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Neural Networks for Onboard Maneuver Design

Authors:
  • Advanced Space LLC

Abstract and Figures

This paper presents a general neural network framework, Neural Networks for Electric Propulsion (NNEP), for spacecraft autonomous onboard maneuver design and its application to several astrodynamics regimes. A previous version of this work applied a neural network (NN) model to making a single low-thrust trajectory correction in cislunar space. This paper extends the prior work to allow any number of low-thrust trajectory corrections in cislunar space and implements it in a general framework. This framework also allows space missions to offload the computational "heavy lifting" to ground-based computers. Ground systems generate training data (consisting of tens of thousands of off-nominal maneuver designs) and train a series of NNs, where each NN is applicable to a predefined range of states and/or epochs. The framework's computational burden for the spacecraft is minimal and easily fits within most current flight computers. The NN framework is also implemented in prototype flight software as a coreFlight System (cFS) app with minimal external dependencies. Simulation results show accuracy and propellant use comparable to or better than the best human-in-the-loop ConOps. This framework is extended further to apply to interplanetary low-thrust trajectory correction, geostationary orbit (GEO) station keeping, and station keeping in Gateway's near-rectilinear halo orbit (NRHO).
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IAC-22-C1,3,3,x73038
Neural Networks for Onboard Maneuver Design
Nathan Parrish Réa*, Timothy M. Sullivanb, Matthew D. Popplewellc,
Kirk S. Roerigd, Clayton Michaele, Tyler Hanff, Tyler Presserg
a Advanced Space, LLC, 1400 W 122nd Ave, Westminster, CO 80234, nathan.re@advancedspace.com
b Advanced Space, LLC, 1400 W 122nd Ave, Westminster, CO 80234
c Advanced Space, LLC, 1400 W 122nd Ave, Westminster, CO 80234, matthew.popplewell@advancedspace.com
d Advanced Space, LLC, 1400 W 122nd Ave, Westminster, CO 80234, kirk.roerig@advancedspace.com
e Advanced Space, LLC, 1400 W 122nd Ave, Westminster, CO 80234
f Advanced Space, LLC, 1400 W 122nd Ave, Westminster, CO 80234, tyler.hanf@advancedspace.com
g Advanced Space, LLC, 1400 W 122nd Ave, Westminster, CO 80234, tyler.presser@advancedspace.com
* Corresponding Author
Abstract
This paper presents a general neural network framework, Neural Networks for Electric Propulsion (NNEP),
for spacecraft autonomous onboard maneuver design and its application to several astrodynamics regimes. A previous
version of this work applied a neural network (NN) model to making a single low-thrust trajectory correction in cislunar
space. This paper extends the prior work to allow any number of low-thrust trajectory corrections in cislunar space and
implements it in a general framework. This framework also allows space missions to offload the computational “heavy
lifting” to ground-based computers. Ground systems generate training data (consisting of tens of thousands of off-
nominal maneuver designs) and train a series of NNs, where each NN is applicable to a predefined range of states
and/or epochs. The framework’s computational burden for the spacecraft is minimal and easily fits within most current
flight computers. The NN framework is also implemented in prototype flight software as a coreFlight System (cFS)
app with minimal external dependencies. Simulation results show accuracy and propellant use comparable to or better
than the best human-in-the-loop ConOps. This framework is extended further to apply to interplanetary low-thrust
trajectory correction, geostationary orbit (GEO) station keeping, and station keeping in Gateway’s near-rectilinear halo
orbit (NRHO).
Keywords: Neural Networks, Maneuver Design, Autonomous, Onboard
Acronyms/Abbreviations
AI = Artificial Intelligence
AoL = Argument of Latitude
BLT = Ballistic Lunar Transfer
CAPS = Cislunar Autonomous Positioning System
cFS = coreFlight System
CRTBP = Circular-Restricted Three Body Problem
ECEF = Earth-centered Earth-fixed
ECI = Earth-centered inertial
EP = Electric Propulsion
FSW = Flight Software
GEO = Geostationary Orbit
Isp = Specific Impulse
LM = Levenberg-Marquardt
MLP = Multilayer Perceptron
NIM = NRHO Insertion Maneuver
NN = Neural Network
NNEP = Neural Networks for Electric Propulsion
NRHO = Near Rectilinear Halo Orbit
OMM = Orbit Maintenance Maneuver
TCM = Trajectory Correction Maneuver
TPBVP = Two Point Boundary Value Problem
TIP = Trajectory Interface Point
VNB = Velocity-Normal-Binormal
1. Introduction & Background
1.1 Motivation
The state of the art for deep space mission navigation
and operation is to have a dedicated team of 2-3 people
for a single spacecraft, plus a crew to run the ground
station. The deep space navigation process consists of
spacecraft downlink, state estimation, trajectory re-
optimization, hardware sequencing, and data uplink
require two to three days. Ground station tasking
constraints delay the process further, so spacecraft
instructions are often days to weeks old by execution
time. Aggressive ground scheduling reduces the delay to
several days but further reduction requires automation.
Two examples of this are the Dawn and BepiColombo
missions. The BepiColombo mission (arriving at
Mercury in 2025) uses ground timelines of 1 week [1].
