station-keeping for quasi-periodic orbits marcel duering · massimiliano vasile university of...

65nd International Astronautical Congress, Toronto, Canada. Copyright ©2014 by the International Astronautical Federation. All rights reserved.

IAC-14.C1.1.1

STATION-KEEPING FOR QUASI-PERIODIC ORBITS

Marcel Duering

University of Strathclyde, Advanced Space Concepts Laboratory

email: [email protected]

Markus Landgraf

Mission Analysis Section, ESA/European Space Operations Centre


Massimiliano Vasile

University of Strathclyde, Advanced Space Concepts Laboratory


Abstract The shallow gravity gradient in the libration point regions enables manoeuvring at low ∆v expenses,but implicates a sensitivity to small perturbations. A variety of bounded orbits can be determined aroundeach libration point and station-keeping is required to maintain them for multiple revolutions. In this paper,a station-keeping algorithm based on the orbital lifetime expectancy is proposed for so-called quasi-periodicsolutions. The method introduced is based on the identification of a manoeuvre maximising the lifetime of anorbit within defined boundaries. The manoeuvre direction and magnitude is finally optimised with a differentialevolution algorithm. The novelty of the method presented here is the identification of the downstream centremanifold by the lifetime analysis to preserve the orbit with its properties forward in time. The study shows thatthe manoeuvre direction is directly correlated to stability information that is provided by the Floquet modaltheory. Finally, numerical calculations were carried out for trajectories around the far-side libration point in theEarth-Moon system to show the effectiveness of this station-keeping approach. The robustness is proven by theintroduction of errors and the evaluation of their impact.

I INTRODUCTION

Operational orbits near the co-linear librationpoints offer interesting opportunities and usingthem might enable many space mission scenarios.The balanced gravitational forces of the Earth andMoon build a prone dynamical environment result-ing that small perturbation have a large effect onthe spacecraft and their operational orbits. Thisbehaviour, in particular, is beneficial for spacecraftprojects that rely on regular orbital manoeuvresand transfers. Another advantage of the libra-tion point regions is the location with respect toMoon and Earth. These orbits enable access tothe surface of the Moon, and it is feasible to ob-serve the Moon in a close distance without chang-ing the gravity potential as it is required to landon the Moon. Furthermore, libration point anddistant retrograde orbits require less ∆v than lowlunar orbits to maintain them. The libration pointregions provide a variety of bounded orbits. Thementioned dynamical instabilities and the unstablebehaviour of most of these orbits prevent the ref-erence path from being followed precisely for sev-

eral revolutions and frequent manoeuvres are de-manded to remain on orbit. Strategies required tocalculate those are discussed in this paper. Regard-less of the approach, the fundamental objective ofany station-keeping algorithm is to design manoeu-vres that maintain a spacecraft orbit for some de-sired length of time. This is taken as starting pointof this research and a methodology based on thelifetime parameter is applied and proven.

Station-keeping strategies have been studied forseveral dynamical frameworks and missions in thepast. Methods using theoretical knowledge basedon modal Floquet theory have been explored by[1]. With the help of modes the motion can be de-scribed by centre components, stable and unstablemodes. The cancellation of the unstable compo-nents is used to maintain a spacecraft in a libra-tion point orbit. A similar methodology is alreadyapplied in the past utilising a linearisation of theequation of motion [2]. For periodic orbits domi-nant manoeuvre directions can be determined bythe eigenvectors if the monodromy matrix, see [3]and [8]. An optimal control approach is used tooptimise the station-keeping manoeuvres over the

IAC-14.C1.1.1 1

mailto:[email protected]




lifetime of the orbits including the fulfilment ofconstraints during the mission, see [6] and [7]. Inaddition, research focus on the usage of a referencepath that is pre-calculated. In those cases controlcan be applied assuring that the spacecraft followsa desired orbit. The drawback of this method isthat precise states on the reference orbit are re-quired during the station-keeping. They have to bestored e.g. utilising truncated Fourier series. Fur-thermore, the integration of the variational equa-tions is time consuming in the station-keeping pro-cess.

