A METHOD TO DEFINE LOW-THRUST, INVARIANT

(Preprint) AAS 12-137

ATTAINABLE SETS IN SPACE MISSION DESIGN: A METHOD TO DEFINE LOW-THRUST, INVARIANT MANIFOLD TRAJECTORIES G. Mingotti∗, F. Topputo†, and F. Bernelli-Zazzera‡

A method to incorporate low-thrust propulsion into the invariant manifolds technique for space trajectory design is presented in this paper. Low-thrust propulsion is introduced by means of attainable sets that are used in conjunction with invariant manifolds to define first guess solutions in the restricted-three body problem. They are optimized in the restricted four-body problem where an optimal control problem is formalized. Several missions are investigated in the Earth– Moon system: transfers to libration point orbits and to periodic orbits around the Moon. Attainable sets allow the immediate design of efficient complex space trajectories.

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 non-Keplerian orbits may even serve the same purpose of conic orbits with a considerable reduction of propellant mass. This feature has been demonstrated by a class of Earth– Moon transfers that require less propellant than Hohmann transfers.2 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-body problem is considered. 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 the 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;3 the coupled restricted three-body problems approximation is instead used to design low energy transfers to the Moon.4 Although these methods permit to efficiently use the dynamics governing the spacecraft motion, they are based on the application of instantaneous velocity changes, or ∆v, and therefore ∗

Post-Doctoral Research Associate, Distributed Space Systems Lab, Faculty of Aerospace Engineering, Technion–Israel Institute of Technology, 32000 Haifa, Israel † PhD, Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Via La Masa 34, 20156 Milano, Italy. ‡ Full Professor, Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Via La Masa 34, 20156 Milano, Italy.

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they are not suitable when low-thrust propulsion is considered. Indeed, according to the rocket propulsion theory, the propellant mass fraction spent to achieve a ∆v is   mp ∆v = 1 − exp − , (1) mi Isp g0 where mp is the propellant mass, mi is the initial mass, Isp is the engine’s specific impulse, 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. Nevertheless, the methods described in2–5 model spacecraft maneuvers with instantaneous velocity changes. These impulses can be realized through high-thrust, low-Isp propulsion. On the other hand, equation (1) indicates that low-thrust, high-Isp systems are an appealing means when minimizing propellant mass. As an example, the Isp of ion engines is approximately ten times higher than that of chemical engines. However, the schemes based on instantaneous ∆v fails with low-thrust propulsion. The existing methods needs to be re-formulated to include the lowthrust propulsion, so combining the benefits of flying over an n-body vector field with those of having an elevate Isp . In this work, a method to design low-thrust, invariant-manifold trajectories in the Sun–Earth–Moon–Spacecraft four-body problem6 is presented. This is applied to compute transfers 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 specified 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 using 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 patchedconics 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.7–12 THE CIRCULAR RESTRICTED THREE-BODY PROBLEM In three dimensions, the circular restricted three-body problem (SCRTBP) studies the motion of the spacecraft, P3 , 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 are13 ∂Ω3 ∂Ω3 ∂Ω3 x ¨ − 2y˙ = , y¨ + 2x˙ = , z¨ = (2) ∂x ∂y ∂z where 1 1−µ µ 1 + + µ(1 − µ), (3) Ω3 (x, y, z, µ) = (x2 + y 2 ) + 2 r1 r2 2 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 r12 = (x + µ)2 + y 2 + z 2 , r22 = (x + µ − 1)2 + y 2 + z 2 . (4) For fixed µ, the Jacobi integral reads J(x, y, z, x, ˙ y, ˙ z) ˙ = 2Ω3 (x, y, z) − (x˙ 2 + y˙ 2 + z˙ 2 ),

