Halo Orbit design

Pegasus-AIAA Student Conference, Poitiers, France

April 11 – 13, 2012

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Combining Low-Thrust and Manifold Dynamics for Vertical Lyapunov Orbits Missions

C. Finocchietti1. University of Pisa, Pisa, 56122, Italy

Abstract: With the aim at designing innovative space missions both in terms of operative orbits and transfer trajectories, this study deals with the Circular Restricted Three Body Problem and its applications. The vertical Lyapunov orbits, peculiar solutions of the model are computed together with the ballistic trajectories (manifolds) associated with them. Moreover, with the aim at designing low-fuel consumption transfer trajectories toward vertical Lyapunov orbits, the combination of the three body model dynamic and low-thrust transfer strategies is investigated. Finally, the optimization problem of the low-thrust transfer trajectory is considered and a direct multiple shooting method is presented for its solution.

Nomenclature

m = spacecraft mass ! = spacecraft mass variation in time M1 = bigger primary mass M2 = smaller primary mass µ = mass parameter l = distance between the primaries ! = angular velocity of the primaries around their center of mass t = time Li = Lagrange point (i=1,…,5) Ω = effective potential Ω!,!,! = derivatives of the effective potential respect to the x, y or z coordinate E = Energy of the system i = Lagrange point index C = Jacobi costant !, !, ! = synodic coordinates of the spacecraft velocity vector x, y, z = synodic coordinates of the spacecraft position vector !, !, ! = synodic coordinates of the spacecraft acceleration vector r1, r2 = distances between spacecraft and primaries c2 = constant greater than one Ax, Az = in-plane and out-of-plane amplitudes of the linearized periodic solutions !!" ,!! = in-plane and out-of-plane frequencies of the linearized periodic solutions !,! = in-plane and out-of-plane phases of the linearized periodic solutions k2 = constant !! = switch function !!", !!" e !!" = constants !! = function of the time !, ! = state vectors without and with the spacecraft mass respectively ! = state transition matrix 1 Graduate Student, Aerospace Engineering, University of Pisa, 56122, Pisa, Italy e.mail: [email protected].


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Φ = flux !! = eigenvectors of the monodromy matrix !!/!!"#$ = opportunely normalized eigenvectors of the monodromy matrix d = perturbation for the manifold computation ! = thrust of the engine !!" = specific impulse of the thruster !! = value of the acceleration due to gravity at the Earth’s surface !! = acceleration vector !!" , !!" , !!" = components along the reference system axis of the acceleration vector ! = thrust efficiency ! = power to the thruster ! = performance index for the optimal control problem ! = semimajor axis of the orbit w.r.t the Earth centered inertial system ! = eccentricity of the orbit w.r.t the Earth centered inertial system ! = number of segment for the transfer time discretization !! = fraction of maximum thrust at the time !! !!,!! = in-plane and out-of-plane angles that specify the orientation of the thrust direction at the time !!

I. Introduction HE interest of scientific community toward particular telecommunication and scientific space missions (e.g., the exploration of magnetic fields, plasma sheet and energy particles around a celestial body, the collection of solar

wind samples, the lunar poles coverage, etc…) has given the impulse to study dynamical multi-body models. These models, which are more complex than the Keplerian model, allow new orbits, different from classical conical orbits, which are more accurate and permit to satisfy specific mission constraints. Indeed, a n-body problem, with n>2, better describes the solar system dynamics than a two-body model. Using a non-Keplerian dynamical model in the preliminary mission design also results in low energy transfer1,2.

In particular, analyzing the motion of a spacecraft in the contest of the Circular Restricted Three-Body Problem3 (CRTBP), that is considering the simultaneous action on the spacecraft of the gravitational forces of two different celestial bodies (the primaries), it is possible to compute innovative trajectories advantageous both for transfer and operative orbit applications.

The CRTBP dynamical model has five equilibrium points, called Lagrange or libration points which are denoted by Li with i=1,…,5. The equilibrium point orbits and the associated ballistic solutions (i.e. manifolds), peculiar to the CRTBP, are attractive for space mission design and they have already been used successfully in the past4. The first libration point mission has been the International Sun-Earth Explorer mission4 (ISEE-3). It was launched on 12 August 1978 and then it was directly transferred into a Sun-Earth Halo orbit near the first libration point (L1) of the system.

More recently, in 2010, NASA operated for the first time the spacecraft ARTEMIS-P15 (Acceleration, Reconnection, Turbulence and Electrodynamics of the Moon’s Interaction with the Sun) into a orbit around the Earth-Moon L1 equilibrium point. At the present time, a large number of other libration point missions in both the Earth-Moon system and the Sun-Earth system are under study (see e.g., Rif. 6 and Rif. 7).

The aim of this work is to study the effectiveness of the CRTBP in the preliminary space mission design. In particular, it focuses on the transfer design to a vertical Lyapunov periodic orbit (one of the two basic families emanating from the collinear equilibrium points) in the Earth-Moon system. In order to design low fuel consumption transfer, we chosen to combine low-thrust transfer and manifolds dynamics. Indeed, the usage of a high specific impulse thruster, such as an Electric Propulsion device, allows for a significant propellant mass saving if compared to chemical solution. But, an electric thruster allows only low-thrust, in the order of some tens of milliNewton, due to the limited amount of power available on a spacecraft nowadays (typically up to few kW). Thus, it is necessary a longer transfer time compared to chemical transfer and continuous thrust arcs, so it is required to define the appropriate steering program to implemented the transfer toward the target orbit.