The paper starts with a short introduction tolibration point orbits along with their invariantmanifolds followed by the introduction of thestation-keeping process. The proposed algorithmexploits the fact that the nature of the dynamics al-ready provides the required information on the de-sired trajectory although the unstable componentsprohibits the propagation of long times spans. Itconsists of three main steps, the first is the de-termination of a manoeuvre magnitude and direc-tion to maximise the orbital lifetime within somegiven boundaries. The arising optimisation prob-lem is solved by a differential evolution algorithm,see [9]. The next step is the manoeuvre executingsending the spacecraft towards a trajectory that isideally not part of neither the invariant stable norunstable manifold. A forward propagation to thenext xy-plane crossing finalises the station-keepingstep. The previous two steps are repeated until thedesired lifetime is assured. The station-keeping isapplied once per revolution at the crossing of thexy-plane in positive direction as described above,but can be applied at arbitrary positions and num-ber of revolutions to cope with perturbations orafter a certain thrust is accumulated.

II BACKGROUND

The space around the Moon in particular if thedistance to the Moon exceeds 60000 km providesa dynamical environment that is influenced by thegravity of multiple bodies. This distance can bedetermined by the mass ratio of the Earth and theMoon. The station-keeping is studied in a sim-ple dynamical model to prove the concept, an ad-vanced model fed by ephemeris data can be utilisedin a later stage to more accurately predict station-keeping costs. In order to take the gravitationaleffect of the two bodies into account, the dynam-ics are modelled as circular restricted three-bodyproblem (CR3BP). The model describes a vehiclemoving under the gravitational forces of a primaryand a secondary body. It is restricted in the sensethat the particle is massless, and circular indicatesthat the motion of the secondary with respect to

the primary is idealised as a circular orbit. Themotion of the spacecraft is described in terms ofrotating coordinates relative to the barycentre ofthe system primaries leading to equations of mo-tion that are autonomous and the solutions aretime-invariant. A so-called synodic reference frameis introduced with a rotating x-axis directed fromthe primary to the secondary body. The two pri-maries are stationary in this frame. The refer-ence system rotates with a constant angular ve-locity about the barycentre at the same rotationrate as the secondary body rotates around the pri-mary. The position of the centre of mass can bedetermined from initial conditions because of theconstant angular velocity. The conservation of an-gular momentum allows a computation of the an-gular velocity from initial conditions. The secondorder differential equation of motion is introducedin the Euler-Lagrange form as

x = 2x+ x− 1− µr31

(x+ µ)− µ

r32(x− 1 + µ) (1)

y = −2x+ y − 1− µr31

y − µ

r32y (2)

z = −1− µr31

z − µ

r32z (3)

where r21 = (x − µ)2 + y2 + z2 represents thedistance from the spacecraft to the larger primary,and r22 = (x + 1 − µ)2 + y2 + z2 to the larger pri-mary. A known integral of motion in the circularrestricted three-body problem is the Jacobi con-stant C which is directly related to the conservedenergy given by

E =− C

2=

1

2(x2 + y2 + z2)

−{

1

2(x2 + y2) +

1− µr1− µ

r2+µ(1− µ)

2

}(4)

Later the Jacobi constant is used to classify pe-riodic and quasi-periodic trajectories. Equation 3defines the motion of the spacecraft in a normalisedcoordinates such that the gravitational parameterG is equal to one. The non-dimensional orbital pe-riod is normalised to 2π by factor t∗, which is theinverse of the mean motion of the primaries. Thefactor for distance quantities is the characteristiclength l∗, which is the distance between the pri-maries. For weights the characteristic mass m∗ isthe total mass of the system. This study focus onthe Earth/Moon three-body problem assuming avalue for the mass ratio of µ = 3.0406 · 10−6.

IAC-14.C1.1.1 2


0.7 0.8 0.9 1 1.1 1.2

−0.2

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0.1

0.2

x [adim]

y [a

dim

]

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−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

x [adim]

z [a

dim

]

Figure 1: Periodic orbit around the far-side libration point with its stable and unstable hyperbolic manifoldbranches. Size of Moon plotted with a factor of 3. The red spheres indicate the libration points L1 and L2.

