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AAS 21-547

ECLIPSE-CONSCIOUS TRANSFERS TO LUNAR GATEWAY

USING EPHEMERIS-DRIVEN TERMINAL COAST ARCS

Sandeep K. Singh*, Brian D. Anderson†, Ehsan Taheri‡and John L. Junkins§

A novel indirect-based trajectory optimization framework is proposed that leverages ephemeris-

driven, “invariant manifold analogues” as long-duration asymptotic terminal coast arcs while

incorporating eclipses and perturbations during the optimization process in an ephemeris

model; a feature lacking in state of the art softwares like MYSTIC and Copernicus. The end-

to-end trajectories are generated by patching Earth-escape spirals to a judiciously chosen set

of states on pre-computed manifolds. The results elucidate the efﬁcacy of the proposed tra-

jectory optimization framework using advanced indirect methods and by leveraging a Com-

posite Smooth Control (CSC) construct. Multiple representative cargo re-supply trajectories

are generated for the Lunar Orbital Platform-Gateway (LOP-G). The results quantify accu-

rate ∆Vcosts required for achieving efﬁcient eclipse-conscious transfers for several launch

opportunities in 2025 and are anticipated to be used for analogous uncrewed lunar missions.

INTRODUCTION

Low-thrust electric propulsion systems are suitable for long-duration deep space missions. A variety of

engines have been developed to cater to low-thrust trajectories like solar electric propulsion (SEP) [1], and

beamed power [2]. Although the working concept may be derived from different underlying principles of

physics, low-thrust engines are typically characterized by high speciﬁc impulse (Isp) values, which directly

corresponds to high propulsion efﬁciency, and therefore less propellant is required for delivering the same

amount of ∆V. Trajectory design using low-thrust propulsion systems is a complex process mainly due to a

departure from natural dynamics caused by the continuous forcing function incorporated in the mathematical

model. Additionally, planet-centric phases of trajectories can be characterized by many orbital revolutions

(revs) and numerical convergence becomes a challenge for numerical solution methods. Local extremals

often occur, which can complicate the trajectory optimization process [3].

Optimization methods can be broadly classiﬁed into indirect and direct methods [4, 5], where indirect

methods require numerical solution of analytically derived necessary and sufﬁcient optimality conditions [6,

7], while direct methods involve an iterative procedure leading to sequentially improving some parameterized

approximation of the optimal solution. Indirect methods have undergone signiﬁcant recent enhancements

to alleviate associated challenges, with concepts like homotopy [8, 9, 10, 11] and arc length continuation

methods employed to make the problem more amenable to numerical treatment. Often times, a judicious

choice of the element space used to deﬁne the dynamical model also improves convergence [11, 12].

Most state of the art trajectory design software tools provide robust high-ﬁdelity solutions for optimal

transfers (e.g., Copernicus [13], with its built-in direct and indirect solvers and MYSTIC [14] that uses a

non-linear static/dynamic optimal control algorithm). However, Copernicus and MYSTIC do not incorporate

eclipse-induced coast arcs constraints within the optimization. Copernicus and MYSTIC do not currently

incorporate eclipses during the optimization process.

*PhD Candidate, Department of Aerospace Engineering, Texas A&M University, College Station, Texas 77843-3141, AAS Member.

†Current employment is Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109.

‡Assistant Professor, Department of Aerospace Engineering, Auburn University, Auburn, AIAA Senior Member.

§Distinguished Professor, Department of Aerospace Engineering, Texas A&M University, College Station, Texas 77843-3141, Honorary

Fellow AIAA.

1

Another vibrant research area in the realm of low-thrust transfers leverages insights from invariant man-

ifolds of the many families of periodic orbits existing in simpliﬁed dynamical models like the Circular Re-

stricted Three-Body Problem (CR3BP). For instance, Anderson and Lo [15] studied the role of invariant

manifolds in the dynamics of low-thrust trajectories passing through unstable regions of the three-body prob-

lem. Dellnitz et al. [16] used the reachable sets concept coupled with invariant manifolds of libration orbits

to solve for a low-thrust Earth-Venus transfer. Vaquero and Howell [17] studied the leverage provided by

resonant-orbit manifolds for designing transfers between Earth-Moon libration-point orbits. More recently,

Singh et al. [18] demonstrated a methodology to leverage stable/unstable manifold pairs of a halo orbit

around the Earth-Moon L1for end-to end transfers in the cis-lunar space. They also extended their work

to study the behavior of the invariant manifolds of Southern L2Near Rectilinear Halo Orbits (NRHOs) in a

multi-body system for a more accurate representation of the manifolds, and using them as terminal coast arcs

for trajectory design [19]. Similar approaches to low-thrust trajectory design also appear in [20, 21].

Although trajectory design in cislunar space seems well studied, real mission design requires eclipse-

conscious trajectories especially for spacecraft equipped with SEP engine for analyzing impact of eclipses

on mission cadence due to battery depth of discharge. Typically, engines can only be ON in ‘full-Sun’ and

power is consumed during eclipses to operate the spacecraft. Moreover, the use of invariant manifolds as

long-terminal coast arcs (LTCAs) reduces the problem complexity signiﬁcantly, where the resulting two-

point boundary-value problems (TPBVP) are easier to solve rather than an end-to-end transfer. The pre-

computation of ‘invariant manifold analogues’ for an ephemeris-corrected quasi-periodic libration-point orbit

provides more accurate patch-point states than the ones provided in [19].

In this paper, we ﬁrst present a methodology for pre-computation of stable/unstable manifold analogues for

an ephemeris-corrected quasi-periodic orbit, and demonstrate this procedure using the ephemeris-corrected

9:2 NRHO of the Southern Earth-Moon L2family. Secondly, we present an in-depth analysis towards selec-

tion of patch-points, which determines the judicial initial state for the LTCA and the terminal state anchor

vector for the ﬁrst phase of the transfer. Fuel-optimal transfers are formulated from two ‘super Synchronous

Geostationary Transfer Orbits’ (sGTOs) to a selected patch-point using an indirect optimization method. The

Composite Smooth Control (CSC) [22] method is exploited to cover multi-functional discontinuous switches

to both the thrust proﬁle due to eclipse and fuel-optimality conditions. The resulting trajectories patched with

the LTCA, provide accurate, efﬁcient and eclipse-conscious transfer trajectories with an ephemeris-driven

LTCA. Transfer opportunities have been listed for transferring a spacecraft from the considered sGTOs to the

quasi-periodic 9:2 NRHO of the southern L2family in the year 2025 with their respective fuel cost and eclipse

behavior. Finally, the impact of an early departure for one such transfer has been quantiﬁed elucidating the

differences between a typical fuel-optimal transfer and eclipse-conscious fuel-optimal transfers.

