# Machine Learning in Orbit Estimation: a Survey

Francisco M. Caldas 111PhD. Student, NOVA LINCS<EMAIL_ADDRESS>and
Cláudia Soares 222Professor, NOVA LINCS<EMAIL_ADDRESS>NOVA School
of Science and Technology, Caparica, Portugal

###### Abstract

Since the late ’50s, when the first artificial satellite was launched, the
number of resident space objects (RSOs) has steadily increased. It is
estimated that around 1 Million objects larger than 1 cm are currently
orbiting the Earth, with only 30,000, larger than 10 cm, presently being
tracked. To avert a chain reaction of collisions, termed Kessler Syndrome [1],
it is indispensable to accurately track and predict space debris and
satellites’ orbit alike. Current physics-based methods have errors in the
order of kilometres for 7 days predictions, which is insufficient when
considering space debris that have mostly less than 1 meter. Typically, this
failure is due to uncertainty around the state of the space object at the
beginning of the trajectory, forecasting errors in environmental conditions
such as atmospheric drag, as well as specific unknown characteristics such as
mass or geometry of the RSO. Leveraging data-driven techniques, namely machine
learning, the orbit prediction accuracy can be enhanced: by deriving
unmeasured objects’ characteristics, improving non-conservative forces’
effects, and by the superior abstraction capacity that Deep Learning models
have of modelling highly complex non-linear systems. In this survey, we
provide an overview of the current work being done in this field.

##  1 Introduction

It is estimated that more than 36,000 objects larger than 10 centimetres, and
millions of smaller pieces, exist in Earth’s orbit [2]. To safeguard active
spacecraft, it is necessary to accurately determine where each Resident Space
Object (RSO) is, and where it will be, at all times. To create such a complex
body of knowledge, the larger problem of orbit estimation is divided into
smaller problems, each one largely complex but with a specific goal in mind.
In this review we observe three main sub-problems: Orbit Determination, Orbit
Prediction and Thermospheric Mass Density.

  * •

Orbit Determination (OD): the OD is the determination of the orbit of the
object based on observations. The Extended Kalman Filter [3] (EKF) is the de
facto standard for orbit determination in real-world scenarios. The accuracy
of this process depends on the amount of sequential observations used to
determine the orbit, and the type of observation, e.g., laser ranging and GPS
tracking is far more precise than optical observations. The output of this
method is commonly a Gaussian probability distribution of the orbit of the
object, usually represented through a vector consisting of the object’s
estimated position and velocity and a covariance matrix reflecting the
uncertainty. Currently, this process is limited by the assumptions of the EKF,
by lack of knowledge in RSO’s shape and attitude and dynamic model
simplifications, which we will further examine in Section 3.

  * •

Orbit Prediction (OP): Orbit Prediction is the process of predicting the
future position and associated uncertainty of any given RSO. Two method
families exist for OP: one pursuing analytical solutions, with the other
exploiting numerical approximations. Numerical methods are time-consuming but
precise, while the analytical methodologies are simpler and faster. To be
tractable, these algorithms hold simplifying assumptions that hinder accuracy
[4]. Each method is limited by OD in the sense that a orbital state with high
uncertainty will necessarily evolve to have an inaccurate orbit prediction.
When used in a Collision Avoidance scheme, this process is used to propagate
the state of the RSO until the time of closest approach (TCA) to any actively
monitored satellite. The current limitations in orbit prediction are the
Gaussian assumption — which does not hold against the true distribution [5] —
the simplified modelling of perturbation forces, the unknown information
regarding RSO characteristics and the uncertainty over space weather forecast
data.

  * •

Thermospheric Density Mass: A set of force models determine the acceleration
of any RSO. Earth’s gravity potential is the most meaningful, but solar and
lunar gravitational attraction and Earth/ocean tides all have an effect on the
RSO [6]. Of the non-conservative forces, the most relevant for objects in LEO
is the satellite drag. Being applied in the opposite direction of the RSO’s
velocity, this force is the largest source of uncertainty for most RSO’s in
LEO [7] and correctly determining atmospheric density is key to calculate
satellite drag. All state-of-the-art models currently used for density
estimation are empirical (data-driven) models that have been consistently
updated since the last century. The lack of predictive capability and
uncertainty estimation are the two main drawbacks of these methods.

Table 1:  Outlining of input and output for each task Task | Input | Output
---|---|---
Orbit Determination | ground stations observations | orbital states at time $t_{0}$
Orbit Propagation | orbital states | orbital states at time $t_{1}$
Thermospheric Density Mass | Space Weather $(F_{10.7},F_{30},ap,..)$ | local density and temperature

In Table 1 we have a summary of the input/output for each task. In the
following sections, we will do a thorough review of published work using
machine learning techniques to help increase space safety, protecting
spacecraft, while working to diminish the amount of debris orbiting the Earth.
Each section will start with a brief description of the task, followed by a
state-of-the-art review of the classical methods currently in operation. The
machine learning body of work will be presented in semi-chronological order,
from the initial use of historical data to improve physical models to the most
recent developments in the area, that use highly complex models. The next
section will briefly define the physical forces that determine the path, or
orbit, of a satellite or space debris.

