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Rapid Accessibility Evaluation for Ballistic Lunar Capture via Manifolds: A Gaussian Process Regression Application

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Abstract

A supervised machine learning approach called the Gaussian Process Regression (GPR) is applied to approximate the optimal bi-impulse rendezvous maneuvers in cis-lunar space. The use of GPR approximation of the optimal bi-impulse transfer to patch-points associated with various invariant manifolds in the cis-lunar space is demonstrated. The proposed method advances preliminary mission design operations by avoiding the computational costs associated with repeated solution of the optimal bi-impulsive Lambert transfer because the learned map is efficient to compute. This approach promises to be useful for aiding preliminary mission design. The use of invariant manifolds as part of the transfer trajectory design offers unique features in reducing propellant consumption while facilitating the solution of the trajectory optimization problems. Long ballistic capture coasts are also very attractive for mission guidance, navigation and control robustness. A multi-input single-output GPR model is shown to efficiently represent the fuel costs (in terms of the $\Delta$V magnitude) associated with the class of orbital transfers of interest. A multi-input multi-output GPR model is developed and shown to provide efficient approximations. Multi-resolution use of local GPRs over smaller sub domains, and their use to construct a global GPR model is also demonstrated. One of the unique features of GPRs is to provide an estimate on the quality of the approximations in the form of covariance, which is shown to provide statistical consistency to the optimal trajectories generated from the approximation process. Numerical results demonstrate a basis for optimism for the utility of the proposed method.
Rapid Accessibility Evaluation for Ballistic Lunar Capture via
Manifolds: A Gaussian Process Regression Application
Sandeep K. Singh , John L. Junkins and Manoranjan Majji
Texas A&M University, College Station, TX 77843.
Ehsan Taheri§
Auburn University, Auburn, AL 36849.
A supervised machine learning approach called the Gaussian Process Regression (GPR)
is applied to approximate the optimal bi-impulse rendezvous maneuvers in cis-lunar space.
The use of GPR approximation of the optimal bi-impulse transfer to patch-points associated
with various invariant manifolds in the cis-lunar space is demonstrated. The proposed method
advances preliminary mission design operations by avoiding the computational costs associated
with repeated solution of the optimal bi-impulsive Lambert transfer because the learned map
is efficient to compute. This approach promises to be useful for aiding preliminary mission
design. The use of invariant manifolds as part of the transfer trajectory design offers unique
features in reducing propellant consumption while facilitating the solution of the trajectory
optimization problems. Long ballistic capture coasts are also very attractive for mission
guidance, navigation and control robustness. A multi-input single-output GPR model is shown
to efficiently represent the fuel costs (in terms of the ΔV magnitude) associated with the class of
orbital transfers of interest. A multi-input multi-output GPR model is developed and shown to
provide efficient approximations. Multi-resolution use of local GPRs over smaller sub domains,
and their use to construct a global GPR model is also demonstrated. One of the unique features
of GPRs is to provide an estimate on the quality of the approximations in the form of covariance,
which is shown to provide statistical consistency to the optimal trajectories generated from the
approximation process. Numerical results demonstrate a basis for optimism for the utility of
the proposed method.
I. Introduction
Cis-lunar space is an extremely important domain for mankind to explore and exploit in the quest to make humans a
multi-planet species. As the Earth’s nearest neighbor heavenly body, the Moon represents an excellent natural test-bed
PhD Candidate, Department of Aerospace Engineering, Student Member AIAA.
Distinguished Professor, Department of Aerospace Engineering, Honorary Fellow AIAA.
Assistant Professor, Department of Aerospace Engineering, Senior Member AIAA.
§Assistant Professor, Department of Aerospace Engineering, Senior Member AIAA
for technology development and demonstration to enable missions to other solar system bodies, as well as serve as
an intermediate way-point. Mankind has always yearned for pastures anew and their innate survival instincts along
with emerging commercial incentives has led space exploration to gain unprecedented traction in recent years. NASA
outlines their primary objective regarding a gateway in lunar orbit [
1
,
2
] as “NASA shall establish a Gateway to enable a
sustained presence around and on the Moon and to develop and deploy critical infrastructure required for operations
on the lunar surface and at other deep space destinations.” Re-supply cargo missions to the gateway orbit (which can
tolerate moderately long coasts) are of particular interest and help motivate the current study.
A vibrant research area in the field of trajectory optimization is the design of low-thrust transfers that leverage
insights from invariant manifolds associated with the many families of periodic orbits that exist in simplified dynamical
models like the Circular Restricted Three-Body Problem (CR3BP). Anderson and Lo [
3
], for instance, studied the role
of invariant manifolds in the dynamics of low-thrust trajectories passing through unstable regions of the three-body
problem. Dellnitz et al. [
4
] used the reachable sets concept coupled with invariant manifolds of libration orbits to solve
for a low-thrust Earth-Venus transfer. Vaquero and Howell [
5
] studied the leverage provided by resonant-orbit manifolds
for designing transfers between Earth-Moon libration-point orbits. More recently, Singh et al. [
6
] demonstrated a
methodology to leverage stable/unstable manifold pairs of a halo orbit around the Earth-Moon
𝐿1
for end-to end transfers
in the cis-lunar space. They also extended this work to study the behavior of the invariant manifolds of southern
𝐿2
Near
Rectilinear Halo Orbits (NRHOs) in a multi-body system for a more accurate representation of the manifolds, and using
them as terminal coast arcs for trajectory design [
7
]. Similar approaches to low-thrust trajectory design also appear
in [
8
10
]. Additionally, transfers in the lunar domain especially involving periodic orbits like NRHOs and Distant
Retrograde Orbits (DROs) have also been studied extensively by various researchers [11–13].
While planning a manifold-aided mission to the Moon, the primary step is to pick target patch-points for insertion
which allows a ballistic capture transfer to the target via these invariant space curves. A number of heuristic methods
[
6
,
7
,
14
] for a priori evaluation of the patch points have been studied previously. These approaches primarily use
osculating element-space phase portraits of the whole set of piercing points, defined as the points where the periodic-orbit
manifolds intersect the Earth plane in the Earth-Moon synodic frame. This enables an informed choice by the mission
designer while designing piece-wise optimal trajectories. In addition to simplifying the solution process by splitting the
end-to end trajectory into phases, the enforced terminal coasts lead to fuel savings when compared to a time-optimal
transfer albeit at the expense of flight time. Additionally, the long coast also enables precise navigation and course
corrections. The methodology described above works well but requires trajectory analysis of a dense set of piercing
points in order to find the optimum patch-conditions. This requires repeated solution of ‘N’ two-point boundary value
problems (TPBVPs) and then comparing the fuel costs, where ‘N’ represents the number of discrete points considered
on the periodic orbit. Numerical solution of these TPBVPs is time-consuming, especially for low-thrust type transfers.
A framework to train an input/output model covering an appropriate domain in the cis-lunar space (which can later be
2
used to interpolate the fuel-cost based on the element-space states of the patch points) can be instrumental in solving
trajectory optimization problems in the cis-lunar space.
