Probabilistic Model for Aircraft in Climb

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A PROBABILISTIC MODEL FOR AIRCRAFT IN CLIMB USING

MONOTONIC FUNCTIONAL GAUSSIAN PROCESS EMULATORS

A PREPRINT

Nick Pepper

The Alan Turing Institute

The British Library

London, UK

npepper@turing.ac.uk

Marc Thomas

NATS

George De Ath

Department of Computer Science

University of Exeter

Exeter, UK

Enrico Oliver

Department of Computer Science

University of Exeter

Exeter, UK

Richard Cannon

NATS

Richard Everson

Department of Computer Science

University of Exeter

Exeter, UK

Tim Dodwell

Department of Computer Science

University of Exeter

Exeter, UK

digiLab

Exeter, UK

October 5, 2022

Keywords

Trajectory Prediction, Probabilistic Machine Learning, Functional Data Analysis, Gaussian Process

Emulators, Monotonicity

ABSTRACT

Ensuring vertical separation is a key means of maintaining safe separation between aircraft in

congested airspace. Aircraft trajectories are modelled in the presence of signiﬁcant epistemic

uncertainty, leading to discrepancies between observed trajectories and the predictions of deterministic

models, hampering the task of planning to ensure safe separation. In this paper a probabilistic model

is presented, for the purpose of emulating the trajectories of aircraft in climb and bounding the

uncertainty of the predicted trajectory. A monotonic, functional representation exploits the spatio-

temporal correlations in the radar observations. Through the use of Gaussian Process Emulators,

features that parameterise the climb are mapped directly to functional outputs, providing a fast

approximation, while ensuring that the resulting trajectory is monotonic. The model was applied as a

probabilistic digital twin for aircraft in climb and baselined against BADA, a deterministic model

widely used in industry. When applied to an unseen test dataset, the probabilistic model was found to

provide a mean prediction that was 21% more accurate, with a 34% sharper forecast.

1 Introduction

Trajectory prediction (TP) plays an important role in the safe management of airspaces and is used by air trafﬁc

controllers to inform decision-making by predicting arrival times or detecting potential conﬂicts [

]. To ensure safety,

air trafﬁc controllers strive to maintain both longitudinal and vertical separation of aircraft [

]. In this context vertical TP,

the prediction of an aircraft’s altitude with time, is especially important as controllers must devise plans that maintain

separation at all times. State-of-the-art TP methods project the future path of an aircraft through models of the ﬂight

arXiv:2210.01445v1 [cs.CE] 4 Oct 2022

A Probabilistic Model for Aircraft in Climb A PREPRINT

Figure 1: An uncertainty bound for an aircraft climbing between ﬂight levels 200 and 300, obtained from runs of a

deterministic TP code with maximal and minimal mass, compared to the observed radar data (left). An illustration of

the desired data-driven bounds, that better reﬂects the observed trajectories (right).

mechanics of aircraft, in which an estimate of the present state of the aircraft is used as an initial condition for the

numerical solution of a set of equations approximating the physics governing the aircraft’s ﬂight (see, e.g. [

]).

However, there is a high level of epistemic uncertainty inherent in this approach due to: the simpliﬁcations necessarily

made by the model; an uncertain knowledge of the aircraft’s state; lack of knowledge of the pilot’s intentions [

]; and

the unknown inﬂuence of environmental effects on the aircraft trajectory [

]. As a consequence a mismatch between

the predictions of physics-based TP methods and the actual path followed by aircraft can be observed, especially during

climbs and descents [8, 9].

The observed mismatch between physics-based methods and real trajectories has motivated the application of machine

learning-based methods to TP. In some respects TP is a very amenable problem for machine learning due to the large

quantity of radar observations available to train models [

]. In recent years, methods based on Neural

Networks [

] have been proposed for TP. However, there are several challenges facing machine learning methods

for TP that must be addressed. Firstly, given that high levels of epistemic uncertainty contribute so signiﬁcantly to

model-mismatch in state-of-the-art equations-based models, a probabilistic approach would appear to be essential. Next

generation TP models must be able to efﬁciently handle uncertainties and to clearly express the uncertainty in the

model predictions [

]. This is the motivation for approaches based on Sequential Monte Carlo sampling [

] and

Gaussian Mixture Models in the TP literature [

]. Indeed, several recent reviews of Machine Learning methods have

expressed this same point, that moving away from deterministic black-box machine learning models is necessary for

the massive industrial application of machine learning [

], particularly in risk-averse industries such as aeronautics.

