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Eclipse-Conscious Transfers to Lunar Gateway Using

Ephemeris-Driven Terminal Coast Arcs

Sandeep Singh ∗and John Junkins †

Texas A & M University, College Station, TX, 77843

Brian Anderson ‡

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, 91109

Ehsan Taheri§

Auburn University, Auburn, AL, 36849

A novel methodology is proposed for designing low-thrust trajectories to quasi-periodic,

near rectilinear Halo orbits that leverages ephemeris-driven, “invariant manifold analogues”

as long-duration asymptotic terminal coast arcs. The proposed methodology generates end-

to-end, eclipse-conscious, fuel-optimal transfers in an ephemeris model using an indirect for-

mulation of optimal control theory. The end-to-end trajectories are achieved by patching

Earth-escape spirals to a judiciously chosen set of states on pre-computed manifolds. The

results elucidate the eﬃcacy of employing such a hybrid optimization algorithm for solving

end-to-end analogous fuel-optimal problems using indirect methods and leveraging a compos-

ite smooth control construct. Multiple representative cargo re-supply trajectories are generated

for the Lunar Orbital Platform-Gateway (LOP-G). A novel process is introduced to incorpo-

rate eclipse-induced coast arcs and their impact within optimization. The results quantify

accurate Δ𝑉costs required for achieving eﬃcient eclipse-conscious transfers for several launch

opportunities in 2025 and are anticipated to ﬁnd applications for analogous uncrewed missions.

Nomenclature

𝑹𝑖= position of body i with respect to Solar System Barycenter in inertial frame (km)

𝑹𝑐= position of chosen center body c with respect to Solar System Barycenter in inertial frame (km)

𝑹= position of spacecraft with respect to Solar System Barycenter in inertial frame (km)

𝝆𝑖= position of body i with respect to chosen center in inertial frame (km)

𝝆= position of spacecraft with respect to chosen center in inertial frame

𝒓𝑖= position of body i with respect to chosen center in rotating frame (km)

∗PhD Candidate, Department of Aerospace Engineering, Texas A&M University, AIAA Student Member.

†Distinguished Professor, Department of Aerospace Engineering, Texas A & M University, Honorary Fellow AIAA.

‡Mission Design Engineer, Jet Propulsion Laboratory, California Institute of Technology.

§Assistant Professor, Department of Aerospace Engineering, Auburn University, AL, 36849, AIAA Senior Member.

𝒓= position of spacecraft with respect to chosen center in rotating frame (km)

𝛀= angular rotation vector, expressed in inertial frame (rad/s)

𝝎= angular rotation vector, expressed in rotating frame (rad/s)

𝑡∗= reference epoch for rotating frame deﬁnition (s)

I. Introduction

Low-thrust propulsion technology, since its conception, has had the potential to revolutionize robotic exploration of

the solar system. Low-thrust electric propulsion systems are, often times, the only feasible option for long-duration deep

space missions due to propellant constraints. There exist a variety of engines 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 physics, low-thrust engines are typically characterized by high speciﬁc impulse (

Isp

) values. The

high

Isp

directly corresponds to high

Δ𝑉

eﬃciency, and therefore less propellant is required for delivering the same

amount of

Δ𝑉

. It is emphasized that the same amount of

Δ𝑉

is typically less eﬃcient with low-thrust systems due to

ﬁnite burn loss. The

Isp

is usually high enough to oﬀset these ﬁnite burn losses when optimizing propellant mass. In

addition to the propellant eﬃciency, the mass and volume, compared to the usual chemical high-thrust engines are also

greatly reduced, so the useful payload mass delivered to the target ﬁnal orbit may be vastly increased.

Trajectory design using low-thrust propulsion systems, which incorporate continuous low-thrust arcs instead of

more nearly impulsive thrust provided by their chemical counterparts, is however a complex process. The complexity

stems from a departure from natural dynamics due to the continuous forcing function incorporated in the mathematical

model as well as the ‘bang-oﬀ-bang’ optimal control proﬁle for fuel-optimal problems and other events such as shadow

entry and exit conditions. Additionally, due to the transfers typically being long-duration, they can be characterized by

multiple revolutions (revs) and numerical convergence becomes a challenge for the optimizer. Local extremals often

occur, which further complicates 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

], while direct

methods involve an iterative procedure leading to sequentially improving some parameterized approximation of the

optimal solution. Due to their ease of formulation, larger domain of convergence and being adept at handling inequality

constraints, direct methods have been used for solving many optimization problems [

7

]. On the other hand, indirect

methods have also undergone signiﬁcant recent enhancements to alleviate associated challenges, with concepts like

homotopy [

8

–

10

] 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 [8, 10].

2

Most state of the art trajectory design software like Copernicus [

11

], which has built-in direct and indirect solvers

and MYSTIC [

12

], which uses a non-linear static/dynamic optimal control algorithm provide robust high-ﬁdelity

solutions for optimal transfers. However, Copernicus and MYSTIC do not incorporate eclipse-induced coast arcs

constraints within the optimization. In particular, Copernicus falls short in cases where solution for fuel-optimal

transfers with additional eclipse-induced coast arcs due to the spacecraft passing through shadow regions using indirect

formulation of the optimal control problem (OCP) are desired. MYSTIC also does not currently incorporate eclipses

during the optimization process. Both tools have post-optimization capability to analyze shadow events and provide

a summary of the number and duration of eclipses, but they do not yet incorporate eclipses within the optimization

procedure. The Q-law guidance algorithm [

13

] is already incorporated within MYSTIC for initial trajectory generation

of many-orbital-revolution phases of ﬂight [

12

]. Recently, the Q-law was leveraged to design optimal trajectories

while including eclipsing in the cislunar trajectory using direct optimization methods [

14

]. Other works also study

the inclusion of eclipses during trajectory design in the cislunar domain, especially during the earth-spiralling phase

[15, 16].