##  2 Forces Model

The equation of motion of a space object in a Cartesian ECI (Earth-Centered
Inertial) coordinate system [5] can be written in the form

$f(x,t)=\ddot{r}=-\frac{\mu_{\oplus}}{|r|_{2}^{3}}r+\textbf{a}_{pert}(r,\dot{r},t),$
(1)

where $r$ is the position vector, $[r\quad\dot{r}]^{T}=x$, and $\mu_{\oplus}$
is the gravitational constant multiplied by the masses of the Earth and the
RSO, with the perturbing forces being

$\textbf{a}_{pert}(r,\dot{r},t)=\textbf{a}_{NS}+\textbf{a}_{S/M}+\textbf{a}_{drag}+\textbf{a}_{SRP}+\textbf{a}_{tides}+\textbf{a}_{others}.$
(2)

For a given initial condition $x(t_{0})=x_{0}$, the position of a satellite at
any point in time can be implicitly written as the solution of the equation
flow $\phi$:

$x(t)=\phi(t;x_{0},t_{0}).$ (3)

The perturbation of Earth’s gravity potential, $\textbf{a}_{NS}$, is due to
the fact that Earth is not a perfect sphere. This force is modeled using
spherical harmonic potential equations, with varying level of precision
depending on degree. The most recently published model, the Earth Gravity
Model (EGM2012), has coefficients up to degree 2190 and it is accurate to
$0.1$ m [8]. The $n$-Body perturbations, $\textbf{a}_{S/M}$, affect the
trajectories of RSOs in orbit, with the Solar and Lunar perturbations being
the two most relevant forces on the space object’s body. Other planets in the
solar system, such as Jupiter, have a very minimal impact on satellites. The
atmospheric drag, $\textbf{a}_{drag}$, especially in the Thermosphere layer
($85/125-600/1000$ km), can lead to significant effects, and it is responsible
for the decay of orbits in Low Earth Orbit (LEO). Using the canon-ball model,
the aerodynamic drag can be represented by [9]:

$\textbf{a}_{drag}=\frac{1}{2}\rho\frac{ac_{D}}{m}v_{rel}^{2}\frac{v_{rel}}{|v_{rel}|}$
(4)

where $\rho$ is the local atmospheric mass density, $c_{D}$ is the drag
coefficient, $a$ is the cross-sectional area, $m$ the mass of the space
object, and $v_{rel}$ is the relative velocity of the object with respect to
the rotating atmosphere. For most satellites, the mass and cross-sectional
area is known in advance, and is assumed to be constant for the lifespan of
the satellite, despite slight changes that might occur, due to, for example,
fuel consumption. This is however not the case for the majority of RSOs,
especially space debris that originated from breakups. Taking that into
account, it is common to define a combined parameter, the Ballistic
Coefficient ($BC=\frac{m}{ac_{D}})$, which integrates mass, area and drag
coefficient to be calculated as an extra feature during Orbit Determination.
The local mass density $\rho$ is the variable obtained from the Thermospheric
Density Mass models, which by itself is particularly challenging to model, as
we will see in Section 5.

The Solar Radiation Pressure, $\textbf{a}_{SRP}$, is the force that arises
from the absorption and reflection of light from the Sun. It depends on the
cross-sectional area and mass, but unlike atmospheric drag it does not depend
on altitude. It is the dominant non-gravitational perturbation on higher
altitudes (MEO and GEO) and for High Area to Mass Ratio (HAMR) objects [5]. Of
particular importance to the representation of this force is the determination
of the entry and exit times of a given RSO from the shadow of the Earth, along
with the determination of the force orientation, as the object-Sun vector is
always changing and with it the cross-sectional area.

Tidal effects, $\textbf{a}_{tides}$, are also taken into account in orbital
determination, with the small periodic deformations of the solid body of the
Earth referred to as solid Earth tides, which, alongside ocean tides, impact
Earth’s gravity potential [6]. Others perturbations include General Relativity
theory adjustments, geomagnetic pulls, Earth’s albedo and atomic clock
corrections.

Figure 1: The two main stages of orbital estimation necessary for collision
avoidance and debris and satellite tracking. The information obtained from OD
is used as input for Orbit Prediction (OP). Figure adapted from [4] and [10].