2

(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 emanating from them.14–16 The generic periodic orbit about Li , i = 1, 2, is referred to as γi , whereas its stable and unstable manifolds are labeled Wγsi and Wγui , 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.13 When the motion of P3 is constrained on the plane z = 0, it is possible to derive the planar version of the restricted three-body problem. The dynamics of the planar circular restricted threebody problem (PCRTBP) are represented 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, therefore the planar Lyapunov orbits possess stable and unstable two-dimensional manifolds that act as separatrices for the states of motion.17 LOW-THRUST PROPULSION AND ATTAINABLE SETS To model the controlled motion of P3 under the gravitational attractions of P1 , P2 , and the lowthrust propulsion, the following differential equations are considered x ¨ − 2y˙ =

∂Ω3 Tx + , ∂x m

y¨ + 2x˙ =

∂Ω3 Ty + , ∂y m

z¨ =

∂Ω3 Tz + , ∂z m

m ˙ =−

T Isp g0

,

(7)

q where T = Tx2 + Ty2 + Tz2 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 ∈ [ti , 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. 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. 18. Definition of Attainable Set Let φT (τ ) (xi , ti ; t) be the flow of system (7) at time t under the guidance law T (τ ), τ ∈ [ti , t], and starting from (xi , ti ) with xi = (xi , yi , zi , x˙ i , y˙ i , z˙i , mi ). The generic point of a tangential low-thrust trajectory is x(t) = φT (xi , ti ; t), (8)

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p where T = T v/v, v = (x, ˙ y, ˙ z), ˙ v = x˙ 2 + y˙ 2 + z˙ 2 , 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. 10, 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 (xi , ϕ, τ ) = φT (xi , ti ; τ )|τ ≤ t , (9) where the dependence on the initial state xi 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 guidance law T (τ )) on S(ϕ): [ AT (ϕ, τ ) = γT (xi , ϕ, τ ). (10) xi ∈X

According to the definition in Eq. (10), the attainable set is made up by orbits that reach 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. 12, 19.) 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. FROM ATTAINABLE SETS TO OPTIMAL TRAJECTORIES Once first guess solutions are achieved by using the attainable sets formulation previously described, they are later optimized in a four-body framework, under the perspective of optimal control The Controlled Bicircular Restricted Four-Body Problem The model, in the spatial case, 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 Tx + , ∂x m

y¨ + 2x˙ =

∂Ω4 Ty + , ∂y m

z¨ =

∂Ω4 Tz + , ∂z m

T θ˙ = ωS , m ˙ =− . (11) Isp g0

This is the controlled version of the spatial bicircular restricted four-body problem (SBRFBP)6, 20 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 ms − 2 (x cos θ + y sin θ). rs ρ

(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

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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 rs2 = (x − ρ cos θ)2 + (y − ρ sin θ)2 + z 2 .

(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 controlled planar bicircular restricted four-body problem (PBRFBP) is achieved by setting z = 0 in Eqs. (11)–(13). This model is used in the remainder to design Earth–Moon low-energy, low-thrust transfers. Optimal Control Problem Formulation The optimal control aims at finding the guidance law T (τ ), τ ∈ [ti , tf ], that minimizes Z tf T (τ ) J= dτ. I sp g0 ti

(14)

It is easy to verify through the last of Eqs. (11) that J is the propellant mass; i.e., J = mi − mf , where mi , 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 p p (x + µ)2 + y 2 + z 2 > RE , (x + µ − 1)2 + y 2 + z 2 > RM , (16) where RE and RM are the normalized mean radii of the Earth and Moon, respectively. The initial boundary condition is q q (xi + µ)2 + yi2 = rE , x˙ 2i + y˙ i2 = vE − rE , (17) (xi + µ)(x˙ i − yi ) + yi (y˙ i + xi + µ) = 0, zi = z˙i = 0, which enforces the spacecraft to be at the periapsis of a planar Earth-parking orbit uniquely specified by periapsis altitude and eccentricity, hpE , eE , respectively (rE = RE + hpE is the periapsis radius; p 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.21 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.22 More specifically, a multiple shooting scheme is implemented.23 With this strategy, Eqs. (11) are forward integrated within N − 1 intervals in which [ti , tf ] is split. This