II.I Invariant manifold structure

In the framework of the restricted three-bodyproblem, one-parameter periodic orbit familiesand two-parameter quasi-periodic orbit familiesare found, which are associated with the four-dimensional invariant centre manifold. There aretwo classes: the mentioned centre invariant and thehyperbolic invariant manifold, see [4]. These or-bits are obtained numerically, the periodic orbitsare solved by a two-point boundary value prob-lem, whereas the quasi-periodic solutions requirea calculation scheme based on the approximationby truncated Fourier series or equivalent methods,see [5]. Bounded orbits can be classified as stableand others as unstable through linearisation of theequations of motion about a libration point andbifurcation theory. The hyperbolic invariant man-

1.1 1.12 1.14 1.16−0.12

−0.11

−0.1

−0.09

−0.08

−0.07

−0.06

x [adim]

y [a

dim

]

Figure 2: Periodic orbit around L2 (grey). Thered and violet dots represent the escaping be-haviour here at the xy-plane crossing. The blue(dashed) line is the zero velocity curve at the en-ergy level of the orbit.

ifold consists of a stable and an unstable set oftrajectories associated to the centre invariant man-ifold. The focus in Fig. 1 is set to the proximityof the Moon with its two co-linear libration pointsL1 and L2. Both pictures indicate the manifoldbranches to the inner and outer regions, which canbe divided by the zero velocity curves for a desiredorbital energy. The zero velocity curve enables theintroduction of forbidden regions, that cannot bereached at a given energy level. Fig. 4 indicates vi-sually the escaping behaviour if the periodic orbitis propagated forward (black) and backward (grey)with a manoeuvre executed in the correspondingdirection governed by the linearisation and studyof the eigenvectors of the monodromy matrix. Thezero velocity curves are plotted for three differentvalues, the outer one corresponds to the energyof the halo orbit. Fig. 3 quantifies this deviationwith the euclidean norm of the distance betweenthe initial and returning positions. The solid line

1.11 1.12 1.13 1.14 1.15 1.16

−0.1

−0.095

−0.09

−0.085

−0.08

x [adim]

y [a

dim

]

0.005

0.01

0.015

0.02

Figure 3: Deviation from nominal orbit plottedfor number of downstream xy-plane crossings.

IAC-14.C1.1.1 3


is governed by forward the dashed one by back-wards propagation.

III STATION-KEEPINGMETHODOLOGY

The previously described dynamically sensitive be-haviour of most libration point orbits prohibits thepropagation of multiple revolutions. A techniqueis required to guide the trajectory by introducedmanoeuvre to compensate for perturbations eitherintroduced by gravitational forces or numerical er-rors. The objective of the algorithm introduced inthe following is to identify the a manoeuvre direc-tion and magnitudein such a way that the orbitproperties are maintained and the orbital lifetimeis maximised. The station-keeping methodology isexplained in the next section followed by the ap-plication to periodic and quasi-periodic trajecto-ries. For a periodic orbit the identifying parame-ter is the Jacobian, for quasi-periodic orbit thereis an additional parameter, either the frequencyω2 or the rotational number ρ. Before the station-keeping procedure is explained, the focus is set onthe basic idea of the algorithm, which is the deter-mination and implementation of manoeuvres thatincrease the lifetime of an orbit.

III.I Determination of the orbital lifetimeand manoeuvre directions

Manoeuvres for station-keeping purposes are intro-duced by evaluating the time a trajectory evolvesin certain boundaries around a libration point. As-suming that the initial state is already within theselimits and on the xy-plane in the synodic referenceframe, the state can be perturbed in such a waythat the orbital lifetime changes. The objective isnow to find a manoeuvre that the lifetime within

0.7 0.8 0.9 1 1.1 1.2

−0.2

−0.1

0

0.1

0.2

x [adim]

y [a

dim

]

Figure 4: Boundary definitions (blue) required forthe orbital lifetime definition process. For equa-tions, see 7.