T


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It was chosen to study vertical Lyapunov orbits (Fig. 1) because, due to their wide out-of-plane excursion, they might be useful to explore the physical space around a primary attractor and to observe polar regions of one or both primaries. In the Sun-Earth system vertical Lyapunov orbits with one year period extend above Sun poles and return close to Earth once a year. This scenario provides a useful orbit for Sun polar regions observation and also for the collection of solar wind samples that can even be brought back to Earth. Another useful application of this kind of orbit in the Earth-Moon system is the lunar poles coverage, to be considered, in particular, for communication ports for future lunar stations.

In order to design the whole mission, the computation of the target vertical Lyapunov orbit together with the

associated manifolds and the definition of the appropriate low-thrust transfer strategy are necessary. Moreover, with the aim at minimizing the fuel consumption, it was studied the problem of the trajectory optimization to determine the optimal time history of the thrust magnitude and direction. To solve the optimal control problem a direct method was applied. It discretizes the continuous optimization problem in a parametric problem of large but finite dimension, the Direct Multiple Shooting Method8. The resulting Non-Linear Programming (NLP) problem was then solved numerically using a Sequential Quadratic Programming9 (SQP) technique.

II. Dynamical Model The Circular Restricted Three-Body Problem is the

dynamical model chosen to describe the motion of the spacecraft. In particular, the CRTBP studies the motion of a body (the spacecraft) under the action of the gravitational forces of two more massive bodies, M1 and M2 (the primaries), which are supposed to move on circular orbits around their common center of mass. The gravitational attraction of the spacecraft on these two bodies is neglected because the spacecraft is assumed to have mass (m) very small in comparison with the primaries (m << M2 < M1).

The classical reference frame to study the CRTBP is the synodic frame (Fig. 2). It is a rotating coordinate system such that the origin is centered at the barycenter of primaries, the x-axis is always directed from the larger toward the smaller primary, the z-axis is perpendicular to the primaries plane of motion and the y-axis completes the right-handed system. Thus, in the synodic frame the x-axis rotates uniformly with the primaries, so that these last result always in the same position and the CRTBP equations of motion became autonomous.

Figure 1. Vertical Lyapunov orbits of 80,000 km vertical amplitude around the Lagrange points L1 (left) and L2 (right) of the Earth-Moon system in the dimensionless synodic frame.

Figure 2. Synodic reference frame and Lagrange points of the CRTBP. !=0.1, non-dimensional units.

Earth

Moon L1

Z, D

U

X, DU Y, DU Y, DU X, DU

Z, D

U Earth

L2

Moon

X, DU

Y, D

U


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With the introduction of a non-dimensional set of units, such that the distance between the two primaries (l), their angular velocity (!) and the sum of their masses (M1+M2) are set to one, the CRTBP equations of motion depend only on one physic parameter, called mass parameter, that is3:

! = !! !! +!! , ! ! 0 , 0.5

and are written in the dimensionless form3:

! − 2! = !! ! + 2! = !! ! = !! In Eqs. (2) the subscripts denote the partial derivatives of the effective potential associated to the problem, that is3:

! !, !, ! = 12!! + !! +

1 − !!!

+!!!+12! 1 − !

where !! and !! are the distances of the spacecraft from M1 e M2 respectively and their expression is3:

!! = ! + ! ! + !! + !! !! = ! + ! − 1 ! + !! + !! (4)

The Eqs. (2) have only one first integral of motion, the Jacobi integral !, that is3:

2! !, !, ! − !! + !! + !! = ! (5)

This is in relation with the energy (!) of the system by3:

! = −2! (6)

Therefore, until there are no perturbations of any type, the energy is constantly the same along an orbit. Another important property of the Eqs. (2) is the existence of five stationary points, called Lagrange or libration

points (Fig. 2), in which the acceleration vector field vanishes. In literature, these equilibrium points are denoted by Li with i=1,…,5. The first three (L1, L2 and L3) are on the x-axis, so they are collinear with the primaries, whereas the last two (L4 and L5) are on the primaries’ plane and form equilateral triangles with them, therefore they are called triangular points.

To compute the synodic coordinates of Lagrange points, we refer the reader to Rif. 3. Concerning the nature of the equilibrium in each Lagrange point, we recall that: any collinear point is unstable and any triangular point is stable until ! < 0.03852 (it is true for any system of three bodies of space mission interest). In the rest of the work, only L1 and L2 are considered since L3 has a slow dynamic and a mild instability and it is associated to higher energy level.

A. The Dynamic around the Collinear Libration Points Moser’s theorem10 states that in the vicinity of a collinear point the solutions of the nonlinear system have the

same qualitative behavior than the solutions of the linearized one, thus to study the dynamic near a collinear point it is possible to analyze the linear behavior and then to extend the qualitative results obtained to the non-linear case.

The equations of motion (Eqs. (2)) can be written in a new rotating reference frame centered in the collinear point and rescaled with the distance of the spacecraft from the nearest primary11.Then expanding the nonlinear terms, the resulting linearized equations are11:

where !! is a constant larger than one and depending on the mass parameter (see Rif. 11).

! − 2! − 1 + 2!! ! = 0 ,

! + 2! + !! − 1 ! = 0 ,

! + !!! = 0 .

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The solutions of the Eqs. (7) are given by the product saddle × centre × centre (Fig. 3). The saddle part gives rise to hyperbolic orbits and the two center parts generates periodic orbits. The combined motion of these parts gives rise to the invariant manifold “tubes” associated to the periodic orbits, which are two-dimensional surfaces composed by all the asymptotic one-dimensional trajectories to the resonant orbit. If a spacecraft lies on the stable or unstable manifold “tube”, the dynamic of the manifold lead it to or far, respectively, to the periodic orbit without any further propellant consumption.

