 & L2).

I 3-Body trajectories much richer than 2-body trajectories.

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� Numerical Construction of Orbits

I Developed procedure to construct orbitwith prescribed itinerary.

I Example: An orbit with itinerary (X,J;S,J,X).

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� Construction of (M,X;M, I,M) Orbits

I Invariant mfd. tubes (S × I) separate transit/nontransit orbits.

I Red curve (S1) (Poincare cut of L2 unstable manifold).Green curve (S1) (cut of L1 stable manifold).

I Any point inside the intersection region ∆M is a (X ; M, I) orbit.

∆M = (X;M,I)

Intersection Region

0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

x (Jupiter-Moon rotating frame)

y (

Jupit

er-M

oon r

ota

ting f

ram

e)

ML1 L2

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

-0.06

-0.04

-0.02

0

0.02

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y (������� � ��� �������

otating frame)

(X;M)

(;M,I)

y (

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Forbidden Region

Forbidden Region

Stable

Manifold

Unstable

Manifold

Stable

Manifold Cut

Unstable

Manifold Cut

� Construction of (M,X;M,I,M) Orbits

I The desired orbit can be constructed by

• Choosing appropriate Poincare sections and

• linking invariant manifold tubes in right order.

X

I M

L1 L2M

U3

U2

U1U4

U3

U2

� Petit Grand Tour of Jupiter’s Moon

I Petit Grand Tour can be constructed similarly

• Approximate 4-body system as 2 nested 3-body systems.

• Choose appropriate Poinare section.

• Link invariant manifold tubes in right order.

• Refine initial solution in 4-body model.

Jupiter Europa

Ganymede

Spacecrafttransfertrajectory

∆V at transferpatch point

-1. 5 -1. 4 -1. 3 -1. 2 -1. 1 -1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

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2

Transferpatch point

� Petit Grand Tour of Jupiter’s Moons (Planar Model)

I Used invariant manifoldsto construct trajectories with interesting characteristics:

• Petit Grand Tour of Jupiter’s moons.1 orbit around Ganymede. 4 orbits around Europa.

• A ∆V nudges the SC fromJupiter-Ganymede system to Jupiter-Europa system.

I Instead of flybys, can orbit several moons for any duration.

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Ganymede

���

Jupiter

Jupiter

Europa’sorbit

Ganymede’sorbit

Transferorbit

∆V���

Europa

Jupiter

� Extend from Planar Model to Spatial Model

Ganymede's orbit

Jupiter

0.98

0.99

1

1.01

1.02

-0.02

-0.01

0

0.01

0.02

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

xy

z

0.99

0.995

1

1.005

1.01 -0.01

-0.005

0

0.005

0.01

-0.01

-0.005

0

0.005

0.01

y

x

z

Close approachto Ganymede

Injection intohigh inclination

orbit around Europa

Europa's orbit

(a)

(b) (c)

-1. 5

-1

-0. 5

0

0.5

1

1.5

-1. 5

-1

-0. 5

0

0.5

1

1.5

x

yz

� Look for Natural Pathways to Bridge the Gap

I Tubes of two 3-body systems may not intersect for awhile.May need large ∆V to “jump” from one tube to another.

I Look for natural pathways to bridge the gap

• between z0 where tube of one system (Ganymede) entersand z2 where tube of another system exits (into Europa realm)by “hopping” through phase space (z1).

Jupiter Europa

Ganymede

Spacecrafttransfertrajectory

∆V at transferpatch point

f1

f2

f12

f2

f1z0

z1z2

z3z4

z5 U2

U1

Exit

Entrance

(a) (b)

� Transport in Phase Space via Tube & Lobe Dynamics

I By using

• tubes of rapid transition that connect realms

• lobe dynamics to hop through phase space,

New tour only needs ∆V = 20m/s (50 times less).

f1

f2

f12

f2

f1z0

z1z2

z3z4

z5 U2

U1

Exit

Entrance

Low Energy Tour of Jupiter’s MoonsSeen in Jovicentric Inertial Frame

Jupiter

Callisto Ganymede Europa

(a) (b)

� Lobe Dynamics: Mixed Phase Space

I Poincare section reveals mixed phase space:

• resonance regions and

• ‘‘chaotic sea”.

L1 L2

Exterior Realm

Interior (Jupiter) Realm

Moon Realm

Forbidden Realm(at a particular energy level)

spacecraft

Poincare sectionL1 L2

Identify

Argument of Periapse (radians)Semim

ajo

r Axis

(Neptune =

1)

(a) (b)

� Transport between Regions via Lobe Dynamics

I Invariant manifolds divide phase space into resonance regions.

I Transport between regions can be studied via lobe dynamics.

Identify

Argument of Periapse

Semim

ajo

r Axis

R1

R2

q0

pipj

f -1(q0)q1

L2,1(1)

L1,2(1)

f (L1,2(1))

f (L2,1(1))

� Transport between Regions via Lobe Dynamics

I Segments of unstable and stable manifoldsform partial barriers between regions R1 and R2.

I L1,2(1), L2,1(1) are lobes; they form a turnstile.

• In one iteration, only points from R1 to R2 are in L1,2(1)

• only points from R2 to R1 are in L2,1(1).

I By studying pre-images of L1,2(1),one can find efficient way from R1 to R2.

R1

R2

q0

pipj

f -1(q0)q1

L2,1(1)

L1,2(1)

f (L1,2(1))

f (L2,1(1))

� Hopping through Resonaces in Low Energy Tour

I Guided by lobe dynamics, hopping through resonances(essential for low energy tour) can be performed.

I To get SC captured by secondary (m2),need to decrease semi-major axis passing through resonances.

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50.65

0.7

0.75

0.8

P

m1

m2

Surface-of-section Large orbit changes

� Tube/Lobe Dynamics: Transport in Solar System

I To use tube dynamics/lobe dynamics of spatial 3-body problemto systematically design low-fuel trajectory.

I Part of our program to study transport in solar systemusing tube and lobe dynamics.

f1

f2

f12

f2

f1z0

z1z2

z3z4

z5 U2

U1

Exit

Entrance

Low Energy Tour of Jupiter’s MoonsSeen in Jovicentric Inertial Frame

Jupiter

Callisto Ganymede Europa

(a) (b)

� Low Thrust Trajectories in a Multi-Body Environment

I Incorporation of low thrust.

I Design to take best advantage of natural dynamics.

I See Shane D. Ross.

Spiral out from Europa Europa to Io transfer

� Parking a Satellite near an Asteroid Pair

I Find stable periodic/quasi-periodic orbits for SCto observe binary as it orbits the Sun.

•Model for asteroid pair:sphere and rigid body (3 connected masses).

•Model for SC motion: binary in relative equilibrium.

I See Gabern, Koon, and Marsden [2004].

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0 0.02 0.04 0.06 0.08

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