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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 formulation 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 composite 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 incorporate eclipse-induced coast arcs and their impact within optimization. The results quantify accurate Δ costs required for achieving efficient eclipse-conscious transfers for several launch opportunities in 2025 and are anticipated to find applications for analogous uncrewed missions.
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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׈𝒆1,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(𝑡𝑡))sin(𝜔(𝑡𝑡))0
sin(𝜔(𝑡𝑡))cos(𝜔(𝑡𝑡))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=𝑹𝑖𝑹𝑐
and
𝝆=𝑹𝑹𝑐
. 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(𝑹𝑹𝑖)
|𝑹𝑹𝑖|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(𝝆𝝆𝑖)
|𝝆𝝆𝑖|3Õ
𝑖𝑐
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(𝒓𝒓𝑖)
|𝒓𝒓𝑖|3Õ
𝑖𝑐
𝜇𝑖(𝒓𝑖)
|𝒓𝑖|3
| {z }
𝒇𝑔(𝑡, 𝒓,¤
𝒓)
2𝝎פ
𝒓𝝎× (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
|𝒓𝒓𝑖|31+3(𝑥𝑥𝑖)2
|𝒓𝒓𝑖|2,𝜕¥𝑦
𝜕𝑦 =Õ
𝑖
𝜇𝑖
|𝒓𝒓𝑖|31+3(𝑦𝑦𝑖)2
|𝒓𝒓𝑖|2,
𝜕¥𝑧
𝜕𝑧 =Õ
𝑖
𝜇𝑖
|𝒓𝒓𝑖|31+3(𝑧𝑧𝑖)2
|𝒓𝒓𝑖|2,𝜕¥𝑥
𝜕𝑦 =Õ
𝑖
𝜇𝑖
|𝒓𝒓𝑖|33(𝑥𝑥𝑖)(𝑦𝑦𝑖)
|𝒓𝒓𝑖|2,
𝜕¥𝑥
𝜕𝑧 =Õ
𝑖
𝜇𝑖
|𝒓𝒓𝑖|33(𝑥𝑥𝑖)(𝑧𝑧𝑖)
|𝒓𝒓𝑖|2,𝜕¥𝑦
𝜕𝑧 =Õ
𝑖
𝜇𝑖
|𝒓𝒓𝑖|33(𝑧𝑧𝑖)(𝑦𝑦𝑖)
|𝒓𝒓𝑖|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
𝜆𝑈𝜆1
and
𝜆𝑆𝜆2
represent the unstable and stable eigenvectors, respectively. These are labeled as,
𝝃𝑈 ,0𝝃𝑈(𝑡0)
and
𝝃𝑆,0𝝃𝑆(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=𝐸𝑒sin 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+1𝑒𝑖sin 𝐸𝑗+1=2𝜋
𝑇𝑖
(𝑡𝑗+1𝑡𝑝),
𝑡𝑗+1=𝑡𝑝+𝑇𝑖
2𝜋(𝐸𝑗+1𝑒𝑖sin 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=𝜏𝑇𝑖
2,
𝐸1=
0,Peri-Peri
𝜋, Apo-Apo
,
𝑡𝑝=
𝜏, Peri-Peri
𝜏𝑇𝑖
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 =sin1𝑅𝑆
||𝒓𝑆/𝑠𝑐 | | ,ABr =sin1𝑅𝐵
||𝒓𝐵/𝑠𝑐 | | ,AD =cos1𝒓𝑇
𝐵/𝑠𝑐 𝒓𝑆/𝑠𝑐
||𝒓𝐵/𝑠𝑐 | | | | 𝒓𝑆/𝑠 𝑐 || ,(21)
where
𝒓𝑆/𝑠𝑐 =𝒓𝑆𝒓𝑠𝑐
and
𝒓𝐵/𝑠𝑐 =𝒓𝐵𝒓𝑠𝑐
. 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=|𝜔0𝜔pp |
, 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
𝑨R6×1
denotes the unforced vector part of the dynamics and
BR6×3
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 , 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"1tanh 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+𝜆𝑚1,(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"1tanh 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 𝜖do
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
𝑓𝛾 > 1;Initial Guess:𝝀2b
0(𝜌𝑠=1)
while 𝜌𝑠𝜌
𝑠do
while 𝜖2𝑏
MF 𝜖do
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=𝝀2b𝜌𝑠=𝜌
𝑠
while 𝜌𝑜𝜌
𝑜do
while 𝜖2𝑏𝑆
MF 𝜖do
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 𝜌𝑝1do
while 𝜖hfS
MF 𝜖do
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.
