2021

These authors contributed equally to this work.

These authors contributed equally to this work.

[1]Koki Ho

[1]Daniel Guggenheim School of Aerospace Engineering, Georgia Institute of
Technology, 620 Cherry St NW, Atlanta, 30332, GA, USA

# Optimizing Multi-Spacecraft Cislunar Space Domain Awareness Systems via
Hidden-Genes Genetic Algorithm

Lois Visonneau<EMAIL_ADDRESS>Yuri Shimane<EMAIL_ADDRESS><EMAIL_ADDRESS>*

###### Abstract

This paper proposes an optimization problem formulation to tackle the
challenges of cislunar Space Domain Awareness (SDA) through multi-spacecraft
monitoring. Due to the large volume of interest as well as the richness of the
dynamical environment, traditional design approaches for Earth-based
architectures are known to have challenges in meeting design requirements for
the cislunar SDA; thus, there is a growing need to have a multi-spacecraft
system in cislunar orbits for SDA. The design of multi-spacecraft-based
cislunar SDA architecture results in a complex multi-objective optimization
problem, where parameters such as number of spacecraft, observability, and
orbit stability must be taken into account simultaneously. Through the use of
a multi-objective hidden genes genetic algorithm, this study explores the
entirety of the design space associated with the cislunar SDA problem. A
demonstration case study shows that our approach can provide architectures
optimized for both cost and effectiveness.

###### keywords:

Space Domain Awareness, Cislunar, CR3BP, Hidden Gene, Periodic Obits

##  1 Introduction

The cislunar space is gaining traction as increasing numbers of both human and
robotic exploration missions are being planned and launched in the coming
years. With greater traffic in the cislunar space, the need for space domain
awareness (SDA) extended further out of the Earth-orbit region is primordial.
While the Low-Earth Orbit (LEO), Medium Earth Orbit (MEO), and
Geosynchronous/Geostationary Earth Orbit (GSO/GEO) regimes are well-understood
and regulated through existing infrastructures, these existing techniques are
not always extendable to the cislunar domain Holzinger2021_report ; one
primary contributor to this difficulty is the nature of the dynamics
significantly affected by both the Earth and the Moon’s gravitational forces;
the motion of objects in the three-body problem, even in the case of
simplified models, are well-known to be chaotic Frueh2021 . From the
perspective of sensor tasking, the challenge also rise from the vast volume
that a cislunar SDA architecture must cover compared to an Earth-orbit SDA
architecture. In terms of orders of magnitude, the Earth-Moon distance is
roughly 10 times the GEO altitude, while low-energy transfers may have apogees
extending to around 1.5 million km, roughly five times the Earth-Moon
distance. Traditional Earth-based sensors have difficulty covering such a
large volume, particularly the region near the direction of the Moon; rather,
spacecraft tasked with performing observations of targets need to be placed
into appropriate orbits.

Several previous studies proposed approaches for studying architectures tasked
for cislunar SDA. While earlier works have concentrated on the coverage of the
lunar surface Grebow2008 , the past few years saw a rapid growth in the number
of works focusing on detecting, tracking, and maintaining custody of both
cooperative and non-cooperative assets in the cislunar region Fowler2020 ;
Cunio2020 ; Bolden2020 ; Dao2020 ; Thompson2021 ; Frueh2021b ; Vendl2021 ;
Wilmer2022 ; Dahlke2022 ; Fedeler2022 ; Klonowski2022 . The vastness of the
cislunar space necessitates the use of orbits that extend beyond the region
between LEO and GEO, where traditional surveillance and tracking architectures
currently reside. The region outside GEO, commonly referred to as xGEO in the
US Defense sector, typically results in two types of orbital locations that
may be used: the prior are highly elliptic Earth orbits that exhibit resonance
with the Earth-Moon system, and the latter are libration point orbits (LPOs)
that revolve around any of the five Lagrange points of the Earth-Moon system.

Some variation in the definition of the cislunar “domain” of interest is seen
in literature, and this has a direct impact on the type of orbit(s) that may
be useful. For example, the works by Wilmer et al Wilmer2022  and Dahlke et al
Dahlke2022  study SDA activities for the vicinity of Earth-Moon L1, L2, or L3
points; hence, these are done by employing LPOs about these points. In
contrast, there are also numerous studies considering vast, cislunar regions
of interest. For example, Vendl and Holzinger Vendl2021  considered a
trapezoidal region of interest lying in the Earth-Moon orbital plane,
extending from GEO to the far side of the Moon, in this case, the authors have
considered the use of LPO families in the vicinity of L1 and L2. An even
larger area of interest is considered in the works of Cunio et al Cunio2020 ,
Bolden et al Bolden2020  and Frueh et al Frueh2021b , where the entire region
engulfed by the orbit of the Moon is to be monitored. In this case, Earth-Moon
resonant orbits are promising candidates as their tracks span over large
volumes of the cislunar space. Some other works have considered simulating the
tracking performance of actual, flown, planned, or hypothetical translunar
trajectories or cislunar spacecraft. The works of Dao et al Dao2020 , Fowler
et al Fowler2020 , and Thompson et al Thompson2021  combine sensor models with
candidate observer orbits to conduct a high-fidelity simulation of tracking
cislunar objects.