The Dawn mission used timelines of 1-5 weeks during its
the interplanetary transfer, 3 days during normal
operations at Vesta, and 36 hours while passing through
unstable resonances with Vesta’s gravity field [2].
Regardless of the length of the timeline, the spacecraft
always used thruster instructions based on out-of-date
navigation results.
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In addition to deep space missions, the space industry
is trending towards a larger number of small spacecraft,
enabling new capabilities at lower costs. This trend will
also necessitate automating operations.
In response to the growing need for real-time and
onboard maneuver planning for spacecraft, this paper
presents a general framework using neural networks for
the maneuver planning process. The framework
leverages the powerful fundamental principles of optimal
control and a rich field of recent advancements in the area
of artificial intelligence (AI) to automate spacecraft
maneuver correction, resulting in improved spacecraft
maneuver accuracy, lowered operations complexity, and
cost savings. Specifically, NNs are used as function
approximators, learning the complex relationship
between spacecraft state and the costates defining the
optimal control to return to a reference trajectory.
Typically, when optimizing spacecraft maneuvers, a
mission designer is tasked with solving a two-point
boundary value problem (TPBVP). Solving the TPBVP
is a numerically sensitive task that takes significant
computational resources to solve. Solving such a problem
also requires large numerical optimization software
libraries and long runtimes. Both requirements make it
unappealing to solve the TPBVP in flight software
(FSW) running on a limited-performance flight
computer. The framework presented in this paper solves
these problems by “learning” the optimal solution as a
function of the state which can then be uploaded and
evaluated onboard the spacecraft. In this deployment, the
computationally challenging training is done on the
ground and the evaluation of the NN is done onboard the
spacecraft. To demonstrate the viability of the framework
for onboard spacecraft computations, it is implemented
in prototype flight software as a cFS app.
This paper provides an overview of the neural
network architecture as well as the framework built
around it to develop maneuvers. Several applications of
this framework are also analyzed with simulation results
provided for each. Results of tests using the framework
as a cFS app are also discussed.
1.2 Background: optimal control
Optimal control theory forms the basis of the neural
network framework. An optimal control problem within
the presented framework is defined as: Minimize the
performance index given as:
󰇛󰇜 󰇛󰇜
(1)
where is the cost of the endpoints and is the cost of
the path. In addition, is the state, is the control, and
is time. The subscript represents initial, and the
subscript represents final. The state vector is subject to
differential constraints (the state dynamics) given as:
󰇗󰇛󰇜 󰇛󰇛󰇜󰇛󰇜󰇜
(2)
where is a vector of constant parameters. Terminal
constraints are applied as:

(3)
As trajectory optimization is nearly always an
underdetermined problem, additional constraints must be
introduced to find an optimal trajectory. The constraints
are derived via the Hamiltonian as follows:
󰇛 󰇜 󰇗󰇛󰇜
(4)
where are the costates. The costate dynamics are then
given by

(5)
The Hamiltonian Minimization Condition readily yields
the optimal control policy
󰇛 󰇜 󰇛󰇜
(6)
For the minimum fuel transfer, the term of the
objective function is chosen as
󰇛󰇜
(7)
However, this results in a numerically sensitive problem.
To make the indirect optimal control problem easier to
solve, the term of the objective function is chosen
differently as:
󰇟 󰇛 󰇜 󰇛 󰇜󰇠
(8)
where is a homotopy parameter. When is ~1, the
optimal control is very smooth. As ϵ is reduced to, say,
10-4, the optimal control becomes nearly
indistinguishable from the minimum fuel solution. This
modification to the objective function was studied by
Bai, Turner, and Junkins [3], then further refined by
Bertrand and Epenoy [4], and studied in the context of
NN optimal control by Parrish [5]. To keep the problem
formulation simple and generic to different dynamical
environments, additional path constraints can be
introduced.
For low thrust trajectories, the NN is tasked with
learning the relationship between a spacecraft’s current
state, which is perturbed from a nominal reference
trajectory, and the costates, , for an optimal maneuver
that returns to the reference trajectory.
1.3 Background: neural networks
Neural networks are a mathematical tool capable of
approximating arbitrarily complex functional
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relationships. NNs are inspired by biological brains and
consist of a network of “neurons”, where each neuron
performs a basic mathematical operator on its inputs,
passing its output to the next neuron.
The NNs used in this paper are simple feedforward
networks consisting of an input layer, a series of hidden
layers, and an output layer. This network formulation is
also known as a multilayer perceptron (MLP). A generic
feedforward neural network with a single hidden layer is
shown below in Figure 1.
Figure 1. A simple neural network with one hidden
layer.
One of the advantages of NNs is their ability to
approximate arbitrarily complex functions given
sufficient network sizes and training samples.
Additionally, while the training process takes significant
time, the evaluation of the network given a set of inputs
is extremely fast and requires few computational
resources, making NNs suitable for flight-approved
hardware.
2. Methods
2.1 Training data generation
Much of the complexity of the framework is in the
generation of quality training data. This section describes
how those simulated data are created for each type of
application.
2.1.1 Low thrust transfer general approach to
training data generation
A reference trajectory must be available for the
spacecraft to follow. The reference trajectory can be
generated by any means, as the training sample
generation only requires the position and velocity over
time. In this work, we use a proprietary tool to build an
optimal trajectory with the following high-level steps:
Direct multiple shooting in circular-restricted
three body problem (CRTBP) dynamics, with
mesh refinement to add more nodes near lunar
close approaches.
