Quantification of trajectory prediction uncertainty

EXPLORATORY RESEARCH

Quantification of trajectory prediction uncertainty DD2.2 COPTRA Grant: Call: Topic: Consortium coordinator: Edition date: Edition:

699274 H2020-SESAR-2015-1 Sesar-09-2015 CRIDA A.I.E. 07 FEBRUARY 2017 01.00.00

D2.2 QUANTIFICATION OF TRAJECTORY PREDICTION UNCERTAINTY

Authoring & Approval Authors of the document Name/Beneficiary

Position/Title

Date

E. CASADO / BR&T-E

WP02 Leader

07/02/2017

M. UZUN / ITU

WP02 Contributor

07/02/2017

J. BOUCQUEY / EUROCONTROL

WP02 Contributor

07/02/2017

S. RODRÍGUEZ / CRIDA

WP02 Contributor

07/02/2017

Name/Beneficiary

Position/Title

Date

R. JUNGERS / UCL

WP03 Leader

01/02/2017


WP04 Leader

01/02/2017

E. KOYUNCU / ITU

WP05 Leader

04/02/2017


WP06 Leader

05/02/2017

Reviewers internal to the project

Approved for submission to the SJU By — Representatives of beneficiaries involved in the project Name/Beneficiary

Position/Title

Date

R. JUNGERS / UCL

WP03 Leader

07/02/2017


WP04 Leader

07/02/2017

E. KOYUNCU / ITU

WP05 Leader

07/02/2017


WP06 Leader

07/02/2017

N. SUAREZ / CRIDA

Project Coordinator

07/02/2017

Rejected By - Representatives of beneficiaries involved in the project Name/Beneficiary

Position/Title

Date

N/A

Document History Edition

Date

Status

Author

Justification

0.1

22/11/2016

REVISED

E. CASADO

Creation

0.2

27/01/2017

REVISED

E. CASADO

Improved description of PC theory

0.3

01/02/2017

REVISED

N. SUAREZ

Minor updates and corrections

2

© – 2017 – COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions


0.4

01/02/2017

REVISED

J. BOUCQUEY

Minor updates and corrections

0.5

05/02/2017

REVISED

E. CASADO

Added Section 4

1.0

07/02/2017

REVISED

E. CASADO

Document closure

© – 2017 – COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions.

3


COPTRA COMBINING PROBABLE TRAJECTORIES This document is part of a project that has received funding from the SESAR Joint Undertaking under grant agreement No 699274 under European Union’s Horizon 2020 research and innovation programme.

Abstract This document defines the Uncertainty Quantification (UQ) technique that will be used throughout the COPTRA project to assess the uncertainty associated to individual trajectory predictions. The approach described herein is classified within the forward uncertainty propagation methods that aim at determining the statistical distribution of a model output obtained by assessing the impact on them of the considered random inputs. It is based on the Polynomial Chaos (PC) theory, which relies on a polynomial description of the random inputs and a mathematical formulation of the polynomial expansions that characterize the model outputs built upon the input’s expansions. This is a nonintrusive approach that only requires the computation of the original model a number of times to obtain the polynomial descriptions of the outputs. Among all possible applications of the PC theory, this deliverable focuses on the so-called Arbitrary PC Expansions (aPCE) that takes advantage of actual data to characterize the inputs uncertainty.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-ncnd/4.0/.

4



Table of Contents 1

2

3

Introduction ............................................................................................................... 7 1.1

Purpose and Scope ........................................................................................................ 7

1.2

Intended Readership ..................................................................................................... 7

1.3

Acronyms and Terminology ........................................................................................... 8

Trajectory Prediction Infrastructure .......................................................................... 11 2.1

Introduction ................................................................................................................ 11

2.2

Trajectory Prediction Approach .................................................................................... 11

2.3

The Aircraft Intent Description Language...................................................................... 12

Formal Quantification of Trajectory Prediction Uncertainties .................................... 15 3.1

Introduction ................................................................................................................ 15

3.2

Polynomial Chaos Theory ............................................................................................. 15

3.2.1 3.2.2

4

aPCE application to the TP infrastructure ................................................................. 20 4.1

Introduction ................................................................................................................ 20

4.2

Application of aPCE to Trajectory Prediction ................................................................. 20

4.3

Example of application ................................................................................................ 21

4.3.1 4.3.2 4.3.3 4.3.4 4.3.5

5

Generalized PCE .............................................................................................................................. 16 Arbitrary PCE ................................................................................................................................... 17

AI uncertainty .................................................................................................................................. 21 Initial Condition uncertainty ........................................................................................................... 22 Weather forecast uncertainty ......................................................................................................... 23 APM uncertainty ............................................................................................................................. 23 Predicted Entry and Exit times uncertainty ..................................................................................... 23

Final Remarks .......................................................................................................... 26

References ...................................................................................................................... 27


5


List of Figures Figure 1 Aircraft Trajectory Prediction Architecture ________________________________________________ 11 Figure 2 AIDL Alphabet and Grammar Rules _____________________________________________________ 14 Figure 3 AIDL Alphabet and Grammar Rules _____________________________________________________ 22

List of Tables Table 1: Acronyms and Terminology ____________________________________________________________ 10 Table 2: Germs’ Distributions and related Polynomial Basis _________________________________________ 17 Table 3: Probability Distribution that Characterize the AI Uncertainty _________________________________ 22 Table 4: Initial Conditions ____________________________________________________________________ 23 Table 5: PCE Definitions related to the Entry and Exit Times _________________________________________ 24 Table 6: Univariate Polynomial Basis ___________________________________________________________ 25 Table 7: Entry & Exit Times Statistics ___________________________________________________________ 25

6



1 Introduction1

1.1 Purpose and Scope This document intends to expose the mathematical formulation of the UQ framework to be used within WP02 to quantify the uncertainty of individual trajectory predictions. This document presents the trajectory prediction infrastructure that will implement a 3 degrees of freedom (3 DOF) model representing the Aircraft Motion Model (AMM). Based on this formulation of the prediction problem and the identification of sources of prediction uncertainty referred within the COPTRA deliverable D2.1 [1], an uncertainty quantification method will be proposed. This document includes a mathematical formulation of the PC theory, with a special focus on its application to the problem of quantifying trajectory prediction uncertainty. This formulation will be used within COPTRA to provide measurements of the uncertainty associated to individual trajectories. This prediction uncertainty quantification will be used by WP03 as input to assess the uncertainty of traffic predictions.

