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AAS 23-228
PERIODIC TRANSFERS THAT DEPART AND RETURN TO AN
OPERATING ORBIT USING RESONANT ORBIT STRUCTURES IN
THE PLANAR THREE-BODY PROBLEM
Noah I. Sadaka*and Kathleen C. Howell†
Many future satellite applications in cislunar space require periodic transfers that
shift away from some operational orbit but return. Numerous resonant orbit fami-
lies in the Earth-Moon Circular Restricted Three-Body Problem (CR3BP) possess
a ratio of orbital period to lunar period that is sufficiently close to an integer ratio
and can be exploited to uncover periodic transfers. By locating homoclinic con-
nections associated with the operating orbit that incorporate resonant structures,
and potentially linking them to resonant orbits, transfers are available for in-orbit
refueling/maintenance as well as surveillance/communications applications that
depart and return to the same phase in the operating orbit.
INTRODUCTION
As interest in developing infrastructure for cislunar space applications increases, additional ca-
pabilities are necessary to create a robust transportation network for spacecraft. One important
future capability is refueling or servicing spacecraft already in orbit, an especially critical element
for infrastructure development. Spacecraft operating in near-Earth orbits such as Northrup Grum-
man’s Mission Extension Vehicle missions (MEV-1 and MEV-2) have demonstrated that on-orbit
maintenance is feasible,1and considerable effort is currently being invested to develop component
standards and solve hardware design challenges to enable refueling and maintenance by servicing
spacecraft.2However, the trajectory design aspect for cislunar operations still requires much analy-
sis.
Space surveillance and communications is also an important challenge, i.e. the ability to track
the locations of spacecraft in cislunar space and enable inter-satellite communications. Pasquale et
al.3demonstrate an optimization method for creating constellations of satellites around the Moon
based on desired areas of lunar surface coverage. Such a constellation could be paired with a relay
satellite, to store data sent from the constellation and to downlink to Earth.
For such applications, an operating orbit serves as a “home base”; departures and arrivals then
offer access to wide-ranging destinations. For the servicing and refueling applications, the operating
orbit may represent the base orbit for a depot, from which smaller servicing spacecraft (termed
servicers) depart to rendezvous with a spacecraft to be serviced (termed the customer). After the
servicing operation is complete, the servicer returns and a rendezvous with the depot occurs in the
*Graduate Student, School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, 47907;
nsadaka@purdue.edu.
†Hsu Lo Distinguished Professor of Aeronautics and Astronautics, School of Aeronautics and Astronautics, Purdue Uni-
versity, West Lafayette, IN, 47907; howell@purdue.edu. Fellow AAS; Fellow AIAA.
1
operating orbit. For a surveillance and/or communications satellite, some flexibility is introduced
such that a vehicle remains in an operating orbit for some time and then, periodically, returns to
the near-Earth region to downlink acquired data or otherwise engage near-Earth spacecraft. This
vehicle then returns to the operating orbit and the same phase at which it departed.
Given the complex gravitational environment in cislunar space, it is reasonable to consider tra-
jectories that leverage these gravitational dynamics in planning scenarios for various applications.
Thus, the Circular Restricted Three-Body Problem (CR3BP) is assumed for the preliminary model-
ing to construct these trajectories as it includes the gravitational influences of both the Earth and the
Moon, i.e., the dominant contributors to the gravitational forces in cislunar space. This model is ap-
propriate for the Earth-Moon system during initial trajectory design assessments as the Moon’s orbit
is nearly circular relative to the Earth.4Thus, the dynamical structures in this model are available
to be leveraged for real-life mission design in the cislunar region.
Resonant orbits and invariant manifolds with resonant structures emerge as a clear type of bounded
motion that serve as a basis to create periodic transfers. Resonant structures offer a variety of ge-
ometries and traverse the Earth-Moon system to many destinations. Several previous investigations
have also successfully demonstrated the practicality and advantages of using resonant structures for
trajectory design in the cislunar environment.5,6
THE CIRCULAR RESTRICTED THREE-BODY PROBLEM
The Circular Restricted Three-Body Problem (CR3BP) is a time-independent model that de-
scribes the motion of a particle with negligible mass, labelled P3, in a gravitational potential formed
by two spherically-symmetric massive bodies. In this model, the massive bodies, termed the pri-
maries and denoted P1and P2for the larger and smaller mass, respectively, move in circular orbits
about their common barycenter. Any CR3BP system is characterized by the mass ratio µ=M2
M1+M2
for the primaries, where M1and M2are the masses of P1and P2, respectively. For the Earth-Moon
CR3BP, this mass ratio is approximately µ= 0.0121506. The motion in this model is governed
by three second-order differential equations, expressed in a barycenter-centered frame that rotates
with the primaries. The coordinate axes for this rotating frame are defined by an ˆx−ˆy−ˆzdextral
orthonormal triad, viewed in teal in Figure 1, where the ˆxaxis is directed from P1to P2, the ˆz
axis is parallel to the orbital angular momentum vector for the orbit of the primaries, and ˆycom-
pletes the right-handed triad. The orientation of the rotating frame with respect to an inertial frame
at a certain time is defined by the angle θin Figure 1. The CR3BP equations of motion are non-
dimensionalized to generalize the model and reduce numerical error. The characteristic quantities
are defined as the distance between the primaries l∗, the total mass of the system, m∗=M1+M2,
and the characteristic time t∗is selected such that the mean motion ˙
θfor the primaries relative to
the barycenter Bis equal to 1. This non-dimensionalization of the problem variables results in the
distance between the primaries being equal to 1 and the locations of P1and P2on the ˆx-axis being
equal to −µand 1−µ, respectively. The resulting equations of motion for the six-dimensional state
for P3,r = [x, y, z, ˙x, ˙y, ˙z], are
¨x= 2 ˙y+ Ωx(1)
¨y=−2 ˙x+ Ωy(2)
¨z= Ωz(3)
where Ωiis a partial derivative of the pseudo-potential function Ωwith respect to each of the position
2
components, i,
Ω = 1−µ
∥
R13∥+µ
∥
R12∥+x2+y2
2(4)
as evident in Figure 1,
R12 and
R13 are the respective position vectors from P1and P2to P3. (Note
that arrow overbars indicate vectors.)