##  3 Orbit Determination

This is the process of using observations to obtain a state vector
$x\in\mathbb{R}^{p}$ of a given space object [11]. In this survey space
objects will be all objects, either satellites or space debris, currently
orbiting the Earth. A state vector of $p=6$ is sufficient to represent the
velocity and position of the satellite at a given time, but others variables,
such as the Ballistic Coefficient (BC), or clock bias can also be calculated
as part of the state vector [12]. This state vector can also be represented by
the traditional Keplerian elements $(e,a,i,\Omega,\omega,\theta)$ or by
equinoctial elements $(a,h,k,p,q,\ell)$. The type of observations used for
this purpose are radio, radar, laser or image; with observations usually
divided between cooperative or non-cooperative observations [10]. As the name
suggests cooperative observation is when the object in question, e.g., a
satellite, has a built-in capability to respond to tracking. This gives the
Ground Station (GS) the ability to know with detail the position and ID of the
object. In this case we have a significant amount of observations across the
period in which the GS, or GS’s, are communicating with the satellite. The
non-cooperative observations are necessary to track defunct satellites or
space debris. In this case the identification of the satellite is a necessary
part of the process, and the number of observations, usually radar, are
relatively small for each space object.

The state-of-the-art approach for this problem is the Extended Kalman Filter
(EKF) [4]. This method is an approximation of the Kalman Filter to non-linear
systems, such as the equations of motion of a space object. Its limitations
include the use of only the first order Taylor series expansion to approximate
the system to a linear system, as well as the assumption that the noise,
measurement and process, is Gaussian [13]. Despite that, the applicability of
the filter for sequential data, alongside providing a probability distribution
of the estimate state, proves to very useful and justifies the current use.
Because of the relevance of the EKF method to orbit determination, we will
briefly explain the algorithm.

The intended goal of this method is to combine knowledge of the dynamical
system with observations, therefore the dynamical and observation equations
can be represented as [13, 14]:

$\begin{split}\dot{x}(t)&=f(x,t)+w(t)\\\ y_{k}&=h(x_{k})+v_{k}\end{split}$ (5)

where $w(t)\sim N(0,Q(t))$ is the process noise, $y_{k}$ is the measurement,
$h(x_{k})$ the measurement function and $v_{k}$ the measurement noise
following a zero-mean Gaussian with covariance $R_{k}$, $v_{k}\sim
N(0,R_{k})$. Through this general formula we can observe that the formula
combines discrete observations with continuous dynamical equations. When
$f(x,t)$ and $h(x_{k})$ are linear functions, and the initial distribution is
Gaussian, with $w(t)$ and $v_{k}$ expressing white noise (uncorrelated
Gaussian distributions with zero-mean), the Kalman Filter is the optimal
filter [15]. A known and much studied filter in the automatic control and
time-series research fields, a great deal of books [16, 17, 18] present in
depth explanations of the Kalman filter, which can be summarized as a two-step
algorithm (predict and filter) as illustrated in Figure 2.

Figure 2: General Kalman Filter Diagram. $A_{k}$ is the state transition
matrix at time $k$, $K_{k}$ is the Kalman Gain at each observational moment,
$H_{k}$ is the linear measurement mode at time step $k$, and the output is the
estimated mean ($\hat{x}_{k}$) and covariance ($\Sigma_{k}$) at time step $k$.
$\hat{x}_{k|k-1}$ and $\Sigma^{-}_{k}$ are internal components representing
the a priori prediction before the observation $y_{k}$ is introduced.

The Extended Kalman Filter is applied when the linearity assumption does not
hold, in particular the linearity of $f(x,t)$ and/or $h(x_{k})$. To do so, and
before the KF algorithm in Figure 2 ought to be used, both equations are
linearized using the first order of Taylor’s approximation, at each
observational moment $k$:

$\begin{split}f(x_{k})&\approx f_{k}(\hat{x}_{t|k})+\frac{\partial
f(x_{k},t)}{\partial x_{k}}\Bigr{|}_{x_{k}=\hat{x}_{t|k}}+...\\\
h(x_{k})&\approx h_{k}(\hat{x}_{t|k})+\frac{\partial h(x_{k})}{\partial
x_{k}}\Bigr{|}_{x_{k}=\hat{x}_{t|k}}+...\\\ \end{split}$ (6)

Note in particular in Figure 2 that the prediction step for the a priori mean
($\hat{x}_{k|k-1}$) does not need a linear function, it is the covariance
matrix and the Kalman Gain matrix that require the simplification of the
dynamical and measurement equations. This is exactly the intuition behind
improvements over the base method such as the Unscented Kalman Filter (UKF)
[3], the Second Order Kalman Filter [19] and others methods referred to in Lee
[20]. These methods more closely represent the non-linearity of the problem,
but are, nonetheless, Gaussian or linear approximations. Sequential Monte
Carlo methods such as the Unscented Particle Filter [21] or the Divided
Difference Particle Filter [20] are also popular, and shown to be superior to
the EKF by [22] and [23], with the disadvantage of high computational costs
[23]. The UKF in particular sits on the border between Monte Carlo and linear
methods, with deterministically chosen sigma-points ($\mathcal{X}$) [3], made
to represent the Gaussian distribution, being independently propagated across
time through $x_{k+1}=\phi(t_{k+1};x_{k},t_{k})$ and then used to derive the
mean and covariance at the $k+1$ time step.

