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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

𝑿0∈R6

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 10−6as 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

𝑖−𝒗𝑖

and

Δ𝒗𝑓=𝒗𝐿

𝑓−𝒗𝑓

. 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 f∗conditioned on f,𝑋and 𝑋∗is 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(x𝑝−x𝑞)𝑇𝑄(x𝑝−x𝑞)),(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

2f𝑇𝐾−1f−1

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

𝐷−𝑑𝑖𝑚

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+(x𝑝−x𝑞)𝑇Q (x𝑝−x𝑞)

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 𝐷−dim 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:R6−→ R,

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:R6−→ R6,

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(1−𝑒2)−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 ×10−3VE 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 ×10−3VE 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

A1.5×10−3

100%1.4×10−3

100%7.7×10−4

100%9.4×10−4

92%2.8×10−3

25.2%1.4×10−3

100%

B9.2×10−4

100%8.6×10−4

100%5.1×10−4

100%8.8×10−4

86.6%2.1×10−3

45.4%1.1×10−3

100%

C7.9×10−4

100%9.0×10−4

100%7.1×10−4

100%1.3×10−3

60.5%1.7×10−3

63.2%1.1×10−3

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

References

[1]

Bobskill, M. R., and Lupisella, M. L., “The role of cis-lunar space in future global space exploration,” Proc. Global Space

Exploration Conf, 2012.

[2]

Crusan, J. C., Smith, R. M., Craig, D. A., Caram, J. M., Guidi, J., Gates, M., Krezel, J. M., and Herrmann, N. B., “Deep space

gateway concept: Extending human presence into cislunar space,” 2018 IEEE Aerospace Conference, IEEE, 2018, pp. 1–10.

[3]

Anderson, R. L., and Lo, M. W., “Role of invariant manifolds in low-thrust trajectory design,” Journal of guidance, control, and

dynamics, Vol. 32, No. 6, 2009, pp. 1921–1930. URL https://doi.org/10.2514/1.37516.

[4]

Dellnitz, M., Junge, O., Post, M., and Thiere, B., “On target for Venus–set oriented computation of energy eﬃcient

low thrust trajectories,” Celestial Mechanics and Dynamical Astronomy, Vol. 95, No. 1-4, 2006, pp. 357–370. URL

https://doi.org/10.1007/s10569-006- 9008-y.

[5]

Vaquero, M., and Howell, K. C., “Leveraging resonant-orbit manifolds to design transfers between libration-point orbits,” Journal

of Guidance, Control, and Dynamics, Vol. 37, No. 4, 2014, pp. 1143–1157. URL https://doi.org/10.2514/1.62230.

[6]

Singh, S. K., Anderson, B. D., Taheri, E., and Junkins, J. L., “Exploiting manifolds of L1 halo orbits for end-to-end Earth–Moon

low-thrust trajectory design,” Acta Astronautica, Vol. 183, 2021, pp. 255–272.

[7]

Singh, S. K., Anderson, B. D., Taheri, E., and Junkins, J. L., “Low-Thrust Transfers to Southern

𝐿2

Near-Rectilinear Halo

Orbits Facilitated by Invariant Manifolds,” Journal of Optimization Theory and Applications, 2021, pp. 1–28.

[8]

Qu, Q., Xu, M., and Peng, K., “The cislunar low-thrust trajectories via the libration point,” Astrophysics and Space Science, Vol.

362, No. 5, 2017, p. 96.

[9]

Cox, A. D., Howell, K. C., and Folta, D. C., “Trajectory design leveraging low-thrust, multi-body equilibria and their manifolds,”

The Journal of the Astronautical Sciences, Vol. 67, No. 3, 2020, pp. 977–1001.

29

[10]

Topputo, F., Vasile, M., and Bernelli-Zazzera, F., “Low energy interplanetary transfers exploiting invariant manifolds of the

restricted three-body problem,” Journal of the Astronautical Sciences, Vol. 53, No. 4, 2005, pp. 353–372.

[11]

Capdevila, L. R., and Howell, K. C., “A transfer network linking Earth, Moon, and the triangular libration point regions in the

Earth-Moon system,” Advances in Space Research, Vol. 62, No. 7, 2018, pp. 1826–1852.