ˆ
X
ˆ
Y
ˆx
ˆy
R13
R23
θ
P1
P2
B
P3
r
Figure 1: The CR3BP rotating frame
This investigation is focused on trajectories in the Planar CR3BP (PCR3BP), where all motion
occurs in the Earth-Moon orbital plane and thus z= 0 in the equations of motion. Five equilibrium
solutions, denoted the Lagrange points, exist in this model. The collinear Lagrange points, i.e., L1,
L2, and L3, are all located along the ˆx-axis. The equilateral Lagrange points L4and L5are off the
ˆx-axis but form equilateral triangles with the primaries.
No analytical solution exists in the CR3BP, however, one integral of motion is available i.e.,
the Jacobi Constant (JC). This integral of the motion represents an energy-like quantity and is
evaluated for any state as
JC = 2Ω −( ˙x2+ ˙y2+ ˙z2)) (5)
of course, JC remains constant along any ballistic arc. Equation (5) is rearranged to solve for the
resulting velocity magnitude at a given position and for a specified JC level. Locations with zero
rotating velocity define the zero-velocity curves (ZVC) for the PCR3BP. These curves bound the
possible motion of a trajectory.
PERIODIC ORBITS
Periodic orbits are a type of repeating motion in the CR3BP where a trajectory returns to its initial
state after some period, T. These solutions provide structure in an otherwise complex dynamical
environment and are frequently leveraged as predictable behavior for trajectory design. While the
CR3BP is a lower-fidelity model, only including the two largest contributors to the gravitational
force on a spacecraft, analysis in the CR3BP is adequate as orbits and trajectory arcs are generally
successfully transitioned to a high-fidelity N-body model.7Leveraging CR3BP-based structures
served as the basis for design in numerous space missions, e.g.,the Earth-Moon distant retrograde
orbit incorporated during the Artemis I mission8and the 2:1 spatial resonant orbit that is the current
orbit for the Transiting Exoplanet Survey Satellite (TESS).9
3
Periodic orbits are constructed by propagating an appropriate initial state and using a corrections
algorithm to deliver periodicity. Given one periodic orbit, families of orbits are generated by vary-
ing one parameter (such as the x-position associated with a perpendicular crossing) and using a
continuation process.
Useful insights result from the application of dynamical systems theory to the CR3BP, including
orbital stability information. The State Transition Matrix (STM) associated with the equations of
motion provides a linear estimate of the variation in the final states due to a perturbation in the initial
states. For a periodic orbit, the STM evaluated after exactly one period is termed the monodromy
matrix; that is, a stroboscopic map of the variation in the states analyzed using discrete-time systems
theory. As this model is a Hamiltonian system, the six eigenvalues λiof the monodromy matrix
occur in three reciprocal pairs and describe the linear stability of an orbit. An orbit is considered
stable if all |λi| ≤ 1and unstable if any eigenvalue |λi|>1. The eigenvectors associated with an
unstable eigenvalue and its reciprocal pair are exploited to numerically generate invariant manifolds.
Invariant manifolds are successfully employed in numerous other investigations to generate low-cost
transfers and to explore the dynamical relationships between periodic orbits.10, 11
Resonant Orbits
In the Earth-Moon CR3BP, a vehicle in a resonant orbit possesses an orbital period that is com-
mensurate with the period of the lunar orbit.12 The ratio of orbital periods is typically expressed
as a p:qresonance, where in the Earth-Moon system, prepresents the number of revolutions of
the spacecraft around the Earth and qis the number of revolutions of the Moon around the Earth.
In an Earth-centered Kepler problem, the Moon does not impart any gravitational influence and,
therefore, a resonant orbit is defined by an exactly integer resonance ratio. However, the addition of
the lunar gravitational effects for resonant orbits to the CR3BP results in the resonance ratio being
close to, but not exactly, a precisely integer ratio.12
Exterior resonant orbits are defined when p < q and, for interior resonant orbits, p > q. The
variety of resonance ratios and the different characteristics of interior and exterior resonant orbits
provides a wide range of orbit geometries. One of the advantages of using resonant structures for
transfer design is that the pathways collectively traverse through large swaths of space as apparent
in Figure 2, where a set of resonant orbits is plotted at several energy levels along with the corre-
sponding ZVCs. Resonant orbits reach a variety of useful locations throughout Earth-Moon space,
including regions near the primaries, the collinear Lagrange points, the equilateral Lagrange points,
and the exterior region of the system. Notably, the availability of different resonant orbits depends
on the selected energy level, with many exterior resonant orbits only available at lower values of
Jacobi Constant. These values of JC are associated with ZVC bounds that restrict access to cer-
tain regions, for example the Moon may not be accessible if the L1gateway is closed. However,
the large variety of geometries offered by resonant orbits are an attractive option for many space
infrastructure applications and as operating orbits for satellites.
The selection of a specific resonant orbit is based on information represented by the resonance
ratio. The values for pand qare expressed as the number of revolutions around the Earth as viewed
by an inertially-fixed observer. Therefore, the orbit is propagated for prevolutions to yield an
orbit that is periodic in the rotating frame. Once an appropriate conic orbit is generated, different
strategies are employed to transition the orbit to the CR3BP.13 A converged p:qresonant orbit
in the CR3BP then seeds a differential corrections process to construct a family of p:qresonant
orbits.