For cislunar SDA scenarios involving coverage of regions in the cislunar
space, observation limitations using single spacecraft have been identified
Vendl2021 . Recent work Badura2022  has shown the importance of having
sufficient geometric diversity between multiple observers in order to obtain
high observability. Nonetheless, questions regarding the optimal number and
distribution of observers in space still remain unanswered.

This paper’s focus is on optimizing the multi-satellite architecture for
cislunar SDA. Optimizing the design of such an architecture that ensures
performance while also minimizing the cost involved in its maintenance is a
challenging task, where a varying number of spacecraft must be considered
online. In particular, the main complexity in choosing the appropriate
architecture is in the combinatorial nature of the problem, as the number of
spacecraft considered and the choice of orbits have infinite possibilities.

To allow for a variable number of observers trading off the overall
architecture cost and effectiveness, an algorithm capable of handling a
variable-length decision vector must be considered. In this work, a multi-
objective variant of the hidden-genes genetic algorithm (HGGA) Gad2011 ;
Abdelkhalik2016 ; Abdelkhalik2018  is employed. HGGA allows for the
exploration of the entire design space in one search, simultaneously varying
the continuous variables, the periodic orbits periods and phases angles, as
well as the discrete variables such as the choice of orbits considered and the
number of spacecraft required. Through the use of the multi-objective HGGA, we
are able to obtain Pareto front solutions corresponding to a variety of
cislunar SDA architectures, providing insights into the trade-offs between
performance and costs for this problem.

This paper is organized as follows: firstly, Section 2 introduces the
simulation environment, including the dynamical system, definitions of the
target distribution, and measures to quantify the performance of a cislunar
SDA architecture. Then, in Section 3, the multiobjective optimization problem
formulation, along with the multi-objective HGGA algorithm, are introduced.
This is followed by Section 4, where the proposed formulation is employed to
obtain Pareto front solutions of cislunar SDA architectures. Finally, Section
5 concludes this work.

##  2 Simulation of Cislunar Space Domain Awareness Architecture

The problem of cislunar space domain awareness requires the development of a
dedicated simulation environment that tracks the location of the Earth, Moon,
Sun, observer(s), and target(s). To this end, the Earth-Moon circular
restricted three-body problem (CR3BP), along with the associated Earth-Moon
rotating frame, is particularly advantageous as it provides a reference frame
in which the Earth and Moon may be assumed to be constant. In this work, the
targets are considered to be stationary points distributed in cislunar space,
while

###  2.1 Dynamical System

The study of motion in the Earth-Moon system using the CR3BP has been highly
successful due to the relatively low eccentricity of the Moon’s orbit and the
small effect of the Sun’s gravitational effect, especially in the vicinity of
the Earth or Moon Frueh2021 . The CR3BP equations of motion are given by

$\displaystyle\ddot{x}-2\dot{y}$ $\displaystyle=\dfrac{\partial U}{\partial
x}$ (1) $\displaystyle\ddot{y}+2\dot{x}$ $\displaystyle=\dfrac{\partial
U}{\partial y}$ $\displaystyle\ddot{z}$ $\displaystyle=\dfrac{\partial
U}{\partial z}$

where $U$ is the pseudo-potential given by

$U=\dfrac{x^{2}+y^{2}}{2}+\dfrac{1-\mu}{r_{1}}+\dfrac{\mu}{r_{2}}$ (2)

and $\mu$ is the mass-parameter of the CR3BP system, given by

$\mu=\dfrac{{M}_{2}}{M_{1}+M_{2}}$ (3)

where $M_{1}$ is the mass of the primary body, and $M_{2}$ is the mass of the
secondary body. The corresponding state-transition matrix (STM) can be
propagated along with the states through the initial value problem

$\left\\{\begin{array}[]{l}\dot{\Phi}(t)=\boldsymbol{A}(\boldsymbol{x})\Phi(t)\\\
\Phi(0)=\mathbf{I}_{6\times 6}\end{array}\right.$ (4)

where $\boldsymbol{A}$ is the Jacobian of the dynamics.

###  2.2 Periodic Orbit and Database

Periodic orbits about libration points of the CR3BP (LPOs) are promising
candidates for cislunar SDA architectures due to their positioning within the
cislunar space. Specifically, the L1 and L2 Lyapunov and halo families, as
well as the distant retrograde orbit (DRO) family offer both favorable
geometric coverage within the space of interest and are known to have
acceptable or high stability properties suitable for long-term use.

