Quantification of trajectory prediction uncertainty

Transcription

1 EXPLORATORY RESEARCH Quantification of trajectory prediction uncertainty DD2.2 COPTRA Grant: Call: H2020-SESAR Topic: Sesar Consortium coordinator: CRIDA A.I.E. Edition date: 07 FEBRUARY 2017 Edition:

2 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 Reviewers internal to the project Name/Beneficiary Position/Title Date R. JUNGERS / UCL WP03 Leader 01/02/2017 J. BOUCQUEY / EUROCONTROL WP04 Leader 01/02/2017 E. KOYUNCU / ITU WP05 Leader 04/02/2017 S. RODRÍGUEZ / CRIDA WP06 Leader 05/02/2017 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 J. BOUCQUEY / EUROCONTROL WP04 Leader 07/02/2017 E. KOYUNCU / ITU WP05 Leader 07/02/2017 S. RODRÍGUEZ / CRIDA 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 /11/2016 REVISED E. CASADO Creation /01/2017 REVISED E. CASADO Improved description of PC theory /02/2017 REVISED N. SUAREZ Minor updates and corrections COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions

3 0.4 01/02/2017 REVISED J. BOUCQUEY Minor updates and corrections /02/2017 REVISED E. CASADO Added Section /02/2017 REVISED E. CASADO Document closure 2017 COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions. 3

4 COPTRA COMBINING PROBABLE TRAJECTORIES This document is part of a project that has received funding from the SESAR Joint Undertaking under grant agreement No 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 COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions

5 Table of Contents 1 Introduction Purpose and Scope Intended Readership Acronyms and Terminology Trajectory Prediction Infrastructure Introduction Trajectory Prediction Approach The Aircraft Intent Description Language Formal Quantification of Trajectory Prediction Uncertainties Introduction Polynomial Chaos Theory Generalized PCE Arbitrary PCE apce application to the TP infrastructure Introduction Application of apce to Trajectory Prediction Example of application AI uncertainty Initial Condition uncertainty Weather forecast uncertainty APM uncertainty Predicted Entry and Exit times uncertainty Final Remarks References COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions. 5

6 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 COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions

7 1 Introduction 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). 1 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 COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions. 7

8 1.3 Acronyms and Terminology Term Definition AI AIDL AMM apce APM ATM BADA BR&T-E CAS CDM COPTRA CRIDA D DAE DCB DOF DST EM EUROCONTROL F FL FMS GFS GMT gpce h HA HC HHL Aircraft Intent Aircraft Intent Description Language Aircraft Motion Model Arbitrary Polynomial Chaos Expansion Aircraft Performance Model Air Traffic Management Base of Aircraft Data Boeing Research & Technology Europe Calibrated Airspeed Collaborative Decision Making Combining Probable Trajectories Centro de Referencia de Investigación, Desarrollo e Innovación Drag Differential Algebraic Equation Demand & Capacity Balance Degree of Freedom Decision Support Tools Earth Model European Organization for the Safety of Air Navigation Fuel Consumption Flight Level Flight Management System Global Forecast System Greenwich Mean Time Generalized Polynomial Chaos Expansion Geodetic altitude Hold Altitude Hold Course Hold High Lift devices COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions

9 HL HLG Hp HS HSB ISA ITU LG LIDL M m NOAA ODE PC PCE PDF PMM SB SESAR SJU T TAS TBO TCE TCI TL TLP TM TOC TOD TP UCL High Lift devices Hold Landing Gear Pressure altitude Hold Speed Hold Speed Breaks International Standard Atmosphere Istanbul Technical University Landing Gear Low Idle Engine Regime Mach Number mass National Oceanic and Atmospheric Administration Ordinary Differential Equation Polynomial Chaos Polynomial Chaos Expansion Probability Density Functions Point-Mass Model Speed Breaks Single European Sky ATM Research Programme SESAR Joint Undertaking (Agency of the European Commission) Thrust True Airspeed Trajectory Based Operations Trajectory Computation Engine Trajectory Computation Infrastructure Throttle Law Track Lateral Path Trajectory Management Top of Climb Top of Descent Trajectory Predictor Université Catholique de Louvain 2017 COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions. 9

10 UQ WP Wx Wy φ λ χ ψ Uncertainty Quantification Work Package North wind component West wind component Latitude Longitude Bearing Heading TABLE 1: ACRONYMS AND TERMINOLOGY COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions

