Optimal Low-Thrust, Invariant Manifold Trajectories via ... · of diﬀerent three-body systems.46 In Ref. 18, Earth–Mars transfers with ballistic escape and low-thrust capture

Optimal Low-Thrust, Invariant Manifold

Trajectories via Attainable Sets∗

Giorgio Mingotti†

Universitat Paderborn, Paderborn, 33098, Germany

Francesco Topputo‡

Politecnico di Milano, Milano, 20156, Italy

Franco Bernelli-Zazzera§

Politecnico di Milano, Milano, 20156, Italy

A method to incorporate low-thrust propulsion into the invariant man-

ifolds technique is presented in this paper. The low-thrust propulsion is

introduced by means of special attainable sets that are used in conjunction

with invariant manifolds to define a first guess solution. This is later opti-

mized in a more refined model where an optimal control formalism is used.

Planar low-energy, low-thrust transfers to the Moon as well as spatial low-

thrust, stable-manifold transfers to halo orbits in the Earth–Moon system

are presented. These solutions are not achievable via neither patched-conics

methods nor standard invariant manifolds technique. The aim of the work is

to demonstrate the usefulness of the proposed method in delivering efficient

solutions, which are compared to known examples.

Nomenclature

A Attainable set

∗Part of the work in this paper has been presented at “CelMec V: The fifth international meeting oncelestial mechanics”, Viterbo, Italy, 6–12 September, 2009.

†Post Doctoral Fellow, Institute fur Industriemathematik, Warburger Str. 100. E-mail: [email protected]

‡Post Doctoral Fellow, Dipartimento di Ingegneria Aerospaziale, Via La Masa, 34. E-mail:[email protected]

§Full Professor, Dipartimento di Ingegneria Aerospaziale, Via La Masa, 34. AIAA Senior Member. E-mail: [email protected]

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Low-Thrust, Invariant Manifold Trajectories via Attainable Sets – Mingotti, Topputo, Bernelli

Az Out-of-plane amplitude of halo orbits

C Jacobi energy

c Vector of equality constraints

E Set of escape conditions

e Orbital eccentricity

g0 Gravitational acceleration at sea level

g Vector of inequality constraints

hp Periapsis altitude

Isp Specific impulse

J , J Jacobi integral

J Objective function

K Set of capture conditions

Li Lagrangian points, i = 1, . . . , 5

N Number of multiple shooting intervals

M Number of multiple shooting subnodes

m0,p Spacecraft initial mass, propellant mass

m1,2 Mass of P1,2

mS Mass of the Sun

m Spacecraft mass

m Mass flow rate

P1,2 Larger, smaller primary

P3 Third body

RE,M Normalized Earth, Moon radius

r1,2 Distance of P3 from P1,2

rS Distance of P3 from the Sun

S Surface of section

W s,uγi

Stable, unstable manifold of periodic orbit about Li, i = 1, 2

T Transfer points set

T , T Low-thrust vector, thrust magnitude

t Time

v, v Velocity vector, speed

X Set of admissible initial conditions

x Spacecraft state

x, y, zCoordinates of P3

γ Orbit of differential equations

γi Periodic orbit about Li, i = 1, 2

∆t Transfer time

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∆v Instantaneous velocity change

θ Angular phase of the Sun

η Vector of multiple shooting defects

µ Mass parameter

ρ Distance of the Sun from the Earth–Moon barycenter

τ Argument of T , duration of low-thrust arc

φ Flow of differential equations

ϕ Surface of section angle

Ω3,4 Potential of the three-, four-body problem

ωs Angular velocity of the Sun

ω Argument of perigee

Superscript

(˙) Differentiation with respect to time

E,M Earth, Moon

HT High-Thrust

LT Low-Thrust

Subscripts

0, f Initial, final

1, N Initial, final (direct transcription)

E,M Earth, Moon

i Lagrange point, i = 1, . . . , 5

TLI Trans-Lunar Injection

x, y, zVector component along x, y, z

I. Introduction

Non-Keplerian orbits have proven to be a viable solution to accomplish more and more

demanding mission requirements that cannot be achieved with conic orbits, solution of the

two-body problem. This is the case, for instance, of periodic orbits about equilibrium points

of three-body systems. These orbits offer unique opportunities as the spacecraft is at rest

with respect to a pair of primaries.1 It has been shown that using non-Keplerian orbits in

place of conic orbits may involve possible reductions of propellant mass. This is demonstrated

by a class of Earth–Moon transfers that require less propellant than Hohmann transfers.2, 3

In principle, a non-Keplerian orbit is a solution defined in the vector field generated

by n mass particles, n ≥ 3. When studying the motion of a spacecraft, the restricted n-

3 of 27


body problem is considered, and the orbit of the spacecraft is sought while the orbits of

the primaries are given. Designing space trajectories in this framework is not trivial, as the

analyticity, typical of the two-body problem, is lost. Dynamical system theory has recently

been proposed as a valuable means to fill this gap. It has been used to design trajectories

that exploit the phase space structure of the restricted n-body problem in a natural way.

This includes using stable and unstable manifolds associated to Lagrange points and periodic

orbits around them. As a result, methodologies to design both libration point missions and

low energy interplanetary transfers have been formulated. In short, to access a libration

point orbit, it is sufficient to place the spacecraft on the stable manifold associated to that

orbit;4–7 the coupled restricted three-body problem approximation is instead used to design

low energy transfers to the Moon.8–11 The term “invariant manifolds technique” is used in

the remainder to label these methodologies.

In this work, a method to design low-thrust, invariant-manifold trajectories in the Sun–

Earth–Moon–Spacecraft four-body problem12 is presented. This is applied to compute trans-

fers to the Moon and to the Lagrange point orbits. Low-thrust propulsion is handled by

means of suitably devised sets labeled as “attainable sets”. In essence, an attainable set

is a collection of orbits propagated from a set of admissible initial conditions with a spec-

ified time and with a prescribed thrust profile. Managing attainable sets means handling

many candidate solutions at once — rather than a single low-thrust orbit. The idea of us-

ing attainable sets consists in imitating the role played by invariant manifolds in trajectory

design. Attainable sets can be intersected with invariant manifolds to define non-Keplerian

orbits that are not achievable with neither patched-conics methods nor standard invariant

manifolds technique. This paper elaborates on previous works by the same authors aimed at

integrating together knowledge coming from dynamical system theory and optimal control

problems for the design of efficient low-energy, low-thrust trajectories.13–18

Motivations

The standard invariant manifolds technique, used to design either trajectories to libration

point orbits4–7 or low energy transfers,8–11, 19 is used to derive efficient solutions that take

advantage of the three-body dynamics. Nevertheless, these methods are based on the ap-

plication of instantaneous velocity changes (∆v). These impulses can be realized through

high-thrust, low-Isp propulsion systems.