the geometric limits is maximised not leaving a de-fined region. For orbits with larger amplitudes,when the limits include the secondary body it isrequired to prevent the spacecraft falling into theprimary. For this reason the region around the sec-ondary body is added as an exclusion zone, see Eq.7. The forbidden region defined by the zero veloc-ity curves enables a introduction of the limits thatare purely defined on the x-component of the statevector. In order to determine the correction veloc-ity, the trajectory is propagated forward from theinitial state or from the previous correction point.The numerical integration stops when the trajec-tory passes through the boundaries set as

x < xL2 + δ1 (5)

x > xL2 − δ1 (6)

δ2 <√

(x− xs)2 + (y − ys)2 + (z − zs)2 (7)

where δ1 is the maximal amplitude in x-directionof the orbit and δ2 = 10−4. Restricting the station-keeping manoeuvre to the in-plane direction andallowing it only at the crossing of the x-y planereduces the number of parameters to be optimised.In this case z = 0 and z is evaluated as

z2 = v2 − x2 − z2 (8)

The starting point for the evaluation of the or-bital lifetime are state vectors on the xy-planecrossing in positive direction. The initial states arepropagated forward until a boundary condition isreached. The time until this event is defined asTm which is maximised by a differential evolutionalgorithm. In order to extend the lifetime of an or-bit, either the velocity, the position or both can bemodified. The implementation of a velocity changeis straight forward. Changing the position wouldleave a discontinuity in the trajectory. To copewith this problem a position change is implementedwith two velocity changes, the first one instantlyto vary the position of a point downstream of thetrajectory, where the second manoeuvre is applied.The re-directing manoeuvre guiding escaping tra-jectory towards longer lifetime, non-escaping solu-tions is now used to design a station-keeping algo-rithm.

A variety of algorithms are tested to achieve theidentification of the next centre manifold, for thispurpose an initial state vector on a periodic or-bit is chosen as targeting point on the xy-plane.The centre manifold is described as the phase spacethat is neither affected by the attraction of the sta-ble manifold nor the repulsion of the unstable one.The following three cases are considered:

IAC-14.C1.1.1 4


−1 −0.5 0 0.5 1

x 10−3

−1

−0.5

0

0.5

1x 10

−3

∆ vx [adim]

∆ v y [a

dim

]

86

88

90

92

94

96

98

4 6 8 10 12 14

1.85

1.9

1.95

Tmax

[adim]

θ

−1 −0.5 0 0.5 1

x 10−3

−1

−0.5

0

0.5

1x 10

−3

∆ vx [adim]

∆ v y [a

dim

]

84

86

88

90

92

94

96

98

5 10 151.18

1.2

1.22

1.24

1.26

1.28

1.3

Tmax

[adim]

θ

Figure 5: Results of the optimisation process. Optimal manoeuvre and its direction downstream (forward)(top). For the upstream (backward propagation) (bottom).

1. Fix position vector x, optimal ∆v appliedat xy-plane crossing to maximise the orbitallifetime Tm downstream. The fixed positionvector prohibits a movement along the stablemanifold direction.

2. Fixed position vector x, v chosen to maxi-mize the mean lifetime mean(T+

m , T−m) both

upstream and downstream. This only leadsto feasible solutions if the velocity vector isaccurate.

3. Adaptation of x and v aiming a maximal or-bital lifetime Tm downstream. In this casethe position can walk along the stable mani-fold increasing the lifetime on the one side butmoving away from the desired solution.

III.II Station-keeping strategy

The determination of the next downstream centremanifold is significant for a station-keeping algo-rithm that on the one side should keep the orbital

properties of a particular orbit and on the otherside is not relying on a reference trajectory. Tar-geting the next downstream centre manifold im-plies that no stable or unstable component is addedand the spacecraft evolves still on the ’same’ or-bit. From the previously mentioned strategies, thefirst one has been proven as the most effective ap-

Figure 6: Definition of the lifetime extension ma-noeuvre on the velocity plane. The manoeuvredirection is defined as θ, |∆v| the magnitude.