More recently, the non-linearity of the problem as been tackled through the
use of machine learning and data-driven techniques. Hartikainen et al. [24]
used Latent Force Models (LFMs) to combine the dynamic system with a more
freely defined measurement component, implementing Gaussian Processes to model
measurement noise, and using a machine learning approach to define the
covariance structure of the Gaussian Process [25, 26]. This method improves
upon the Kalman Filter by removing the uncorrelated white noise assumption and
replacing it by a time-varying GP. In [27], it is presented a general approach
to OD using distribution regression. It is shown that under a known and
continuous dynamic system, with known spacecraft characteristics and
observable orbital parameters, there is a possible mapping from the
probability distribution of the observed samples to the orbital parameters.
This method can be used either to estimate the posterior distribution of the
orbital parameters based on full batch observations, or to identify the ID of
the spacecraft associated with each sample. In [28], Sharma extends this work
and presents a method for collaborative and non-collaborative orbit
classification, by estimating the probability of each sample to represent the
distribution associated with its ID. This technique is particularly useful to
to identify satellites with identical Radio Frequency (RF) transmissions or
very close orbits, but was not extended to more than 2 spacecrafts. The
distribution regression approach was also followed in [29], extending the
process to observations from more than one Ground Station (GS).

Different authors [30, 31, 32] independently presented an improvement over the
current methods for OD by using Gaussian Mixture Models (GMMs) to model the
estimated distribution. As a GMM is known to approximate any distribution
given a sufficient number of Gaussian distributions [33, 34], it is used to
approximate the real distribution of the state during the OD period. In
particular, this allows for a more accurate characterization of the
distribution of $X$, being $X$ the state at any given time during OD, while
also being particular useful for the step of Orbit Propagation (OP), as all
current methods are capable of propagating a Gaussian distribution across
time, and are trivially scalable to a finite number of independent Gaussian
distributions. This work was extended by [35] and [36] by adapting the GMM
weights across time, and further enhanced by Vittaldev [37, 38] who presented
an univariate splitting library along arbitrary directions that is used to
generate a Multidirectional Gaussian Mixture Model. In [39], Horwood et al.
also proposed a new distribution to improve OD — the Gauss von Mises (GVM) —
using this distribution in tandem with a particular set of orbital elements,
the $J_{2}$ equinoctial orbital elements, to reduce the non-linearity of the
process. The GVM is the counterpart distribution of the Gaussian distribution
for circular support. In a series of publications [40, 41, 5] Horwood et al.
compare their approach with the UKF, EKF and GMM methods using a Particle
Filter (PF) as the ground truth model. Both the GMM ($k=49$) and the GVM are
shown to faithfully represent the orbit distribution for a period 8 times
longer than the UKF and the EKF.

##  4 Orbit Prediction

Orbit Prediction can usually be seen as the process of propagating a known
orbital state to a future orbital state. The current physics-based methodology
is to solve the differential equations of motion either analytically or
numerically to predict the mean state at a future time. The numerical methods
are time consuming and the sheer number of space debris objects orbiting the
Earth makes them only usable for academic or non-real time applications [42].
In general, the non-linear stochastic dynamic system can be defined using the
Itô stochastic differential equation [43]:

$\dot{x}=f(x,t)dt+G(t)d\beta(t),$ (7)

where $f(x,t)$ is the deterministic part of system, as seen in Equation (1),
$G(t)$ is the diffusion of the process, and $\beta(t)$ is a Brownian motion,
also known as a Wiener process, with zero mean and covariance $Q$. Given an
initial probability distribution of $x$, $p(x_{0},t_{0})$, the pdf at a future
time $p(x,t)$ is governed by the Fokker-Planck-Kolmogorov equation (FPKE)
[44]:

$\frac{dp}{dt}=-\nabla^{T}_{x}(pf)+\frac{1}{2}\text{tr}[\nabla_{x}\nabla_{x}^{T}(pG(t)Q(t)G(t)^{T})],$
(8)

where $\nabla_{x}$ is the gradient in $x$ viewed as a column operator. Solving
this equation would provide a complete description of the position and
uncertainty of the orbit at any time $t>0$. However, there is no analytical
solution to most FPKE, with orbital mechanics being particularly hard to solve
due to the non-linearity of the perturbed dynamics in a state space with six
dimensions. The orbit prediction numerical approximations of this solution
usually entail a set of assumptions to allow the problem to be solvable. In
this review, before jumping to the advances in the field, we are going to
present the more common approach for Orbit Prediction, where the output from
the EKF, a Gaussian Distribution with mean $\mu(t_{0})=\mu_{0}$ and covariance
$\Sigma_{0}$ is propagated according to a first-order systems of SDEs. Using
the SDE in Equation (8), assuming the noiseless case $(Q(t)=0)$ and the flow
solution (3), the mean and covariance at time $t$ is:

$\begin{split}\mu(t)&=\phi(t;\mu_{0},t_{0})\\\
\Sigma(t)&=\Phi(t;\mu_{0},t_{0})\Sigma_{0}\Phi(t;\mu_{0},t_{0})^{T},\end{split}$
(9)

where the flow solution $\phi(t;\mu_{0},t_{0})$ and State Transition Matrix
(STT) $\Phi(t;\mu_{0},t_{0})$ are solved by:

$\begin{split}\phi^{\prime}(t;\mu_{0},t_{0})&=f(\mu_{0},t)\\\
\Phi^{\prime}(t;u_{0},t_{0})&=F(\phi(t;\mu_{0},t_{0}),t)\Phi(t;u_{0},t_{0}),\end{split}$
(10)

where $\Phi(t_{0};u_{0},t_{0})=I$ and, in the case of the EKF, the linearly
propagated covariance is obtained with $F(\phi(t;\mu_{0},t_{0}),t)$ as the
partial derivative of the function $f(x(t),t)$ in Equation (1):

$F(x,t)=\frac{\partial f(\epsilon,t)}{\partial\epsilon}\Bigr{|}_{\epsilon=x}.$
(11)

To solve the equations in (10) there are many numerical methods, with some
numerical integration methods specialized for orbit propagation, as seen in
[6] and [45]. The most relevant numerical integrators used for orbit
propagation found in the literature include explicit Runge-Kutta methods such
as Dormand-Prince 8(7) (DP8) or Runge-Kutta-Nystrom 12(10) (RKN12) and
predictor-corrector methods, namely Adams-Bashforth-Moulton (ABM) and Gauss-
Jackson (GJ) [5]. In particular, implicit Runge-Kutta methods such as the ones
developed by Bradley et al. [46] (GL-IRK) and Bai and Junkins [47] (GC-IRK)
are quite interesting algorithms for practical applications as they allow for
parallelization, are A-stable [48] at all orders and are more accurate than
DP8 [49]. Analytical methods are more common, and of those, the widely used
method is the SGP4 (Simplified General Perturbations Model 4) [50]. Made
available to the public in 2006 [51], it has since them become the most used
propagator for orbit prediction, despite the lack of accuracy for long-time
predictions. The input data for SGP4 is a Two Line Element (TLE) that includes
information about the object and its orbit, such as satellite number, mean
orbital elements, drag, ballistic coefficient, revolution number and a time
stamp. One information it does not provide, crucially, is the initial
uncertainty, so we can deduce the fact that the information on Resident Space
Objects made publicly available through TLEs at space-track.org is one of the
reasons for the ample use of SGP4 [52].

Using the current methodology two types of errors exist when doing Orbit
Propagation: satellite force model’s errors and measurement errors. The first
error type relates to simplifications in the model, such as Earth’s
Gravitational harmonics model, or unknown information specific to the RSO,
such as shape, attitude or cross-sectional area. Measurement errors are all
errors associated with the difference between the real state of the satellite
(such as position and velocity) and the measured state, including numerical
truncation errors, as well as other navigation errors caused by clock accuracy
or coordinate systems precision. In Figure 3 we can see a summary of the error
sources in orbit prediction, as presented by [10] and [53].

Figure 3: Error Sources in Orbital Prediction, adapted from [10].

In recent years, data-driven techniques were used to tackle both sides of the
problem, either employing machine learning to improve the model’s
representation of reality or to mitigate the errors caused by numerical
solutions and linear assumptions. Levit and Marshall [54] showed that when
using a sufficient number of TLE’s, they could be used as pseudo-observations
to fit a high-precision special perturbations numerical propagator. This
method was applied to a set of satellites, and the OP error was reduced from
1.5 km/day to 0.1 km/day. This work shows that past information can be
leveraged to improve orbit prediction, which is a key factor for any machine
learning setup. This work was closely followed by [55], in which the same
method was applied, alongside a bias correction function. Similarly, other
authors [56, 57] have shown improvement of the SGP4 prediction using error
correction functions, based on prediction error periodicity.

A second approach is to remove the physics-based propagator altogether. In
[58], it was proposed a data-driven model that was able to approximate the
SGP4 algorithm using a polynomial fit. A different method was introduced in
[24] using Latent Force Models (LFMs) to combine the physical models
principles with non-parametric data-driven components. By assuming Gaussian
process (GP) priors on unknown forces, this method allows to determine future
orbit positions and corresponding uncertainty. This work was by extended
Rautalin et al. [59] who obtained positive results for a set of satellite
constellations in Medium Earth Orbit (MEO) and Geo-Synchronous Orbit (GEO).