According to the rocket propulsion theory, the propellant mass fraction spent to achieve

a given ∆v ismp

m0= 1− exp

(

−∆v

Isp g0

)

, (1)

where mp is the propellant mass, m0 is the initial spacecraft mass, Isp is the specific impulse,

4 of 27


and g0 is the gravitational acceleration at sea level. Trajectories defined within n-body

problems may involve lower ∆v than patched-conics transfers, and therefore they may require

less propellant mass, but their efficiency can be further improved. Eq. (1) indicates that low-

thrust, high-Isp systems represent an appealing solution when minimizing propellant mass.

As an example, the Isp of ion engines is approximately ten times higher than that of chemical

engines. However, there exist no standard approaches to incorporate the low-thrust into the

invariant manifold technique as a simple part of generating an initial solution without some

type of continuation process. Thus, the invariant manifold technique needs to be rethought

to include the low-thrust propulsion in its formulation. Such approach would combine the

benefits of flying over an n-body vector field with those of having a large Isp.

Background

Transfers to Libration Point Orbits. Designing transfers to libration point orbits

dates back to the ISEE-3 mission in 1978.20, 21 Since then, the phase portrait about collinear

points of the restricted three-body problem has been fully understood.22, 23 In particular,

as periodic orbits possess stable and unstable manifolds,4, 24, 25 some effort has been spent

in exploiting these features.26 Modern methods face the problem of designing transfers

to libration point orbits under the perspective of dynamical system theory.4–6 The well

established method to design transfers to libration point orbits can be shortly stated: in

order to reach the final orbit at a zero cost, the spacecraft has to be placed on the stable

manifold associated to the periodic orbit. Once on this manifold, the dynamical system

provides at bringing the spacecraft to its nominal orbit. Recent Genesis’ trajectory has been

designed with this procedure.7

Low Energy Transfers Using Invariant Manifolds. Low energy transfers exploit

the ballistic capture upon arrival to reduce the excess velocity, typical of hyperbolic ap-

proaches. This kind of transfer has been formulated at the end of ’80s and has been success-

fully used in 1991 to rescue the Japanese spacecraft Hiten.2, 3 Based on observations made in

Ref. 27, the ballistic capture mechanism has been reviewed under the perspective of dynami-

cal system theory.9 As invariant manifolds of planar libration point orbits act as separatrices

for the states of motion,28 ballistic capture orbits are defined in the region of the phase space

enclosed by the stable manifold emanating from a “source” periodic orbit. This observation,

together with the coupled restricted three-body approximation, has allowed reformulating

Earth-to-Moon low energy transfers,9, 11 and designing novel moon-to-moon transfers about

outer planets.8, 10 This method has also been extended to design transfers in systems with

non-intersecting manifolds.19

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Low-Thrust Propulsion in n-Body Problems. Low-thrust propulsion has been used

in conjunction with an n-body coast arc in the definition of the trajectory for the LGAS

mission.29 In this context, a spiral arc is patched with a transit orbit to reach the Moon.

The resulting trajectory recalls the one performed by SMART-1.30 In recent years, some

effort has been spent to derive optimal low-thrust orbits in n-body dynamical frameworks.

In particular, the use of invariant manifolds as first guess solutions for low-thrust trajectory

optimization within accurate dynamical models is described in Ref. 31. In these models,

capture and escape orbits have been obtained with sophisticated optimization algorithms.32

In addition, low-thrust propulsion has been used within the restricted three-body problem to

design both interplanetary transfers33–35 and transfers to the Moon.14, 15, 17, 36–40 Concerning

transfers to libration point orbits, low-thrust arcs from high altitude Earth orbits to halos of

the Earth–Moon system have been presented in Ref. 41. To take advantage of the three-body

dynamics, low-thrust orbits have to target a piece of stable manifold associated to the final

orbit. This is shown in Ref. 13, 14, 42, 43 for the Earth–Moon system, and in Ref. 44, 45 for

the Sun–Earth system. The same concept has been used to design transfers between halos

of different three-body systems.46 In Ref. 18, Earth–Mars transfers with ballistic escape and

low-thrust capture are computed.

II. The Spatial Circular Restricted Three-Body Problem

In the spatial circular restricted three-body problem (SCRTBP) the motion of the space-

craft, P3, is studied in the gravitational field generated by the mutual circular motion of two

primaries, P1, P2 of masses m1, m2, respectively, about their common center of mass. The

equations of motion are47

x− 2y =∂Ω3

∂x, y + 2x =

∂Ω3

∂y, z =

∂Ω3

∂z(2)

where

Ω3(x, y, z, µ) =1

2(x2 + y2) +

1− µ

r1+

µ

r2+

1

2µ(1− µ), (3)

and µ = m2/(m1+m2) is the mass parameter of problem. Eqs. (2) are written in barycentric

rotating frame with normalized units: the angular velocity of P1, P2, their distance, and the

sum of their masses are all set to 1. It is easy to verify that P1, of mass 1− µ, is located at

(−µ, 0, 0), whereas P2, of mass µ, is located at (1 − µ, 0, 0); thus, the distances between P3

and the primaries are

r21 = (x+ µ)2 + y2 + z2, r22 = (x+ µ− 1)2 + y2 + z2. (4)

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For fixed µ, the Jacobi integral reads

J(x, y, z, x, y, z) = 2Ω3(x, y, z)− (x2 + y2 + z2), (5)

and, for a given energy C, it defines a five-dimensional manifold

J (C) = (x, y, z, x, y, z) ∈ R6|J(x, y, z, x, y, z)− C = 0. (6)

The projection of J (C) onto the configuration space (x, y, z) defines the Hill’s surfaces

bounding the allowed and forbidden regions of motion associated to C.

The vector field defined by Eqs. (2) has five well-known equilibrium points, the Euler–

Lagrange points, labeled Lk, k = 1, . . . , 5. This study deals with the portion of the phase

space surrounding the two collinear points L1 and L2. In a linear analysis, these two points

behave like the product saddle × center × center. Thus, in their neighborhood there exist

families of periodic orbits together with two-dimensional stable and unstable manifolds em-

anating from them.4, 24, 25 The generic periodic orbit about Li, i = 1, 2, is referred to as γi,

whereas its stable and unstable manifolds are labeled W sγiand W u

γi, respectively.