INVARIANT MANIFOLD ANALOGUES IN AN EPHEMERIS MODEL

In this section, an approach is presented for computing high-ﬁdelity perturbed analogues for CR3BP invari-

ant manifolds in an ephemeris-driven point-mass gravitational model (“Ephemeris Model”). These “invariant

manifold analogues” are not invariant in the strict sense, because the dynamical model is non-autonomous.

Using nearly-periodic segments of orbits in a high-ﬁdelity model and approximating them as periodic in

order to compute their invariant manifold analogues has been employed with success in missions such as

Genesis [23]. The advantage of this approach is that the structure of the manifolds is driven by the stability

properties of the orbit in the Ephemeris Model, and the trajectories “on” the manifold satisfy the Ephemeris

Model equations of motion. This method assumes the user has a “near-periodic” high-ﬁdelity orbit solution

available. An exact periodicity is unlikely due to the already discussed complexities of the Ephemeris Model.

Equations of Motion and Variational Equations

Without loss of generality, i= 1,2is chosen to represent the primary and secondary bodies from the

CR3BP. Naturally, the chosen bodies should approximate the behavior of bodies orbiting a common barycen-

ter. When deﬁning the simpliﬁed rotating frame, we will choose a ﬁxed angular rate and axis of rotation and

allow this axis to deﬁne the z-axis of the frame. Thus, Ω= [0,0, ω]>and ˙ω= 0, where the angular rate (ω)

is chosen to be the assumed constant mean motion of the primaries with the axis of rotation along the angular

2

momentum vector at a chosen reference epoch (t∗). Considering, three basis vectors listed in Eq. (1), Q0in

Eq. (2) is deﬁned as the rotation matrix from the inertial frame to the rotating frame at t∗. The triad of unit

vectors, with components in the Earth-Moon barycentric inertial frame are

ˆe1=R2(t∗)−R1(t∗)

|R2(t∗)−R1(t∗)|,ˆe2=ˆe3×ˆe1,ˆe3=(R2(t∗)−R1(t∗)) ×(˙

R2(t∗)−˙

R1(t∗))

|(R2(t∗)−R1(t∗)) ×(˙

R2(t∗)−˙

R1(t∗))|,(1)

Q0=ˆeT

1ˆeT

2ˆeT

3>,ˆei=unit vectors with Cartesian non-rotating components (2)

Using an additional simple time-varying rotation matrix Qz(t), deﬁned in Eq. (3),

Qz(t) =

cos(ω(t−t∗)) sin(ω(t−t∗)) 0

−sin(ω(t−t∗)) cos(ω(t−t∗)) 0

0 0 1

,(3)

the complete rotation matrix Q(t)as Q(t) = Q0Qz(t)can be formed. The coordinates in this rotating

frame to inertial coordinates can be related using Eq. (4). The displacement vectors relative to the chosen

center care ρi=Ri−Rcand ρ=R−Rc. The relative displacement and angular velocity vectors with

rotating components are

ri=Q(t)ρi,r=Q(t)ρ,ω=Q(t)Ω.(4)

The inertial point mass gravity forcing function is well known and leads to the inertial equations of motion

(Eq. (5)),

¨

R=F(t, R) = −X

i

µi(R−Ri)

|R−Ri|3,(5)

where µiis the gravitational parameter of body i. The gravitational forcing function can alternatively be

expressed as a function of the displacements relative to an accelerating center (which can serve as an idealized

inertial origin), as is shown in Eq. (6) as

φ(t, ρ) = −X

i

µi(ρ−ρi)

|ρ−ρi|3−X

i6=c

µi(ρi)

|ρi|3.(6)

Finally, the forcing function, when the time derivative of the displacement vectors are considered (relative

to the rotating frame) requires additional Centrifugal and Coriolis acceleration terms due to the simpliﬁed

deﬁnition of the frame and is shown in Eq. (7) as

¨

r=f(t, r,˙

r) = −X

i

µi(r−ri)

|r−ri|3−X

i6=c

µi(ri)

|ri|3

| {z }

fg(t,r,˙

r)

−2ω×˙

r−ω×(ω×r)

| {z }

fr(t,r,˙

r)

,(7)

which can be split into the gravitational forcing terms, fg(t, r,˙

r)and the kinematic terms due to the frame

rotation, fr(t, r,˙

r). The state vector and its time derivative are deﬁned in Eq. (8) as

x=r

˙

r,˙

x=˙

r

f(t, r,˙

r).(8)

In order to compute the variational equations, we require the Jacobian of Eq. (8), expressed with simpliﬁed

3x3 blocks in Eq. (9).

J(t, r,˙

r)≡∂˙

x

∂x=03x3I3x3

∂f

∂r

∂f

∂˙

r.(9)

3

Finally, the variational equations in the form of the well-known State Transition Matrix (STM) differential

equation, Eq. (10), is used for propagation, with initial value problem initialized with Φ(t0, t0)as a 6×6

identity matrix. ˙

Φ(t, t0) = J(t, r,˙

r)Φ(t, t0).(10)

Orbit Manifold Parameters

Let the nearly-periodic orbit have a period of Ti, where iis the rev number in consideration. The ﬂow

around a “ﬁxed point” can be characterized by studying the Monodromy matrix M(t0), which serves as a

local linearization of the ﬂow and can be computed as

M(t0) = Φ(t0+Ti, t0),(11)

using Eq. (10). In the current context, the apsidal period was chosen to deﬁne the period for each quasi-

periodic revolution. Then, computing the eigenvalues of M(t0)would ideally yield the properties: λ1=

1/λ2,λ3=λ4= 1,λ5=λ∗

6, where |λ1|>|λ2|, and λ∗

6indicates the complex conjugate of λ6. Since

the orbit is “near-periodic”, the eigenvalue set only approximates these characteristics. The eigenvectors

associated with the real eigenvalues λU≡λ1and λS≡λ2represent the unstable and stable eigenvectors,

respectively. These are labeled as, ξU,0≡ξU(t0)and ξS,0≡ξS(t0). Initial states for the unstable or stable

“invariant manifold analogue” at t0, assuming the state of the orbit is x0at t0, can be generated using the

following relation

xU(t0) = x0+sξU,0,xS(t0) = x0+sξS,0,(12)

where s=±1is the “sense” of the manifold and is the desired small offset magnitude along the manifold.