B. The Families of Periodic Orbits Near a Collinear Libration Points The linearized periodic solutions around a collinear point are:

where !!, !!" and ! are the in-plane amplitude, frequency and phase respectively, and !!, !! and ! are the out-of-plane amplitude, frequency and phase. The in-plane and out-of-plane frequencies are different for any value of the mass parameter (i.e., !!" ≠ !!) and if !!"/!! is a rational number Eqs. (8) represent periodic solutions. Due to the fact that the CRTBP is a Hamiltonian system, isolated periodic orbits do not exist and the presence of a periodic orbit involves the characterization of the whole family.

Different families of orbit can be obtained depending on the choices of the two constants !! and !!: if !! ≠ 0 and !! = 0 planar Lyapunov periodic orbits are originated; if !! = 0 and !! ≠ 0 vertical Lyapunov periodic orbits are generated; if !! ≠ 0 and !! ≠ 0 linear Lissajous quasi-periodic trajectories are obtained. Near a collinear point there is also the Halo family of periodic orbit characterized by equal eigenfrequencies (!!" = !!). It is generated by the nonlinear contributions when the in-plane and the out-of-plane amplitudes are large enough.

To identify a single orbit in the family one parameter must be specified (e.g. !! for planar Lyapunov orbits, !! for vertical Lyapuonv and Halo orbits).

III. Vertical Lyapunov Orbits and Associated Manifolds Computation

A. Vertical Lyapunov Orbits Characteristics The linear behavior of a vertical Lyapunov orbit is a harmonic oscillation perpendicular to the primaries plane of

motion (! ! = !!cos (!!! + !)). In the rotating frame, the nonlinear solution is a three dimensional periodic orbit, almost vertical, with typical

eight-shape geometry and doubly symmetric: axisymmetric respect to the x-axis and symmetric respect to the (x, z)-plane (Fig. 1). The time step along the trajectory between two successively symmetric configurations is equal to a quarter of the orbit’s period.

B. Vertical Lyapunov Orbits Computation Since the CRTBP has no analytic solutions, the periodic orbit computation is a challenging problem. It is

possible only using analytical and numerical approximation techniques.

Figure 3. Solutions of the linearized CRTBP equations of motion near a collinear point12.

! ! = !!cos (!!"! + !)

! ! = −!!!!sin (!!"! + !)

! ! = !!cos (!!! + !)

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In this study, with the aim at computing a-priori chosen vertical Lyapunov orbit a third order analytic approximation was calculated and used as first guess for suitable numerical correction and continuation schemes. On one hand, the applied analytic method follows the work of D. L. Richardson, who used these techniques successfully in designing Halo periodic orbits for the ISEE-3 mission13-15. The numerical approximation procedures, on the other hand, are based on the symmetry proprieties characteristic of the periodic orbit in the synodic reference frame and on Newton method schemes.

1. Analytic Approximation: Richardson’s Method

In the late 1970s, D. L. Richardson developed a third-order local analytical solution for halo-type periodic orbits about the collinear points of the CRTBP14. The method he applied is based on the manipulation of the CRTBP Lagrangian into a compact series of Legendre polynomials. The equations of motion are then derived from this Lagrangian using Euler-Lagrange equations13. Approximated third-order solutions of the resulting equations of motion are constructed using techniques similar to the Lindstedt-Poincaré method16. In particular, non-linear terms in the equations of motion are truncated at degree three and some frequency corrections together with amplitudes and phase adjustments are employed to eliminate the secular terms in order to obtain periodic orbits14. For the complete derivation of the third-order analytical solutions the reader is referred to Rif. 17. The Richardson’s analytic approximation of Halo-type orbits in the neighborhood of any of the collinear points, opportunely corrected by R. Thurman and P. A. Worfolk17, is given by:

Where !! and !! are respectively the in-plane and the out-of-plane amplitude of the linearized motion solutions (Eqs.(8)). !! is a switch function (!! = 2 − !, ! = 1,3), !!", !!" e !!" are constants and !! is function of the time (see Rif. 17).

The third-order analytical approximation of a-priori chosen vertical Lyapunov orbit (i.e., the value of !! is fixed.) was computed setting in Eqs. (9) !! = 0 and !! equal to the chosen value. Figure 4 shows the results obtained applying this method for 57,000 km out-of-plane vertical Lyapunov orbits in the Earth-Moon system.

The energy along the analytically approximated orbit results variable in time with sinusoidal behavior of period equal to one half the orbit period (Fig. 5). The energy is not constant, as required for a CRTBP solution, because the computed orbit is only an approximated solution. In particular, increasing the out-of-plane amplitude of the vertical Lyapunov orbit, the energy oscillation amplitude grows and the analytical approximation becomes worse and worse (Fig. 5). Moreover, taking the third-order analytical approximations as initial conditions to propagate the system of Eqs. (2), no periodic motion can be obtained (Fig. 5), thus some iterative correction methods of the analytical first-guess solution are mandatory.

Figure 4. Analytic third order approximations of 57,000 km out-of-plane amplitudes vertical Lyapunov orbits about L1 and L2. Synodic reference frame, !=0.01215.

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! = !!"!!! + !!!!!! − !! cos !! + !!"!!! − !!"!!! cos 2!! + !!"!!! − !!"!!!!! !"#(3!!)