Eqs. (2) are used in this paper alternatively to describe the motion in the Sun–Earth

(SE) or Earth–Moon (EM) system. The mass parameters used for these models are µSE =

3.0359× 10−6 and µEM = 1.21506683× 10−2, respectively.47

The Planar Circular Restricted Three-Body Problem This is a version of the

restricted three-body problem in which the motion of P3 is constrained on the plane z = 0.

The dynamics of the Planar Circular Restricted Three-Body Problem (PCRTBP) are repre-

sented by the first two equations in Eqs. (2) (with z = 0 in Eqs. (3)–(4)). In this problem,

the Jacobi integral is a four dimensional manifold, and its projection on the configuration

space (x, y) defines the Hill’s curves. The linear behavior of L1, L2 is saddle × center, there-

fore the planar Lyapunov orbits possess stable and unstable two-dimensional manifolds that

act as separatrices for the states of motion.28

III. Low-Thrust Propulsion and Attainable Sets

To model the controlled motion of P3 under the gravitational attractions of P1, P2, and

the low-thrust propulsion, the following differential equations are considered

x− 2y =∂Ω3

∂x+

Tx

m, y + 2x =

∂Ω3

∂y+

Ty

m, z =

∂Ω3

∂z+

Tz

m, m = −

T

Isp g0, (7)

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where T =√

T 2x + T 2

y + T 2z is the present thrust magnitude. Continuous variations of the

spacecraft mass, m, are taken into account through the last of Eqs. (7). This causes a

singularity arising when m → 0, beside the well-known singularities given by impacts of P3

with P1 or P2 (r1,2 → 0). The guidance law, T (t) = (Tx(t), Ty(t), Tz(t)), t ∈ [t0, tf ], in Eqs. (7)

is not given, but rather it is found through an optimal control step where objective function

and boundary conditions are specified (see Section IV). However, in order to construct a

first guess solution, the profile of T over time is prescribed at this stage. Using tangential

thrust, attainable sets can be defined in the same fashion as reachable sets are defined in

Ref. 33.

Definition of Attainable Set

Let φT (τ)(x0, t0; t) be the flow of system (7) at time t under the guidance law T (τ), τ ∈

[t0, t], and starting from (x0, t0) with x0 = (x0, y0, z0, x0, y0, z0, m0). The generic point of a

tangential low-thrust trajectory is

x(t) = φT(x0, t0; t), (8)

where T = Tv/v, v = (x, y, z), v =√

x2 + y2 + z2, and T is a given, constant thrust

magnitude. With given T , tangential thrust maximizes the variation of Jacobi energy, which

is the only property that has to be considered when designing trajectories in a three-body

framework. (The thrust tangential to the inertial velocity maximizes variation of the orbit’s

semi-major axis; in Ref. 16, a comparison between tangential thrust in either rotating or

inertial frame shows negligible differences in the final optimal solution).

Let S(ϕ) be a surface of section perpendicular to the (x, y) plane and forming an angle

ϕ with the x-axis. The low-thrust orbit, for a chosen angle ϕ, is

γT(x0, ϕ, τ) = φ

T(x0, t0; τ)|τ ≤ t , (9)

where the dependence on the initial state x0 is kept. In Eq. (9), τ is the duration of the

low-thrust law, whereas t is the time at which the orbit intersects S(ϕ). The orbit γT

is

entirely thrust when τ = t; a thrust arc followed by a coast arc can be achieved by setting

τ < t.

The attainable set is a collection of low-thrust orbits (all computed with the same guid-

ance law T (τ)) on S(ϕ):

AT(ϕ, τ) =

⋃

x0∈X

γT(x0, ϕ, τ). (10)

According to the definition in Eq. (10), the attainable set is made up by orbits that reach

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S(ϕ) at different times, although all orbits have the same thrust history and therefore the

same mass. (This definition improves that given in Ref. 18, 35.)

Attainable set in Eq. (10) is associated to a generic domain of admissible initial conditions

X ; it will be shown how X can be defined for the two case studies. Attainable sets can be

used to incorporate low-thrust propulsion in a three-body frame with the same methodology

developed for the invariant manifolds. More specifically, invariant manifolds are replaced by

attainable sets, which are manipulated to find transfer points on a surface of section.

IV. From Attainable Sets to Optimal Trajectories

The Controlled Spatial Bicircular Restricted Four-Body Problem

First guess solutions achieved by using attainable sets are optimized in a four-body framework

under the perspective of optimal control. The model used to take into account the low-thrust

propulsion and the gravitational attractions of the Sun, the Earth, and the Moon is

x− 2y =∂Ω4

∂x+

Tx

m, y + 2x =

∂Ω4

∂y+

Ty

m, z =

∂Ω4

∂z+

Tz

m, θ = ωS, m = −

T

Isp g0. (11)

This is the controlled version of the spatial bicircular restricted four-body problem (SBRFBP)12, 48

and, in principle, it incorporates the perturbation of the Sun into the EM model. The four-

body potential Ω4 reads

Ω4(x, y, z) = Ω3(x, y, z, µEM) +ms

rs−

ms

ρ2(x cos θ + y sin θ). (12)

The physical constants introduced to describe the Sun perturbation have to be in agreement

with those of the EM model. Thus, the distance between the Sun and the Earth–Moon

barycenter is ρ = 3.88811143× 102, the mass of the Sun is ms = 3.28900541× 105, and its

angular velocity with respect to the EM rotating frame is ωS = −9.25195985 × 10−1. The

Sun is located at (ρ cos θ, ρ sin θ, 0), and therefore the Sun-spacecraft distance is

r2s = (x− ρ cos θ)2 + (y − ρ sin θ)2 + z2. (13)

It is worth noting that this model is not coherent because all three primaries are assumed

to move in circular orbits. Nevertheless, the SBRFBP catches basic insights of the real four-

body dynamics as the eccentricities of the Earth’s and Moon’s orbits are 0.0167 and 0.0549,

respectively, and the Moon’s orbit is inclined on the ecliptic by just 5 deg.

The Planar Bicircular Restricted Four-Body Problem. The controlled planar

bicircular restricted four-body problem (PBRFBP) is achieved by setting z = 0 in Eqs.

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(11)–(13). This model is used in the remainder to design Earth–Moon low-energy, low-thrust

transfers.