1.2 Intended Readership This document is to be used by the COPTRA consortium to guide the research and development to be conducted within WP02. It introduces the theory of Polynomial Chaos and exposes how to apply it to quantify the uncertainty associated to an individual trajectory prediction. An example of application of this theory to assess the stochastic variability of the entry and exit times into an airspace sector has been also included in this deliverable. COPTRA addresses a very specific aspect of TBO related with the ability to help demand-capacity and complexity management as well as planning through the identification and management of uncertainty (both at trajectory and traffic levels) as expressed in the S2020 advanced Demand & Capacity Balance (DCB) concept. The added value that this deliverable brings into the SESAR 2020 programme is mainly the provision of a probabilistic trajectory predictor to S2020 PJ09.01 “Advanced Demand and Capacity Balance” and orientation to the assessment of integrating trajectory uncertainty models into existing tools. Furthermore, the final value of the project should be the provision of a traffic prediction based on probabilistic traffic situations to S2020 PJ09 (Network Prediction and Performance).

The opinions expressed herein reflect the author’s view only. Under no circumstances shall the SESAR Joint Undertaking be responsible for any use that may be made of the information contained herein. 1


7


1.3 Acronyms and Terminology

8

Term

Definition

AI

Aircraft Intent

AIDL

Aircraft Intent Description Language

AMM

Aircraft Motion Model

aPCE

Arbitrary Polynomial Chaos Expansion

APM

Aircraft Performance Model

ATM

Air Traffic Management

BADA

Base of Aircraft Data

BR&T-E

Boeing Research & Technology – Europe

CAS

Calibrated Airspeed

CDM

Collaborative Decision Making

COPTRA

Combining Probable Trajectories

CRIDA

Centro de Referencia de Investigación, Desarrollo e Innovación

D

Drag

DAE

Differential Algebraic Equation

DCB

Demand & Capacity Balance

DOF

Degree of Freedom

DST

Decision Support Tools

EM

Earth Model

EUROCONTROL

European Organization for the Safety of Air Navigation

F

Fuel Consumption

FL

Flight Level

FMS

Flight Management System

GFS

Global Forecast System

GMT

Greenwich Mean Time

gPCE

Generalized Polynomial Chaos Expansion

h

Geodetic altitude

HA

Hold Altitude

HC

Hold Course

HHL

Hold High Lift devices



HL

High Lift devices

HLG

Hold Landing Gear

Hp

Pressure altitude

HS

Hold Speed

HSB

Hold Speed Breaks

ISA

International Standard Atmosphere

ITU

Istanbul Technical University

LG

Landing Gear

LIDL

Low Idle Engine Regime

M

Mach Number

m

mass

NOAA

National Oceanic and Atmospheric Administration

ODE

Ordinary Differential Equation

PC

Polynomial Chaos

PCE

Polynomial Chaos Expansion

PDF

Probability Density Functions

PMM

Point-Mass Model

SB

Speed Breaks

SESAR

Single European Sky ATM Research Programme

SJU

SESAR Joint Undertaking (Agency of the European Commission)

T

Thrust

TAS

True Airspeed

TBO

Trajectory Based Operations

TCE

Trajectory Computation Engine

TCI

Trajectory Computation Infrastructure

TL

Throttle Law

TLP

Track Lateral Path

TM

Trajectory Management

TOC

Top of Climb

TOD

Top of Descent

TP

Trajectory Predictor

UCL

Université Catholique de Louvain


9


UQ

Uncertainty Quantification

WP

Work Package

Wx

North wind component

Wy

West wind component

φ

Latitude

λ

Longitude

χ

Bearing

ψ

Heading TABLE 1: ACRONYMS AND TERMINOLOGY

10



2 Trajectory Prediction Infrastructure

2.1 Introduction This Section describes the trajectory prediction process upon which the UQ techniques will be applied to quantify trajectory prediction uncertainty. The Trajectory Predictor (TP) architecture used for the proposed analysis is also described. This approach facilitates the identification of the uncertainty sources because it decouples the different data models required for predicting an aircraft trajectory.

2.2 Trajectory Prediction Approach Aircraft trajectory prediction is a well-known problem that has been studied for years. Although there are different alternatives to address the problem, some elements are common among TP implementations [2]. Regardless of the approach followed to obtain a prediction, the aircraft motion is usually expressed as a function of the current aircraft state, an estimation of pilot or Flight Management System (FMS) intent, meteorological forecasts, and the knowledge of the aircraft performance. The main difference between current TP implementations and those envisioned in the future Trajectory Based Operations (TBO) environment is the availability of aircraft intent (AI) information. The synchronisation of such information between onboard and ground systems will increase predictions reliability and accuracy because of a better awareness about aircraft behaviour in the short and medium timeframes [3]. FIGURE 1 AIRCRAFT TRAJECTORY PREDICTION ARCHITECTURE TRAJECTORY COMPUTATION INFRASTRUCTURE

Trajectory Computation Engine

INITIAL CONDITIONS

COMPUTED TRAJECTORY

Motion Profiles Configuration Profiles

AIRCRAFT INTENT

Pilot event

1st DOF

M=0.78 HA (P)

CAS=280kt HS (M)

ST (T)

VSL (ROC)

SBA

TLP (GC)

HS

Vertical

TA

M .78 280 KCAS 4500ft

180 KCAS

R?

N370945.72 W0032438.01

Time

HS (ABS)

d

d

? TLP (CRT)

SBA

Capture of target bank

Roll-in anticipation

HL

CAS=210kt HA (GEO)

TOD

Capture of target bank

HT (T) ?