References
[1]
Williams, S. N., and Coverstone-Carroll, V., “Benefits of solar electric propulsion for the next generation of planetary
exploration missions,The Journal of the Astronautical Sciences, Vol. 45, No. 2, 1997, pp. 143–159. URL
https:
//doi.org/10.1007/BF03546373.
[2]
Forward, R. L., “Advanced Space Propulsion Study-Antiproton and Beamed Power Propulsion,” Tech. rep., HUGHES
RESEARCH LABS MALIBU CA, 1987.
[3]
Pan, X., and Pan, B., “Practical Homotopy Methods for Finding the Best Minimum-Fuel Transfer in the Circular Restricted
Three-Body Problem,” IEEE Access, Vol. 8, 2020, pp. 47845–47862. URL doi:10.1109/ACCESS.2020.2978246.
[4]
Betts, J. T., “Survey of numerical methods for trajectory optimization,Journal of guidance, control, and dynamics, Vol. 21,
No. 2, 1998, pp. 193–207. URL https://doi.org/10.2514/2.4231.
[5]
Conway, B. A., “A survey of methods available for the numerical optimization of continuous dynamic systems,Journal of
Optimization Theory and Applications, Vol. 152, No. 2, 2012, pp. 271–306. URL DOI10.1007/s10957-011-9918- z.
34
[6]
Olympio, J. T., “Optimal control problem for low-thrust multiple asteroid tour missions,” Journal of guidance, control, and
dynamics, Vol. 34, No. 6, 2011, pp. 1709–1720. URL https://doi.org/10.2514/1.53339.
[7]
Betts, J. T., and Erb, S. O., “Optimal low thrust trajectories to the moon,” SIAM Journal on Applied Dynamical Systems, Vol. 2,
No. 2, 2003, pp. 144–170. URL https://doi.org/10.1137/S1111111102409080.
[8]
Pan, B., Lu, P., Pan, X., and Ma, Y., “Double-homotopy method for solving optimal control problems,” Journal of Guidance,
Control, and Dynamics, Vol. 39, No. 8, 2016, pp. 1706–1720. URL https://doi.org/10.2514/1.G001553.
[9]
Pérez-Palau, D., and Epenoy, R., “Fuel optimization for low-thrust Earth–Moon transfer via indirect optimal control,” Celestial
Mechanics and Dynamical Astronomy, Vol. 130, No. 2, 2018, p. 21. URL
https://doi.org/10.1007/s10569-017- 9808-2
.
[10]
Singh, S. K., Taheri, E., Woollands, R., and Junkins, J., “Mission Design for Close-Range Lunar Mapping by Quasi-Frozen
Orbits,” 70th International Astronautical Congress, Washington DC, USA, 2019.
[11]
Williams, J., Senent, J. S., Ocampo, C., Mathur, R., and Davis, E. C., “Overview and software architecture of the copernicus
trajectory design and optimization system,” 2010.
[12]
Whiffen, G., “Mystic: Implementation of the static dynamic optimal control algorithm for high-fidelity, low-thrust trajectory
design,” AIAA/AAS Astrodynamics Specialist Conference and Exhibit, 2006, p. 6741.
[13]
Petropoulos, A., “Low-thrust orbit transfers using candidate Lyapunov functions with a mechanism for coasting,AIAA/AAS
Astrodynamics Specialist Conference and Exhibit, 2004, p. 5089.
[14]
Shannon, J., Ozimek, M., Atchison, J., and Hartzell, C., “Rapid Design and Exploration of High-Fidelity Low-Thrust Transfers
to the Moon,” 2020 IEEE Aerospace Conference, IEEE, 2020, pp. 1–12.
[15]
Yang, G., “Earth-moon trajectory optimization using solar electric propulsion,” Chinese Journal of Aeronautics, Vol. 20, No. 5,
2007, pp. 452–463.
[16] Feistel, A., and Ranieri, C., “Modeling Perturbations and Operational Considerations when Using Indirect Optimization with
Equinoctial Elements,” AAS/AIAA Space Flight Mechanics Meeting, 2009, pp. 1737–1756.
[17]
Anderson, R. L., and Lo, M. W., “Role of invariant manifolds in low-thrust trajectory design,” Journal of guidance, control, and
dynamics, Vol. 32, No. 6, 2009, pp. 1921–1930. URL https://doi.org/10.2514/1.37516.
[18]
Dellnitz, M., Junge, O., Post, M., and Thiere, B., “On target for Venus–set oriented computation of energy efficient
low thrust trajectories,Celestial Mechanics and Dynamical Astronomy, Vol. 95, No. 1-4, 2006, pp. 357–370. URL
https://doi.org/10.1007/s10569-006- 9008-y.