If the input/output (I/O) model can use a modest number ‘M
<<
N’ of training solutions, this map will be extremely
useful for making preliminary accessibility type analysis of these patch-points and also provide starting iteratives for
an infinity of neighboring trajectories. Gaussian Process Regression (GPR) is one such tool, which can be used for
supervised learning to predict target value(s) given some observations. The approach is recursive and adaptive with
the implicit location and shape of the interpolation functions being adapted through training data. The method also
produces an uncertainty estimate for the predictions which enables intelligent and adaptive input sampling. GPR was
first proposed by Danie G. Krige [
15
] to estimate gold distribution. Since then it has been used in several other fields like
airfoil design [
16
] and aerodynamic coefficient of a spaceplane [
17
]. Some other works demonstrate the effectiveness of
GPR as an I/O training model for regression in a wide variety of applications ranging transmission spectroscopy [
18
],
evolution of micro-structure statistics in super alloys [
19
], probabilistic modelling of wind-turbine power curves [
20
]
and bearing degradation assessment [
21
]. The use of learning methods in the field of astrodynamics is not new either
[
22
,
23
]. Recently Izzo et al. [
24
] used deep neural networks (DNNs) for real time guidance of a spacecraft during
Earth-Venus mass-optimal interplanetary transfer. Sánchez-Sánchez and Izzo [
25
] also used DNNs to study landing
problems. Izzo et al. [
26
] provide a survey of Artificial Intelligence (AI) trends in spacecraft guidance, dynamics
and control. Li et al. [
27
] used neural networks for time-optimal low-thrust interplanetary transfers as well as for
time-optimal orbit raising for Electric Propulsion GEO satellites [
28
]. More recently, Shang and Liu [
29
] used GPR to
assess accessibility of Main-Belt asteroids.
GPR considers inference directly in function space. It is an adaptive, probabilistic regression method and is
computationally efficient while handling I/O relations in regression problems. Due to being based on Bayesian inference,
it implicitly provides an estimate of the mean as well as the covariance of the predicted outputs, a measure that other
learning methods like neural networks fail to provide. Other learning-based approximation methods can provide an
explicitly computed statistical covariance measure of the predictions. However, due to the advantage of statistical
inference with adaptive I/O representation, in this paper, a GPR-based method is chosen to predict
Δ
V costs for
bi-impulse transfers from a geocentric orbit to a set of manifold patch-points (see Figure 1). First, training samples
are generated by discretizing the domain spanning probable location of the piercing points and solving for bi-impulse
transfer trajectories. Next, a multiple-input single-output (MISO) GPR is trained using the generated dataset to estimate
Δ
V for achieving such a transfer with the element-space of the patch-point as inputs. Next, a multi-input multi-output
(MIMO) GPR is trained to predict the departure and arrival velocity vectors on the transfer trajectory and used to
compute
Δ
V. The MIMO GPR takes advantage of correlation in the outputs to produce the final approximation. In
terms of the manifolds, prediction results are presented for three
𝐿1
halo orbits in the Earth-Moon CR3BP with different
Jacobi constants. This paper presents a novel approach, to the best of our knowledge, for estimating accessibility of
3
periodic orbits via manifolds using GPRs.
Fig. 1 Optimal Bi-impulsive Lambert Solution from GEO to a Stable Manifold patch-point.
The remainder of the paper is organized as follows, Section II presents the details of generation of manifolds in the
CR3BP for three candidate Halo orbits around the Earth-Moon
𝐿1
Lagrange point. Section III introduces the generation
of an optimal multiple-revolution, bi-impulse transfer trajectory for a given value of the maximum allowed number of
orbital revolutions
𝑁rev
around the Earth. In Section IV we discuss the GPR and outline some performance metrics
for the trained model. Finally, Section V discusses the numerical results from the trained GPR models. This section
highlights the importance of domain discretization as well as the advantages in terms of prediction accuracy while
utilizing the correlation among outputs through a Multiple-Input Multiple-Output (MIMO) GPR model as compared to
a Multiple-Input Single-Output (MISO) GPR.
II. Manifolds in the CR3BP
The points of equilibrium along the line joining two massive bodies, the so-called Lagrangian points
𝐿1
,
𝐿2
and
𝐿3
, and their local linear stability properties have been known for more than two centuries, since the developments by
Euler and Lagrange. Farquhar [
30
,
31
] did extensive research studying periodic motion near
𝐿1
and
𝐿2
, covering topics
related to halo orbits and their application in the cislunar space and beyond. The equations of motion of a spacecraft in
the CR3BP system can be expressed in a synodic frame using the effective potential,
𝑈
, of the system with the Jacobi
constant (C) being the only conserved quantity.
Halo orbits belong to a class of periodic orbits that bifurcate from the planar Lyapunov orbits in the simplified
CR3BP model. Halo orbits are computed by differential correction of initial conditions obtained from Richardson’s
4
third-order expansion. The reader is referred to [
32
] for more details on periodic orbit computation. The methodology
for computing stable and unstable manifolds associated with a particular halo orbit is based on numerical methods. The
Monodromy matrix,
𝑀(𝑡)
, which is the state transition matrix after one period (P) of the halo orbit, plays an important
role. For any point along the halo orbit, the Monodromy matrix serves as the linearization of the Poincaré map near
the fixed point at time
𝑡
. The characteristics of the local geometry of the phase space can be determined from the
eigenvalues and eigenvectors of 𝑀(𝑡).
The state transition matrix,
Φ(𝑡, 𝑡 P)
, is evaluated along a halo orbit and can be obtained numerically by integrating
the variational equations at the same time that the halo orbit is being computed. Following this, the local eigenvalues
and eigenvectors of
𝑀(𝑡)
can also be computed numerically. Since halo orbits are periodic, their Monodromy matrices
will always have two unity eigenvalues [
33
]. For many halo orbits of interest, the remaining 4 eigenvalues include a real
pair and a complex conjugate pair as
𝜆1>1, 𝜆2=1
𝜆1
, 𝜆3=𝜆4=1, 𝜆5=¯
𝜆6,k𝜆5k=1,(1)
where
𝜆5
and
𝜆6
are complex conjugates, and
𝜆1
and
𝜆2
are real. The pair (
𝜆1, 𝜆2
) are associated with the unstable and
stable directions, (
𝜆3, 𝜆4
) are associated with neutral directions, whereas (
𝜆5, 𝜆6
) are associated with a rotation direction
and related to existence of quasi-periodic orbits around the halo orbit. While there are naturally stable periodic orbits
possible where there is no real pair of eigenvalues, this case is not of interest for our current discussion because the
orbits considered in this work are unstable halo orbits in the Earth-Moon CR3BP in a specific energy regime, where it
has been well established that the orbits are unstable and have real eigenvalue pairs of 𝑀(𝑡).
Building on the work described in [
6
] in order to maintain consistency, the three orbits of interest in the context
of the current work belong to the Halo orbit family around/near the Earth-Moon
𝐿1
. In particular, the converged
initial (𝑿>=[𝒓>,𝒗>]) conditions for the halo orbits considered in this paper are provided in Table 1. Note that DU =
386274
.
56245094
km
and VU = 1
.