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

the many families of periodic orbits existing in simpliﬁed dynamical models like the Circular Restricted Three-Body

Problem (CR3BP). Anderson and Lo [

17

], for instance, studied the role of invariant manifolds in the dynamics of

low-thrust trajectories passing through unstable regions of the three-body problem. Dellnitz et al. [

18

] used the reachable

sets concept coupled with invariant manifolds of libration orbits to solve for a low-thrust Earth-Venus transfer. Vaquero

and Howell [

19

] studied the leverage provided by resonant-orbit manifolds for designing transfers between Earth-Moon

libration-point orbits. More recently, Singh et al. [

20

] demonstrated a methodology to leverage stable/unstable manifold

pairs of a halo orbit around the Earth-Moon

𝐿1

for end-to end transfers in the cis-lunar space. They also extended this to

study the behavior of the invariant manifolds of Southern

𝐿2

Near 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

[

21

]. Similar approaches to low-thrust trajectory design also appear in [

22

–

24

]. Additionally, transfers in the lunar

domain especially involving periodic orbits like NRHOs and Distant Retrograde Orbits (DROs) have also been studied

extensively by various researchers [25–28].

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

trajectories especially for spacecraft equipped with SEP systems for analyzing impact of eclipses on mission cadence

due to battery depth of discharge. Typically, thrust can only be on in ‘full-Sun’ and power is consumed during eclipses

to operate the spacecraft. In addition to this, the use of invariant manifolds as long-terminal coast arcs (LTCAs) reduces

the problem complexity signiﬁcantly, where the resulting two-point boundary-value problem (TPBVP) is easier to

solve 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 [

21

]. These

3

considerations, if taken into account in the optimization procedure, would provide more realistic trajectories than the

previous works.

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

𝐿2

family. Secondly, we present an in-depth analysis towards selection 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 transfer design from two ‘super Synchronous Geostationary Transfer Orbits’ (sGTOs) to

a selected patch-point is then formulated using an indirect optimization method. The Composite Smooth Control

(CSC) [

29

] 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. Several transfer opportunities have been listed

for transferring a spacecraft from the considered sGTOs to the quasi-periodic 9:2 NRHO of the southern

𝐿2

family 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.

II. Invariant Manifold Analogues in an Ephemeris Model

Examination of states’ ﬂow near an equilibrium point in the continuous-time, autonomous, non-linear dynamical

systems reveal locally stable, center and unstable sub-spaces and eventually, the corresponding global manifolds. These

topological spaces constrain the natural nonlinear trajectories approaching and departing equilibrium solutions of the

dynamical system (e.g., periodic orbits in the Circular Restricted 3-Body Problem (CR3BP)).

From the perspective of mission design, the dynamical system of interest is one that describes the motion of a

spacecraft subject to gravitational forces from celestial bodies, as well as non-gravitational perturbing forces. The

dynamical systems used can vary from simple point-mass restricted 2-Body problem to more realistic models such as an

ephemeris-driven, multi-body gravitational model including non-conservative perturbations from spherical harmonics,

atmospheric drag and solar radiation pressure. An intermediate model that is deemed simple enough to analyze particle

motion, yet complex enough to include a major perturbation while providing useful insights is the CR3BP. Invariant

manifolds associated with the plethora of periodic orbit families which exist in this model are available for use by

mission designers. Moreover, using invariant manifolds can provide fuel-eﬃcient transfers to these orbits. In addition to

the cost-savings for common missions, it can also be mission-enabling for more complex, deep-space missions that

would otherwise be infeasible due to current limitations of launch vehicle and in-space propulsion technologies as

shown in [30, 31].

Thus, leveraging invariant manifolds in the CR3BP for mission design has recently become an exciting research topic.

4

However, while computing manifolds in this simple dynamical system is relatively straightforward, the solutions are

inherently chaotic and as such may behave very diﬀerently when transferred to a higher-ﬁdelity model. In this section,

an approach for computing high-ﬁdelity perturbed analogues for CR3BP invariant manifolds in an ephemeris-driven

point-mass gravitational model (“Ephemeris Model”) are highlighted. These “invariant manifold analogues” are not

invariant in the strict sense, because the dynamical model is non-autonomous and therefore, does not have those ideal

structures. Instead, the outputs serve as a useful qualitative approximation of CR3BP periodic orbit manifolds to

design trajectories in a higher-ﬁdelity model. 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 [

32

]. 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.

The methodology is explained in the following sections. First, section II.A describes the equations of motion used

as well as the associated variational equations. Secondly, the method in which these equations are used to compute the

orbit’s invariant manifold properties is described in section II.B, a necessary step for producing the manifold initial

conditions. A discussion on computation of the initial conditions as well as their propagation is also provided.

A. Equations of Motion and Variational Equations

Without loss of generality,

𝑖=

1

,

2is chosen to represent the two primaries from the CR3BP. Naturally, the chosen

bodies should approximate the behavior of bodies orbiting a common barycenter. 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 momentum vector at a chosen reference epoch (

𝑡∗

). Considering,

three basis vectors listed in Eq.

(1)

,

𝑄0

in 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)

5

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)

we can form the complete rotation matrix 𝑄(𝑡)in Eq. (4),

𝑄(𝑡)=𝑄0𝑄𝑧(𝑡).(4)

With this we can relate coordinates in this rotating frame to inertial coordinates using Eq.

(5)

. The displacement

vectors relative to the chosen center

𝑐

are

𝝆𝑖=𝑹𝑖−𝑹𝑐

and

𝝆=𝑹−𝑹𝑐

. The relative displacement and angular velocity

vectors with rotating components are

𝒓𝑖=𝑄(𝑡)𝝆𝑖,

𝒓=𝑄(𝑡)𝝆,(5)

𝝎=𝑄(𝑡)𝛀.