4
Figure 2: Several resonant orbits plotted at different energy levels. The locations of the Lagrange
points are marked with crosses and the ZVCs appear as black curves. Regions that are inaccessible
appear with grey fill
Phasing is an important consideration in the design of periodic transfers and resonant orbits
emerge as an appropriate type of trajectory for phasing constraints. Some families of resonant
orbits possess a range over which the period of various members within the family do not vary sig-
nificantly.13 This period is also frequently close to the expected resonant period of Tres = 2πq. For
example, the 2:3, 5:3, and 2:1 resonant orbit families are plotted in Figure 3, and colored by the
fraction of actual period Tto the resonant period Tres. A large proportion of each family possess
periods that are very close to the resonant period as illustrated by the T/Tres ratios that are close
to one, thus, the periods of these members are predictable and the orbits are applicable for periodic
transfer design. However, as noted by Anderson et al.,13 orbits that loiter near the smaller primary
over extended intervals results in larger deviations between the orbit period and the resonant period,
a phenomenon that is apparent for the 2:3 resonant orbit family plotted in Figure 3(a).
(a) 2:3 resonant orbit family (b) 5:3 resonant orbit family (c) 2:1 resonant orbit family
Figure 3: Members of resonant orbit families colored by the fraction of orbital period to resonant
period T
Tres
5
An attribute of many prograde resonant orbits, including those appearing in Figure 3, are the
“loops” at apses that appear when viewed in the CR3BP rotating frame. These loops are a result
of the variation in the orbit’s rotating velocity with respect to the angular velocity of the primaries.
Interior resonant orbits have loops at their apoapses pointing outwards, and exterior resonant orbits
have loops at their periapses pointing inwards. The locations and velocity directions of these loops
are exploited to reach desired locations in space when designing periodic transfers.
Lyapunov Orbits
An advantage of multi-body gravitational models, such as the CR3BP, is the emergence of equi-
librium solutions that serve as a source of periodic behaviors. Various libration point orbits (LPO)
exist in the vicinity of their respective Lagrange points, and include the planar Lyapunovs and their
spatial counterpart, the halo families of simply symmetric orbits. These orbits are successfully lever-
aged for mission design in the Earth-Moon system and are candidate orbits for lunar surveillance
missions14 and cislunar depot locations.2This investigation is focused on members of the L1and
L2planar Lyapunov orbit families; a subset is plotted in Figure 4 from which one is selected as the
operating orbit. The L1and L2Lyapunov orbits that are plotted in Figure 4 are linearly unstable and
their invariant manifolds appear in previous investigations as the basis to design low-cost transfers.5
The presence of manifolds implies that L1and L2orbits are attractive for an operating orbit as many
natural departure and arrival paths are available to be leveraged to explore cislunar space.
(a) Members of the L1Lyapunov family (b) Members of the L2Lyapunov family
Figure 4:L1and L2Lyapunov orbit families in the Earth-Moon system
The third collinear Lagrange point, L3, is also the source for a family of Lyapunov orbits as
plotted in Figure 5. Notably, these L3Lyapunov orbits are in 1:1 resonance with the Moon, and
are constructed in the Earth-Moon system using either the typical linearization technique employed
for the L1and L2Lyapunov orbits4or by generating an appropriate 1:1 resonant conic orbit and
transitioning it to the CR3BP. As this orbit family is located on the opposite side of the Earth as
compared to the Moon, the members plotted in Figure 5 do not spend significant time (if any) near
the Moon and, therefore, its period does not vary significantly from the 1:1 resonance ratio.
6
Figure 5: Members of the L3Lyapunov family colored by the period over 2π
GENERATING PERIODIC TRANSFERS
Invariant manifolds provide natural transfers paths that asymptotically depart from the operating
orbit and return without any propellant expenditure; an attractive option for a servicing spacecraft
as propellant can be conserved for a refueling operation. Homoclinic connections, a special type
of invariant manifold arcs where an unstable arc flows into a stable arc, are investigated as a basic
framework to freely shuttle spacecraft throughout space to accomplish mission objectives.
Figure 6 includes a schematic for a transfer design strategy. A symmetric homoclinic connection
with propagation time 2Tman is generated from the operating orbit with period TOand with a
manifold insertion location at time τ, as measured from a perpendicular crossing of the orbit. The
homoclinic connections with resonant structures are explored, offering potential transfers into a
periodic orbit at a common perpendicular crossing with period TP O . Transferring into a periodic
orbit allows for greater flexibility in designing periodic trajectories originating from the operating
orbit. For example, a servicing or on-orbit refueling could occur in such an orbit, where both the
servicer and customer rendezvous and remain for one or more periods along the orbit before the
return of the servicer to the operating orbit. To identify a periodic transfer path, the fraction Qis
defined as the sum of the time-of-flight for each segment along the transfer over the period of the
operating orbit. Periodic transfers are produced when Qis an integer, that is,
Q=nP OTP O + 2Tman + 2τ
TO
(6)
where nP O is an integer representing the number of revolutions to remain in the periodic orbit before
returning to the operating orbit via the homoclinic connection. For a scenario where no transfer to
a periodic orbit is planned, nP O is zero.
Constructing Natural Transfers to/from the Operating Orbit with Resonant Structures
The use of homoclinic connections to depart from and return to the operating orbit is desirable
as these trajectories supply propellant-free transfers. Invariant manifold structures in the CR3BP
also provide insight into the dominant dynamical flows in a gravitational environment dominated
7
by two massive bodies. Once a suitable unstable L1or L2Lyapunov is identified for the operating
orbit, its invariant manifolds are generated and homoclinic connections with desirable structures are
identified for the construction of periodic transfers.