! = !!!!"# !! + !!"!!! + !!!!!! !"# 2!! +

+ !!"!!! − !!"!!!!! !"#(3!!) + !!!!!! + (!!" − !!")!!!!! sen !!

! = !!!! cos !! + !!!!"!!!! (cos 2!! − 3)+!! !!"!!!!! − !!"!!! cos (3!!)

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Z, D

U


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2. Numeric Approximation: Differential Correction Procedure and Continuation Method

In this section, we show how to numerically correct the first guess solution obtained with the analytic approximation and how to compute a family of vertical Lyapunov orbits, parameterized by their z-excursion Az.

The Differential Correction is the Newton-like procedure usually used to compute the periodic orbit of the CRTBP (see e.g., Rif. 17 and 18 where this procedure has been implemented). It is an iterative method based on the Mirror Theorem of Roy and Ovenden19. The theorem states that any trajectory in which two mirror configurations occur at distinct epochs is periodic. In the restricted three-body problem there are two possible types of mirror configuration: type-A is one in which the third body crosses the x-axis with its instantaneous velocity vector perpendicular to the axis, whereas type-P is one in which crosses the y=0 plane with velocity vector perpendicular to the plane. If two consecutive mirror configurations are of the same type, the orbit is simply-symmetric (e.g. planar Lyapunov orbits, Halo orbits), whereas if are of different type the orbit is doubly-symmetric (e.g. vertical Lyapunov orbits).

Thus, the applied Differential Correction procedure for computing a vertical Lyapunov orbit consists in determining a type P mirror configuration on the analytic approximation to use as initial condition !! = !!, 0, !!, 0, !!, 0 for integrating the system of Eqs. (2) until an intersection !∗ with the z=0 plane occurs:

!∗ = Φ !!, !∗ = !∗, 0, !∗, !∗, !∗,!∗

Then, Newton’s method consists in adjusting the values of !! through a first order expansion imposing the periodicity (i.e., imposing a type-A mirror configuration setting !∗ = 0 and !∗ = 0). The resulting equations system is:

! ∙

∆!!!∆!!!∆!!!

+ !(!∗)∆!∗ =

!∗!!!!∗!∗

− !∗

where ! is the state transition matrix (! ∶= !Φ/!!), ∆!!, ∆!!, and ∆!! are the corrections for the initial guess coordinates, ∆!∗ is the correction for the time step between the two mirror configurations and ! ∶= !Φ/!". The system of Eqs. (11) gives three equations with four unknowns (∆!!, ∆!!, ∆!! and ∆!∗). The indeterminacy is due to the fact that the family is parameterized by the out-of-plane amplitude, so taking ∆!!=0 the Eqs. (11) can be resolved and an iterative technique can be set with new initial condition X0new= (x0+∆!!, 0, z0 , 0, v0+∆!!, 0). The method is applied iteratively until the final accuracy is obtained.

Figure 5. a) Energy values along some analytically computed vertical Lyapunov orbits around L2 of the Earth-Moon system. Az=1,000 km to 100,000 km with a step of 3,000 km, !=0.01215. b) Solution, in the synodic frame, of the integration of the CRTBP’s equations of motion (blue line) starting from the analytical third-order approximation (green line) of a vertical Lyapunov orbit. Az=57,000 km, L1, !=0.01215.

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Y, DU X, DU

Z, D

U

Time, gg

Ene

rgy,

MU

*DU

2 /TU

2

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The third-order analytical approximations provide initial conditions allowing for the convergence of the Differential Correction procedure only for limited values of Az (e.g., Az < 139,000 km and Az < 154,000 km for vertical Lyapunov orbits around L1 and L2 respectively, in the Earth-Moon system). Therefore, in order to compute wider orbits a continuation method on the parameter Az was implemented.

Starting from the analytical approximation of a vertical Lyapunov orbit of small amplitude Az0, the continuation method consists in making the z-excursion vary from Az0 to Az according to an appropriated subdivision (i.e., the step ∆Az is fixed a-priori.), and applying at each iterations the Differential Correction procedure with an initial condition given by the final point obtained from the previous step modified by increasing the z-coordinate of ∆Az.

In this study, an amplitude step of 1,000 km was chosen and up to 250,000 km out-of-plane excursion orbits in the Earth-Moon system were constructed using the continuation method. Figure 6 shows some of the orbits computed in the case of the first and second libration points of the Earth-Moon system.

Moreover, comparing the differential correction results and the continuation method solutions, the last method gives better approximations of the a-priori chosen (i.e., fixed a nominal value for Az.) periodic orbit. Indeed, the amplitude error, computed as the difference between the value of Az at the end of the applied numerical method and the nominal value of Az, is smaller for the continuation method. In particular, for the differential correction procedure the amplitude error increases with Az, instead it decreases with Az using the continuation method (Table 1).

Az

Nominal Value [km]

Amplitude Error Differential Correction

Amplitude Error Continuation Method

[km] [%] [km] [%] 1000 0.0063 0.0006 - - 5000 0.7913 0.0158 - -

10,000 6.3304 0.0633 - - 13,000 13.9079 0.1070 0.0063 0.000048 15,000 21.3652 0.1424 0.0063 0.000042 20,000 50.6434 0.2532 0.0063 0.000031 35,000 271.4170 0.7755 0.0063 0.000018 56,000 1,111.7 1.9852 0.0063 0.000011 70,000 2,171.3 3.1019 0.0063 0.000009 85,000 3,887.7 4.5737 0.0063 0.000007

100,000 6,330.4 6.3304 0.0063 0.000006 139,000 17,001 12.2310 0.0063 0.000004 150,000 Not convergent Not convergent 0.0063 0.000004 200,000 Not convergent Not convergent 0.0063 0.000003 250,000 Not convergent Not convergent 0.0063 0.000002

Table 1. Amplitude errors using the differential correction procedure and the continuation method to compute the vertical Lyapunov orbits of nominal amplitudes Az (L1, !=0.01215). The continuation method has been applied only when the differential correction error was larger than 0.1 % (Az0 =1,000 km, ∆Az=1,000 km).