End of engine transient to idle

3rd DOF

h=4500ft HS (CAS)

Roll-in anticipation

d

2nd DOF HS (M)

Weather Model

Horizontal

AIRCRAFT INTENT

Aircraft Performance Model

AIRCRAFT TRAJECTORY

FL320 M .88

HHL

yBFS HC (GEO)

? SHL

HHL

Auto beginning of HL devices deployment

SB

HSB

LG

HLG

xBFS

zBFS


11


A typical architecture of kinetic TP approach is depicted in Figure 1, where input datasets are clearly identified. This solution considers that at least a basic knowledge of aircraft intent is available. The process of generating an aircraft intent instance from a flight intent instance or a flight plan is out of the scope of this study. The main advantage of this approach is the capability to decouple the uncertainty sources, leading to separate and uncorrelated analyses of their individual influences. The required inputs to the prediction process are:    

Initial conditions. Aircraft initial state (e.g., position, speed, and mass) from which the trajectory will be predicted. AI description. An instance of aircraft intent comprises a set of instructions that describe, in an unambiguous manner, how the aircraft is to be operated during the time interval. Atmospheric conditions. A model of the wind field and atmosphere conditions (temperature and pressure deviation with respect to the International Standard Atmosphere [ISA]) is necessary to compute a trajectory. Aircraft performance. Information about the drag (D), thrust (T) and fuel consumption (F) needs to be provided to solve the mathematical formulation of the AMM.

2.3 The Aircraft Intent Description Language As depicted in Figure 1, the considered Trajectory Computation Infrastructure (TCI) is divided into three different modules. Its key component is the trajectory computation engine (TCE), responsible for executing the trajectory computation process by means of the integration of the equations of motion in combination with the AI instructions. The trajectory computation process requires an Aircraft Performance Model (APM) that provides models for the different aircraft performance aspects that appear in the equations. The computation infrastructure is not complete without the presence of an earth model (EM) to provide, among others, the values of the atmospheric conditions that appear in both the equations of motion and the performance models. The trajectory engine relies on a collection of resolution strategies and numerical recipes to integrate the equations describing the aircraft motion into the predicted trajectory, which resembles the actual trajectory of the real flight. The deviations between the actual and the predicted trajectory result from the differences between the real AI, aircraft performance and weather conditions and the models used to represent them. There is an association between the aircraft intent that can be accepted by a certain TCE and the equations of motion that it integrates to obtain the resulting aircraft trajectory, as the TCE must be able to interpret any valid instance of aircraft intent and integrate it to obtain the predicted trajectory. The equations of motion shown below are considered sufficient to accurately describe the aircraft motion in an ATM context and contain the following assumptions:   

thrust force is parallel with the airspeed; the variation with time of the aircraft path angle is small when compared with the other terms in the dynamic equations, so the path angle rate can be removed from the corresponding differential equation, converting it into an algebraic expression; symmetric flight, in which the airspeed vector is contained within the aircraft plane of symmetry. 𝑑𝑣𝑇𝐴𝑆 𝑑𝑡

12

−

̇ 𝑇−𝐷−𝑊𝑠𝑖𝑛𝛾 𝑇𝐴𝑆 𝑚


+

𝑑𝑤1𝑊𝐹𝑆 𝑑𝑡

=0


𝑑𝛾𝑇𝐴𝑆 𝑑𝑡

𝐿𝑐𝑜𝑠𝜇̇ 𝑇𝐴𝑆 −𝑊𝑐𝑜𝑠𝛾𝑇𝐴𝑆 𝑚 𝑇𝐴𝑆

−𝑣

𝑑𝜒𝑇𝐴𝑆 𝑑𝑡

1

[

1 ̇ 𝐿𝑠𝑖𝑛𝜇 [ 𝑚𝑇𝐴𝑆 𝑇𝐴𝑆 𝑐𝑜𝑠𝛾𝑇𝐴𝑆

−𝑣

𝑑𝑚 𝑑𝑡

𝑑𝑤3𝑊𝐹𝑆 𝑐𝑜𝑠𝜇𝑇𝐴𝑆 𝑑𝑡

+(

𝑑𝑤3𝑊𝐹𝑆 𝑠𝑖𝑛𝜇𝑇𝐴𝑆 𝑑𝑡

+(

−

+

𝑑𝑤2𝑊𝐹𝑆 𝑐𝑜𝑠𝜇𝑇𝐴𝑆 𝑑𝑡

)] = 0

)] = 0

+𝑓 =0

𝑑𝜆 𝑑𝑡

−

𝑣𝑇𝐴𝑆 𝑐𝑜𝑠𝛾𝑇𝐴𝑆 𝑠𝑖𝑛𝜒𝑇𝐴𝑆 +𝑤2𝑊𝐹𝑆 (𝑁+ℎ)𝑐𝑜𝑠𝜑

=0

𝑑𝜑 𝑑𝑡

−

𝑣𝑇𝐴𝑆 𝑐𝑜𝑠𝛾𝑇𝐴𝑆 𝑐𝑜𝑠𝜒𝑇𝐴𝑆 +𝑤1𝑊𝐹𝑆 (𝑀+ℎ)

=0

𝑑ℎ 𝑑𝑡

𝑑𝑤2𝑊𝐹𝑆 𝑠𝑖𝑛𝜇𝑇𝐴𝑆 𝑑𝑡

− 𝑣𝑇𝐴𝑆 𝑠𝑖𝑛𝛾𝑇𝐴𝑆 = 0

The mathematical problem has one independent variable [t], 10 dependent variables: true airspeed [vTAS]. Aerodynamic path angel [γTAS], aerodynamic heading [χTAS], aircraft mass [m ], longitude [λ], latitude [φ], geodetic altitude [h], throttle parameter [δT], aerodynamic bank angle [μTAS] and lift [L]; and 7 equations. This is a 3 degrees of freedom (DOF) problem whose output, for a given set of 3 input variables [δT μTAS L], is the 7 state variables [vTAS γTAS χTAS m λ φ h]. The described system of equations is valid for a given aerodynamic configuration, and so then, it is required to define it in advance to be able to solve the system. The described system of equations is valid for a given aerodynamic configuration, and so then, it is required to define it in advance to be able to solve the system. The equations shall be combined with those of the APM and the EM to form the mathematical problem whose solution provides the aircraft trajectory. The main purpose of the APM is to provide the aircraft response (forces and fuel consumption) to given command and control inputs based on its position, airspeed, and the atmospheric pressure and temperature. It also provides the operational envelope that ensures that these expressions are valid, as well as the transients involved in those manoeuvres that cannot be modelled by the equations of motion. The EM in turn provides the atmospheric pressure and temperature, wind, gravity, and magnetic deviation based on the aircraft position and time. Finally, an AI instance describes six algebraic equations that complement the above differential equations univocally defining an aircraft trajectory. This six algebraic equations or constraints transform the Ordinary Differential Equation (ODE) system into a Differential Algebraic Equation (DAE) system. The rules that ensure that a set of six constraints define a solvable DAE system with a unique solution are stated by the Aircraft Intent Description Language (AIDL) [4]. The AIDL is a formal language intended to express aircraft intent in a univocal, rigorous, and standardized manner. As a formal language, the AIDL is defined over the finite set of instructions that can be executed by a certain TCE [5]. The instructions belonging to that set, which comprise the AIDL alphabet, are said to be executable by the TCE in question. In addition to being the alphabet symbols, the AIDL instructions also have certain features employed by the language grammar. The AIDL grammar comprises the set of rules according to which the instructions (or alphabet symbols) can be combined into valid instances of aircraft intent (or language strings). It contains rules governing how to combine instructions both sequentially (instructions with contiguous, non-overlapping execution intervals) and simultaneously (instructions with overlapping action intervals). The AIDL rules are based