Figure 6: Periodic transfer schematic
To approximate the invariant manifold surfaces, manifold arcs from many fixed points are prop-
agated. Visualizing manifold surfaces is challenging, therefore, a Poincar´
e section is defined with a
hyperplane at y= 0 and the impinging of trajectories on that map is analyzed. This dimensionality-
reduction technique allows the full four-dimensional state to be visualized on a two-dimensional
map; by constraining one dimension with the location of the hyperplane, and with all manifold arcs
possessing the same Jacobi Constant value as their generating orbit, points that appear close by on
the Poincar´
e section are near each other in the four-dimensional space. This property is especially
useful in converging homoclinic connections; stable and unstable manifold arc intersections on the
map that are close to one another are used as initial guesses in a robust differential corrections
scheme and corrected for continuity.
Symmetric homoclinic connections yield perpendicular crossings (PC) in a solution and reduce
the complexity for locating candidate homoclinic connections for periodic transfers. Exploiting
symmetric structures reduces the computation time to converge to a homoclinic connection, given
that only one half of the path requires correction. The corrections scheme appears in Figure 7, where
the step-off offset distance dto numerically construct a manifold occurs after time τalong the orbit
as measured from a perpendicular crossing along the generating orbit. A manifold arc constraint
developed by Haapala and Howell10 ensures that the arc remains on the operating orbit’s manifold.
The manifold arc selected from the Poincar´
e map is discretized into nsegments with respective
propagation times Tiand intermediate arcs are constrained for continuity. The final state x′
nalong
the terminal arc is constrained to be a perpendicular crossing.
Figure 7: Symmetric homoclinic connection targeter schematic. Adapted from Haapala and How-
ell10
8
Predicting the existence of homoclinic connections that incorporate resonant structures are demon-
strated by Vaquero and Howell;15 numerous examples are produced as well. By identifying mani-
fold arcs for the operating orbit that pass near the fixed points representing the resonant orbits on a
Poincar´
e section, homoclinic connections with structures similar to nearby resonant orbits are often
observed. Invariant manifolds for the operating orbit that are aligned with resonant structures take
advantage of both the natural departure and arrival properties of invariant manifolds and the geo-
metrical and timing properties of resonant orbits. The resulting transfer reflects a resonant structure
where clear geometrical elements from the associated resonant orbit are apparent, e.g., the number
and location of apses. The osculating period for the transfer is also close to the resonant orbital
period. An example highlighting these properties is plotted in Figure 8 for a transfer corresponding
to JC = 3.02261. An intersection of the stable and unstable manifolds for the L2Lyapunov orbit
that pass near a fixed point of the 5:3 resonant orbit at the same energy level are identified on the
map in Figure 8(a). This initial guess is corrected to result in the homoclinic connection plotted in
red in Figure 8(b) that shares many geometrical elements with the 5:3 resonant orbit plotted with
a dashed green line. The ratio of osculating period Tosc to Lunar period Tmoon for the transfer is
plotted in Figure 8(c) with the black dashed line highlighting the precise resonant ratio of 3
5for a
5:3 resonant orbit. Note that the osculating period tends to infinity at the start and end of the transfer
as the osculating orbit is hyperbolic near the L2Lyapunov orbit.
One advantage of employing L1and L2Lyapunov orbits as an operating orbit is their existence
across many energy levels that then yield many different homoclinic connections that exhibit reso-
nant structures.16 The homoclinic connections for these orbits flow in close vicinity to the structure
existing in the periodic orbits. Given innumerable unstable resonant orbit families, many homoclinic
connections emanating from L1and L2orbits also incorporate resonant structure.
Continuing Homoclinic Connections and Converging Periodic Transfers
A given homoclinic connection is unlikely to be exactly periodic with the motion in the operating
orbit. Prior knowledge of the time-of-flight for the homoclinic connection, as well as a known
value for the period of an orbit, is required to evaluate Qin Equation (6) and assess if it is an
integer value. However, the necessity for the numerical correction and propagation of trajectories to
evaluate Equation (6) results in a homoclinic connection as identified from a Poincar´
e section that
is not necessarily periodic.
With the challenges in constructing the appropriate links, the homoclinic connections are con-
tinued into a family of transfers that all possess the same underlying structures to enable the de-
termination of periodic members. Casoliva et. al.16 show that homoclinic connections exist in
one-parameter families. For this analysis, natural parameter continuation is sufficient to generate
such a family. The corrections scheme in Figure 7 employs a fixed Jacobi Constant value for the
operating orbit, thus, by monotinically varying the Jacobi Constant value for the operating orbit
and using the previously-converged homoclinic connection as an initial guess, the same corrections
strategy is applied, producing a new homoclinic connection for this new operating orbit. Each
member in the family is, therefore, a ballistic transfer at some Jacobi Constant value to and from
an operating orbit defined at the same Jacobi Constant value and yet conserves the desired resonant
structure apparent in the initial homoclinic connection. For example, the transfer in Figure 8(c) is
continued and plotted in Figure 9(a), where the 5:3 structure of all the members of the family is
maintained.