C. Invariant Manifolds Associated to the Vertical Lyapunov Orbits Computation In order to design transfer trajectories exploiting manifold dynamics, in the following we introduce the

numerical method used to approximate the stable and unstable manifold “tubes” of the vertical Lyapunov orbits and the obtained results. Each trajectory on the stable manifold “tube” of a periodic orbit is composed by the set of points which converges to the periodic orbit forward in time. Conversely, each trajectory on the unstable manifold “tube” consists of those points which converge to the orbit backward in time.

The idea of the algorithm implemented is to find a linear approximation of the manifolds in the neighborhood of the periodic orbit and than to globalize this approximation by flowing points under the CRTBP field17,18.

The one-dimensional stable/unstable manifold associated with a point of a periodic orbit is tangent to the eigenvector of the monodromy matrix (i.e., the principal matrix along a periodic orbit evaluated at a time equal to one period.), computed in that point, whose eigenvalue is real and has magnitude smaller/greater than one. So a small step in the direction of the appropriated eigenvector (!! = (!!, !!, !!, !!, !!,!!) or !! = (!!, !!, !!, !!, !!,!!)) provides a linear approximation of a point on the stable or unstable manifold.

Figure 6. Vertical Lyapunov orbits around L1 (red) and L2 (blue) computed with the continuation method. Az=1,000 km to 190,000 km, !=0.01215.


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Therefore, applying the globalization method17,18,20, the stable/unstable one-dimensional manifolds associated to each point !! of the orbit can be computed propagating forward/backward the initial condition20:

!!,!/! = !! ± !!!/!!"#$ where !!!"#$ and !!!"#$ are the normalized stable and unstable eigenvectors and the parameter d represents the distance in DU between the point of the orbit and the initial condition in the direction of the eigenvector. In this study, the globalization algorithm has been used to compute manifold tubes associated to vertical Lyapunov orbits around L1 and L2 of both the Sun-Earth (Fig. 7) and Earth-Moon (Fig. 8). In both systems the manifold tubes of a given vertical Lyapunov orbit look like eight-shape tubes with significant z-coordinates. The most important result we obtained for space mission design toward the collinear points of the Earth-Moon system is the fact the part of stable manifold tubes in the Earth region stay too far from Earth (the distance is greater than 100,000 km in position space.) to plane a mission using it with a direct departure from Earth.

Figure 7. Stable (green and blue) and unstable (black and red) manifold tubes of the vertical Lyapunov orbits with Az=300,000 km around L2 in the Sun-Earth system. Synodic reference frame.

Figure 8. Stable (green and blue) and unstable (black and red) manifold tubes of the vertical Lyapunov orbit with Az =57,000 km around L1 in the Earth-Moon system. Synodic reference frame, !=0.01215.

1 0.5 0 0.5 1

1

0.8

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0.2

0.4

0.6

0.8

1

M2

Varietà invarianti di un’orbita Lyapunov verticale di L2 Sole Terra [Az=300.000 km]

X [DU]

M1

Y [D

U]

0.995 1 1.005 1.01 1.015 1.02

0.01

0.005

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0.005

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Y [D

U]

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Y, DU

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IV. Preliminary Study of Space Missions combining Low-Thrust and Manifold Dynamics In this section we focus on the design of a transfer strategy to target a vertical Lyapunov orbit near the first

collinear point of the Earth-Moon system starting from a Geocentric Transfer Orbit (GTO). In order to limit the fuel consumption, we chose to insert the spacecraft on a specific trajectory of the target

Lyapunov stable manifold, so that the spacecraft was naturally led toward the periodic orbit by the manifold dynamics. However, due to the large distance between the Earth and the stable manifolds (i.e, greater than 100,000 km.) an initial propulsive phase was mandatory to reach the manifold. Moreover, at the insertion point on the manifold, other than the position vector also the velocity vector of the spacecraft has to match the one of the manifold.

With the aim at further saving the mass propellant, an Electric Propulsion device was chosen as propulsive system. Thus, the transfer trajectory from the GTO to the stable manifold is a low-thrust spiraling-out phase.

Choosing an initial low-thrust phase, the control vector (i.e., the acceleration vector !! given by the engine.) has to be included in the Eqs. (2). Furthermore, it is necessary to add the mass variation equation. Consequently, the complete system of equations of motion becomes:

where ! is the spacecraft’s mass time variation, ! is the thrust magnitude, !!" is the specific impulse of the thruster !! is the value of the gravity acceleration at the Earth’s surface and !!" , !!" , !!" are the components along the reference system axis of the instantaneous acceleration vector !!(!) = !(!)/!(!). In the case of an electric thruster the thrust magnitude is given by:

! =2!"!!"!!

in which ! is the thrust efficiency and ! is the power to the engine. Using a backward procedure, the transfer trajectory combining an initial low-thrust arc and a final costing phase along the stable manifold of the target vertical Lyapunov orbit was design. In particular, we fixed eccentricity (e=0.7234±0.002) and semimajor axis (a=24,510±10 km) of a GTO parking orbit, the maximum propulsive transfer time (200 days), the stable manifold departure angle, the mass of the spacecraft at the insertion point on the manifold (m!=500 kg) and the thrust strategy. This was assumed continuous with constant maximum thrust magnitude along the spacecraft’s instantaneous velocity vector:

!!" = !!!