Optimal Control Problem Definition

The optimal control aims at finding the guidance law T (τ), τ ∈ [t0, tf ], that minimizes

J =

∫ tf

t0

T (τ)

Isp g0dτ. (14)

It is easy to verify through the last of Eqs. (11) that J is the propellant mass; i.e., J =

m0 −mf , where m0, mf are the initial, final spacecraft mass. The thrust magnitude must

not exceed a maximum threshold given by technological constraints. This is imposed along

the whole transfer through

T (t) ≤ Tmax, (15)

where Tmax is the maximum available thrust. In addition, the following path constraints are

imposed to avoid impacts with the Earth and Moon along the transfer

√

(x+ µ)2 + y2 + z2 > RE ,√

(x+ µ− 1)2 + y2 + z2 > RM , (16)

where RE and RM are the normalized mean radii of the Earth and Moon, respectively. The

initial boundary condition is

√

(x0 + µ)2 + y20 = rE,√

x20 + y20 = vE−rE , (x0+µ)(x0−y0)+y0(y0+x0+µ) = 0, z0 = z0 = 0,

(17)

which enforces the spacecraft to be at the periapsis of a planar Earth-parking orbit uniquely

specified by periapsis altitude and eccentricity, hEp , e

E , respectively (rE = RE + hEp is the

periapsis radius; vE =√

(1− µ)(1 + eE)/rE is the periapsis velocity).

Solution by Direct Transcription and Multiple Shooting

The optimal control problem is transcribed into a nonlinear programming problem by means

of a direct approach.49 This method generally shows robustness and versatility, and does not

require explicit derivation of the necessary conditions of optimality; it is also less sensitive

to variations of the first guess solutions.50 More specifically, a multiple shooting scheme is

implemented.51 With this strategy, Eqs. (11) are forward integrated within N − 1 intervals

in which [t0, tf ] is split. This is done assuming N points and constructing the mesh t0 =

t1 < · · · < tN = tf . The solution is discretized over these N grid nodes; i.e, xj = x(tj).

The matching of position, velocity, Sun phase, and mass is imposed at the endpoints of the

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intervals in the form of defects as

ηj = xj − xj+1 = 0, j = 1, . . . , N − 1 (18)

with xj = φT (τ)(xj , tj; tj+1), τ ∈ [tj , tj+1]. To compute T (τ) a second-level time discretiza-

tion is implemented by splitting each of the N − 1 interval into M − 1 subsegments. The

control is discretized over the M subnodes; i.e., T j,k, j = 1, . . . , N , k = 1, . . . ,M . A

third-order spline interpolation is achieved by selecting M = 4. Initial and final time t1,

tN , are included into the nonlinear programming variables, so allowing the formulation of

variable-time transfers. The transcribed nonlinear programming problem finds the states

and the controls at mesh points (xj and T j,k) in the respect of Eqs. (18) while also satisfy-

ing both boundary and path constraints (Eqs. (15)–(17)), and minimizing the performance

index (Eq. (14)). (The final boundary condition is specified in Sections V and VI for the two

case studies). It is worth stressing that not only the initial low-thrust portion, but rather

the whole transfer trajectory is discretized and optimized, so allowing the low-thrust to act

also in regions preliminary made up by coast arcs. The optimal solution found is assessed a

posteriori by forward integrating the optimal initial condition using an eighth-order Runge–

Kutta–Fehlberg scheme (tolerance set to 10−12) by cubic interpolation of the discrete optimal

control solution.

V. Case Study 1: Planar Low-Energy, Low-Thrust Transfers to

the Moon

Impulsive Low Energy Transfers to the Moon

In literature, planar low energy transfers to the Moon are designed by decoupling the four-

body problem into two PCRTBP: the SE and EM models. Two different portions of the

transfer trajectory are designed apart in each of these two models by exploiting the knowledge

of the phase space about the collinear Lagrange points. The two legs are then patched

together in order to define the whole trajectory. This procedure is referred to as the coupled

restricted three-body problems approximation. It is briefly recalled. (See Ref. 8–11, 15, 16,

27, 33, 34, 46 for more details.)

In the planar SE model, the Jacobi energy, CSE, is chosen such that CSE . C2, where

C2 is the energy of L2. (The Earth-escape leg is constructed considering the dynamics

about L2; using L1 instead is straightforward.) The planar periodic orbit, γ2, and its stable

manifold, W sγ2(SE), for given CSE, are computed. The solution space is studied with the

aid of Poincare section. As these cuts represent two-dimensional maps for the flow of the

PCRTBP, it is possible to assess whether an orbit lies on the stable manifold or not. Orbits

11 of 27


lying on W sγ2(SE) asymptotically approach γ2 in forward time, orbits inside W s

γ2(SE) are

transit orbits (that pass from the Earth region to the exterior region), whereas orbits outside

W sγ2(SE) are non-transit orbits.24, 25, 28 Candidate trajectories to construct the Earth-escape

portion are those non-transit orbits close to both W sγ2(SE) and W u

γ2(SE). The set ESE is

the set of Earth-escape orbits that intersect the departure orbit (Fig. 1(b)).

Analogously, the Jacobi energy CEM , CEM . C2, is chosen in the planar EM model.

For an exterior Moon capture to occur, the dynamics about L2 is considered. The periodic

orbit γ2 associated to CEM and its stable manifold W sγ2(EM) are computed. Using again the

separatrix property, typical of the PCRTBP, the set leading to Moon capture, KEM , is defined

as the set of orbits inside W sγ2(EM). It is possible to represent this set on the same surface of

section used for the SE model (Fig. 1(b)). Low energy transfers to the Moon are then defined

as those orbits originated by ESE∩KEM . These two sets are characterized by different values

of the Jacobi constant, CSE and CEM , respectively, and therefore an intermediate impulsive

maneuver is needed to remove the discontinuity in velocity. In addition, two other impulsive

maneuvers are needed at both ends of the trajectory: the first one is needed to leave the

parking orbit and to place the spacecraft into a translunar trajectory; the second is instead

used to place the spacecraft into a stable, final orbit about the Moon.

x [SE frame]

y [S

E fr

ame]

0.995 1 1.005 1.01−8

−6

−4

−2

0

2

4

6

8x 10

−3

L2

γ2

Wu(SE)

Ws(SE)

Earth

Surface ofsection

Ws(EM)

(a) Intersection of manifolds.

1 2 3 4 5 6 7 8

x 10−3

−0.09

−0.08

−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

y [SE frame]

y [S

E fr

ame]

.

