1

Dynamical Systems,and Space Mission Design

Jerry Marsden

Martin Lo (JPL), Wang-Sang Koon and Shane Ross (Caltech)

Control and Dynamical Systems and JPLCalifornia Institute of Technology

CDS 270-1, Lecture 1, March 27, 2000

marsden@cds.caltech.eduhttp://www.cds.caltech.edu/˜marsden/

Control and Dynamical Systems

2

Structure of the Talk

• Part 1 : Homoclinic and heteroclinic structures related to thecollinear libration points in the the three body problem and itsrelation to resonant transition of comets and the Genesis mission.

• Part 2 : Petit Grand Tour of the Moons of Jupiter (Ganymedeand Europa) and Lunar Capture (Efficient missions to the Moon).

• Part 3 : Optimal control for halo orbit insertion. (Collaborationwith Radu Serban, Linda Petzold and Roby Wilson)

3

Acknowledgments

• H. Poincare, J. Moser

• C. Conley, R. McGehee

• R. Farquhar et al.

• C. Simo, J. Libre, R. Martinez

• G. Gomez, J. Masdemont

• E. Belbruno, B. Marsden, J. Miller

• K. Howell and the Purdue group

• P. Chodas, D. Yeomans, A. Chamberlin, R. Wilson

• Genesis mission design team

• M. Dellnitz & Paderborn dynamical systems group

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More Information and References

• Part 1:

Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross [1999]Heteroclinic connections between periodic orbitsand resonance transitions in celestial mechanics ,

available fromhttp://www.cds.caltech.edu/ ˜marsden/

• Parts 2 and 3 are covered in papers in preparation.

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Connecting Orbits

� Simple Pendulum

• Equations of a simple pendulum are θ + sin θ = 0.

• Write as a system in the plane ;

dθ

dt= v

dv

dt= − sin θ

• Solutions are trajectories in the plane.

• The resulting phase portrait shows some important basic fea-tures:

connecting (homoclinic) orbit

saddlepoint

upright pendulum (unstable point)

downward pendulum (stable point)

θ

θ = v

7

� Higher Dimensional Versions are Invariant Manifolds

Stable Manifold

Unstable Manifold

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� Periodic Orbits

• Can replace fixed points by periodic orbits and do similar things.For example, stability means nearby orbits stay nearby.

x0

periodic orbit

nearby trajectory winding towards the periodic orbit

9

� Invariant Manifolds for Periodic Orbits

• Periodic orbits have stable and unstable manifolds .

Unstable Manifold (orbits move away from the periodic orbit)

Stable Manifold (orbits move toward the periodic orbit)

10

Resonant Transitions

� Jupiter Comets–such as Oterma

• Comets moving in the vicinity of Jupiter do so mainly under theinfluence of Jupiter and the Sun–i.e., in a three body problem.

• These comets sometimes make a rapid transition from out-side to inside Jupiter’s orbit.

• Captured temporarily by Jupiter during transition.

• Exterior (2:3 resonance) → Interior (3:2 resonance).

• The next figure shows the orbit of Oterma (AD 1915–1980) in aninertial frame

-8 -6 -4 -2 0 2 4 6 8

-8

-6

-4

-2

0

2

4

6

8

x (AU, Sun-centered inertial frame)

y (A

U, S

un-c

ente

red i

nert

ial

fram

e)

Jupiter’s orbit

Sun

END

START

First encounter

Second encounter

3:2 resonance

2:3 resonance

Oterma’s orbit

Jupiter

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• Next figure shows Oterma’s orbit in a rotating frame (so Jupiterlooks like it is standing still) and with some invariant manifolds inthe three body problem superimposed.

-8 -6 -4 -2 0 2 4 6 8

-8

-6

-4

-2

0

2

4

6

8

4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6

-0.8

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0

0.2

0.4

0.6

0.8

y (A

U, S

un-J

upite

r ro

tatin

g fr

ame)

x (AU, Sun-Jupiter rotating frame)

L1

(b)

y (A

U, S

un-J

upite

r ro

tatin

g fr

ame)

x (AU, Sun-Jupiter rotating frame)(a)

L2

Jupiter

Sun

Jupiter

Forbidden Region

Forbidden Region

1980

1910

Oterma’sTrajectory

Comet Oterma’s Trajectory (1910 to 1980)