A periodic orbit must obey the relation

$\boldsymbol{x}(t+P)=\boldsymbol{x}(t)$ (5)

where $\boldsymbol{x}$ are the states, and $P$ is the period. In the context
of the LPOs of the selected families, they are known to exhibit a symmetry
about the $xz$-plane in the Earth-Moon rotating plane; leveraging this result,
the well-known single-shooting scheme Howell1984

$\displaystyle X_{k+1}=X_{k}-\left[\dfrac{\partial F(X_{k})}{\partial
X}\right]^{-1}F(X_{k})$ (6)

where the residual $F$ is defined by

$F=\begin{bmatrix}y(P/2)\\\ \dot{x}(P/2)\\\ \dot{z}(P/2)\end{bmatrix}$ (7)

and the design vector $X$ contain 3 out of the 4 free variables, namely
$x(0)$, $z(0)$, $\dot{y}(0)$, and the period $P$, can be employed. The
Jacobian $\dfrac{\partial F(X_{k})}{\partial X}$ can be simply constructed
from combinations of the entries of the STM and the equations of motion if $T$
is included in $X$. The choice of variables to be included in $X$ is
application dependent; in the context of creating a database of LPOs, the
choice is made based on the ease of convergence to the LPO in the desired
range of $X$. For example, low-amplitude halos are difficult to obtain unless
$z(0)$ is fixed to a desired non-zero value.

Once an LPO is obtained, its linear stability index $\nu$ can be defined by
evaluating the eigenvalues of its monodromy matrix, which corresponds to the
STM at $t=P$. Specifically, $\nu$ is given by Grebow2008

$\nu=\dfrac{1}{2}\left(\lambda_{\max}+\dfrac{1}{\lambda_{\max}}\right)$ (8)

where $\lambda_{\max}$ is the largest eigenvalue of $\Phi(P)$.

A finite number of periodic orbits belonging to the selected families are
first calculated using a single-shooting method. Figure 1 shows a sample set
of the LPOs constructed for this work. This results in a database that can be
interpolated against properties such as the $x$-axis crossing points on the
$xz$-plane, or the periods, of each LPO. This database can then be linearly
interpolated to yield an initial guess of an arbitrary $x$-axis crossing or
period, which can then rapidly converge to the corresponding LPO.

Figure 1: Example Libration Point Orbits for the Earth-Moon system studied in
this work

When considering the location of a spacecraft in LPOs, its position along the
orbit at a given time must also be considered due to the revolution of the
Sun. We define an initial phase angle $\phi\in[0,2\pi]$ such that the
spacecraft’s initial position along an LPO is given by

$\boldsymbol{x}(t_{0})=\boldsymbol{x}\left(\dfrac{\phi}{2\pi}P\right)$ (9)

###  2.3 Sun-Direction

Along with the state of the observer spacecraft and the location of the
observing targets, the location of the Sun is a fundamental quantity that must
be identified when simulating SDA activities. In the context of trajectory
design, the bi-circular restricted four-body problem (BCR4BP), which considers
the Sun’s motion to be along a circular path on the $xy$-plane of the Earth-
Moon rotating frame, is an insightful model to generate transfers that
leverage gravitational effects of all three bodies, such as low-energy
transfers Koon2001 ; belbruno2002analytic ; jpl-monograph-series12 ;
Boudad2020 . However, for the purpose of analyzing SDA architectures, it is
important to consider the offset between the Earth-Moon and Sun-Earth planes.
As such, in this work, JPL NAIF’s SPICE data is used to obtain the vector from
the Earth-Moon barycenter to the Sun. During the simulation, an arbitrary
reference epoch is chosen, and this Sun-direction vector is extracted at each
time step where the observation is to be simulated.

As an example, the Sun-direction is plotted for a 180 days period in the
Earth-Moon rotating frame in Figure 2, starting on 2024 MAR 25 07:12:36.633.
This reference epoch has been chosen simply as a future date where the Sun
lies on the $xz$-plane of the Earth-Moon rotating frame. Note that the
scattered circles do not represent the actual position of the Sun, but rather
its direction in the Earth-Moon rotating frame. It is possible to observe the
Sun’s helical path, which results in a variation of the direction of the
illuminating source in cislunar space.

Figure 2: Sun directions over 180 days, in the Earth-Moon rotating frame

###  2.4 Observation Targets and Metric

Comparing different cislunar architectures requires the definition of metrics.
Based on Vendl2021 , we consider an observation metric allowing for the
comparison of the persistent detection of the cislunar space for a given
mission horizon. To this end, we define a spatial observation zone as global
as possible so that any trajectory of a spacecraft leaving earth for the
vicinity of the moon or the surrounding would have its trajectory entirely
included in the volume defined. Within this zone, a discrete number of targets
are placed in a gridded fashion. Then, based on the locations of the
observer(s), targets, and the Sun, the observability metric $\Gamma$ is
defined as a scalar value that quantifies the viability of a given SDA
architecture.