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is done assuming N points and constructing the mesh ti = 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 intervals in the form of defects as ¯ j − xj+1 = 0, ηj = x

j = 1, . . . , N − 1

(18)

¯ j = φT (τ ) (xj , tj ; tj+1 ), τ ∈ [tj , tj+1 ]. To compute T (τ ) a second-level time discretization with x 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 satisfying both boundary and path constraints (Eqs. (15)–(17)), and minimizing the performance index (Eq. (14)). (The final boundary condition is specified in the following Sections 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. SCENARIO 1: PLANAR LOW-ENERGY, LOW-THRUST 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. (See Ref. 4, 5, 9, 10, 18, 24–28 for more details.) Impulsive Low Energy Transfers to the Moon 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, Wγs2 (SE), for given CSE , are computed. The solution space is studied with the aid of Poincar´e 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 lying on Wγs2 (SE) asymptotically approach γ2 in forward time, orbits inside Wγs2 (SE) are transit orbits (that pass from the Earth region to the exterior region), whereas orbits outside Wγs2 (SE) are non-transit orbits14, 15, 17 (Fig. 1(a)). Candidate trajectories to construct the Earth-escape portion are those non-transit orbits close to both Wγs2 (SE) and Wγu2 (SE). The set ESE on a suitable chosen Poincar´e section (SB ) is the set of Earth-escape orbits that intersect the departure orbit. 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 Wγs2 (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 Wγs2 (EM ) (Fig. 1(a)). It is possible to represent this set on a proper Poincar´e section

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(a) ESE Earth-escape trajectory in the SE model.

(b) KEM Moon-capture trajectory in the EM model.

Figure 1. Decoupling the four-body problem into two PCRTBP: the SE and EM models.

(SC ); when a coordinate transformation is introduced, both sets ESE and KEM can be plotted on the same surface (Fig. 2(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. 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: q q (xi + µ)2 + yi2 = rE , x˙ 2i + y˙ i2 = vE − rE , (xi + µ)(x˙ i − yi ) + yi (y˙ i + xi + µ) = 0. (19) 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, hpM , prescribed by the mission requirements. The transfer terminates when the spacecraft is at the periapsis of this orbit. While both eM and hpM 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 (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 cos ωM , yf = rM sin ωM ,

where rM = RM

x˙ f = (rM − vM ) sin ωM , y˙ f = (vM − rM ) cos ωM , p + hpM and vM = µ(1 + eM )/rM . The domain of admissible final states is X M = {xf (ωM )|ωM ∈ [0, 2π]},

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(20)

(21)

(a) Sample low-thrust capture portion.

(b) Transfer point determination.

Figure 2. Preliminary determination of low-energy, low-thrust transfers to the Moon. The transfer point is found by intersecting the attainable set AM (ϕ, −τ ) and the T Earth-escape set ESE .

and the attainable set, for some ϕ, τ > 0, is ATM (ϕ, −τ ) =

[

γT (xf (ωM ), −τ ).

(22)

xf ∈X M

Since the first part of the transfer is defined on ESE , the transfer points, if any, that generate lowenergy, low-thrust transfers are contained in the set M Tϕ,−τ = ESE ∩ ATM (ϕ, −τ ).

(23)

M Tolerable mismatch can be admitted in Tϕ,−τ as discontinuities are spread in the subsequent optimization step. Fig. 2 shows the transfer point determined for a sample low-energy, low-thrust transfer to the Moon. Eq. (20) is evaluated with varying ωM to construct the set of admissible initial conditions according to Eq. (21). The first guess solutions found through Eq. (23) are optimized in the PBRFBP formalizing an optimal control problem with the following final boundary condition q q (xf + µ − 1)2 + yf2 = rM , x˙ 2f + y˙ f2 = vM − rM , (24) (xf + µ − 1)(x˙ f − yf ) + yf (y˙ f + xf + µ − 1) = 0,