13


on the instructions features, and are necessary to ensure that the resulting aircraft intent defines the trajectory to be computed in an unambiguous manner and according to the model of the aircraft motion upon which the TCE relies. Following Figure 2 shows the list of possible instructions and summarizes the grammar and lexical rules that govern the generation of a valid AI instance. FIGURE 2 AIDL ALPHABET AND GRAMMAR RULES

The AIDL will be used in WP02 to univocally describe the trajectories in a formal manner. This will significantly help the process of identifying and characterizing the uncertainty associated to the description of an AI instance.

14



3 Formal Quantification of Trajectory Prediction Uncertainties

3.1 Introduction Among all possible techniques that can be applied to quantify uncertainty, Polynomial Chaos (PC) theory has been selected due to its capability to provide analytical representations of the uncertainty propagation and its computational efficiency, especially, with respect to traditional Monte Carlo approaches. This Section provides a description of the PC theory, considering it as the most suitable technique for trajectory prediction UQ compliant with the expected flexibility (i.e., applicability to all possible prediction within the ATM environment) and efficiency (e.g., capability of providing analytical quantification with low time-consuming processes). A summary of the PCE theoretical basis, with a special focus on arbitrary PCE (aPCE) is included in the following Section 3.2.

3.2 Polynomial Chaos Theory A Polynomial Chaos Expansion (PCE) is a mechanism to represent a stochastic random variable z by means of a basis of polynomials of another random variable ξ [6]. 𝑧 ~ f(𝜉)



Equation (2) is read as “z is distributed as f(ξ)”, meaning that the probabilistic distribution that represents the stochastic behaviour of z is the same as that representing f(ξ), where the variable is known ξ as the germ of the distribution. Given distributions for z and ξ, there is no a unique function f that satisfies (2). Unlike, there is a variety of functions that could build the random variable z form the selected germ ξ. Furthermore, additional representations are plausible using different germs rather than ξ. PCE is an approach that expands function f in a polynomial series. The polynomial basis (𝜓𝑖 ) is a set of orthogonal polynomials with respect to the probability density (PDF) function Γ(ξ) of the germ ξ, 〈𝜓𝑖 , 𝜓𝑗 〉 = ∫ 𝜓𝑖 (𝜉) ∙ 𝜓𝑗 (𝜉) ∙ Γ(𝜉) 𝑑𝜉 = 𝛿𝑖𝑗



where δij is the Kronecker delta. A very important feature of such polynomial basis is that all polynomials of order i≥1 have zero mean due to their orthogonality with 𝜓0 , and that the covariance between two polynomials of different order is zero (uncorrelated polynomials) because of 〈𝜓𝑖 , 𝜓𝑗 〉 = 0; ∀ i≠j. Thus, it is possible to build an orthonormal basis of polynomials by assuming 〈𝜓𝑖 , 𝜓𝑖 〉 = 1 ∀ i.


15


Making use of the orthonormal basis of polynomials, the relationship stated in (2) can be formulated as follows: ∞



𝑧 = ∑ 𝑎𝑖 𝜓𝑖 (𝜉) 𝑖=1

The PC theory based on Wiener-Askey theory of homogeneous chaos [7], defines the i-th mode as the combination of ai (mode strength) and 𝜓𝑖 (𝜉) (mode function). Since there are many possible functions f that may satisfy (2), there a multiple PCEs for a given z using a determined germ ξ that will only differ in the mode strengths. The exposed formulation is also valid for representing stochastic random variables in which ξ is a vector comprising N multiple germs {ξ1, ξ2, … , ξN}. In this case, the stochastic variable z will be represented by the following expansion: ∞

𝑧 = ∑ 𝑏𝑖 𝜙𝑖 (𝜉1 , 𝜉2 , … , 𝜉𝑁 )



𝑖=1

Where 𝜙𝑖 is a tensor product of the univariate polynomial bases of each ξj. 𝑁

𝛼𝑖

𝜙𝑖 (𝜉1 , 𝜉2 , … , 𝜉𝑁 ) = ∏ 𝜓𝑗 𝑗 (𝜉𝑗 )



𝑗=1 𝑁

∑ 𝛼𝑗𝑖 ≤ 𝑝,

𝑖 = 1, … , 𝑁



𝑗=1

The multivariate index 𝛼𝑗𝑖 represents the combinatory of all possible products of 𝜓𝑗𝑘 (polynomial of order k belonging to the polynomial basis of germ ξj), where p is the number of expansion terms and depends on the number of germs N and the order of the expansion d (order at which the expansion is truncated). 𝑝=

(𝑁 + 𝑑)! 𝑁! 𝑑!



The index 𝛼𝑗𝑖 is a p x N matrix that represents the corresponding expansion degree for germ j in expansion term i. The unknown coefficients ai (univariate) or bi (multivariate) of the PCE can be obtained by projecting each variable on the polynomial basis, such as the Galerkin projection method [8], or by estimating them form a limited number of simulations applying regression techniques, such as the probabilistic collocation method (computationally more efficient in most cases) [9].