Due to the option of the sense, the unstable manifold can evolve in one of two directions, as can the stable

manifold. To produce an approximation of a manifold surface, it is standard practice to compute initial

manifold states on several sampled times (various choices for t0and corresponding states x(t0)) on an orbit

rev. The above process can be repeated for every desired point on the orbit, or alternatively, the state transition

matrix of the orbit itself can be used to approximately propagate the stable and unstable eigenvectors from a

reference state on the orbit to another state on the orbit. This is applied as

ξU(t)≈Φ(t, t0)ξU,0

|Φ(t, t0)ξU,0|,ξS(t)≈Φ(t, t0)ξS,0

|Φ(t, t0)ξS,0|.(13)

Once several initial states have been generated on the chosen manifold these states are then all propagated

using the dynamics in Eq. (8), with unstable manifolds propagated forward in time, and stable manifolds

backward in time.

Discretization of the Near-Periodic Orbit

In order to generate the manifold analogues, several points on the near-periodic rev should be analyzed to

reveal the “optimal” initial boundary condition. This process was carried out by ﬁrst considering a discretiza-

tion of states on the selected near-periodic rev followed by generation of the corresponding Monodromy

matrix. A spectral decomposition of M, reveals the stable/unstable eigenvector directions, thereby enabling

computation of the manifold analogues. Consider a near-periodic rev on an ephemeris-corrected, 9:2 reso-

nant NRHO of the southern L2family. A simple methodology for extraction of the discrete states include

discretization at uniformly spaced time intervals or uniformly spaced intervals in the eccentric anomaly, E.

Discretization in time leads to a sparse scattering of points near the periapse region, which often leads to a

restricted mapping of the manifold analogues. A more uniform spatial scattering achieved by discretization

in Eleads to a more comprehensive and robust mapping in our case.

ECLIPSE MODELING

Eclipse-conscious trajectories are deﬁned as trajectories where the engine throttle is turned ‘OFF’ (Thrust

= 0) whenever the spacecraft is in the shadow of an occulting body. An event-trigger handling capability is

4

required during the numerical optimization, which implicitly constrains the admissible control when applying

Pontryagin’s Minimum Principle (PMP). Several shadow models have been studied extensively [24, 25, 26].

In this paper, a conical shadow [25] model has been adopted for analyzing the manifold analogues as well as

designing eclipse-conscious optimal trajectories.

Figure 1: Schematic for the Conical Shadow Model.

The conical shadow model assumes spherical shapes of the occulting body and the Sun as is depicted in

Figure 1. The two bodies are viewed as overlapping discs by the spacecraft. For the spacecraft to be in at

least a partial eclipse shadow, the occulting body must block some of the angular view of the sun from the

spacecraft. Deﬁning the parameters Apparent Solar radius (ASr), Apparent Occulting Body radius (ABr) and

Apparent Distance (AD) as the following,

ASr = sin−1RS

||rS/sc|| ,ABr = sin−1RB

||rB/sc||,AD = cos−1rT

B/sc rS/sc

||rB/sc|| ||rS/sc|| ,(14)

where rS/sc =rS−rsc and rB/sc =rB−rsc. An implicit time-varying function, fec (rS/sc(t),rB/sc(t))

can be deﬁned, which determines the eclipse condition as: “Eclipse occurs when the sum of the apparent

angular radii exceeds the apparent angular distance.” The function fec is expressed as,

fec(rS/sc (t),rB/sc (t)) = ASr +ABr −AD. (15)

This time-varying function is evaluated at every time-step during state propagation and the sign (+ / -)

triggers the onset or exit from the eclipse event (Thrust: OFF / ON). For simplicity, we do not distinguish

between partial or total eclipse; we adopt the rule that the thrust should be OFF in even a partial eclipse, and

ON only in “full sun”. Note that additional relatively simple conditions on ASr, ABr and AD can be used

to distinguish umbra, penumbra, and antumbra. We use these conditions for post processing, but they do not

apply to the dynamics or optimization in our model.

MANIFOLD PATCH CONDITIONS

Using the procedure described in the preceeding sections, stable invariant manifold-analogues were gener-

ated for all “apo-apo” revs of the ephemeris-corrected 9:2 Southern L2NRHO. Several Earth-periapse states

on the generated trajectories were extracted for further analysis to gauge viability. Figure 2 shows stable

manifold analogues for 100 discrete points on the “apo-apo” rev starting ‘2025 JAN 02 20:02:28.677’.

These points were analyzed using phase portraits to pick the ‘best’ patch-point condition for trajectory

design. The primary consideration for selecting the patch-point, and thereby the manifold, was distance from

5

(a) Earth Mean Equator J2000 (EMEJ2000) frame. (b) Moon-centered rotating (MCR) frame.

Figure 2:Patch-point candidates on stable manifold analogues.

Earth (Rd). Other considerations were maximum eclipse duration (tec) and differences in the argument of

periapse (∆ω), eccentricity (∆e), and inclination (∆i) between the geocentric departure orbit and the patch

condition. While minimizing any combination of [Rd,∆ω, ∆e, ∆i], enable efﬁcient transfers in terms of

fuel consumption, tec is also an important parameter that can determine viability of the LTCAs and render

otherwise valid choices impractical. Once an attractive patch-point is selected, it can either be held ﬁxed or

can be subject to further reﬁnement in an optimization process. Also of signiﬁcance, the nature of temporal

and spatial behavior of manifold trajectories in a high-ﬁdelity model implies that each subsequent state on the

manifold propagates as a part of the original ballistic trajectory, enabling tracking via guidance algorithms.

This provides robustness to missed thrust-arcs in the preceding phases of the converged trajectory. Figure

Jan 25 Feb 25 Mar 25 Apr 25 May 25

Jun 25

Jul 25 Aug 25 Sep 25 Oct25 Nov 25 Dec 25

1 2 3 4 5 6 7 8 9 10 11 12

0

50

100

150

200

250

Max. Eclipse Duration (tec) (mins.)

Figure 3:tec for stable manifold analogues in year 2025.

3 depicts the maximum duration of eclipse (tec) experienced by the spacecraft while coasting on the stable

manifold analogues. Five ‘best’ cases in ascending order of Rdwere considered for each rev in YR 2025.