9
(a) y= 0 Poincar´
e section generated from 5000 fixed points evenly spaced in time around
the orbit, a step-off distance of 40km, and a propagation time of 173.7 days
(b) The transfer (red) viewed in configuration space
along with a 5:3 resonant orbit (green, dashed) and
generating L2Lyapunov (black)
(c) The osculating resonance ratio of the trans-
fer (red) and the 5:3 perfect resonant ratio (black,
dashed)
Figure 8: (a) Poincar´
e section with unstable (pink) and stable (dark blue) manifold arcs. Fixed
points for the 5:3 resonant (orange) and L2Lyapunov (black) orbits are marked. (b, c) Recognizing
the existence of resonant structures in a homoclinic connection
Once a family of homoclinic connections is generated, Equation (6) is evaluated across the family
to identify members that produce a periodic transfer. The time-of-flight for the different segments
comprising the transfer, including the period for a periodic orbit with a perpendicular crossing loca-
tion that is the same as the symmetric homoclinic connection, must be evaluated for each member.
Equation (6) is evaluated across the family in Figure 9(a) and is plotted in Figure 9(b). A periodic
transfer is identified with a period equal to 9 revolutions of the L2Lyapunov operating orbit with
JC = 3.067. As the period of the potential operating orbits, the homoclinic connection, and the
destination periodic orbit all vary smoothly throughout the family, Equation (6) is evaluated for
evenly-spaced values in JC. Such a process occurs along a subset of the family and the periodic
solutions are identified using a bracketing iterative method.
10
(a) Family of ballistic transfers from L2Lyapunov
orbit with 5:3 resonant structure
(b) Equation (6) evaluated along the family. A pe-
riodic transfer is identified with a star
Figure 9: Identifying a periodic transfer with 5:3 resonance structure from a L2Lyapunov operating
orbit
Defining the time-of-flight for a manifold arc Tman is challenging due to the asymptotic nature
of invariant manifolds. The same manifold arc can be reached from any location along the orbit by
varying the step-off distance d, and monotinically decreasing deventually results in the manifold arc
“wrapping” around the orbit for an additional revolution, adding TOto Tman. Therefore, the transfer
time Ttdeparting from the operating orbit is defined as equal to Tt=nTO+ 2Tman + 2τ, where
nis either zero, a positive integer, or a negative integer assuming that the manifold does not depart
the orbit. Then, Tman and τare defined based on the selected step-off distance; for this analysis
d= 40km. Once a periodic transfer is achieved, increasing the numerator in Equation (6) by nTO
does not influence the periodic properties for the transfer, as Qremains an integer. Thus, transfer
time-of-flight is defined to be the time-of-flight using a step-off distance d= 40km, however, nTO
can be added to this time if a smaller offset distance is preferred.
PERIODIC TRANSFERS FOR CISLUNAR INFRASTRUCTURE DEVELOPMENT
Periodic transfers are potentially useful for a variety of important applications in cislunar space
to support future development of the region. Several examples are showcased to demonstrate the
versatility of resonant orbits and resonant structures in transfer design and to evaluate periodic trans-
fers for application to different mission scenarios. Transfers that are comprised solely of homoclinic
connections are first demonstrated followed by periodic transfers that deliver a vehicle into a reso-
nant orbit for a servicing scenario. Finally, sample transfers that reach L3Lyapunov orbits from the
Lunar region are introduced.
Homoclinic Connections as Transfer Options
One prominent example that leverages the dominant dynamical motion in the CR3BP for a space-
craft servicing application was originally proposed by the National Aeronautics and Space Admin-
istration (NASA) Goddard Space Flight Center (GSFC), termed the LOTUS framework.2This
11
framework planned for a space telescope, originally in an orbit in the vicinity of Sun-Earth L2, to
rendezvous with a servicing depot in an Earth-Moon L1Lyapunov operating orbit. Both vehicles
then transfer into a “highly elliptical” orbit around the Earth that requires minimal propellant use
for each; then, a human crewed vehicle meets up with the telescope and depot stack at an apse and
subsequently returns to Earth after the servicing operation is completed.2The space telescope and
depot then transfer back to the operating orbit with a relatively small impulse.2The original transfer
is viewed in Figure 10(a) and appears to be leveraging an 8:3 resonance structure with an energy
level near JC = 3.15; the small maneuvers required to leave from and arrive into the operating
orbit implies the use of invariant manifold structures. A similar structure is successfully constructed
in the CR3BP and is plotted in Figure 10(b), where the transfer time equals 8.74 revolutions of
the operating orbit. Using this initial transfer to construct a family, the periodic transfer shown in
Figure 10(c) emerges, with a Jacobi Constant value of 3.0903 and a transfer time of 110.93 days,
corresponding to exactly eight 13.87-day revolutions of the operating L1Lyapunov orbit.
(a) LOTUS framework, adapted
from 2021, On-Orbit Satellite
Servicing Study: Project Report,
NASA GSFC.2
(b) L1at JC=3.15 homoclinic con-
nection with 8:3 resonance.
(c) Periodic LOTUS-like transfer
with JC=3.0903.
Figure 10: The LOTUS framework
This periodic transfer maintains the mission requirements from the NASA study while permitting
new capabilities and mission types. One of the initially stated objectives for this framework included
an orbit that is easily accessible from Earth. Such a constraint is possible due to the numerous apses
in the 8:3 resonance structure used for an insertion into and departure from the LOTUS, a property
retained by the periodic transfer. The applicability of this trajectory framework is extendable to
create a depot resupply framework by inserting a spacecraft into the LOTUS at one of the apses and
a natural rendezvous occurs with the depot that remains in the operating orbit.
Certain types of missions scenarios benefit from the ability to occasionally depart from their
operating orbit and allow closer passes of the Earth to downlink data or interface with satellites in
that region. For example, science missions near the Moon could return to downlink data closer to the
Earth and reduce their reliance on the Deep Space Network. Similarly, space surveillance satellites
that complete observations in their operating orbit could then potentially transfer to the near-Earth
region for data transmission. Transfers that make close passes of the Earth offer paths for spacecraft
launched from Earth or temporarily stationed near it to originally insert into their operating orbit as
well. Therefore, resonant structures that include close approaches of the Earth are considered for
periodic transfer design.