Orbit of Oterma in a Rotating Frame

-8 -6 -4 -2 0 2 4 6 8

-8

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-4

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0

2

4

6

8

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y (A

U, S

un-J

upite

r ro

tatin

g fr

ame)

x (AU, Sun-Jupiter rotating frame)

L1

(b)

y (A

U, S

un-J

upite

r ro

tatin

g fr

ame)

x (AU, Sun-Jupiter rotating frame)(a)

L2

Sun

Jupiter

Forbidden Region

Forbidden Region

1980

1910

Oterma’sTrajectory

Comet Oterma’s Trajectory (1910 to 1980)

Orbit of Oterma in a Rotating Framewith the Homoclinic-Heteroclinic Chain

L2 Homoclinic Orbit

L1 Homoclinic Orbit

Homoclinic-Heteroclinic Dynamical ChainL1 Liapunov Orbit Homoclinic Trajectory

L2 Liapunov Orbit Homoclinic Trajectory

Heteroclinic Trajectory Connecting L1 and L2 Liapunov Orbits

HeteroclinicConnection

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• Now lets look at two movies of the trajectory of cometOterma , first in an inertial frame and then in a frame ro-tating with the sun and Jupiter .

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Movie: Oterma in inertial

frame

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Movie: Oterma in a

rotating frame

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The Planar Restricted Three Body Problem–PCR3BP

� General Comments

• The two main bodies could be the Sun and Jupiter , or theSun and Earth , etc. The total mass is normalized to 1; theyare denoted mS = 1 − µ and mJ = µ, so 0 < µ < 1.

◦ The two main bodies rotate in the plane in circles counterclock-wise about their common center of mass and with angular ve-locity ω (also normalized to one).

◦ The third body, the comet or the spacecraft , has mass zeroand is free to move in the plane.

• The planar restricted three body problem is used for simplicity.Generalization to the three dimensional problem is of courseimportant, but many of the effects can be described well with theplanar model.

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� Equations of Motion

• Notation: Choose a rotating coordinate system so that

◦ the origin is at the center of mass

◦ the Sun and Jupiter are on the x-axis at the points (−µ, 0)and (1 − µ, 0) respectively–i.e., the distance from the sun toJupiter is normalized to be unity.

◦ Let (x, y) be the position of the comet in the plane relative tothe positions of the Sun and Jupiter.

◦ distances to the Sun and Jupiter:

r1 =√

(x + µ)2 + y2 and r2 =√

(x − 1 + µ)2 + y2.

-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 е mJ = µ

Sun

Jupiter’s orbit

comet

x = — µ

Jupiter

x = 1 — µ

r1

r2

y

x

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• Kinetic energy (wrt inertial frame) in rotating coordinates:

K(x, y, x, y) =1

2

[(x − ωy)2 + (y + ωx)2

]

• The Lagrangian is K.E. − P.E., given by

L(x, y, x, y) = K(x, y, x, y) − V (x, y),

where

V (x, y) = −1 − µ

r1− µ

r2.

• Euler-Lagrange equations :

x − 2ωy = −∂Vω

∂x, y + 2ωx = −∂Vω

∂y

where the augmented potential is

Vω = V − ω2(x2 + y2)

2.

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• Legendre transform to get Hamiltonian form .

• The Hamiltonian ( �= K.E. + P.E.) is

H =(px + ωy)2 + (py − ωx)2

2+ Vω(x, y),

• Relationship between momenta and velocities :

x =∂H

∂px= px + ωy; y =

∂H

∂py= py − ωx.

• Remaining dynamical equations :

px = −∂H

∂x= py − ωx − ∂Vω

∂x

py = −∂H

∂y= −px − ωy − ∂Vω

∂y.

• Jacobi constant is often written C = −2H .

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� Five Equilibrium Points

• Three collinear (Euler, 1750) on the x-axis— L1, L2, L3

• Two equilateral points (Lagrange, 1760)— 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

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� Stability

• Eigenvalues of the linearized equations at L1, L2, L3 have onereal and one imaginary pair. The stable and unstable man-ifolds of these equilibria play an important role.

• Associated periodic orbits are called the Liapunov orbits–Genesis targets a 3D counterpart, the halo orbits . Their sta-ble and unstable manifolds are also important.

• For space mission design, the most interesting equilibria are theunstable ones, not the stable ones!

Consider the dynamics plus the control!