Note that several of the candidate LTCAs are devoid of eclipses and an a priori knowledge enables a judicious

choice of the manifold analogue and associated patch-condition for various launch period scenarios. It is

apparent that the worst case scenario in terms of eclipses occur in June 2025 with a maximum tec = 250

6

mins. Although this includes both ‘UMBRAL’ (full shadow) and ‘PENUMBRAL’ (partial shadow) eclipse

components, we are interested in the total eclipse duration to provide this essential analysis and demonstration

of indirect trajectory optimization subject to eclipse constraints.

Another important consideration, as discussed above is ∆ω=|ω0−ωpp|, where ω0and ωpp are the

arguments of periapse of the initial geocentric orbit and the target patch-point condition in the EMEJ2000

frame. Transfers dealing with a minimal ∆ωhave been found to be efﬁcient and cheaper due to minimal effort

required to rotate the major axis of the osculating orbits, leading to the the ﬁnal optimal osculating orbit before

insertion onto the manifold coast. Typical initial geocentric departure orbits have ω=ωnom

pp = 0° or 180°.

Therefore, in order to minimize ∆ω,ωpp ⊂ {L1, L2}where {L1, L2} ∈ Ris a small neighborhood around

ωnom

pp (±30°), depending upon the value of ω0. Figure 4 shows an ωpp vs. Rdphase portrait of all the candidate

patch-point conditions, segregated by the ‘MONTH’ of the year 2025 when the spacecraft gets inserted into

the NRHO. Note that the colorbar depicts the coast-time in days on the LTCAs.

Figure 4:ωpp vs. Rdfor candidate patch-points in the YR 2025.

In addition to the primary advantage of minimizing ∆ω, analysis of the ωpp vs. Rdphase portrait also

introduces a ﬂexibility in terms of mission launch window analysis and eclipse duration. In essence, it equips

mission designers with multiple patch-points to choose from, depending on the departure epoch and ωnom

pp .

An informed choice enables designers to circumvent many iterations of trajectory design in order to get a

favorable “Spacecraft - Sun - Occulting Body” geometry.

TIME-OPTIMAL AND FUEL-OPTIMAL CONTROL FORMULATIONS

Establishing the target NRHOs and propagation of their associated manifolds are performed using synodic

Cartesian frames. It is known, however, that the set of Cartesian coordinates are not suitable for efﬁcient con-

vergence of indirect optimal planet-centric, many-spiral trajectories during the Earth escape/capture phases

[12, 27, 28, 29]. Therefore, the set of modiﬁed equinoctial elements (MEEs) [30] are used to formulate time-

and fuel-optimal low-thrust trajectory optimization problems.

Let x= [p, f, g, h, k, l]>denote the state vector associate with MEEs. Their dynamics can be written as

˙

x(t) = A(x, t) + B(x, t)a,(16)

where A∈R6×1denotes the unforced vector part of the dynamics and B∈R6×3denotes the control

inﬂuence matrix with their explicit forms given in [12]. The total acceleration vector, a, expressed in the

Local-Vertical Local-Horizontal frame attached to the spacecraft can be written as

a=ap+ρp[aSun +aMoon],(17)

7

where ap,aSun, and aMoon denote accelerations due to the propulsion system, and perturbing accelerations

due to the Sun and Moon, respectively. In Eq. (17), ρp∈[0,1] denotes a continuation parameter that is used

to gradually include the non-linear third-body perturbations due to the Sun, and Moon. Two-body dynamics

corresponds to When ρp= 0, whereas ρp= 1 corresponds to the high-ﬁdelity model. Point-mass gravity

models are used for planetary perturbations and perturbation due to solar radiation pressure is ignored. The

acceleration due to the propulsion system can be written as

ap=T

mδsδoˆ

α,(18)

where δo∈[0,1] is the engine throttling input, ˆ

αdenotes the thrust unit direction vector, and δs∈[0,1]

reﬂects eclipse events. In Eq. (18), Tdenotes the maximum thrust value and mdenotes the spacecraft total

instantaneous mass. The time rate of change of mass of the spacecraft can be written as

˙m=−T

cδsδo,(19)

where c=Ispg0is the constant effective exhaust velocity. In this work, it is assumed that speciﬁc impulse

and the maximum thrust value of the thruster of the spacecraft remain constant during the entire maneuver.

Minimum-Time Formulation

For minimum-time formulation the cost functional is written as

minimize

δo,ˆ

αJ=Ztf

t0

dt, (20)

subject to :

Equations (16),(17) &(19),x(tf)−˜

xd=0,

x(t0) = x0, m(t0) = m0,

where ˜

xddenotes the set of desired (target) MEEs. Let λ= [λp, λf, λg, λh, λk, λl]>denote the costate vector

associated with the MEEs and let λmdenote the costate associated with mass. The Hamiltonian associated

with the minimum-time problem can be written as

HMT = 1 + λ>[A(x, t) + B(x, t)a]−λm

T

cδsδo.

Extremal (denoted by ‘*’ in superscript) control inputs, ˆ

α∗, and throttle factor, δ∗, become

ˆ

α∗=−B>λ

||B>λ||, δ∗

o= 1.(21)

For minimum-time problems, the optimal control strategy is to operate the thruster at its maximum capac-

ity; however, the eclipses have to be considered. Assuming ρsas a smoothing parameter, the eclipse factor

can be written as δs=1

2[1 −tanh(fec/ρs)], where fec (see Eq. (15)) is treated as a “distance measure”

associated with eclipse events. Since ﬁnal time is free, the ﬁnal value of the Hamiltonian has to be zero,

HMT(tf) = 0. The Euler-Lagrange equation is used to derive the costate dynamics using the Hamiltonian

associated with the minimum-time problem as ˙

λ>=−(∂HMT /∂x)and ˙

λm=−∂HMT /∂m.

Minimum-Fuel Formulation

For minimum-fuel formulation the cost functional is written as

minimize

δo,ˆ

αJ=Ztf

t0

T

cδsδodt, (22)

subject to :

Equations (16),(17) &(19),x(tf)−˜

xd=0,

x(t0) = x0, m(t0) = m0.

8

Formulation of the optimal control is straightforward. The Hamiltonian associated with the minimum-fuel

problem can be written as

HMF =T

cδo+λ>[A(x, t) + B(x, t)a]−λm

T

cδo.