[19]
Vaquero, M., and Howell, K. C., “Leveraging resonant-orbit manifolds to design transfers between libration-point orbits,Journal
of Guidance, Control, and Dynamics, Vol. 37, No. 4, 2014, pp. 1143–1157. URL https://doi.org/10.2514/1.62230.
35
[20]
Singh, S. K., Anderson, B. D., Taheri, E., and Junkins, J. L., “Exploiting manifolds of L1 halo orbits for end-to-end Earth–Moon
low-thrust trajectory design,Acta Astronautica, Vol. 183, 2021, pp. 255–272. https://doi.org/https://doi.org/10.1016/j.actaastro.
2021.03.017.
[21]
Singh, S. K., Anderson, B. D., Taheri, E., and Junkins, J. L., “Low-Thrust Transfers to Candidate Near-Rectilinear Halo Orbits
facilitated by Invariant Manifolds,” 2020 AAS/AIAA Astrodynamics Specialist Virtual Lake Tahoe Conference, 2020.
[22]
Qu, Q., Xu, M., and Peng, K., “The cislunar low-thrust trajectories via the libration point,” Astrophysics and Space Science, Vol.
362, No. 5, 2017, p. 96.
[23]
Cox, A. D., Howell, K. C., and Folta, D. C., “Trajectory design leveraging low-thrust, multi-body equilibria and their manifolds,
The Journal of the Astronautical Sciences, Vol. 67, No. 3, 2020, pp. 977–1001.
[24]
Topputo, F., Vasile, M., and Bernelli-Zazzera, F., “Low energy interplanetary transfers exploiting invariant manifolds of the
restricted three-body problem,” Journal of the Astronautical Sciences, Vol. 53, No. 4, 2005, pp. 353–372.
[25]
Capdevila, L. R., and Howell, K. C., “A transfer network linking Earth, Moon, and the triangular libration point regions in the
Earth-Moon system,” Advances in Space Research, Vol. 62, No. 7, 2018, pp. 1826–1852.
[26]
Zhang, R., Wang, Y., Zhang, H., and Zhang, C., Transfers from distant retrograde orbits to low lunar orbits,” Celestial
Mechanics and Dynamical Astronomy, Vol. 132, No. 8, 2020, pp. 1–30.
[27]
Oshima, K., “The use of vertical instability of
𝐿_
1and
𝐿_
2planar Lyapunov orbits for transfers from near rectilinear halo
orbits to planar distant retrograde orbits in the Earth–Moon system,” Celestial Mechanics and Dynamical Astronomy, Vol. 131,
No. 3, 2019, pp. 1–28.
[28]
Trofimov, S., Shirobokov, M., Tselousova, A., and Ovchinnikov, M., “Transfers from near-rectilinear halo orbits to low-perilune
orbits and the Moon’s surface,” Acta Astronautica, Vol. 167, 2020, pp. 260–271.
[29]
Taheri, E., Junkins, J. L., Kolmanovsky, I., and Girard, A., “A novel approach for optimal trajectory design with multiple
operation modes of propulsion system, part 1,” Acta Astronautica, 2020. URL https://doi.org/10.1016/j.actaastro.2020.02.042.
[30]
Gomez, G., Jorba, A., Masdemont, J., and Simó, C., “A dynamical systems approach for the analysis of the SOHO mission,”
ESA, Spacecraft Flight Dynamics p 449-454(SEE N 92-24719 15-12), 1991.
[31]
Koon, W. S., Lo, M. W., Marsden, J. E., and Ross, S. D., “Low energy transfer to the Moon,Celestial Mechanics and
Dynamical Astronomy, Vol. 81, No. 1, 2001, pp. 63–73.
[32]
Lo, M. W., Williams, B. G., Bollman, W. E., Han, D., Hahn, Y., Bell, J. L., Hirst, E. A., Corwin, R. A., Hong, P. E., Howell,
K. C., et al., “Genesis mission design,” The Journal of the astronautical sciences, Vol. 49, No. 1, 2001, pp. 169–184. URL
https://doi.org/10.1007/BF03546342.
[33]
Geffroy, S., and Epenoy, R., “Optimal low-thrust transfers with constraints—generalization of averaging techniques,Acta
astronautica, Vol. 41, No. 3, 1997, pp. 133–149. URL https://doi.org/10.1016/S0094-5765(97)00208-7.
36
[34]
Cerf, M., “Fast solution of minimum-time low-thrust transfer with eclipses,” Proceedings of the Institution of Mechanical
Engineers, Part G: Journal of Aerospace Engineering, Vol. 233, No. 7, 2019, pp. 2699–2714. URL https://doi.org/10.1177/
0954410018785971.
[35]
Woollands, R., and Taheri, E., “Optimal Low-Thrust Gravity Perturbed Orbit Transfers With Shadow Constraints,The 2019
AAS/AIAA Astrodynamics Specialist Conference, Portland, Maine, 2019.