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 1 where three sample solutions are reported. These are compared to some classical solutions, and to reference impulsive and lowthrust 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 maneuver, whose magnitude is ∆vi , 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 ∆vi

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Table 1. Low-energy, low-thrust (LELT) solutions with eE = 0 and eM = 0. They are optimized in the PBRFBP and are compared to a set of classical solutions (WSB: weak stability boundary, BP: bi-parabolic, HO: Hohmann, BE: bi-elliptic;2 MIN: minimum theoretical29 ) as well as to reference impulsive (Ref. 20, 30–35) and low-thrust (Ref. 8, 9, 35–40) solutions.

hpE [km]

hpM [km]

Isp [s]

∆vi [m/s]

mcp /mT LI

mp /mi

∆t [days]

LELT#1 LELT#2 LELT#3

200 200 200

100 100 100

3000 3000 3000

3211 3210 3211

0.050 0.050 0.050

0.681 0.680 0.681

145 149 153

WSB BP HO BE MIN

167 167 167 167 167

100 100 100 100 100

300 300 300 300 300

3161 3232 3143 3161 3099

0.205 0.217 0.253 0.285 0.190

0.729 0.739 0.742 0.756 0.717

90-120 ∞ 5 55-90 —

Ref. 32 Ref. 20 Ref. 33 Ref. 34

167 167 167 167

100 100 100 100

300 300 300 300

3126 3137 3265 n.a.

0.211 0.216 0.192 n.a.

0.727 0.730 0.733 0.731

292 43 255 85

Ref. 36, 37 Ref. 39

315 584

n.a. 1000

n.a. 1673

— —

— —

0.070 0.174

7 ∼500

Type

is reported in Table 1. 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 Z tf mp T (τ ) 1 HT = [1 − exp(−∆vi /(Isp g0 ))] + dτ, (25) LT g mi mi ti Isp 0 where the first part (in square brackets) is the propellant fraction associated to the impulsive, HT = 300 s), whereas the second part is the propellant mass spent in high-thrust maneuver (Isp LT = 3000 s). The initial mass, m , is calculated such that a fixed mass of the low-thrust arc (Isp i mT LI = 1000 kg is placed into translunar orbit. The term mcp /mT LI in Table 1 is introduced to indicate the propellant mass fraction of the Moon capture phase. For low-energy, low-thrust solutions, mcp /mT LI is obtained through Z tf T (τ ) J= dτ, (26) ti Isp g0 whereas for reference impulsive solutions this term takes into account all the maneuvers necessary to carry out the transfer except ∆vi . Eccentricity and periapsis altitudes of initial, final orbits are also reported in Table 1. It can be seen that the overall propellant mass fraction, mp /mi , is lower than that associated to all reference impulsive solutions having comparable initial and final orbits. This LT , that is ten times I HT . More specifically, presented is due to the low-thrust specific impulse, Isp sp 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 addition, when the performance of the Moon capture only is concerned, the presented solutions show a notable reduction of the relative propellant mass

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(a) Trajectory in the Earth-centered frame.

(b) Trajectory in the Moon-centered frame.

(c) Thrust profile.

(d) Mass consumption.

Figure 3. Transfer solution LELT#1 in Table 1.

fraction, mcp /mT LI , with respect to all reference solutions with similar final orbits. As for the lowthrust 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 1 are computed in the Earth–Moon restricted three-body problem, there is evidence that the low-energy, low-thrust solutions exploit the natural four-body dynamics in a more efficient way. Fig. 3 shows the solution LELT#1 of Table 1. In details, the transfer orbit presented in the Earthcentered frame (Fig. 3(a)) shows a capture mechanism similar to that of exterior WSB transfers2 (this is in agreement with the discussions in Ref. 4, 24). 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. 3. SCENARIO 2: SPATIAL LOW-THRUST, STABLE MANIFOLD TRANSFERS TO HALO ORBITS IN THE EARTH–MOON SYSTEM When Wγs1 (SE), the stable manifold of an L1 halo orbit in the SE model with out-of-plane amplitude Az = 105 km, is computed, it is evident that this set approaches the Earth, and it can be