Uncertainty estimates in TP for climbing aircraft are not generally data-driven. For instance, the left panel of Figure 1

illustrates an envelope created by two runs of a deterministic TP model with minimum and maximum nominal mass,

where the nominal mass refers to the aircraft mass expected by the model. However, this is an over-conservative

approach. What is desired from a probabilistic TP model is illustrated on the right panel of Figure 1: a data-driven

credible interval that trades off some conservatism in favour of more realistically representing the uncertainty.

In parallel with the continued reﬁnement of TP methods, there has been a drive in recent years to explore the application

of Artiﬁcial Intelligence (AI) to emulate the role of an Air Trafﬁc Controller (ATCO) that has been motivated by the

maturity of dynamic optimisation and reinforcement learning [

]. In this context, the probabilistic approach to TP

appears to be particularly useful. During training, an AI agent’s plan is evaluated within a Digital Twin of an airspace.

These plans must be robust to variabilities in the trajectories of aircraft, requiring a probabilistic digital twin. This offers

another area of application for a probabilistic TP model [23].

The second challenge facing ML methods is to guarantee that predicted trajectories satisfy physical constraints,

particularly when testing on unseen data. In the case of TP, there is a requirement that these trajectories are achievable

for the performance envelope of an aircraft. Simultaneously, there is a qualitative constraint that predicted trajectories

correspond to modes of aircraft operation that are observed in the real-world. In the case of aircraft in climb, a

reasonable expectation is that the altitude of an aircraft will increase monotonically with time, until a target altitude

is reached. Enforcing a monotonicity constraint is straightforward if the ML approach corrects the parameters of an

equation-based model. For instance, Alligier et al. proposed a ML model to better predict the mass and speed intent

of climbing aircraft, unknown parameters which are then fed into the Base of Aircraft Data (BADA) physics-based

model [

]. However, there is a computational cost associated with the numerical solution of BADA, especially in

A Probabilistic Model for Aircraft in Climb A PREPRINT

a probabilistic approach that might require multiple model evaluations. A fast approximation to BADA is required,

where model inputs are mapped directly to trajectories. However, guaranteeing that such fast approximations will

themselves be monotonic is challenging. Some progress has been made in this regard in the setting of pointwise data,

for example recent work with Gaussian Process Emulators (GPEs) has found correlation functions that correspond to a

monotonic mean function [

]. However, little attention has been paid in the ML literature to fast approximations

with monotonic functional outputs.

Finally, a point prediction approach to TP, for instance by using a neural network to predict future states of an aircraft,

may be limited as it does not fully exploit the spatio-temporal correlations available in the radar observations [

Instead, a functional approach may be a more natural way to express the predicted trajectory. This is the motivating

philosophy in the papers of Nicol et al. [

], in which functional Principal Component Analysis (fPCA) is used to

analyse aircraft trajectories. In this work we propose a method for vertical TP using a functional Gaussian Process

approach in which we marry ideas from Functional Data Analysis (FDA) with Gaussian Process Emulators. The model

produces a functional estimate of a trajectory that is informed by the physics of the problem, that is, the trajectory is

constrained to be monotonic. By exploiting the posterior variance, the predictive uncertainty of the method can be

expressed by sampling the posterior distribution. In the following section we describe the probabilistic model, before

demonstrating its application to a dataset of real ﬂight data in Section 3, where the model is baselined against BADA, a

deterministic TP code used widely in industry.

2 A Probabilistic model for climbing aircraft

We propose a probabilistic model that can express the ﬂight level,

, as a function of time,

, for an aircraft that is

cleared to climb by an Air Trafﬁc Control Ofﬁcer (ATCO). Flight levels refer to the altitude at standard air pressure,

expressed in hundreds of feet. A probabilistic model for the climb is denoted

f(t|x)

, where

x∈ X ⊆ <nx

represents a

set of

features that parameterise the climb. In what follows these features are considered to be time independent,

consisting of the instructions issued to the aircraft by the ATCO and the available data pertaining to the aircraft’s state

when the command was issued. The model is trained using a set of radar observations containing

monotonically

increasing trajectories,

D={(x(i), f(i)(t))}, i = 1 . . . nd

. One radar sweep takes approximately 6s, while the duration

of a typical climb in

is on the order of 5-10 minutes. The method consists of two parts: ﬁrstly, trajectories in

are

described with a functional, monotonic representation that is parameterised by a set of hyper-parameters,

y∈ Y ⊆ <ny

;

and secondly, a set of nyGaussian Process Emulators are deﬁned to accomplish the probabilistic mapping x→y.