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

(6)

),

¥

𝑹=𝑭(𝑡, 𝑹)=−Õ

𝑖

𝜇𝑖(𝑹−𝑹𝑖)

|𝑹−𝑹𝑖|3,(6)

where

𝜇𝑖

is 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. (7) as

𝝓(𝑡, 𝝆)=−Õ

𝑖

𝜇𝑖(𝝆−𝝆𝑖)

|𝝆−𝝆𝑖|3−Õ

𝑖≠𝑐

𝜇𝑖(𝝆𝑖)

|𝝆𝑖|3.(7)

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. (8) as

¥

𝒓=𝒇(𝑡, 𝒓,¤

𝒓)=−Õ

𝑖

𝜇𝑖(𝒓−𝒓𝑖)

|𝒓−𝒓𝑖|3−Õ

𝑖≠𝑐

𝜇𝑖(𝒓𝑖)

|𝒓𝑖|3

| {z }

𝒇𝑔(𝑡, 𝒓,¤

𝒓)

−2𝝎×¤

𝒓−𝝎× (𝝎×𝒓)

| {z }

𝒇𝑟(𝑡, 𝒓,¤

𝒓)

,(8)

6

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. (9) as

𝒙=

𝒓

¤

𝒓

,¤

𝒙=

¤

𝒓

𝒇(𝑡, 𝒓,¤

𝒓)

.(9)

In order to compute the variational equations, we require the Jacobian of Eq.

(9)

, expressed with simpliﬁed 3x3

blocks in Eq. (10).

𝐽(𝑡, 𝒓,¤

𝒓) ≡ 𝜕¤

𝒙

𝜕𝒙=

03𝑥3𝐼3𝑥3

𝜕𝒇

𝜕𝒓

𝜕𝒇

𝜕¤

𝒓

.(10)

The indicated position and velocity partials of the acceleration terms due to frame rotation are expressed in Eq.

(11)

.

𝜕𝒇𝑟

𝜕𝒓=

𝜔20 0

0𝜔20

0 0 0

,𝜕𝒇𝑟

𝜕¤

𝒓=

0 2𝜔0

−2𝜔0 0

0 0 0

.(11)

Note:

𝜕𝒇

𝜕𝑟 =𝜕𝒇𝒓

𝜕𝑟 +𝜕𝒇𝒈

𝜕𝑟 ,(12)

where partials of the gravity forcing terms with respect to position upon exploiting symmetry reduce to 6 instead of 9

diﬀerential equations for the distinct elements and can be expressed as shown in Eq. (13),

𝜕𝒇𝑔

𝜕𝒓=

𝜕¥𝑥

𝜕𝑥

𝜕¥𝑥

𝜕𝑦

𝜕¥𝑥

𝜕𝑧

𝜕¥𝑦

𝜕𝑥

𝜕¥𝑦

𝜕𝑦

𝜕¥𝑦

𝜕𝑧

𝜕¥𝑧

𝜕𝑥

𝜕¥𝑧

𝜕𝑦

𝜕¥𝑧

𝜕𝑧

=

𝜕¥𝑥

𝜕𝑥

𝜕¥𝑥

𝜕𝑦

𝜕¥𝑥

𝜕𝑧

𝜕¥𝑥

𝜕𝑦

𝜕¥𝑦

𝜕𝑦

𝜕¥𝑦

𝜕𝑧

𝜕¥𝑥

𝜕𝑧

𝜕¥𝑦

𝜕𝑧

𝜕¥𝑧

𝜕𝑧

,and (13)

𝜕¥𝑥

𝜕𝑥 =−Õ

𝑖

𝜇𝑖

|𝒓−𝒓𝑖|31+3(𝑥−𝑥𝑖)2

|𝒓−𝒓𝑖|2,𝜕¥𝑦

𝜕𝑦 =−Õ

𝑖

𝜇𝑖

|𝒓−𝒓𝑖|31+3(𝑦−𝑦𝑖)2

|𝒓−𝒓𝑖|2,

𝜕¥𝑧

𝜕𝑧 =−Õ

𝑖

𝜇𝑖

|𝒓−𝒓𝑖|31+3(𝑧−𝑧𝑖)2

|𝒓−𝒓𝑖|2,𝜕¥𝑥

𝜕𝑦 =−Õ

𝑖

𝜇𝑖

|𝒓−𝒓𝑖|33(𝑥−𝑥𝑖)(𝑦−𝑦𝑖)

|𝒓−𝒓𝑖|2,

𝜕¥𝑥

𝜕𝑧 =−Õ

𝑖

𝜇𝑖

|𝒓−𝒓𝑖|33(𝑥−𝑥𝑖)(𝑧−𝑧𝑖)

|𝒓−𝒓𝑖|2,𝜕¥𝑦

𝜕𝑧 =−Õ

𝑖

𝜇𝑖

|𝒓−𝒓𝑖|33(𝑧−𝑧𝑖)(𝑦−𝑦𝑖)

|𝒓−𝒓𝑖|2.

Finally, the variational equations in the form of the well-known State Transition Matrix diﬀerential equation, Eq.

(14)

,

7

are used for propagation to perform an eigenanalysis of the locally linearized ﬂow around a “ﬁxed point”, with the initial

value problem initialized with Φ(𝑡0, 𝑡0)as a 6×6identity matrix.

¤

Φ(𝑡, 𝑡0)=𝐽(𝑡, 𝒓,¤

𝒓)Φ(𝑡, 𝑡0).(14)

While the ﬁxed rotation rate and axis of the frame chosen can be seen as a simplifying assumption, it is a simpliﬁcation

that should only aﬀect the frame, not the underlying physical dynamics. Propagation in an inertial ephemeris model

and the ﬁxed rotating ephemeris model deﬁned here should yield the same solutions to within the eﬀect of numerical

inaccuracies, and have done so in the authors’ testing. The choice of

𝑡∗

is similarly a choice that only aﬀects the chosen

frame, and should not aﬀect the dynamics themselves.