PMP has to be used characterize extremal (denoted by ‘*’ in superscript) control inputs, ˆ

α∗, and throttle

factor, δ∗, as

ˆ

α∗=−B>λ

||B>λ||, δ∗

o= arg min

δo∈[0,1] HMF(x∗, δo,λ∗).(23)

The optimal δodepends on the switching function, SF , deﬁned as

δ∗

o=(1,for SF > 0,

0,for SF < 0,SF =c||B>λ||

m+λm−1,(24)

The Euler-Lagrange equation is used to derive the costate dynamics ˙

λ>=−(∂HMF /∂x)and ˙

λm=

−∂HMF /∂m. An additional admissible control constraint, resulting in eclipse-conscious solutions is δs= 0

during the course of the eclipse event. This can be implemented either by including a logical check on the

sign of fec from Eq. (15), or via PMP by restricting the admissible set δo. Derivation of the costate dynamics

(for both time- and fuel-optimal problems and in the presence of perturbations) is achieved using a symbolic

code as outlined in [22].

Optimization Algorithm

Equation (18) is the total acceleration delivered by the propulsion system and δoand δsdenote the com-

bined engine throttling input due to application of the PMP and encountering of eclipse events, respectively.

Since there are two sources that can lead to discontinuity in the thrust proﬁle, the CSC methodology [22, 31]

is used to alleviate non-smoothness issues. Assuming ρoand ρsas the respective smoothing parameters, the

optimal δ∗can be written as δ∗=δsδowith δs=1

2[1 −tanh(fec/ρs)]. Here, fec (see Eq. (15)) is treated as a

“distance measure” associated with eclipse events and SF is the standard thrust switching function. The pro-

posed high-ﬁdelity, Hybrid Eclipse-Conscious Trajectory Optimization Routine (HECTOR) is summarized

using the two Algorithms given below.

Algorithm 1: Hybrid Eclipse-Conscious Trajectory Optimization Routine (HECTOR) - TIME OPTI-

MAL

Result: Minimum time (t∗

f), Converged co-state vector (λ2b

0)

Departure States on the sGTO : (xo),Arrival States on the LTCA : ( ˜

xd)

while MT ≥∗do

Initial Guess: tf(Time of Flight) & λ(t0)

Propagate: Eqs. (16),(19) & Adjoint equations with ρp= 0, δ∗

o= 1, δs,ˆ

α∗;

MT =||MT|| =||x(tf)−˜

xd, HMT (tf)||;

end

In Algorithm 1 of the HECTOR, the minimum-time problem is solved. The unknown values are the

initial costates and total time of ﬂight. The value of ∗is set to 1.0×10−12 . The resulting TPBVP is

solved using a standard single-shooting method, however, the eclipses are incorporated into the formulation

through the δsfactor. Any coast arc during the minimum-time maneuver is due to (penumbral) eclipse events

and for the rest of the maneuver the thruster is always ON. The next step is to ﬁnd fuel-optimal solutions

as outlined in Algorithm 2 below. However, the time of ﬂight has to be greater than the time of ﬂight

of the minimum-time solution. Thus, a scalar parameter γis considered. The value of γdepends on the

problem. Here, it is considered to lie in a range γ∈[1.2,1.4]. In order to simplify the problem, the initial

9

problem corresponds to restricted two-body Earth-spacecraft dynamics (denoted by superscript ‘2b’). As a

consequence, the associated Hamiltonian, H2b

MF takes a simpler form (since ρp= 0).

In addition, the smoothing parameter ρsis set to 1 to handle the eclipse transitions. The value of ρsis

lowered to below a certain ρ∗

svalue to get sharp transitions at the entry and exit of eclipse-induced coast arcs.

Then, a step is initiated to decrease the value of ρobelow a certain threshold, ρ∗

oin order to obtain bang-off-

bang thrust proﬁles. The value of ρ∗

oin our simulations is set to 1.0×10−6. The ﬁnal step is to introduce

perturbations through ρp. When, ρpis equal to one, the solution associated with the high-ﬁdelity model is

obtained. While the value of ρpis increased, it is ensured that the ﬁnal solution corresponds to ρp= 1. We

emphasize that the Hamiltonian is updated according to the considered dynamics to take into account the

contribution of the perturbing accelerations with the costate differential equations updated accordingly [22].

Algorithm 2: Hybrid Eclipse-Conscious Trajectory Optimization Routine (HECTOR) - FUEL OP-

TIMAL

Result: Eclipse-Conscious Fuel-Optimal Transfer Trajectory in the HFM

Fuel-optimal: tMF

f=γ t∗

f∀γ > 1;Initial Guess:λ2b

0(ρs= 1)

while ρs≥ρ∗

sdo

while 2b

MF ≥∗do

Propagate: Eqs. (16), (19) & Adjoint equations with δ∗,ˆ

α∗;

2b

MF =||2b

MF|| =||x(tMF

f)−˜

xd, λm(tMF

f)||

end

λ2b

0=λ2b∗(Converged Value), ρs= 0.9ρs;

end

Initial Guess :λ2bS

0=λ2b∗ρs=ρ∗

s

while ρo≥ρ∗

odo

while 2bS

MF ≥∗do

Propagate: Eqs. (16), (19) & Adjoint equations with δ∗,ˆ

α∗;

2bS

MF =||2bS

MF || =||x(tMF

f)−˜

xd, λm(tMF

f)||

end

λ2bS

0=λ2bS∗(Converged Value), ρo= 0.9ρo;

Initial Guess :λhfS

0=λ2bS∗|ρo=ρ∗

o;ρs=ρ∗

s

while ρp≤1do

while hfS

MF ≥∗do

Propagate: Eqs. (16), (19) & Adjoint equations with

a=a∗

p(δ∗

s, δ∗

o,ˆ

α∗) + ρp(aSun +aMoon);

hfS

MF =||hfS

MF|| =||x(tMF

f)−˜

xd, λm(tMF

f)||

end

λhfS

0=λhfS∗(Converged Value), ρp= 1.1ρp;

end

A ﬁnal fuel-optimal TPBVP is solved with ρp= 1.

end

RESULTS: SGTO - 9:2 SOUTHERN L2NRHO

A representative problem was solved for transferring a spacecraft from a ‘Super Synchronous GTO’

(sGTO) to the 9:2 NRHO of the Southern L2family, using the stable invariant manifold analogues as LT-

CAs. This problem was aimed at providing accurate and efﬁcient transfer trajectories for resupply cargo

missions to the Lunar Gateway, proposed to be stationed in the said NRHO by 2024 [32]. With consideration

to the timeline, the ephemeris-corrected NRHO revs in the year 2025 were considered to generate manifold

analogues, having the full transfer occur in the year 2025.