2.1 Monotonic representation of functional data

Functional Principal Components Analysis (fPCA) is a popular tool in FDA for performing dimensionality reduction on

functional data. In fPCA a function is expressed as a weighted sum of orthonormal basis functions, φi(t), and a mean

function, µ(t):

f(t)≈µ(t) +

i=1

αiφi(t),(1)

where

α∈ <nc

are referred to as the Principal Component scores and

is the number of Principal Components [

fPCA has proven to provide successful representations of functional data, however, the process of ﬁnding orthogonal

basis vectors is purely data-driven and is oblivious to the physical constraints on the data generating process. For

instance, in the targeted application of vertical TP it is expected that

f(t)

will be a smooth monotonic function, with

constraints imposed on its derivative with respect to time,

∂tf

, by the performance of the aircraft. Since they are

orthogonal, the fPCA modes,

φi(t)

, are oscillatory. These basis functions are chosen such that they commit the smallest

L2error for each ncamong all possible bases.

In theory an inﬁnite number of modes are required to represent monotonic functions. However, in practice the summation

is truncated and it is therefore possible for the set of Principal Component scores,

, to correspond to a trajectory that is

not monotonic even if all the trajectories in

are themselves monotonic [

]. For this reason we, instead, employ the

monotonic representation of functional data from Ramsay [33]. Provided that fsatisﬁes the conditions that:

• log ∂tfis differentiable;

•∂tlog ∂tf=∂2

tf/∂tfis Lebesgue square integrable;

A Probabilistic Model for Aircraft in Climb A PREPRINT

Figure 2: Plot of a trajectory (red) and its reconstructions (blue) using the monotonic representation proposed here with

varying numbers of Fourier modes (left). Associated weighting functions w(t)(right).

then such a monotonic function may be represented using an integral form:

f(t) = β0+β1Zt

τ0

exp Zs

τ0

w(u)duds, (2)

where

β0

and

β1

are coefﬁcients to be determined and

w(t)

represents a square integrable function such that

w=∂2

tf/

∂tf

τ0

represents the time when the manoeuvre begins. The second condition requires that the ratio of the curvature

of the trajectory to its slope are bounded. Practically, satisfying these conditions requires the trajectory to be strictly

monotonic (i.e.

∂tf > 0

) and that

∂2

tf→0

only if

∂tf→0

. Given that we expect

∂tf

to be continuous for an aircraft

in climb, these conditions are considered reasonable for aircraft trajectories. Inspired by the work of Shin et al. [

], in

which fPCA is used to represent this function, we cast w(t)as a Fourier series:

w(t) = a0+

i=1

aicos(2πit) + bisin(2πit),(3)

where the set of coefﬁcients

a,b∈ <nw

β1

, and

are determined through Stochastic Gradient Descent (SGD) for

each trajectory in

, in which the residual sum of squares (RSS) loss between the observations and the functional

representation is minimised. In what follows we enforce the initial condition

f(τ0= 0) = fi

, where

represents

the ﬂight level of the ﬁrst observation after the clearance to climb was issued. As a consequence

β0=fi

. Figure 2

illustrates the reconstruction of a normalised trajectory in

using this formulation and the associated

w(t)

. Note that

there is some rigidity in the representation, in the example shown increasing the number of Fourier modes from 3 to 20

reduces the error in the representation of the data by 62%, which we denote

L

. However, we do not expect

L→0

nw→ ∞ because of the rigidity.

The rationale for the use of the Fourier series is to select a basis that is guaranteed to be orthogonal. This addresses a

disadvantage of the method proposed by Shin, where a transformation for the B-splines used as basis functions must be

found to orthogonalise them (which we note has the effect of adding oscillations into the resulting basis). A further

beneﬁt is that

w(t)

does not need to be integrated numerically as analytical expressions are available. The Ramsay

framework provides a bijective representation of the functional data in

. The developments in Appendix A detail

the Stochastic Gradient Descent (SGD) procedure used to ﬁnd the set of optimal parameters in the representation,

y= [ ˆ

β1,ˆa0,ˆ

a,ˆ

b]>

, that, when repeated for each of the trajectories in

, yields the set

Dy={ˆ

y(i)}, i = 1, . . . , nd

(with ny= 2nw+ 2).

2.1.1 Dimensional Reduction with Principal Component Analysis

Having described the

climbs in

with the monotonic framework of Ramsay, a model is desired to learn the mapping

between the features

and the corresponding

2nw+ 2

parameters in this framework. To simplify this mapping, a

projection of the data is performed using a Principal Component Analysis (PCA) on

. Having performed this

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APROBABILISTICMODELFORAIRCRAFTINCLIMBUSINGMONOTONICFUNCTIONALGAUSSIANPROCESSEMULATORSAPREPRINTNickPepperTheAlanTuringInstituteTheBritishLibraryLondon,UKnpepper@turing.ac.ukMarcThomasNATSGeorgeDeAthDepartmentofComputerScienceUniversityofExeterExeter,UKEnricoOliverDepartmentofComputerScienceUniversity...

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