Control often makes unstable objects, not attractors, of inter-est!1 Under proper control management they are incrediblyenergy efficient .

1This is related to control of chaos ; see Bloch, A.M. and J.E. Marsden [1989] Controlling homoclinic orbits,Theor. & Comp. Fluid Mech. 1, 179–190.

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� Hill’s Regions

• Our main concern is the behavior of orbits whose energy is justabove that of L2.

• The Hill’s region is the projection of this energy region ontoposition space.

• For this case, the Hill’s region contains a “neck” about L1 andL2. This neck region and its relation to the global orbit structureis critical: it was studied in detail by Conley, McGehee andthe Barcelona group .

• Orbits with energy just above that of L2 can be transit orbits,passing through the neck (Jupiter) region between the interiorregion (inside Jupiter’s orbit) and the exterior region (out-side Jupiter’s orbit). They can also be nontransit orbits orasymptotic orbits

-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)

S JL1

Zero velocity curve bounding forbidden

region

x (nondimensional units, rotating frame)

y (n

ondi

men

sion

al u

nits

, rot

atin

g fr

ame)

L2

(a) (b)

exteriorregion

interiorregion

Jupiterregion

forbiddenregion

L2

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� Heteroclinic Connection

• These are located numerically by finding an intersection ofthe stable and unstable manifolds using a Poincare cut.

• The stable and unstable manifolds are those of Liapunov orbitswith the same Jacobi constant .

0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08

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y (rotating frame)y

(rot

atin

g fr

ame)

x (rotating frame)

y (r

otat

ing

fram

e)

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

JL1 L2

x (rotating frame)

y (r

otat

ing

fram

e)

Heteroclinic Connection

JL1 L2

Forbidden Region

Forbidden Region

Stable

Manifold

Unstable

Manifold

Poincare Cuts

HeteroclinicIntersections

Stable

Manifold

Unstable

Manifold

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� Example: Construction of (J,X;J,S,J) Orbits

• Illustrate the construction of orbits with given itineraries

• Closely related to constructions in symbolic dynamics

• Invariant manifold tubes separate transit from nontransitorbits—these tubes and their properties are important!

• Green curve = Poincare cut of L1 stable manifold.

• Red curve = cut of L2 unstable manifold.

• Any point inside the intersection region ∆J gives rise to an(X;J,S) orbit.

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 (rotating frame)

y (r

otat

ing

fram

e)

JL1 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

0.04

0.06

y (rotating frame)

∆J = (X;J,S)

(X;J)

(;J,S)

y (r

otat

ing

fram

e).

Forbidden Region

Forbidden Region

StableManifold

UnstableManifold

UnstableManifold Cut

StableManifold Cut

Intersection Region

30

• Continue with further Poincare sections on the inside or outsideas appropriate to locate points that satisfy the conditions for theremaining itinerary.

JS L2

L1

PoincareSection

JL1 L2

(J,X;J,S,J)

(J,X;J)

(;J,S,J)

Forbidden Region

Forbidden Region

Sun

Jupiter

(;X,J)

(J;X)

(;S,J)

(J;S)

∆X = (J;X,J) (J,X;J,S,J)∆S = (J;S,J)

(J,X;J,S,J)Orbit

UnstableManifold

PoincareSection

Exterior Region Interior Region Jupiter Region

∆X = (J;X,J)

∆S = (J;S,J)

32

• One can numerically implement this procedure.

• Conley [1968] and McGehee [1969] proved the existence ofhomoclinic orbits for both the interior and exteriorregion .

• Llibre, Martinez and Simo [1985] showed analytically the ex-istence of transversal symmetric (1,1)-homoclinic or-bits in the interior region under appropriate conditions.

• For our problem, we need to find transversal homoclinic orbitsin both interior and exterior regions as well as transversal hete-roclinic cycles for the L1 and L2 Liapunov orbits.

• Numerical techniques of Gomez, Jorba, Masdemont and Simo[1993] play an important role.

33

• Comets like Oterma are

◦ mostly heliocentric, with the key perturbations are dominatedby Jupiter’s gravitation.

◦ Motion is very nearly in Jupiter’s orbital plane

◦ Jupiter’s small eccentricity (.0483) plays little role during the fastresonance transition (less than or equal to one Jupiter period induration).

◦ The PCR3BP is an adequate starting model for illuminating theessence of the resonance transition process.