Peng and Bai [60] have an extensive number of publications on this field. In
their first paper within the error correction framework, a Support Vector
Machine (SVM) is used to learn the structural part of the historical error and
to improve the SGP4-based prediction. In a simulated catalogue, they have
shown that the error correcting model can be deployed to improve RSO’s orbit
prediction. This duo of researchers also examined the capabilities of three ML
models, specifically, Artificial Neural Networks (ANN), Gaussian Processes
(GP) and Support Vector Machines (SVM) in [61, 60, 62], coming to the
conclusion that the ANN had the best results, despite being more prone to
overfitting. In [63] the authors evaluated the SVM model on a real dataset and
more recently proposed a data fusion approach to combine uncertainty
information from the EKF OD process with the one obtained from the GP model
[64].

This hybrid approach, that uses a ML algorithm to error correct a physics-
based model is by far the most common, and various neural network
architectures have been leveraged for this purpose. Under the same hybrid SGP4
approach, four independent works show that Convolutional Neural Networks (CNN)
[65], Feed-Forward Neural Networks (FNN) [66], Recurrent Neural Networks (RNN)
[67] and Long-Short Term Memory (LSTM) [68] can be used to improve SGP4 orbit
prediction. Using a Gradient Boosting Decision Tree (GBDT) and a CNN, Li et
al. [69, 70] improved by more than 75% the along-track direction prediction
for five satellites in LEO and GEO. All of this work has been done under the
same data regimen, in particular using publicly available satellite data from
the International Laser Ranging System (ILRS) as the ground truth and using a
set of TLEs for a limited number of satellites as the training data. Two yet
to achieve goals in this line of research is a generalisation of the machine
learning models to unseen RSOs (which Peng and Bai refers as a type III
generalisation [71]), so that a machine learning model is not developed for
each RSO, and the use of exogenous variables outside of the TLE information.

As mentioned before, two error sources exist in Orbit Prediction, and
therefore two lines of work could be developed to enhance a physics-based
model. The previously shown work focus is on correcting numerical errors and
observational errors that occur when using simplifying assumptions,
particularly the limited SGP4 propagator. A different approach is on the use
of ML to improve our knowledge of the Earth, space and RSOs and thus obtaining
a more exact model of reality.

##  5 Thermospheric Density Mass Models

In any orbit prediction process of LEO objects, knowledge of thermospheric
mass density (the $\rho$ in Equation (4)) is essential to accurately predict
future states, since it is known that air drag is the main non-conservative
force affecting satellites and debris alike in LEO [72]. The state-of-the-art
model series currently used are the Jacchia-Bowman 2008 (JB2008) [73], that
has a corrective model in the form of the High Accuracy Satellite Drag Model
(HASDM) [7], the Drag Temperature Model 2020 (DTM) [74] and the Naval Research
Laboratory for Mass Spectrometer and Incoherent Scatter radar 2.0 model
(NRLMSIS 2.0) [9]. These empirical models are derived combining information
from mass spectrometer data, inferred temperature from ground stations, solar
UV measurements and on-board drag-derived density data. They are, however, far
from exact when used in Orbit Prediction. Vallado and Finkleman [75] revealed
an average one-sigma accuracy of 10% to 15% for empirical density models,
depending on model, solar activity and location [76]. These empirical models
lack predictive capacity, and can be further compromised by erroneous forecast
of the driving features of the model. In the work by Emmert et al. [77] it is
shown that a $10\%$ error in Extreme Ultraviolet (EUV) light prediction will
result in more than $200$ km uncertainty for a given satellite after $7$ days.

The use of machine learning for space weather indexes, such as the $F_{10.7}$,
which is a proxy for EUV light, dates back to the 90’s, when a ANN was used to
predict solar flux [78] or Gleisner et al. [79] predicted geomagnetic storms
using Time delay Neural Networks (TDNN). On the whole, due to the public
availability of space weather indexes, and their natural periodic features,
this area has an ample array of scientific publications [80, 81, 82, 83, 84]
and for an extensive review of all machine learning applications in this
particular field we recommend the excellent survey done by Camporeale [85].

The initial approach for improving atmospheric mass density models is to use
calibration data to fine tune the parameters of the original model. This was
the approach taken in [7] to develop the HASDM, and other authors showed also
superior results with calibrated models: Doornbos et al. [86] used TLE data to
calibrate a complex physical model, halving orbit prediction error; with the
same approach later used [87] to calibrate the NRLMSISE-00 model during
periods of increased solar activiy. In [88] a process was suggested to
calibrate any empirical model using the least squares method, which [89] found
to be the most effective calibration method for reducing orbit prediction
error. Using more than one model was found to be helpful in reducing
forecasting errors, when a Multi-model Ensemble [90] combining three well-
known physical and empirical models was able to reduce forecasting error in
the training and test set. A similar approach was presented by [91] in which a
NN was used to combine empirical models in order to reduce density error. [92]
leveraged a Neural Network to calibrate models during magnetic storms, a
period that is known to be particularly challenging for empirical models,
while [93] and [94] used LSTM and Gaussian Processes (GP), respectively, to
calibrate the original models.