11 AIRCRAFT INTENT Motion Profiles Configuration Profiles Pilot event M=0.78 CAS=280kt h=4500ft HS (M) HA (P) d ST (T) HS (M) HS (CAS) VSL (ROC) End of engine transient to idle TLP (GC) Roll-in anticipation HT (T)? d? SBA TLP (CRT) SBA d HS (ABS) HHL SHL HHL HSB HLG Capture of target bank Roll-in anticipation Capture of target bank Auto beginning of HL devices deployment? HA (GEO) HC (GEO) CAS=210kt z BFS AIRCRAFT TRAJECTORY Horizontal Vertical M.88 FL320 TOD M.78 TA 280 KCAS N W Time R? 4500ft 180 KCAS D2.2 QUANTIFICATION OF TRAJECTORY PREDICTION UNCERTAINTY 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 INITIAL CONDITIONS Trajectory Computation Engine COMPUTED TRAJECTORY 1 st DOF AIRCRAFT INTENT Aircraft Performance Model Weather Model 2 nd DOF 3 rd DOF y BFS HL HS SB x BFS LG 2017 COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions. 11

12 D2.2 QUANTIFICATION OF TRAJECTORY PREDICTION UNCERTAINTY 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. dv TAS dt T D Wsinγ TAS m + dw 1 WFS dt = COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions

13 D2.2 QUANTIFICATION OF TRAJECTORY PREDICTION UNCERTAINTY dγ TAS dt dχ TAS dt 1 [ Lcosμ TAS Wcosγ TAS v TAS m 1 [ Lsinμ TAS v TAS cosγ TAS m + ( + ( dm dt + f = 0 dw 3 WFS dt dw 3 WFS dt dλ v TAScosγ TAS sinχ TAS +w WFS 2 dt (N+h)cosφ dφ v TAScosγ TAS cosχ TAS +w WFS 1 dt (M+h) dh dt v TASsinγ TAS = 0 cosμ TAS + dw 2 WFS sinμ TAS )] = 0 dt sinμ TAS dw 2 WFS cosμ TAS )] = 0 The mathematical problem has one independent variable [t], 10 dependent variables: true airspeed [v TAS]. Aerodynamic path angel [γ TAS], aerodynamic heading [χ TAS], aircraft mass [m ], longitude [λ], latitude [φ], geodetic altitude [h], throttle parameter [δ T], aerodynamic bank angle [μ TAS] 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 [δ T μ TAS L], is the 7 state variables [v TAS γ TAS χ TAS 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 = 0 = 0 dt 2017 COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions. 13

14 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 COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions

15 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 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 (ψ i ) is a set of orthogonal polynomials with respect to the probability density (PDF) function Γ(ξ) of the germ ξ, ψ i, ψ j = ψ i (ξ) ψ j (ξ) Γ(ξ) dξ = δ ij where δ ij 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 ψ 0, and that the covariance between two polynomials of different order is zero (uncorrelated polynomials) because of ψ i, ψ j = 0; i j. Thus, it is possible to build an orthonormal basis of polynomials by assuming ψ i, ψ i = 1 i COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions. 15

16 Making use of the orthonormal basis of polynomials, the relationship stated in (2) can be formulated as follows: z = a i ψ i (ξ) i=1 The PC theory based on Wiener-Askey theory of homogeneous chaos [7], defines the i-th mode as the combination of a i (mode strength) and ψ i (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 {ξ 1, ξ 2,, ξ N}. In this case, the stochastic variable z will be represented by the following expansion: z = b i φ i (ξ 1, ξ 2,, ξ N ) i=1 Where φ i is a tensor product of the univariate polynomial bases of each ξ j. φ i (ξ 1, ξ 2,, ξ N ) = ψ j α j i (ξ j ) N j=1 N i α j p, i = 1,, N j=1 The multivariate index α j i represents the combinatory of all possible products of ψ j k (polynomial of order k belonging to the polynomial basis of germ ξ j), 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 α j i is a p x N matrix that represents the corresponding expansion degree for germ j in expansion term i. The unknown coefficients a i (univariate) or b i (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] 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 COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions

17 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 {ξ 1, ξ 2,, ξ N} are independent, identically distributed and can be represented by one of the distributions presented above 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 ξ i 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 ξ i can be expressed as k ψ k (k) i = c m m=0 c k (k) = 1 ξ i m The zero-order polynomial can be immediately obtained: ψ i 0 = c 0 (0) ξi 0 = 1 The basis is successively constructed for all remaining polynomials of the ξ i basis up to the selected order d by solving the following system of equations based on (12): 2017 COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions. 17