[12]

Zhang, R., Wang, Y., Zhang, H., and Zhang, C., “Transfers from distant retrograde orbits to low lunar orbits,” Celestial

Mechanics and Dynamical Astronomy, Vol. 132, No. 8, 2020, pp. 1–30.

[13]

Oshima, K., “The use of vertical instability of

𝐿_

1and

𝐿_

2planar Lyapunov orbits for transfers from near rectilinear halo

orbits to planar distant retrograde orbits in the Earth–Moon system,” Celestial Mechanics and Dynamical Astronomy, Vol. 131,

No. 3, 2019, pp. 1–28.

[14]

Singh, S., Junkins, J., Anderson, B., and Taheri, E., “Eclipse-conscious transfer to lunar gateway using ephemeris-driven

terminal coast arcs,” Journal of Guidance, Control, and Dynamics, 2021, pp. 1–17.

[15]

Krige, D. G., “A statistical approach to some basic mine valuation problems on the Witwatersrand,” Journal of the Southern

African Institute of Mining and Metallurgy, Vol. 52, No. 6, 1951, pp. 119–139.

[16]

Liu, X., Zhu, Q., and Lu, H., “Modeling multiresponse surfaces for airfoil design with multiple-output-Gaussian-process

regression,” Journal of Aircraft, Vol. 51, No. 3, 2014, pp. 740–747.

[17]

Dufour, R., de Muelenaere, J., and Elham, A., “Trajectory driven multidisciplinary design optimization of a sub-orbital

spaceplane using non-stationary Gaussian process,” Structural and Multidisciplinary Optimization, Vol. 52, No. 4, 2015, pp.

755–771.

[18]

Gibson, N., Aigrain, S., Roberts, S., Evans, T., Osborne, M., and Pont, F., “A Gaussian process framework for modelling

instrumental systematics: application to transmission spectroscopy,” Monthly notices of the royal astronomical society, Vol.

419, No. 3, 2012, pp. 2683–2694.

[19]

Yabansu, Y. C., Iskakov, A., Kapustina, A., Rajagopalan, S., and Kalidindi, S. R., “Application of Gaussian process regression

models for capturing the evolution of microstructure statistics in aging of nickel-based superalloys,” Acta Materialia, Vol. 178,

2019, pp. 45–58.

[20]

Rogers, T., Gardner, P., Dervilis, N., Worden, K., Maguire, A., Papatheou, E., and Cross, E., “Probabilistic modelling of wind

turbine power curves with application of heteroscedastic Gaussian Process regression,” Renewable Energy, Vol. 148, 2020, pp.

1124–1136.

[21]

Hong, S., and Zhou, Z., “Application of Gaussian process regression for bearing degradation assessment,” 2012 6th International

Conference on New Trends in Information Science, Service Science and Data Mining (ISSDM2012), IEEE, 2012, pp. 644–648.

[22]

Dachwald, B., “Low-thrust trajectory optimization and interplanetary mission analysis using evolutionary neurocontrol,”

Doktorarbeit, Institut für Raumfahrttechnik, Universität der Bundeswehr, München, 2004.

30

[23]

Carnelli, I., Dachwald, B., and Vasile, M., “Evolutionary neurocontrol: A novel method for low-thrust gravity-assist trajectory

optimization,” Journal of guidance, control, and dynamics, Vol. 32, No. 2, 2009, pp. 616–625.

[24]

Izzo, D., and Öztürk, E., “Real-Time Guidance for Low-Thrust Transfers Using Deep Neural Networks,” Journal of Guidance,

Control, and Dynamics, Vol. 44, No. 2, 2021, pp. 315–327.

[25]

Sánchez-Sánchez, C., and Izzo, D., “Real-time optimal control via deep neural networks: study on landing problems,” Journal

of Guidance, Control, and Dynamics, Vol. 41, No. 5, 2018, pp. 1122–1135.

[26]

Izzo, D., Märtens, M., and Pan, B., “A survey on artiﬁcial intelligence trends in spacecraft guidance dynamics and control,”

Astrodynamics, Vol. 3, No. 4, 2019, pp. 287–299.