B. Orbit Manifold Parameters

Consider a single revolution of a nearly-periodic orbit in the rotating frame deﬁned in the preceding section. Let the

orbit have a period of

𝑇𝑖

, where

𝑖

is the rev number. The ﬂow around a “ﬁxed point” on this rev can be characterized by

studying the 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),(15)

using Eq.

(14)

. It is important to note that the period

𝑇𝑖

can be deﬁned in several ways, such as the time at crossing a

given ﬁxed plane or by using the apsidal period. While the Monodromy matrix,

𝑀(𝑡0)

, has the same period, for all

choices of

𝑡0

on an exactly periodic orbit, the elements of the Monodromy matrix, in reality do have small variations for

each choice of starting state at each

𝑡0

. 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(16)

where

|𝜆1|>|𝜆2|

, and

𝜆∗

6

indicates 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≡𝜆1

and

𝜆𝑆≡𝜆2

represent 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

𝒙0

at

𝑡0

, can

be generated using the following relation

𝒙𝑼(𝑡0)=𝒙0+𝑠𝜖𝝃𝑈 ,0,𝒙𝑺(𝑡0)=𝒙0+𝑠𝜖 𝝃𝑆,0,(17)

8

where

𝑠=±

1is the “sense” of the manifold and

𝜖

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

𝑡0

and 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|.(18)

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

dynamics in Eq. (9), with unstable manifolds propagated forward in time, and stable manifolds backward in time.

C. 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 discretization 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 through

the process outlined in section II.B.

An important consideration in this procedure is the discrete spacing of the points on the quasi-periodic orbit.

Consider a near-periodic rev on an ephemeris-corrected, 9:2 resonant NRHO of the southern

𝐿2

family. Let the orbital

period of this rev be

𝑇𝑖

. A simple methodology for extraction of the discrete states include discretization at uniformly

spaced time intervals or uniformly spaced intervals in eccentric anomaly, 𝐸.

For

𝑁

discrete points, the discretization in time follows a fairly straightforward procedure, where the states are

extracted as

𝒙0(𝑡𝑗+1)=𝒙0(𝑡𝑗+𝛿𝑡) ∀ 𝑗∈ {1,2,3, ...., 𝑁},where 𝛿𝑡 =𝑇𝑖/𝑁. (19)

Discretization in eccentric anomaly (

𝐸

) can be carried out by uniformly spacing in

𝐸

, where

𝐸∈ [

0

,

2

𝜋]

. While an

NRHO does not strictly follow the dynamics of a keplerian orbit, the 9:2 Lunar NRHO under consideration is orbiting

close enough to the Moon that it becomes an adequate approximation for more even spacing of points. A mapping from

𝐸

to

𝑡

occurs by ﬁrst mapping it to the Mean Anomaly (

𝑀

) using Kepler’s equation,

𝑀=𝐸−𝑒sin 𝐸

, where

𝑒

is the

eccentricity. Assuming

𝑇𝑖

as the rev period and

𝑒𝑖

as the rev eccentricity,

𝜏

as the time for a given apoapse/periapse

9

passage, and 𝑡𝑝as the periapsis epoch for the rev, ∀𝑗∈ {1,2,3.... 𝑁},

𝑀𝑗+1=𝐸𝑗+1−𝑒𝑖sin 𝐸𝑗+1=2𝜋

𝑇𝑖

(𝑡𝑗+1−𝑡𝑝),

𝑡𝑗+1=𝑡𝑝+𝑇𝑖

2𝜋(𝐸𝑗+1−𝑒𝑖sin 𝐸𝑗+1),

where, 𝐸𝑗+1=𝐸𝑗+𝛿𝐸 and 𝛿𝐸 =2𝜋

𝑁,

For Peri-Peri rev, 𝐸1=0, 𝑡 𝑝=𝜏,

For Apo-Apo rev, 𝐸1=𝜋, 𝑡 𝑝=𝜏−𝑇𝑖

2,

𝐸1=

0,Peri-Peri

𝜋, Apo-Apo

,

𝑡𝑝=

𝜏, Peri-Peri

𝜏−𝑇𝑖

2,Apo-Apo

.

(20)

Intuitively, a uniform spacing in time leads to a denser distribution of points near apoapse as compared to near

periapse. This is not ideal for analyzing resulting manifold clusters as it is quite possible to miss some viable choices

near periapsis, where the state is rapidly changing. Thus, discretization in eccentric anomaly is an alternative, which

unsurprisingly leads to a more regular “arc-length” dispersion of the discrete near-periodic states and provide a

comprehensive coverage of all stable/unstable manifold analogue clusters.

As an example, consider the apoapsis to apoapsis or “apo-apo” rev for the 9:2 synodic NRHO with the ﬁrst apoapse

passage epoch: ‘2025 JAN 02 20:02:28.677’ with

𝑁=

100. Note that the ‘best’ case manifold conditions are considered

as Earth-periapse points among the cluster of 100 trajectories whose distance from the Earth is minimum and are

henceforth referred to as ‘optimal’ unless stated otherwise. Figures 1a and 1c depict the distribution of ﬁxed points for

time and eccentric anomaly as discretization parameters respectively, with the red markers being the orbital locations

leading to Earth-periapse conditions with

𝑟𝑓≤

3

.

5

×

10

5

km. It is now apparent, that as hypothesized, 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 discrete search for attractive manifold trajectories is more likely to miss favourable manifolds

due to non-uniform spacing. On the other hand, a more uniform spatial scattering achieved by discretization in

𝐸

leads

to a more comprehensive and robust mapping in our case (Figures 1b and 1d). An alternative approach to discretization

can be to employ variable-step integration methods for orbit propagation, which can provide a heuristic yet reasonable

discretization based on the dynamical model sensitivity.