Indirect multiple shooting in the CRTBP with the
smoothed minimum fuel objective function.
Convert solution in CRTBP to an ephemeris
model.
Direct multiple shooting in the ephemeris model.
Indirect multiple shooting in the ephemeris
model with the smoothed minimum fuel
objective function.
Training data are generated by first choosing a time
at random from the time span of the reference
trajectory. At time  the position and velocity are
perturbed by some  and  , drawn from random
uniform distributions with lower and upper bounds
󰇛󰇜 and 󰇛󰇜 .The choice of and
 define the a “training tube” size illustrated in
Figure 2.
Figure 2. Illustration of the “training tube” around
the reference path, used for low-thrust trajectory
corrections.
With the training tube defined, the final step is to
solve the fixed time TPBVP for the transfer back to the
nominal trajectory at time  The time interval 
is fixed and chosen for the problem, typically on the order
of a few days or tens of days. The use of a receding
horizon keeps the spacecraft close to the nominal path at
all times, thus keeping the spacecraft within the training
tube.
A core element of this approach is the use of
checkpoints at which the training samples are required to
return to the reference path. This is illustrated in Figure
3. We found that adding ~2-5 checkpoints with fixed final
times helped guide simulated spacecraft to stay on a fuel-
optimal trajectory. Checkpoints are added immediately
prior to sensitive events such as a flyby (when small state
errors can be hard to recover from) or the start of a long
thrust arc (when the control authority to respond to
anomalies is more limited).
Figure 3. Illustration of checkpoints with the
“training tube”. All training samples leading up to a
checkpoint epoch are required to return to the nominal
at the checkpoint, rather than at a receding time horizon.
Training samples can now be generated with indirect
single shooting. Partial derivatives for the trajectory
solver are computed via automatic differentiation, which
allows the trajectory to be solved to numerical precision.
An example of the training tube is given in Figure 4.









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Figure 4. Example training tube around an arbitrary
section of a nominal transfer.
2.1.2 Cislunar low thrust transfer training
data generation
The Earth-Moon system was chosen due to its
sensitive dynamics which makes learning accurate
corrections more difficult. Algorithms developed for this
dynamical environment will necessarily work for simpler
dynamics. Initial tests were conducted using the CRTBP
and then extended to a point-mass ephemeris model using
the JPL DE430 ephemeris.
The test transfer is from an Earth-Moon NRHO
(Gateway’s planned orbit) to a larger Earth-Moon L2
halo orbit. The nominal transfer’s time of flight is 24
days. The spacecraft has an initial mass of 1,000 kg and
a maximum thrust of 300 mN with an Isp of 2,000 sec.
This hypothetical spacecraft propulsion system is
equivalent to the high end of currently-feasible thrust-to-
weight ratios of electric propulsion (EP) spacecraft. The
maximum thrust is limited to 240 mN for the nominal
transfer to build in margin for recovery from errors. The
nominal transfer requires 5.4 kg of propellant. Spacecraft
thrusting is not allowed within a radius of 20,000 km
from the Moon to avoid the additional operational
complexity of a powered lunar flyby. The trajectory is
defined in the Moon-centered J2000 inertial frame. The
example NRHO to 2 transfer is shown in Figure 5. In
addition, the nominal transfer’s thrust profile is computed
as in Figure 6.
The nominal transfer and all NN training trajectories
are modeled with the point mass gravity of Earth, Moon,
and Sun from the DE430 ephemeris. The spacecraft
specific impulse (Isp) and maximum thrust are assumed
constant throughout the nominal transfer.
The training tube for this transfer is defined as 
 and   relative to the nominal
trajectory. Training samples return to the nominal
trajectory after 4 days or at the next checkpoint epoch,
whichever is earlier.
Figure 5. NRHO to nominal low thrust transfer,
viewed in the Earth-Moon rotating frame.
Figure 6. Nominal thrust profile vs. time for NRHO
to transfer.
2.1.3 Interplanetary low thrust transfer
training data generation
The same method for training sample generation is
also applied to an interplanetary, heliocentric transfer.
For this transfer, a solar electric propulsion model is used
to provide realistic spacecraft control authority. The
nominal transfer consists of an Earth launch, Mars flyby,
and finally Mars rendezvous, of which only the first leg
(Earth to Mars flyby) will be used to train the neural
network. This leg was designed with a duty cycle of 80%,
has a time-of-flight of 290 days, and has a total mass drop
of approximately 64.5 kg. The transfer uses near bang-
bang control to model real-world operations. The force
model used for the nominal trajectory and all NN
trajectories includes the point mass gravities of Earth,
Jupiter, Mars, and the Sun from the DE430 ephemeris.
Figure 7 illustrates a heliocentric inertial view of the
nominal interplanetary trajectory on which the
framework was trained.
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Figure 7. Sun-centered, inertial view of nominal
transfer.
The available thrust is a function of a realistic power
model. The power available to the electric propulsion
engine is inversely proportional to the square of the
distance from the Sun, with a maximum of 3 kW at 1 AU.
The engine model has a fixed Isp and jet efficiency (1500
s and 50%, respectively) and includes losses for regulator
efficiency, off-pointing, solar cell degradation, and bus
power requirements. The thrust curves for the
interplanetary transfer are given in Figure 8.