3.2.1 Generalized PCE Originally PC theory was applied only to Gaussian stochastic processes, although rapidly its applicability was extended to more general stochastic processes. The generalized PC (gPC) approach [10] considers

16



that both inputs and outputs can be represented by known random distributions. For those other nonnormal distributions, there are already known basis of polynomials that allow the identification of the corresponding PCE. Table 2shows the relationship between germs’ distributions and their associated orthogonal polynomials of the Askey scheme. Germ ξ Gaussian Gamma Beta Uniform Poisson Binomial Negative binomial Hypergeometric

Polynomial Basis Hermite Laguerre Jacobi Legendre Charlier Krawtchouk Meixner Hahn

TABLE 2: GERMS’ DISTRIBUTIONS AND RELATED POLYNOMIAL BASIS

This approach considers that the germs {ξ1, ξ2, … , ξN} are independent, identically distributed and can be represented by one of the distributions presented above.

3.2.2 Arbitrary PCE The gPC method requires the exact knowledge of the germs, which is not the case in most real systems. To manage incomplete or/and implicit distributions only defined by their statistical moments, the use of Gram-Schmidt orthogonalisation led to the definition of the arbitrary PC (aPC) method [11]. In aPC, the exact probability description of the germs is not strictly necessary. For a finite-order expansion, only a finite number of statistical moments is required. This method enables data-driven applications of the PC theory, in which data samples with limited size allow the inference of the polynomial description of the system outputs as a result of the impact of uncertain inputs described by arbitrary distributions (e.g., discrete, continuous, or discretized continuous). This approach requires from the construction of the polynomial basis representing the stochastic behaviour of each germ ξi by means of the statistical moments calculated from data. To obtain such basis, orthogonality is imposed. In addition, the coefficient of the term of highest order is set to 1 for of all polynomials of the basis. Considering that, the expansion of the polynomial of order k belonging to the polynomial basis of germ ξi can be expressed as 𝑘

𝜓𝑖𝑘

(𝑘)

= ∑ 𝑐𝑚 𝜉𝑖𝑚



𝑚=0 (𝑘)

𝑐𝑘 = 1



The zero-order polynomial can be immediately obtained: (0)

𝜓𝑖0 = 𝑐0 𝜉𝑖0 = 1



The basis is successively constructed for all remaining polynomials of the ξi basis up to the selected order d by solving the following system of equations based on (12):


17


𝑘 (0) ∫ 𝑐0

(𝑘)

∙ [ ∑ 𝑐𝑚 𝜉𝑖𝑚 ] ∙ Γ(𝜉𝑖 ) 𝑑𝜉𝑖 = 0 𝑚=0 𝑘

1

∫[∑

(1) 𝑐𝑚 𝜉𝑖𝑚 ] ∙

(𝑘)

[ ∑ 𝑐𝑚 𝜉𝑖𝑚 ] ∙ Γ(𝜉𝑖 ) 𝑑𝜉𝑖 = 0

𝑚=0

𝑚=0



⋮ 𝑘−1

∫[∑

𝑘 (𝑘−1) 𝑚 𝑐𝑚 𝜉𝑖 ] ∙

(𝑘)

[ ∑ 𝑐𝑚 𝜉𝑖𝑚 ] ∙ Γ(𝜉𝑖 ) 𝑑𝜉𝑖 = 0

𝑚=0

𝑚=0 (𝑘)

𝑐𝑘 = 1 This system of equations can be rearranged if the first equation is introduced into the second, the first and the second into the third and so on, and considering the condition defined by (9). The system of equations turns into: 𝑘 (𝑘)

∫ ∑ 𝑐𝑚 𝜉𝑖𝑚 ∙ Γ(𝜉𝑖 ) 𝑑𝜉𝑖 = 0 𝑚=0 𝑘 (𝑘)

∫ ∑ 𝑐𝑚 𝜉𝑖𝑚+1 ∙ Γ(𝜉𝑖 ) 𝑑𝜉𝑖 = 0 𝑚=0

⋮

𝑘



(𝑘)

∫ ∑ 𝑐𝑚 𝜉𝑖𝑚+𝑘−1 ∙ Γ(𝜉𝑖 ) 𝑑𝜉𝑖 = 0 𝑚=0

(𝑘)

𝑐𝑘 = 1 The m-th statistical moment of the germ ξi can be defined as: 𝜇𝑘 = ∫ 𝜉𝑖𝑚 ∙ Γ(𝜉𝑖 ) 𝑑𝜉𝑖



Thus, the system of equations posed in (12) turns into: 𝑘 (𝑘)

∑ 𝑐𝑚 𝜇𝑚 = 0 𝑚=0 𝑘 (𝑘)

∑ 𝑐𝑚 𝜇𝑚+1 = 0 𝑚=0

⋮

𝑘 (𝑘)

∑ 𝑐𝑚 𝜇𝑚+𝑘−1 = 0 𝑚=0

(𝑘)

𝑐𝑘 = 1 or alternatively written in a matrix form as:

18





𝜇0 𝜇1 ⋮ 𝜇𝑘−1 [ 0

𝜇1 𝜇2 ⋮ 𝜇𝑘 0

⋯ ⋯ ⋮ ⋯ ⋯

(𝑘)

𝑐0 𝜇𝑘 (𝑘) 𝜇𝑘+1 𝑐1 ⋮ ⋮ = 𝜇2𝑘−1 𝑐 (𝑘) 𝑘−1 1 ] (𝑘) [ 𝑐𝑘 ]

0 0 ⋮ 0 [1]



(𝑘)

The coefficients 𝑐𝑚 of each of the polynomials belonging to the basis of the germ ξi can be computed if and only if all moments of the ξi distribution up to order 2k-1 are finite. This ensures that the moment matrix in (16) is not singular and, therefore, the linear system is solvable. Hence, only the capability of obtaining the 2k-1 moments of the ξi distribution is required to apply aPC, avoiding the need of having an explicit description of the associated probability density function of the ξi distribution.


19


4 aPCE application to the TP infrastructure

4.1 Introduction This section proposes an example of how the PC approach exposed in previous sections will be applied to compute the uncertainty of a sector entry and exit times. These times will be used to quantify of the demand and capacity balance uncertainty of the selected sector. First, a theoretical introduction to the concept of non-intrusive aPCE approaches is presented. This will provide the means to apply this approach to the problem of quantifying trajectory prediction uncertainty.