18 k (0) (k) c 0 [ cm m=0 ξ i m ] Γ(ξ i ) dξ i = 0 1 (1) [ c m m=0 k ξ m (k) i ] [ c m m=0 ξ i m ] Γ(ξ i ) dξ i = 0 k 1 (k 1) [ c m m=0 k ξ m (k) i ] [ c m m=0 c k (k) = 1 ξ i m ] Γ(ξ i ) dξ i = 0 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) c m m=0 k ξ i m Γ(ξ i ) dξ i = 0 (k) c m ξ m+1 i Γ(ξ i ) dξ i = 0 m=0 k (k) c m m=0 ξ i m+k 1 Γ(ξ i ) dξ i = 0 c k (k) = 1 The m-th statistical moment of the germ ξ i can be defined as: μ k = ξ i m Γ(ξ i ) dξ i Thus, the system of equations posed in (12) turns into: k (k) c m μ m = 0 m=0 k (k) c m μ m+1 = 0 m=0 k (k) c m μ m+k 1 = 0 m=0 c k (k) = 1 or alternatively written in a matrix form as: COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions

19 c 0 (k) μ 0 μ 1 μ k (k) μ 1 μ 2 μ k+1 c 1 = μ k 1 μ k μ (k) 2k 1 c [ ] k 1 (k) [ c k ] [ 1] The coefficients c m (k) of each of the polynomials belonging to the basis of the germ ξi can be computed if and only if all moments of the ξ i 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 ξ i distribution is required to apply apc, avoiding the need of having an explicit description of the associated probability density function of the ξ i distribution COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions. 19

20 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 b i 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 ψ i to define a system of equations that returns b i. 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 b i 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 {(ξ 1,1,,ξ N,1),, (ξ 1,q,,ξ N,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 ξ i are obtained as COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions

21 the roots of the polynomial ψ d+1 (ξ i ) of next higher order than the order of the polynomial ψ d (ξ i ) 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 uncertainty 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 CAS 0 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 (CAS d = 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 COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions. 21

22 FIGURE 3 AIDL ALPHABET AND GRAMMAR RULES AIRCRAFT INTNET Motion Profiles Initial Conditions M c = 0.7 CAS d = 260kt Hp = 10,000ft HS(CAS) HS(MACH) HS(CAS) TOC -> FL c = 350 TOD -> FD 2 = 400km TL(MCMB) HA(PRE) TL(LIDL) FD 1 = 300km χ c = -4º TLP(Geodesic) TLP(Circular Arc) HC(GEO) OPERATIONS OP#1 OP#2 OP#3 OP#4 OP#5 OP#6 OP#7 M = 0.77 TOC TRAJECTROY DESCRIPTION Vertical Horizontal A A M = 0.77 TOC FD 1 = 300km B R Magnetic North TOD TOD -4º CAS = 260kt CAS = 260kt Hp = 10,000ft Hp = 10,000ft C The following parameters have been considered sources of AI uncertainty: cruise Mach speed (M c) cruise pressure altitude (FL c), and TOD location (FD 2) Table 3 shows the probability density functions (PDFs) that have been used to characterize the variability of the selected sources of AI uncertainty. M c [-] FL c [FL] FD 2 [km] UNI(0. 71, 0. 69) a 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 Initial Condition uncertainty COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions

23 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. t 0 λ 0 φ 0 m 0 CAS 0 Hp 0 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 t 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 APM uncertainty The selected aircraft type will be a Boeing 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 (d 0) has been chosen as source of APM uncertainty. According to the BADA 4.1 dataset of the referred aircraft type, the triangular distribution TRI( , ) will be used to represent the variability of the aircraft performance. This is a distribution between and with mode 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 b i solving the linear system of equations (16) for both the entry and exit times of the designated sector COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions. 23