Figure 8. Available and nominal thrust used by
interplanetary transfer.
Training samples for the NN are continuously
generated in a defined region around the nominal transfer
according to the “training tube” described previously. For
this problem, we generated 150,000 samples between the
start and end of the nominal transfer in a uniform
distribution up to 5,000 km and 50 m/s in position and
velocity magnitude, respectively. The single shooting
algorithm then solves for the minimum fuel, smoothly-
varying-thrust costate solution that allows each sample to
rendezvous with the nominal transfer either 30 days later
or at the end of the transfer, whichever is sooner, from
the sample epoch.
2.1.4 GEO station keeping training data
generation
The GEO station keeping problem is based on the
idea of keeping a spacecraft within an operational slot
defined by latitude and longitude bounds. Perturbations
from solar and lunar point mass gravities, solar radiation
pressure, and the non-uniformity of Earth’s gravitational
field cause a GEO spacecraft to drift in the longitudinal
and latitudinal directions from its nominal nadir location.
The effects of these perturbations are illustrated in Figure
9.
Figure 9. Natural longitudinal and latitudinal drift
due to perturbations.
Autonomous maneuver planning for GEO spacecraft
has been studied extensively in literature [6], [7].
However, existing strategies rely on linearized dynamical
models or approximate optimization methods. Neural
Networks provide a means for on-board maneuver
correction without over-simplifying the problem.
GEO station keeping using chemical propulsion
consists of two separate maneuver types: an east-west
maneuver that corrects longitudinal drift and a north-
south maneuver that corrects latitudinal drift. The
maneuvers are described in the Velocity-Normal-
Binormal (VNB) local frame. East-west maneuvers are
performed solely in the velocity direction which
increases or decreases the orbit semi-major axis,
consequently decreasing or increasing the orbit angular
speed and causing the spacecraft to drift westward or
eastward, respectively. Eventually, the influence of the
orbit perturbations creates a turnaround point, at which
the spacecraft begins drifting in the opposite direction.
For a spacecraft that experiences a natural eastward
drift, an optimal east-west maneuver places the
turnaround point at the westerly boundary of the
operating slot in order to maximize the time between
maneuvers. North-south maneuvers return the inclination
of the orbit to zero. Maneuvers that only change
inclination require a component in both the velocity and
orbit normal directions. However, for this application, the
north-south maneuvers consist of a burn solely in the
orbit normal direction. Performing such a maneuver
causes a deviation from the nominal orbit semi-major
axis and, consequently, an east-west maneuver frequently
immediately follows a north-south burn. Experience has
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shown that a NN can more easily learn the north-south
maneuvers when the problem is structured this way. An
example EW maneuver is shown in Figure 10.
Figure 10. An example east-west maneuver the
spacecraft (originally blue) approaches the easterly
boundary of the operational box (30.05°) causing a
maneuver to be performed. The spacecraft (now red) is
propagated to the turnaround point near the westerly
boundary of the box (29.95°).
For this problem, the spacecraft is initialized in GEO
with a nadir point of 30° East and 0° North and an epoch
of 29373 UTC MJD. The operational slot is defined with
longitudinal and latitudinal bounds of ±0.1° to model
realistic application. To generate samples, a targeting
scheme in GMAT delivers the optimal station keeping
maneuvers for the nominal GEO spacecraft over ten
years. Additional samples are created by perturbing the
initial state off the nominal and again finding the optimal
station keeping history for ten years. The perturbed initial
states consist of combinations of initial epoch, latitude,
and longitude values in the ranges of [29373, 29391 UTC
MJD], [-0.3, 0.3°], and [29.96,30.02°] respectively.
A separate NN is trained for each maneuver type. The
input layer of the NN takes 14 quantities: the spacecraft’s
latitude, longitude, Cartesian velocity components in an
Earth-centered Earth-fixed (ECEF) frame, longitude rate,
orbit eccentricity, and the Cartesian position components
of the Moon and Sun in an Earth centered inertial (ECI)
J2000 frame. The longitude rate is found by taking the
difference of the Earth’s rotation rate and the spacecraft’s
mean motion. The output layer of the NN is a single
quantity: the burn magnitude of the appropriate
maneuver. A NN with three hidden layers of 30 neurons
each and these input and output layers (total of 3,271
learned NN weights) is capable of learning both
maneuver types well.
To evaluate the performance of the NN, it is desirable
to propagate a GEO spacecraft in GMAT using a high-
fidelity environment. However, the NN is built and
trained using the Julia language and GMAT lacks a built-
in interface with Julia. In order to pass the NN
evaluations to GMAT, a TCP socket is used to send
information between a Julia script, which evaluates the
NN, and a Python script, which interfaces with the
GMAT API.
A GEO spacecraft, with an initial state at 29373 UTC
MJD and a sub-satellite point of 30° East and 0° North,
is propagated using the above simulation scheme for 10
years. The spacecraft is propagated in GMAT until an
EW or NS station keeping maneuver is required. A
Python script collects the end state using the GMAT API
and sends the state to Julia over the TCP socket. A Julia
script then determines which maneuver type is required,
evaluates the correspond NN, and sends the burn
magnitude over the socket to Python. Python then
reinitializes the GMAT simulation with the previous end
state and corresponding burn. GMAT performs the NN
burn before again propagating until the next maneuver.