4.2 Application of aPCE to Trajectory Prediction Once the trajectory prediction model under study is determined and the sources of input uncertainties are identified and characterized (as described in D2.1 – Section 4), it is possible to assess how uncertainty propagates into the outputs by applying the PC theory. The process aims at obtaining the mode strengths bi of the model outcomes. There are basically two approaches [12] that can be followed: 

Intrusive approach, which proposes to substitute the inputs to the model by the related PCE and solve the system of equations to obtain PCE of the outputs. For instance, Galerkin projection takes advantage of the orthogonality of the polynomial basis 𝜓𝑖 to define a system of equations that returns bi. The main drawbacks of this solution are: an explicit mathematical representation of the model is a must; the system of equations to be solved is of higher complexity than the original one; a modification of the original solver is required to obtain the solution; it is usually tailored to a specific model and its implementation cannot be extended to other models; and if additional germs are to be considered in the description of the outputs, a reformulation of the system of equations is required.



Non-intrusive approach, which treats the model as a black-box. Inputs are sampled to obtain the set of corresponding outputs from which the mode strengths bi can be calculated by regression methods. Main advantages of this solution are: an explicit representation of the model is not a must; mode strengths are easily obtained; any modification of the outputs PCE can be straightforward assessed; it does not imply any modification of the original definition of the model specification; and it can be applicable to different models by just obtaining the corresponding outputs of the selected input sampling.

Based on the drawbacks of the intrusive approaches and the advantages of non-intrusive ones, the research presented in this document follows the so-called non-intrusive Probabilistic Collocation Method (PCM) [13]. The PCM establishes at which collocation points {(ξ1,1 ,…,ξN,1), …, (ξ1,q ,…,ξN,q)} the model needs to be evaluated to obtain the intentioned set of outputs that enable the computation of the mode strengths of the PCE outputs. The collocation points related to the germ ξi are obtained as

20



the roots of the polynomial 𝜓𝑑+1 (𝜉𝑖 ) of next higher order than the order of the polynomial 𝜓𝑑 (𝜉𝑖 ) at which the PCE is to be truncated. In the case of multivariate problems with N independent variables, the number of collocation points reaches up to q = (d+1)N corresponding to the N combinations of the (d+1) roots of the univariate polynomial expansions. Evaluating the model at those computed collocation points, the following system of linear equations returns the mode strengths by applying regression techniques (e.g., least square fitting).

4.3 Example of application The main objective of WP02 is to provide a representation of the uncertainty related to an individual aircraft trajectory prediction. The mathematical approach formulated above provides this capability thanks to a polynomial representation of the uncertainty of the trajectory prediction inputs. According to the PC theory, these polynomial expansions can be properly combined to characterize the uncertainty of prediction outputs. The following trajectory prediction problem exemplifies the usage of the aPCE approach to compute the uncertainty of entry and exit times based on a theoretical characterization of inputs uncertainty2.

4.3.1 AI uncertainty Thanks to the use of the AIDL, it is possible to describe the trajectory as a chronologically ordered sequence of operations that univocally represent the trajectory. The main assumption is to consider the aircraft operated in clean configuration throughout the complete trajectory. Hence, it is only required to specify the sequence of AIDL instructions along the lateral and the two longitudinal threads. The lateral thread describes the trajectory in the horizontal plane, while the two longitudinal ones do it in the vertical plane. The aircraft will fly 300 km (sector entry point) on the geodesic between waypoints A and B, describing a circular arc of radius (R = 10 km) around B up to capturing a bearing χc of -4º respect to the magnetic North. Once this bearing is captured, the aircraft will follow the geodesic defined by the circular arc end point and the waypoint C. The vertical profile is described by an initial climb from the initial conditions at maximum climb (MCMB) rating and constant CAS0 up to the transition altitude at which the Mach cruise speed (Mc = 0.7) is reached. From this point the climb is performed at constant Mach speed until reaching the Top of Climb (TOC) at flight level FL350. This condition defines the beginning of the cruise phase, which is executed at a constant Mach number and FL. This phase ends at the Top of Descent (TOD), defined by a point 400km (sector exit point) away from the beginning of the trajectory. Following the TOD, the aircraft initiates the descending phase at a constant Mach speed and low idle engine regime (LIDL) up to the transition altitude at which the descent is performed at constant CAS (CASd = 260kn). The trajectory ends once the aircraft reaches 10,000ft of pressure altitude. For the sake of clarity, Figure 3 only shows the three motion profiles required to describe the considered trajectory out of the six required as by the AIDL grammar rules.

2

This document will be updated in October 2017 constituting the Deliverable 2.3 and will contain a detailed characterization of inputs uncertainty based on the exploitation of real data. In this current release, only theoretical distributions have been applied to provide an example of how to apply the proposed theory. © – 2017 – COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions.

21


Mc = 0.7

Initial Conditions

HS(MACH) TOD -> FD2 = 400km

TL(MCMB)

HA(PRE)

OP#1

M = 0.77

OP#2

TL(LIDL)

χc = -4º

FD1 = 300km TLP(Geodesic)

TLP(Circular Arc)

OP#3

Hp = 10,000ft

HS(CAS)

TOC -> FLc = 350

OPERATIONS

OP#4

HC(GEO)

OP#6

OP#5

OP#7

TOC

Vertical

TOD CAS = 260kt

A Hp = 10,000ft

B

Horizontal

TRAJECTROY DESCRIPTION

CASd = 260kt

HS(CAS)

Motion Profiles

AIRCRAFT INTNET

FIGURE 3 AIDL ALPHABET AND GRAMMAR RULES

M = 0.77

TOC

TOD

A

C

R

FD1 = 300km

CAS = 260kt

-4º Magnetic North

Hp = 10,000ft

The following parameters have been considered sources of AI uncertainty: 

cruise Mach speed (Mc)



cruise pressure altitude (FLc), and



TOD location (FD2)

Table 3 shows the probability density functions (PDFs) that have been used to characterize the variability of the selected sources of AI uncertainty.