10

Target Orbits and Spacecraft Parameters

Geocentric sGTOs are of great commercial value. The near-circular regime of the sGTOs has perigee above

the synchronous altitude, a region termed as the GEO graveyard belt [33]. These orbits ﬁnd use as storage

and disposal location for derelict geosynchronous satellite debris. For the current mission, we assume that the

resupply spacecraft has been initially placed in a geocentric elliptical orbit of the super synchronous domain.

As the name suggests, a spacecraft in this orbit would orbit the Earth at a faster angular rate than the Earth’s

rotation rate near perigee, and slower at apogee. The orbits in this domain typically have a somewhat larger

apogee than a GTO, as shown in Figure 5. Table 1 lists the orbital elements for the two orbits of the sGTO

category selected as the initial orbits for the representative problem.

Figure 5: Schematic depicting the domain of Geocentric orbits.

The two starting orbits (sGTO1and sGTO2) differ only in their argument of periapse as is given in the

Table 1. The choice of sGTO with ω=ωnom

pp = 0° or 180° was made depending on the departure epoch to

enable a favorable eclipse condition and therefore an efﬁcient transfer. Perigee is the ideal position to thrust in

order to raise apogee and vice versa, while on the other, eclipses are frequently longer around apogee than the

perigee, but can be more likely to occur at perigee than apogee depending on the 3D geometry and time of the

year. These considerations are pertinent to designing transfers analogous to the representative problem and

give rise to two competing effects, namely, efﬁcacy of thrusting and eclipse duration. Additionally, fuel-cost

for in-plane rotation of the transfer spirals is another criterion for selection of the departure orbit as discussed

in the previous section.

Table 1: Classical Orbital Elements of the sGTO.

Orbit a(km) e i (°) Ω(°) ω(°) ν(°) Period (hrs.) Center

sGTO144364 0.65 27 11.3044 180 0 25.832 Earth

sGTO244364 0.65 27 11.3044 0 0 25.832 Earth

The terminal orbit for the mission is the ephemeris-corrected 9:2 resonant NRHO of the Southern L2

family. Cartesian states for the Deep Space Gateway (DSG) platform in this orbit are available in the SPICE

kernels (BODY ID ‘-60000’) with respect to the Earth’s center (BODY ID ‘399’) [34]. The available states

span 15 years from ‘JAN 2, 2020’ to ‘FEB 11, 2035’, with the trajectory being continuous in position but has

repeated, small corrective velocity adjustments (∼1.86 mm/s). The dynamical model used for differential

correction was an n-body gravity model, with eclipse avoidance properties achieved by a judicious choice of

initial condition. Since the orbit is quasi-periodic, every rev has a slightly different orbital period with the

average being ∼6.562 days.

Additionally, the resupply spacecraft was assumed to have an initial mass (m0) of 1000 kg, with the engine

11

generating a maximum thrust (Tmax) of 0.5 N (approximately equivalent to 2 NASA NEXT engines [35])

resulting in a maximum thrust acceleration of 5.0×10−4m/s with a constant speciﬁc impulse of Isp = 1500 s.

Efﬁcient transfers from sGTO1/sGTO2to the NRHO via pre-computed stable invariant manifold analogues,

leveraging them as LTCAs were designed for the assumed spacecraft properties. The transfer from the sGTO

to the patch point was solved using the indirect formulation of the OCP, described in the previous section, in

an eclipse-conscious, fuel-optimal sense. While sGTO1was the choice for departure orbit for transfers with

insertion dates in January, February, March, October, November and December, for insertion dates in May,

June, July, August and September, sGTO2was the departure orbit.

Transfer : Eclipses on Fuel-optimal Coast Arcs

Eclipse-conscious transfers are mindful of all shadow encounters of the spacecraft. The formulation using

CSC enables the algorithm to recognise onset and termination of eclipses on all intermediate spirals and

enforces an ‘OFF’ condition for the engine. These eclipse regions on the transfer spirals can occur in between

two consecutive thrust arcs ‘THRUST - ECLIPSE - THRUST’, between a thrust and a coast arc ‘THRUST

- ECLIPSE - COAST’ or between consecutive coast arcs ‘ COAST - ECLIPSE - COAST’. In this section, a

transfer opportunity is presented, where eclipses occurr in between consecutive coast arcs.

The spacecraft on this transfer trajectory gets inserted in the NRHO on ‘25 JAN 2025 18:39:50.103’. The

total transfer time was 106.59 days with the spacecraft coasting on the LTCA for 50.59 days. The total ∆V

for the transfer was 1641.867 m/s. The transfer trajectory is shown in Figure 6, plotted in the EMEJ2000

frame. Other important transfer events are : departure on ‘11 OCT 2024 04:17:37.145’ and patching with the

LTCA on ‘06 DEC 2024 04:17:37.145’.

Figure 6: Full Transfer : sGTO1- 9:2 NRHO (EMEJ2000).

Note that highlighted ‘green’ regions of the spirals depict the part of the trajectory when the spacecraft

is in Earth’s umbral shadow and the ‘cyan’ regions depict the penumbral shadow regions. Here, all the

shadow regions coincide with the fuel-optimality-driven coast arcs (due to PMP). For the given departure and

insertion epochs, the time-varying Sun-Earth-Spacecraft geometry results in such a favourable situation from

a fuel-optimality point of view. Since, the eclipse-induced coast arcs which would have been forced coasts in

the midst of neighboring thrust arcs, are on fuel-optimal coast arcs, the optimal thrusting sequence remains

intact. This efﬁcacy would be elucidated upon comparing the ∆Vcost with the former.