028146820425093
km/s
. For a representative state vector
𝑿0R6
on a halo orbit,
Table 1 Initial states for Halo orbits under consideration.
C3.128 3.143 3.158
Orbit ID A B C
𝑥(DU) 0.825590193734960 0.824466089718698 0.823674016784454
𝑦(DU) 0 0 0
𝑧(DU) 0.078139515054458 0.063014128770865 0.044598842157494
𝑣𝑥(VU) 0 0 0
𝑣𝑦(VU) 0.191593557061357 0.174303386773383 0.153934501615348
𝑣𝑧(VU) 0 0 0
let
ˆ
𝒀𝑠(𝑿0)
denote the normalized stable eigenvector and
ˆ
𝒀𝑢(𝑿0)
denote the normalized unstable eigenvector. Let
𝜖
5
(a) 𝐶=3.158, 𝑠 =+1
(b) 𝐶=3.143, 𝑠 =+1
(c) 𝐶=3.128, 𝑠 =+1
Fig. 2 Stable manifolds and associated piercing-points for the candidate Halo Orbits, s = +1.
6
denote the displacement scaling factor along the eigenvectors from
𝑿0
. The magnitude of
𝜖
should be small enough
to lie within the validity of the linear estimate, yet not so small that the time of flight becomes too large due to the
asymptotic nature of the manifolds. These eigenvectors are used to generate approximate manifolds. The displacement
scaling factor, 𝜖 > 0, was chosen to be 106as suggested by Gomez et al. [34]. Then,
𝑿𝑠(𝑿0)=𝑿0+𝑠 𝜖 ˆ
𝒀𝑠(𝑿0),(2)
is the initial state-space for the stable manifold, where
𝑠∈ [−
1
,
1
]
denotes the sense of perturbation. The stable manifold
branches are generated by propagating the initial conditions backwards in time. Figure 2 shows the manifold piercing
points, where the stable manifolds intersect the Earth-plane at
𝑋=𝜇
, (
𝜇=𝑀moon/𝑀Earth
), in the Earth-Moon CR3BP
system with a positive sense (
𝑠=+
1) perturbation. Note that the maximum coast time on the manifold is limited to
81.965 days and the orbit is discretized into
𝑁=
1000 points. The piercing points with negative
𝑌
values are closer to
the Earth as elucidated in the figures above. Therefore, it is heuristically appropriate to only consider this subset of all
piercing points while analysing their accessibility from any geocentric orbit. Now, in order to complete our data-set of
heuristically appropriate piercing points, the other perturbation direction i.e.,
𝑠=
1also need to be analyzed. These
piercing point sets are pictorially represented in Figure 3.
Transformation of the piercing point Cartesian states in the CR3BP synodic frame to the classical element space in
the Earth-Centred Equatorial inertial (ECEI) frame is required to gauge the aptness of these piercing points as patch
states for the departure leg from the geocentric orbit of choice. Figures 5 and 4 depict the distances from geocenter in
km for the piercing-point IDs as well as the inertial inclination (
𝑖
) vs. eccentricity (
𝑒
) phase plots for the candidate Halo
orbits. Note that a heuristic limit has been put on the piercing points such that
𝑖∈ [
15
,
30
]°
and
𝑒∈ [
0
,
0
.
6
]
with respect
to the ECEI frame. As is evident from the figures above, for the Halo Orbit family around
𝐿1
the negative perturbation
(
𝑠=
1) direction in the initial fixed point condition on the periodic orbit leads to a more regular/ less chaotic distribution
of piercing points when compared to the positive perturbation direction (
𝑠=+
1). Another interesting graphical insight
is that the geocentric distance of the piercing points are smaller for
𝑠=
1which make them more suitable for further
investigation as patch-points, more so for low-thrust type transfers. Therefore, in this paper, accessibility analysis will
cover these points for all three candidate Halo orbits.
III. Bi-Impulse Transfer-Trajectory Generation
In order to generate bi-impulsive trajectories from an initial geosynchronous orbit to the family of patch-points
associated with stable manifolds of fixed points on an Earth-Moon
𝐿1
Halo orbit, we use a multi-revolution variant
of the Lambert’s problem [
35
]. Lambert problem is one of the most important two-point boundary value problems
in orbital mechanics and plays a significant role in preliminary design phases of many space missions [
36
39
]. The
7
(a) 𝐶=3.158, 𝑠 =1
(b) 𝐶=3.143, 𝑠 =1
(c) 𝐶=3.128, 𝑠 =1
Fig. 3 Stable manifolds and associated piercing-points for the candidate Halo Orbits, s = -1.
8
0 200 400 600 800 1000
Piercing Point ID
0
2
4
6
8
10
12
Distance from Geocenter (km)
105
C = 3.128
C = 3.143
C = 3.158
(a) Geocentric distance in km for piercing points.
0 0.1 0.2 0.3 0.4 0.5 0.6
e
15
20
25
30
i (deg)
inclination vs. eccentricity, s = +1
C = 3.128
C = 3.143
C = 3.158
(b) i vs. e for piercing points.
Fig. 4 Candidate Halo Orbits, s = +1.
0 200 400 600 800 1000
Piercing Point ID
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Distance from Geocenter (km)
105
C = 3.128
C = 3.143
C = 3.158
X 238
Y 94000.1
(a) Geocentric distance for piercing points.
0.1 0.2 0.3 0.4 0.5 0.6
e
15
20
25
30
i (deg)
inclination vs. eccentricity, s = -1
C = 3.128
C = 3.143
C = 3.158
(b) i vs. e for piercing points.
Fig. 5 Candidate Halo Orbits, s = -1.
9
problem statement is as follows: given two position vectors denoted by
𝒓𝑖
and
𝒓𝑓
, respectively, and a specified time of
flight between the two points, Time of flight (TOF), what are the connecting arcs that connects the two position vectors?
Figure 6 depicts a scenario in which a spacecraft is to depart from point ‘A‘ on orbit 1 and rendezvous at point
‘B’ on orbit 2. Here, both orbits 1 and 2 are assumed to be elliptical orbit. It is also assumed that all motions are
counter-cloclwise. A representative connecting arc is denoted as a sold black line that directly connects points ‘A’ and
’B’, which means that this trajectory makes zero orbital revolution around the planet. However, it is a well-known
theoretical result that there are at most 2
𝑁rev +
1trajectories that qualify for solutions to the Lambert problem [
40
]. The
TOF-
𝑎max
plot in Figure 6 shows all possible (elliptical) trajectories for different number of revolutions. Here, solutions
up to
𝑁rev =
3are shown. There is a unique value for the semi-major axis value below which it is not possible to find
elliptical orbits, which is denoted by
𝑎min
. For trajectories with
𝑁rev =
0, all solutions form two branches, i.e., a lower
branch (with a negative slope) and an upper branch (with a positive slop). However, for
𝑁rev
1, the profile of the
lower branch has a distinct difference compared to the lower branch of the
𝑁rev =
0. More specifically, the lower branch
of trajectories with
𝑁rev
1has a decreasing-increasing slope. As an example, a Lambert problem is considered with
a fixed time of flight corresponding to the horizontal line. This line has three intersections: two with the solutions
corresponding to
𝑁rev =
1and one with the upper branch of solutions with
𝑁rev =
0. This last intersection is the ‘1’
that is considered in the 2
𝑁rev +
1formula. The scales on the TOF-
𝑎min
plot will change for different parameters of
the central body and for different values of
𝒓𝑖
and
𝒓𝑓
, but the overall profile of the curves remains the same. Thus, a
systematic procedure for finding all possible solutions can be followed. The procedure for finding all possible solutions
Fig. 6 Bi-impulsive trajectories using Lambert algorithm and depiction of all possible solutions.