10

(a) Insertion Locations: 𝐸Discretization (b) ‘5’ Best Manifolds: 𝐸Discretization

-6

-5

-4

-2

-3

Z (km)

104

-2

Discretization in 't', Epoch: 2025 JAN 02 20:02:28.677

2

0

-1

104

X (km)

0

104

Y (km)

0

2

-2

4

all insertion locations

insertion locations with rf<3.5e+05 km

(c) Insertion Locations: 𝑡Discretization (d) ‘5’ Best Manifolds: 𝑡Discretization

Fig. 1 Discretization Parameter - 𝐸vs. 𝑡

11

III. Eclipse Modeling

In this paper, 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. In order to optimally design such trajectories,

an event-trigger handling capability is 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 [

33

–

37

]. In this paper, a conical shadow [

38

] model has been adopted for analyzing the manifold analogues

as well as designing eclipse-conscious optimal trajectories. The conical shadow model assumes spherical shapes of the

occulting body and the Sun as is depicted in Figure 2.

Fig. 2 Schematic for the Conical Shadow Model.

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/𝑠 𝑐 || ,(21)

where

𝒓𝑆/𝑠𝑐 =𝒓𝑆−𝒓𝑠𝑐

and

𝒓𝐵/𝑠𝑐 =𝒓𝐵−𝒓𝑠𝑐

. An implicit time-varying function,

𝑓𝑒𝑐 (𝒓𝑆/𝑠𝑐 (𝑡),𝒓𝐵/𝑠 𝑐 (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/𝑠 𝑐 (𝑡)) =𝐴𝑆𝑟 +𝐴𝐵𝑟 −𝐴𝐷. (22)

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

12

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. In this

work, we only consider the Earth as the occulting body and ignore eclipses that could be caused by the Moon. This is a

fair assumption, since lunar induced eclipses are more relevant for transfers exclusively in the lunar domain, for instance

a selenocentric transfer from a High Lunar Orbit (HLO) to a Low Lunar Orbit (LLO) or vice versa.

IV. Manifold Patch Conditions

Based on the methodology discussed in the preceding sections, stable invariant manifold analogues were generated

for all “apo-apo” revs of the ephemeris-corrected 9:2 Southern

𝐿2

NRHO. Several Earth-periapse states on the generated

trajectories were extracted for further analysis to gauge viability. Figure 3 shows stable manifold analogues for 100

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

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

Fig. 3 Patch-point candidates on stable manifold analogues.

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 Earth (

𝑅𝑑

). Other

considerations were maximum eclipse duration (

𝑡𝑒𝑐

) and diﬀerences in the argument of periapse (

Δ𝜔

), eccentricity (

Δ𝑒

),

and inclination (

Δ𝑖

) between the geocentric departure orbit and the patch condition. While minimizing any combination

of

[𝑅𝑑,Δ𝜔, Δ𝑒, Δ𝑖]

, 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

13

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 4 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

𝑅𝑑

were considered for each rev in YR 2025. Note that

several of the candidate LTCAs are devoid of eclipses and this knowledge enables a judicious choice of the manifold

analogue and associated patch-condition for various launch period scenarios.

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.)

Fig. 4 𝑡𝑒𝑐 for stable manifold analogues in year 2025.

It is apparent that the worst case scenario in terms of eclipses occur in June 2025 with a maximum

𝑡𝑒𝑐 =

250 minutes.

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

Δ𝜔=|𝜔0−𝜔pp |

, where

𝜔0

and

𝜔pp

are the arguments

of periapse of the initial geocentric orbit and the target patch-point condition in the EMEJ2000 frame, respectively.

Transfers with a minimal

Δ𝜔

have 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

𝑝 𝑝 =

0

°or

180

°

. Therefore, in order to minimize

Δ𝜔

,

𝜔pp ⊂ {𝐿1, 𝐿 2}

where

{𝐿1, 𝐿2} ∈ R

is a small neighborhood around

𝜔nom

pp

(

±

30

°

), depending upon the value of

𝜔0

.

Figure 5 shows an

𝜔pp

vs.

𝑅𝑑

phase portrait of all the candidate patch-point conditions, segregated by the ‘MONTH’ of

the year 2025 when the spacecraft gets inserted into the NRHO. The colorbar depicts the coast-time on the LTCAs.

In addition to the primary advantage of minimizing

Δ𝜔

, analysis of the

𝜔pp

vs.

𝑅𝑑

phase portrait also introduces a

14

Fig. 5 𝜔pp vs. 𝑅𝑑for candidate patch-points in the YR 2025.

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.

V. Time-Optimal and Fuel-Optimal Control Formulations

Electric propulsion systems oﬀer highly desirable propellant-eﬃciency features. However, low-thrust trajectories can

frequently consist of many revolutions, which complicates the task of trajectory design. The presence of perturbations

and many possible eclipses and incorporation of these factors within the optimization formulation will further complicate

the task of trajectory design.

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 is not suitable for eﬃcient convergence of indirect

optimal planet-centric, many-spiral trajectories during the Earth escape/capture phases [

39

–

42

]. Therefore, the set of

modiﬁed equinoctial elements (MEEs) [

43

] is 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,(23)

15

where

𝑨∈R6×1

denotes the unforced vector part of the dynamics and

B∈R6×3

denotes the control inﬂuence matrix

with their explicit forms given in [

42

]. 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].(24)

where

𝒂𝑝

,

𝒂Sun

, and

𝒂Moon

denote accelerations due to the propulsion system, and perturbing accelerations due to the

Sun and Moon, respectively. In Eq.

(24)

,

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. When

𝜌𝑝=

0, a two-body dynamics is considered,

whereas

𝜌𝑝=

1corresponds 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,(25)

where

𝛿𝑜∈ [

0

,

1

]

is the engine throttling input,

ˆ

𝜶

denotes the thrust unit direction vector, and

𝛿𝑠∈ [

0

,

1

]

reﬂects eclipse

events. In Eq.

(25)

,

𝑇

denotes the maximum thrust value and

𝑚

denotes the spacecraft total instantaneous mass. The

time rate of change of mass of the spacecraft can be written as

¤𝑚=−𝑇

𝑐𝛿𝑠𝛿𝑜,(26)

where

𝑐=𝐼sp𝑔0

is 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. The control inputs

are (𝛿,ˆ

𝜶).