The bang-off-bang throttle sequence for the transfer trajectory is depicted in Figure 7. The ‘black’ colored

discontinuous function represents the eclipse condition (0: No Eclipse, 1: Umbral, 2: Penumbral), whereas

the ‘red’ plot depicts the corresponding thrust sequence. The ‘blue’ curve represents the switching function

used for tangent hyperbolic smoothing. The blown-up image in Figure 7 clearly shows the eclipse arcs being

12

0 10 20 30 40 50 60 70 80 90 100 110

TOF (Days)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

SF,Thrust, Eclipse condition

SF

Thrust (N)

Eclipse Condition

Figure 7: Thrusting sequence, eclipse condition, and switching function vs. time.

out of phase with the thrust arcs, and lying on the intermediate fuel-optimal-driven coast arcs. Thus, there are

no eclipse-induced coast arcs in this transfer trajectory. An eclipse on the selected LTCA is also evident from

the ﬁgure, which occurs 65.46 days after departure. Finally, as mentioned before, the frequency of eclipses is

Figure 8: Thrusting Sequence and Eclipse Condition.

also an important parameter along with their duration. Figure 8 depicts this data displayed in a histogram plot

with separate bins depending on eclipse duration. The spacecraft encounters a total of 19 eclipses enroute to

the NRHO from the sGTO1for the mentioned departure and arrival epochs. It encounters 7 eclipses in the

130 - 140 mins range, 6 in the 120 - 130 mins range, 5 in the 110 - 120 mins range and 1 in the 80 - 90 mins

range. The duration of the longest eclipse encountered was 136 mins. We mention that even when eclipses

do not prevent optimal thrust-on arcs, sufﬁciently long eclipses for solar electric propulsion systems lead to

deep battery discharge (mission dependent) and may require a sunshine coast to re-charge the battery upon

exit from the shadow. Note the spacecraft computer, sensors, communications system and attitude control

system must continue operations in shadow. Long eclipses are generally to be avoided. While “long” eclipse

duration is mission dependent, more than 150 minutes hours in shadow is frequently considered unacceptable,

and greater than 120 minutes is frequently undesirable. These issues must be studied in a mission-speciﬁc

context to ﬁnalize the trajectory design during the preliminary phases of mission/vehicle design.

13

Figure 9: Full Transfer: sGTO1- 9:2 NRHO (EMEJ2000).

Transfer : Eclipses on Fuel-optimal Thrust Arcs

A representative transfer is presented in this section, where the eclipse arcs occur in between consecutive

thrust arcs. Theoretically, eclipse-driven coast arcs result in loss of thrusting efﬁciency, more so, if such coast

arcs appear near the periapse region of the intermediate transfer spirals, where the spacecraft velocity is high.

The resulting throttling sequence represents a signiﬁcant departure from the ideal fuel-optimal behavior,

where the optimality conditions manifest into a regular distribution of the thrust arcs around the periapse

region for maximum efﬁciency. This concept is elucidated in the numerical solution presented.

The spacecraft on this transfer trajectory departs from sGTO1and is inserted in the NRHO on ‘19 MAR

2025 01:45:00.464’. The total transfer time was 135.39 days with the spacecraft coasting on the LTCA for

79.39 days. The total ∆V for the transfer was 1991.916 m/s. The transfer trajectory is shown in Figure 9, plot-

ted in the EMEJ2000 frame. Other important transfer events are : Departure on ‘03 NOV 2024 16:23:36.467’

and Patching with the LTCA on ‘29 DEC 2024 16:23:36.467’.

The bang-off-bang throttle sequence is depicted in Figure 10. Since the eclipse-driven coast arcs happen

during thrust arcs, it leads to forced coasts for the eclipse duration. Comparing it with Figure 7, it is apparent

that there are no eclipses on the selected LTCA. The blown up image in Figure 10 clearly depicts the switch in

the throttle sequence due to the eclipse-driven coast arcs. The frequency of ‘Earth - occulted’ eclipses, being

an important consideration for accurate mission design, has been portrayed using a histogram plot in Figure

11 with separate bins depending on the eclipse duration. The spacecraft encounters 15 eclipses enroute to the

NRHO from the sGTO1for the mentioned departure and arrival epochs. It encounters 1 eclipse each in the

80 - 90 mins range and 100 - 110 mins range, 2 eclipses in the 110 - 120 mins range and 11 eclipses in the

120 - 130 mins range. The duration of the longest eclipse encountered was 126 mins.

Summary : Transfer Opportunities in 2025

In this section, opportunities for feasible and efﬁcient transfers of a spacecraft with the assumed parameters

from a sGTO to the 9:2 NRHO have been listed. The transfers were solved for one insertion opportunity

every month of the year 2025. Figure 12 depicts all the spatial trajectories with respect to the Earth Mean

Equator (EMEJ2000) frame. Note the counter-clockwise movement of the eclipse arcs as the insertion date

is swept throughout the year. All candidate trajectories shown in Figure 13, were solved for using the same

methodology with the LTCAs and associated patch-points pre-computed and identiﬁed. The ‘best’ LTCA

for each transfer was identiﬁed by picking the one with a patch-point (periapse point) having an argument of

periapse as close to the sGTO ωas possible, among the top ﬁve cases having the least radius with respect to

14

0 20 40 60 80 100 120

TOF (Days)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

SF, Thrust (N), Eclipse Condition

SF

Thrust (N)

Eclipse Condition

Figure 10: Thrusting sequence, eclipse condition, and switching function.

Figure 11: Thrusting sequence and eclipse condition.

15

the Earth. The nature of eclipses encountered by the spacecraft enroute has been plotted as binned histograms

in Figure 13.

(a) Trajectories for January - July 2025.

(b) Trajectories for August - December 2025.

Figure 12:Eclipse-conscious trajectories: YR 2025.

Table 2 summarizes important transfer parameters for all the mission scenarios. The duration of the longest

eclipse as well as the number of eclipses encountered depend on the ‘Sun-Earth-Spacecraft’ geometry for the

duration of the transfer. These results present mission designers with efﬁcient eclipse-conscious trajectories

for future resupply missions to the Lunar Gateway, which is beyond the scope of state-of the art trajectory

design tools. The fuel cost presented is comparable to analogous results presented in [19].

16

Figure 13: Eclipse binning for Mission Scenarios in YR 2025.

Table 2: Summary of departure and arrival times, ∆V, maximum eclipse duration and number of eclipses

for different mission scenarios in YR 2025.

