transfer to l3

Spacecraft transfers to the L3 point ofthe Sun-Earth system

Celestial Mechanics Working SeminarUniversitat de Barcelona - 18 Nov. 2009

E.Fantino, M.Tantardini, Y.Ren, P.Pergola, G.Gomez, J.Masdemont

The solution to a new flight dynamics problem

as a journey

into orbital mechanics

and multi-body dynamics

E.Fantino - CMWS -18 Nov. 09

Contents

I Statement of the problem

I Scientific objectives of a mission to L3

I L3: a hard place to live

I Classical, high-thrust two-body maneuvers (HT)

I Gravity assisted patched conics with optimization (GA)

I Invariant manifold strategy (IM)

I Low-thrust optimal problem (LT)


Statement of the problem

I L3 is the third collinear libration point in the Sun-Earth CR3BP

I Position in synodical barycentric coordinates obtained bysolution of quintic Lagrange equation: x = 1.0000012668,r1 = 0.9999982264 ⇒ slightly inner orbit wrt Earth’s

I Getting there is essentially a re-phasing problem = add 180◦ intrue anomaly


Whereas both L1 and L2 have been exploited for a long timeas host places for space probes,no space mission has ever been sent to L3


What science from L3?

solar physics relativity asteroids

I Space weather / solar activity monitoringI supplementary observations (wrt L1 or Earth)I part of a formation of s/c around the Sun

I Fundamental physics experiments: gravitational light bendingI follow-up of Cassini/Huygens radio science measurements

I Tracking of hidden NEOs: blind spot at superior conjunctionwith the Sun


The adversities at L3

I Communicating with the Earth

⇒ large orbits (Halo or Lyapunov ≥ 0.1 AU in y) or relaysatellites at L4/L5

I Gravitational perturbations (Jupiter, Venus)

⇒ station keeping strategy


HT in Sun-s/c 2BP

• High-thrust engines ⇒ impulsive ∆V ’s

• Keplerian (two-body), heliocentric orbits

• Planar approximation

Connecting conic arcs with maneuvers at the patch points in orderto eliminate discontinuities in the velocity vectors

Note that the Hohman transfer is not applicable here: transferfrom/to the same circular orbit with one 180o elliptical arc and twotangential burns can only be made through the circular orbit itself⇒ at arrival the s/c encounters the Earth, not L3!


Bi-elliptic transfer (1/2)

I three-burns, two half elliptic arcs, total transfer angle = 2π

I requirement: arrival point be in opposition to the Earth ⇒ arelation between sum of transfer times on the two elliptic orbitsand Earth’s orbital period:

π

√(ra + rb)3

8GM�+ π

√(rb + rc)3

8GM�=

n

2T⊕, n = 1, 3, 5, ...

thus providing the distance rb.


Bi-elliptic transfer (2/2)

Note that ∆Vb <<

Best option: n = 3, TOF = 1.5 years, ∆V = 6.7 km/s


Two-tangent burn transfer

At the limit in which the difference between departure and arrivalorbits is neglected

⇒ the apoapsis maneuver (∆Vb) disappears

⇒ the transfer covers one full ellipse


Bi-elliptic one-tangent burn transfer (1/2)

I two elliptic arcs (e1 and e2), total transfer angle < 2π

I three-burns: one tangential (∆Va)


Bi-elliptic one-tangent burn transfer (2/2)

Timing/phasing requirement:

π

√(ra + rb)3

8GM�+ Te2 =

n

2T⊕, n = 1, 3, 5, ...

Given n and rb, Te2 allows to solve the Lambert problem for thesecond arc.

For ∆φ = 270◦, the best option flies to 1.7 AU, costs 16.9 km/s,and takes 1 year and 3 months


Multi-revolution transfer (1/2)

I many (m) revolutions on one and the same ellipse

I at each rev. a phase difference ∆Ψ wrt the Earth is gained:∆Ψ = π/m

I two burns: one to insert, one to leave

I transfer time Tm: Tm =2m + 1

2T⊕, m = 1, 2, 3, ...

