Numerous Earth-Moon trajectory and lunar orbit options are available for Cubesat missions. Given the limited Cubesat injection infrastructure, transfer trajectories are contingent upon the modification of an initial condition of the injected or deployed orbit. Additionally, these transfers can be restricted by the selection or designs of Cubesat subsystems such as propulsion or communication. Nonetheless, many trajectory options can b e considered which have a wide range of transfer duration, fuel requirements, and final destinations. Our investigation of potential trajectories highlights several options including deployment from low Earth orbit (LEO) geostationary transfer orbits (GTO) and higher energy direct lunar transfer and the use of longer duration Earth-Moon dynamical systems. For missions with an intended lunar orbit, much of the design process is spent optimizing a ballistic capture while other science locations such as Sun-Earth libration or heliocentric orbits may simply require a reduced Delta-V imparted at a convenient location along the trajectory

Clark, Pamela E.

Dichmann, Donald James

Folta, David

Haapala, Amanda

Howell, Kathleen

NASA Technical Reports Server

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25th AAS/AIAA Space Flight Mechanics Meeting , Jan 11-15, 2015, Williamsburg, VA
LunarCube 
Transfer Trajectory Options
David Folta, NASA Goddard Space Flight Center
Donald Dichmann, NASA Goddard Space Flight Center
Pamela Clark, Catholic University 
Amanda Haapala, Purdue University
Kathleen Howell, Purdue University
25th AAS/AIAA Space Flight Mechanics Meeting
Jan 11-15, 2015, Williamsburg, VA
https://ntrs.nasa.gov/search.jsp?R=20150020472 2019-08-31T05:39:14+00:00Z
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25th AAS/AIAA Space Flight Mechanics Meeting , Jan 11-15, 2015, Williamsburg, VA
Introduction
• Numerous Earth-Moon (EM) trajectory and lunar orbit options are available for 
LunarCube missions
• Our investigation of potential trajectories highlights several transfer and lunar 
capture scenarios
o Low Earth orbit (LEO); Geostationary transfer orbits (GTO); Higher energy direct 
lunar transfer orbits (EM-1)
o Lunar elliptical and circular orbits with minimal capture requirements
o Yield a wide range of transfer durations, fuel requirements, and final destinations 
including Sun-Earth and Earth-Moon libration orbits, and heliocentric designs 
• Given the limited injection infrastructure, many designs are contingent upon 
the modification of an initial condition of the injected or deployed orbit
• Restricted by subsystems selection such as propulsion or communication
• Application Earth-Moon dynamical system design approach
o Apply natural trajectory flow and take advantage of system perturbations 
o For missions with an intended lunar orbit, much of the design process is spent 
optimizing a ballistic capture
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25th AAS/AIAA Space Flight Mechanics Meeting , Jan 11-15, 2015, Williamsburg, VA
Introduction
• Trajectory - Propulsion trades drive many mission design options
o Both low and high-thrust transfers are feasible assuming sufficient power or fuel mass
• For the EM-1 injected initial design, modify the lunar flyby distance to alter the 
system energy, matching that of a typical Sun-Earth system heteroclinic 
manifold
o Option uses dynamics similar to the ARTEMIS mission design
o Manifold and maneuvers raise perigee to that of a lunar orbit, adjust the timing wrt 
the Moon, rotate the line of apsides, and target a ballistic lunar encounter. 
o Orbital energy (C3) with respect to the Moon is targeted to < -0.1 km2/s2
• LEO or GTO design options use impulsive maneuvers to phase onto a local 
Earth-Moon manifold, which then transfers LunarCube to a lunar encounter
• Investigation concludes with several design options which provide estimated V 
requirements, achieved lunar orbit parameters, and associated transfer trajectory 
information