Wsγ2

(EM)

Transfer pointEM

ε

Wuγ2

(SE)

SE(Earth−escape orbits)

K

(Moon−capture orbits)

(b) Poincare section.

Figure 1. The coupled restricted three-body problems approximation as presented in Ref. 9. Once themanifolds of SE and EM model are computed, their intersections are properly used to define the transferpoint. The first guess so derived is shown bold.

The intersection ESE ∩ KEM defines the transfer point. The whole trajectory design is

reduced to the definition of this point, so indicating the conciseness of the method. The

mechanism is summarized in Fig. 1. The use of impulsive maneuvers is evident and intrinsic

in this methodology. Fig. 2 reports a sample solution as derived with the restricted three-

body problems approximation. The performances of two sample solutions are reported in

Table 1 for the purpose of comparison (optimizing these transfers is out of the scope of

12 of 27


the present work)a. The rocket equation (1) is evaluated with Isp = 300 s. Although it is

demonstrated that these solutions outperform the patched-conics trajectories in terms of

propellant mass,2, 3 their cost could be further reduced with low-energy, low-thrust transfers.

−1 0 1 2 3

−1

−0.5

0

0.5

1

1.5

2

X [EM length units]

Y [E

M le

ngth

uni

ts]

Earth

Moon Orbit

Transfer Point

Figure 2. A typical solution derived with the standard coupled restricted three-body approximation (Earth-centered inertial frame). The transfer trajectory is similar to those presented in Ref. 2, 3, 9, 11, 15, 16.

Sol. ∆t [days] ∆v [m/s] Isp [s] mp/m0

#1 95 3847 300 0.729

#2 90 3850 300 0.731

Table 1. Performances of the impulsive low energy transfers to the Moon in terms of time-of-flight (∆t), totalvelocity change (∆v), and propellant mass fraction (mp/m0). The costs refer to a 100km circular orbit aboutthe Moon and a 200km circular parking orbit around the Earth. These two solutions outperform patched-conictransfers (the cost for an Hohmann transfer between the same orbits is about 4000m/s; see Ref. 2).

Attainable Sets for Transfers to the Moon

Low-energy, low-thrust transfers to the Moon are defined as follows. The spacecraft is

assumed to be initially on a planar Earth-parking orbit as defined by Eq. (17). An impulsive

maneuver, carried out by the launch vehicle, places the spacecraft on a translunar trajectory;

from this point on, the spacecraft can only rely on its low-thrust propulsion to reach the final

orbit around the Moon. This orbit has moderate eccentricity, eM , and periapsis altitude,

hMp , prescribed by the mission requirements. The transfer terminates when the spacecraft is

at the periapsis of this orbit. While both eM and hMp are given, the orientation ωM of the

final orbit around the Moon is not fixed.

To build a first guess solution, the low-thrust term is assumed to act in the EM model

only, whereas the coast arc belongs to the Earth-escape set, ESE, defined in the SE model

aNext NASA’s GRAIL mission will use a 3.5-month low-energy transfer similar to that represented inFig. 2; see http://moon.mit.edu/design.html.

13 of 27


http://moon.mit.edu/design.html

(this assumption is removed in the trajectory optimization phase). The attainable set is

made up by tangential low-thrust orbits that are integrated backward from the final transfer

point. More specifically, the final state is function of the argument of periapsis, xf = xf (ωM),

through

xf = 1−µ+rM cosωM , yf = rM sinωM , xf = (rM−vM ) sinωM , yf = (vM−rM ) cosωM ,

(19)

where rM = RM + hMp and vM =

√

µ(1 + eM )/rM . The domain of admissible final states is

XM = xf (ωM)|ωM ∈ [0, 2π], (20)

and the attainable set, for some ϕ, τ > 0, is

AMT(ϕ,−τ) =

⋃

xf∈XM

γT(xf(ω

M),−τ). (21)

Since the first part of the transfer is defined on ESE, the transfer points, if any, that generate

low-energy, low-thrust transfers are contained in the set

T Mϕ,−τ = ESE ∩AM

T(ϕ,−τ). (22)

Tolerable mismatch can be admitted in T Mϕ,−τ as discontinuities are spread in the subsequent

optimization step. Fig. 3 shows the transfer point determined for a sample low-energy,

low-thrust transfer to the Moon. The attainable set AMT(ϕ,−τ) has been obtained with

T = 0.5N, τ = 6.90 EM time units, ϕ = π/2.

1 1.1 1.2 1.3 1.41.4−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

x [EM frame]

y [E

M fr

ame]

γ2

L2

ωM

(a) Sample low-thrust capture portion.

3 4 5 6 7 8

x 10−3

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

y [SE frame]

y [

SE

fram

e]

Wuγ2

(SE)

Wsγ2

(EM)

Transfer point

εSE

(Earth−escape orbits)

AMT

(Attainable set)

.

(b) Transfer point determination.

Figure 3. Preliminary determination of low-energy, low-thrust transfers to the Moon. The transfer point isfound by intersecting the attainable set and the Earth-escape set. The section in Fig. 3(b) is made at the samelocation of that reported in Fig. 1(b).

14 of 27


Final Boundary Conditions and Trajectory Optimization. Eq. (19) is evaluated

with varying ωM to construct the set of admissible initial conditions according to Eq. (20).

The first guess solutions found through Eq. (22) are optimized in the PBRFBP with the

procedure shown in Section IV. The optimal control problem is defined by Eqs. (14)–(17)

and by the following final boundary condition

√

(xf + µ− 1)2 + y2f = rM ,√

x2f + y2f = vM−rM , (xf+µ−1)(xf−yf )+yf(yf+xf+µ−1) = 0,

(23)

which enforces the transfer to end at the periapsis of the final orbit about the Moon.

Low-Energy, Low-Thrust Transfers to the Moon

Optimal low-energy, low-thrust solutions are presented in Table 2 where three sample so-

lutions are reported. These are compared to low-energy, high-thrust in Table 1, to some

classical solutions, and to reference impulsive and low-thrust solutions. Impulsive solutions

are compared to low-energy, low-thrust transfers in terms of propellant mass ratio. This is

achieved through Eq. (1) using the total ∆v in literature and assuming Isp = 300 s.