IAC-14.C1.1.1 5


proach for the identification of the centre manifold.The extension of the lifetime causes the return ofthe spacecraft to the vicinity of the original or-bit around the libration point. During the optimi-sation process a constant Jacobian is assured byadjusting the ∆vz component, see Eq. 8. This en-ables full steering capabilities during the optimi-sation. This leaves two optimisation parameters,which are ∆vx and ∆vy. Alternatively the direc-tion θ and magnitude |∆v| can be defined. Two op-timisation runs will show the correlation betweenthese two parameters, one conducted for a forwardand the other one for a backward propagation, in-dicating the stable and unstable (dominant) direc-tions. The size of the population is set to N = 20and 200 generations are calculated. The differen-tial evolution algorithm has two parameter subject

10−10

10−8

10−6

10

15

20

25

|∆ v| [adim]

Tm

ax

10−10

0.5

0.52

0.54

0.56

0.58

|∆ v| [adim]

θ

10−10

10−15

10−10

10−5

100

|∆ v| [adim]

d cmf

Figure 7: Correlation between the maximal or-bital lifetime Tmax, the direction θ and the mag-nitude of the manoeuvre, here for a precise initialstate (unperturbed).

10−8

10−6

10−4

5

10

15

20

25

|∆ v| [adim]

Tm

ax

10−8

10−6

10−4

0

0.2

0.4

0.6

0.8

|∆ v| [adim]θ

10−5

10−8

10−6

10−4

10−2

|∆ v| [adim]

d cmf

Figure 8: Correlation between the maximal or-bital lifetime Tmax, the direction θ and the mag-nitude of the manoeuvre, here for a state with anerror that is gained by propagating the state fromstudy in Fig. 5 for two revolutions downstream.(perturbed).

to adaptation, the differential weight F = [0, 2]and CR = [0, 1] representing the crossover proba-bility. The tuning parameters CR and F are setto [0.5, 0.8] for our purpose.

Fig. 5 show all evaluated members during thedifferential evolution process. The ∆vx-∆vy plotsshow the correlation between the manoeuvre andthe orbital lifetime. The best 10% of the solutionin Fig. 5 (right) are taken to produce the Tmax - θplots, shown the manoeuvre direction. Significantin Fig. 5 (left) is the chaotic behaviour for smallorbital lifetimes. With increasing Tmax the graphsbifurcate into a dominant solution.

The relation between manoeuvre directions,magnitudes and their corresponding orbital life-times Tmax are further studied as they represent

IAC-14.C1.1.1 6


a major step towards the design of the station-keeping algorithm. Two initial state vectors, onethat belongs to a periodic orbit, which is the out-come of the solved two-point boundary value prob-lem, and the second one afflicted with errors intro-duced by the propagation of two revolutions down-stream are used. Fig. 7 shows the outcome of theoptimisation for the unperturbed, whereas Fig. 8for the perturbed case. For visualisation purposesthe lifetime, manoeuvre direction, and the associ-ated error on the next xy-plane crossing are evalu-ated over a wide range manoeuvre magnitudes. Nooptimisation is required for the results in Fig. 7 andFig. 8. The optimisation results are marked with avertical dashed lines. There is a precise error mea-surement available for periodic orbit as the stateon the xy-plane maintains on every return. Theeuclidean distance dcmf between the initial statean each return is shown in the |∆v| - dcmf plots.Remarkable is the smooth behaviour for the errorfunction dcmf and manoeuvre direction θ, whereasthe orbital lifetime indicates a chaotic pattern. Areason for this is the fact that the location wherethe trajectory escapes moves along the boundaryplanes. In some cases the is also the switching be-tween the two boundary planes visible.

The methodology of this part of the algorithmis explained in the previous. In the following theperformance of the algorithms is tested. Perturbed

1.0925 1.093 1.0935 1.094

0.115

0.1155

0.116

0.1165

x [adim]

y [a

dim

]

1.0925 1.093 1.0935 1.094

0.115

0.1155

0.116

0.1165

x [adim]

y [a

dim

]

Figure 9: Return map on the xy-plane for an ini-tial state on a periodic orbit. Unperturbed state(top), perturbed state (bottom).