Mc [-] 𝐔𝐍𝐈(𝟎. 𝟕𝟏, 𝟎. 𝟔𝟗)a

FLc [FL]

FD2 [km]

𝓝(𝟑𝟓𝟎, 𝟓)b

𝓝(𝟒𝟎𝟎, 𝟏𝟎)

a. UNI(a,b) - Uniform distribution between a and b b. 𝓝(μ,σ) - Gaussian distribution of mean μ and standard deviation σ

TABLE 3: PROBABILITY DISTRIBUTION THAT CHARACTERIZE THE AI UNCERTAINTY

4.3.2 Initial Condition uncertainty

22



Considering the selected aircraft type and the AI description, the set of initial conditions that describe the first aircraft state, from which the trajectory is to be predicted, is included in following Table 4.

t0

λ0

φ0

m0

CAS0

Hp0

5:20 GMT

9º N

58º E

70 ton

245 kt

3,000 ft

TABLE 4: INITIAL CONDITIONS

The initial time has been considered as source of IC uncertainty. The normal distribution 𝒩(100, 10) represents the variability in seconds of the departure time referred to the nominal t0.

4.3.3 Weather forecast uncertainty The selected day of operation is 2016 February 14. Weather forecasts for such date downloaded from the National Oceanic and Atmospheric Administration (NOAA) website will be used as a representation of the weather conditions affecting the trajectory. In addition to the numerical weather predictions, weather forecast providers include, as a product, the associated standard deviations of the atmospheric variables at each grid point. They are computed by intentionally perturbing the initial atmosphere state to assess stochastic effects on the forecast. Considering that all potential weather conditions are a priori equi-probable, then the variability of the atmospheric variables can be represented by uniform distributions. Normalizing all distributions to a uniform distributions in the interval [-1,1], it is possible to reduce the weather uncertainty characterization to the definition of a unique parameter.

4.3.4 APM uncertainty The selected aircraft type will be a Boeing 737-800 equipped with CFM56-7B26/27 engines developed by CFMI, joint-owned company of Safran Aircraft Engines and GE Aviation. The B738W26 dataset included in the release 4.1 of BADA (Base of Aircraft Data) [14] provides the required performance models. The independent coefficient of the BADA 4.1 drag polar coefficient model (d0) has been chosen as source of APM uncertainty. According to the BADA 4.1 dataset of the referred aircraft type, the triangular distribution TRI(−5.5335, −5.5331) will be used to represent the variability of the aircraft performance. This is a distribution between -5.5335 and -5.5331 with mode -5.5335.

4.3.5 Predicted Entry and Exit times uncertainty The aPCE method only requires the computation of the statistic moments of the PDFs that represent each individual source of uncertainty. According to (8), the number of mode strengths (p) will depend on the selected order of the PCE (d) and the number of uncertainty sources (i.e., germs). In the proposed example, the number of considered inputs N is 6, while d has set to 2. Thus, it is necessary to compute 28 coefficients bi solving the linear system of equations (16) for both the entry and exit times of the designated sector.


23


The following Table 5 collects the polynomials and related coefficients that characterize the variability of the entry and exit times as defined by (5), (6) and (7).