A different approach has appeared recently, using Reduced Order Models (ROM)
as a quasi-physical model. In [95] a ROM was used to approximate the TIE-GCM
complex physical model, with simulated orbit ephemerides used for model
calibration. Reduced Order Models are models that take the high dimension
space to a lower dimensional while maintaining most of the information. In
this case, the order reduction is done using Proper Orthogonal Decomposition
(POD), which is usually referred to in the machine learning world as Principal
Component Analysis (PCA). This work was extended using accelerometer-derived
density estimates from the Challenging Minisatellite Payload (CHAMP) and
Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) [96]
satellites and TLE data [97] for model calibration. Gondelach and Linares [98]
showed that a ROM model with TLE-estimated densities provides more precise
Orbit Determination than the NRLMSISE-00 and JB2008 models, especially in the
along-track coordinate. This ROM approach was improved upon using an anto-
encoder for dimensionality reduction [99]. This work was further refined by
using Sparse Identification of Nonlinear Dynamical systems techniques (SINDy)
to derive an explicit non-linear differential equation that defined the
atmospheric density in the encoded (low-dimension) space [100].

In 2017, Mehta et al. [101] published a global density estimate derived from
the GRACE and CHAMP satellites, using a physical model with high-fidelity drag
coefficient modelling. This data was then used to develop an LSTM-based Neural
Network for global density prediction, with promising results [102].
Similarly, Bonasera et al. [103] used two different machine learning
techniques to derive uncertainty over the estimated density, Monte-Carlo
Dropout and Deep Ensemble, with both methods surpassing JB2008 and
NRLMSISE-00; Both techniques use a sampling method to obtain a probability
distribution of the estimated quantity. Monte-Carlo Dropout "turns-off" a
percentage of the neurons in a Neural Network for each sample, so as to obtain
a slightly different result [34]. Deep Ensemble is a technique that combines
the output of a series of Neural Networks to obtain an estimated mean and
uncertainty of the prediction [104]. The same group of researchers also
developed a method to forecast with uncertainty estimation the input features
for empirical models, combining solar data images with time-series information
[105]. In 2020, SET (Space Enviroment Techonologies) made public 20 years of
density data derived from HASDM, the model used by the United States Air Force
[106]. With this data it was possible to train a NN to replicate the HASDM
model [107] using the same input, so as to add uncertainty information through
the use of Monte-Carlo Dropout. Further advancements were accomplished by the
use of Bayesian Neural Networks on the global HASDM dataset and the local
CHAMP dataset [108]. Bayesian Neural Networks have recently garnered more
attention in the machine learning world, as they allow for a more principled
understanding of uncertainty, when compared to Monte-Carlo Dropout or
Ensembles. Bayesian Neural Networks replace the neurons in a network, which
are unknown parameters, or weights, with random variables, with the training
process learning the probability distribution function of each neuron [109].

##  6 Future Challenges and Conclusion

In this Section, we will discuss some of the current and future lines of
research. Globally, we see a shift from linear and physical models to more
data-driven non-linear models across the entire field. Some of these linear
models started being used in the 1960s [110], and many of the computational
limitations that justify the linear approach have been surpassed. We now know
that the Gaussian assumption used in the Extended Kalman Filter (EKF) is not
verified [5], and the new approaches presented using Gaussian Mixture Models
(GMM) [37] and the Gauss Von Mises (GVM) distribution [40] are highly
promising, however two main drawbacks are yet to be addressed: the GMM
necessarily needs a set of $k$ Gaussian distribution that is not constant for
all RSOs, depending on shape, orbit, altitude and solar activity. Determining
the exact number of Gaussian distributions necessary for sufficiently accurate
orbit determination is an open area of investigation. A second hurdle
regarding this area is the Probability of Collision (PoC). One of the key
motivations for precise Orbit Determination is the protection of satellites
and the space environment, and calculating the PoC is necessary to determine
orbital avoidance manoeuvres. In [37] the author is the first to evaluate the
GMM for PoC using Monte Carlo simulation. However, in most cases, the
probability of collision is closer to $10^{-30}$, which makes Monte Carlo too
time consuming for general use. A PoC formula that can be applied for all
distributions is a very interesting path of research, with some initial work
done by [111], who generalized the PoC equation to any distribution and also
hypothesises the use of GMMs. The authors of this survey found that, in this
particular area, most machine learning is focused on distribution regression,
the fitting of a distribution to a set of observations, with very limited use
of Deep Learning techniques, with the exception of [112], who used a ANN-EKF
hybrid. On the contrary, many Deep Learning architectures have been
experimented with in Orbit Prediction, most of then with the focus on
improving upon the SGP4 propagator. While this work can be relevant to improve
publicly available catalogues, a necessary development for industrial use is
the introduction of uncertainty in the ML-OP scheme. Most Neural Network have
Bayesian counterparts, meaning Neural Networks that predict a distribution,
however limited work has been done in this front. The most recent works in
this area validate this trend, with [64] combining Gaussian Process (GP)
uncertainty with EKF-derived uncertainty, and [67] using a Deep Ensemble to
perform variance estimation.