A. Minimum-Time Formulation

For minimum-time formulation the cost functional is written as

minimize

𝛿𝑜,ˆ

𝜶𝐽=∫𝑡𝑓

𝑡0

𝑑𝑡, (27)

subject to :

Equations (23),(24) &(26),𝒙(𝑡𝑓) − ˜

𝒙𝑑=0,

𝒙(𝑡0)=𝒙0, 𝑚(𝑡0)=𝑚0,

where

˜

𝒙𝑑

denotes the set of desired (target) MEEs. Let

𝝀=[𝜆𝑝, 𝜆 𝑓, 𝜆𝑔, 𝜆 ℎ, 𝜆𝑘, 𝜆𝑙]>

denote the costate vector associated

with the MEEs and let

𝜆𝑚

denote the costate associated with mass. The Hamiltonian associated with the minimum-time

16

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.(28)

For minimum-time problems, the optimal control strategy is to operate the thruster at its maximum capacity; 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!#,(29)

where

𝑓𝑒𝑐

(see Eq.

(22)

) 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>

,¤

𝜆𝑚=−𝜕𝐻MT

𝜕𝑚 .(30)

The TPBVP associated with the minimum-time problem is formed by the set of state diﬀerential equations from

Eq.

(23)

, mass diﬀerential equation from Eq.

(26)

and the derived co-state diﬀerential equations from Eq.

(30)

with the

full state vector deﬁned as

X7×1=[𝒙>𝑚]>

, the full costate vector deﬁned as

𝚲7×1=[𝝀>𝜆𝑚]>

. The full state vector

X

at

𝑡0

is known and the seven unknown initial costates,

𝚲

, and the unknown time of ﬂight,

𝑡𝑓

, are solved numerically

satisfying the following set of eight boundary conditions,

𝒙−˜

𝒙𝑑=0&[𝜆𝑚(𝑡𝑓)𝐻𝑀𝑇 (𝑡𝑓)]>=0(transversality and Hamiltonian conditions). (31)

B. Minimum-Fuel Formulation

For minimum-fuel formulation the cost functional is written as

minimize

𝛿𝑜,ˆ

𝜶𝐽=∫𝑡𝑓

𝑡0

𝑇

𝑐𝛿𝑠𝛿𝑜𝑑𝑡, (32)

subject to :

Equations (23),(24) &(26),𝒙(𝑡𝑓) − ˜

𝒙𝑑=0,

𝒙(𝑡0)=𝒙0, 𝑚(𝑡0)=𝑚0,

17

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 must be used to 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∗).(33)

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,(34)

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

¤

𝝀=−𝜕𝐻MF

𝜕𝒙>

,¤

𝜆𝑚=−𝜕𝐻MF

𝜕𝑚 .(35)

The TPBVP associated with the minimum-fuel problem is formed by the set of state diﬀerential equations, Eq.

(23)

,

mass diﬀerential equation, Eq.

(26)

, and the derived costate diﬀerential equations, Eq.

(35)

, with the full state vector

deﬁned as

X7×1=[𝒙>𝑚]>

, the full costate vector deﬁned as

𝚲7×1=[𝝀𝜆𝑚]𝑇

. The full state vector

X

at

𝑡0

is

known and the seven unknown initial costates,

𝚲

, are solved numerically satisfying the following set of seven boundary

conditions,

𝒙−˜

𝒙𝑑=0, 𝜆𝑚(𝑡MF

𝑓)=0(transversality condition).(36)

An additional admissible control constraint, resulting in eclipse-conscious solutions is

𝛿𝑠=

0during the course

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

𝑓𝑒𝑐

from Eq.

(22)

,

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 [

29

]. The following

section presents the algorithm elucidating the former approach, which has been used in this work.

C. Optimization Algorithm

Equation

(25)

is the total acceleration delivered by the propulsion system and

𝛿𝑜

and

𝛿𝑠

denote the combined

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

18

are two sources that can lead to discontinuity in the thrust proﬁle, the CSC methodology [

29

,

44

] is used to alleviate

non-smoothness issues. Assuming

𝜌𝑜

and

𝜌𝑠

as the respective smoothing parameters, the optimal

𝛿∗

can be written as

𝛿∗=𝛿𝑠𝛿𝑜, 𝛿𝑠=1

2"1−tanh 𝑓𝑒𝑐

𝜌𝑠!#, 𝛿𝑜=1

2"1+tanh 𝑆𝐹

𝜌𝑜!#,(37)

where

𝑓𝑒𝑐

(see Eq.

(22)

) is treated as a “distance measure” associated with eclipse events. On the other hand,

𝑆𝐹

is the standard switching function derived from necessary optimality condition. The proposed high-ﬁdelity, Hybrid

Eclipse-Conscious Trajectory Optimization Routine (HECTOR) is summarized using two Algorithms given below.

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

Result: Minimum time (𝑡∗

𝑓), Converged co-state vector (𝜆2𝑏

0)

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

𝒙𝑑)

while 𝜖MT ≥𝜖∗do

Initial Guess: t 𝑓(Time of Flight)&𝝀(𝑡0)

Propagate: Eqs. (23),(26) &(30) with 𝜌𝑝=0, 𝛿∗

𝑜=1, 𝛿𝑠,ˆ

𝜶∗;

𝜖MT =||𝝐MT | | =| |𝒙(𝑡𝑓) − ˜

𝒙𝑑, 𝐻𝑀 𝑇 (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

𝜖∗

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

𝛿𝑠

factor. 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

𝛾

is considered. The value of

𝛾

depends 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 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

𝜌𝑠

is set to 1 to handle the eclipse transitions. The value of

𝜌𝑠

is lowered

to below a certain

𝜌∗

𝑠

value to get sharp transitions at the entry and exit of eclipse-induced coast arcs. Then, a step is

initiated to decrease the value of

𝜌𝑜

below a certain threshold,

𝜌∗

𝑜

in order to obtain bang-oﬀ-bang thrust proﬁles. The

value of

𝜌∗

𝑜

in our simulations is set to 1

.