To make the simulation more realistic, navigation and
thruster errors are simulated. The navigation error is
modeled as Gaussian noise (0, σ2) where σ is the
standard deviation according to the values given in Table
1. Table 1. GEO simulation setup.
5e-5
5e-5
1
1e-6
0.5
The thruster error is modeled at 2% for this
simulation. The Moon and Sun Cartesian positions in the
ECI frame are obtained through JPL’s SPICE toolkit and
thus no error is applied to these values.
2.1.5 NRHO station keeping training data
generation
Within the Cislunar system, the station keeping
problem was also analyzed using the framework. The
authors specifically chose to test the framework to
simulate station keeping within an NRHO due to its
relevance for the upcoming NASA Lunar Gateway.
NASA’s upcoming Lunar Gateway is planned to operate
in a southern L2 NRHO. The orbit has close approaches
to the lunar surface allowing low-cost access to the lunar
surface and consistent line-of-sight with Earth ground
stations. The orbit is phased such that it avoids all total
Earth eclipses [8]. Advanced Space’s CAPSTONE
mission launched on June 28, 2022 and will demonstrate
operations within the same NRHO that will be used by
the Gateway.
Existing station keeping strategies for the designated
NRHO target a velocity vector component in the Earth-
Moon rotating frame and a time of perilune passage
several revolutions downstream of a given state.
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Targeting the velocity component in the direction ()
at perilune passage 6.5 revolutions downstream is
currently favored as it delivers a minimum Δ and
maintains the NRHO near its reference. The planned
station keeping strategy for CAPSTONE follows the
strategy planned for Gateway: perform an orbital
maintenance maneuver (OMM) up to once every
revolution that targets both and at an osculating
true anomaly of 200° [9]. The used for targeting is the
velocity in the direction of the Earth-Moon rotating
frame at perilune. This and the time of perilune
passage () are compared to those of the reference orbit.
When is within a given threshold, the maneuver only
targets .
Figure 11. Maneuver location on NRHO viewed
from the Y-Z plane of the Earth-Moon rotating frame.
Training samples for the NN are generated by
perturbing the six-dimensional state of the orbit at each
OMM epoch for the full reference period (15 years, up to
2035). At each OMM epoch, the reference state is
determined from ephemeris and 500 samples are
generated by perturbing the position and velocity vectors
such that they form a uniform spherical distribution with
of 3,000 km and 30 cm/s respectively. For each
sample, the corresponding station keeping maneuver that
targets VX and TP 6.5 revolutions downstream is
designed using the Jet Propulsion Laboratory’s Monte
library. The OMM is designed using a realistic dynamical
model with the forces modeled in Table 2 through Table
4.
Table 2. Point masses used for NRHO dynamics.
Point Masses
Sun
Mercury
Venus
Mars Barycenter
Jupiter Barycenter
Saturn Barycenter
Neptune Barycenter
Uranus Barycenter
Pluto Barycenter
Table 3. Spherical harmonics used for NRHO
dynamics.
Body
Dataset
Order
Earth
GGM03C
16
Moon
GL900D
16
Table 4. Solar Radiation Pressure parameters for
NRHO dynamics.
Parameter
Value
Flat Area (󰇜
0.6
Mass (󰇜
25
1.5
A single feedforward neural network is trained for
all OMM epochs. The NN consists of two hidden layers
each with 40 neurons. The input layer has a size of 42
consisting of:
the six-dimensional perturbed sample state, six-
dimensional reference state,
six-dimensional error vector between the
perturbed and reference states,
three-dimensional relative spacecraft-Sun
position vector in the synodic frame,
three-dimensional relative spacecraft Earth
position vector in the synodic frame,
three-dimensional vector of Euler angles
describing the Moon orientation in inertial space,
six-dimensional perturbed state 6.5 revolutions
downstream,
six-dimensional reference state 6.5 revolutions
downstream,
the difference between the perturbed state and
reference 6.5 revolutions downstream, and
the difference between the perturbed state and
reference 6.5 revolutions downstream.
The output layer has a size of three consisting of the
impulsive three-dimensional maneuver vector in the
Earth-Moon rotating frame. In total, there are 3,483
learned NN weights.
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2.1.6 TCM training data generation
As mentioned in section 2.1.5, Advanced Space’s
CAPSTONE mission will demonstrate operations within
the same NRHO that will be used by the Gateway. In
order to reach and eventually insert into the target
NRHO, CAPSTONE follows a ballistic lunar trajectory
(BLT), a fuel-efficient path that leverages the Sun’s
gravity to increase radius of periapsis and inclination.
Along the BLT, several trajectory correction maneuvers
(TCMs) occur to clean up launch vehicle insertion errors,
maneuver errors, and navigation errors. Nominally, five
TCMs will occur along the BLT but the exact number can
increase to eight depending on vehicle parameters,
insertion errors, maneuver errors, and navigation errors.