12
Several resonant orbits have members that pass near the Earth, such as the 2:1 family plotted
in Figure 3(a). Gupta et. al6identify cislunar surveillance applications for this family of orbits,
as spacecraft in a 2:1 resonant orbit potentially interface with satellites in LPOs near the Moon,
observe the Lunar region including the Lunar far side, and periodically reach low Earth altitudes.
The members of the 2:1 resonant orbit family with periapses very close to the Earth possess high
energies with values of JC =≈2.2−2.6, i.e., much higher than those for the L1and L2Lyapunov
orbits typically considered for Lunar operations and plotted in Figures 4(a) and 4(b), thus, it is
unlikely that homoclinic connections with a 2:1 structure exist for these Lyapunov orbits such that
a pass very close to the Earth occurs. However, homoclinic connections can still be constructed
that retain the 2:1 resonance structure and pass near the Earth, as apparent in the periodic transfer
from a L1Lyapunov plotted in Figure 11. The Jacobi Constant for this transfer is JC = 2.9135, a
relatively high energy level for the L1Lyapunov family, but this transfer includes two periapses at
an Earth altitude of 88239km. The time-of-flight for this transfer is 140.72 days, significantly longer
than the expected 2:1 resonance period of 1 sidereal month, i.e. 27.28 days, due to the manifold arcs
wrapping several revolutions of the operating orbit (a period of 28.14 days), to depart and arrive.
These details are consistent with the higher-energy L1Lyapunov orbits that are less unstable than
the lower-energy members of the family.14
Figure 11: Periodic transfer with 2:1 structure from L1Lyapunov at JC =2.9135
Figure 12: Earth closest approaches
for 2:1 and 3:1 resonant orbit fami-
lies
If closer approaches of the Earth are desired, 3:1 reso-
nance structures are available. As apparent in Figure 12, for
the same Jacobi Constant value, a 3:1 resonant orbit more
closely approaches the Earth than a 2:1 resonant orbit, such
that lower altitude Earth orbits are accessible at a lower en-
ergy level. For this example, a L2Lyapunov orbit is se-
lected as the operating orbit, although similar 3:1 geometries
also exist for a L1Lyapunov. The periodic transfer with a
3:1 resonance structure is plotted in Figure 13(a). This ho-
moclinic connection includes some additional structure that
links the L2Lyapunov to the 3:1 resonant structure, but the
resonant geometry is clearly present and comparable to the
3:1 resonant orbit plotted at the same Jacobi Constant level
of JC = 3.03258 in Figure 13(b). This transfer includes
three close approaches of the Earth, with the periapse on
13
the x−axis at an altitude of 42555km above the Earth and the off-axis periapses with altitudes
of 44658km, that is, around 10000km above the GEO altitude. The time-of-flight for this transfer is
176.53 days, and the operating orbit has a period of 17.65 days.
(a) Periodic transfer with 3:1 structure
from L2Lyapunov at JC = 3.03258
(b) 3:1 resonant orbit at JC = 3.03258
Figure 13: Transfers for near-Earth approaches
Resonant Orbits as Destinations
Homoclinic connections with resonant structure can enable low-cost periodic transfers into pe-
riodic orbits. For a servicer and depot application, assume that the depot remains in the operating
orbit and the servicer departs to service a customer on a different periodic orbit. The homoclinic
connection serves as the mechanism for the servicer to freely depart and return to the operating orbit
while shuttling the servicer to an appropriate location in space such that it transfers onto and then
off from the destination orbit after some specified nP O revolutions.
Figure 14: Periodic transfer from
an L2Lyapunov orbit (black)
to a 5:3 resonant orbit (blue,
dashed) using a homoclinic con-
nection with 5:3 resonance struc-
ture (green)
For example, in Figure 14, a periodic transfer from an L2
Lyapunov operating orbit, with J C = 3.0352, into a 5:3 res-
onant orbit using a homoclinic connection with 5:3 resonance
structure is plotted. The common perpendicular crossing be-
tween the homoclinic connection and the destination orbit,
identified with a black star, is used to insert onto the orbit and
then to depart it to return to the operating orbit along the homo-
clinic connection. The cost to transfer between the homoclinic
connection and the resonant orbit is ∆v= 12.35m/s, for a
total cost of ∆v= 24.7m/s over the entire transfer. The ser-
vicer spends one revolution along the 5:3 resonant orbit, with
a Jacobi Constant of JC = 3.0593 and a period of 81.4 days,
before returning to the operating orbit, for a total time-of-flight
of 227.86 days or 13 revolutions of the 17.52-day operating or-
bit.
14
The low transfer cost is enabled by the similarity in structure between the homoclinic connection
and the resonant orbit. Both the homoclinic connection and resonant orbit possess similar osculat-
ing periods at their common perpendicular crossing – a distance rfrom the Earth – where a conic
approximation for the velocity is v2bp =pc(2/r −1/(c(Tosc/(2π))2)1/3), with cdefined as the
Earth’s gravitational parameter and v2bp is expressed in an Earth-centered inertial frame. The ve-
locity in the CR3BP does not exactly equal v2bp due to the Lunar gravitational influence, however,
the similarity in the osculating period ensures that the homoclinic connection and resonant orbit
compare similar velocities; a burn at a perpendicular crossing ensures that the velocity directions
are aligned, contributing to a low-cost transfer.