1.08 1.09 1.1 1.11 1.120.08

0.1

0.12

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x [adim]

y [a

dim

]1.08 1.09 1.1 1.11 1.12

0.08

0.1

0.12

0.14

x [adim]y

[adi

m]

Figure 10: Return map on the xy-plane for an ini-tial state on a quasi-periodic orbit. Unperturbedstate (top), perturbed state (bottom).

and unperturbed state vectors provide the entrypoint, the optimal manoeuvre is determined andexecuted the trajectory is propagated to the nextxy-plane crossing (full revolution). This procedureis repeated until the required number of station-keeping manoeuvres for the trajectory lifetime isdetermined. Fig. 9 and Fig. 10 show the xy-planemap with several returns for a initial state on a pe-riodic and quasi-periodic orbit. The grey (dotted)rings present the centre manifold structure aroundthe initial state vector, which is indicated as blackmarker. The representation of the centre manifoldby those curves is widely used and enables the vi-sualisation on the plane with z = 0. The blackdot that is for the unperturbed case in the centreof the marker. The red markers are the return-ing points after each station-keeping manoeuvre.Once a return point is identified it is used as start-ing condition for the next manoeuvre evaluation.

IV PRELIMINARY RESULTS

Among the existing bounded libration point or-bits station-keeping costs are computed for peri-odic and quasi-periodic cases following the sameapproach. In general, the station-keeping algo-rithm is applicable to all libration point orbits thatpossess an invariant hyperbolic manifold structure.The numerical algorithm is used to continue and

IAC-14.C1.1.1 7


extend the orbital lifetime of orbits. Applying thealgorithm in a dynamical regime modelled by thecircular restricted three-body problem accountingneither for uncertainties nor a navigation budgetleads to very small manoeuvres that simply com-pensate numerical errors introduced by the propa-gation.

IV.I Station-keeping for periodic orbits

The previously explained station-keeping algo-rithm is applied to a quasi-periodic trajectory. Themaintenance effort in terms of ∆v is determined forhalo orbits with an orbital period between 13.3 to14.8 days. About 10-15 station-keeping manoeu-vres are evaluated leading to an orbital lifetimeof about half a year. The maximal manoeuvremagnitude is limited during the optimisation. Fora better assessment of the manoeuvre magnitudeand directions the station-keeping are applied ev-ery second crossing of the xy-plane.

#5 10 15 20 25 30

" v

[mm

/s]

0.5

1

1.5

2

2.5

#5 10 15 20 25 30

3 [r

ad]

2.3

2.4

2.5

2.6

2.7

2.8

Figure 11: Station-keeping manoeuvres for a pe-riodic orbit, magnitude (top), direction (bottom).

For the periodic case the station-keeping errorcan be easily evaluated as the trajectory return tothe same point on the xy-plane. The euclideandistance between the returns serve as error func-tion. The order of magnitude of the error functionin this case is 10−9. The optimisation shows thatall the station-keeping manoeuvre are conducted inthe same direction (or 180 deg shifted). Additionalthe vectors are align with the direction of the sta-ble mode eigenvectors for the periodic case. This

1.05 1.1 1.15

−0.05

0

0.05

0.1

0.15

x [adim]

z [a

dim

]

Figure 12: Resulting trajectories for the halo or-bit family, mean propagation time of orbits is 180days.

is not surprising station-keeping methods incorpo-rating Floquet analysis utilise manoeuvres that arealigned with the unstable mode aiming for a can-cellation of the unstable component. The manoeu-vre history is shown in Fig. 11. The resulting haloorbits are plotted in Fig. 12.

IV.II Station-keeping forquasi-periodic orbits

The station-keeping method for quasi-periodic or-bits is straight forward. The periodic and quasi-periodic case differ only in the return map on thexy-plane, which is now described as invariant curveinstead of a point for the periodic case. This meansthat the optimal-station keeping direction vary de-pending on the return direction on the invariantcurve, see Fig. 13 as comparison. For this pur-

1.08 1.09 1.1 1.11 1.12

0.08

0.1

0.12

0.14

x [adim]

y [a

dim

]

Figure 13: Station-keeping results visualised onthe return map (xy-plane) for a family of quasi-periodic orbits rising from a halo orbit (quasi-halotype).