is as follows: Given TOF, the first step is to determine the maximum
𝑁rev
value, which is determined through comparing
the TOF with the lowest time on the lower branch of trajectories with
𝑁rev
1. For instance, if
TOF 𝑡min,Nrev=1
, then,
we need to check if
TOF 𝑡min,Nrev=2
. This search continues until we find the maximum
𝑁rev
value. Then, one starts
10
a systematic search to determine all intersections of the TOF line. Once semi-major axis values associated with all
possible connecting arcs are known, it is possible to determine the initial and final velocity vectors of the connecting
Lambert arcs (i.e., 𝒗𝐿
𝑖and 𝒗𝐿
𝑓). The bi-impulsive minimum-Δ𝑣objective can be defined
minimize Δ𝑣total =||Δ𝒗𝑖| | + | |Δ𝒗𝑓||,(3)
where
Δ𝒗𝑖=𝒗𝐿
𝑖𝒗𝑖
and
Δ𝒗𝑓=𝒗𝐿
𝑓𝒗𝑓
. Here,
𝒗𝑖
and
𝒗𝑓
denote the initial departure orbit and final target orbit
velocity vectors of points ‘A’ and ‘B’, respectively. So far, the selection of Lambert arcs has been considered from a
pure total impulse point of view (i.e.,
Δ𝑣total
). However, there are additional considerations that must be considered
when analyzing all feasible solutions. For instance, it is possible to discard some (or many) of the Lambert arcs from
a geometrical standpoint. More specifically, for trajectories with
𝑁rev =
0, if the initial and final positions are such
that the periapse point of the connecting arc is below a certain limit, the trajectory is already infeasible and can be
discarded. On the other hand, if the trajectory does not pass through periapse, it is a feasible trajectory. For trajectories
with 𝑁rev 1, the analysis of the periapse altitude is a simple test for discarding the arcs.
In this paper, optimal bi-impulse transfers from an initial geostationary orbit to a domain of possible spacecraft states
in the cis-lunar domain were solved using the methodology described above. Note that all possibilities for
𝑁rev
5were
considered and the optimal solution in terms of
Δ
V was included in the database. The classical element set defining the
initial orbit are provided in Table 2.
Table 2 Initial orbit classical elements.
a (km) e i (°)Ω(°)𝜔(°)𝜈(°)
42378 0 0 0 0 0
IV. Gaussian Process Regression
A Gaussian process is defined as a collection of random variables, any finite number of which have a joint
Gaussian distribution [
41
]. It is completely specified by its mean function and the covariance function. For
instance, consider a real process
𝑓(x)
. The mean function
𝑚(x)=E[𝑓(x)]
and the covariance function
𝑘(x,x0)=
E[( 𝑓(x) − 𝑚(x) ( 𝑓(x0) − 𝑚(x0))]. The Gaussian process is then written as 𝑓(x) ∼ GP (𝑚(x), 𝑘 (x,x0)).
Gaussian Process Regression (GPR) is a supervised learning method, which makes use of the marginalization
property of Gaussian Processes. It is primarily a non-parametric, Bayesian approach, which infers a probability
distribution over all possible functions, instead of determining exact parameters of some set of basis function as is the
case with traditional approximation techniques. Random functions are typically drawn from a prior distribution, which
is a zero mean Gaussian distribution with the covariance function evaluated at input points. The basis functions of
11
interest are parameterized not merely by a multiplicative amplitude, but also non-affine parameters embedded in the
argument list. This allows the admissible basis functions to be tailored in a recursive Bayesian approach. The zero
mean assumption does not lead to any loss of generality. There are many valid choices for covariance function, with the
Squared Exponential (SE) and the Rational Quadratic (RQ) functions being the most popular choices covering a vast
majority of applications.
GPR uses Bayes’ rule to incorporate training data to compute a posterior distribution using Likelihood and the
assumed prior distribution. This procedure is known as conditioning the joint Gaussian prior distribution on observations.
Therefore, the joint distribution of the training outputs,
f
, and the testing outputs
f
according to the prior is expressed as
f
f
∼ N 0,
𝐾(𝑋, 𝑋)𝐾(𝑋, 𝑋)
𝐾(𝑋, 𝑋)𝐾(𝑋, 𝑋)
!,(4)
where
N (.)
denotes the normal distribution notation and if there are
𝑛
training points and
𝑛
testing points then
𝐾(𝑋, 𝑋)
denotes the
𝑛×𝑛
matrix of covariances evaluated at all training and test pair of points. The other entries, i.e.,
𝐾(𝑋, 𝑋)
,
𝐾(𝑋, 𝑋)
and
𝐾(𝑋, 𝑋)
are evaluated similarly. Note that while making predictions, we consider the joint distribution
of both the training and prediction data-sets.
In order to generate the posterior distribution over functions, the joint prior distribution above must be restricted
to contain only those functions which agree in a statistical sense with the observed data-points, i.e., incorporate the
information from the training set. The distribution of the fconditioned on f,𝑋and 𝑋is expressed as
f|f, 𝑋, 𝑋∼ N ( 𝐾(𝑋, 𝑋 )𝐾(𝑋 , 𝑋 )1f, 𝐾 (𝑋, 𝑋) − 𝐾(𝑋, 𝑋 )𝐾(𝑋 , 𝑋 )1𝐾(𝑋, 𝑋)).(5)
Thus,
f
can be sampled from this joint posterior distribution. This gives the statistically inferred predicted outputs
at some desired inputs when the prediction inputs are used instead of the testing set.
A. Optimization of Hyperparameters
As mentioned before, a multitude of possible covariance functions exist. These families of functions are typically
characterized by a number of non-affine free hyperparameters, which need to be determined to shape the basis functions
to capture the training data. The determination of an appropriate covariance function followed by computation of the
associated hyperparameters falls under the training of a Gaussian process.