Figure 15: Periodic transfer from
an L2Lyapunov orbit (black)
to a 2:3 resonant orbit (blue,
dashed) using a homoclinic con-
nection with 2:3 resonance struc-
ture (green)
An even lower cost periodic transfer appears in Figure 15
where a L2Lyapunov operating orbit reaches a customer on
a 2:3 resonant orbit by means of a homoclinic connection
with a 2:3 resonance structure. This transfer has a total cost
of ∆v= 14.24m/s with a time-of-flight of 258 days, cor-
responding to 10 revolutions of the 25.8-day operating orbit.
The energy level for the operating orbit is JC = 2.9492 and
the energy level for the customer orbit is JC = 2.9533, a very
small energy gap that contributes to the low cost for this trans-
fer. The servicer spends one revolution, or 71.75 days, along
the customer orbit before returning to the depot on the oper-
ating orbit via the inbound leg of the homoclinic connection.
A periodic transfer with an L2Lyapunov operating orbit at
JC = 2.9532 and a customer orbit at JC = 2.9573 is avail-
able if two revolutions of the customer orbit are planned, cor-
responding to 13 revolutions of the 25.17-day operating orbit.
Additional transfers are available if more revolutions along the
customer orbit are desired.
Figure 16: Periodic transfer from
aL2Lyapunov (black) to a 1:2
resonant orbit (blue, dashed) using
a homoclinic connection with 2:3
resonance structure (green)
Homoclinic connections with resonant structure can be
leveraged to transfer to different resonant orbits as well. As
an illustrative example, in Figure 16 a transfer from a L2Lya-
punov operating orbit to a customer’s 1:2 resonant orbit using a
homoclinic connection with 2:3 resonance structure is shown.
The transfer cost is ∆v= 102.7m/s, higher than the previous
examples due to the homoclinic connection and resonant orbit
no longer sharing a common resonant structure. The operating
orbit has an energy level of JC = 2.961 and the customer or-
bit is at an energy of JC = 2.932 where the servicer stays in
the customer orbit for two revolutions for a servicing time of
109.7 days and a total time-of-flight of 288.9 days.
Periodic transfers to stable orbits are also enabled, that is of
particular interest as several spacecraft are placed into stable
resonant orbits to leverage of their long-term stability. One
such set of orbits is the stable 2:1 resonant orbits, that differ
from the unstable members of the 2:1 family plotted in Figure
3(c) by being aligned with the y-axis rather than the x-axis.
15
Notably, Broucke17 has demonstrated that both the stable and
unstable 2:1 families are, in fact, part of the same family. However, for this analysis, they are
differentiated based on their stability and geometry. A spatial stable 2:1 resonant orbit is the selected
operating orbit for TESS, providing motivation for a servicing mission that can access such an orbit.
Figure 17: Periodic transfer from
an L2Lyapunov orbit (black) to
a 2:1 stable resonant orbit (blue,
dashed) using a homoclinic con-
nection with 3:1 resonance struc-
ture (green)
A periodic transfer from an L2Lyapunov to a stable 2:1
resonant orbit appears in Figure 17. The stable 2:1 orbits in-
clude periapses on the x-axis and some members include close
approaches with the Earth, thus, a 3:1 resonant orbit struc-
ture is leveraged to transfer from the operating orbit to the
2:1 stable resonant orbit. The operating orbit has an energy
level of JC = 2.9954, and the customer orbit energy equals
JC = 2.5152, for a total transfer cost of ∆v= 129.3m/s.
The servicer spends 1 revolution along the 2:1 orbit, corre-
sponding to a servicing time of 27.31 days, and the transfer
offers a time-of-flight of 221.23 days, corresponding to 11 rev-
olutions of the 20.1-day operating orbit.
L3Lyapunov Orbits as Destinations
Compared to the L1and L2Lyapunov families, orbits in the
vicinity of L3are less frequently the focus. However, Davis et
al.18 describe some advantages for using L3Lyapunov orbits
for mission design, e.g., the ability to view nearly 50% of the
Earth’s surface at one time and low station-keeping require-
ments. Reaching L3Lyapunov orbits from the Lunar region also serves as a relevant case study
to demonstrate the use of resonant structures to design periodic transfers that reach a variety of
destinations in the Earth-Moon system.
An exterior resonant orbital structure for the homoclinic connection is advantageous for reaching
L3, as many of these orbits include motion in the vicinity of L3with a perpendicular crossing that
matches the velocity direction for an L3Lyapunov orbit at that location. For example, a 4:5 resonant
structure is used to create a periodic transfer from an L2Lyapunov orbit to an L3Lyapunov orbit
and is plotted in Figure 18(a). The operating orbit energy level equals J C = 2.941 and the customer
orbit energy is 2.991, with a total cost ∆v= 150.82m/s. The servicer spends 1 revolution along
the customer orbit, with a period of 27.01 days, and the time-of-flight for the transfer is 270.3
days, equal to 10 revolutions of the 27.03-day operating orbit. Note that both the customer orbit,
the operating orbit, and the homoclinic connection transfer time occur in a nearly 1:1:8 resonance
between each segment and the Moon. As a result, the JC value for the operating orbit is nearly
unchanged if additional revolutions of the customer orbit are incorporated, as Qremains very close
to an integer if nP O is varied. Figure 18(b) illustrates this phenomenon; as nP O is varied, the
energy level of the operating orbit remains nearly unchanged, with small variances in J C required
to accommodate that the operating orbit, customer orbit, and homoclinic connection are not in
perfect resonance with one another. The resonance between the different elements of the transfer
adds versatility to a spacecraft servicing scenario, as the servicer could potentially spend an arbitrary
number of revolutions on the customer orbit before returning due to the fact that the transfer is nearly
unchanged as nP O is increased.