I time/rev. Te = Tm/m and apoapsis distance rb = 2a− ra


Multi-revolution transfer (2/2)

Trade-off between TOF and ∆V ⇒ m = 2÷ 4

⇒ TOF = 2.5÷ 4.5 yrs; ∆V = 4÷ 2.5 km/s


Optimized GA patched conics

Assumptions:

I 3D Patched Conics method

I Two-Body models: Sun-s/c, planet - s/c


... assumptions

I The planetary motion is modelled by means of ephemerides (inthe form of interpolating polynomials)

I L3 geometrically defined: Earth’s true anomaly + 180o

I Arrival event: L3 is modeled as a planet with zero mass

I Impulsive maneuvers only (= no low-thrust)

I Maneuvers are only allowed at the swingby pericenters


Procedure (1/6)

Given:

• start date (and position), end date (and position),

• names and order (= sequence) of n planets to be encounteredand dates of encounters,

a trajectory is designed by

• solving n + 1 3D Lambert problems,

• estimating one ∆V for each planetary encounter: it has to beprovided by planet + engine,

• computing the swingby parameters (orbital elements of thehyperbolas)


Procedure (2/6)A few words on the swingby:

V∞i = vi − vP

V∞o = vo − vP

If V∞i = V∞o ⇒ the swingby is natural.In general, this does not occur, and incoming and outgoing hyperbolas aredifferent: ν− 6= ν+


Procedure (3/6)But we can impose that they pass through a common pericenter (rm) where thedifference in the velocity vectors is supplied by the engine (if possible)

sin(ν− + ν+) =V∞o × V∞i

V∞iV∞o

sin ν+ =1

1 + V 2∞i/V

2cm

; sin ν− =1

1 + V 2∞o/V 2

cm

V 2cm =

GMP

rm

Hence, rm is the unknown to be determined as solution of a nonlinear equation

sin−1

(V∞o × V∞i

V∞iV∞o

)= sin−1

(1

1 + V 2∞i/V

2cm

)+ sin−1

(1

1 + V 2∞o/V 2

cm

)Finally:

∆V = Vm+ − Vm− =√

V 2∞o + 2GMP/rm −

√V 2∞i + 2GMP/rm


Procedure (4/6)Optimization problem with:

I n planetary swinbgys

I n + 2 dates ( = n swingbys + departure + arrival)

I n + 2 (potential) maneuvers

I a set of nonlinear constraints wrt minimum swinbgy altitude


Procedure (5/6)

Objective function to be minimized:

C = ∆Vd + ∆Va +n∑

i=0

[∆VGAi + Wi ·

(RPi + hmin i − rπi )2

R2Pi

]

At the i th swingby:

∆VGAi = magnitude of periapsis maneuverRPi = equatorial radius of the planetrπi = periapsis distance of the current solutionhmin i = minimum allowed swingby altitudeWi = weight (0 or 10 ÷ 100)


Procedure (6/6)

Optimization strategy:

1. Initial guesses (= dates) are given by a global optimizer:genetic algorithm (when n >>) or grid search on a discreterange of dates (when n = 1, 2).

2. Look for local optimization based on varying the dates, aimingat minimum of C with SQP/Simplex algorithm: at eachiteration a full trajectory from departure to arrival is computed,and the objective function is evaluated.


Example of grid search: EVL3


Sequences

Pl. sequence ∆VTot ∆VB TOF

EEL3 3.80 2.80 586

E4r1E2r2L3 4.72 4.13 2674

E2r1E2r2L3 4.95 4.31 2321

EL3 6.57 – 548

EML3 6.26 0.01 560

EVVEL3 6.94 18.57 1248

EVEML3 7.89 10.77 1080

EMEL3 9.85 2.40 1236

EMVL3 11.65 8.18 757

EVEL3 11.80 13.01 859

EVML3 12.82 9.27 737

EVVL3 14.40 10.22 964


Earth-Mars-L3

TOF = 1 y 195 d ∆Vd = 3.22 km/s ∆Va = 3.04 km/s ∆VTot = 6.26 km/s

Mars swingby: TOF to encounter = 341 d ∆Vπ = 0 km/s


Earth-Venus-Venus-Earth-L3

TOF = 3 y 122 d ∆Vd = 3.0 km/s ∆Va = 3.7 km/s ∆Vtot = 6.9 km/sVenus swingby: TOF to encounter = 172 d ∆Vπ = 0.07 km/sVenus swingby: TOF to encounter = 449 d ∆Vπ = 0 km/s

Earth swingby: TOF to encounter = 84 d ∆Vπ = 0.12 km/s


Earth-Earth-L3

TOF = 1 y 221 d ∆Vd = 0.0014 km/s ∆Va = 3.29 km/s ∆Vtot = 3.8 km/sEarth swingby: TOF to encounter = 39 d ∆Vπ = 0.51 km/s

Very low (< 0.5 km/s) incoming relative speed ⇒ TO BE VERIFIED!