For a noise-free model, the squared exponential function for instance can be parameterized in terms of hyperparameters
as
𝑘(x𝑝,x𝑞)=𝜎2
𝑓exp (− 1
2(x𝑝x𝑞)𝑇𝑄(x𝑝x𝑞)),(6)
12
where,
𝚽=({𝑄}, 𝜎2
𝑓)
is the vector of hyperparameters,
𝜎2
f
regulates the amplitude of the output distribution and
xp
,
xq
are samples of input vectors individually spanning the whole input space. The most common choice for the matrix
𝑄
is
diag(l)2𝐼
where
l
is a vector of positive values and are typically analogous with characteristic length scales. In general,
a numerical optimization methodology is incorporated to determine these set of hyperparameters with the objective that
the likelihood of the training outputs given the regression model is maximized. The log likelihood is defined as
L(𝚽)=log 𝑝(f|𝑋, 𝚽)=1
2f𝑇𝐾1f1
2log |𝐾| − 𝑛
2log 2𝜋. (7)
Note that the log-likelihood is a function of
𝚽
and (
𝑋, f
). Thus, given the training data and a choice of covariance
function, the optimal set of hyperparameters (𝚽) can be computed as
𝚽=arg max𝚽L(𝚽)| 𝑋 , f.(8)
For instance, MATLAB routine fminsearch can be used with an initial guess for the set of hyperparameters to
minimize the negative log likelihood in order to optimize the hyperparameters. It is hereby noted that, the initial guess is
key due to the existence of various local extrema for a multivariable optimization problem. It is therefore recommended
to use a ‘population-based algorithm’ like particle-swarm optimization (PSO) or evolutionary algorithms in conjunction
with fminsearch to find the global optimum in a specified domain. The results presented in the paper are a result of a
hybrid global-local algorithm, ‘PSO-fminsearch’ optimization algorithm.
The choice of covariance function for the problems considered in this paper is a modified RQ function. The classical
RQ function is stationary, i.e., depends solely on distances between samples in
𝐷𝑑𝑖𝑚
Euclidean space. In order to
cover the non-stationarity due to the different orbital elements, which serve as inputs to the problems discussed later,
an automatic relevance determination (ARD) distance measure is integrated into the covariance function and is now
expressed as,
𝑘(x𝑝,x𝑞)=𝜎2
𝑓exp 1+(x𝑝x𝑞)𝑇Q (x𝑝x𝑞)
2𝛼!𝛼
,(9)
where
𝜎2
𝑓
is analogous to an amplitude measure of the output distribution,
𝛼
represents the shape of the signal and
Q
is
a symmetric matrix of the characteristic length scale in the different input dimensions and determines the relevance of a
particular input to the covariance function and is expressed for a 𝐷dim input system as,
Q=diag 1
𝑙2
1
,1
𝑙2
2
,1
𝑙2
3
, ....., 1
𝑙2
𝐷!and 𝚽=[𝑙2
1, 𝑙2
2, 𝑙2
3, ...., 𝑙 2
𝐷, 𝜎2
𝑓, 𝛼].(10)
13
B. Multiple-Input Single-Output (MISO) GPR
The primary parameter associated with quantification of accessibility of the manifold patch-points is the total
impulsive
Δ
V required to depart the initial geocentric orbit on the transfer trajectory as well as that required at the time
of arrival to the final target state. As discussed in the preceding sections, the optimal bi-impulse Lambert solution was
used to generate the data-set used for training the regression model.
The first data-set defines the mapping function associating the optimal bi-impulse velocity increment to the orbital
elements associated with the patch-points. This can be mathematically expressed as,
𝑓:R6R,
XD=[𝑎, 𝑒 , 𝑖, Ω, 𝜔, 𝜈]Δ𝑉 .
(11)
The GPR model corresponding to the mapping defined above takes multiple inputs and provides a single output as
its prediction. Hence, it is termed as a MISO GPR in all future references to this method of determining the accessibility
metric.
C. Multiple-Input Multiple-Output (MIMO) GPR
A higher-fidelity prediction of the departure and arrival impulsive velocity vectors is more pertinent for mission
designers as it would nullify the need to re-solve the whole trajectory and characterize the transfer trajectory completely.
In addition, the accessibility metric, i.e.,
Δ
V would also be available as a byproduct. In order to train a regression model,
which is able to predict all components of Vdep and Varr, the mapping function can be expressed as
𝑓:R6R6,
XD=[rPP,vPP ]→ [Vdep,Varr],
(12)
where,
[rPP,vPP ]
denote the position and velocity vectors associated with the patch-point. This GPR model is termed as
a MIMO GPR and referred to as such henceforth. In addition to providing higher-fidelity predictions, another advantage
of using a MIMO GPR is using the correlations among the outputs for training purposes. As discussed before, the
covariance functions are solely dependent on the inputs. On the other hand, in case the desired prediction include
multiple outputs, the knowledge of correlation between the outputs is lost while using multiple MISO GPR models.
The same scenario occurs, albeit implicitly, when the desired ‘single’ output is a composite function of the multiple
individual outputs, as is the case discussed above. Therefore, using a MIMO GPR instead of multiple MISO GPRs is
expected to provide more accurate predictions as shown in [42, 43].
14
V. Domain Discretization Approach
From the discussion in the preceding sections, a domain is defined for distributing target states for computing the
bi-impulse Lambert solutions from the selected initial orbit and populate the database. The maximum number of
revolutions for candidate Lambert solutions is (
Nrev =
5). The spacecraft departs from a geosynchronous orbit (GEO)
initially en-route to a manifold patch points using a departure and arrival impulse. Table 3 shows the range of values
considered as target states,
Table 3 Range of Classical Elements for defining the domain of patch states.
Element Minimum Maximum
𝑎(km) 90000 220000
𝑒0 0.6
𝑖(deg) 15 30
Ω(deg) 0 360
𝜔(deg) 0 360
𝜈(deg) 0 360
Note that only the cases where the 90000
km <||R| | <
220000
km
are included in the database and the other target
states falling outside the prescribed range for geocentric distance are discarded for consistency with the domain that the
manifold patch-points span. Here,
DE =
6378
km
and
VE =q𝜇𝐸
DE km/s
are the distance and velocity values used for
scaling purposes. The metrics for characterizing the accuracy of the predictions using a trained GPR model is primarily
the Mean Absolute Error (MAE) defined as,
MAE =1
𝑁𝑡
𝑛
Õ
𝑖=1
|Δ𝑉𝐿,𝑖 Δ𝑉GPR,𝑖 |,(13)
where
𝑁𝑡
denotes the number of testing points,
Δ𝑉𝐿,𝑖
is the true output of the
𝑖th
testing point and
Δ𝑉GPR,𝑖
is the
prediction of the output of the
𝑖th
testing point. While MAE is a standard measure of accuracy for trained GPR models,
it is not a robust measure of prediction accuracy, especially for this application. Any testing point, which lies on or near
the boundary of the full data-set is likely to skew the results and exaggerate the numerical value of the MAE. Therefore,
in this paper the MAE is coupled with Relative Error Percentage (
RE
%) as a secondary metric, which quantifies the
number of testing points within an acceptable bounds around the truth. RE%is defined as
RE%𝑖=|Δ𝑉𝐿,𝑖 Δ𝑉GPR,𝑖 |
Δ𝑉𝐿,𝑖
×100,(14)
and captures the local prediction behavior of the discrete testing/prediction points rather than quantifying a mean
measure. A GPR model was trained using uniformly distributed points in the entire domain described in Table 3. The
model used 80% of the data-set for training while the other 20% were used for testing the trained model. The domain
15
is depicted in the Figure 7. The accuracy metrics for the trained full domain GPR model is: MAE = 0.0373 VE and
around 41
.
8% of the predictions had
RE
%
<
25%. The results have been plotted in Figure 8 along with the predicted
±3𝜎bounds.