• The use of Goddard’s dynamical systems mission design tool, Adaptive 
Trajectory Design (ATD), and operational software (GMAT, Astrogator) are 
utilized to generated results
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25th AAS/AIAA Space Flight Mechanics Meeting , Jan 11-15, 2015, Williamsburg, VA
Constraints  
Low thrust and impulsive maneuvers concepts
• Low thrust level investigated vary from N to mN, 
o Limits the control authority and trajectory modifications 
o Power limited, with power < 100W(?) 
• Attitude control and pointing constraints may impede use or drive designs
• Impulsive designs drive fuel mass, deterministic Vs, or timing
Launch vehicle and related primary trajectories
• Secondary payloads cannot drive primary mission goals but can provide a 
minimal cost approach
• Constrain the mission design wrt launch/ injection parameters
o Injection energy can vary over launch period or window
o Number of launch opportunities can be limited
• Three injection options limitations
o LEO – launch dates, inclination and accelerations (Nodal precession and 
atmospheric drag)
o GTO – launch dates and line of apsides alignment
o EM-1 – launch dates, varying injection energy over window, unknown 
trajectory (apoapsis) direction
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25th AAS/AIAA Space Flight Mechanics Meeting , Jan 11-15, 2015, Williamsburg, VA
• Describes long-term qualitative behavior of complex dynamical systems
• Employs differential equations (continuous) / difference equations (discrete) to 
model system behavior
• Nonlinearity lead to complexity but not necessarily a loss of predictability.
• Focus not on precise solutions, but on general exploration of space (periodic 
orbits, quasi-periodic motion, chaos, …)
Dynamical Systems Theory
• Poincaré maps and invariant manifolds useful to locate long-term capture trajectories about 
the smaller primary in CR3BP
• Images from Howell, Craig Davis, and Haapala, Journal of Mathematical Problems in Engineering, Special Issue: 
Mathematical Methods Applied to the Celestial Mechanics of Artificial Satellites, 2012.
EML2
EML1
Moon
Lorentz Attractor – a non linear system 
sensitivity
Earth-Moon Poincare map of 
manifolds to hyperplane 
Periapsis Poincare map
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25th AAS/AIAA Space Flight Mechanics Meeting , Jan 11-15, 2015, Williamsburg, VA
Sun-Earth-Moon (GMAT)Earth-Moon CR3BP (ATD©)
• Simplified model, autonomous system
• Provides useful information about fundamental solutions (libration point 
orbits, stable/unstable invariant manifolds, retrograde orbits, …)
• Solutions from CR3BP transitioned to ephemeris model, generally, maintain 
orbit characteristics
**  Images from Haapala, Vaquero, Pavlak, Howell, and Folta, AAS/AIAA Astrodynamics Specialist Conference, 2013.
Circular Restricted Three-Body Problem
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25th AAS/AIAA Space Flight Mechanics Meeting , Jan 11-15, 2015, Williamsburg, VA
The ARTEMIS Connection 
• In 2009, two small spacecraft where transferred from low elliptical Earth orbits 
to lunar elliptical orbits
o Use of a dynamical system (manifold) approach with numerical targeting 
o Lower thrust propulsion system (4N) with constrained thrust direction on a spinning 
spacecraft
o Orbit-Raising maneuvers performed near periapsis to raise apoapsis to lunar distance
o Lunar Gravity Assists (LGAs) to align trajectory for Earth-moon libration insertion 
and to raise periapsis
Sun‐Earth rotating frame
P1 Design P2 Design
Sun‐Earth rotating frame
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25th AAS/AIAA Space Flight Mechanics Meeting , Jan 11-15, 2015, Williamsburg, VA
The ARTEMIS Connection - Manifolds
P1 Planned Stable Sun-Earth P1 Pre and Post TCM5 Stable Sun-Earth Manifold 
• In an ARTEMIS example, consider only the outbound arc of P1
• Follow the original outbound path to the location of a correction maneuver which 
shifted the spacecraft onto a different path, (orange) manifold