𝒏

𝒐𝒌𝒋

𝒌

𝒆𝒏𝒕𝒓𝒚

𝝓𝒊

∑ 𝒐𝒌𝒋

𝒃𝒊

∙ 𝟏𝟎−𝟑

𝒃𝒆𝒙𝒊𝒕 ∙ 𝟏𝟎−𝟑 𝒊

𝒋=𝟏

0 0

0 0

0 0

0 0

0 1

0 0

1 2

0

𝜓10 ∙ 𝜓20 ∙ 𝜓30 ∙ 𝜓40 ∙ 𝜓50 ∙ 𝜓60

1

𝜓10

∙

𝜓20

∙

𝜓20

∙

𝜓30

∙

𝜓30

∙

𝜓40

∙

𝜓41

∙

𝜓51

∙

𝜓50

0.000581382273221

0.000553463337982

∙

𝜓60

-0.000000000025871

-0.000000000045291

0

0

1

0

0

3

1

𝜓10

0

0

1

0

0

0

4

1

𝜓10 ∙ 𝜓20 ∙ 𝜓31 ∙ 𝜓40 ∙ 𝜓50 ∙ 𝜓60

1

𝜓10

1

𝜓11

0 1

1 0

0 0

0 0

0 0

0 0

5 6

∙ ∙

𝜓20

∙

𝜓20

∙

𝜓30

∙

𝜓30

∙

𝜓30

∙

𝜓40

∙

𝜓40

∙

𝜓40

∙

𝜓50

∙

𝜓50

∙

𝜓50

2.157499790357981

∙

0

𝜓21

1.674819942061835

𝜓60

0.009781741719567

0.009423029609186

∙

𝜓60

-0.000000000028927

-0.000000000050128

∙

𝜓60

0.007672991681747

0.007390661289936

∙

𝜓61

0

0

0

0

0

1

7

1

𝜓10

-0.000000000007048

0.024102456749993

0

0

0

0

0

2

8

2

𝜓10 ∙ 𝜓20 ∙ 𝜓30 ∙ 𝜓40 ∙ 𝜓50 ∙ 𝜓62

-0.000000000023286

-0.000000000040769

0

0

0

0

2

0

9

2

𝜓10 ∙ 𝜓20 ∙ 𝜓30 ∙ 𝜓40 ∙ 𝜓52 ∙ 𝜓60

2

𝜓10

0

0

0

2

0

0

10

∙

𝜓20

∙

𝜓20

∙

𝜓30

∙

𝜓32

∙

𝜓42

∙

𝜓40

∙

𝜓50

∙

𝜓50

0.000000000022925

-0.000000000232408

∙

𝜓60

-0.000000000024737

0.000006706705933

∙

𝜓60

-0.000217953073596

-0.000264447331640

-0.016117761224563

-0.015280471381999

∙

𝜓60

0.000000115878506

0.000000115776009

∙

𝜓60

0.000000000018054

-0.000010325443852

∙

𝜓60

0

0

2

0

0

0

11

2

𝜓10

0

2

0

0

0

0

12

2

𝜓10 ∙ 𝜓22 ∙ 𝜓30 ∙ 𝜓40 ∙ 𝜓50 ∙ 𝜓60

2

𝜓12

2

𝜓11

2 1

0 1

0 0

0 0

0 0

0 0

13 14

∙

𝜓20

∙

𝜓21

∙

𝜓20

∙

𝜓30

∙

𝜓30

∙

𝜓31

∙

𝜓40

∙

𝜓40

∙

𝜓40

∙

𝜓50

∙

𝜓50

∙

𝜓50

1

0

1

0

0

0

15

2

𝜓11

0.000153821701091

0.000196468421278

1

0

0

1

0

0

16

2

𝜓11 ∙ 𝜓20 ∙ 𝜓30 ∙ 𝜓41 ∙ 𝜓50 ∙ 𝜓60

-0.021558847108579

-0.026046567563358

1

0

0

0

1

0

17

2

𝜓11 ∙ 𝜓20 ∙ 𝜓30 ∙ 𝜓40 ∙ 𝜓51 ∙ 𝜓60

0.019345804121815

0.018644414438870

2

𝜓11

∙

𝜓61

0.000002843183415

0.000002842470793

∙

𝜓60

1

0

0

0

0

1

18

∙

𝜓20

∙

𝜓21

∙

𝜓30

∙

𝜓30

∙

𝜓40

∙

𝜓40

∙

𝜓51

∙

𝜓51

0

1

0

0

1

0

19

2

𝜓10

-0.000000000022903

-0.000241418369369

0

0

1

0

1

0

20

2

𝜓10 ∙ 𝜓20 ∙ 𝜓31 ∙ 𝜓40 ∙ 𝜓51 ∙ 𝜓60

0.000008529523572

0.000004862533904

2

𝜓10

∙

𝜓60

0.009200638263392

0.008913003906892

2

𝜓10

∙

𝜓61

-0.004641950569511

-0.004480290238112

∙

𝜓60

0 0

0 0

0 0

1 1

1 0

0 1

21 22

∙

𝜓20

∙

𝜓20

∙

𝜓20

∙

𝜓30

∙

𝜓30

∙

𝜓31

∙

𝜓41

∙

𝜓41

∙

𝜓41

∙

𝜓51

∙

𝜓50

∙

𝜓50

0

0

1

1

0

0

23

2

𝜓10

-0.000000000022875

-0.000000000040046

0

1

0

1

0

0

24

2

𝜓10 ∙ 𝜓21 ∙ 𝜓30 ∙ 𝜓41 ∙ 𝜓50 ∙ 𝜓60

0.015278956756757

0.014717279812421

2

𝜓10

∙

𝜓60

0.000000000018128

0.000000500564078

2

𝜓10

∙

𝜓60

-0.012899033013469

-0.012126258683645

∙

𝜓61

-0.015058074823713

-0.014512969290543

∙

𝜓61

0.000000000023872

-0.000000535950556

0 0

0 1

0 1

1 0

0 0

1 0

25 26

0

1

0

0

0

1

27

2

𝜓10

0

0

0

0

1

1

28

2

𝜓10

∙

𝜓21

∙

𝜓21

∙

𝜓21

∙

𝜓20

∙

𝜓30

∙

𝜓31

∙

𝜓30

∙

𝜓30

∙

𝜓41

∙

𝜓40

∙

𝜓40

∙

𝜓40

∙

𝜓50

∙

𝜓50

∙

𝜓50

∙

𝜓51

TABLE 5: PCE DEFINITIONS RELATED TO THE ENTRY AND EXIT TIMES

24



The univariate polynomial basis used to represent the distributions of the selected sources of uncertainty are shown in next Table 6. 𝒙𝟎

𝒙𝟏

𝒙𝟐

𝝍𝟎𝟏

1

0

0

𝝍𝟎𝟐

1

0

0

𝝍𝟎𝟑

1

0

0

𝝍𝟎𝟒

1

0

0

𝝍𝟎𝟓

1

0

0

𝝍𝟎𝟔

1

0

0

𝝍𝟏𝟏

-10.0644

0.1006

0

𝝍𝟏𝟐

-97.9231

0.0140

0

𝝍𝟏𝟑

-70.4711

0.0066

0

-4

𝝍𝟏𝟒

-78.91

1.97e

0

𝝍𝟏𝟓

-1.425

4.1774

0

𝝍𝟏𝟔

-0.0084

1.7204

0

𝝍𝟐𝟏

69.8058

-1.4102

0.0071

𝝍𝟐𝟐

1051.4

-3.0155

2.16e-4

𝝍𝟐𝟑

3.5394e+3

-0.6636

3.11e-5

𝝍𝟐𝟒

4.3956e+3

-0.022

2.75e-8

𝝍𝟐𝟓

1.7348

-13.6027

16.7291

𝝍𝟐𝟔

-1.1313

-0.0047

3.3482

TABLE 6: UNIVARIATE POLYNOMIAL BASIS

The variability of entry and exit times to the selected airspace volume can be quantify through the mean (μ) and standard deviation (σ) computed from the corresponding distribution. Table 7 shows the obtained results for the example presented herein.

𝑵

𝝁 = 𝒃𝟏

𝝈 = √∑ 𝒃𝟐𝒊 𝒊=𝟐

Entry Time

1,674.8 [sec]

44.57 [sec]

Exit time

2,157.5 [sec]

51.55 [sec]

TABLE 7: ENTRY & EXIT TIMES STATISTICS


25


5 Final Remarks

To address the problem of trajectory prediction uncertainty quantification, a novel approach based on the aPCE methodology has been described. PCE has been applied in dynamic systems to evaluate the uncertainty of parameters that characterise the system by means of a polynomial description of the variability of inputs. The advantage of using aPCE relies on the capability of describing the input distributions driven by data. This method does not require an analytical description of the input distributions, it is only necessary to have the possibility of computing the statistics moments up to a certain order (which will determine the maximum order of the polynomial expansion). With this information, it is possible to identify the polynomial expansion of the inputs and, based on this, to obtain the expansion of the outputs. The benefits that can be obtained from the application of the aPCE-based trajectory prediction uncertainty quantification are manifold:

26



This solution can potentially be applied to any trajectory predictor without modifying its native implementation.



It is a fast and computationally efficient procedure, especially when compared with classical approaches like Monte Carlo, and can be considered as a pseudo-real time process taking into account typical look-ahead trajectory prediction times.



It facilitates the individual quantification of prediction uncertainties of a traffic sample of thousands flights thanks to its low computational performance requirements.



It is a data-driven process, that is, analytical representations of the probability distributions characterizing the sources of uncertainty are not required.



It provides analytical representations of prediction uncertainties formed by polynomial expansions that can be easily processed by computer-based Collaborative Decision Making (CDM) processes.



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