Most research relies on the learning of historical error patterns, and it is
possible to obtain improvements by introducing variables that improve the
representation of the physical forces being applied on the RSOs. Some
exploration in the estimation of unknown physical properties of RSO has also
been performed recently by Guthrie et al. [113] who performed attitude
estimation of unknown space debris using siamese CNNs, or with the use of CNNs
combined with a 3D image generator to determine the pose of a series of
satellites [114, 115]. Using light curves data, the work in [116] leveraged
deep learning techniques to determine RSOs shape. The retrieval of physical
characteristics is particularly relevant for space debris, however it suffers
from a lack of labeled data, with real information on space debris’ shape
seldom available. Some preliminary work on Meta-Learning Models [117]
indicates that this area is primed for the use of transfer learning (using
models trained on some type of data to solve a different problem) and few-shot
learning techniques (learning with very limited data). Currently, the shape
and orientation of RSOs is not yet used in Orbit Determination or Prediction,
meanwhile, atmospheric density and air drag forces are considered in any
current force model, thus it is relevant to have accurate density prediction
models. The classical models (e.g., JB2008, DTM) referenced before are
empirical models, which in practice means that they are, in a sense, machine
learning algorithms: physics-aware equations, mostly linear or sinusoidal,
with coefficients regressed from data. When we observe the limitations of the
current algorithms, with 10 to 15% 1-sigma accuracy [75], it becomes a
question if the empirical models can be completely replaced by more modern
deep learning models, and that is exactly what recent papers [107, 103, 100]
have proposed, using different methods. During this review we observed that
most machine learning papers in this subject where published since 2019, as
can be seen in Figure 4, which speaks to the excitement around this area and
to the results currently being achieved. As with everything else in this
field, model uncertainty and forecast confidence intervals need to be a key
feature in all models designed in the future, so to integrate atmospheric
density model uncertainty in Orbit Propagation. Furthermore, the introduction
of grey-box models, such as the one proposed by [100], will become more
common, since these types of models are easier to evaluate and can implement
physics aware restrictions within them.

Figure 4: Number of publications between 2000 and 2022 cited in this review,
with applications divided by three categories: Orbital Determination, Orbital
Prediction and Atmospheric Density Mass models.

In conclusion, we see a field where classical algorithms can eventually be
replaced, but that will only occur if we use machine learning techniques to
better represent and understand the underlying physical forces. Across the
board, the predictions will become more focused on probability distributions
than point-wise estimations, and the push to create grey-box models, simpler
physics-aware algorithms that can be understood by physicists and machine
learning specialists alike will gain prominence, as that is the real problem —
How can we use machine learning to gain a better understanding of a system,
or, in this case, the physical forces that govern space?

## Funding Sources

This work was partially supported by FCT/PT Space under the PhD grant
I-2021-03954 and the strategic project NOVA LINCS (UIDB/04516/2020).

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Francisco Caldas holds a M.Sc. and B.Sc. in Applied Mathematics in Probability
and Statistics from Instituto Superior Técnico. He is a Ph.D. student in
Computer Science with a defined interest in Machine Learning applied to space
safety. His research interest include the application of Explainable ML to
time-series, and the application of Deep Learning techniques to various space
related problems. Combining his academic background in Mathematics with
practical experience working in a space start-up, his goal is to leverage the
modelling ability of neural networks to uncover the complex relations between
variables that can be derived from data, and apply those models to areas so
diverse as healthcare, air quality, orbit determination and space weather.

Cláudia Soares holds a Ph.D., M.Sc. and B.Sc. in Electrical and Computer
Engineering from Instituto Superior Técnico, Portugal, and a diploma in modern
languages and literature, from Nova University of Lisbon. Prof. Soares is an
Assistant Professor at the Computer Science department, NOVA School of Science
and Technology, and a researcher at NOVA LINCS, Portugal. She uses real-world
data problems to identify shortcomings of current machine learning, data
science, and Big Data methods. She applies optimization, statistics, and
probability theory to address those gaps, developing robust and interpretable
learning methods that can be trusted in real life. Her application areas are
in environmental and urban sciences, healthcare, transportation, and space.