[27]

Li, H., Baoyin, H., and Topputo, F., “Neural networks in time-optimal low-thrust interplanetary transfers,” IEEE Access, Vol. 7,

2019, pp. 156413–156419.

[28]

Li, H., Topputo, F., and Baoyin, H., “Autonomous Time-Optimal Many-Revolution Orbit Raising for Electric Propulsion GEO

Satellites via Neural Networks,” arXiv preprint arXiv:1909.08768, 2019.

[29]

Shang, H., and Liu, Y., “Assessing accessibility of main-belt asteroids based on Gaussian process regression,” Journal of

Guidance, Control, and Dynamics, Vol. 40, No. 5, 2017, pp. 1144–1154.

[30]

Farquhar, R. W., and Kamel, A. A., “Quasi-periodic orbits about the translunar libration point,” Celestial mechanics, Vol. 7,

No. 4, 1973, pp. 458–473.

[31]

Farquhar, R., Muhonen, D., and Richardson, D., “Mission design for a halo orbiter of the Earth,” Astrodynamics Conference,

1976, p. 810.

[32] Howell, K., and Breakwell, J., “Almost rectilinear halo orbits,” Celestial mechanics, Vol. 32, No. 1, 1984, pp. 29–52.

[33]

Gomez, G., Koon, W. S., Lo, M. W., Marsden, J. E., Masdemont, J., and Ross, S. D., Invariant manifolds, the spatial three-body

problem and space mission design, 109, American Astronautical Society, 2001.

[34]

Gómez, G., Jorba, A., Masdemont, J., and Simó, C., “Study of the transfer from the Earth to a halo orbit around the equilibrium

pointL 1,” Celestial Mechanics and Dynamical Astronomy, Vol. 56, No. 4, 1993, pp. 541–562.

[35]

Gooding, R., “A procedure for the solution of Lambert’s orbital boundary-value problem,” Celestial Mechanics and Dynamical

Astronomy, Vol. 48, No. 2, 1990, pp. 145–165.

[36]

Wagner, S., Wie, B., and Kaplinger, B., “Computational solutions to Lambert’s problem on modern graphics processing units,”

Journal of Guidance, Control, and Dynamics, Vol. 38, No. 7, 2015, pp. 1305–1311.

[37]

Arora, N., Russell, R. P., Strange, N., and Ottesen, D., “Partial derivatives of the solution to the Lambert boundary value

problem,” Journal of Guidance, Control, and Dynamics, Vol. 38, No. 9, 2015, pp. 1563–1572.

31

[38]

Woollands, R. M., Read, J. L., Probe, A. B., and Junkins, J. L., “Multiple revolution solutions for the perturbed lambert problem

using the method of particular solutions and picard iteration,” The Journal of the Astronautical Sciences, Vol. 64, No. 4, 2017,

pp. 361–378.

[39]

Shimoun, J., Taheri, E., Kolmanovsky, I., and Girard, A., “A study on GPU-enabled lambert’s problem solution for space

targeting missions,” 2018 Annual American Control Conference (ACC), IEEE, 2018, pp. 664–669.

[40]

Ochoa, S. I., and Prussing, J. E., “Multiple revolution solutions to Lambert’s problem,” Spaceﬂight mechanics 1992, 1992, pp.

1205–1216.

[41] Williams, C. K., and Rasmussen, C. E., Gaussian processes for machine learning, Vol. 2, MIT press Cambridge, MA, 2006.

[42]

Zeng, Z., Li, J., Huang, L., Feng, X., and Liu, F., “Improving target detection accuracy based on multipolarization MIMO GPR,”

IEEE Transactions on Geoscience and Remote Sensing, Vol. 53, No. 1, 2014, pp. 15–24.

[43]

Liu, H., Long, Z., Tian, B., Han, F., Fang, G., and Liu, Q. H., “Two-dimensional reverse-time migration applied to GPR with a

3-D-to-2-D data conversion,” IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, Vol. 10,

No. 10, 2017, pp. 4313–4320.

[44]

González-García, A. C., “Lunar extremes, lunar cycles and the minor standstill,” Journal of Skyscape Archaeology, Vol. 2,

No. 1, 2016, pp. 77–84.

32