Fig. 7 Full domain of piercing points (RIGHT) and bi-impulse transfers from GEO (LEFT).
Fig. 8 Testing predictions for the GPR model trained on the full domain.
While the results are quite encouraging in terms of prediction accuracy, the obvious question that arises is - Will
discretization of the domain and training individual GPR models for each sub-domain lead to an improved accuracy?
Heuristically, smaller sub-domains, which include highly-correlated transfer trajectories should lead to a better trained
16
GPR model. For this paper, the full domain is divided into sub-domains with discretization in three dimensions -
semi-major axis (𝑎), right ascension (Ω), and argument of periapse (𝜔).
The discretization was performed uniformly across the three dimensions mentioned above while the other classical
elements namely, eccentricity (
𝑒
), inclination (
𝑖
) and true anomaly (
𝜈
) were picked from the full domain on the respective
axis. If “
𝑛grid
” represents the number of grid points along a single axes, the total number of sub-domains are
(𝑛grid
1
)3
with a GPR model to be trained in each sub-domain. Figure 9 depicts the meshed
{𝑎, Ω, 𝜔}
domain for
𝑛grid =
4with
the {1,1,1}grid highlighted.
Fig. 9 Meshed {𝑎, Ω, 𝜔 }sub-domain.
It is immediately apparent that the domain cannot be sub-divided indefinitely as it will add to the training time to a
point when the improved accuracy in predictions will lead to diminishing returns as a whole. The reader is hereby
reminded of the fact that the accessibility analysis aims at only providing a preliminary quantification of
Δ
V to make the
trip to the patch-point and gauge the apt merging conditions on the stable manifold families. Figure 10 depicts the
{1,1,1}grid specifically for different values of 𝑛grid.
Table 4 shows the improvement in MAE as well as an increased number of predictions on the test-data having
RE%
25% achieved by considering a larger value of
𝑛grid
, i.e., finer meshing of the domain. The resolution considered
for a mesh is a choice of the mission designer and the methodology works for any level of mesh finesse with the choice
being at the user’s discretion. Figure 11 shows the mean prediction data along with the
±
3
𝜎
bounds along with the
discrete prediction errors with respect to the truth.
It is extremely cumbersome to visualize the complexity of the problem at hand. The trend plots, for instance, in
Figures 8 and 11 have piercing point ID on the abscissa which is not the independent parameter of interest and these plots
do not entirely capture the dependence of the output on the multiple independent parameters e.g., (IP =
{𝑎, 𝑒 , 𝑖, Ω, 𝜔, 𝜈 }
)
in the case of a MISO regression model. For a more representative graphical portrayal, an ingenious way is utilised
17
(a) Grid = {1,1,1}, 𝑛grid =3
(b) Grid = {1,1,1}, 𝑛grid =9
Fig. 10 Discrete domains on the {a,Ω,𝜔} for different 𝑛grid.
Table 4 A comparison of trained GPR model accuracy with increasing 𝑛grid.
ngrid RE%25% MAE (VE) Mesh#
3 75.0%0.0190 8
9 77.3%0.0171 512
21 88.0%0.0108 8000
51 95.1%0.0046 125000
18
Fig. 11 Prediction errors and uncertainty bounds for the sub-domain: {1,1,1}grid, 𝑛grid =21.
where initially we reduce the input space dimension to
R2
by only considering a subset of IP :
{Ω, 𝜔}
. The next step is
to fix the remaining elements in IP at some fixed value and generate a mesh-grid (
𝜒×𝜒
) in the
R2
space in appropriate
bounds. Finally, solve the Lambert Problem and extract the optimal solution for spacecraft transfer from the specified
initial orbit to the generated input-vectors (
𝜒2
). Now, consider
{𝑎, Ω, 𝜔}
. This subset has one distance element (‘
𝑎
’)
and two angle elements (‘
Ω
&
𝜔
’), which is analogous to spherical coordinates. Therefore, the mesh-grid considered
earlier with a fixed value of 𝑎are basically uniformly sampled points on a sphere of radius 𝑎. In order to visualize the
dependent output, which in this case is the optimal
Δ
V, the perturbation in the semi-major axis corresponding to
Δ
V
magnitude at each sample point (with appropriate normalization) is computed in accordance with the perturbation
vis-viva equation, assuming fixed values of {𝑒, 𝜈 }as shown below.
Δ𝑉2=𝜇 2(1+𝑒cos 𝜈)
Δ𝑎(1𝑒2)1
Δ𝑎!.(15)
The portrayal of perturbed surface of the uniform sphere contains topographical information commensurate with the
respective optimal
Δ
V and is more illuminating in terms of problem complexity even in a significantly lower dimensional
space. As an example, Figure 12 depicts the initial and deformed spheres for a (50
×
50) mesh-grid in
{Ω, 𝜔}
where
{𝑎=90000 km, 𝑒 =0, 𝑖 =15°, 𝜈 =0°}.
19
Fig. 12 ΔV encoded in spherical topography for visualisation of problem complexity.
VI. Numerical Results and Discussion
A. Manifold Patch-Points in the ECEI frame
In order to make predictions and quantify accessibility, the piercing points from the Earth-Moon synodic frame
need to be converted to classical elements in the ECEI frame. Here, the Ecliptic is defined as the plane containing the
Earth and Moon, i.e., the synodic frame datum. This is carried out by assuming that the inertial frame coincides with
the Earth-Moon synodic frame at the time of insertion into the periodic orbit at a particular fixed point, whereas the
assumed constant rotation rate of the Earth-Moon synodic frame defines the current orientation of the ECEcI frame.
Finally, conversion to the ECEI frame is achieved by a rotation about the ECEI
𝑥
-axis by the angle between the Ecliptic
plane and the Earth’s equatorial plane.
It is also noted that in reality, the angle between the Moon’s orbit, i.e., our Ecliptic plane and the Earth’s equator
varies between a maximum of (28
°
36
0
) and a minimum of (18
°
20
0
) on a 18.6 year cycle corresponding to the major and
minor lunar standstill respectively [
44
]. The last lunar standstill was a minor standstill in October 2015. Assuming, a
linear variation, the angle between the Ecliptic and Earth’s orbit in 2025 is estimated to be
5
.
462
°
. This is a crude
approximation but since we are dealing with a simplified dynamical system (CR3BP), this estimate does not reduce the
fidelity of the results.
B. Comparison of Predictions and Truth
Using the state-space representation of the set of patch-points associated with candidate halo orbits, and the trained
GPR models, predictions are made using both MISO and MIMO GPR models for
Δ
V. We emphasize that
Δ
V is a
20
post-processing product of the MIMO GPR model predictions, whereas for the MISO GPR model it is obtained as
a direct prediction. Figure 13 depicts the overlayed polar phase-plots (
𝑒vs. {𝑖, 𝜔, Ω, 𝜈}
) for the patch-points of the
candidate Halo Orbits.
0
30
60
90
120
150
180
210
240
270
300
330
0
0.2
0.4
0.6
e vs. (i, , , )
C = 3.158, i
C = 3.158,
C = 3.158,
C = 3.158,
C = 3.143, i
C = 3.143,
C = 3.143,
C = 3.143,
C = 3.128, i
C = 3.128,
C = 3.128,
C = 3.128,
-15
0
15
30
0
0.2
0.4
0.6
e vs. i
e vs.
e vs.
e vs.