Departure Epoch Arrival Epoch ∆V (m/s) Max. Eclipse (mins) #Eclipses

11 OCT 2024 04:17:37.15 25 JAN 2025 18:39:50.11 1641.867 135 19

05 OCT 2024 04:00:27.53 21 FEB 2025 00:19:30.41 1729.059 120 17

03 NOV 2024 16:23:36.47 19 MAR 2025 01:45:00.47 1991.916 126 15

16 JAN 2025 14:12:43.83 04 MAY 2025 00:41:05.43 1956.159 64 8

01 FEB 2025 23:50:15.69 19 JUN 2025 02:58:55.25 2189.803 176 27

02 APR 2025 01:36:17.10 21 JUL 2025 21:10:58.86 1979.373 164 19

01 MAY 2025 06:28:26.86 17 AUG 2025 01:44:34.07 1817.332 124 17

26 MAY 2025 23:20:11.50 12 SEP 2025 01:44:34.07 2054.525 86 17

15 JUN 2025 15:44:17.11 28 OCT 2025 17:18.58.24 2103.743 124 29

11 JUL 2025 04:23:01:44 23 NOV 2025 10:55:04.80 1900.061 130 27

19 AUG 2025 13:18:07.01 19 DEC 2025 14:58:07.03 1772.081 242 28

Early Departure

Purely fuel-optimal trajectories typically result in a lower ∆V cost for a larger time of ﬂight for the same

rendezvous type transfer. The limiting case of a barely reachable terminal state is, however, simultaneously

a minimum-time and minimal-fuel trajectory, if the time of ﬂight established from a minimum-time transfer

is the speciﬁed ﬁnal time. As an aside, this duality between minimum-fuel and minimum-time extremals is

exploited in the recent paper by Taheri and Junkins [36] to construct optimal switching surfaces considering

an inﬁnite family of maximum thrust speciﬁcations. In this work, we are dealing with eclipse-conscious

trajectories, a part of which is solved using the fuel-optimal formulation. Due to the optimizer being aware

of shadow regions the spacecraft encounters, it is important to perform a “launch-period” study to analyze

the trade off between fuel-cost and time of ﬂight for such eclipse-conscious trajectories. Note that in the

results presented below, the family of trajectories has the same number of revolutions with a larger time of

ﬂight due to sliding the departure date backwards in time. For this analysis, the ‘best’ transfer opportunity

among the candidate trajectories presented in the section above in terms of fuel cost, frequency of eclipse

and maximum eclipse duration was picked as the nominal trajectory. The spacecraft departs sGTO2on ‘16

JAN 2025 14:12:43.83’ and inserts into the NRHO on ‘04 MAY 2025 00:41:05.433’, incurring a ∆Vcost of

17

(a) Nominal trajectory. (b) Trajectory for departure on et9P.

Figure 14:Effect of early departure on transfer trajectories.

1956.159 m/s. Enroute to the NRHO, the spacecraft encounters 8 distinct eclipses with the duration of the

longest encountered eclipse being 64 mins. A family of transfers were solved by sliding the departure date

backwards in time with a step size of the orbital period of sGTO2(TsGTO2). Left plot in Figure 14 depicts

the nominal transfer trajectory and the right plot depicts the trajectory for the analogous transfer but an early

departure (et9P =etdep −9TsGTO2). Here, etdep is the ephemeris time associated with the nominal departure

epoch (‘16 JAN 2025 14:12:43.83’). Notice the longer ﬁnal spiral for the early departure trajectory on the

right plot in Figure 14. The trajectory is nearly identical in terms of the thrust and coast arcs up to the last but

one spiral. This is because of the eclipse-induced coast arcs occurring in the same region of the trajectory,

thereby causing a near-identical loss of thrusting efﬁciency. Since the number of revs are ﬁxed, the optimizer

forces the spacecraft to accommodate the residual time of ﬂight on the ﬁnal spiral. Table 3 lists the impact of

early departure on fuel cost and eclipses.

Table 3: Early Departure Results: sGTO - 9:2 NRHO.

Departure etiP Departure Epoch ∆V (m/s) Max. Eclipse (mins) #Eclipses

et1P JAN 15 12:22:49.39 2001.921 64 8

et2P JAN 14 10:32:54.95 2037.897 62 9

et3P JAN 13 08:43:00.51 2063.117 64 9

et4P JAN 12 06:53:06.06 2084.662 64 9

et5P JAN 11 05:03:11.62 2114.921 64 9

et6P JAN 10 03:13:17.18 2167.921 64 10

et7P JAN 09 01:23:22.74 2219.465 62 10

et8P JAN 07 23:33:28.30 2255.739 64 10

et9P JAN 06 21:43:33.85 2278.431 62 10

It was observed that, eclipse-conscious, fuel-optimal transfers for this case, countered the intuition of the

inverse trade-off between time and fuel cost for purely fuel-optimal class of transfers. It is evident from Table

3 that, an early departure or a longer time of ﬂight, for eclipse-conscious transfers with the same number

of revs, lead to a larger fuel cost. For an approximately 9-day sliding of the departure date from nominal,

the ∆Vcost increased by 322.272 m/s. The nature of eclipses encountered changed marginally, with the

maximum eclipse duration reducing to 62 mins, whereas the number of eclipses increased to 10.

18

CONCLUSION

A methodology to generate efﬁcient, eclipse-conscious transfers leveraging ephemeris-driven asymptotic

long terminal coast arcs has been presented. Using favorable periapse states as patch-points on a set of pre-

computed manifolds, an algorithm is devised to solve end-to end transfers using indirect formalism of optimal

control. The resulting solutions present accurate transfer trajectories, with a knowledge of eclipse transitions

of the spacecraft, solved in a high-ﬁdelity model.

The algorithm was applied to generate transfers from a Super Synchronous GTO to the 9:2 Near Rectilinear

Halo Orbit of the southern L2family, with a focus on providing accurate trajectories for “cargo re-supply”

type missions to the Lunar Gateway. Speciﬁcally, efﬁcient transfer trajectories have been presented for 11

distinct launch dates in the year 2025. Interestingly, the interaction of “fuel-optimal” thrust-coast-thrust

sequence with “shadow-induced” coast arcs were found to be illuminating and captured using two separate

examples. It was observed that in the cases where eclipses interfere with the optimal thrust-coast-thrust

sequence, the thrust and coast arcs are forced to redistribute, due to the loss of thrusting efﬁciency vis-a-vis

the “shadow-induced” coast arcs. The fuel cost was found to depend signiﬁcantly on the position of shadow-

regions on the intermediate spirals, which typical fuel-optimal trajectories are oblivious to. The frequency

of such regions along with the duration of the maximum eclipse, important considerations towards an actual

mission design for charging of on-board batteries, have been presented. It is anticipated that these results will

ﬁnd advantageous applications, especially for future un-crewed missions.

FUNDING SOURCES & ACKNOWLEDGMENTS

We are pleased to acknowledge the Jet Propulsion Laboratory, Air Force Research Laboratory, Dzyne,

Inc., and Texas A & M University for sponsorship of various aspects of this research. This work was com-

pleted at Texas A&M University. A part of this research was carried out at the Jet Propulsion Laboratory,

California Institute of Technology, under a contract with the National Aeronautics and Space Administration

(80NM0018D0004).

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