####  2.4.1 Target Distribution

A set of static targets that encompasses the cislunar space are considered in
this work. The targets are equispaced in Cartesian space within a truncated
cone of interest that encompasses the region between the GEO belt and Earth-
Moon L2. The choice of this set of targets is motivated by the so-called “cone
of shame”, which is the lunar exclusion zone identified by the Air Force
Research Laboratory to be an area of high interest for monitoring afrl2020 .
Targets used in this work are shown in Figure 3.

Figure 3: Equispaced targets distributed in cislunar space in the Earth-Moon
rotating frame

####  2.4.2 Observation Metric

The visibility of an object is measured by the apparent magnitude, which is a
relative scale for quantifying the brightness of objects. This work follows
the model used by Vendl and Holzinger Vendl2021 , which is presented here for
completeness. Suppose the target has a diameter $d$ and is at a range $\zeta$
from the observer. The apparent magnitude of a target is given by

$m_{\mathrm{target}}=m_{\mathrm{S}}-2.5\log_{10}\left(\dfrac{d^{2}}{\zeta^{2}}\left[\dfrac{a_{\mathrm{spec}}}{4}+a_{\mathrm{diff}}p_{\mathrm{diff}}(\psi)\right]\right)$
(10)

where $m_{\mathrm{S}}=-26.74$ is the apparent magnitude of the Sun, $\psi$ is
the solar phase angle given by

$\psi=\arccos{\left(\dfrac{\boldsymbol{r}_{\mathrm{OT}}\cdot\boldsymbol{r}_{\mathrm{ST}}}{\|\boldsymbol{r}_{\mathrm{OT}}\|\|\boldsymbol{r}_{\mathrm{ST}}\|}\right)}$
(11)

where $\boldsymbol{r}_{\mathrm{OT}}$ is the relative position vector from the
observer to the target, $\boldsymbol{r}_{\mathrm{ST}}$ is the relative
position vector from the Sun to the target, and $p_{\mathrm{diff}}(\psi)$ is
the diffuse phase angle function for a diffuse sphere, given by

$p_{\mathrm{diff}}(\psi)=\dfrac{2}{3\pi}\left[\sin(\psi)+(\pi-\psi)\cos(\psi)\right]$
(12)

Evaluating the apparent magnitude of a target at all time steps over the
mission duration yields a distribution function. The value of apparent
magnitude $m(\alpha)$ in the distribution function is such that for $\alpha$%
of the time steps, the target is brighter than this value is extracted and
compared to a cutoff magnitude $m_{\mathrm{co}}$. The target is then deemed
visible if

$m(\alpha)\geq m_{\mathrm{co}}$ (13)

Using the dynamics presented in the previous sections, candidate observers are
placed in LPO families; examples of LPOs considered in this work are shown in
Figure 1.

Given a set of observers $\mathcal{O}$, a set of targets $\mathcal{T}$, the
duration of the observation campaign $\Delta T$, and a time-step $\Delta t$,
the simulation computes an observability metric $\Gamma$, which corresponds to
the percentage of targets inside the grid that are deemed visible by (13),
using Algorithm 1.

Algorithm 1 Computation of observability metric for a given architecture

Observer set $\mathcal{O}$, target set $\mathcal{T}$, observation campaign
duration $\Delta T$, time-step $\Delta t$

$N=\Delta T/\Delta t$

for $i=1$ to $N$ do $\ignorespaces\triangleright$ for each time-step

for $o$ in $\mathcal{O}$ do $\ignorespaces\triangleright$ for each observer

for $t$ in $\mathcal{T}$ do $\ignorespaces\triangleright$ for each target

Update position of observer $o$

Compute visible magnitude of $t$ from $o$

end for

end for

end for

Compute $\Gamma$ as average number of visible targets over $N$ time-steps

In practice, the algorithm is implemented via vectorization to avoid
performance degradation due to the nested for-loops.

##  3 Optimization of Cislunar Monitoring Architecture

Leveraging the simulation environment for cislunar SDA, we consider a
multiobjective optimization problem that simultaneously seeks to maximize the
coverage and minimize the operational cost of the SDA architecture. First,
this section introduces the optimization problem that is to be solved. Then,
we introduce the HGGA, which enables the optimization of the SDA architecture
without prescribing a constant number of observers a priori.

###  3.1 Optimization Problem Formulation

The multiobjective optimization problem is formulated as follows:

$\min_{\mathbf{X}}\left(-\Gamma,\,\,\sum_{j=1}^{n}i_{j}\nu_{j},\,\,N_{\mathrm{spacecraft}}\right)$
(14)

Here, the objectives are to maximize the coverage, minimize the total
stability index of all LPOs used, and minimize the total number of spacecraft.