Low-energy, low-thrust solutions formulated in this work use an initial impulsive maneu-

ver, whose magnitude is ∆v0, that is supposed to be performed by the launch vehicle’s upper

stage when the spacecraft is on a 200 km circular parking orbit about the Earth. The value

of ∆v0 is reported in Table 2. For the sake of a fair comparison, the propellant mass spent

in this maneuver has to be considered together with that spent by the low-thrust system.

The complete propellant mass fraction used to assess the transfer efficiency is therefore

mp

m0= [1− exp(−∆v0/(I

HTsp g0))] +

1

m0

∫ tf

t0

T (τ)

ILTsp g0dτ, (24)

where the first part (in square brackets) is the propellant fraction associated to the impulsive,

high-thrust maneuver (IHTsp = 300 s), whereas the second part is the propellant mass spent

in the low-thrust arc (ILTsp = 3000 s). The initial mass, m0, is calculated such that a fixed

mass of mTLI = 1000 kg is placed into translunar orbit; this value has been used to integrate

the last of Eq. (11). The term mcp/mTLI in Table 2 is introduced to indicate the propellant

mass fraction of the Moon capture phase. For low-energy, low-thrust solutions, mcp/mTLI

is obtained through Eq. (14), whereas for reference impulsive solutions this term takes into

account all the maneuvers necessary to carry out the transfer except ∆v0. Eccentricity and

periapsis altitudes of initial, final orbits are also reported in Table 2.

It can be seen that the overall propellant mass fraction, mp/m0, is lower than that as-

sociated to all reference impulsive solutions having comparable initial and final orbits. This

is due to the low-thrust specific impulse, ILTsp , that is ten times IHTsp . More specifically,

15 of 27


Initial Orbit Final Orbit

Type hEp [km] eE hM

p [km] eM Isp [s] ∆v0 [m/s] mcp/mTLI mp/m0 ∆t [days]

LELT#1 200 0 100 0 3000 3211 0.050 0.681 145LELT#2 200 0 100 0 3000 3210 0.050 0.680 149LELT#3 200 0 100 0 3000 3211 0.050 0.681 153LEHT#1 200 0 100 0 300 3210 0.194 0.729 95LEHT#2 200 0 100 0 300 3211 0.195 0.731 90WSB 167 0 100 0 300 3161 0.205 0.729 90-120BP 167 0 100 0 300 3232 0.217 0.739 ∞HO 167 0 100 0 300 3143 0.253 0.742 5BE 167 0 100 0 300 3161 0.285 0.756 55-90MIN 167 0 100 0 300 3099 0.190 0.717 —Ref. 52 200 0 100 0.824 300 3165 0.022 0.666 93Ref. 53 200 0 100 1 300 3124 0 0.654 83Ref. 54 167 0 100 0 300 3126 0.211 0.727 292Ref. 48 167 0 100 0 300 3137 0.216 0.730 43Ref. 55 167 0 100 0 300 3265 0.192 0.733 255Ref. 56 167 0 100 0 300 n.a. n.a. 0.731 85Ref. 36 167 0 100 0 300 3134 0.238 0.737 14Ref. 37, 38 315 0 n.a. 0 n.a. 0 — 0.070 7Ref. 39 35908 0 1738 0 5000 0 — 0.058 33Ref. 30 584 0.716 1000 0.621 1673 0 — 0.174 ∼500Ref. 40 493 0.001 198 0.001 3300 0 — 0.791 253Ref. 14 400 0.723 100 0 3000 0 — 0.125 124Ref. 15 200 0 1000 0.650 3000 3195 0.031 n.a. 236Ref. 36 400 0.723 100 0 3000 0 0.051 0.150 116

Table 2. Comparison between low-energy, low-thrust (LELT) and low-energy, high-thrust (LEHT) solutions inTable 1. The former are optimized in the PBRFBP, the latter are derived with the RTBP approximation. LELTsolutions are also compared to a set of classical solutions (WSB: weak stability boundary, BP: bi-parabolic, HO:Hohmann, BE: bi-elliptic;2 MIN: minimum theoretical57) as well as to reference impulsive (Ref. 36,48,52–56)and low-thrust (Ref. 14, 15, 30, 36–40) solutions. LELT solutions departing from GTO (with hE

p = 200 km) areachieved by subtracting 2.453km/s from ∆v0; the size of the initial impulse changes, whereas the remaining partof the transfer is similar.

presented solutions have about 5% less propellant mass than a standard WSB transfer and

6.2% less than a Hohmann transfer. Moreover, comparing these results to the low-energy,

high-thrust transfers, an average reduction of 5% of propellant mass is also achieved. In ad-

dition, when the performance of the Moon capture only is concerned, the presented solutions

show a notable reduction of the relative propellant mass fraction, mcp/mTLI , with respect

to all reference solutions with similar final orbits. As for the low-thrust reference solutions,

presented results outperform those in Ref. 40 in both cost and transfer time (this is the only

low-thrust reference work with comparable initial and final orbits). Although all reference

low-thrust transfers reported in Table 2 are computed in the Earth–Moon restricted three-

body problem, there is evidence that the low-energy, low-thrust solutions exploit the natural

16 of 27


four-body dynamics in a more efficient way.

Fig. 4 shows the solution LELT#1 of Table 2. The solution has been obtained with

N = 1500; the computational time is about 28 hours b. In details, the transfer orbit

presented in the Earth-centered frame (Fig. 4(a)) shows a capture mechanism similar to

that of exterior WSB transfers2 (this is in agreement with the discussions in Ref. 9,27). The

most distant point of the trajectory from the Earth is approximately four times the Earth–

Moon distance. The low-thrust capture and the thrust and mass profiles are also shown in

Fig. 4.

−1 0 1 2 3

−1

−0.5

0

0.5

1

1.5

2

2.5

3

x [EM length units]

y [E

M le

ngth

uni

ts]

(a) Earth-centered frame.

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1

−0.1

−0.05

0

0.05

0.1

x [EM length units]

y [E

M le

ngth

uni

ts]

(b) Moon-centered frame.

(c) Thrust profile. (d) Mass consumption.

Figure 4. Solution LELT#1 in Table 2.

bThe computational time reported here and below refers to the optimization only. Computing the attain-able sets takes about 1 hour with a standard pc.