10

shown that this happens in the Sun–Earth system for a wide class of orbits about both L1 and L2.41 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 3200 m/s is sufficient to insert the spacecraft onto these stable manifolds departing from low-Earth orbits.3, 16, 41 Impulsive Transfers to Earth–Moon Halo Orbits 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 where Wγs1 (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. The stable manifold of an L1 halo orbit in the SE model with out-of-plane amplitude Az = 105 km 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 L2.41 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 3200 m/s is sufficient to insert the spacecraft onto these stable manifolds departing from low-Earth orbits.3, 16, 41 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. 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. 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. (19). 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 lowthrust system is used to raise the orbit up to target a point on the stable manifold Wγsj , j = 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 upon the argument of perigee, xi = xi (ω E ), through xi = rE cos ωE − µ, yi = rE sin ωE , x˙ i = (rE − vE ) sin ωE , y˙ i = (vE − rE ) cos ωE , (27) p where rE = RE + hpE and vE = (1 − µ)(1 + eE )/rE . The domain of admissible initial states can be written as X E = {xi (ωE )|ωE ∈ [0, 2π]}, (28)

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E (ϕ, τ ) (x, y coordinates). (a) AT

(b) Wγs2 , (x, y coordinates).

(c) AE (ϕ, τ ) ∩ Wγs2 , (r, z coordinates). T

(d) AE (ϕ, τ ) ∩ Wγs2 , (r, vr coordinates). T

Figure 4. Preliminary determination of low-thrust, stable-manifold transfers to halo orbits. The transfer point is found by intersecting the attainable set and the stable p ˙ vr = r, ˙ vz = z.) ˙ manifold. (r = x2 + y 2 , vt = rθ,

and therefore the attainable set, for some ϕ, τ > 0, is AE (ϕ, τ ) = T

[ xi

γT (xi (ωE ), τ ).

(29)

∈X E

Once the halo orbit γj , j = 1, 2, is given, its stable manifold Wγsj can be generated. The transfer points that generate low-thrust, stable-manifold transfers, are given by E Tϕ,τ = AE (ϕ, τ ) ∩ Wγsj . T

(30)

Time τ in Eq. (31) 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 E = ∅. It is worth mentioning that first guess solutions are being generated with Eq. these cases Tϕ,τ (31). Thus, small discontinuities can be tolerated when looking for the transfer point. Two states are deemed as intersecting if |xA − xW | ≤ ε, where xA ∈ ATE (ϕ, τ ), xW ∈ Wγsi , and ε is a given

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Table 2. Low-thrust, stable manifold (LTSM) solution for transfers from low-Earth and GTO orbits to L1 and L2 halos, optimized in the RFBP. A set of low-thrust references (Ref. 8,42–44) is reported for the sake of comparison.

Type LTSM#1 LTSM#2 LTSM#3 LTSM#4 Ref. 43 Ref. 44 Ref. 42 Ref. 8

hpE [km]

eE

Az [km]

Li

Isp [s]

mp /mi

∆t [days]

200 400 200 400

0 0.72 0 0.72

8000 8000 8000 8000

L1 L1 L2 L2

3000 3000 3000 3000

0.171 0.090 0.183 0.091

178 91 195 114

20000 400 488 400

0 0.72 n.a. 0.72

13200 8000 7000 16000

L1 L1 L2 L2

2000–3700 3000 425 3000

0.119 0.096 0.058 0.089

84 89 365 107

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 sets ATE (ϕ, τ ) and Wγsi have different dimensions, being the first planar 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 ∈ Wγsi with z = 0. As the spatial flow of the (ϕ, τ ) ∩ Wγsi ∩ {z = 0} SCRTBP is not tangential to the {z = 0} plane, possible intersection AE T 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 4 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 SB (T = 0.5 N, τ = 14.80 EM time units, and ϕ = −π/6). Note that according to definitions os the dynamics of the problem, all low-thrust orbits reach S(−π/6) at different times, although they have the same thrust duration τ . The optimal control problem for low-thrust transfers to halo orbits is defined with the following final boundary condition xf = xW ,

xW ∈ Wγsj ,

(31)