• We were motivated by the paper of Belbruno and (Brian) Mars-den [1995]; related reference is that of Liao, Saari and Xia [1996],etc. (In our explanations, we don’t invoke anything about Arnolddiffusion—resonant transition is much simpler than this).

34

The Genesis Discovery Mission

� Some General Comments

• Mission Purpose. To gather solar wind samples and to returnthem to Earth for analysis.

• Mission Constraint. Must return in Utah during the daytime

• Will descend with a parachute for a helicopter snatch

• Must have a lunar swingby contingency in case of bad weather

• Highly energy efficient (very small ∆v required).

• One wants to make use of the dynamical sensitivity to design lowcost trajectories—the Genesis trajectory is one example.

35

� Mission Trajectory

• Final trajectory closely resembles the following four part missiondesign:

◦ 1. insertion onto an L1 halo orbit stable manifold .

◦ 2. using saddle point controllers, remain on the halo orbit forabout 2 years (4 revolutions)

◦ 3. return to a near halo orbit around L2 via a near hetero-clinic connection

◦ 4. return to Earth on a near impact orbit (unstable manifold ofa halo orbit around L2.

• Final trajectory uses three dimensional problem and takes intoaccount all the major bodies in the solar system.

0 0.5 1 1.5 2 2.5 3

x 106

6

4

2

< SUN

L1EARTH

LUNAR ORBIT

RETURN PARKING ORBIT

HETEROCLINICRETURN ORBIT

L2*0

2

4

6

8

GENESIS TRAJECTORY IN ROTATING COORDINATESx 10

5

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1

-0.8

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0

0.2

0.4

0.6

0.8

1

0.985 0.99 0.995 1 1.005 1.01 1.015

-0.01

-0.005

0

0.005

0.01

y (A

U, S

un-E

arth

rot

atin

g fr

ame)

x (AU, Sun-Earth rotating frame)

L1

(b)

y (A

U, S

un-E

arth

rot

atin

g fr

ame)

x (AU, Sun-Earth rotating frame)(a)

L2EarthSun

Earth

Genesis Mission Trajectory

Homoclinic-Heteroclinic Dynamical ChainL1 Liapunov Orbit Homoclinic Trajectory

L2 Liapunov Orbit Homoclinic Trajectory

Heteroclinic Trajectory Connecting L1 and L2 Liapunov Orbits

Homoclinic-Heteroclinic Dynamical Chains Reveal Dynamics Underlying Genesis Mission Trajectory

Forbidden Region

Forbidden Region

HeteroclinicConnection

L2 Homoclinic Orbit

L1 Homoclinic Orbit

L1 L2

x (km)

y(k

m)

-1E+06 0 1E+06

-1E+06

-500000

0

500000

1E+06

L1 L2

x (km)

z(k

m)

-1E+06 0 1E+06

-1E+06

-500000

0

500000

1E+06

y (km)

z(k

m)

-1E+06 0 1E+06

-1E+06

-500000

0

500000

1E+06

0

500000

z

-1E+06

0

1E+06

x

-500000

0

500000

1E+06

y XY

Z

41

Genesis movie

42

• Potential planet-impacting asteroids may utilize dynamical chan-nels as a pathway to Earth from nearby heliocentric orbits. Thisphenomena has been observed recently in the impact of cometShoemaker-Levy 9 with Jupiter.

• These ideas apply to any planet or moon system . Missionflexibility is achieved post-launch making use of dynamical sensitiv-ity — miniscule fuel expenditures can lead to dramatically differenttrajectories. One could turn a near-Earth mission into an asteroidrendezvous and return mission in situ with an appropriately placedsmall thrust. Rather than being a hindrance to orbital stability,sensitivity facilitates mission versatility .

• Use homoclinic and heteroclinic structures to understand and com-pute transport rates of various objects around the solar system(eg, between Mars and Earth).

43

• Zodiacal Dust Cloud. Numerical simulations of the orbitalevolution of asteroidal dust particles show that the Earth is embed-ded in a circumsolar ring of asteroidal dust known as the zodiacaldust cloud (Dermott et al. [1994]). Simulations and observationsreveal that the zodiacal dust cloud has structure. Viewed in theSun-Earth rotating frame, there are several high density clumps(∼10% greater than the background) which are mostly evenly dis-tributed throughout the Earth’s orbit. The simulations consid-ered the gravitational effects of the actual solar system and non-gravitational forces: radiation pressure, Poynting-Robertson lightdrag, and solar wind drag. The dust particles are believed to spiralin towards the Sun from the asteroid belt, becoming trapped tem-porarily in exterior mean motion resonances with the Earth. Theyare then scattered by close encounters with the Earth leading tofurther spiraling towards, and eventual collision with, the Sun.