The decision vector $\mathbf{X}$ consists of a list of observers:

$[\mathrm{Observer}_{1},...,\,\mathrm{Observer}_{N_{\mathrm{spacecraft}}}]$
(15)

where an observer is defined by the 3 following variables:

$[\mathrm{Orbit\,Family},P,\phi]$ (16)

and orbit families considered are

$[\mathrm{L1\,\,Halo_{S},L2\,\,Halo_{N},DRO,L2\,\,Lyap,L1\,\,Lyap}]$ (17)

A more detailed description of the variables of this problem is given in Table
1. Design variables that are traded off in this work are the total number of
satellites, their respective allocated orbit family, their orbit period, and
the initial phase angle along the orbit. We can further decompose the
variables into two distinct categories, discrete variables which are used to
represent the orbit families. The rest of the variables will be treated as
continuous variables.

Table 1: Variables for the Cislunar SDA problem Discrete Variables | Continuous Variables
---|---
Orbit Family: $L1\,Halo_{S}^{1},\cdots,L1\,Halo_{S}^{i_{1}}$ | Orbit Period: $P_{L1\,Halo_{S}^{i_{1}}},\cdots,P_{L1\,Halo_{S}^{i_{1}}}$
⋮ ⋮ |  ⋮ ⋮
$L1\,Lyap^{1},\cdots,L1\,Lyap^{i_{n}}$ |  $P_{L1\,Lyap^{1}},\cdots,P_{L1\,Lyap^{i_{n}}}$
| Phase Angle:
$\phi_{L1\,Halo_{S}^{i_{1}}},\cdots,\phi_{L1\,Halo_{S}^{i_{1}}}$
|  ⋮ ⋮
|  $\phi_{L1\,Lyap^{1}},\cdots,\phi_{L1\,Lyap^{i_{n}}}$
|

As a consequence of this formulation, the total number of spacecraft is an
intrinsic variable comprised in the orbit family variable as follows.

$N_{\mathrm{spacecraft}}=\sum\limits_{j=1}^{n}i_{j}\,\,\,\,(n=5)$ (18)

###  3.2 Hidden-Genes Genetic Algorithm

Given the above formulation, discrete variable values are directly influencing
the number of continuous variables, creating what’s called a Variables-Size
Design Space (VSDS) problemGad2011 ; Abdelkhalik2018 . Traditional algorithms
are not suited for the search of this type of design space.

A relatively novel type of optimization algorithm, Hidden-Genes Genetic
Algorithm (HGGA)Gad2011 ; Abdelkhalik2016 ; Abdelkhalik2018 ; Ellithy2022
allows in this work for the integer variable to enter as a design variable
thus exploring the VSDS.

Studies on metaheuristic algorithms with variable-length chromosomes go back
to the late 1990s Kajitani1996 ; Kim2005 . Specifically, the hidden-genes
genetic algorithm (HGGA) have been recently investigated to use in the Multi
Gravity Assist trajectory optimization problem tackling the complexity of
assessing the optimal number of planets and the sequence of flybys.

The HGGA method considers a fixed length of chromosome $L_{max}$, which
contains the maximum number of design variables. In this case the maximum
length would correspond to two times the total number of spacecraft given that
each spacecraft is assigned two variables. As seen on Figure 4, a tag,
associated to each gene, assesses if the gene is active or hidden. Hidden
genes are not considered in the evaluation of the objective function.
Chromosomes with lower number of spacecraft are designed by switching genes to
hidden genes in order to have the right number of active design variables that
will be used to calculate the objectives to minimize.

Figure 4: Chromosome with hidden genes Abdelkhalik2018

In practice, tags evolve using stochastic processes. Some mechanisms can be
found in Abdelkhalik2018 , while this works uses classical probability-based
one-point crossover and bit-flip mutation operations for the tags. It is
important to note that those operations are independent from the genes’
mutation and crossover operations that take place in the regular genetic
algorithm as can be seen in 2.

The multi-objective HGGA in this study is built upon the Non-dominated Sorting
Genetic Algorithm II (NSGA-II) multi-objective genetic algorithm Deb2002 ,
using simulated binary crossover and polynomial mutations, akin to the
implementation by Ellithy et al Ellithy2022 . Each operation on the genes and
the tags is performed sequentially starting with the genes when creating a new
generation of chromosomes, following Algorithm 2.

Algorithm 2 Overview of the Hidden-Genes NSGA-II crossover and mutation
mechanisms

Parent population $\mathcal{P}$

Children population $\mathcal{C}$ = [ ]  $\ignorespaces\triangleright$
Initialize list of children

while Number of children $\neq$ Number of parents do

Select two parents randomly

if random(0,1) $>$ Gene Crossover Probability then

Perform gene crossover

end if

if random(0,1) $>$ Tags Crossover Probability then

Perform tags crossover

end if

if random(0,1) $>$ Gene Mutation Probability then

Perform gene mutation

end if

if random(0,1) $>$ Tags Mutation Probability then

Perform tags mutation

end if

for Both children do

for Genes in children chromosome do

if Tag associated to gene $\neq$ 0 then

Use gene in objective function evaluation

end if

end for

end for

Add both children to $\mathcal{C}$

end while

Return $\mathcal{C}$

##  4 Numerical Results

The proposed Cislulnar SDA simulation framework is solved using the HGGA.
Table 2 presents the parameters used for the simulation and the optimization.
We first conduct parametric studies for multiple values of $m_{\mathrm{co}}$
then analyze some of the best architectures extracted from the design space.
The $m_{\mathrm{co}}$ limits were picked based on two reasons, the upper limit
was chosen as the first apparent magnitude where literature’s architectures’
observability begin to decrease Holzinger2021_report . The lower limit is the
highest limiting magnitude attainable by state-of-the-art Cubesats telescopes
for which the apparent magnitude usually ranges from 12 to 15
ashcraft2021versatile ; aperture_optical_science_2021 .