The TCMs are designed using a three-burn
optimization process [10]. When designing any given
TCM n, the backpropagated state from the NRHO at the
next TCM n+1 epoch is targeted and the total ΔV of
TCM n, TCM n+1, and the NRHO Insertion Maneuver
(NIM) is minimized. For example, when designing
TCM1, the total ΔV of TCM1, TCM2, and NIM is
minimized while the targeted state is that of the reference
trajectory at the time of TCM2. Nominally, TCM1
through TCM5 are designed in this manner with TCM5
also designing an optional TCM6. Additionally, if
TCM1, which cleans up initial launch errors, is above a
threshold defined by the spacecraft parameters then
optional maneuvers TCM1b and TCM1c are also
designed. The approximate locations of the TCMs for a
representative BLT are given in Figure 12.
Figure 12. Approximate locations of nominal TCMs
for a given BLT [10].
TCM training data is generated via Monte Carlo
simulation of the operational maneuver design process.
Realistic state errors are sampled relative to the reference
trajectory at each pre-planned TCM epoch, then the 3-
burn optimization described above is carried out to
generate the  vector corresponding to the state error.
For this test case, the training dataset only includes
the nominal TCM1 through TCM5. Any TCM1
maneuver that is over 20 m/s, and would normally require
a secondary TCM1 burn, is filtered from the dataset
before training. A BLT reference trajectory
corresponding to a previously designed Trajectory
Interface Point (TIP) of June 2, 2022, and the
corresponding TCM maneuver schedule is used to
determine the reference TCM states. Sample states are
generated by uniformly perturbing each TCM reference
state in position and velocity such that they form a
spherical distribution. At each sample state the
corresponding TCM is designed using the
aforementioned process. The TCM design, and later
simulation, is performed in the realistic dynamical model
identical to the NRHO dynamics outlined in Table 2
through Table 4.
A separate feedforward NN is used to learn each of
the five nominal TCMs. The NN consists of three hidden
layers with 30 neurons each. The input layer is of
dimension 30 consisting of the six-dimensional perturbed
sample state, six-dimensional reference state, six-
dimensional error vector between the perturbed and
reference states, six-dimensional relative spacecraft-Sun
state vector in inertial space, and six-dimensional relative
spacecraft-Earth state vector in inertial space. The output
layer is of size four consisting of the TCM burn duration
and three-dimensional unit direction vector. In total,
there are 2,914 learned NN weights.
2.2 Machine learning framework
Based on experience from Parrish [5], we chose to use
the Levenberg-Marquardt (LM) optimization algorithm.
NNs are trained using a custom version of the Levenberg-
Marquardt (LM) algorithm. At each iteration, the LM
algorithm computes the Jacobian of all training samples
with respect to all trainable parameters and uses this to
adjust the trainable parameters to minimize the sum of
squared errors. When the objective is to minimize the
sum of squared errors, the Jacobian can be used to
accurately approximate the Hessian as well, leading to
second-order convergence on the optimal NN weights.
The LM algorithm is suited for regression problems with
NNs containing up to tens of thousands of weights. It is
not used for classification problems such as natural
language processing or image recognition because those
require NNs with millions or even billions of weights.
Use of the LM training algorithm on such a large NN
would require computing an excessive number of partial
derivatives and is thus computationally intractable. For
small NNs, such as those commonly used in regression
problems, the LM algorithm has been found to converge
orders of magnitude faster than methods that only
compute a vector gradient.
The LM algorithm makes updates to the weights as
follows:
󰆒 󰇛󰇜󰇛󰇜
(9)
where is the vector of model weights, is the Jacobian
of model outputs with respect to model weights, is an
appropriately-sized identity matrix, is an adjustment
factor to smoothly vary between 1st and 2nd order
convergence, and is the vector of residuals (the
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difference between the model output and the desired
model output).
At the start of this work, the only performant,
publicly-available LM implementation was in MATLAB,
and even that implementation was not capable of GPU
acceleration. We traded all the available implementations
and ultimately decided to write our own in the Julia
language [11] with GPU acceleration via the CUDA.jl
package [12]. Our implementation uses analytically-
defined partial derivatives that we developed and
implemented with great regard to speed. The Jacobian
matrix is strategically broken into sections, and each
section is computed by a separate GPU thread. The
performance bottleneck is the low-level cuBLAS matrix
pseudo-inverse in equation 9.
Our implementation is based on the work by
Tomislav et al [13], particularly with regard to the
automatic weight decay variation. Testing on a variety of
problems has found this implementation to converge
more quickly and robustly than naïve choice of the
parameter.
2.3 ConOps for onboard implementation
The proposed framework would be implemented on
board a spacecraft as follows. First, ground operators
design a nominal transfer from the current state to some
target state, taking into consideration all necessary
constraints. Then, the same ground software is used to
solve tens of thousands of similar, perturbed transfers.
The solution of each perturbed transfer results in a
pair of input-output vectors: given the current state vector
(and problem-specific additional context), return the
primer vector (for continuous-thrust) or the  vector
(for impulsive maneuvers) corresponding to the optimal
path to rendezvous with the target. The ground software
then trains a NN to approximate the relationship between
the input and the output.
Since each of the perturbed trajectories is short and
has a good guess from the nominal path, the perturbed
optimal trajectories can be generated quickly. Proprietary
trajectory optimization tools at Advanced Space can
generate these training samples in a few hours on a
modern workstation computer. If needed, the task can be
parallelized to hundreds or even thousands of CPUs in a
distributed computing environment to generate sufficient
training samples within minutes.