-1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0

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0

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x (AU, Sun-Earth Rotating Frame)

Earth

Sun

y (A

U, S

un-E

art

h R

ota

ting F

ram

e)

Rela

tive P

art

icle

Densi

ty

Earth’sOrbit

45

• Variational and Symplectic Integrators. Symplectic in-tegrators for the long time integrations of the solar system is wellknown through the work of Tremaine, Wisdom and others. Inmany problems in which the dynamics is delicate or where thereare delicate controls, geometric integrators can be useful.

• Other Structures in the Solar System.

As Lo and Ross [1997] suggested, further exploration of the phasespace structure as revealed by the homoclinic-heteroclinic struc-tures and their association with mean motion resonances may pro-vide deeper conceptual insight into the evolution and structure ofthe asteroid belt (interior to Jupiter) and the Kuiper Belt (exteriorto Neptune), plus the transport between these two belts and theterrestrial planet region.

The figure plots the (local) semi-major axis versus the orbital ec-

46

centricity. We show the L1 (green) and L2 (black) manifolds foreach of the giant outer planets. Notice the intersections betweenmanifolds of adjacent planets, which leads to chaotic transport.Also shown are the asteroids (blue dots), comets (red circles), andKuiper Belt objects (green circles).

Comets

Ast

ero

ids

Kuiper Belt Objects

Pluto

Neptu

ne

Ura

nus

Satu

rn

Jupite

rT

roja

ns

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Orb

ital

Eccentr

icit

y

Semi-major Axis (AU)20 25 30 3515105 40 45 50

48

� “Petit Grand Tour” of Jupiter’s moons

• Construction of some new trajectories that visit Europa and Ganymede.

• Example :

◦ 1 orbit around Ganymede.

◦ 4 orbits around Europa, etc.

L2

Ganymede

L1

Jupiter

Jupiter

Europa’sorbit

Ganymede’sorbit

Transferorbit

∆V

L2

Europa

Jupiter

• Idea is to use burns that enable a transfer from one three bodysystem to another.

Semimajor Axis (Jupiter Radii)

Orb

ital E

ccen

tric

ity

L1 (light lines) and L2 (dark lines) manifolds for Io, Europa, Ganymede, and Callisto

Euro

pa

Io

Ganymede

Callisto

0.6

0.5

0.4

0.3

0.2

0.1

010 20 30 40 50 60

49

Petit Grand Tour movie

50

� Lunar Capture: How to get to the Moon Cheaply

• In 1991, the failed Japanese mission, Muses-A (Uesugi [1986]), wasgiven new life with a radical new mission concept and renamedas the Hiten Mission (Tanabe et al. [1982], Belbruno [1987],Belbruno and Miller [1993]).

• We present an approach to the problem of the orbital dynamics ofthis interesting trajectory by implementing in a systematicway the view (Barcelona group and Belbruno) that the Sun-Earth-Moon 4 body system can be modelled as two coupled 3 bodysystems .

• Within each 3 body system, using our understanding of the in-variant manifold structures associated with L1 and L2, we trans-fer from a 200 km Earth orbit into the region where the invariantmanifold structure of Sun-Earth Lagrange points interacts with the

51

invariant manifold structure of the Earth-Moon Lagrange points.

• One utilizes the sensitivity of the “twisting” near the invariantmanifold tubes to target back to a suitable Earth parking orbit.

• This interaction permits a low energy transfer from the Sun-Earthsystem to the Earth-Moon system. The invariant manifold tubesof the Earth-Moon system provide the dynamical channels in phasespace that enable ballistic captures of the spacecraft by the Moon.

• The results are then checked by integration in the bicircular 4-bodyproblem. It works!

• This technique is somewhat cheaper (18% less ∆V ) than the usualHohmann transfer (jumping onto an ellipse that reaches to theMoon, then accelerating to catch it, then circularizing). However,it also takes longer (6 months as opposed to 5 days).