Table 2: Cislunar SDA Simulation and Optimization Parameters Category | Parameter | Value
---|---|---
Simulation | Initial epoch | 2024 MAR 25 07:12:36.633
Campaign duration $\Delta T$, day | 1825
Time-step $\Delta t$, day | 2
Number of targets | 1107
Target diameter, $\mathrm{m}$ | $1$
Target specular reflectance $a_{\mathrm{spec}}$ | $0$
Target diffuse reflectance $a_{\mathrm{diff}}$ | $0.2$
Cut-off magnitude $m_{\mathrm{co}}$ | 18, 17, 16, 15
Fractional visibility requirement, $\alpha$ | 0.9
Optimizer | Number of generations | 100
Chromosomes per generation | 100
Maximum chromosome length, $L_{max}$ | 30
Gene crossover probability | 0.95
Distribution index for gene crossover | 10
Gene mutation probability | 1/15
Distribution index for gene mutation | 50
Tag crossover probability | 0.8
| Tag mutation probability | 0.2

###  4.1 Overview of Pareto Front

As aforementioned, it is known there exists a gap in single spacecraft
observability access Holzinger2021_report , particularly for low apparent
magnitudes. Figure 5 shows that below a cutoff magnitude of 18, performances
start degrading and $\Gamma$ drops to 0% for magnitudes of 16 and below, when
considering a single LPO.

The multi-spacecraft architectures in this work aim at filling this gap. The
multi-objective HGGA extracts the Pareto front for the given objectives from
all the possible solutions. As an example, Figure 6 presents the evolution of
this Pareto front along the 100 iterations of the algorithm for the
$m_{\mathrm{co}}=16$ case, underlining the apparent competitiveness between
the stability index and $\Gamma$ objectives. It is also possible to note the
nature of the optimization problem with respect to the three objectives; the
projections indicate the optimizer is able to reduce both the total stability
index and $\Gamma$ while maintaining the same number of spacecraft.

Figure 5: Pareto solutions of SDA architectures for $m_{\mathrm{co}}$ between
$18$ and $15$ Figure 6: Evolution of the Pareto front for 3 objectives
optimization with HGGA, $m_{\mathrm{co}}$ = 16. Black dots correspond to the
projection of each solution across the three 2D planes.

As previously mentioned, for $m_{\mathrm{co}}$ of 16 and 15, the single
spacecraft solutions only minimize the stability index metric.

The set of nondominated solutions for each $m_{\mathrm{co}}$ comprises many
points. Some patterns are nonetheless visible. For $m_{\mathrm{co}}$ = 18,
single spacecraft solutions are already performing well, and adding new
observers shifts the $\Gamma$ towards better solutions without greatly
modifying the overall stability index. This shift continues as the number of
spacecraft increases but full observability is quickly attained and the
nondominated set, for the larger numbers of observers, therefore, contains
$\Gamma=1$ solutions with lower stability indices.

Looking at the $m_{\mathrm{co}}$ = 17 plot, the shift is visible again. An
observability of around 95% is attained for 2 observers, representing an
improvement of 24% compared to single spacecraft architectures. The gap to
reach full observability is almost filled again while being larger. However,
architectures seem unable to attain full observability. This is due to a
certain percentage of points of the grid of targets never reaching an apparent
magnitude of 17 since some of those points are simply too far away to be
visible to the observers. This phenomenon becomes even more visible for lower
cutoff magnitudes.

Figure 5 shows, for $m_{\mathrm{co}}$ = 16, a sizeable augmentation of the
$\Gamma$ between 1 and 2 observers, going from 1% observability to
approximately 44%. This improvement is also seen from 2 to 4 observers,
increasing to 63%. Again, the flattening of the Pareto front, described for
$m_{\mathrm{co}}$ = 17, is visible. In fact, for a higher number of
spacecraft, the gain in $\Gamma$ is reduced, while the stability indices rise
rapidly.

Finally, with $m_{\mathrm{co}}$ = 15, increasing the number of spacecraft has
limited improvement on $\Gamma$. The maximum $\Gamma$ attained is 33% and the
total stability index is 2267.48, highlighting fairly unstable orbits. This
drop in $\Gamma$ between $m_{\mathrm{co}}$ of 16 and 15 is once again due to
the percentage of targets that never exceed an apparent magnitude of 15 given
the observers position becomes too important, in particular in the near-Earth
areas of the grid where the LPO become too distant to perceive the target
spacecraft.