• Subsequent to and along the outbound trajectory two outbound manifold arcs emerge
• Represent potential outcomes from flow along the optimal path and the alternative that 
incorporates a possible correction maneuver  
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25th AAS/AIAA Space Flight Mechanics Meeting , Jan 11-15, 2015, Williamsburg, VA
Deployment options
Figure 8.  Example of Earth-Moon Local Manifold
• The local Earth-Moon manifold has a particular geometry and design that is based on the Earth 
and moon dynamics (CRTB)
• This manifold as illustrated provides a background on the types of trajectories desired for a 
natural flow towards either the moon or the Earth-Moon libration point orbits, EML1 or EML2. 
• The premise is that a spacecraft is inserted onto an intermediate orbit which asymptotically 
converges onto the manifold or intersects with the manifold 
• A manifold matching DV places the spacecraft onto one of the manifold trajectories which 
then flows to the region of lunar interest. 
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25th AAS/AIAA Space Flight Mechanics Meeting , Jan 11-15, 2015, Williamsburg, VA
• Initial orbit assumed 200 km LEO Final lunar orbit 1000 km, 
• Trajectories designed using ATD©
LEO to the Moon
Design-1 
(figure 9)
Design-2 
(figure 10)
Design-3 
(figure 11)
Design-4 
(figure 12)
Insertion DV (m/s) 3137 3047 3046 3099
DV 1 (m/s) 500 553 1183 899
DV 2 (m/s) 192 192 507 524
S/C DV total (m/s) 692 745 1690 1423
Manifold duration (days) 10 27 25 32
Transfer Duration (days) 13 30 27 36
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25th AAS/AIAA Space Flight Mechanics Meeting , Jan 11-15, 2015, Williamsburg, VA
GTO to the Moon
Design-1
(figure 15)
Design-2
(figure 16)
Design-3
(figure 17)
Design-4
(figure 18)
DV 1 (m/s) 2507 676 719 679
DV 2 (m/s) 0 0 824 421
DV  (m/s) (Lunar) 593 731 517 598
S/C DV total (m/s) 3100 1407 2060 1698
Transfer Duration (days) 5.4 4.7 20.3 16.2
• Initial orbit assumed 200 km, 24 deg inclination, LEO Final lunar orbit 1000 km, 
• Trajectories designed using ATD©
• Insertion from GTO Peripais and intermediate transfer reduces manifold matching DV cost
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25th AAS/AIAA Space Flight Mechanics Meeting , Jan 11-15, 2015, Williamsburg, VA
• Without altering the EM-1 injection energy, a LunarCube would perform a close lunar 
flyby and depart into heliocentric space
• Options to alter LGA energy include changing the flyby distance and orientation, 
permit trajectories to Sun-Earth L1/L2, Earth-moon L1/L2 , and lunar orbits
• Slow down from EM-1 injection approaching lunar flyby 
o Immediately after injection from EM1, thrust against velocity vector (relative to Earth) for 
several days 
o Option-1: Enter highly eccentric orbit around Earth and gradually raise perigee and lower 
apogee to approach Moon, in both orbit and phase
o Option-2: Achieve LGA to enter onto Manifold to raise perigee and approach moon
o Thrust against velocity vector (relative to Moon) to capture / spiral into a distant lunar orbit
o or change elliptical eccentricity
• Speed Up from EM-1 injection approaching lunar flyby 
o Immediately after injection from EM1, thrust along velocity vector (relative to Earth) 
o Achieve LGA to insert into a highly eccentric Earth orbit, with inclination close to
o Moon orbit. 
o Raise perigee and lower apogee to approach Moon, in both orbit and phase
o Thrust against velocity vector (relative to Moon) to capture / spiral into a distant lunar orbit
o or change elliptical eccentricity
EM-1 to the Moon
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25th AAS/AIAA Space Flight Mechanics Meeting , Jan 11-15, 2015, Williamsburg, VA
EM-1 to the Moon, A low Thrust Option
EM1 
Transfer  
Low Thrust 
(red)
Low Thrust 
arc (red)
Low Thrust 
arc (red)
Lunar 
Encounter
Lunar Orbit 
(Blue Circle)
Capture Lunar 
Encounter
Lunar 
Gravity 
Assist
Coast arc (Blue)
Sun – Earth Line
Coast arcs 
(Blue)
Low Thrust 
arc (green)
Coast arc (Blue)
Low Thrust 
Periapsis arc 
(green)