0

×

10

−6

. The ﬁnal step is to introduce perturbations through

𝜌𝑝

. When,

𝜌𝑝

is

equal to one, the solution associated with the high-ﬁdelity model is obtained. While the value of

𝜌𝑝

is increased, it is

ensured that the ﬁnal solution corresponds to

𝜌𝑝=

1. It is emphasized that the Hamiltonian is updated according to the

considered dynamics to take into account the contribution of the perturbing accelerations. Thus, the costate diﬀerential

19

equations are updated accordingly [

29

]. Please note that a time-ﬁxed, fuel-optimal problem is solved. In fact, time-free,

fuel-optimal problems are not well-deﬁned problems since for such maneuvers the time of maneuver increases to inﬁnity.

Convergence robustness of indirect optimization methods are known to be inferior to direct optimization methods if

the initial guesses are not “close” to those that correspond to an extremal solution. In addition, there are many local

extremal solutions. There are actually four issues that impact convergence robustness of indirect optimization methods:

1) the choice of coordinates, 2) non-smoothness of certain events (e.g., throttle switches and eclipse entry and exist

conditions), 3) the adopted homotopy/continuation method, and 4) proper scaling of states [

45

]. All the aforementioned

items play important roles in the solution of challenging optimal control problems. Our experience indicates that

appropriate choice of coordinates (i.e., item 1) has enormous impact on the convergence robustness of indirect methods

when it comes to solving many-revolution low-thrust trajectories. In addition, the idea of smoothing (item 2) enlarges

the domain of convergence of the resulting boundary-value problems to an extent that random initialization of the

missing initial costate values is suﬃcient to obtain convergence.

Random initialization of the costates is used to ﬁnd the solution of the ﬁrst member of the family of optimal

control problems. The limit of achievable number of orbital revolutions depends on many factors such as the strength

of disturbances and the choice of solution methodology (e.g., single-shooting vs. collocation-based method). Ref

[

37

], for instance, provides cases with up to 500 revolutions using a single-shooting method and using MATLAB’s

fsolve and ode45 built-in functions and using the set of modiﬁed equinoctial elements. In this paper, we have used

the same methodology (and solvers), but a principled continuation methodology is followed and is described in the

proposed Algorithm. We have been able to achieve convergence without using the state transition matrix. The default

ﬁnite-diﬀerence method (of fsolve) is used to calculate the sensitivity of the constraints with respect to the design

variables.

20

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

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 𝜖2𝑏

MF ≥𝜖∗do

Propagate: Eqs. (23), (26) & (35) with 𝛿∗,ˆ

𝜶∗;

𝜖2𝑏

MF =||𝝐2b

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 𝜖2𝑏𝑆

MF ≥𝜖∗do

Propagate: Eqs. (23), (26) & (35) with 𝛿∗,ˆ

𝜶∗;

𝜖2bS

MF =||𝝐2bS

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 𝜖hfS

MF ≥𝜖∗do

Propagate: Eqs. (23), (26) & (35) with 𝒂=𝒂∗

𝑝(𝛿∗

𝑠, 𝛿∗

𝑜,ˆ

𝜶∗) + 𝜌𝑝(𝒂Sun +𝒂Moon);

𝜖hfS

MF =||𝝐hfS

MF || =||𝒙(𝑡MF

𝑓) − ˜

𝒙𝑑, 𝜆𝑚(𝑡MF

𝑓)||

end

𝝀hfS

0=𝝀hfS∗(Converged Value), 𝜌 𝑝=1.1𝜌𝑝;

end

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

end

VI. 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

𝐿2

family, using the stable invariant manifold analogues as LTCAs. 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 [

46

]. With consideration to the timeline, the ephemeris-corrected NRHO revs in

21

the year 2025 were considered to generate manifold analogues, having the full transfer occur in the year 2025.

A. 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 [

47

]. 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 6. Table 1 lists the

orbital elements for the two orbits of the sGTO category selected as the initial orbits for the representative problem.

Fig. 6 Schematic depicting the domain of Geocentric orbits.

The two starting orbits (

sGTO1

and

sGTO2

) diﬀer only in their argument of periapse as is given in the Table 1. The

choice of sGTO with

𝜔=𝜔nom

𝑝 𝑝 =

0

°or

180

°

was made depending on the departure epoch to enable a favorable eclipse

condition and therefore an eﬃcient transfer. Note that thrusters aboard the spacecraft are switched ‘OFF’ during the

duration of every encountered eclipse. On the one hand, 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. It is also worthwhile to note that a multi-objective

optimization process where radiation dose is considered along with the fuel-optimal costs, is usually characterized by

22

several impulsive periapse raises immediately after insertion into a GTO and prior to other maneuvers in order to escape

the Van Allen Belt. This solution causes a penalty in terms of fuel-optimality but is essential for mission survival [

48

].

In this paper, we ignore this factor and only concentrate on fuel-optimality in conjunction with eclipses.

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

As discussed before, 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’) [

49

]. 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. The NRHO has been

plotted in the EMEJ2000 frame between ‘2025 JAN 02 20:02:28.677’ and ‘2025 DEC 23 04:37:56.003’ in Figure 7.

Fig. 7 9:2 𝐿2Southern NRHO in the EMEJ2000 frame.

Additionally, the resupply spacecraft was assumed to have an initial mass (

𝑚0

) of 1000 kg, with the engine generating

23

a maximum thrust (

Tmax

) of 0.5 N (approximately equivalent to 2 BPT-4000 Hall-thruster engines (now named XR5)

operating at 4.839 kW [

50

]) resulting in a maximum thrust acceleration of 5

.