!! + !! + !! !!" = !!

!!! + !! + !!

!!" = !!!

!! + !! + !!

The propulsive system that was chosen for implementing the simulations was the PPS®-1350-G Hall Effect thruster21. This was preferred because of its medium level performances compared to those of the other existing electric thrusters and because it has been flight-proven with more than 5,000 hours of operation in space during the ESA’s SMART-1 mission. Table 2 shows the values of the engine’s characteristics used in this study.

Integrating backward in time Eqs. (13) and Eqs. (15) starting from different points on the chosen stable manifold, we evaluated the possibility to reach the selected GTO. In Fig. 9, on the left, the red points along the stable manifolds (blue trajectory) of a 56,000 km vertical Lyapunov orbit represent the initial conditions able to

! !!" ! ! 1425 W 1500 s 0.5 96,8 mN

Table 2. Implemented values of the PPS®-1350-G performances.

Figure 8. PPS-1350-G Hall Effect thruster[X] .

! − 2! = Ω! + !!" ,

! + 2! = Ω! + !!" ,

! = Ω! + !!",

! = −!

!!"!! .

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match, within the maximum propulsive transfer time, both the values of eccentricity and semimajor axis of the a-priori chosen GTO at the end of the backward integration. The results are low-thrust transfers to the stable manifold taking between 183 and 193 days and consuming a quantity of propellant between 104 and 110 kg. That is, transfers starting from the selected GTO to the 56,000 km vertical Lyapunov orbit around L1 that take about 220 days consuming less than the 20 % of the initial spacecraft mass (see Fig. 9, right).

V. Transfer Trajectory Optimization In order to minimize the propellant required for the transfer from the GTO to the Lyapunov vertical orbit, in the

following we faced the trajectory optimization problem. We aim at finding the control vector time history !! = !!(!) during the propulsive phase that minimizes the

initial mass !! of the spacecraft at the parking orbit while satisfies specific mission constraints (i.e., system Eqs. (13), initial conditions, terminal conditions, simple bounds on the state variables and on the control variables). Minimizing the initial mass of the spacecraft corresponds to minimize the propellant mass required for the transfer because the final mass of the spacecraft at the insertion point on the stable manifold is a-priori fixed (m!=500 kg) in our backward analysis.

Setting the trajectory optimization problem in agreement with the standard optimal control theory22, the following scalar performance index ! has to be minimized:

! = !!

Numerical methods are used to compute approximated solutions for continuous optimal control problem. Essentially, two approaches are mostly applied23,24: direct and indirect. In the indirect approach, the necessary conditions for optimality are derived using calculus-of-variation techniques. This requires the addition of costate variables (i.e., Lagrange multipliers), equal in number to the state variables, and their governing equations. The resulting two-point boundary-value problem is solved numerically. On the other hand, the direct approach is to transcribe the continuous optimal control problem into a parameter optimization problem. The parameters are the discrete representation of the state and control time histories. The system of dynamical equations is satisfied by integrating them stepwise, using either implicit or explicit rules, and generating nonlinear constraint equations that must be accomplished by the parameters. In this way, the problem is converted into a NonLinear Programming problem that can then be numerically solved by a NLP problem solver23. In literature, a variety of direct and indirect methods have been proposed and implemented (see e.g., Rif. 23-25). In this study, we developed a direct multiple shooting method following the work of H. G. Bock8. The whole transfer time is divided in ! − 1 intervals according to a defined grid (e.g., uniform, logarithmic):

Figure 9. a) The red points along the a-priori chosen manifold (blue trajectory) are the initial condition able to reach, within the maximum integration time, the selected GTO at the end of the backward procedure. Vertical Lyapunov orbit with Az=56,000 km, L1, Earth-Moon system. b) Typical tangential continuous low-thrust transfer (red line) from a GTO to a stable manifold (blue line) of a 56,000 km vertical Lyapunov orbit around the first Earth-Moon collinear point, in the synodic frame.

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!! = !! < !! < !! < ⋯ < !!

where !! with ! = 0,… ,! are nodes. Using a backward integration procedure, !! is the time at the insertion point of the spacecraft on the stable manifold (!!) and !! is the initial time of the optimal trajectory on the parking orbit. The state and control vectors are discretized on each node (!!,!!! with ! = 0,… ,!). In particular, the state vector is composed by seven elements: the position and velocity vector coordinates and the mass of the spacecraft, ! = (!, !, !, !, !,!,!). The control vector (i.e., the acceleration vector) in each node is identified by its magnitude and direction:

!!! = ( !!!!"#/!!,!!,!! ) in which !!! 0,1 gives the fraction of maximum thrust !!"# at the time !!, !! is the value of the spacecraft’s mass at the time !!, !!! −!,! and !!! −!/2,!/2 are the in-plane and the out-of-plane angles that specify the orientation of the thrust direction with respect to the instantaneous velocity vector and plane of motion. Thus, we defined the following set of NLP variables:

! = !!,!!,… ,!!,!!,!!,… ,!!!!,!!,!!,… ,!!!!,!!,!!,… ,!!!!, !!, !! The control vector magnitude and directional angles time histories are approximated on every subinterval by a linear law. Whereas, Eqs. (13) are integrated stepwise from the initial value !! of the dynamical state at the beginning of each segment using an explicit four-stages Runge-Kutta formula25. Than, the following constraints are imposed in each node to ensure continuity of the solution trajectory (defects):