Fig. 13 e vs. {𝑖, 𝜔, Ω, 𝜈}in classical element space for piercing points: ECEI frame.
The phase plot is extremely illuminating for specific applications of supervised learning. The advantages in terms of
higher accuracy achieved by discretizing a larger domain in
𝑛
dimensions into smaller sub-domains has been discussed
in the preceding sections. Figure 13 enables us to shrink the initial larger domain at the onset. Following this up
by further sub-divisions, the accuracy of predictions in individual sub-domains is expected to improve even more as
compared to the trained GPR models described in Section V. The shrunk domain is described in Table 5. Considering
the data-points in the entire “shrunk” domain for training-testing, the trained single GPR model exhibits better prediction
accuracy than before. Figure 14 shows the mean prediction and uncertainty bounds around them along with the domain
in position sub-space. MAE for predictions was 1.373 ×103VE whereas 96.2% of the predictions had RE%5%.
Table 5 Range of Classical Elements for defining the “shrunk” domain.
Element Minimum Maximum
𝑎(km) 90000 220000
𝑒0 0.6
𝑖(deg) 15 30
Ω(deg) 150 210
𝜔(deg) 60 120
𝜈(deg) 340 30
21
(a) Trajectories and the “Shrunk” Domain in Position Sub-Space.
(b) Prediction errors and uncertainty bounds.
Fig. 14 Pictorial representation of the “shrunk” domain and GPR prediction performance.
22
The performance metrics even with a single GPR model across the “shrunk” domain shows immense promise.
The improvement due to discretization in MAE is evident when we compare these results with a GPR trained on
the grid
{
1
,
1
,
1
}
with
ngrid =
3(
MAE =
8
.
2127
×
10
5VE
). It is hereby reiterated that increasing
𝑛grid
is strongly
affected by the trade-off (accuracy vs. computation time/memory) as mentioned before. Table 6 shows the converged
hyperparameters for the MISO GPR model trained on the “shrunk” domain for making predictions for the Halo orbit
patch-points generated by stable manifolds (𝑠=1) piercing the Earth-plane in the Earth-Moon synodic frame.
Table 6 Converged Hyperparameters - MISO GPR Model “shrunk” domain.
Hyperparam. log(𝜎𝑓) log(𝑙𝑎) log(𝑙𝑒) log(𝑙𝑖) log(𝑙𝜔) log(𝑙Ω) log(𝑙𝜈) log(𝛼)
Conv. Value -8.8346 0.3655 0.4649 -0.5270 -0.4955 -0.8716 -0.8863 18.8801
Figure 19a shows the predicted and true values of the optimal bi-impulse
Δ
V required for a spacecraft to merge onto
the family of
𝐶=
3
.
128 stable manifold patch-points. Figure 19f depicts the residuals and the uncertainty in terms of
±3𝜎bounds.
0 100 200 300 400 500 600 700 800 900 1000
Patch-Point ID
0.34
0.36
0.38
0.4
0.42
0.44
0.46
0.48
0.5
0.52
V (VE)
Predictions - Stable Manifold Patch-Points (C = 3.128)
Real value
Predicted value
(a) True and Predicted ΔV.
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
Residual, 3 bounds
Residuals and Uncertianty Bounds - Predictions (C = 3.128)
0 100 200 300 400 500 600 700 800 900 1000
Patch-Point ID
Residual
+ 3
- 3
(b) Residual and Uncertainty in ΔV predictions.
Fig. 15 MISO GPR Predictions for ΔV for Halo Orbit ‘A’, s = -1
The converged minimum of the ‘negative’ log likelihood was -6.0409 for the trained GPR. The MAE for the
predictions on the 1000 discrete patch points was 3
.
1
×
10
3
VE with 91
.
4% of the predictions having a
RE
%
2% and
all predictions having a
RE
%
5%. Predictions were also made using the same GPR for patch-points associated with
the Halo Orbits ‘B’ and ‘C’. The following plots (Figure 16 and 17) depict the predictions and the respective uncertainty
associated with these predictions.
Note that for predictions associated with Halo Orbit ‘B’, the MAE was 4
.
640
×
10
3
VE with 97
.
2% of the predictions
having a
RE
%
2% and all predictions having a
RE
%
5%. On the other hand for Halo Orbit ‘C’ predictions, MAE
was 5.2439 ×103VE with all of the predictions having a RE%2%.
23
0 100 200 300 400 500 600 700 800 900 1000
Patch-Point ID
0.4
0.42
0.44
0.46
0.48
0.5
0.52
0.54
0.56
0.58
V (VE)
Predictions - Stable Manifold Patch-Points (C = 3.143)
Real value
Predicted value
(a) True and Predicted ΔV.
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
Residual, 3 bounds
Residuals and Uncertainty Bounds - Predictions (C = 3.143)
0 100 200 300 400 500 600 700 800 900 1000
Patch-Point ID
Residual
+ 3
- 3
(b) Residual and Uncertainty in ΔV predictions.
Fig. 16 MISO GPR Predictions for ΔV for Halo Orbit ‘B’, s = -1.
0 100 200 300 400 500 600 700 800 900 1000
Patch-Point ID
0.48
0.5
0.52
0.54
0.56
0.58
0.6
0.62
0.64
V (VE)
Predictions - Stable Manifold Patch-Points (C = 3.158)
Real value
Predicted value
(a) True and Predicted ΔV.
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
Residual, 3 bounds
Residuals and Uncertainty Bounds - Predictions (C = 3.158)
0 100 200 300 400 500 600 700 800 900 1000
Patch-Point ID
Residual
+ 3
- 3
(b) Residual and Uncertainty in ΔV predictions.
Fig. 17 MISO GPR Predictions for ΔV for Halo Orbit ‘C’, s = -1.
24
C. Departure and Arrival Predictions: MIMO GPR
A MIMO GPR based on the same training and testing points with inputs in Cartesian position and velocity space
instead of classical elements space, was trained in order to make predictions for the full velocity vectors at departure
from the initial orbit and arrival at the terminal orbit. Table 7 shows the converged hyperparameters for the MIMO
GPR model trained on the “shrunk” domain. Note that the patch-points are still the Earth-plane piercing points of
stable-manifolds perturbed in the 𝑠=1direction.
Table 7 Converged Hyperparameters - MIMO GPR Model “shrunk” domain.
Hyperparam. log(𝜎𝑓) log(𝑙𝑎) log(𝑙𝑒) log(𝑙𝑖) log(𝑙𝜔) log(𝑙Ω) log(𝑙𝜈) log(𝛼)
Conv. Value -8.5349 -8.5589 -8.6806 1.3017 1.8040 1.4879 -0.7649 -2.2141
The converged minimum of the ‘negative’ log likelihood was -5.7393 for the trained GPR. Figure 18 depicts the
predictions of
Vdep
components on the optimal bi-impulse transfer trajectory whereas Figure 19 depicts the components
of
Varr
for patch-points associated with stable manifolds of Halo Orbit ‘B’. Table 8 summarizes the MIMO prediction
results for all candidate Halo Orbits. As is evident from the table, the GPR model predicts the individual components to
a high degree of accuracy. The y-component of
Varr
is the worst prediction in terms of
MAE
as well as
RE
%made by
the GPR for all three candidate orbits.