Feasible alternative: resonant swingbys

Introduce two intermediate elliptical orbits with semimajor axis ' 1AU, so that they are cheap to reach and leave, and resonant withthe Earth: after a number of revolutions on the first ellipse, the s/cencounters the Earth where the swingby occurs BUT it is fast.

After the swingby, the s/c and enters the second ellipse and afterthe appropriate number of revolutions returns to the orbit of theEarth to encounter L3.

Rather cheap (4 km/s), feasible, BUT long (>> 2000 days).


An inner resonant swingby

TOF = 7 yrs 4 months ∆Vd = 2.5 km/s ∆Va = 2.1 km/s ∆Vtot = 4.7 km/s

Earth swingby: TOF to encounter = 4 yrs ∆Vπ = 0.08 km/s


An outer resonant swingby

TOF = 6 yrs 6 months ∆Vd = 2.6 km/s ∆Va = 2.3 km/s ∆Vtot = 4.9 km/s

Earth swingby: TOF to encounter = 4 yrs ∆Vπ = 0.08 km/s


IM transfers

Genesis Herschel/Planck

I Genesis and Herschel/Planck efficiently used the IMs ofperiodic orbits around L1 and L2

I How about using the IMs of periodic orbits around L3 to send as/c there?

I How do these objects look like?


Horseshoe motionWe know that in the range 0 < µ < 0.01174 the IMs of L3 have ahorseshoe shape [Barrabes & Olle (2006)].In particular, it holds for µ�⊕ = 3.0404234 · 10−6.And it can be extended to the IMs of planar Lyapunov orbits of L3.

Examples of natural objects:

• Saturn’s co-orbital satellites Janus and Epimetheus (Voyager).

• some near-Earth asteroids (e.g., Asteroid 2002 AA29)


Planar Lyapunov orbits around L3

74 planar Lyapunov orbits of L3 with J in [2.9855538, 3.0000061]

x-amplitude in [10−4, 10−1] AU and y -amplitude up to 0.25 AU


IMs

The time to approach the progenitor Lyapunov orbit from the pointof closest approach to the Earth depends on µ1/3 [Font (1990)](µ1/2 for L1 and L2) and this justifies the large times foundthroughout the family wrt the typical times covered by IMs of L1

and L2:

700 yrs 800 yrs


An alternative? Unstable IMs of L1/L2


Propellant and time budget (1/2)We chose a Poincare section at y = 0 (one might obtain betterresults on other sections).

There from each pair [manifold trajectory, Lyapunov orbit]intersecting at a given x coordinate, the insertion maneuver ∆Vi iscomputed:

∆Vi =√

x2IM + (y2

IM − y2Ly )

The time of flight varies with the trajectory on the manifold. Thevariations are approx. the same on the two manifolds.The TOF can be represented as a function of the location of thestarting point (IC) on the progenitor L1/L2 Lyapunov (given as atime in units of the period).


Propellant and time budget (2/2)


LT transfers

Assumptions:

I Planar Sun-s/c two-body model

I Electrical engine always ON, providing constant thrust

I Departure from the surface of Earth’s sphere of influence, fromeither L1 or L2

I Arrival: s/c at rest at L3


Dynamical equations

x = −GM�r3

x +T

mαx

y = −GM�r3

y +T

mαy

m = − T

g0Isp(1)

(2)

where:T = thrust (force) provided by the engine = 90 mNm = s/c mass = 500 kgg0 = gravitational acceleration at the Earth’s surfaceIsp = specific impulse of the engine = 3100 sα = (αx , αy ) = direction of thrust (unit vector)


Optimal control

The trajectory is solved as an optimal control problem (see Yuan’s lecture onoptimal control, 2008): find the thrust direction α which minimizes the massconsumption = the transfer time

λr = −∂H

∂r=

(λv

GM�r 3− 3GM�λ

Tv r

r 5

)r

λv = −∂H

∂v= −λr

λm = −∂H

∂m= −λv

T

m2

H = Hamiltonian of the system:

H = λTr v + λT

v

(−GM�

r 3r +

T

mα

)− λm

T

g0Isp

λTr , λT

v , λm = Lagrange multipliers (or costates) associated with position r,

velocity v and mass m.