24 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). o j k k o j k n j=1 φ i b i entry 10 3 b i exit ψ 0 1 ψ 0 2 ψ 0 3 ψ 0 4 ψ ψ ψ 0 1 ψ 0 2 ψ 0 3 ψ 0 4 ψ ψ ψ 0 1 ψ 0 2 ψ 0 3 ψ 1 4 ψ ψ ψ 0 1 ψ 0 2 ψ 1 3 ψ 0 4 ψ ψ ψ 0 1 ψ 1 2 ψ 0 3 ψ 0 4 ψ ψ ψ 1 1 ψ 0 2 ψ 0 3 ψ 0 4 ψ ψ ψ 0 1 ψ 0 2 ψ 0 3 ψ 0 4 ψ ψ ψ 0 1 ψ 0 2 ψ 0 3 ψ 0 4 ψ ψ ψ 0 1 ψ 0 2 ψ 0 3 ψ 0 4 ψ ψ ψ 0 1 ψ 0 2 ψ 0 3 ψ 2 4 ψ ψ ψ 0 1 ψ 0 2 ψ 2 3 ψ 0 4 ψ ψ ψ 0 1 ψ 2 2 ψ 0 3 ψ 0 4 ψ ψ ψ 2 1 ψ 0 2 ψ 0 3 ψ 0 4 ψ ψ ψ 1 1 ψ 1 2 ψ 0 3 ψ 0 4 ψ ψ ψ 1 1 ψ 0 2 ψ 1 3 ψ 0 4 ψ ψ ψ 1 1 ψ 0 2 ψ 0 3 ψ 1 4 ψ ψ ψ 1 1 ψ 0 2 ψ 0 3 ψ 0 4 ψ ψ ψ 1 1 ψ 0 2 ψ 0 3 ψ 0 4 ψ ψ ψ 0 1 ψ 1 2 ψ 0 3 ψ 0 4 ψ ψ ψ 0 1 ψ 0 2 ψ 1 3 ψ 0 4 ψ ψ ψ 0 1 ψ 0 2 ψ 0 3 ψ 1 4 ψ ψ ψ 0 1 ψ 0 2 ψ 0 3 ψ 1 4 ψ ψ ψ 0 1 ψ 0 2 ψ 1 3 ψ 1 4 ψ ψ ψ 0 1 ψ 1 2 ψ 0 3 ψ 1 4 ψ ψ ψ 0 1 ψ 1 2 ψ 0 3 ψ 1 4 ψ ψ ψ 0 1 ψ 1 2 ψ 1 3 ψ 0 4 ψ ψ ψ 0 1 ψ 1 2 ψ 0 3 ψ 0 4 ψ ψ ψ 0 1 ψ 0 2 ψ 0 3 ψ 0 4 ψ ψ TABLE 5: PCE DEFINITIONS RELATED TO THE ENTRY AND EXIT TIMES COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions

25 The univariate polynomial basis used to represent the distributions of the selected sources of uncertainty are shown in next Table 6. x 0 x 1 x 2 ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ e -4 0 ψ ψ ψ ψ e -4 ψ e e -5 ψ e e -8 ψ ψ 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. μ = b 1 N σ = b i 2 i=2 Entry Time 1,674.8 [sec] [sec] Exit time 2,157.5 [sec] [sec] TABLE 7: ENTRY & EXIT TIMES STATISTICS 2017 COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions. 25

26 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: 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 COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions

27 References [1] COPTRA D2.1. Techniques to determine trajectory uncertainty and modelling. Version 1, January [2] Mondoloni, Stephane, and Sip Swierstra. "Commonality in disparate trajectory predictors for air traffic management applications." Digital Avionics Systems Conference, DASC The 24th. Vol. 1. IEEE, [3] Bronsvoort, Jesper, et al. "Real-Time Trajectory Predictor Calibration through Extended Projected Profile Down-Link." Eleventh USA/Europe Air Traffic Management Research and Development Seminar [4] Lopez Leones, Javier. Definition of an aircraft intent description language for air traffic management applications. Diss. University of Glasgow, [5] Lopez-Leones, Javier, et al. "The aircraft intent description language: A key enabler for airground synchronization in trajectory-based operations." Digital Avionics Systems Conference, DASC'07. IEEE/AIAA 26th. IEEE, [6] O'Hagan, Anthony. "Polynomial chaos: A tutorial and critique from a statistician s perspective." SIAM/ASA J. Uncertainty Quantification 20 (2013): [7] Xiu, Dongbin, and George Em Karniadakis. "The Wiener--Askey polynomial chaos for stochastic differential equations." SIAM journal on scientific computing 24, no. 2 (2002): [8] Xiu, Dongbin, and George Em Karniadakis. "Modeling uncertainty in flow simulations via generalized polynomial chaos." Journal of computational physics (2003): [9] Li, Heng, and Dongxiao Zhang. "Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods." Water Resources Research 43, no. 9 (2007). [10] Wan, Xiaoliang, and George Em Karniadakis. "Multi-element generalized polynomial chaos for arbitrary probability measures." SIAM Journal on Scientific Computing 28.3 (2006): [11] Oladyshkin, S., et al. "A concept for data-driven uncertainty quantification and its application to carbon dioxide storage in geological formations." Advances in Water Resources (2011): [12] Onorato, G., et al. "Comparison of intrusive and non-intrusive polynomial chaos methods for CFD applications in aeronautics." V European Conference on Computational Fluid Dynamics ECCOMAS, Lisbon, Portugal [13] Shi, Liangsheng, et al. "Probabilistic collocation method for unconfined flow in heterogeneous media." Journal of Hydrology (2009): COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions. 27

28 [14] EUROCONTROL. Base of Aircraft Data (BADA). Eurocontrol Research & SESAR website Last accessed on January 29, COPTRA Consortium. All rights reserved. Licensed to the SESAR Joint Undertaking under conditions