IAC-14.C1.1.1 8


1.12 1.13 1.14 1.15

−0.02

−0.01

0

0.01

0.02

x [adim]

y [a

dim

]

Figure 14: Station-keeping results visualised onthe return map on the xy-plane for a family ofquasi-periodic orbits rising from a vertical lya-punov orbit (lissajous type).

pose 50 crossing are evaluated to resolve the in-variant curve. The station-keeping is applied totwo families of quasi-periodic orbits with an en-ergy of E = −C/2 = −1.548. Quasi-periodic orbitfamilies share the orbital energy but differ in thesecond parameter which is the rotation numberρ. The manoeuvre magnitude is below 0.1mm/sfor all cases having only the purpose to cancel out

1.08 1.1 1.12 1.14 1.16 1.18

−0.05

0

0.05

0.1

x [adim]

z [a

dim

]

1.1 1.12 1.14−0.15

−0.1

−0.05

0

0.05

0.1

0.15

x [adim]

z [a

dim

]

Figure 15: Quasi-periodic trajectory with an or-bital lifetime extended to 180 days.

the numerically introduced error. Fig. 13 and Fig.14 show the locations of the crossings on the xy-plane. For two cases the corresponding trajectoriesare plotted in Fig. 15, the section representation ishighlighted in red in Fig. 13 and Fig. 14.

IV.III Evaluation of manoeuvre directions

The purpose of the previously introduced station-keeping approach is to maintain an orbit for an ar-bitrary lifetime preserving orbital properties. Theresults, in particular, the obtained manoeuvre di-rection can be compared with results from a Flo-quet mode analysis. Subspaces can be determinedthat correspond either to unstable or stable partsof the motions. Fig. 16 highlights the identifieddominant manoeuvre direction for the halo orbitfamily with parameters as in Fig. 12. The opti-misation results in this study for the halo orbitfamily are shown for an escape towards the stable(red, bold) and unstable (blue, bold) hyperbolicmanifold. The results plotted with thin markerscorrespond to the in-plane (x and y components ofthe position vector) determined by this subspaces.The discrepancies between the Floquet mode ap-proach and the station-keeping algorithm proposedin this paper increase with growing orbital ampli-tudes. The explanation for this is the linearisationthat is required for the Floquet approach. Withgrowing amplitudes, here in Fig. 16 increasing Ja-cobian C, non-linearities are introduced.

Another outcome of this study is the dominantmanoeuvre direction that pushes the spacecraft to-wards the invariant stable or unstable manifoldbranches. These trajectories are required for plan-ning e.g. transfer and rendez-vous scenarios inspace mission design. Fig. 7 and Fig. 8 show the

3.05 3.06 3.07 3.08 3.090

0.5

1

1.5

2

2.5

3

C [adim]

θ [r

ad]

Figure 16: Comparison of optimal manoeuvre di-rections with results obtained from Floquet sub-spaces, evaluated for the halo orbit family foran escape towards the stable (red) and unstable(blue) hyperbolic manifold.

IAC-14.C1.1.1 9


transition between the region with a manoeuvremagnitude at an order of 10−6 with a directionincreasing to 0.58 and the region where the direc-tion stabilises to a fix values, but higher manoeuvremagnitude. For the evaluation of this dominant di-rections, the manoeuvre magnitude is set to 10−3.Forward (unstable) or backward (stable) propaga-tion is required in the lifetime assessment.

V SUMMARY AND CONCLUDINGREMARKS

The proposed station-keeping algorithm is basedon an orbital lifetime analysis paired with a dif-ferential evolution algorithm for optimisation ex-ploiting the natural interaction between the centreand hyperbolic invariant manifolds. It has beendemonstrated that the manoeuvre determinationby lifetime measurements along with its optimisa-tion designing station-keeping manoeuvre is a ro-bust method, fast in implementation and flexiblein its application to a variety of orbits. For a pe-riodic orbit the results of the proposed algorithmare comparable to the modal control that can bederived from the Floquet modes. This is correctfor orbits with small amplitudes. Non-linearitiesare introduced with growing amplitudes and opti-misation approach performs better than the Flo-quet procedure. Another advantage is that thereis no intermediate linearisation required. A de-tailed study is required to accounting for naviga-tion and manoeuvring uncertainties. This station-keeping approach is easily transferable into a non-autonomous dynamical model fed by ephemerisdata. Furthermore, is can be used to generatefamilies of orbits by introducing a simple familycontinuation method.

REFERENCES

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