17 of 27


VI. Case Study 2: Spatial Low-Thrust, Stable Manifold

Transfers to Halo Orbits in the Earth–Moon System

Impulsive Transfers to Earth–Moon Halo Orbits

Fig. 5(a) shows W sγ1(SE), the stable manifold of an L1 halo orbit in the SE model with

out-of-plane amplitude Az = 105 km. It is evident that this set approaches the Earth, and

it can be shown that this happens in the Sun–Earth system for a wide class of orbits about

both L1 and L2.58 Direct transfers from low Earth orbits are therefore possible with a single-

impulse maneuver. This impulse fills the energy gap between the departure orbit and the

stable manifold. The cost required to reach an halo orbit in the Sun–Earth system slightly

depends upon Az. A typical ∆v of about 3200m/s is sufficient to insert the spacecraft onto

these stable manifolds departing from low-Earth orbits.4, 5, 58

When transfers to halos in the Earth–Moon system are considered, the picture is different.

Although the SE model and the EM model and their behaviors are similar, transfers to halos

in the EM model represent a different design problem as the Earth is the largest primary in

this model. This causes the stable manifold to not approach the Earth (see Fig. 5(b) where

W sγ1(EM) with Az = 8 × 103 km is shown). Thus, a direct, single-impulse transfer from a

low Earth orbit is not permitted in the Earth–Moon frame. An intermediate arc from low

Earth orbit up to a point on the stable manifold has to be used. This leads to a two-impulse

strategy. The total cost depends upon both the transfer arc and the state targeted on the

stable manifold, as well as on Az and departure orbit. For the sake of subsequent comparison,

Table 3 reports the total cost of two sample transfers (optimizing these two-impulse transfers

would be out of the scope of this paper). The cost is presented in terms of total ∆v and

propellant mass fraction, computed through Eq. (1) with Isp = 300 s. The cost is relative

to a circular, 200 km departure orbit. The two orbits with Az = 8 × 103 km are chosen as

they have been proposed for space applications.1, 41 The reported costs slightly varies with

the size of the libration point orbit (a detailed study can be found in Ref. 58, 59). The two

solutions are shown in Fig. 6. The propellant mass fraction needed for this kind of transfers

can be significantly reduced with stable-manifolds, low-thrust transfers are formulated.

Type Az [km] ∆t [days] ∆v [m/s] Isp [s] mp/m0

L1 8000 67.3 3659 300 0.711

L2 8000 65.0 3676 300 0.713

Table 3. Cost needed to transfer a spacecraft from a 200km low-Earth orbit to two different halo orbits in theEarth–Moon system.

18 of 27


0.99 1

−4

−2

0

2

4

6

x 10−3

Earth

x

y

Transfer Trajectory

SE L1 Halo Orbit (Az = 100000 km)

(a) W sγ1, Sun–Earth system.

−0.5 0 0.5 1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

x

EM L1 Halo Orbit (Az = 8000 km)

y EarthMoon

(b) W sγ1, Earth–Moon system.

Figure 5. Stable manifold of two L1 halo orbits in the SE and EM systems (the orbits have out-of-planeamplitude Az = 105 km and Az = 8× 103 km, respectively).

−1 −0.5 0 0.5 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

EM L1 Halo (Az = 8000 km)

x

y

Manifold insertion

(a) Transfer to EM L1.

−1 −0.5 0 0.5 1 1.5−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

EM L2 Halo (Az = 8000 km)

x

yManifold insertion

(b) Transfer to EM L2.

Figure 6. Transfer to halo orbits about L1 and L2 in the Earth–Moon system (Az = 8× 103 km for both orbits;departure from a 200km circular parking orbit). ‘Manifold insertion’ indicates the point where the secondmaneuver is performed.

Attainable Sets for Transfers to Halo Orbits

Low-thrust transfers to halo orbits are defined as follows. The spacecraft is assumed to be

initially on a planar Earth-parking orbit as defined by Eq. (17). The argument of perigee,

ωE, of this orbit is not fixed. The transfer begins when the spacecraft is at the perigee.

From this point on, the low-thrust system is used to raise the orbit up to target a point on

the stable manifold W sγi, i = 1, 2. The out-of-plane amplitude, Az, of the final halo orbit is

assumed prescribed by mission requirements.

As both eccentricity and apsidal altitudes are prescribed, this initial state depends only

19 of 27


upon the argument of perigee, x0 = x0(ωE), through

x0 = rE cosωE −µ, y0 = rE sinωE, x0 = (rE − vE) sinωE, y0 = (vE − rE) cosω

E, (25)

where rE = RE+hEp and vE =

√

(1− µ)(1 + eE)/rE . The domain of admissible initial states

can be written as

XE = x0(ωE)|ωE ∈ [0, 2π], (26)

and therefore the attainable set, for some ϕ, τ > 0, is

AET(ϕ, τ) =

⋃

x0∈XE

γT(x0(ω

E), τ). (27)

Once the halo orbit γi, i = 1, 2, is given, its stable manifold W sγi

can be generated. The

transfer points that generate low-thrust, stable-manifold transfers, are given by

T Eϕ,τ = AE

T(ϕ, τ) ∩W s

γi. (28)

Time τ in Eq. (28) stands for the duration of tangential low-thrust. Typically, for short

times the low-thrust is not able to sufficiently raise the initial orbit such that the stable

manifold is reached; in these cases T Eϕ,τ = ∅.

It is worth mentioning that first guess solutions are being generated with Eq. (28). Thus,

small discontinuities can be tolerated when looking for the transfer point. Two states are

deemed as intersecting if |xA − xW | ≤ ε, where xA ∈ AET(ϕ, τ), xW ∈ W s

γi, and ε is a given

tolerance. The greater ε is, the higher number of first guess solutions is found; however, ε

should be kept sufficiently small to permit the convergence of the subsequent optimization

step.

The two setsAET(ϕ, τ) andW s

γihave different dimensions, being the first planar (according

to Eqs. (17),(26)–(27)) and the second three-dimensional. In detail, the only possibility for

the points xA, xW to match in configuration space is to restrict the search to only those

points xW ∈ W sγi

with z = 0. As the spatial flow of the SCRTBP is not tangential to the

z = 0 plane, possible intersection AET(ϕ, τ) ∩W s

γi∩ z = 0 would still produce solutions

with out-of-plane velocity discontinuity. If this mismatch is moderate, the discontinuity is

eliminated in the subsequent optimal control step.

Figure 7 shows the attainable set, a portion of the stable manifold (L2 halo with Az =

8000 km), and the transfer point all reported on a common surface of section (T = 0.5N,

τ = 14.80 EM time units, and ϕ = −π/6). Note that according to definitions in Eqs. (9)–(10)

all low-thrust orbits reach S(−π/6) at different times, although they have the same thrust

duration τ .