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.8, 45, 46 Low-Thrust, Stable-Manifold Transfers to Halo Orbits Optimal low-thrust, stable-manifold solutions are presented in Table 2. Some sample solutions 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 a GTO with 400 km perigee altitude. Table 2 reports also some known low-thrust reference solutions. The final mass is LT = 3000 s have instead a state of dynamical system in case of low-thrust; mi = 1000 kg and Isp been considered in this case. The dramatic reduction of propellant mass ratio (mp /mi ) 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.8, 44 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,

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(a) Trajectory in the Earth–Moon frame.

(b) Thrust profile.

Figure 5. Transfer solution LTSM#4 in Table 2.

departing from GTO requires about half of the propellant mass associated to low-Earth orbits, and about half transfer time. A sample low-thrust, stable-manifold solution (LTSM#4 in Table 2) is reported in Fig. 5. Fig. 5(a) shows the trajectory in the configuration space; Fig. 5(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. REFERENCES [1] R. Farquhar, “Lunar Communications with Libration-Point Satellites,” Journal of Spacecraft and Rockets, Vol. 4, 1967, pp. 1383–1384. [2] E. Belbruno and J. Miller, “Sun-Perturbed Earth-to-Moon Transfers with Ballistic Capture,” Journal of Guidance, Control, and Dynamics, Vol. 16, 1993, pp. 770–775. [3] K. Howell, B. Barden, and M. Lo, “Application of Dynamical Systems Theory to Trajectory Design for Libration Point Missions,” The Journal of the Astronautical Sciences, Vol. 45, 1997, pp. 161–178. [4] W. Koon, M. Lo, J. Marsden, and S. Ross, “Low Energy Transfer to the Moon,” Celestial Mechanics and Dynamical Astronomy, Vol. 81, 2001, pp. 63–73. [5] G. G´omez, W. Koon, M. Lo, J. Marsden, J. Masdemont, and S. Ross, “Invariant Manifolds, the Spatial Three-Body Problem and Space Mission Design,” Advances in the Astronautical Sciences, Vol. 109, 2001, pp. 3–22. [6] C. Sim´o, G. G´omez, A. Jorba, and J. Masdemont, “The Bicircular Model near the Triangular Libration Points of the RTBP,” From Newton to Chaos, Vol. 1, Plenum Press, New York, 1995, pp. 343–370. [7] F. Topputo, Low-Thrust Non-Keplerian Orbits: Analysis, Design, and Control. PhD thesis, Politecnico di Milano, Milano, Italy, 2007. [8] G. Mingotti, F. Topputo, and F. Bernelli-Zazzera, “Combined Optimal Low-Thrust and Stable-Manifold Trajectories to the Earth–Moon Halo Orbits,” American Institute of Physics Conference Proceedings, Vol. 886, 2007, pp. 100–110, 10.1063/1.2710047. [9] G. Mingotti, F. Topputo, and F. Bernelli-Zazzera, “Low-Energy, Low-Thrust Transfers to the Moon,” Celestial Mechanics and Dynamical Astronomy, Vol. 105, 2009, pp. 61–74. [10] G. Mingotti, F. Topputo, and F. Bernelli-Zazzera, “Numerical Methods to Design Low-Energy, LowThrust Sun-Perturbed Transfers to the Moon,” Proceedings of the 49th Israel Annual Conference on Aerospace Sciences, Tel Aviv – Haifa, Israel, 2009. [11] G. Mingotti, Trajectory Design and Optimization in Highly Nonlinear Astrodynamics. PhD thesis, Politecnico di Milano, Milano, Italy, 2010.

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