Final Science Orbit 
(Red)
Transfer Trajectory with Low Thrust
(Sun-Earth Rotating Coordinate Frame)
Lunar Capture with Low Thrust
(Lunar Inertial Coordinate Frame)
• Launch Dec 15, 2017
• Lunar Capture in ~ 231 days
• Total DV of ~ 869  m/s
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25th AAS/AIAA Space Flight Mechanics Meeting , Jan 11-15, 2015, Williamsburg, VA
EM-1 to the Moon, A low Thrust Option
• Launch Dec 15, 2017
• Lunar Capture in ~ 171 days
• Total DV of ~ 1554  m/s
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25th AAS/AIAA Space Flight Mechanics Meeting , Jan 11-15, 2015, Williamsburg, VA
Slow Down Approaching Flyby
Earth Inertial View
Slow Down Approaching Flyby
Earth-Moon Rotating View
Speed Up Approaching Flyby
Earth Inertial View
EM-1 to the Moon
Slow 
down
Speed up
Injection 
date
15-Dec-
2017
15-Dec-
2017
Science
orbit 
insertion
6-Aug-
2018
31-Jul-
2018
Transfer 
time (days)
234 228
Delta-V 
(m/sec)
1142 1315
Other options to maintain apoapsis near lunar orbit distance and then 
raise periapsis for a minimal lunar orbit capture
Speed Up Approaching Flyby
Earth-Moon Rotating View
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25th AAS/AIAA Space Flight Mechanics Meeting , Jan 11-15, 2015, Williamsburg, VA
Lunar Cube Transfer Trajectory Options 
Sample EM-1 Transfer Comparisons
Decreased 
Velocity
Decreased 
Velocity
Decreased 
Velocity
Increased 
Velocity
Increased 
Velocity
Related Fig 16 17 18 19 20
Initial Mass (kg) 9 12 Need 12 Need
Thrust Level (mN) 0.5 2 3 2 3
Total DV (m/s) 869 629 1314 1595 1141
Transfer DV (m/s) 673 190 1082 557 860
Lunar Capture DV (m/s) 196 439 41 1038 25
Lunar Flyby Radius (km) 6763 5025 4696 2510 6318
Max Transfer Range (Km) 1,524,000 1,719,925 447,959 1,154,950 467,698
Total Transfer Duration to 
Capture (days)
231 250 41 171 214
Lunar Capture Duration 
(days)
60 27 11 65 15
Maximum Lunar Eclipse 
Duration (hrs)
1.0 4.6 0.9 4.0 3.3
Lunar Orbit apoapsis x 
periapsis (km)
6800 x 100 9993 x 1545 tbd 350 x 50 tbd
Lunar Orbit Inclination 
(deg)
20 144 32 165 139
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25th AAS/AIAA Space Flight Mechanics Meeting , Jan 11-15, 2015, Williamsburg, VA
Lunar Missions
Low Thrust 
Arc in Green
Lunar Capture with Low Thrust
(Lunar Inertial Coordinate Frame)
A variety of lunar science orbits can be achieved from any of these analyzed 
transfers
• Low thrust capture and insertion using a ballistically captured lunar orbit
• Perform an alignment of periapsis (apsides) with science goals
• Target a given periapsis altitude or periapsis decay over time
• Target various eccentricity, semi-major axis, inclinations
• Achieve various science parameters, e.g. Solar angles
Science 
Orbit in red
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25th AAS/AIAA Space Flight Mechanics Meeting , Jan 11-15, 2015, Williamsburg, VA
Conclusions
• There are numerous Lunar Cube Transfer Trajectory Options available
• The deployment strategy, as a secondary payload, drives the available designs 
options 
• Both low thrust and high performance propulsion systems can be used
o High thrust can result in mass / volume considerations
o Low thrust ranging from -N to m-N can augment the trajectory given the proper 
initial conditions
o Power level will drive low thrust capabilities and the ensuing trajectory design
• Transfer and lunar capture into science orbit durations can be time-consuming
• Use of dynamical systems, aka manifolds, can aid in the design and provide an 
intuitive approach in addition to optimization
• Combining dynamical systems techniques with high or low thrust propulsion 
systems provides versatile, efficient techniques for transfers to the Moon, 
especially for low-thrust options on high energy deployment trajectories. 
• With a lower cost and many secondary payload opportunities, Lunar Cubes can 
be the next step for flexible trajectory designs, to the Moon and beyond


Lunar Cube Transfer Trajectory Options

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