Table 8 MIMO predictions - All Halo Orbits {MAE (VE),RE%2%}.
Orbit ID 𝑉dep𝑥𝑉dep𝑦𝑉dep𝑧𝑉arr𝑥𝑉arr𝑦𝑉arr𝑧
A1.5×103
100%1.4×103
100%7.7×104
100%9.4×104
92%2.8×103
25.2%1.4×103
100%
B9.2×104
100%8.6×104
100%5.1×104
100%8.8×104
86.6%2.1×103
45.4%1.1×103
100%
C7.9×104
100%9.0×104
100%7.1×104
100%1.3×103
60.5%1.7×103
63.2%1.1×103
100%
The advantage in terms of computational efficiency is elucidated by the fact that, on a Intel Core i7-10700 CPU
2.90GHz with a 32.0 GB RAM, computation of the optimal bi-impulsive maneuver to the 1000 manifold patch-points
associated with the Halo Orbit ‘A’ using Lambert’s solver for maximum
𝑁rev =
5required a run-time of 160.8 seconds.
On the other hand, GPR training which is a one-time operation required 13.4 seconds for the MISO GPR and 219.6
seconds for the MIMO GPR. After training is complete, the predictions at the same 1000 manifold patch-points required
only 0.06 seconds for MISO predictions and 0.31 seconds for MIMO seconds.
The results indicate that characterizing accessibility to manifold patch-points for a single periodic orbit by itself is
much faster by using a trained GPR model instead of repeatedly solving the Lambert problem and extracting the optimal
solution, but this disadvantage in terms of computational time is further exacerbated in-case the analysis needs to cover
multiple periodic orbits and their associated manifold patch-points in the Earth-Moon domain.
25
0 100 200 300 400 500 600 700 800 900 1000
Patch-Point ID
-0.3
-0.28
-0.26
-0.24
-0.22
-0.2
-0.18
-0.16
Vdepx
(VE)
Real value
Predicted value
(a) True and Predicted 𝑉dep𝑥.
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
Residual (Vdepx
) 3 bounds
0 100 200 300 400 500 600 700 800 900 1000
Patch-Point ID
(b) Residual and Uncertainty in 𝑉dep𝑥predictions.
0 100 200 300 400 500 600 700 800 900 1000
Patch-Point ID
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
Vdepy
(VE)
Real value
Predicted value
(c) Residual and Uncertainty in 𝑉dep𝑦predictions.
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
Residual (Vdepy
) 3 bounds
0 100 200 300 400 500 600 700 800 900 1000
Patch-Point ID
(d) Residual and Uncertainty in 𝑉dep𝑦predictions.
0 100 200 300 400 500 600 700 800 900 1000
Patch-Point ID
-0.2
-0.19
-0.18
-0.17
-0.16
-0.15
-0.14
-0.13
Vdepz
(VE)
Real value
Predicted value
(e) Residual and Uncertainty in 𝑉dep𝑧predictions.
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
Residual (Vdepz
) 3 bounds
0 100 200 300 400 500 600 700 800 900 1000
Patch-Point ID
(f) Residual and Uncertainty in 𝑉dep𝑧predictions.
Fig. 18 MIMO GPR Predictions for Vdep for Halo Orbit ‘A’, s = -1.
26
0 100 200 300 400 500 600 700 800 900 1000
Patch-Point ID
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
Varrx
(VE)
Real value
Predicted value
(a) True and Predicted 𝑉arr𝑥.
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
Residual (Varrx
) 3 bounds
0 100 200 300 400 500 600 700 800 900 1000
Patch-Point ID
(b) Residual and Uncertainty in 𝑉arr𝑥predictions.
0 100 200 300 400 500 600 700 800 900 1000
Patch-Point ID
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Varry
(VE)
Real value
Predicted value
(c) Residual and Uncertainty in 𝑉arr𝑦predictions.
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
Residual (Varry
) 3 bounds
0 100 200 300 400 500 600 700 800 900 1000
Patch-Point ID
(d) Residual and Uncertainty in 𝑉arr𝑦predictions.
0 100 200 300 400 500 600 700 800 900 1000
Patch-Point ID
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
Varrz
(VE)
Real value
Predicted value
(e) Residual and Uncertainty in 𝑉arr𝑧predictions.
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
Residual (Varrz
) 3 bounds
0 100 200 300 400 500 600 700 800 900 1000
Patch-Point ID
(f) Residual and Uncertainty in 𝑉arr𝑧predictions.
Fig. 19 MIMO GPR Predictions for Varr for Halo Orbit ‘A’, s = -1
27
VII. Conclusion
A supervised learning methodology based on Gaussian Process Regression is presented to assess accessibility of
stable manifold patch-points associated with three candidate Halo orbits around the Earth-Moon
𝐿1
. A priori analysis of
the patch-point position and velocity states converted to classical elements in an Earth-centred inertial frame is shown to
be illuminating in terms of the domain for training of the regression model. The effect on prediction accuracy of domain
discretization is elucidated by considering a mesh structure of the entire domain and splitting into sub-domains based on
grid points.
Two separate regression models, i.e., multi-input single-output and multi-input multi-output were trained and
used to make predictions of optimal bi-impulsive
Δ
V requirements and the complete departure and arrival velocity
vectors respectively. The results show remarkable prediction accuracy with the worst prediction mean absolute error
being 1
.
1
×
10
3
VE (VE = 7.904 km/s). The trained regression model was effectively used to predict accessibility
to the patch-points and can be utilised as a preliminary mission-design tool. The major advantage of using Gaussian
Process Regression is in terms of computation time when evaluating accessibility to points in a particular domain on
which the model was trained. In particular, for evaluating the accessibility for all patch points in our domain using
Lambert’s solver, the run-time is
500 seconds, whereas using a multi-input multi-output regression model the run time
is
219
.
6
seconds (training) +
0
.
93
seconds (predictions)
. Obviously, the computational disadvantage of repeated
solutions of the Lambert Problem is further exacerbated when accessibility of a larger number of patch-points belonging
to different periodic orbit families is sought. Due to this efficiency of the learned map, this approach could also aid near
real-time applications like re-planning/re-targeting of an active spacecraft.
Declarations
Funding
We are pleased to acknowledge the Air Force Research Laboratory, Dzyne, Inc., and Texas A&M University for
sponsorship of various aspects of this research.
Conflicts of Interests/Competing Interests
The authors have no conflicts of interest to disclose.
Ethics Approval
Not Applicable
Consent to Participate
Not Applicable
28
Consent for Publication
Not Applicable
Availability of Data and Material
Not applicable
Code Availability
All readers interested in the code are invited to contact the corresponding author.
Authors’ Contribution
SKS - Conceptualization, Coding, Manuscript, JLJ - Conceptualization, Manuscript, MM - Conceptualization and
ET - Coding, Manuscript
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