!!!! − !!!! = ! ! = 0,… ,! − 1 where !!!! is the result of the integration from !! and !!!!. At the initial time !! = !! we impose the insertion of the spacecraft on the stable manifold !!(!) a-priori selected:

! !! − !! !! = 0 and at the final time !! we impose the matching of the semimajor axis and eccentricity values of the parking orbit (!!"# =24.510 km, !!"# = 0,7234):

! !! − !!"# = 0 ! !! − !!"# = 0 The resulting NPL problem has dimension !! = 7(! + 1) + 3! + 2, so increasing the number ! of nodes increases the size of the problem and thus the computing time. We implemented the optimization problem for a transfer to a 57,000 km vertical Lyapunov orbit around L1 in the Earth-Moon system using as first guess the transfer trajectory obtained with a continuous tangential thrust strategy, as described in the section IV. Choosing ! such that the time step between two nodes was 3 days, the dimension of the relative NPL problem resulted of some hundreds of variables because of the long transfer time necessary to reach the stable manifold (about 190 days). Consequently, using a standard laptop with 2.66 GHz processor, the computing time was too long, of the order of tens of hours; thus to reduce the dimension of the NPL problem, the maximum acceleration value was set at 3.9e-3 m/s2 so that ! = 20 was proved to be sufficient. Additionally the out-of-plane angle ! was constantly set equal to zero. Figures 10 and 11 show the results obtained in the cases of a uniform (!!!! − !! = 1/!) and a logarithmical (!!!! − !! = log!" (! + 1)/! temporal grid respectively. The optimized transfer using a uniform grid takes about 8.5 days consuming 77 kg of propellant, whereas the result with a logarithmical grid is a transfer spending a slightly shorter time (8.1 days.) requiring a slightly larger quantity of propellant (78 kg). Hence, facing the optimization problem about a 30% reduction in the propellant mass consumption and an increase of about 4-9% in the transfer time were obtained if compared with the same transfer computed using a continuous tangential thrust strategy (motorized phase time: 7.8 days, propellant consumption: 110 kg).

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Moreover, the maximum defects’ value of the solution at the end of the optimization process was greater in the case of the logarithmical computational grid (~10-4) than in that of the uniform grid (~10-5). This is due to the fact that the logarithmical nodes subdivision allows a greater number of nodes in the initial phases of the motorized transfer where rapid changes in the state and control occur. This approach allows for more accurate solutions.

VI. Conclusion In this study we studied the dynamical model of the CRTBP in order to use its peculiar orbit to design innovative

space missions. In particular, we focused on the vertical Lyapunov periodic orbits around the libration points because these orbits might be useful to either scientific and telecommunication space missions, due to their potentially large out-of-plane excursion.

Figure 10. a) Optimized low-thrust transfer with uniform temporal grid (blue line) and constant tangential low-thrust transfer (orange line), which was used as first guess for the direct multiple shooting method, from a GTO to the stable manifold of a vertical Lyapunov orbit. Az=57,000 km, L1, Earth-Moon system. b) Optimized time histories of the thrust direction and magnitude for the blue transfer in the figure on the left. ! = 20.

Figure 11. a) Optimized low-thrust transfer with logarithmical temporal grid (blue line) and constant tangential low-thrust transfer (orange line), which was used as first guess for the direct multiple shooting method, from a GTO to the stable manifold of a vertical Lyapunov orbit. Az=57,000 km, L1, Earth-Moon system. b) Optimized time histories of the thrust direction and magnitude for the blue transfer in the figure on the left. ! = 20.

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We computed vertical Lyapunov periodic orbits around the collinear points of the Sun-Earth and Earth-Moon systems together with the computation of the bi-dimensional manifolds associated with the orbits. Both analytical (third-order analytical approximation) and numerical methods (differential correction procedure and continuation method) were used.

Moreover, we designed preliminary low-thrust space missions from a parking orbit around the Earth towards the above mentioned target periodic orbits. Transfer strategies that combine low-thrust arcs with the stable manifolds associated with the periodic orbit were applied. In this way, low propellant consumption transfers from a Geostationary Transfer Orbit towards a Lyapunov vertical periodic orbit around the first libration point between the Earth and the Moon were assessed. Such transfers take about 220 days consuming a quantity of propellant lower than 20% of the initial spacecraft mass.

Finally, a preliminary approach to the optimization problem of the low-thrust transfer trajectory was considered. The aim was to determine the time histories of magnitude and direction of the propulsive acceleration vector able of minimizing the propellant consumed. In particular, a direct multiple shooting method was implemented for the solution of this problem. Results obtained in the transfers from a GTO to a 57,000 km Lyapunov vertical orbit in the Earth-Moon system shown a 30% reduction in the propellant mass consumption and an increase of about 4-9% in the transfer time if compared with the same transfers computed using a continuous tangential thrust strategy.

The study conducted was a preliminary analysis, thus in depth examinations are the natural following. In particular, higher order approximation techniques of computing the periodic orbits and their manifold tubes are necessary to increase the accuracy of the resulting trajectories. Furthermore, specific mission constraints have to be considered in order to design more realistic space mission.

Finally, with the aim at reducing the computing time the direct multiple shooting method implemented in the resolution of the optimization problem needs to be refined and other methods to solve the optimization problem have to be investigated.