Once training samples are generated, the NN model
is trained. The model is saved as a binary file dictating
the shape of the model and the model weights, and the
binary file is either loaded on the spacecraft prior to
launch or uplinked to the spacecraft during flight.
Trajectory corrections are made continuously over
the course of an orbit transfer. The frequency of
corrections is tunable for the mission, with tests here
using a 1060-minute update cadence.
The spacecraft is assumed to perform onboard orbit
determination which returns a state error signal. Example
onboard orbit determination technologies include the
Cislunar Autonomous Positioning System (CAPS) which
is under development at Advanced Space [10] , GNSS,
optical navigation, one-way ranging with an atomic
clock, and other technologies in development. The
spacecraft passes the best estimate of the current state
into the NN model, which returns the costate vector or
 vector. The spacecraft numerically integrates the state
forward in time to the next tick, with control held
inertially fixed. Other elements of the FSW turn the
control vector into thruster instructions which are
executed to fly the trajectory. A schematic of the
framework’s concept of operations is shown below in
Figure 13.
Figure 13. Schematic of onboard ConOps.
3. Results: Low Thrust Control
To evaluate the performance of the NNs, a spacecraft
is flown along the reference trajectory using the NNs to
deliver trajectory corrections. The simulation propagates
two spacecraft concurrently, a “truth” and a “predicted”
spacecraft. The predicted spacecraft’s state is
periodically updated to that of the truth, plus some
navigation noise, according to a navigation frequency
parameter. At a defined NN evaluation frequency, the
predicted spacecraft’s state is used to evaluate the
appropriate NN and deliver the desired control. To
translate the costate outputs of the NNs to thrust
commands, the NN is evaluated frequently, and the
spacecraft thruster is only fired when the magnitude of
the costate control law is above a defined threshold
parameter. The truth spacecraft executes the desired
control commands from the predicted spacecraft plus
some thruster error.












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3.1 Earth-Moon 3-body transfer
The results from Monte Carlo simulation of a NRHO
to transfer with the proposed framework and
simulated ground control are provided in Table 5, for a
range of navigation-update cycle speeds. Historically,
achieving a 2-day navigation and control update loop
with human-in-the-loop control has required 24-hr
coverage from operations teams and is only practical for
short periods of critical operations. However, the speed
of the control loop in space is limited only by the onboard
navigation requirements. Once a state estimate is
available for NNEP, it can update the thruster commands
immediately.
Table 5. Comparison of neural net framework and
simulated ground control.
Simulation type
(nav update cycle)
Final Error
(km)
Average
Propellant (kg)
NNEP 2-day
39 ±15
6.03±0.24
NNEP 4-day
54±23
6.16±0.33
NNEP 6-day
129±84
6.22±0.32
Ground 2-day
22±11
6.1±1.1
Ground 4-day
75±63
6.8±1.3
3.2 Interplanetary transfer
Table 6 shows the results for a 200-trial Monte Carlo
simulation of the NNEP and ground control strategies for
different navigation update frequencies. NNEP
outperforms the ground control in terms of final error
from the reference trajectory but uses slightly more fuel
in the process. The NNEP software makes up for larger
navigation errors with a nearly instantaneous navigation
solution to control update cadence.
Table 6. Neural net framework and human control
comparison (N = 200). Data provided as
mean ± standard deviation.
Simulation type
(nav update cycle)
Final Error
(km)
Propellant
use (kg)
NNEP 8-day
167±19
64.46±0.09
NNEP 16-day
205±60
64.50±0.10
NNEP 28-day
184±62
64.49±0.10
Ground 8-day
364±252
64.22±0.03
Ground 16-day
525±321
64.25±0.06
Ground 28-day
879±516
64.28±0.13
4. Results: Impulsive Control
4.1 GEO station keeping
Results for the GEO station keeping trial show that
the NN framework can deliver necessary station
keeping maneuvers such that the spacecraft maintains
its operational slot for the entire 10-year simulation
time. The total  for all maneuvers over 10 years using
NNEP is 521.365 m/s. In comparison, a ground
simulation, in which GMAT propagates the spacecraft
and calculates optimal station keeping burns, results in a
total  of 520.481 m/s over 10 years. The NN delivers
near optimal maneuvers instantaneously without the
need for a human operator or optimization software.
Figure 14. Motion of a GEO spacecraft in the ECEF
frame propagated for 10 years using the NN for station
keeping predictions.
4.2 NRHO station keeping
To evaluate the performance of the NN, a spacecraft
in the defined NRHO is propagated for the full reference
period (15 years) using the NN to deliver OMMs every
revolution. The spacecraft is propagated in JPL’s Monte
Python package. Since the NN is constructed and trained
in the Julia language but the simulation occurs in Python,
a TCP socket is used to pass information between a Julia
script running concurrently with the Python simulation.
Two spacecraft are simulated, a “NN” spacecraft that
evaluates the trained NN for OMMs and a “truth”
spacecraft that uses SNOPT and the aforementioned
station keeping strategy to deliver OMMs. Both are
initialized with the same state which is randomly
generated and slightly off-reference. To make the
simulation more realistic, navigation and thruster errors
are simulated. The navigation noise is modeled as a
uniform spherical perturbation to the six-dimensional
spacecraft state with  of 1 km and 1 cm/s for position
and velocity respectively. The thruster error is modeled
as a uniform spherical distribution in direction with a
random magnitude according to a uniform distribution
with an upper bound of 2% of the maneuver magnitude.