0

×

10

−4

m/s with a constant speciﬁc

impulse of Isp =1500 s.

Eﬃcient transfers from

sGTO1/sGTO2

to 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

sGTO1

was 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.

B. 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

Δ

V for the transfer was

1641.867 m/s. The transfer trajectory is shown in Figure 8, 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’.

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 Δ𝑉cost with the former.

The bang-oﬀ-bang throttle sequence for the transfer trajectory is depicted in Figure 9. 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 9 clearly shows the eclipse arcs being 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

24

Fig. 8 Full Transfer : sGTO1- 9:2 NRHO (EMEJ2000).

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

Fig. 9 Thrusting sequence, eclipse condition, and switching function vs. time.

25

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 also an important parameter along with their duration.

Figure 10 depicts this data displayed in a histogram plot with separate bins depending on eclipse duration. The spacecraft

Fig. 10 Thrusting Sequence and Eclipse Condition.

encounters a total of 19 eclipses enroute to the NRHO from the

sGTO1

for 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, greater than 150 minutes 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.

C. 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

26

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

sGTO1

and 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

Δ

V for the transfer was 1991.916 m/s. The transfer trajectory is shown in Figure 11, plotted 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’.

Fig. 11 Full Transfer: sGTO1- 9:2 NRHO (EMEJ2000).

The bang-oﬀ-bang throttle sequence is depicted in Figure 12. Since the eclipse-driven coast arcs happen during

thrust arcs, it leads to forced coasts for the eclipse duration. Comparing it with Figure 9, it is apparent that there are no

eclipses on the selected LTCA. The blown up image in Figure 12 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 13 with separate bins depending on the eclipse duration. The spacecraft

encounters 15 eclipses enroute to the NRHO from the

sGTO1

for 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.

27

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

Fig. 12 Thrusting sequence, eclipse condition, and switching function.

Fig. 13 Thrusting sequence and eclipse condition.

28

D. 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 14 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 14, 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

𝜔

as possible, among the top ﬁve

cases having the least radius with respect to the Earth. The nature of eclipses encountered by the spacecraft enroute has

been plotted as binned histograms in Figure 15.

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 [21].

Table 2 Summary of departure and arrival times, Δ𝑉, maximum eclipse duration and number of eclipses for

diﬀerent mission scenarios in YR 2025.

Departure Epoch Arrival Epoch ΔV (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

29

(a) Trajectories for January - July 2025.

(b) Trajectories for August - December 2025.

Fig. 14 Eclipse-conscious trajectories: YR 2025.

30

Fig. 15 Eclipse binning for Mission Scenarios in YR 2025.

E. Early Departure

Purely fuel-optimal trajectories typically result in a lower

Δ

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 [

51

] 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

sGTO2

on ‘16 JAN 2025 14:12:43.83’ and inserts into the NRHO on ‘04 MAY 2025 00:41:05.433’,

incurring a

Δ𝑉

cost of 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.

As mentioned above, 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 16 depicts the nominal transfer trajectory and the right

plot depicts the trajectory for the analogous transfer but an early departure (

𝑒𝑡9P =𝑒𝑡dep −

9

TsGTO2

). Here,

𝑒𝑡dep

is the

31

ephemeris time associated with the nominal departure epoch (‘16 JAN 2025 14:12:43.83’).

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

Fig. 16 Eﬀect of early departure on transfer trajectories.

Notice the longer ﬁnal spiral for the early departure trajectory on the right plot in Figure 16. 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 ΔV (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

32

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

Δ𝑉

cost 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.

It is worthwhile to note that, for this discussion the departure epoch was assumed to be the only variable in the

problem. A more comprehensive analysis is possible by considering the rendezvous LTCA patch state as an additional

variable and solving for all possible combinations. This search is beyond the scope of this work, however, it is likely that

optimization of the patch state would indeed modify the

Δ𝑉

cost for each case. An inverse viewpoint can be taken: If the

Δ𝑉

variations are within the vehicle’s capability, then we see that the methodology gives rise to neighboring trajectories

with small and comparable maximum eclipse duration. The methodology of pre-computing the LTCAs provides an

inherent ﬂexibility during trajectory design, where several viable transfer trajectories can be readily computed between

boundary conditions that lie in a neighborhood of the nominal trajectory boundaries. It is also worth mentioning that

a pragmatic approach to launch period analysis should include an ‘anchor point’ (ﬁxed wrt. epoch) to drive future

operational requirements of the mission, therefore making it an important consideration along with the vehicle capability.

It is important to emphasize that in practice there are navigational challenges associated with ﬂying a manifold trajectory.

In particular, highly precise position and velocity tolerances are likely needed for delivery onto the manifold coast

portion of the NRHO. In addition to radiometric state estimation uncertainties, maneuver execution errors are usually a

major source of error for SEP and are factors that would need to be characterized. Additionally, the many-revolution

segments of trajectories have to be continually re-optimized in ﬂight because spacecraft state uncertainty in the fast

variable quickly drifts from the reference (within a week or two). For these reasons, orbital averaging techniques [

52

]

have proved a valuable tool for the orbit raising phase since an end-to-end reference trajectory becomes necessary in this

regime.

VII. Conclusion

In this paper, a methodology to generate eﬃcient, eclipse-conscious transfers leveraging ephemeris-driven asymptotic

long terminal coast arcs has been presented. A rigorous treatment of computation and subsequent analysis of approximate

stable manifold analogues for a quasi-periodic orbit in a high-ﬁdelity model was provided. Using favorable periapse

states as patch-points on these 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 Geosynchronous Transfer Orbit to the

9:2 Near Rectilinear Halo Orbit of the southern

𝐿2

family, with a focus on providing accurate trajectories for “cargo

33

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

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.

Acknowledgments

This work was completed 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). The authors’ would like to thank Jon Sims and Daniel Grebow of Jet Propulsion Laboratory,

California Institute of Technology for participating in illuminating discussions throughout the course of this work.

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