Optimal control

Optimal thrust vector α is obtained by setting ∂H/∂α = 0 with thenormalization constraint αT · α = 1.⇒ the optimal control α? is:

α∗ = −λv

λv

Nonlinear constrained optimization problem (solved with SNOPT).The optimization parameters are the initial values (t = 0) of thecostates and the time of flight.The performance index is the time of flight. A set of constraints acton the final value of the states, i.e., rf and vf .


Two examples

L1, TOF = 1 y 201 days L2, TOF = 1 y 264 days

∆m/m0 = 0.29 ∆m/m0 = 0.32


Thrust direction history

Angles between the thrust direction and

• the velocity vector (cont. line)

• the vector perpendicular to the radius vector (dashed line)

L1 L2


Integration in RTBP

The ICs found for the two cases have been integrated in theSun-Earth CR3BP

L1 L2

Refinement is necessary if the Earth’s gravity is taken into account


Conclusions

Comparison in terms of ∆V budget versus TOF can easily be madeamong the three techniques that use impulsive maneuvers:

• multi-revolution transfer: 2.2 km/s in 4.5 years

• patched conics with multiple swingbys are more expensive: onlythe standard EEL3 case is cheap and fast but requires furtherverification.

• IM transfers are cheap (0.5 - 1.7 km/s) but longer (> 6 years)

A different concept, based on electrical engines, allows directtransfers from either L1 or L2 to L3 in ' 1.5 years with 30% massconsumption.


Acknowledgments

E. Fantino, Y. Ren and P. Pergola have been supported by theMarie Curie Actions Research and Training Network AstroNetMCRTN-CT-2006-035151.

G. Gomez and J.J. Masdemont have been partially supported by theMCyT grants MTM2006-05849/Consolider and MTM2006-00478,respectively.


Publication

Presented at the Fifth International Celestial Mechanics Meeting(CELMEC V), San Martino al Cimino (Viterbo, Italy), 6 - 12September 2009

Recently submitted to Celestial Mechanics & Dynamical Astronomy


References (1/2)

• Barrabes, E., Olle, M., Invariant Manifolds of L3 and horseshoemotion in the restricted three-body problem, Nonlinearity, 9,2065-2090, 2006

• Brison, A.E., Ho, Y.-C., Applied optimal control, BlaisdellPublishing Company, Waltham (Massachusetts), 1969

• Font, J., The role of homoclinic and heteroclinic orbits intwo-degrees of freedom Hamiltonian systems, Ph.D.dissertation, Departament de Matematica Aplicada i Analisi,Universitat de Barcelona, 1990

• Gill, P.E., Murray, W., Saunders, M.A., Snopt: an sqpalgorithm for large-scale constrained optimization, SIAM J.Optim., 12, 979-1006, 2002


References (2/2)

• Hechler, M., Yanez, A., Herschel/Planck Consolidated Reporton Mission Analysis FP-MA-RP-0010-TOS/GMA Issue 3.1,2006

• Lo, M.W., Williams, B.G., Bollman, W.E., et al., GenesisMission Design, AIAA Space Flight Mechanics, Paper No.AIAA 98–4468, 1998.

• Vallado, D.A., Fundamentals of astrodynamics andapplications, Microcosm Press, Hawthorne (California), 2007

• Senent, J., Ocampo, C., Capella, A., Low-thrustvariable-specific-impulse transfers and guidance to unstableperiodic orbits, J. Guid. Contr. Dyn. 28, 280-290, 2005

• Szebehely, V., Theory of orbits, Academic Press, New York(Massachusetts), 1967


Merci !!


Earth-Venus-L3:


transfer to l3

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