[36] Aziz, J. D., “Low-Thrust Many-Revolution Trajectory Optimization,” Ph.D. thesis, University of Colorado at Boulder, 2018.
[37]
Taheri, E., “Optimization of Many-Revolution Minimum-Time Low-Thrust Trajectories Using Sundman Transformation,” AIAA
Scitech 2021 Forum, 2021, p. 1343. URL https://arc.aiaa.org/doi/abs/10.2514/6.2021-1343.
[38]
Aziz, J., Scheeres, D., Parker, J., and Englander, J., “A smoothed eclipse model for solar electric propulsion trajectory
optimization,” Transactions of the Japan Society for Aeronautical and Space Sciences, Aerospace Technology Japan, 2019, pp.
17–181.
[39]
Haberkorn, T., Martinon, P., and Gergaud, J., “Low thrust minimum-fuel orbital transfer: a homotopic approach,” Journal of
Guidance, Control, and Dynamics, Vol. 27, No. 6, 2004, pp. 1046–1060. URL https://doi.org/10.2514/1.4022.
[40]
Liu, H., and Tongue, B. H., “Indirect spacecraft trajectory optimization using modified equinoctial elements,Journal of
Guidance, Control, and Dynamics, Vol. 33, No. 2, 2010, pp. 619–623. URL https://doi.org/10.2514/1.45498.
[41]
Taheri, E., Kolmanovsky, I., and Atkins, E., “Enhanced smoothing technique for indirect optimization of minimum-fuel
low-thrust trajectories,Journal of Guidance, Control, and Dynamics, Vol. 39, No. 11, 2016, pp. 2500–2511. URL
https://doi.org/10.2514/1.G000379.
[42]
Junkins, J. L., and Taheri, E., “Exploration of alternative state vector choices for low-thrust trajectory optimization,” Journal of
Guidance, Control, and Dynamics, Vol. 42, No. 1, 2019, pp. 47–64. URL https://doi.org/10.2514/1.G003686.
[43]
Walker, M., Ireland, B., and Owens, J., “A set modified equinoctial orbit elements,Celestial mechanics, Vol. 36, No. 4, 1985,
pp. 409–419. URL https://doi.org/10.1007/BF01227493.
[44]
Taheri, E., Junkins, J. L., Kolmanovsky, I., and Girard, A., “A novel approach for optimal trajectory design with multiple
operation modes of propulsion system, part 2,” Acta Astronautica, 2020. URL https://doi.org/10.1016/j.actaastro.2020.02.047.
[45]
Ross, I. M., Gong, Q., and Sekhavat, P., “Low-thrust, high-accuracy trajectory optimization,” Journal of Guidance, Control,
and Dynamics, Vol. 30, No. 4, 2007, pp. 921–933.
[46]
Whitley, R. J., Davis, D. C., Burke, L. M., McCarthy, B. P., Power, R. J., McGuire, M. L., and Howell, K. C., “Earth-moon near
rectilinear halo and butterfly orbits for lunar surface exploration,AAS/AIAA Astrodynamics Specialists Conference, Snowbird,
Utah, 2018.
[47]
Flohrer, T., Choc, R., and Bastida, B., “Classification of geosynchronous objects,GEN-DBLOG-00086-OPS-GR, , No. 14,
2012.
37
[48]
Lorenz, S. L., C., and Guzman, “MULTI-OBJECTIVE OPTIMIZATION OF LOW THRUST TRAJECTORIES FOR
PROPELLANT MASS, TIME OF FLIGHT, AND RADIATION DOSE,Paper No. AAS 19-398, 29th AAS/AIAASpace Flight
Mechanics Meeting, Ka’anapali, Hawaii, 2019, 2019.
[49]
Lee, D. E., “White Paper: Gateway Destination Orbit Model: A Continuous 15 year NRHO Reference Trajectory,JSC-E-DAA-
TN72594, NTRS - NASA Technical Reports Server, 2019.
[50]
Hofer, R., “High-specific impulse operation of the BPT-4000 Hall thruster for NASA science missions, 46th
AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, 2010, p. 6623.
[51]
Taheri, E., and Junkins, J. L., “How many impulses redux,” The Journal of the Astronautical Sciences, 2019, pp. 1–78. URL
https://doi.org/10.1007/s40295-019- 00203-1.
[52]
Kluever, C. A., and Conway, B., “Low-thrust trajectory optimization using orbital averaging and control parameterization,”
Spacecraft Trajectory Optimization, 2010, pp. 112–138.
38
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... Because of the definition of Chebyshev nodes in Eqs.(10) and(11). ...
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