VII. Acknowledgments The author greatly thanks Professor Mariano Andrenucci not only for his scientific support during the whole

study period but also for his human attitude. Special thanks for supervising this study are directed to Doctor Pierpaolo Pergola, his helpful advises and comments were precious.

VIII. References 1 Belbruno, E., and Miller, J., “Sun-Perturbed Earth-to-Moon Transfers with Ballistic Capture,” Journal of Guidance,Control

and Dynamics, Vol. 16, 1993, pp. 770-775. 2 Vasile, M., Topputo, F., and Bernelli-Zazzera, F., “Low Energy Interplanetary Transfers Exploiting Invariant Manifolds of

the Restricted Three-Body Problem,” Journal of Astronautical Sciences, Vol. 53, 2005, pp. 353-372. 3Szebehely, V., Theory of orbits The Restricted Problem of Three Bodies, Academic Press, New York and London, 1967. 4Farquhar, R. W., “The flight of ISEE-3/ICE: origins, mission history and a legacy,” Journal of Astronautical Sciences, Vol.

49, No. 1, Jan. 2001, pp., 23-73. 5 Woodard, M., Folta, D., and Woodfork, D., “ARTEMIS: The First Mission to the Lunar Libration Points,” presented at the

21st International Symposium on Space Flight Dynamics (ISSFD), Toulouse, France, September 28 - October 2, 2009. 6 NASA Goddard Space Flight Center, “Mission Summary: James Webb Space Telescope,”, 1st May 2006 [online database],

URL: http://www.ipac.caltech.edu/irspace/JWST.pdf [cited 10 January 2012]. 7 Wallner, O., Ergenzinger, K., and Brouard, L., “EUCLID Mission Assessment Study Observing the Dark Universe with

EUCLID,” ESTEC Noordwijk Conference, 18 November 2009. 8Bock, H. G., and Plitt, K. J., “A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems,”

Proceedings of the 9th IFAC World Congress, edited by Colloquia, Vol. 9, Budapest, New York, 2-6 July 1984, pp. 242-247. 9Enright, P. J., and Conway, B. A., “Discrete Approximations to Optimal Trajectories Using Direct Transcription and

Nonlinear Programming”, Journal of Guidance, Control and Dynamics, Vol. 15, 1992, pp., 994-1002. 10Moser, J., “On the Generalization of a Theorem of A.Lyapunov”, Communications on Pure Applied Math, Vol. 11, 1958,

pp. 257-271. 11Jorba, A., and Masdemont, J., Dynamics in the Centre Manifold of the Collinear Points of the Restricted Three Body

Problem, Physica D, Vol. 132, No. 1-2, Jul. 1999, pp. 189-213. 12Gomez, G., Koon, W. S., Lo, M. W., Marsden, J. E., Masdemont, J., and Ross, S. D., Invariant Manifolds, the Spatial

Three-Body Problem and Space Mission Design, AIAA/AAS Astrodynamics Specialist Meeting, Quebec City, Canada, August 2001, AAS 01-301.

13Richardson, D. L., “A note on a Lagrangian Formulation for Motion about the Collinear Points”, Celestial Mechanics and Dynamical Astronomy, Vol. 22, No. 3, 1980, pp. 231-236.


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14Richardson, D. L., “Analytic Construction of Periodic Orbits about the Collinear Points”, Celestial Mechanics, Vol. 22, No. 3, 1980, pp. 241-253.

15Richardson, D. L., “Halo Orbit Formulation for the ISEE-3 Mission”, Journal of Guidance and Control, Vol. 3, No. 6, Nov. Dec. 1980, pp. 543-548.

16Nayfeh, A. H., Perturbation Methods, Pure and applied math, John Wiley & Sons, 1973. 17Thurman, R., and Worfolk, P. A., “The Geometry of Halo Orbits in the Circular Restricted Three-Body Problem”,

Geometry Center Research Report, GCG95, University of Minnesota, 1996. 18Bernelli Zazzera, F., Topputo, F., and Massari, M., “Assessment of Mission Design Including Utilization of Libration

Points and Weak Stability Boundaries”, Politecnico di Milano, ACT-RPT-MAD-ARI-03-4103b (unpublished). 19Roy, A. E., and Ovenden, M. W., “On the Occurrence of Commensurable Mean Motions in the Solar System. II The Mirror

Theorem”, Monthly Notices of the Royal Astronomical Society, Vol. 115, London, 1955, p.296-309. 20Ozimek, M. T., and Howell, K. C., “Low Thrust Transfers in the Earth-Moon System Including Application to Libration

Point Orbits”, Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, Mackinac Island, MI, 2007, Paper AAS07-43. 21Snecma Groupe SAFRAN, “PPS®-1350-G Stationary Plasma Thruster”, June 2009 [online database], URL:

http://www.snecma.com/IMG/pdf/PPS1350_ang-2.pdf [cited 10 January 2012]. 22Bryson, A., and Ho, Y., Applied Optimal Control, Wiley, New York, 1975. 23Betts, J., “Survey of Numerical Methods for Trajectory Optimization”, Journal of Guidance, Control and Dynamics, Vol.

21, No. 2, March-April 1998, pp. 193-207. 24Conway, B. A., “Spacecraft Trajectory Optimization”, Cambridge University Press, 2010. 25 Enright, P. J., and Conway, B. A., “Discrete Approximations to Optimal Trajectories Using Direct Transcription and

Nonlinear Programming”, Journal of Guidance, Control and Dynamics, Vol. 15, No. 4, July-August 1992, pp. 994-1002.

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