20 of 27


−0.5 0 0.5 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x [EM frame]

y [E

M fr

ame]

First guess solution

AET (Attainable set)

Portion of WSγ2

S(−π/6)

(a) AE

T(ϕ, τ) and W s

γ2(x, y coordinates). (b) AE

T(ϕ, τ) ∩W s

γ2, (r, z coordinates).

(c) AE

T(ϕ, τ) ∩W s

γ2, (r, vr coordinates). (d) AE

T(ϕ, τ) ∩W s

γ2, (r, vz coordinates).

Figure 7. Preliminary determination of low-thrust, stable-manifold transfers to halo orbits. The transfer pointis found by intersecting the attainable set and the stable manifold. (r =

√

x2 + y2, vr = r, vz = z.)

Final Boundary Conditions and Trajectory Optimization. The optimal control

problem for low-thrust transfers to halo orbits is defined by Eqs. (14)–(17) and by the

following final boundary condition

xf = xW , xW ∈ W sγi, (29)

which enforces xf to lie on the target stable manifold. This state can be described by

means of two parameters: one defined along the halo orbit and the other defined along the

manifold.14, 45, 60

Low-Thrust, Stable-Manifold Transfers to Halo Orbits

Optimal low-thrust, stable-manifold solutions are presented in Table 4. Four sample solu-

tions to halos around both L1 and L2 are reported. For each libration point, two different

initial orbits about the Earth have been considered: a circular, 200 km parking orbit and

21 of 27


a GTO with 400 km perigee altitude. Moreover, optimal high-thrust, stable-manifold solu-

tions in Table 3 have been reported. Table 4 reports also some known low-thrust reference

solutions. High-thrust solutions are compared to low-thrust solutions in terms of propellant

mass consumption, calculated through Eq. (1) with IHTsp = 300 s. The final mass is instead

a state of dynamical system (11) in case of low-thrust; m0 = 1000 kg and ILTsp = 3000 s have

been considered in this case. The dramatic reduction of propellant mass ratio (mp/m0) is

due to this difference in specific impulse as well as to the design strategy, the dynamical

model, and the transfer optimization. The presented solutions show costs and transfer times

that are close to the low-thrust reference solutions with similar departure orbits.14, 43

Analyzing the low-energy, low-thrust solutions only, it can be seen that the propellant

mass required to reach the halos around L2 is slightly higher than that needed for L1. This

is in agreement with the different Jacobi energy of the two libration point orbits. The flight

time needed to reach L2 is longer than that necessary to go to L1. Moreover, departing from

GTO requires about half of the propellant mass associated to low-Earth orbits, and about

half transfer time.

Initial Orbit Final Orbit

Type hEp [km] eE Az [km] Li Isp [s] mp/m0 ∆t [days]

LTSM#1 200 0 8000 L1 3000 0.171 178

LTSM#2 400 0.72 8000 L1 3000 0.090 91

LTSM#3 200 0 8000 L2 3000 0.183 195

LTSM#4 400 0.72 8000 L2 3000 0.091 114

HTSM#1 200 0 8000 L1 300 0.711 67

HTSM#2 200 0 8000 L2 300 0.713 65

Ref. 41 488 n.a. 7000 L2 425 0.058 365

Ref. 14 400 0.72 16000 L2 3000 0.089 107

Ref. 42 20000 0 13200 L1 2000–3700 0.119 84

Ref. 43 400 0.72 8000 L1 3000 0.096 89

Table 4. Low-thrust, stable manifold (LTSM) solution for transfers from low-Earth and GTO orbits to L1 andL2 halos. Optimal high-thrust, stable-manifold (HTSM) solutions of Table 3 have also been reported. Thelatter are derived in the RTBP whereas LTSM are optimized in the RFBP. A set of low-thrust references (Ref.14, 41–43) is reported for the sake of comparison.

A sample low-thrust, stable-manifold solution (LTSM#4 in Table 4) is reported in Fig. 8.

This solution has been obtained with N = 2250; the computational time is about 42 hours.

Fig. 8(a) shows the trajectory in the configuration space; Fig. 8(b) presents the guidance

law. The engine is on duty at the maximum level during the first part of the transfer. This is

the signature of the attainable set and first guess used to initialize the optimization. In the

second part of the transfer a small amount of control is needed to match the stable manifold

conditions. From this point on, no propulsion is needed to reach the L2 orbit.

22 of 27


−0.5 0 0.5 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x [EM frame]

y [E

M fr

ame]

Moon

L1 L2

(a) Optimal transfer trajectory.

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

Time [days]

T [N

]

T (t)T

max

(b) Optimal guidance law.

Figure 8. Solution LTSM#4 in Table 4.

VII. Conclusions

This work presents a method to include the low-thrust propulsion into the existing in-

variant manifolds techniques. Low-thrust orbits are handled with the definition of attainable

sets. Preliminary solutions are defined by intersecting attainable sets and invariant man-

ifolds. These are characterized by a thrust arc (that exploits the high specific impulse of

low-thrust systems) and a coast arc (that exploits the natural dynamics of the restricted

three-body problem). The preliminary solutions are optimized in the controlled four-body

problem. This modified version of the four-body model has been selected as an intermediate

step required for convergence of the algorithm. The convergence of the problem in the four-

body problem can serve as a continuation step as the solution moves toward a full model.

However, it is expected that the cost will not be affected too much in the transition from a

four-body problem to the full ephemeris model. The optimal control problem is solved via a

direct transcription and multiple shooting procedure. Thus, the optimal solutions presented

are local minima, as a local optimization scheme is used (the optimization converges to the

optimal solution in whose basin of attraction the first guess lies).

Planar low-energy, low-thrust transfers to the Moon have been formulated. It is shown

that these solutions outperform standard impulsive low-energy transfers as well as reference

low-thrust transfers computed in the Earth–Moon problem. This indicates that the low-

energy, low-thrust transfers exploit the gravitational field generated by the Sun, the Earth,

and the Moon in a more natural way.

Spatial low-thrust, stable-manifold transfers to halo orbits of the Earth–Moon system

have also been presented. These solutions considerably reduce the propellant mass when

compared to their analogous high-thrust version. The usefulness of having introduced an

attainable set formulation for these transfers is demonstrated by the fact that they show

23 of 27


performances close to reference solutions, which are computed with other means.

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Optimal Low-Thrust, Invariant Manifold Trajectories via ... · of diﬀerent three-body systems.46 In Ref. 18, Earth–Mars transfers with ballistic escape and low-thrust capture