The best results in terms of $\Gamma$ for each $m_{\mathrm{co}}$ and the
related number of spacecraft and stability indices are given in Table 3.

Table 3: Cislunar SDA results | Single Spacecraft | Multi Spacecraft
---|---|---
$m_{\mathrm{co}}$ | $N_{\mathrm{spacecraft}}$ | $\Gamma$ | Total $\nu$ | $N_{\mathrm{spacecraft}}$ | $\Gamma$ | Total $\nu$ | Average $\nu$
18 | 1 | 0.95 | 55.18 | 2 | 0.98 | 56.29 | 28.14
3 | 1.0 | 57.33 | 19.11
17 | 1 | 0.73 | 55.18 | 2 | 0.92 | 100.17 | 50.08
3 | 0.94 | 119.66 | 39.89
7 | 0.96 | 163.25 | 23.32
16 | 1 | 0.03 | 1.00 | 2 | 0.44 | 51.11 | 25.55
4 | 0.63 | 58.06 | 14.52
6 | 0.67 | 1419.99 | 236.67
8 | 0.72 | 1095.21 | 136.90
10 | 0.76 | 604.79 | 60.48
12 | 0.77 | 720.87 | 60.07
15 | 1 | 0.01 | 536.80 | 2 | 0.04 | 1790.04 | 895.02
4 | 0.15 | 701.61 | 175.40
6 | 0.25 | 1137.78 | 189.63
8 | 0.30 | 2261.87 | 282.73
10 | 0.31 | 2181.53 | 218.15
13 | 0.33 | 2267.48 | 174.42

###  4.2 Analysis of Architectures

Among all the architectures selected by the HGGA, an evolution in the
allocation of spacecraft to orbit families is clearly visible when varying the
$m_{\mathrm{co}}$. Figure 7 shows the number of times a particular orbit is
utilized by the Pareto front solutions for all values of $m_{\mathrm{co}}$
considered.

Overall, apart from an $m_{\mathrm{co}}$ value of 18, the L1 Lyapunov family
is used by any value of $m_{\mathrm{co}}$ at roughly equal frequencies. This
can be explained by the family being at the center of the target grid, thus
offering an effective position for covering large portions of the region of
interest. For $m_{\mathrm{co}}$ = 18, single spacecraft architectures are
already nearly optimized using a single L1 Lyapunov orbit and closing the gap
only requires DROs, which are the most stable option. Meanwhile, it is also
possible to observe that as the $m_{\mathrm{co}}$ is reduced, the architecture
begins to also require an observer at the vicinity of L2 dedicated to targets
on the far end. Also, the low $\nu$ of DROs make them a particularly
attractive family vis-a-vis the second objective of the problem from (14),
resulting also in their relative frequent use.

Figure 7: Relative distribution of spacecraft among orbit families for
different $m_{co}$

For each value of $m_{\mathrm{co}}$, it is also possible to observe intuitive
variations of family allocations as the number of spacecraft increases. As
$m_{\mathrm{co}}=16$ offers the largest changes in $\Gamma$ when adding new
spacecraft and overall the most important increases in performance, we focus
further on analyzing different architectures for this particular cutoff
magnitude. We select architectures encompassing various total number of
spacecrafts, while having reasonable total $\nu$. For those reasons, 2-, 3-,
5- and 7-spacecraft architectures with the highest $\Gamma$ are selected from
the Pareto front, at the exception of the 7-spacecraft solution where a more
stable one is selected with a slightly lower observability metric.

Table 4 summarizes the information of the LPOs used by these architectures.

Architecture A and B on Figure 8 shows the low number of spacecraft solutions
that achieved the highest $\Gamma$. Directly, as opposed to higher
$m_{\mathrm{co}}$ minimal spacecraft number architectures, the L1 Lyapunov
orbits have been replaced by L2 Lyapunov orbit which confirms the trend
discussed previously.

Another interesting thing to notice is that from 2 to 3 spacecraft, the orbits
family and orbits periods variables are close to unchanged and the optimized
solution simply is derived by the optimizer using an offset of the initial
phasing of the L2 Lyapunov orbit observers while enabling an increase in
approximately 24% of the observability.

Also, in both cases, the L2 Lyapunov orbits use 1:1 synodic resonant orbits,
combined with an initial phase angle close or equal to zero, hence aligned
with the original sun-earth-moon positions as discussed in Section 2.
Resonance with the synodic period has been proven to provide particularly
beneficial light conditions throughout the mission horizon
Holzinger2021_report .

When augmenting the number of spacecraft per architecture, the variability in
the combinations of orbit families assigned to spacecraft augments as well.