16
It is possible to reach L3Lyapunov orbits of different sizes by adjusting the type of resonant
structure that serves as the basis for the homoclinic connection. A homoclinic connection using
a resonant structure with a smaller loop around L3, such as a 2:3 resonant structure, delivers the
servicer to a smaller L3Lyapunov orbit. Two transfers linking an L2Lyapunov operating orbit to
an L3Lyapunov customer orbit using a homoclinic connection with an 2:3 resonant structure appear
in Figures 19(a) and 19(c). The first transfer, in Figure 19(a), also exploits the near 1:1:6 resonance
of the operating orbit, customer orbit, and homoclinic connection, as their respective periods are
27.57 days, 27.01 days, and 166.0 days, respectively. The transfer cost is ∆v= 205.04m/s and
the time-of-flight for a transfer incorporating 1 revolution along the customer orbit is 220.58 days,
corresponding to 8 revolutions of the operating orbit. The required operating orbit J C value, if nP O
is varied, is plotted in Figure 19(b), where small variations in JC are required if nP O is adjusted to
maintain the periodicity of the transfer. Larger variations are required as compared to the transfer
in Figure 18 due to the fact that the resonance between the operating orbit, customer orbit, and
homoclinic transfer are not as close of an integer ratio.
(a) Periodic transfer from an L2Lya-
punov orbit (black) to an L3Lyapunov
orbit (blue, dashed) using a 4:5 resonant
structure (green)
(b) Variation in the operating orbit JC
required for a periodic transfer with
nP O revolutions of an L3Lyapunov
Figure 18: Periodic transfer from L2to L3Lyapunov orbit using 4:5 resonant structure
The second transfer, in Figure 19(c), uses an operating orbit with a higher energy level to reach
an even smaller L3Lyapunov customer orbit. Recall that the periods along the L3Lyapunov family
do not exhibit much variation relative to the Lunar period Tmoon due to the 1:1 resonance with the
Moon, therefore, the customer orbit results in a period of roughly one sidereal month. However,
a periodic transfer with a smaller JC value for the operating orbit results in a period no longer
close to a sidereal month as the period of the L2Lyapunov orbits varies with Jacobi Constant. The
periodic transfer is, therefore, more sensitive to the number of revolutions of the customer orbit,
as varying nP O in Equation (6) results in a larger change in the operating orbit’s Jacobi Constant
value (and, therefore, the customer orbit’s Jacobi Constant value) to converge to a periodic transfer.
The energy level for the 21.98-day operating orbit and the customer orbit are JC = 2.977 and
JC = 3.01, respectively, and this transfer results in a cost of ∆v= 220.7m/s. The servicer
spends one revolution along the 27.0-day customer orbit and the total time-of-flight for the transfer
is 197.97 days.
17
(a) Transfer where the operating
orbit and customer orbit are in a
near 1:1 resonance
(b) Operating orbit JC required
for periodic transfers with nP O
revolutions of an L3Lyapunov
for an L2Lyapunov orbit with
near 1:1 Lunar resonance
(c) Transfer to a smaller L3Lya-
punov
Figure 19: Periodic transfers from an L2Lyapunov orbit (black) to an L3Lyapunov orbit (blue)
using some 2:3 resonant structure (green)
CONCLUDING REMARKS
Low-cost periodic transfers from an L1or L2Lyapunov operating orbit are generated to support
crucial mission applications for the development of cislunar space. A transfer framework is pro-
posed that uses ballistic transfers from the operating orbit with resonant structures for propellant-
free departures from and arrivals into an operating orbit, with an optional insertion into a periodic
resonant orbit at a perpendicular crossing. Natural parameter continuation for homoclinic connec-
tions yields a family of transfers with similar underlying resonant structures whose members are
evaluated for the existence of a periodic transfer. The use of resonant structures is advantageous for
access to many regions of cislunar space from the Lunar vicinity with low variance in their peri-
ods across the family, nearly ensuring the construction of a periodic transfer that reaches a desired
region of space.
Numerous examples of periodic transfers with applications to space infrastructure are provided.
Periodic transfers are potentially composed of a single homoclinic connection to permit a spacecraft
to depart its operating orbit and eventually return without expending any propellant. Applications
that may benefit from such a transfer are spacecraft collecting data in the Lunar region and a near-
Earth pass to downlink data more easily.
Periodic transfers for a spacecraft servicing application are also created. Such a transfer is nec-
essary for a framework that maintains a depot in the operating orbit with a servicer that departs the
depot, offers services to a spacecraft, and returns to the depot to be replenished for another mission
foray. Transfers that leverage a homoclinic connection to transfer the servicer to the customer’s orbit
quire low ∆vmagnitudes if the homoclinic connection and the customer orbit employ the same res-
onance structure. Transfers to orbits with different resonance structures are also possible, including
transfers into stable resonant orbits. Finally, transfers to L3Lyapunov orbits are also constructed,
and the low variation in the 1:1 resonance across the family is used to create transfers where the
18
operating orbit, homoclinic connection, and customer orbit are in a near-perfect resonance with one
another, creating versatile transfer opportunities where the servicer potentially remains in the cus-
tomer orbit for a nearly arbitrary number of revolutions, without the energy level of the transfer
shifting significantly.
ACKNOWLEDGEMENTS
The authors appreciate the financial support provided by the Purdue University School of Engi-
neering Education and the computational facilities in the Rune and Barbara Eliasen Visualization
Laboratory. The first author would like to thank the Fonds de Recherche du Qu´
ebec – Nature et
Technologies for support through grant number 303482. Special thanks to Nicholas B. LaFarge and
Rolfe J. Power for creating and sharing C++ integration tools. Finally, insightful discussions with
fellow members of the Purdue Multi-Body Dynamics Research Group were greatly appreciated.
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