Results in Table 7 show that the NN can deliver
sufficient OMMs such that the NN spacecraft remains in
the vicinity of the reference NRHO for the entire 15-year
reference period. A Monte Carlo analysis was performed
by running the above simulation 100 times and collecting
the results.
The NN delivers OMMs to remain within the vicinity
of the NRHO over 15 years without the need for a ground
team to design maneuvers. The NN, while using slightly
more total DV over the simulation time, outperforms the
ground control in terms of error from the reference orbit.
It should be noted that the amount of difference in 
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between the NN control and ground control is negligible
compared to effect of momentum desaturation
maneuvers, which were not modeled here.
Table 7. NN and ground truth control comparison (N =
100). Data provided as mean ± standard deviation.
Performance Metric
NN Control
Ground
Control
Annual V (m/s)
0.31 ± 0.04
0.25 ± 0.02
Maximum position
deviation (km)
554 ± 33
780 ± 12
4.3 TCM’s for CAPSTONE
To evaluate the performance of the NN, a spacecraft
is flown on the reference BLT using the NNs to deliver
each nominal TCM. The spacecraft is propagated in
JPL’s Monte Python package from TIP to TCM6. Since
the NN is constructed and trained in the Julia language
but the simulation occurs in Python, a TCP socket is used
to pass information between a Julia script running
concurrently with the Python simulation. To make the
simulation more realistic, navigation error is simulated.
The navigation noise is modeled as a uniform spherical
perturbation to the six-dimensional spacecraft state with
 of 5 km and 5 cm/s for position and velocity
respectively.
Table 8 shows the results of a 200 trial Monte Carlo
simulation of the CAPSTONE BLT with NNs delivering
the necessary TCMs. The NNs can provide sufficient
maneuvers such that the spacecraft remains close to the
reference trajectory autonomously.
Table 8. NN control Monte Carlo results (N = 200).
Data provided as mean ± standard deviation.
Performance Metric
NN Control
Total V (m/s)
83 ± 5
Final position error (km)
372 ± 20
5. Results: FSW implementation
The on-board neural network inferencing
functionality is designed to be platform independent.
Thus, NASA’s cFS framework is used to host the neural
network software. To maintain low-compute
requirements, trained models are serialized and then
uplinked to the on-board system. As model integrity and
model relevance are key factors to nominal operations,
the FSW is designed to allow for model uplinks to occur
at any time to replace the current model. To handle off-
nominal exceptions during flight, the FSW is
implemented with a standby mode as shown in Figure 15,
allowing operators to provide recovery commands as
needed without model evaluations.
Figure 15. NNEP cFS App Order of Operations
The FSW implementation is designed to be minimally
intrusive for any spacecraft. We developed a custom
inference engine that is highly memory-efficient.
Profiling was carried out on a NN model representative
of a low-thrust trajectory corrector. Profiling of model
loading, de-serializing, and inference showed a peak
memory use of 86.5 kB. Software dependencies are also
minimal: the entire cFS app is dependent only on
NASA’s CSPICE.
The FSW has been simulated using a mock ground-
station to uplink models, send commands, and to receive
telemetry. Tests were conducted with nominal and off-
nominal initial conditions, as well as with varying
simulation durations. Furthermore, standby mode and in-
flight FSW reinitialization were also tested to ensure
correct recovery behavior. The simulation results were
compared with the non-FSW implementation simulation
results and proved to be equivalent. We hope to be able
to share more details of the FSW implementation in a
future publication.
6. Conclusion
This paper presents a framework for autonomous and
onboard maneuver planning in both low thrust and
impulsive thrust spacecraft built upon NNs. To illustrate
the NN’s ability to generate optimal control response in
different dynamical regimes, simulations were conducted
with both two- and three- body trajectories. To support
the framework’s usability in flight, a concept of
operations is designed to utilize the framework with data
that would traditionally be available to the spacecraft and
to ground operators.
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Copyright ©2022 by the International Astronautical Federation (IAF). All rights reserved.
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The next contribution is the sample generation
method for the NN. To generate training samples for the
NN, the sampling tube method is proposed where a
nominal trajectory is designed and thousands of
perturbed states are used to compute optimal return
maneuvers to the reference trajectory. With the neural
networks trained for different tasks, simulation results are
then presented for each maneuver design task in each
dynamical environment.
Finally, this paper also briefly summarizes a process
by which the framework can be implemented onboard a
spacecraft as a cFS application. Altogether, the proposed
framework and findings from subsequent analysis aim to
provide a highly generalizable architecture based on
neural networks for automated and onboard maneuver
planning in multiple dynamical environments.
Acknowledgements
This research was sponsored by the NASA SBIR
program, contract number 80NSSC20C0139.
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The Cislunar Autonomous Positioning System Technology Operations Navigation Experiment
  • E W Kayser
  • J S Parker
  • M Bollinger
  • T Gardner
  • B Cheetham
E. W. Kayser, J. S. Parker, M. Bollinger, T. Gardner and B. Cheetham, "The Cislunar Autonomous Positioning System Technology Operations Navigation Experiment," in Ascend 2020, Virtual Event, 2020.