0.996 0.998 1 1.002 1.004 1.006 1.008 1.01 1.012

-0.008

-0.006

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-0.002

0

0.002

0.004

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0.008

x

y

Earth

L2 orbitForbidden

Region

Sun

Moon

-4 -3 -2 -1 0 1 2 3 4

-3

-2

-1

0

1

2

x (Earth-Moon distances)

Ballistic Capture of a Spacecraft by the Moon from 200 km Altitude Earth Parking Orbit (Shown in an Inertial Frame)

∆V2=34 m/s

∆VL=3211 m/s

Transfer time is about 6 months(5 day tick marks)

y (E

arth

-Moo

n di

stan

ces)

Earth

Moon'sOrbit

BallisticCapture

0.99 0.995 1 1.005 1.01

-0.010

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0

0.005

0.010

x (AU)

y (A

U)

∆VL

∆V2

L2L1

Sun-Earth Rotating Frame

Earth

Moon'sOrbit

Sun

BallisticCapture

Integrated in 4-body system:Sun-Earth-Moon-Spacecraft

0.8 0.9 1 1.1 1.2

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0

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x (Earth-Moon distances)

L1 L2

Earth

Ballistic Captureby the Moon

Moony (E

arth

-Moo

n di

stan

ces)

Earth-Moon Rotating Frame

54

Movie: Ballistic capture

inertial

55

Movie: Ballistic capture

rotating

56

� Optimal Insertion into a Halo Orbit

• Halo Orbit Insertion goes back to the early days of L1 halo-orbit missions (eg, Farquhar et al [1980] for the ISEE-3 missionlaunched in 1978).

• We study optimal control in the context of mission design withthe aid of dynamical systems and invariant manifolds. In particu-lar, we consider optimizing burns for halo orbit insertion ofGenesis type missions, although the methods are rather general.

• Low thrust and impulsive burn contexts are both importantand the techniques can handle either case.

• Optimization software coopt (COntrol–OPTimization) is usedto do an optimization of the cost function (minimizing ∆V )subject to the constraint of the equations of motion . Wevary the number of impulses and also consider the effect of delaying

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the first impulse.

• coopt rather general software for optimal control and opti-mization of systems modeled by differential-algebraicequations (DAE), developed by the Computational Science andEngineering Group at University of California Santa Barbara. Ithas been designed to control and optimize a general class of DAEsystems which may be quite large. It uses multiple shooting andSQP techniques to do the optimization.

• Due to the problem sensitivity and the instability of the haloorbit, an accurate first guess is essential—provided by a highorder analytic expansion of minimum 3rd order using the LinstedtPoincare method (Simo, Howell and Pernicka, Chua and Parker).

• Simo et al, in conjunction with the SOHO mission in the 1980’swere the first to study invariant manifolds of the halo orbit.

THE SOLAR AND HELIOSPHERIC OBSERVATORY March 09, 2000 21:47:27 UT - Mission Day: 1560 - DOY: 69

http://sohowww.nascom.nasa.gov/

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• Stable manifold of the halo orbit–used to design the trans-fer trajectory which delivers the Genesis spacecraft from launch toinsertion onto the halo orbit (HOI). Unstable manifold–usedto design the return trajectory which brings the spacecraft and itssamples back to Earth via the heteroclinic connection .

• Expected error due to launch is approximately 7 m/s for a boost ofapproximately 3200 m/s from a 200 km circular altitude Earth or-bit. This error is then optimally corrected using impulsive thrusts.Halo orbit missions are very sensitive to launch errors .

• Objective: Find the maneuver times and sizes to minimizefuel consumption for a trajectory starting at Earth and endingon the specified halo orbit around the Lagrange point L1 of theSun-Earth system at a position and with a velocity consistentwith the HOI time.

x

y

z H(t)

∆vi+1Ti+1

Ti

T1

∆v1

∆vi

Tn1

∆vn

Tn

HOI

∆vn1

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Halo Insertion Movie

Computational Science and Engineering

Parametric Study of the Optimal Solution

Influence of: Delay in TCM1 Perturbation in launching velocity

Optimal solutions found for all cases

Number of maneuvers: Unperturbed injection velocity: 1 Perturbed injection velocity: 2 •

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End

Internet site :

http://www.cds.caltech.edu/˜marsden/

Email :

marsden@cds.caltech.edu