This is particularly noticeable for architectures C and D on Figure 8 where
the 5-spacecraft architecture already differs considerably from the low number
of spacecraft combinations in terms of orbit families considered. Although the
architecture C only considers two orbit families, we can notice a 120° off-set
between the 3 DRO observers that allows for continuous monitoring, even in the
varying light conditions, as there are higher chances that one of the three
will have good alignment with regards to the sun. This can be seen as similar
mechanism as the resonant period, adapted to the DROs for which resonant
orbits are not particularly suited, given the great variability in size for
adjacent orbit periods. Again, it is possible to observe 2:1 and 1:1 resonance
with close to 0° initial phasing for the L1 Lyapunov families. This
architecture on the other hand presents poor stability characteristics,
especially when compared to other solutions with more spacecraft.

The 7-observers configuration offers a great geometrical diversity in terms of
orbit, while resulting in a lower total $\nu$ and an even more reduced average
$\nu$ than some of the other combinations considering fewer spacecraft. The
particular 3 DROs configuration has been preserved for lower number of
spacecraft solutions as well as the 1:1 resonant L1 Lyapunov. Added to this is
an approximate 2:1 south L2 Halo. The last two orbits don’t exhibit any
particular geometry characteristics of their own but are efficient when
associated to the rest of the orbits.

Figure 8: Example Pareto solutions with 2-, 3-, 5- and 7-spacecraft architectures for $m_{\mathrm{co}}$ = 16 Table 4: Period and initial phase angle from selected solutions Solution | Orbit Family | Orbital period $P$, days |  | Fraction of
---
synodic period
Phase angle $\phi$, deg
A | L2 Lyapunov | 29.54 | 1.00 | 0.00
DRO | 2.54 | 0.09 | 0.00
B | L2 Lyapunov | 29.48 | 1.00 | 347.44
L2 Lyapunov | 29.50 | 1.00 | 322.05
DRO | 2.51 | 0.09 | 353.20
C | L1 Lyapunov | 14.78 | 0.50 | 328.72
L1 Lyapunov | 29.48 | 1.00 | 341.01
DRO | 16.55 | 0.56 | 117.14
DRO | 17.14 | 0.58 | 240.06
DRO | 16.55 | 0.56 | 353.83
D | L1 Lyapunov | 29.47 | 1.00 | 344.40
L2 Lyapunov | 23.03 | 0.78 | 245.00
L1 Halo (S) | 11.62 | 0.39 | 204.20
L2 Halo (N) | 14.32 | 0.49 | 226.08
DRO | 16.47 | 0.56 | 120.85
DRO | 16.47 | 0.56 | 239.79
DRO | 16.50 | 0.56 | 357.07

As can be observed, when apparent magnitude conditions become more strict,
diversity in the allocations of observers begins to be necessary to achieve
performing $\Gamma$. Performing L1 Lyapunov combinations are still possible at
the cost of a higher overall stability index.

##  5 Conclusion

In this work, we have studied the design optimization of multispacecraft
cislunar SDA architectures. The architecture considered is tasked with a
static, grid-based region of interest observation based on a metric $\Gamma$
defined as the percentage of this region of interest visible over a fraction
$\alpha$. Two other metrics, the total stability index and the total number of
spacecraft are introduced to compete with the first objective and allowed us
to obtain a nondominated set of architectures solutions. Given the variability
in the size of the design space, due to the fact that the number of spacecraft
and the orbit family allocation both enter as design variables, a Hidden-Genes
Non-dominated Sorting Genetic Algorithm II is implemented.

Solutions obtained aim at filling some of the gaps in space domain awareness
monitoring for lower value of limiting magnitude, allowing for less advanced,
and hence cheaper space telescope constellations to complete the mission.
Results showed that it was possible to reduce the limiting magnitude of the
monitoring structures, greatly, increasing the $\Gamma$ for a reasonable
number of spacecraft and total $\nu$. In particular, diversity in the choice
of orbit families is key to achieving better observability, as shown in Figure
7. Another common solution extracted by the optimizer is the use of a small
phase difference between observers from the same orbit family with relatively
similar periods, in order to achieve almost constant ideal light conditions on
the whole cislunar space.

The framework and optimization formulation utilized in this work are very
flexible, allowing for example to add new types of observers without changing
the formulation. These new optical structures could be Low Lunar Orbits, Lunar
Surface Telescopes or even Earth-orbiting observers such as geosynchronous
orbits Klonowski2022  or Highly Eccentric Orbit (HEO) Cunio2020  which have
been proven to be particularly useful for cislunar trajectory monitoring. It
is also possible to change the grid of targets, creating denser areas in
potential congestion points or along the most likely trajectories for lunar
transfer orbits. Finally, the introduction of more concrete cost metrics such
as transfer costs or station-keeping may become interesting in the coming
years as cislunar space domain awareness carry on its rapid development.

Acknowledgments

This work has been funded by the Georgia Tech Research Institute (GTRI)’s
Independent Research and Development (IRAD) program.

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